Customized Porosity in Ceramic Composites via Freeze Casting - CaltechTHESIS

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Customized Porosity in Ceramic Composites via Freeze Casting
Citation
Kuo, Claire Taijung
(2021)
Customized Porosity in Ceramic Composites via Freeze Casting.
Dissertation (Ph.D.), California Institute of Technology..
doi:10.7907/88p1-5v79.
Abstract
Freeze casting is a facile pore-forming technique for ceramics as it affords great tunability in pore structure including size, morphology, wall thickness, tortuosity, and alignment. Nevertheless, similar to any other pore-forming techniques, it has limitations in terms of the range of accessible properties. For example, a porous lamellar structure is highly permeable but easily fractures, while the dendritic structure is the opposite. This research seeks to provide strategies used with freeze casting to achieve a combination of properties that go beyond the current limitations and create optimized pore structures with a specific focus on three properties: strength, permeability, and surface area.
Such strategies utilize two composite material principles. First, particle reinforcement was implemented to optimize the mechanical and transport properties. Second, surface area was increased with hierarchical design for enhanced capture or catalysis applications. To optimize the mechanical and transport properties, we reinforced high-permeability lamellar structures with reinforcement fillers of silicon carbide (SiC) whiskers and carbon nanotubes (CNTs). The two fillers afford two different mechanisms of reinforcement: structural and material reinforcement.
Additions of 30 vol.% SiC whiskers increased the compressive strength by 325% at a small expense in permeability. Shear failure, common in lamellar structures, was prevented by the interwall bridges produced via particle engulfment during freezing. These bridges were demonstrated by the change in microstructure, stress-strain behavior, and fracture surfaces. A 2D
in-situ
solidification experiment was conducted to observe solidification and particle engulfment directly. We proposed a modified engulfment model to account for the complexity stemming from high-aspect ratio particles and non-planar freezing fronts. Reasonable agreement was found between the model, the simulation based on the model, and the experimental values from the freeze-casting and 2D-solidification experiments.
Freeze-casting with CNTs was explored as an alternative reinforcement strategy, but one which maintains the original pore structure. CNTs were pushed aside by the freezing front to pore walls due to their small diameters for low CNT concentration composites (<4.5 wt.%) such that the original pore structures remained. The compressive strength increased, albeit by smaller percentages (118% for 4.3 wt.%) than those with SiC whiskers. The increase was attributed to the toughening of pore walls with no diminishing effect on permeability. In addition, CNTs changed the electrical conductivity by ten orders of magnitude with the addition of 8.2 wt.% of the reinforcement.
Finally, conformal coatings via self-assembly of block copolymers (BCP) were produced by infiltration into a freeze-cast lamellar structure and significantly increased the surface area of the underlying scaffold. A bimodal pore size distribution with nanometer-size pores from the BCP self-assembly and micron-size pores from freeze casting was observed. An increase in compressive strengths was achieved with the introduction of pore hierarchy while retaining permeability of the macroporous structure due to enlarged lamellar spacings from the infiltration process.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Freeze casting; Ceramic composites; Preceramic polymer; Carbon nanotubes; SiC whisker; Porous materials; Block copolymer
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Faber, Katherine T.
Thesis Committee:
Fultz, Brent T. (chair)
Kornfield, Julia A.
Johnson, William L.
Faber, Katherine T.
Defense Date:
15 December 2020
Funders:
Funding Agency
Grant Number
National Science Foundation
DMR-1411218
J Yang & Family Foundation
UNSPECIFIED
Record Number:
CaltechTHESIS:12242020-000539491
Persistent URL:
DOI:
10.7907/88p1-5v79
Related URLs:
URL
URL Type
Description
DOI
Article adapted for Chapter 4.
DOI
Article adapted for Chapter 5.
ORCID:
Author
ORCID
Kuo, Claire Taijung
0000-0002-3720-968X
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
14042
Collection:
CaltechTHESIS
Deposited By:
Tai Jung Kuo
Deposited On:
05 Jan 2021 19:34
Last Modified:
02 Nov 2021 16:55
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Customized porosity in ceramic composites via freeze
casting

Thesis by

Claire Taijung Kuo

In Partial Fulfillment of the Requirements for the
Degree of
DOCTOR OF PHILOSOPHY

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2021
Defended December 15, 2020

ii

Claire Taijung Kuo
ORCID: 0000-0002-3720-968X

iii

ACKNOWLEDGEMENTS
So many people have helped me along the way and without them, I could not reach
where I am today. It has been a wonderful journey.
First, I would like to thank my advisor, Prof. Katherine Faber, who has been the
best mentor anyone could ask for. I have grown so much as a researcher under her
guidance. I especially appreciate her vision that as a researcher, we are here to do
good and sound science. There are rough bumps along the way when experiments
were just not working but her encouragement and support have always made me
want to try again.
Next, I would like to thank Prof. Brent Fultz, Prof. William Johnson, and Prof.
Julie Kornfield for graciously agreeing to be on my thesis committee, encouraging
me when I first started, reviewing my thesis, and giving invaluable advice.
Many thanks also go out to my research group mates in the Faber group: Dr.
Matthew Johnson, Dr. Sarah Miller, Dr. Maninpat Naviroj, Dr. Xiaomei Zeng, Dr.
Wei-Lin Tan, Dr. Neal Brodnik, Dr. Noriaki Arai, Ben Herren, Celia Chari, Vince
Wu, Laura Quinn, and Carl Keck. I would like to thank Putt, Matt, and Wei-Lin
for their mentorships and for helping me navigate research and life. I value Nori’s
5+ years of friendship, helpful suggestions, and support. My gratitude also goes to
Carl for his work on SiC whiskers. I would also like to thank Xiaomei, Neal, Ben,
Celia, Vince, Laura, and Carl for all their research suggestions, support, and sense
of humor that got me through the PhD program. It wouldn’t be the same without
you.
Special thanks to my collaborators, Lisa M. Rueschhoff, Matthew B. Dickerson,
and Tulsi A. Patel at Air Force Research Laboratory for providing expertise in the
block copolymer, analyzing the gel structures, conducting XCT, bouncing ideas off
of, and revising the manuscript. I would also like to thank the administrative staff at
Caltech, Angie, Katie, Jennifer, and Christy, who have always been one email away
and ready to save the day.
I’m infinitely grateful for the support of all my friends at Caltech and in Taiwan, the
Yangi crew, the Craft & Tea group, Hannah and Oscar, Pai, and many more. Finally,
I would like to thank my family for their unwavering support throughout the years.
Their belief in me motivates me to keep trying and reach higher every day.

iv

ABSTRACT
Freeze casting is a facile pore-forming technique for ceramics as it affords great tunability in pore structure including size, morphology, wall thickness, tortuosity, and
alignment. Nevertheless, similar to any other pore-forming techniques, it has limitations in terms of the range of accessible properties. For example, a porous lamellar
structure is highly permeable but easily fractures, while the dendritic structure is
the opposite. This research seeks to provide strategies used with freeze casting
to achieve a combination of properties that go beyond the current limitations and
create optimized pore structures with a specific focus on three properties: strength,
permeability, and surface area.
Such strategies utilize two composite material principles. First, particle reinforcement was implemented to optimize the mechanical and transport properties. Second,
surface area was increased with hierarchical design for enhanced capture or catalysis
applications. To optimize the mechanical and transport properties, we reinforced
high-permeability lamellar structures with reinforcement fillers of silicon carbide
(SiC) whiskers and carbon nanotubes (CNTs). The two fillers afford two different
mechanisms of reinforcement: structural and material reinforcement.
Additions of 30 vol.% SiC whiskers increased the compressive strength by 325%
at a small expense in permeability. Shear failure, common in lamellar structures,
was prevented by the interwall bridges produced via particle engulfment during
freezing. These bridges were demonstrated by the change in microstructure, stressstrain behavior, and fracture surfaces. A 2D in-situ solidification experiment was
conducted to observe solidification and particle engulfment directly. We proposed
a modified engulfment model to account for the complexity stemming from highaspect ratio particles and non-planar freezing fronts. Reasonable agreement was
found between the model, the simulation based on the model, and the experimental
values from the freeze-casting and 2D-solidification experiments.
Freeze-casting with CNTs was explored as an alternative reinforcement strategy, but
one which maintains the original pore structure. CNTs were pushed aside by the
freezing front to pore walls due to their small diameters for low CNT concentration composites (<4.5 wt.%) such that the original pore structures remained. The
compressive strength increased, albeit by smaller percentages (118% for 4.3 wt.%)
than those with SiC whiskers. The increase was attributed to the toughening of pore

walls with no diminishing effect on permeability. In addition, CNTs changed the
electrical conductivity by ten orders of magnitude with the addition of 8.2 wt.% of
the reinforcement.
Finally, conformal coatings via self-assembly of block copolymers (BCP) were produced by infiltration into a freeze-cast lamellar structure and significantly increased
the surface area of the underlying scaffold. A bimodal pore size distribution with
nanometer-size pores from the BCP self-assembly and micron-size pores from freeze
casting was observed. An increase in compressive strengths was achieved with the
introduction of pore hierarchy while retaining permeability of the macroporous
structure due to enlarged lamellar spacings from the infiltration process.

vi

PUBLISHED CONTENT AND CONTRIBUTIONS

[1] C. T. Kuo and K. T. Faber. Permeable carbon nanotube-reinforced silicon oxycarbide via freeze casting with enhanced mechanical stability.
Journal of the European Ceramic Society, 40(6):2470–2479, 2020. doi:
10.1016/j.jeurceramsoc.2019.12.059. URL https://doi.org/10.1016/j.
jeurceramsoc.2019.12.059.
T. Kuo fabricated and characterized CNT composites, and wrote the manuscript.
[2] Taijung Kuo, Carl H. Keck, and Katherine T. Faber. Porous SiC-SiOC composites through particle engulfment of high-aspect ratio particles during freeze
casting. Manuscript in preparation, 2021.
T. Kuo fabricated and characterized SiC composites using SEM and MIP, their
permeabilities, and compressive strengths, developed the model and simulated
solidification process for freeze casting high-aspect ratio particle, and wrote the
manuscript.
[3] Taijung Kuo, Lisa M. Rueschhoff, Matthew B. Dickerson, Tulsi A. Patel, and
Katherine T. Faber. Hierarchical porous SiOC via freeze casting and selfassembly of block copolymers. Scripta Materialia, 191:204–209, 2021. doi:
10.1016/j.scriptamat.2020.09.042. URL https://linkinghub.elsevier.
com/retrieve/pii/S1359646220306412.
T. Kuo fabricated and characterized hierarchical ceramics using SEM, MIP,
and N2 adsorption isotherms, characterized weight loss, volume shrinkage,
compressive strengths, and permeabilities, conducted image analysis on X-ray
computed tomography (XCT) datasets, and wrote the majority of the manuscript.

vii

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vi
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter II: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Freeze Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Solid Contents . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Solvents and Additives . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Solidification and Post-freezing Treatments . . . . . . . . . 11
2.1.4 Particle Engulfment . . . . . . . . . . . . . . . . . . . . . . 15
2.1.5 Sintering/pyrolysis and Post-freeze Casting Treatments . . . 16
2.2 Self-assembled Block Copolymers . . . . . . . . . . . . . . . . . . 17
2.3 Preceramic Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Overview of Polymer-derived Ceramics (PDCs) . . . . . . . 20
2.3.2 Polysilsesquioxane . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Polycarbosilane . . . . . . . . . . . . . . . . . . . . . . . . 23
Chapter III: SiC Whisker-reinforced Composites . . . . . . . . . . . . . . . . 25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 Freeze Casting . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1.1 Materials and Synthesis . . . . . . . . . . . . . . 26
3.2.1.2 Characterization . . . . . . . . . . . . . . . . . . 28
3.2.2 Two-dimensional Solidification Experiments and Simulations 29
3.2.2.1 Experimental Setup . . . . . . . . . . . . . . . . 29
3.2.2.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2.3 Surface Energy . . . . . . . . . . . . . . . . . . . 30
3.2.2.4 Freezing Simulations . . . . . . . . . . . . . . . . 31
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 Freeze-cast Samples . . . . . . . . . . . . . . . . . . . . . 32
3.3.1.1 Pore Structures . . . . . . . . . . . . . . . . . . . 32
3.3.1.2 Mechanical Properties and Permeability . . . . . . 37
3.3.2 Two-dimensional in-situ Solidification Experiments . . . . . 39
3.3.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . 39

viii
3.3.2.2

Engulfment-related Forces and Critical Freezing
Front Velocity . . . . . . . . . . . . . . . . . . . 40
3.3.2.3 Torque and Whisker Rotation . . . . . . . . . . . 45
3.3.2.4 Freezing Simulations and Bridge density . . . . . 47
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter IV: Carbon Nanotube-reinforced Composites . . . . . . . . . . . . . 51
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Materials and Processing . . . . . . . . . . . . . . . . . . . 52
4.2.2 X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.3 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . 54
4.2.4 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.5 Mechanical Properties . . . . . . . . . . . . . . . . . . . . 56
4.2.5.1 Compressive Strength . . . . . . . . . . . . . . . 56
4.2.5.2 Fracture Toughness . . . . . . . . . . . . . . . . . 56
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.1 Carbon Nanotube Dispersion in Solution and Freeze-cast
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . 61
4.3.3 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.4 Mechanical Properties . . . . . . . . . . . . . . . . . . . . 64
4.4 Conclusions and Implications . . . . . . . . . . . . . . . . . . . . . 71
Chapter V: Hierarchical Composites via Self-assembly of Block Copolymer . 73
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.1 Materials and Processing . . . . . . . . . . . . . . . . . . . 74
5.2.2 Pore Structure . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.3 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.4 Mechanical Properties . . . . . . . . . . . . . . . . . . . . 76
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4 Conclusions and Implications . . . . . . . . . . . . . . . . . . . . . 87
Chapter VI: Summary, Conclusions, and Future Work . . . . . . . . . . . . . 89
6.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Appendix A: Stress-Strain Curves of SiC Whisker Composites and Pure SiOC 120
Appendix B: Surface Energy Difference as the Origin of the Repulsive Force . 121
Appendix C: Calculation of Torques . . . . . . . . . . . . . . . . . . . . . . 123
C.1 Torque from Fluid Motion . . . . . . . . . . . . . . . . . . . . . . . 123
C.2 Torque from Frictional Force . . . . . . . . . . . . . . . . . . . . . 124
Appendix D: Examples of Rotation From the Freezing Video . . . . . . . . . 125
Appendix E: Lamellar Spacing and Freezing Front Velocity . . . . . . . . . . 126

ix

LIST OF ILLUSTRATIONS

Number
Page
2.1 Schematic of the freeze casting process showing solution/suspension
preparation, solidification, sublimation, and sintering/pyrolysis steps
[125]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Critical freezing front velocity, above which particles are engulfed
and below which particles are pushed by the freezing front, as a function of particle size and the respective resultant pore structures formed
from particles pushed or engulfed by the freezing front. Adapted from
[30] with permission from Elsevier. . . . . . . . . . . . . . . . . . . 7
2.3 (a) Non-faceted [113] (Copyright (1998) National Academy of Sciences, USA) and (b) faceted [81] (Reprinted with permission from
Elsevier) cellular arrays. . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 (a) Schematic of a faceted solid-liquid interface showing A. a kinksite, B. an atom/molecule adding to the kink-site, C. an interfacial
vacancy, D. an ad-atom, E. exposed underlying lattice plane, and
F. an island of ad-atoms (b) System energy as a function of adatom coverage, 𝜉. The change in the relative position of the energy
minima demonstrate a transition from an atomically rough interface
(𝛼 < 2) to an atomically smooth interface (𝛼 > 2) [67]. Reprinted
by permission from Springer Nature. . . . . . . . . . . . . . . . . . 9
2.5 Optical images of the crystal structures formed from different solvents
and the SEM images of their respective freeze-cast pore structure.
Adapted from [125]. . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Microstructure selection map of crystal morphology and size depending on temperature gradient and growth rate [93] © 2003 John Wiley
& Sons, Inc.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7 Microstructural transition from planar to cellular to dendritic freezing
front (from top to bottom) in a succinonitrile and coumarin 152 binary
alloy [113]. Copyright (1998) National Academy of Sciences, USA. . 13
2.8 Solvent nucleation control via a wettability gradient enables a longrange pore alignment from [201]. Reprinted with permission from
AAAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.9

2.10

2.11

2.12

2.13
2.14
3.1
3.2
3.3
3.4

3.5

SEM images showing (a) the effects on alignment of a magnetic
field on TiO2 scaffolds containing 3 wt% Fe3 O4 (reprinted from
[137] with permission from Elsevier) (b) the effects on alignment
of an electric field on aqueous slurries (reprinted from [173] with
permission from Elsevier), and (c) the effects of acoustic waves on
mechanical properties and pore structure [129]. . . . . . . . . . . . . 15
Schematic of the structure and the mechanical property of PMMAPnBA-PMMA triblock copolymer gel as a function of temperature.
Reproduced from [51]. . . . . . . . . . . . . . . . . . . . . . . . . . 18
(A) Chemical structures of PMMA-PnBA-PMMA, polycarbosilane,
and PMMA. (B) Schematic of resultant morphologies of the gel
system and (C) the corresponding atomic force microscopy (AFM)
images. Scale bars are 100 nm [21]. . . . . . . . . . . . . . . . . . . 19
Typical classes of silicon-based preceramic polymers as precursors
of polymer derived-ceramics [36] © 2010 The American Ceramic
Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Schematic of the processing and chemical structures of preceramic
polymers to polymer-derived ceramics [68, 125]. . . . . . . . . . . . 22
Two proposed models of the microstructure of polysiloxane-derived
SiOC. Reprinted from [183] with permission from Elsevier. . . . . . 23
An SEM image of as-received faceted SiC whiskers. . . . . . . . . . 27
A schematic of (a) freeze casting setup and (b) 2D solidification setup. 30
A schematic of input variables used in freezing simulation. . . . . . . 32
SEM images of freeze-cast lamellar structures with (a) 0, (b) 10,
(c) 20, (d) 30 vol.% SiC whiskers and (e) 20 vol.% SiC whiskers
freeze-cast lamellar structures with SiC-SiOC bridges (blue circles)
and SiC whiskers (yellow circles) bridges (f) Example images of
bridge density measurements. Green lines mark the counted whisker
bridges. (g) Bridge density as a function of the whisker concentration.
Error bars represent ± 1 standard deviation. . . . . . . . . . . . . . . 33
Pore size distributions from MIP and the corresponding SEM images
of freeze-cast pore structures of (a) SiOC using DMC and cyclohexane, adapted from [126], and (b) SiOC and 30 vol.% SiC whisker
composites using DMC in this work. . . . . . . . . . . . . . . . . . 35

xi
3.6

3.7

3.8
3.9

3.10
3.11

3.12
4.1

4.2

Compressive strength as a function of (a) permeability and (b) relative
density. Error bars represent ± 1 standard deviation. (c) Representative stress-strain curves for freeze-cast pure SiOC (dotted black line)
and for a freeze-case 30 vol.% SiC composite (blue line). The insets
show SEM images of fracture surfaces of pure SiOC ceramics and 30
vol.% SiC composites with arrows pointing toward the corresponding
the stress-strain curves. . . . . . . . . . . . . . . . . . . . . . . . . . 37
(a) A schematic of the acting forces during the engulfment process.
(b) An example of a dendrite tip angle measurement. (c) Dendrite tip
angle as a function of freezing front velocity. Error bars represent ±
1 standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A schematic of solute pileup during the freezing process and the
resulting osmotic pressure difference. . . . . . . . . . . . . . . . . . 43
Freezing front velocity versus whisker length with experimental data;
blue circles represent pushed whiskers; orange circles represent engulfed whiskers. The theoretical critical freezing front velocity (grey
line) was calculated with engulfment model described in Section
3.3.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
The torque acting on whiskers as a function of (a) offset distance, and
(b) intersecting angle. . . . . . . . . . . . . . . . . . . . . . . . . . 46
(a) Visualization of simulation (SiC concentration = 2 vol.%, 𝑑 = 2
𝜇m, 𝐿 = 30 𝜇m, 𝑉 𝑓 𝑓 = 45 𝜇m/s). Blue lines represent freezing
dendrites. Whiskers are randomly distributed and orientated. (b)
Probability of engulfment fraction from the simulation for 10, 20,
and 30 vol. % SiC solidification. . . . . . . . . . . . . . . . . . . . . 48
Averaged simulated engulfed whisker fraction as a function of (a)
Freezing front velocity and (b) whisker length. . . . . . . . . . . . . 49
(a) Schematic of the experimental setup for permeability measurement [125]. (b) Schematic of the experimental setup for diametral
compression (Brazilian disk) test. Disks were placed between compression platens with an aligner before the start of the compression
testing to ensure the load was applied along the long axis of pre-crack. 55
Suspensions with 3 mg/ml CNTs and 300 mg/ml MK (a) without,
and (b) with KD1. Representative SEM micrographs of the resulting
freeze-cast structures with 1.3 wt.% CNTs (c) without and, (d) with
KD1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

xii
4.3
4.4

4.5

4.6

4.7

4.8

4.9

Representative SEM micrographs of freeze-cast structures (a) pure
SiOC, (b) 1.3, (c) 4.3, (d) 8.2 wt.% CNT composites. . . . . . . . . . 59
Representative SEM micrographs of freeze-cast walls of (a) pure
SiOC, (b) 1.3 (c) 4.3, (d) 8.2 wt.% CNT composites. The white
dots, present in 1.3, 4.3, 8.2 wt.% samples, are carbon nanotubes.
Sub-micron agglomerates are highlighted with green circles. . . . . . 60
(a) XRD pattern of pure SiOC, 4.3, 8.2 wt.% CNT composites and
MWCNTs. (b) Background subtraction and peak fitting performed
on XRD pattern of 8.2 wt.% CNT composite to identify phases present. 61
Electrical conductivity, calculated without adjustment for porosity,
as a function of CNT concentration of freeze-cast composites. The
resistance of one pure SiOC sample was above the equipment limit
(200 GW) of Keithley 614 Electrometer and its conductivity was
calculated using the limit. This data point was plotted with a circle
and a downward arrow. . . . . . . . . . . . . . . . . . . . . . . . . . 62
(a) Pressure vs. flow velocity to establish permeability of pure SiOC
and CNT-SiOC composites. (b) Permeability (circles with lefthand
axis) and porosity (triangles with righthand axis) as a function of
CNT concentration of freeze-cast composites. Error bars represent
± one standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . 64
(a) Representative stress-strain curves of freeze-cast samples. The
first load drop around 1 MPa in most samples is suspected to be from
the realignment of the washer. Examples of fractured samples after
compression with (b) pure SiOC and (c) 8.2 wt.% CNTs. The blue
disk-like capping is a cured mineral-filled acrylic applied to avoid
crushing behavior common in compression tests of porous ceramics. . 65
SEM micrographs of freeze-cast and pyrolyzed walls of (a) pure SiOC
and (b) 8.2 wt.% CNTs. In pure SiOC, the lamellae are smooth with
few features and bridges. In the 8.2 wt.% composite, the lamellae
are bridged with CNT agglomerates. SEM micrographs of fracture
surfaces with (c) pure SiOC and (d) 8.2 wt.% CNTs. The green
arrows in (d) point to the agglomerate bridges between lamellae. . . . 67

xiii
4.10

4.11

5.1

5.2

5.3

5.4

5.5

(a) Compressive strength and fracture toughness as a function of CNT
concentration. (b) Compressive strength vs. permeability constant of
freeze-cast CNT-SiOC composites compared with that of pure SiOC
both in this work and the values reported by Naviroj [125]. Error
bars represent ± one standard deviation. . . . . . . . . . . . . . . . .
Representative SEM micrographs of fracture surfaces of Brazilian
disks of (a) pure SiOC and (b) 4.3 wt.% CNTs. The insets in (b)
shows a magnified image of a fracture surface of 4.3 wt.% CNT
composites. The rough patches, appearing in 4.3 wt.% composites,
are clusters of CNT agglomerates. . . . . . . . . . . . . . . . . . . .
(a) AFM phase view of as-deposited films (left) and SEM image of
pyrolyzed films (right) made with BCP/PCP gel. (b) Schematic of
the synthesis process for Hierarchical20. SEM images of pyrolyzed
freeze-cast structures of (c) Freezecast20 and (d) Hierarchical20, with
(e) enlarged lamellar spacings and (f) porous conformal coating on
the freeze-cast pore wall of Hierarchical20. . . . . . . . . . . . . . .
Pore size distribution of Hierarchical20 and Freeezecast20 pore structure from MIP. Smoothing of the data, indicated by solid lines, was
performed by averaging sets of five consecutive points minus outliers.
Stages of the lamellar spacing analysis from X-ray CT (A tomogram of Hierarchical20 is used as an example) (a) Original 2D image
and segmented image (b) Euclidean distance map (c) 2D skeleton of
lamellar pores (d) Graphic representation of lamellar spacing measurements (e) Average lamellar spacings from image analysis of 2D
X-ray tomograms. . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) Permeability as a function of open porosity and (b) compressive
strength and Young’s modulus as a function of relative density of
all Freezecast and Hierarchical20 samples. Error bars represent ±1
standard deviation. Dotted lines and the dashed line are linear regressions on log-log plots of baseline materials, included to guide
the reader. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Strain at maximum stress as a function of relative density of all
Freezecast and Hierarchical20 samples. Error bars represent ±1
standard deviation. Dotted lines is a linear regressions of baseline
materials, included to guide the reader. . . . . . . . . . . . . . . . .

68

69

78

80

82

84

84

xiv
5.6

6.1

6.2

6.3

A.1
B.1

D.1

E.1

Representative stress–strain curves of (a) Freezecast20 and Hierarchical20 and (b) Freezecast15, Freezecast25, and Freezecast35. SEM
micrographs of fracture surfaces of (c) Freezecast15 and (d) Freezecast35. Examples of fractured samples after compression of (e)
Freezecast15 and (f) Freezecast35. . . . . . . . . . . . . . . . . . . . 86
Compressive strength as a function of relative density of freeze-cast
pure SiOC from the CNT-reinforced, the SiC-reinforced, and the
BCP-hierarchical porous solid projects. . . . . . . . . . . . . . . . . 91
Compressive strength vs. permeability constant of CNT-reinforced,
SiC whisker-reinforced, and hierarchical composites, and corresponding fracture surfaces and macro-images of fractured samples. The
inset includes the stress-strain curves of freeze-cast pure SiOC of
various densities from BCP-hierarchical project for reference. . . . . 92
Compressive strength vs. permeability constant of CNT-reinforced,
SiC whisker-reinforced, and hierarchical composites compared with
that of pure SiOC reported by Naviroj [125]. . . . . . . . . . . . . . 94
Stress-strain curves of freeze-cast pure SiOC ceramics (black lines)
and 30 vol.% SiC composites (blue lines) . . . . . . . . . . . . . . . 120
A schematic of (a) the surface energy difference, and (b) the surface
area of the dendrite in contact with the whisker during the engulfment
process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Predicted rotations based on the model and examples of the rotation
in the solidification video. Each image sequence is arranged chronologically with (a) original images and (b) images with green lines
marking the whisker configuration. . . . . . . . . . . . . . . . . . . 125
(a) Freezing front velocity and (b) lamellar spacing as functions of
sample height. (c) Lamellar spacing as a function of freezing front
velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xv

LIST OF TABLES
Number
Page
2.1 Melting point, heat of fusion, entropic term of Jackson alpha-factor,
and pore morphology of common solvents. Adapted from [125]. . . . 9
3.1 Composition of suspension and porosity of the pyrolyzed porous
composites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Values of input variables for simulation. . . . . . . . . . . . . . . . . 31
3.3 Comparison of the engulfed whisker ratio between the freeze-cast
samples and freezing simulation . . . . . . . . . . . . . . . . . . . . 49
4.1 Compositions of suspensions and pyrolyzed samples. . . . . . . . . . 54
5.1 Chemical composition (in weight percent) of MK- and SMP-10derived ceramics. The elemental analyses were conducted with a
thermobarometrical redox analysis [154, 185] and by a carbon analyzer [86]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Average porosity after pyrolysis of all Freezecast and Hierarchical20
samples. Six samples were tested for each polymer concentration. . . 75
5.3 Volume and weight changes (in percent) after each processing step of
Freezecast20 and Hierarchical20. At least five samples are measured
for each step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Chapter 1

INTRODUCTION
1.1

Motivation and Objectives

Pores in ceramic materials have an immense influence on their mechanical properties due to their brittle nature compared to their counterparts in polymers and metals.
Nevertheless, ceramics have unique chemical resistance, high-temperature stability,
and high strength otherwise unachievable by other materials, making porous ceramics attractive candidates for filters, electrodes, bio-scaffolds, and catalyst supports
[42, 46, 58, 71]. Each of these applications has different property requirements,
such as its mechanical strength, surface area, or fluid permeability. These properties
are determined by pore characteristics—pore size, morphology, and connectivity,
etc. Therefore, it is crucial to have the capability of tuning the pore structure to
optimize the performance of a final product for a given application.
For transport or filter applications, good mechanical properties ensure product robustness against the fluid backpressure during the operation; high permeability
increases product throughput; high surface area creates more reaction or capture
sites. However, to increase surface area, structures with micropores (<2 nm) or
mesopores (2-50 nm) are preferable but these create high fluid backpressure that
demands higher filter strength and more energy to operate compared to the structures
with macropores (>50 nm). Furthermore, macroporous materials often have better
mechanical and chemical stability [37, 76, 114].
The aforementioned challenges can be mitigated with pores at multiple length-scales
by constructing a hierarchical porous structure. For example, a high surface-area
mesoporous coating for catalytic functions and an underlying macroporous frame
that provides the required mechanical strength and transport property can be coupled.
In this case, an optimal combination of mechanical strength and transport property
requires the capability of tuning the underlying macroporous scaffold.
Common methods to create macropores (>50 nm) in ceramics include additive
manufacturing, direct foaming, the replica technique, and sacrificial templating [54,
169]. Additive manufacturing is capable of creating complex structures precisely,
but the process currently lacks industrial scalability. Direct foaming is inexpensive,
fast, and easy, but only isotropic, spherical pore structures are achievable. The

replica technique is simple and well-established, but the mechanical properties are
usually poor due to the hollow struts in the structures. Sacrificial templating methods
provide a variety of pore morphologies and sizes, but extra care is needed during
the removal of template to avoid cracking.
Among sacrificial template methods, freeze casting is particularly attractive as it
offers tailorability in pore size, morphology, wall thickness, tortuosity, and pore
alignment by controlling solidification [32, 47, 48, 61, 146]. Furthermore, freeze
casting is readily scalable for the production of porous ceramics with complex
structures [36].
Freeze casting utilizes the phase separation between a solvent and dispersed solid or
solute during solidification to template the pore structure. The solvent crystallizes
in different sizes, shapes, and alignment based on the nature of the solvent and
the conditions of solidification. The solvent crystals are subsequently removed
by sublimation, leaving behind a pore network. The chemistry and the physical
properties of the solids, the dispersing or dissolving component, affect both the
solidification process and the surface chemistry of the final product. Polymers,
metals, and ceramics have all been used as solids in freeze casting. Moreover,
more than one type of solid can be incorporated to fabricate composites. The
versatility and tunability make freeze casting an attractive technique for tuning the
pore structure to meet the requirements of each application.
In particular, freeze casting with preceramic polymers to fabricate porous ceramic
has garnered increasing interest as it offers many processing advantages over that
with ceramic powders, including reduced temperatures needed for densify the ceramic walls, better control over freezing front velocity, and greater alignment of the
pore structure [4, 124–127, 193].
While freeze casting is useful for tuning the pore structure, optimizing the structure
for adequate mechanical and transport properties remains a challenge, as they are
often inversely related to each other through porosity. Highly porous structures have
low strength and vice versa [4, 125].
Therefore, the goal of this research is to provide strategies for creating optimized
pore structures that provide sufficient strength, permeability, and surface area under
the framework of freeze-casting. Such strategies are based upon composite material
principles of both particle reinforcement and hierarchical design. First, to optimize
the mechanical and transport properties, we reinforced a high-permeability pore

structure by reinforcement fillers of silicon carbide (SiC) whiskers and carbon nanotubes (CNTs). The two fillers afford two different mechanisms of reinforcement of
pore structures: structural reinforcement across pore walls and material reinforcement within the pore walls. In the former, we explore the phenomenon of whisker
engulfment by the freezing solvent to produce interwall bridges. Second, to increase
the pore surface area, a high-surface-area conformal coating was created on a pore
scaffold. This scaffold can be tuned by the methods outlined in the first part of the
thesis to achieve the desired mechanical and transport properties, and its surface
area can be increased by the high-surface-area coating. With these strategies, a pore
structure can be tuned to meet the requirements for specific applications.
1.2

Thesis Organization

This document is organized as follows: Chapter 2 provides the necessary background on freeze casting, self-assembly block co-polymers, and preceramic polymers. Chapter 3 introduces the structural route of reinforcement with silicon carbide
whisker fillers, a modified particle engulfment model for freeze casting high-aspect
ratio particles, and a simulation based on the proposed engulfment model. The properties of resultant pore structures, including permeability and compressive strength,
are examined. The experimental results are compared with the theoretical results
based on the proposed model. Chapter 4 investigates the material route of reinforcement with carbon nanotubes as reinforcement fillers. Requirements of reinforcement
fillers for the material route are discussed. The processing and dispersion methods
are described, as well as the effect of CNTs on the microstructure, electrical conductivity, permeability, compressive strength, and toughness. Chapter 5 discusses
the design and processing of a hierarchical structure via self-assembly of the block
copolymer. The processing, and resultant microstructures, compressive strength,
permeability and surface area of the hierarchical pore structures are featured. Chapter 6 summarizes the results, compares the mechanical and transport properties of
the three porous composites in this work, and concludes with suggestions for future
study.

Chapter 2

BACKGROUND
2.1

Freeze Casting

Freeze casting utilizes phase separation and particle rejection during solidification
to create pore structures. The process of freeze casting contains four steps, as shown
in Fig. 2.1.
In the first step of the process, powders/polymers are dispersed/dissolved in a dispersing medium/solvent (1a and 1b). The solution or suspension is then cast into a
mold and frozen under a thermal gradient. Phase separation between the dispersing
medium/solvent and the dispersed/dissolved phase occurs. The solvent crystals grow
along the direction of the thermal gradient, rejecting the dispersed/dissolved phase
to form what will ultimately become pore walls (2). The frozen suspension/solution
is sublimed under vacuum, leaving behind a porous body, which is a negative of the
solvent crystals (3). Finally, the porous body requires sintering or pyrolyzation to
densify or convert the polymer into ceramics if necessary (4a and 4b).

Figure 2.1: Schematic of the freeze casting process showing solution/suspension
preparation, solidification, sublimation, and sintering/pyrolysis steps [125].

Although the process of freeze casting requires only four steps, there are plentiful
opportunities for tuning the resultant pore structure and its properties [46, 48, 71,
118, 125, 133, 178]. For example, solid contents and solvent/dispersing medium
are readily adjustable in the solution/suspension preparation step. Freezing front
velocity and thermal gradient during solidification strongly influence the resultant
pore structures. In addition to those two processing parameters, external forces
and nucleation-controlled tools may be used during the process of freezing. Postfreezing treatments such as coarsening change the pore morphology and pore size.
In the sintering/pyrolysis step, firing temperature and atmosphere alter the porosity
and chemistry of the pore structure and can facilitate the growth of nanowires
via carbothermal reduction. Post-freeze casting processes can also be applied to
introduce a functionalized surface, or to increase the surface area. These processing
parameters are discussed in detail in the following sections.

2.1.1

Solid Contents

For ceramic processing, both ceramic powders and preceramic polymers are suitable
ceramic precursors. Freeze casting with powders allows a wider variety of starting
materials for use in bone scaffolds [102], sensors [190], or fuel cells [110]. Magnetic
powders are used along with an external magnetic field for a better pore alignment
[137].
Freeze casting with preceramic polymers produces pore structures with smoother
pore walls, due to the amorphous character of the pyrolyzed ceramic, and a higher
anisotropy. Preceramic polymers also have more processing advantages compared
to powders, including a lower sintering/pyrolysis temperature and transparency upon
dissolution, and therefore enable a precise control of the freezing front velocity as a
result of good contrast between the frozen solvent and liquid solution [126].
It should also be noted that metal or polymer starting materials, though less popular
than ceramics, are also employed in freeze casting [103, 191]. Freeze-casting
with metal powders often produces structures with high oxygen content (sometimes
intentionally), which are later reduced with an additional reduction step [137].
Increasing solids loading decreases the porosity, decreases the pore size, and increases the width of pore walls. These three effects all contribute to better mechanical properties [156]. However, the decrease in porosity has an adverse effect on
transport properties [160]. Higher solids loading decreases pore connectivity [125]
and also results in high viscosity, posing a challenge to obtain stable dispersions or
suspensions for the processing [29, 105].
Smaller particles replicate the features of solvent crystals in higher resolution [46]. In
addition, freeze casting with small particles generally creates stronger pore structures
as the small interparticle pores are more likely to close during sintering [204]. Large
particles are also more likely to be engulfed by the freezing front, not pushed away
to form pore walls. Upon engulfment, the phase separation phenomenon that freeze
casting relies upon ceases to happen, resulting in isotropic pore structures shown in
Fig. 2.2. However, small particles are difficult to disperse and raise the viscosity of
suspension rapidly as they have a high surface area to volume ratio, which can be
detrimental to the processing [105].
Faceted particles or high-aspect ratio particles are used in freeze casting for their
ability to enhance mechanical, thermal or electrical properties. Carbon nanotubes
(CNTs) are used to create conductive and bio-compatible pore structures, ideal for

Figure 2.2: Critical freezing front velocity, above which particles are engulfed and
below which particles are pushed by the freezing front, as a function of particle
size and the respective resultant pore structures formed from particles pushed or
engulfed by the freezing front. Adapted from [30] with permission from Elsevier.

a microbial fuel cell [73]. Alumina platelets introduce reinforcing interpore bridges
[63]. Silicon carbide (SiC) whisker freeze-cast structures are strong, lightweight,
and semi-conductive, and have good thermal conductivity [57].
2.1.2

Solvents and Additives

Since the pore structure is a negative of the solvent crystals, pore morphology is
controlled by the solvent crystal structure. The shape of the freezing front, i.e. nonfaceted (Fig. 2.3(a)) versus faceted (Fig. 2.3(b)), is largely determined by whether
the freezing interface is able to accommodate a larger number of ad-atoms. If only
specific sites such as kink sites at a step edge on the interface are energetically
favorable, the interface grows laterally, advancing one layer at a time, as shown in
Fig. 2.4. This type of growth mechanisms results in an atomically smooth interface
and a faceted crystal that has a preferred growth orientation [67].
An important parameter to predict whether a melt solidifies with faceted or nonfaceted interface is the Jackson alpha-factor, 𝛼. Jackson and Hunt [80] developed a
simple two-layer Bragg–Williams interface model, depicting a solid-liquid interface
that is composed of two layers. The equilibrium configuration of ad-atoms and

Figure 2.3: (a) Non-faceted [113] (Copyright (1998) National Academy of Sciences,
USA) and (b) faceted [81] (Reprinted with permission from Elsevier) cellular arrays.

interfacial vacant sites is determined by the energy of the system, a function of the
ad-atom coverage, 𝜉 and the Jackson alpha-factor, 𝛼.
Melt systems with 𝛼 ≤ 2 have an energy minimum at 𝜉 = 0.5, indicating equal
densities of ad-atoms and interfacial vacant sites. The interface is then atomically
rough. Energy minima occur near 𝜉 = 0 and 𝜉 = 1 for materials with 𝛼 > 2. The
configuration with 𝜉 ∼ 0 describes an interface with sparse ad-atoms and 𝜉 ∼ 1
implies an interface almost completely covered with only a few vacant sites. Both
situations depict an atomically smooth surface.
The Jackson alpha-factor is composed of two terms, the entropic terms from thermodynamic and the geometric term from crystallography [5].
𝛼≡

𝑘 𝐵𝑇𝑚 𝑍

(2.1)

where 𝜂 is the number of in-plane nearest neighbor sites of an atom on the interface,
Z is the total number of nearest neighbors of an atom in the crystal, L is the
latent heat, 𝑘 𝐵 is the Boltzmann constant, and 𝑇𝑚 is the melting temperature. The
geometric term depends on the crystal structure (or the coordination number) and the
orientation of the interface, ranging from 0.25 to 1. The entropic term is calculated
from the melting point and the latent heat of the solvent.

Figure 2.4: (a) Schematic of a faceted solid-liquid interface showing A. a kink-site,
B. an atom/molecule adding to the kink-site, C. an interfacial vacancy, D. an adatom, E. exposed underlying lattice plane, and F. an island of ad-atoms (b) System
energy as a function of ad-atom coverage, 𝜉. The change in the relative position
of the energy minima demonstrate a transition from an atomically rough interface
(𝛼 < 2) to an atomically smooth interface (𝛼 > 2) [67]. Reprinted by permission
from Springer Nature.

Table 2.1: Melting point, heat of fusion, entropic term of Jackson alpha-factor, and
pore morphology of common solvents. Adapted from [125].
Melting point Heat of fusion Entropic term,
(℃)
(kJ/mol)
𝐿/𝑘 𝐵𝑇𝑚
Cyclooctane
14.6
2.41
1.01
Cyclohexane
6.5
2.68
1.15
t-Butanol
25.3
6.70
2.70
Dimetheyl carbonate
3.5
13.22
5.75
Solvent

Pore
morphology
Isotropic
Dendritic
Prismatic
Lamellar

Naviroj [125] revealed a striking correlation between Jackson alpha-factor, solvent
crystal structures, and the resultant freeze-cast pore structures. Table 2.1 shows that
cyclooctane has the lowest entropic term among the chosen solvents, followed by
cyclohexane, t-butanol, and dimethyl carbonate. Fig. 2.5 shows optical images of
the crystal structures and SEM images of the freeze-cast pore structure from these
solvents. Solvents with a particularly low entropic term such as cyclooctane produce
seaweed-like crystal structures. The anisotropy of the seaweed-like structure is even

10

Figure 2.5: Optical images of the crystal structures formed from different solvents
and the SEM images of their respective freeze-cast pore structure. Adapted from
[125].

lower than that of the dendritic structures, resulting in an almost isotropic pore
structure with freeze casting [2]. Freezing with low-entropic term solvents like
cyclohexane results in a dendritic freezing front during solidification and dendritic
pore structures [91].
As the anisotropy of the solvent increases, the crystal structure shows more faceted
features and the resultant pores transitions from dendritic to lamellar. The prismatic
structure from freezing t-butanol can be seen as a transitional state between dendritic
and lamellar structures. The structure is neither unidirectional as dendritic structure
nor is it bi-directional like the lamellar structure, reflecting a medium Jackson
alpha-factor.
Finally, the highest-anisotropy solvent forms highly faceted crystals under larger
supercooling to overcome the sparsity of the atom attachment site on the interface.
The pore structure created by a high-anisotropy solvent such as dimethyl carbonate
exhibits a series of smooth, parallel, and two-dimensional lamellae. The Jackson

11
alpha-factor is a good indicator of the growth mechanism and the anisotropy of the
solidifying interface/crystals. In this thesis, dimethyl carbonate is used predominantly. However, if another pore morphology is desired, a different solvent can be
chosen based on the Jackson alpha-factor to obtain that specific pore structure.
Additives and co-solvents are often used to alter the crystal formation, and consequently, the pore structure. Glycerol is a common cryoprotectant to reduce the crystal
size, create smaller pores and increase interlamellar connections [200]. Polystyrene
serves as a structuring agent and improves pore alignment of camphene freeze-cast
structure [92]. Various alcohols, such as methanol, ethanol, and isopropanol (IPA),
have been added to aqueous freeze-casting systems. Complex freeze-cast structures
with more interlamellar bridges are obtained as a result of multi-phase systems
and the interaction between alcohol additives and other functional additives such
as dispersants and binders [119]. Large freezing point depression is present in
co-solvents near the eutectic point [103]. A reduction of the pore size from 20 𝜇m
to sub-microns is achieved with a more than 30℃ freezing point depression via a
dioxane and dimethyl sulfoxide co-solvent system [87].
2.1.3

Solidification and Post-freezing Treatments

Solidification has direct influences on the resultant pore structure as the size and
alignment of the solvent crystal links to those of the pore structure. In Section 2.1.2,
common pore morphologies—dendritic, prismatic, and lamellar— were discussed.
The major difference between them is the anisotropy of the solvent crystals.
Here, a distinction between the anisotropy and the instability of a crystal structure
should be made. The anisotropy determines whether the crystal is faceted while the
instability categorizes the crystal into planar, cellular, and dendritic regimes shown
in the microstructure selection map (Fig. 2.6). Both faceted and non-faceted crystals
can be in any of those regimes. For example, Fig. 2.3 (a, b) are both cellular arrays
compared to a planar freezing front, yet (a) is non-faceted while (b) is faceted,
which is related to the anisotropy. The nomenclature for pore structures refer to
pores templated by faceted freezing dendrites as prismatic and lamellar structures,
and those by non-faceted dendrites as "dendritic".
The microstructure selection map (Fig. 2.6) shows that different morphologies and
crystal sizes can be accessed by two processing parameters, temperature gradient
and crystal growth rate (freezing front velocity). Higher freezing front velocity
gives rise to finer solidification structures, resulting in finer freeze-cast structures,

12

Figure 2.6: Microstructure selection map of crystal morphology and size depending
on temperature gradient and growth rate [93] © 2003 John Wiley & Sons, Inc..

and ultimately, smaller pore sizes [49, 156]. Although increasing freezing front
velocity and solids loading both reduce pore size, the resultant pore structures are
different in porosity, pore wall thickness, and pore connectivity [124].
In a perfectly precise world, the freezing front is planar and at the freezing-point
isotherm. However, thermal fluctuations cause interfacial instabilities, exacerbated
by the constituent supercooling or undercooling arising from dissolved solutes in
the alloy or multi-component system.
In such systems, solubility is higher in the melt than in the crystal. Solutes are ejected
in front of the freezing front when the solution solidifies into crystals, piling up into
a concentration gradient in front of the advancing freezing front. Due to freezing
point depression from solutes, the freezing point increases with distance from the
interface. If the gradient of the freezing point exceeds the temperature gradient, the
freezing front becomes unstable and transitions from planar, to cellular, and finally
to dendritic (Fig. 2.7). This phenomenon has been thoroughly discussed by Mullins
and Sekerka [121] and Losert et al. [113].

13

Figure 2.7: Microstructural transition from planar to cellular to dendritic freezing
front (from top to bottom) in a succinonitrile and coumarin 152 binary alloy [113].
Copyright (1998) National Academy of Sciences, USA.

The direction and magnitude of the thermal gradient during solidification determines
the pore alignment. Solvent crystals freeze in a direction influenced by the thermal
gradient and the preferred growth orientation of dendrites. For two-dimensional
lamellar structures obtained by freeze casting with water or dimethyl carbonate, a
dual thermal gradient is often employed to obtain pore alignment in both directions.
Bai et al. [8] introduces a dual thermal gradient by placing a wedge between the
solution and the cooling device. The resultant pore structure shows a long-range
pore alignment up to 5 mm.
Wang et al. [181] created a highly radial and centrosymmetric graphene oxide pore
structure by prescribing a radial thermal gradient combined with a second gradient

14

Figure 2.8: Solvent nucleation control via a wettability gradient enables a long-range
pore alignment from [201]. Reprinted with permission from AAAS.

perpendicular to the first one. Pore alignment is further improved by limiting
nucleation during solidification via a grain selector [127] or a wettability gradient
[201].
In addition to the processing parameters discussed previously, external forces such
as magnetic, electrostatic, or mechanical (acoustic) force have been used in freeze
casting [128]. Fig. 2.9 (a) shows lamellar structures with a long-range alignment
by applying a static or rotating magnetic field on the ceramic slurries containing
iron oxide (Fe3 O4 ) powders during freezing [137]. Tang et al. [173] demonstrates
that with an aqueous alumina slurry without magnetic particles, the magnetic field
reduces the pore size while the electric field creates aligned lamellae by influencing
polar water molecules (Fig. 2.9 (b)). Acoustic waves from ultrasound create a
tree-like composite structure with alternating dense and porous concentric rings,
resistant to crack propagation (Fig. 2.9 (c)) [129].

15

Figure 2.9: SEM images showing (a) the effects on alignment of a magnetic field
on TiO2 scaffolds containing 3 wt% Fe3 O4 (reprinted from [137] with permission
from Elsevier) (b) the effects on alignment of an electric field on aqueous slurries
(reprinted from [173] with permission from Elsevier), and (c) the effects of acoustic
waves on mechanical properties and pore structure [129].

2.1.4

Particle Engulfment

As previously stated in Section 2.1.1, small particles are more resistant toward
engulfment by the freezing front. The two most common variables in particle
engulfment are the particle size and the freezing front velocity. For a given particle
size, there exists a critical freezing front velocity, above which particles are engulfed
and below which particles are pushed by the freezing front.
During the process of solidification of a suspension, particles experience forces
including a drag force from fluid motion, a repulsive force from surface energy, a
buoyancy force, the gravitational force, an osmotic force, and other applied forces
(magnetic, electric, mechanic, etc.) [128, 135]. The balance between the forces
determines the critical freezing front velocity for engulfment.
Early engulfment models explored the solidification conditions for spherical particles and planar freezing fronts [175]. The planar freezing front develops a slight
curvature when the thermal conductivity of the particle differs from that of the
solvent [89] due to the change in heat flux. Shangguan et al. [161] established
the mathematical relation between the curvature of freezing front and the relative

16
thermal conductivity between the particle and the solvent. The shape of the freezing
front could cause order-of-magnitude differences in the drag force.
Particles near the freezing front could obstruct the ejection of solutes or impurities
into suspensions, resulting in a solute pile-up. The planar freezing front is concave
near the particle as a result of freezing point depression caused by the pile-up [138].
Other factors, such as particle roughness, the Gibbs-Thomson effect, and particle
wettability, may also influence particle engulfment [19, 26, 134].
If particles are continuously pushed by a planar freezing front, a compacted particle
layer builds up in front of the freezing front, exerting an extra viscous force on the
particles from particle-particle interaction [151].
If the freezing front is dendritic, a unique dendritic tip radius can be determined
for a given freezing front velocity and a thermal gradient along with the solute
concentration [14, 17, 33, 98]. The relative location of particles and dendrites
affects the outcome of the particle-freezing front interaction. Particles between
dendrites are trapped in the inter-dendritic space rather than being engulfed or
pushed by the freezing front, termed particle entrapment [52, 53].
Bouville et al. [20] investigates the interaction between high-aspect ratio particles
and a lamellar freezing front by modeling faceted particles as clusters of spheres.
They deduce that a self-aligning effect on particles comes from compacting motion
of thickening dendrites after particle entrapment. The proposed mechanism cannot
explain interlamellar bridges seen in freeze-cast structures of faceted particles, nor
can it reproduce particle engulfment. The model simplifies the drag force of faceted
particle to be the sum of the drag force of individual spheres regardless of the
different flow motion arising from having faceted particles. While much effort
has been expended on particle engulfment theory, freeze casting high-aspect ratio
particles has yet to be properly modeled. This topic is a promising area for research.
2.1.5

Sintering/pyrolysis and Post-freeze Casting Treatments

The porosity of the pore wall structure from ceramic powders is controlled by the
sintering temperature and time. Increasing sintering temperature promotes closure
of closed pores in the freeze-cast pore walls, decreasing the overall porosity and
increasing the mechanical strength of the walls, and therefore, the structure [59]. In
contrast, partial sintering or sintering at lower temperatures contributes to higher
porosity and surface area [142, 159].
During pyrolysis, preceramic polymers start to degrade around 400℃, opening mi-

17
cropores (<2 nm) in the structure, as described in detail in 2.3.1. As the temperature
increases to 800-1200℃ polymers are completely transformed into ceramics and the
micropores close up. Wilhelm et al. [186] utilize this property of preceramic polymer and fabricates a polymer-ceramic hybrid material, ceramer, raising the surface
area of ceramers close to that of activated carbons.
Catalysts in preceramic polymers promote the formation of nanowires on the surface of pore walls. The pyrolysis atmosphere is shown to affect the type of
nanowires formed. Pyrolysis in nitrogen produces silicon nitride nanowires while
SiC nanowires are obtained pyrolyzed in argon [177, 178]. Post-sintering processes,
such as deposition of a mesoporous (2-50 nm) layer, increase the surface area of the
underlying pore scaffold [39].
2.2

Self-assembled Block Copolymers

A block copolymer (BCP) is a copolymer composed of blocks of repeated units. In
the melt, diblock and triblock copolymers often self-assemble into structures with
nanosized features due to the energy balance between the interfacial energy and the
core-chain stretching. The size and the morphology of such features depend on the
relative fraction of each block and the block length [25, 116].
However, in a dilute solution, the polymer and solvent interaction is dominant
[82]. A symmetric ABA triblock copolymer system is especially of interest for
its capability to generate a thermoreversible gel [120, 157]. In the process of
preparing a thermoreversible gel, an ABA triblock copolymer is dissolved in a
midblock selective solvent at an elevated temperature where both A and B blocks
are dissolved in the solvent. As the temperature decreases to the critical micelle
temperature (CMT), the A endblocks aggregate into micelles while midblocks being
well-solvated and connecting micelles, forming an interconnected network. If the
temperature decreases further, the system goes through a glass transition. The
endblocks are no longer free to exchange between micelles. The block copolymer
system goes from the viscous to the viscoelastic, and finally to the elastic mechanical
state as the temperature decreases from above CMT, to below CMT, and to below the
glass transition temperature, as shown in Fig. 2.10 [51, 157]. The process is reversed
upon heating, hence the name thermoreversible gel. The thermally reversible nature
of thermoreversible gelcasting allows for a longer processing window—the system
can remain in the liquid state as long as needed above CMT—and for recyclablility
from failed casts.

18

Figure 2.10: Schematic of the structure and the mechanical property of PMMAPnBA-PMMA triblock copolymer gel as a function of temperature. Reproduced
from [51].

Common triblock copolymer systems include styrene- and acrylic-based block
copolymers. Styrene-based systems are used as pressure-sensitive adhesives [88],
substrates for microfluidic systems [170], and gate insulators [31]. In addition to
aforementioned applications, acrylic-based systems find uses in porous ceramic processing [56]. Poly(methyl methacrylate)-poly(n-butyl acrylate)-poly(methyl methacrylate) or PMMA-PnBA-PMMA is particularly advantageous among acrylic-based
systems because its CMT is above room temperature but lower than 100℃, so a water bath is sufficient to induce the whole range of transition [157]. In the process of
creating a porous ceramic with the block copolymer, ceramic powders are dispersed
in the block copolymer solution above CMT. The ceramic-block copolymer solution
is then cast into the desired bulk shape. As the temperature decreases below CMT,
the solution gels and retains the shape of the mold. Finally, the porous ceramic is
achieved after subsequent drying, burnout of block copolymer, and sintering [162].
In theory, the block copolymer can be used as a sacrificial template for pores less
than 100 nm. In reality, the relative size of the template features to that of the
precursor particles determines the resolution of the negatives/replicates [46]. It is
difficult to obtain well-dispersed suspensions with nanoparticles small enough to
replicate the features of the self-assembled block copolymer where the size of the

19

Figure 2.11: (A) Chemical structures of PMMA-PnBA-PMMA, polycarbosilane,
and PMMA. (B) Schematic of resultant morphologies of the gel system and (C) the
corresponding atomic force microscopy (AFM) images. Scale bars are 100 nm [21].

features is often in the range of 10 to 100 nm [105]. Porous structures created
with thermoreversible gel casting with ceramic powders typically require additional
sacrificial fillers to serve as templates [56, 162].
One could overcome this obstacle by choosing preceramic polymers as precursors
instead of powders. Preceramic polymers have a radius of gyration of less than 10
nm [55, 105, 192], small enough to replicate the features of the self-assembled block
copolymer. Rueschhoff et al. [147] create porous ceramic films by depositing a
thin film of a PMMA-PnBA-PMMA and polycarbosilane blend. The self-assembled
BCP acts as a template for the polycarbosilane and burns away while polycarbosilane is converted into SiC after pyrolysis. Pore morphologies such as interconnected
micelles, worm-like, and bicontinuous structures (shown in Fig. 2.11) are obtained
by varying the PMMA monomer concentration and relative block length of PMMA
to PnBA. The PMMA-PnBA-PMMA and polycarbosilane blend that produces interconnected micelles is used for the study in this thesis.

20
2.3
2.3.1

Preceramic Polymers
Overview of Polymer-derived Ceramics (PDCs)

Traditionally, ceramic precursors are ceramic powders that can be molded and
sintered into dense ceramics. However, since 1960, ceramics have also been synthesized by the pyrolysis of preceramic polymers. Preceramic polymers, as the name
suggests, are polymers that are converted into ceramics at elevated temperatures
[18, 36, 123].
Preceramic polymers have numerous advantages compared to ceramic powders or
other molecular precursor routes. Since preceramic polymers are in the polymeric
state before the final conversion process, they open avenues of polymer processing
techniques, such as injection molding, extrusion, fiber drawing, and additive manufacturing, that are difficult to use with ceramic powders. They are also readily
machinable and prevent tool wear and fracture from machining finishing ceramic
components. The radii of gyration of polymers are often smaller than the sizes
of commercial ceramic powders, increasing the resolution of replicating templates
[55, 105, 192]. Pyrolysis temperatures are generally lower than the sintering temperatures [126]. Moreover, reducing the residual porosity often requires high pressures
during ceramic powder sintering, increasing the processing cost.
Preceramic polymers do not have drying problems common to the sol-gel process,
which allows almost net-shape forming with fillers [69]. The commercial preceramic
polymers are relatively stable and safe to use, one part due to low toxicity compared
to monomers used in gelation and the other part because of flammable solvents are
not necessary for the processing like ceramic slurries. Theoretically, preceramic
polymers have an infinite processing window as they can be processed in the melt
and do not require gelation to hold the final shape [36].
These advantages make preceramic polymers a facile precursor for ceramic composites [136, 152], coatings [40, 174], and porous ceramics [35, 95, 179], and in
tape casting [68, 75], and additive manufacturing [29, 108, 182].
Preceramic polymers can be grouped into different categories shown in Fig. 2.12
based on the group in the polymer backbone and the sides groups in the polymer.
The following sections focus on the two preceramic polymers used in this study,
polysilsesquioxane (a subtype of polysiloxane, CH3 -SiO1.5 ) and polycarbosilane.
Typically, the processing of preceramic polymers to polymer-derived ceramics
includes three steps—shaping and cross-linking, pyrolysis, and crystallization—

21

Figure 2.12: Typical classes of silicon-based preceramic polymers as precursors of
polymer derived-ceramics [36] © 2010 The American Ceramic Society.

shown in Fig. 2.13. Most polymer shaping techniques are conducted at temperatures below 300℃. Before or during the shape-forming step, the polymers are
often cross-linked to form a thermoset and retain its shape via condensation or
addition. Initiation of cross-linking can be achieved by heating or by adding catalysts [10, 72, 180]. Other cross-linking methods include oxidative, e-beam, and
UV curing but they often come with size limitations or unfavorable side-effects
[36]. Cross-linking determines the rheology of the melt and the yield of PDCs
[123, 172, 188].
Preceramic polymers (PCPs) start decomposition at about 400-600℃, emitting carbon and hydrogen-based oligomers. Pyrolysis is accompanied by a volume shrinkage, a weight loss, and the formation of micropores [176]. These are problematic
for fabricating dense bulk materials as a shrinkage and porosity often facilitate crack
formation. Fillers can be used to ameliorate this problem [75]. Pyrolysis of PCPs
completes at 800-1100℃, converting polymers into amorphous ceramics, often with
short-range orders or nanodomains. Both polysilsesquioxane and polycarbosilane
are converted into silicon oxycarbide (SiOC) after pyrolysis at temperatures below 1100℃. Pyrolysis or heat-treatment at temperatures above 1100℃ promotes
crystallization and converts amorphous SiC to 𝛽-SiC with large volume shrinkage.

22

Figure 2.13: Schematic of the processing and chemical structures of preceramic
polymers to polymer-derived ceramics [68, 125].

2.3.2

Polysilsesquioxane

Cross-linking of polysilsesquioxane occurs at 100-150℃ without a catalyst [10,
189]. Harshe et al. [72] showed that mass loss occurs between 500℃ and 1100℃ in
the pyrolysis of polysilsesquioxane polymer, accompanied by the emission of water,
ethanol, and methanol. The loss of Si-CH3 and C-H bonds at 1000℃ analyzed from
Fourier-transform infrared spectroscopy (FTIR) [125] signals the completion of the
polymer-to-ceramic transition. SiOC pyrolyzed at 1300℃ are shown to be X-ray
amorphous and forms crystalline 𝛽-SiC at temperatures beyond 1300℃.
Two models of the microstructure of polysiloxane-derived SiOC have been proposed
as shown in Fig. 2.14. Model 1 describes the SiOC microstructure as silica
nanodomains of silica tetrahedra surrounded by interdomain boundaries of graphene
layers and the SiC𝑥 O4−𝑥 mixed bonds between silica and graphene layers [150, 153].
Model 2 depicts an interconnecting network of the oxygen-rich SiC𝑥 O4−𝑥 units with
the carbon-rich SiC𝑥 O4−𝑥 units filling the space between the network and the free
carbon nanodomains. In Model 2, free carbon may form spatially bicontinuous
nanodomains if there is more excess carbon as in carbon-rich SiOCs. Recent
evidence of experimentally characterized the mass-fractal dimension of the oxygenrich SiC𝑥 O4−𝑥 units is more consistent with Model 2 [184].
Silica in the SiOC remains amorphous up to 1500℃ despite the fact that pure silica
crystallizes into cristobalite at 1000℃ [144, 183]. The inhibition of crystallization

23

Figure 2.14: Two proposed models of the microstructure of polysiloxane-derived
SiOC. Reprinted from [183] with permission from Elsevier.

is likely due to the presence of free carbon [94, 144, 166].
2.3.3

Polycarbosilane

The majority of mass loss in the pyrolysis of polycarbosilane, arising from the
escape of hydrogen, methane, and silanes, occurs from 200 to 800℃. Mass loss
and volume shrinkage remain fairly constant between 800-1200℃ with no gas
evolution, indicating the completion of pyrolysis of polycarbosilane at 800℃ [100].
Comparing FTIR peaks of cross-linked polycarbosilane and that of its PDCs at
800℃ demonstrates that C-H, Si-H, CH2 peaks disappear, confirming the transition
to ceramic state [147].
The microstructure of amorphous SiOC from polycarbosilanes (with oxygen contamination during processing) consists of amorphous SiC nanodomains and free
carbon nanodomains. Oxygen contamination results in SiC𝑥 O4−𝑥 units filling the
space between these two nanodomains [183]. Above 1300℃, precipitation of
nanoscale 𝛽-SiC crystallites starts. However, the complete transformation into bulk

24
SiC requires the decomposition of O-rich regions and the destruction of the free
carbon network, occurring at 1400-1500℃ [136]. Chemical compositions of PDCs
from polysilsesquioxane and polycarbosilane are listed in Table 5.1.

25
Chapter 3

SIC WHISKER-REINFORCED COMPOSITES
Material in this chapter is reproduced in part from "Porous SiC-SiOC composites
through particle engulfment of high-aspect ratio particles during freeze casting",
C.T. Kuo, C.H. Keck, K.T. Faber; Manuscript in preparation. The work was done
in collaboration with Carl H. Keck and Katherine T. Faber. T. Kuo fabricated
and characterized SiC composites using SEM and MIP, their permeabilities, and
compressive strengths, developed the model and simulated solidification process
for freeze casting high-aspect ratio particle, and wrote the manuscript. C.H. Keck
calculated the bridge density and measured the viscosity of the polymer solution.
K. Faber supervised this work.
3.1

Introduction

As previously mentioned in Section 1.1, two of the most critical properties for most
applications of porous solids are strength and permeability. In freeze casting, the
most common directional pore structures are lamellar and dendritic [156]. Lamellar
structures have a higher permeability, but a lower strength compared to dendritic
structures [125]. They often fail in shear with lamellae slipping past one another.
Attempts have been made to increase compressive strength while retaining high
permeability of lamellar structures by introducing bridges between lamellar walls
to prevent shear failure [64, 97]. Bridges, in this case, were a result of large
Al2 O3 platelets in a fine Al2 O3 powder suspension or of carbon nanotube (CNT)
agglomerates in a CNT-polysiloxane polymer suspension.
The current theories behind the formation of interlamellar bridges are not sufficiently sophisticated to describe actual solidification conditions encountered in
freeze casting high-aspect ratio particles. Some early models adopted spherical
particles and planar freezing fronts or planar freezing fronts with a slight curvature
as a result of the differences in thermal conductivity between particles and the solvent [89, 161, 175]. In more recent models, impurity concentrations, particle size,
and the Gibbs-Thomson effect are included in determining the shape of freezing
fronts [26, 27, 138]. However, the curvature of the freezing front has always been a
response to the freezing front’s proximity to a particle. The modeled solidification
conditions in these works are common in zone refining, where a planar freezing

26
front is often characteristic.
In contrast to planar freezing fronts in zone refining, lamellar or dendritic freezing
fronts are necessary in freeze casting to induce phase separation to create pore
structures [30]. The curvature of the freezing front is not a response to the proximity
of the particles to one another in freeze casting, but is rather uniquely determined
by solidification conditions. Microstructure selection maps show the freezing front
morphology (planar, cellular, and dendritic freezing front) as a function of freezing
front velocity and temperature gradient [77, 99]. In the dendritic regime, a unique
dendritic tip shape can be determined for a given freezing front velocity and a
thermal gradient along with the solute concentration [14, 17, 33, 98].
As a result, the previous models often fall short in describing the solidification
conditions of freeze casting. Recently, Bouville et al. modeled the interaction
between high-aspect ratio particles and a lamellar freezing front [20]. However, the
high-aspect ratio particles were treated as a cluster of spherical particles. The force
acting on the high-aspect ratio particles was a summation of the forces on individual
spherical particles without considering overall fluid dynamics, the origin of the
aforementioned force. Therefore, we propose a new model, specifically designed to
accurately describe the solidification conditions with high-aspect ratio particles and
a lamellar freezing front in freeze casting.
In this chapter, the freeze casting of high-aspect ratio particles is examined by experiments, modeling and simulation to develop a firmer understanding of freezing
front-particle interactions. First, interlamellar bridges are introduced via freeze
casting with silicon carbide (SiC) whisker-polysiloxane preceramic polymer suspension. The resultant microstructure, permeability, and compressive strength of
SiC whisker-silicon oxycarbide (SiOC) porous composites are examined. A 2D
freezing setup is implemented to observe solidification in situ and a new particle
engulfment model is proposed to describe the phenomenon in the 2D solidification
experiment. Finally, the experimental results are compared with the calculated and
the simulated results based on the proposed model.
3.2

Experimental Procedures

3.2.1
3.2.1.1

Freeze Casting
Materials and Synthesis

Silicon carbide whiskers (purity: >90% whisker, diameter: <2.5 𝜇m, length: 50-80
𝜇m, US Research Nanomaterials, Inc., TX, USA) were chosen as the high-aspect

27
ratio particles in this study for their relatively simple shape and negligible thermal
expansion mismatch with the SiOC matrix. As-received SiC whiskers are confirmed
to have a high aspect ratio (>10) and are shown to be faceted in Fig. 3.1.

Figure 3.1: An SEM image of as-received faceted SiC whiskers.

Twenty vol.% preceramic polysiloxane polymer (Silres® MK Powder, CH3-SiO1.5,
Wacker Chemie, Munich, Germany) and 3 wt.% (of SiC) dispersant (Hypermer
KD1, Croda, NJ, USA) were dissolved in dimethyl carbonate (DMC, Sigma-Aldrich,
MO, USA). Then, 10, 20, 30 vol.% SiC whiskers of the solid loading (MK+SiC)
were dispersed in MK-DMC solution and sonicated for 20 minutes. Two wt.%
of crosslinking agent (N-(2-Aminoethyl)-3-aminopropyltrimethoxysilane, SigmaAldrich, MO, USA) was added to the SiC-MK-DMC suspension before freezing.
The suspension was poured into a 24-mm diameter glass mold and quenched at
-35℃. Fig. 3.2(a) shows a schematic of the freeze setup. Each frozen sample was
moved into the freeze dryer until the solvent was removed, leaving a porous polymer
ceramic composite behind. The SiC-MK composites were then pyrolyzed at 1100℃
under argon for 4 hours and converted into porous SiC-SiOC ceramic composites. A
porous MK-derived SiOC ceramic was fabricated as a control sample using a similar
procedure without adding SiC whiskers and KD1, or sonicating. The compositions
of the suspensions and solution are listed in Table 3.1.

28
Table 3.1: Composition of suspension and porosity of the pyrolyzed porous composites.
SiC/Solid
(vol.%)
10
20
30

3.2.1.2

MK (ml)

SiC (ml)

KD1 (g)

DMC (ml)

Porosity (%)

1.00
1.00
1.00
1.00

0.00
0.11
0.25
0.43

0.00
0.01
0.02
0.04

4.00
3.89
3.75
3.57

76.00
76.88
76.03
73.60

Characterization

Microstructure and bridge density were characterized using scanning electron microscopy (ZEISS 1550VP FESEM, Carl Zeiss Microscopy GmbH, Jena, Germany)
and a mercury intrusion porosimeter (MIP, AutoPore IV 9500, Micromeritics Instrument Corp., Norcross, GA, US). Pyrolyzed samples were sectioned 3 mm from the
base of the sample, perpendicular to the freezing direction. For bridge density measurement, samples were similarly prepared, but infiltrated with epoxy (EpoColor,
Buehler, IL, USA), and polished.
Freeze-cast cylinders, 13-mm in diameter and 8-mm tall, were drilled from pyrolyzed
samples along the freezing axis for compressive strength measurements. The top and
the bottom sides were capped with a high-stiffness acrylic (VariDur 3003, Buehler,
IL, USA) to prevent crushing during mechanical testing, common in porous ceramics
[117]. Two sets of self-aligning washers were used to ensure the force was applied
along the freezing axis of the samples. A pair of SiC blocks were placed between the
sample and the washers to ensure an even distribution of force on the sample. The
samples were compressed at a displacement rate of 0.05 mm/min and the maximum
load in the load-displacement curve was used to calculate the compressive strength.
At least three samples were tested for each condition.
Permeability measurements were performed on 13-mm diameter and 5-mm tall
pyrolyzed cylinders. The perimeters of the samples were enclosed with VariDur
3003 acrylic to prevent leakage. The samples were clamped inside a tube with a
pump forcing water through the samples. A pressure gauge was used to measure
fluid pressures. Further details of the experimental setup can be found in [97]. Flow
rates were measured at several pressures from 6 to 70 kPa. The permeability was

29
determined with the Darcy-Forchheimer equation [45, 203]:
Δ𝑃
= 𝑣 + 𝑣2
𝑘1
𝑘2

(3.1)

where ΔP is the pressure drop across the sample, 𝑙 is the height of the sample, 𝑣, 𝜇
and 𝜌 are the flow velocity, dynamic viscosity, and density of water, respectively,
and 𝑘 1 and 𝑘 2 are intrinsic and inertial permeabilities, respectively. Given the
linearity of the data, the inertia permeability (𝑘 2 ), accounting for the non-linear part
of permeability, is ignored. Three samples were tested for each condition.
3.2.2
3.2.2.1

Two-dimensional Solidification Experiments and Simulations
Experimental Setup

To observe the freezing process in real-time, a two-dimensional (2D) solidification
setup was designed, and is shown schematically in Fig. 3.2(b). A 30 𝜇L suspension
was placed between two glass slides to produce a 7-𝜇m liquid film to observe
dendrite/whisker interactions. The left side of the bottom glass slide was in contact
with the cold finger to create a thermal gradient parallel to the glass slides. The
suspension was composed of 0.2 vol.% SiC whisker and 20 vol.% MK preceramic
polymer solvated with 79.8 vol.% DMC. The concentration of whiskers was dilute to
minimize the interparticle interaction and facilitate observation. The concentration
of MK polymer was kept constant so that the freezing front shape, size, and dendritic
spacings, would be similar to those in freeze casting. The copper cold finger was
quickly cooled to and held at -13℃, the temperature producing lamellar spacings
and freezing front velocities closest to those producing bulk samples in this freeze
casting study. A digital microscope (Digital USB Microscope Camera OT-V1,
Opti-Tekscope) was used to record solidification.

30

Figure 3.2: A schematic of (a) freeze casting setup and (b) 2D solidification setup.

3.2.2.2

Viscosity

Before the viscosity measurement, 2 wt.% cross-linking agent was added into the
solution of 20 wt.% MK-DMC solution at room temperature (∼25℃). The viscosity
measurements were conducted within 20 minutes of the cross-linker addition. This
ensured that the viscosity was similar to that displayed during freeze casting. After
∼4 hours, viscosity changed drastically. The viscosities were determined at several
strain rates from 1 to 100 s−1 using ARES RFS rheometer (TA Instruments, New
Castle, DE, USA) with a cone-and-plate fixture, with a 50-mm diameter parallel
cone-plate with a 1° canted surface. The solution displayed shear thinning and the
viscosity at the highest strain rate (100 s−1 ) was used in this study.
3.2.2.3

Surface Energy

To determine the surface energy between the liquid MK-DMC solution and a SiC
whisker, 𝛾𝑊 𝐿 , a droplet experiment was performed. SiC whiskers are assumed to
have an oxide layer; SiO2 was confirmed by X-ray powder diffraction (not shown)
of as-received SiC whiskers. A slab of fused silica was cleaned with acetone and
DI water and then heated in a drying oven at 100℃ for six days. Twenty wt.% MKDMC solution was dropped onto the silica slab. Contact angles from five optical
images were analyzed.

31
3.2.2.4

Freezing Simulations

Freezing the SiC whisker-polysiloxane-DMC suspension was modeled with a MATLAB script consisting of a 2D simulation cell of 100×300 𝜇m2 . The cell comprised
randomly distributed whiskers and a set of dendrites. The freezing front advanced
from left to right in 25-ms time intervals. Each whisker had a randomly generated
starting angle and position. The freezing front velocity and the lamellar spacing
used in the simulation were measured from a series of SEM images of pore structures
and solidification videos of freeze casting. The number of dendrites and whiskers
were determined by the lamellar spacing and SiC concentration, respectively. The
position offsets were calculated from the distance between each whisker and its
nearest dendrite. The values of the input variables are shown schematically in Fig.
3.3, and their values are listed in Table 3.2.
The torque on whiskers were calculated along with the angles that SiC whiskers
changed as the freezing front advanced and time progressed. If a whisker was
not pushed aside before the freezing front reached within the minimal distance
between a particle and freezing front (h), the drag and repulsive force were calculated
to determine if the whisker was pushed or engulfed by the freezing front. A
whisker would be engulfed if the drag force was larger than the repulsive force and
pushed aside if not. Due to the variance in simulation results addressed in Section
3.3.2.4, each engulfed whisker fraction is averaged over 2000 times through repeated
simulations for each condition to gain an overall sense of engulfment.
Table 3.2: Values of input variables for simulation.
SiC whisker height, width, d
SiC whisker length, L
Molecular distance, a0
Minimal distance, h
Density of SiC whisker, 𝜌
Dynamic viscosity, 𝜂
*Dendritic tip slope, a

2 𝜇m
70 𝜇m
2×10−10 m
2×10−10 m
3216 kg/m3
0.02179 Pa· s
2.83
10 vol.%
20 vol.%
SiC concentration in suspension
2.22%
5.00%
*Freezing front velocity, V 𝑓 𝑓
22.45 𝜇m/s 32.05 𝜇m/s
*Lamellar spacing
26.66 𝜇m
43.81 𝜇m
* = determined by experiment

30 vol.%
8.57%
31.9 𝜇m/s
41.52 𝜇m

32

Figure 3.3: A schematic of input variables used in freezing simulation.

3.3

Results and Discussion

3.3.1

Freeze-cast Samples

3.3.1.1

Pore Structures

Shown in Fig. 3.4(a) through (d) are cross-sections of freeze-cast lamellar structures
with different concentrations of whiskers. Without added whiskers, few SiOC
bridges form during pyrolysis as a result of the polysiloxane being phase-separated
into the spaces between the secondary arms of the dendrites during solidification.
In contrast, the structures with 20 and 30 vol.% of whiskers have many bridges.
In addition to the SiC-SiOC bridges (blue circles), there are many bridges of SiC
whiskers (yellow circles), shown in Fig. 3.4(e). SiOC bridges are thicker and
continue along the lamellar walls, whereas whisker bridges are thinner and act
as individual pillars. We argue that these two bridges were formed via different
mechanisms, as evidenced by their different morphologies. SiC-SiOC bridges
formed by polysiloxane are phase-separated into the spaces between secondary
arms, whereas whisker bridges formed by SiC whiskers are engulfed by the freezing
front.

33

Figure 3.4: SEM images of freeze-cast lamellar structures with (a) 0, (b) 10, (c)
20, (d) 30 vol.% SiC whiskers and (e) 20 vol.% SiC whiskers freeze-cast lamellar
structures with SiC-SiOC bridges (blue circles) and SiC whiskers (yellow circles)
bridges (f) Example images of bridge density measurements. Green lines mark
the counted whisker bridges. (g) Bridge density as a function of the whisker
concentration. Error bars represent ± 1 standard deviation.

34
An important observation is that the whisker bridges are largely perpendicular to
the lamellae, indicating the presence of an angular-bias during particle engulfment,
leaving behind whisker bridges that are almost all perpendicular to the freezing
dendrites, and hence, the lamellae. Whiskers that are parallel to the freezing
dendrites during solidification are embedded in the pore walls after freeze casting.
The same phenomena can also be seen in freeze casting of SiC whiskers [57],
CNTs [70], alumina platelets [63], and graphene sheets [181], showing that the
alignment of high-aspect ratio particles is the norm rather than an anomaly. This
demonstrates two important phenomena unique to freeze casting high-aspect ratio
particles, alignment and partial engulfment of particles.
To quantify the number of bridges, freeze-cast samples were infiltrated with epoxy
and sectioned for imaging. After infiltration, only bridges on the sectioned plane
were visible (Fig. 3.4(f)). Only SiC whisker bridges were used to calculate the
bridge density. SiC-SiOC bridges formed by phase separation into secondary arms
were not counted. The number of bridges increases in proportion to the increasing
concentration of whiskers with an R2 value of 0.9661 (Fig. 3.4(g)).

35

Figure 3.5: Pore size distributions from MIP and the corresponding SEM images of
freeze-cast pore structures of (a) SiOC using DMC and cyclohexane, adapted from
[126], and (b) SiOC and 30 vol.% SiC whisker composites using DMC in this work.

Naviroj [126] has shown that lamellar structures naturally have a wider pore size

36
distribution compared to dendritic structures. The pore size of lamellar pores in his
work ranges from 10 to 100 𝜇m and the maximum pore volume (intrusion volume)
remains less than 5%, while the peak of the dendritic pore size is in the range of
20 𝜇m and the maximum pore volume can be as high as 12% (Fig. 3.5(a)). With
the added whiskers as interlamellar bridges, the pore size distribution is narrowed
and the maximum pore volume increases from 4% to 17% for 30 vol.% SiC whisker
composites, as demonstrated in Fig. 3.5(b).

37
3.3.1.2

Mechanical Properties and Permeability

Figure 3.6: Compressive strength as a function of (a) permeability and (b) relative
density. Error bars represent ± 1 standard deviation. (c) Representative stress-strain
curves for freeze-cast pure SiOC (dotted black line) and for a freeze-case 30 vol.%
SiC composite (blue line). The insets show SEM images of fracture surfaces of
pure SiOC ceramics and 30 vol.% SiC composites with arrows pointing toward the
corresponding the stress-strain curves.

Incorporation of SiC whiskers decreases permeability and increases strength (Fig.
3.6(a)). Although addition of 10 vol.% whiskers might have a stronger effect
on permeability (decreases by a factor of 3, albeit with a large uncertainty) than

38
on strength (doubles), the overall trend is approximately an inverse relationship
(approximate slope of -1 on log-log plot). Compared to prior literature on pure
SiOC lamellae [125], the range of permeability overlaps with the present study,
while 30 vol.% whisker reinforcements achieve strength in the present lamellar
structures that is an order of magnitude greater than previously reported.
High-aspect ratio particles serve as reinforcements while maintaining the advantageous high permeability (> 10−12 m2 ) of lamellar structures. However, an increase
in compressive strength could simply result from higher solids loading [156]. If this
were the case, the same reinforcement could be more easily achieved by increasing
the MK polymer concentration. Therefore, it is crucial to compare the composites
with pure SiOC of similar porosity to shed light on its reinforcement mechanism.
It is well-known that compressive strength increases with increasing relative density
following a power-law relation for porous materials [65]:
= 𝐶 ( )𝑛
𝛼0
𝜌0

(3.2)

where C and n are constants for a given material and pore structure (independent of
the porosity of the material), 𝛼 and 𝛼0 are the strength of the porous and the dense
structure, respectively, and 𝜌 and 𝜌0 are the bulk density and the true density of the
structure, respectively. Pure freeze-cast SiOC ceramics with various porosities, and
therefore densities, were used to establish the power-law relation and it was found
that compressive strength scales with the square of relative density (n≈2), indicated
by the grey dashed line (Fig. 3.6(b)).
Although strengthening with added SiC whiskers was expected, this strengthening
was much higher than predicted from an increase in density alone (indicated by
dotted baseline established from freeze-cast pure SiOCs with various density). The
compressive strength is shown in Fig. 3.6(b) to increase by 76%, 191%, and
325% with 10, 20, and 30 vol.% SiC whiskers, respectively. We hypothesize the
strengthening largely comes from two different effects of adding SiC whiskers. The
fraction of SiC whiskers in the freeze-cast walls creates fiber-reinforced composites
walls, likely stiffer and tougher than the pure SiOC pore walls. The other effect
stems from a change in fracture mode arising from SiC whiskers spanning lamellar
pores.
As previously mentioned, lamellar structures often fail in shear fracture due to
insufficient interlamellar connections [111]. This is evidenced by the signature
porous ceramic stress-strain curve. The stress increases linearly to a peak value at

39
which point the lamellar walls start to slip over one another, resulting in a large drop
in stress. This is followed by a gradual increase in stress due to the densification of
the fractured pore structure (dotted black line in Fig. 3.6(c)). In contrast, for SiC
whisker composites, interlamellar bridges formed by the added whiskers prevent
shear failure. The stress-strain curve of the 30 vol.% whisker composite (blue
curve in Fig. 3.6(c)) is interpreted as follows: a series of micro-fracture events
begin at about 3% strain and progress cell-by-cell, leading to a stress maximum at
approximately 30% strain followed by softening. The change in fracture mode with
whisker additions can also be observed from the fracture surfaces (insets in Fig.
3.6(c) with arrows pointing toward the corresponding the stress-strain curves). In
pure SiOC ceramics, the fracture surfaces are clean cleavage between lamellar walls,
leaving behind intact walls. In contrast, in SiC-reinforced composites, the fracture
surface propagates through lamellar walls as the whisker bridges act as interlamellar
connections. These two effects from the addition of SiC whiskers (reinforced pore
walls and prevention of shear fracture) increase the strength of freeze-cast structures,
demonstrating the reinforcing mechanism of high-aspect ratio particles in freezecast systems. The stress-strain curves of all 30 vol.% SiC composites and pure SiOC
tested are included in Appendix A to demonstrate the consistency in the results.
3.3.2
3.3.2.1

Two-dimensional in-situ Solidification Experiments
Theory

To observe the solidification process in situ, low particle concentrations are necessary to prevent obstructions in the freezing front views. However, reducing the
solids loading dramatically can change the solidification parameters (e.g., rheological properties, heat capacity, and thermal conductivity) and, consequently, change
the shape, size and spacings of the freezing dendrites relative to actual freeze-casting
conditions [112]. Therefore, using polysiloxane polymer as the major fraction of the
"solids loading" has an advantage over using SiC whiskers. Polysiloxane is transparent upon dissolution, and therefore, provides a clear avenue to observe solidification
without interference. The SiC whisker concentration is reduced to 0.2 vol.% so that
dendrite-whisker interactions can be observed. Since polysiloxane polymer is the
major component of the solids loading, reducing SiC whisker concentration does
not drastically change the total solids loading. This allows us to observe particle
engulfment by freezing dendrites similar to those in conventional freeze-casting
systems.

40
We present a model of an actual freeze-casting system. A schematic of the particle
engulfment process is shown in Fig. 3.7(a). The whisker experiences two major
forces as the dimethyl carbonate crystals solidify and grow in the direction of the
temperature gradient—a drag force and a repulsive force. The single whisker and
growing dendrite intersect at an angle 𝜃, and an offset distance b. Theoretically, the
model can be extended to 3D.

Figure 3.7: (a) A schematic of the acting forces during the engulfment process.
(b) An example of a dendrite tip angle measurement. (c) Dendrite tip angle as a
function of freezing front velocity. Error bars represent ± 1 standard deviation.

3.3.2.2

Engulfment-related Forces and Critical Freezing Front Velocity

The drag force on a spherical or cylindrical particle with a planar interface is wellknown [26, 27, 62, 101]:
𝑅 2𝑝
𝐹𝑑 = −6𝜋𝜂𝑉 𝑓 𝑓
for a sphere, and
(3.3)
  32
𝑅𝑃
𝐹𝑑 = −3 2𝜋𝜂𝑉 𝑓 𝑓
𝐿 for a cylinder

(3.4)

41
where 𝜂 is the dynamic viscosity of the solvent, R𝑃 is the particle radius, h is
the distance between the particle and the freezing front, V 𝑓 𝑓 is the freezing front
velocity, and L is the length of the cylinder. To account for the non-planar shape
of the interface, some researchers use the relative thermal conductivity between the
particle, K𝑃 , and the solvent, K 𝐿 , to calculate the particle-induced curvature of an
otherwise planar growth front, R 𝐼 [161, 168]
!
! 
𝑅 2𝑝
𝑅 2𝑝 𝐾 𝑃
𝑅𝐼
𝐹𝑑 = −6𝜋𝜂𝑉 𝑓 𝑓
= −6𝜋𝜂𝑉 𝑓 𝑓
(3.5)
𝑅𝐼 − 𝑅𝑃
𝐾𝐿
However, in the dendritic regime, the interface shape is largely determined by the
freezing front velocity, thermal gradient, and solute concentration in the solution
[14, 17, 33, 98]. The theoretical determination of the freezing front (the shape, size
and spacings of the freezing dendrites) is not the focus of this study. Instead, we
neglect curvature and use the measured angle of the faceted freezing dendrite tips
from the 2D solidification experiment for the model: the angles are found to be 39°
(Fig. 3.7(b)), largely invariant with respect to the freezing velocity (Fig. 3.7(c)).
When the ice front advances, fluid flows continuously into the gap to maintain the
liquid film. This flow creates a pressure difference in the gap and the pressure gives
rise to the drag force, which is calculated from the Navier-Stokes equation. In 2D insitu solidification, the flow is restrained to the directions parallel to the glass slides
that confine the SiC whisker suspension. We assume that the volume difference
between the liquid and solid DMC is negligible. The flow becomes unidirectional
and the Navier-Stokes equation can be solved analytically for boundary conditions
where the fluid velocity is zero at the solid surfaces, and by applying conservation
of mass.
𝜕 2 𝑣 𝑥 1 𝜕𝑃
𝜂 𝜕𝑥
𝜕𝑦 2
where v𝑥 is the fluid velocity in the x-direction, and 𝜂 and 𝑃 are the viscosity and
the pressure of the fluid, respectively.
The boundary conditions are as follows:
𝑣 𝑥 (𝑦 = 0) = 0
𝑣 𝑥 (𝑦 = ℎ) = 0,
and the conservation of mass:
∫ ℎ
𝑣 𝑥 (𝑦) × 𝑑 𝑑𝑦 = 𝐿 × 𝑑 × 𝑉 𝑓 𝑓

42
By solving the Navier-Stokes equation, we arrive at the following expression for the
drag force:
(1 − 2𝑏 csc 𝜃) 2
𝐹𝑑 = −6𝜂𝐿 𝑑𝑉 𝑓 𝑓
ℎ[𝐿(𝑎 − cot 𝜃) + 2(ℎ − 𝑎𝑏𝐿 + 𝑏𝐿 cot 𝜃)] 2
(3.6)
(2𝑏 + sin 𝜃) 2
ℎ[𝐿(𝑎 sin 𝜃 + cos 𝜃) + 2(ℎ + 𝑎𝑏𝐿 + 𝑏𝐿 cot 𝜃)] 2
where 𝑑 is the width and height of the whisker and the dendritic tip slope, 𝑎, is equal
to cot 𝜃.
When a planar freezing front (𝑎 = 0, 𝑏 = 0) is assumed and the whisker is parallel
3𝐿 3 𝑑𝜂𝑉 𝑓 𝑓
to the freezing front (𝜃 = 𝜋/2), the drag force reduces to
. The force scales
ℎ3

as (𝐿/ℎ) 3 , similar to two flat surfaces moving toward each other (ℎ−3 ), instead of
a circular cylinder moving toward a flat surface. In the present case, we adopt a
square cylinder morphology instead of the circular cylinders used in previous models
because SiC whiskers used in this study are faceted as shown in Fig. 3.1.
The dynamic viscosity, 𝜂, of 20 wt.% polysiloxane polymer solution was determined
to be 21.79 mPa·s. Other parameters used in the calculation of the drag force include
the minimal distance, ℎ, between the freezing front and the whisker, where ℎ =
2×10−10 m [19, 89, 161] and atomic distance 𝑎 0 = 2×10−10 m [19, 89, 151, 161, 175].
The exact origin of the repulsive force is still in dispute. Approaches using the
surface energy differences [52, 138] or the disjoining force [66, 143] as the origin
have been discussed. We argue that the energy difference before and after the
engulfment cannot be "experienced" by the particles before the engulfment and
therefore, cannot provide the repulsive forces. (The surface energy approach is
discussed in Appendix B for comparison.) Here, we proposed an alternative origin
of the repulsive force, the osmotic pressure , Π, from the concentration gradient of
the polysiloxane polymers in front of the freezing front.
The solubility of the polymers is higher in the liquid than in the solid. Therefore, as
the freezing front advances, rejected solutes may pile up, creating a concentration
gradient in front of the freezing front. The concentration difference and, therefore,
the osmotic pressure difference across the whisker width produce a repulsive force
pushing the whisker forward as shown in Fig. 3.8.

43

Figure 3.8: A schematic of solute pileup during the freezing process and the resulting
osmotic pressure difference.

The difference in the osmotic pressure was estimated to be ∼3000 Pa following
ΔΠ = Δ𝐶 × 𝑅𝑇

(3.7)

where Δ𝐶 is difference of solute molar concentration across the whisker, R is the
ideal gas constant, and T is the temperature in Kelvin.
Polymers of similar molecular weights to the MK polymer (M𝑊 = 9400 g/mol),
such as poly(vinyl alcohol) (M𝑊 = 9000 g/mol) and poly(ethylene glycol) (M𝑊 =
8000 g/mol), are shown to have a solute pileup of 40 to 80 𝜇m and an increase in
concentration of 1.5 to 9 times in aqueous solutions with freezing front velocities of 2
to 10 𝜇m/s [24]. The highest concentration of solute pileup decreases with increasing
freezing front velocity. The length scale varies as diffusivity over velocity, and
diffusivity scales inversely with viscosity. Given that the ratio of viscosities of DMC
to water is 0.9, the field of solute pileup and maximum increase of concentration are
estimated to be ∼40 𝜇m and 2 times for freezing front velocities used in this study
(∼20 to 70 𝜇m/s), respectively.

44
The repulsive force was calculated following
𝐹𝑟 = ΔΠ × 𝑑 × 𝐿 𝑒 𝑓 𝑓 𝑒𝑐𝑡𝑖𝑣𝑒

(3.8)

where 𝐿 𝑒 𝑓 𝑓 𝑒𝑐𝑡𝑖𝑣𝑒 is the length of whisker that is in close contact with the dendrite
tip, and is estimated to be about 40 𝜇m. With a dendritic tip slope of 2.83, the far
end of the whisker is far away from the dendrite, and unlikely to experience any
force.
From freezing videos, we can see whiskers interact with a translucent area around the
opaque whisker. If the translucent area is due to the change in reflective index from
the solute pileup, the origin of the repulsive force is likely osmotic pressure as the
distance is a few microns away from the dendrite tip. In contrast, if the translucent
area is the tapered edge of the dendrite, the origin of the repulsive force may be the
disjoining force, since the distance is in submicron or smaller. A higher-resolution
microscope would be required to determine the origin of the repulsive force.
Critical freezing front velocities were calculated from 𝐹𝑑 = 𝐹𝑟 . Engulfment occurs
when the freezing front velocity exceeds the critical freezing front velocity for a
given whisker length. To compare the model with experiments, critical freezing
front velocities were determined from solidification videos by measuring the length
of the whiskers that are pushed away or engulfed for a given freezing front velocity.
Various freezing fronts were achieved by recording the solidification process from
different distances to the cooling plate. The measurement error of the freezing front
velocities is estimated to be ±5 𝜇m/s. Only whiskers that are roughly perpendicular
to the freezing dendrites and centered were counted. Fig. 3.9 shows reasonable
agreement between experimental data and the theoretical critical freezing front
velocity (marked by a grey line) for conditions of 𝑏 = 0 (whisker bisected by
the freezing front) and 𝜃 = 𝜋/2 (whisker perpendicular to freezing dendrites) in
calculating the drag force. Asthana et al. [6] reviewed several engulfment models
and found that the theoretical predictions and experimental results models often
differ by several orders of magnitude. This makes the agreement between the
theoretical and experiment results here especially compelling. This agreement
indicates that the model is a sufficiently accurate description of the experiment.

45

Figure 3.9: Freezing front velocity versus whisker length with experimental data;
blue circles represent pushed whiskers; orange circles represent engulfed whiskers.
The theoretical critical freezing front velocity (grey line) was calculated with engulfment model described in Section 3.3.2.2.

3.3.2.3

Torque and Whisker Rotation

The fluid motion that gives rise to the drag force also creates torque on the whisker.
The intersecting angle 𝜃, and offset distance 𝑏, determine the orientation and the
magnitude of the torque. The torque is integrated from the pressure difference inside
the gap between the whisker and the dendrite. The rotating whisker also experiences
a frictional force from the viscous liquid around it. This frictional force always acts
against the rotating movement and slows down or prevents rotation. A detailed
calculation of torque can be found in Appendix C.

46

Figure 3.10: The torque acting on whiskers as a function of (a) offset distance, and
(b) intersecting angle.

From the model, we find that intersecting angles and offset distances give rise to
torque and the subsequent rotation. The rotation arises from the pressure difference
caused by the fluid motion that results in the drag force. Only when a whisker is
centered (𝜃 = 𝜋/2) and bisected (𝑏 = 0) by the dendrite is the torque zero and the
whisker does not rotate. If the whisker is not bisected, the whisker rotates toward
the longer side of the whisker (3.10(a)), as is expected.
Fig. 3.10(b) shows that whiskers that are not perpendicular to the freezing front
(𝜃 ≠ 𝜋/2) experience torque and rotate out of the way of the freezing dendrites,
leaving behind whiskers that are mostly parallel or perpendicular to the freezing front
and the lamellar walls. These results were observed in the solidification videos with
slower freezing front velocities (shown in Appendix D) from the 2D solidification
experiments and could explain the aligning effect observed in freeze casting with Al
platelets, SiC whiskers, and CNTs [57, 63, 70]. With faster freezing front velocities,
long whiskers rarely rotate before being engulfed by the freezing front. This rotation
is not limited to large particles that will be engulfed but any particle that encounters

47
freezing dendrites. The particle rotating to the inter-dendritic space will be further
aligned by the compacting motion of thickening dendrites [20]. Interestingly, we
also observed fluid backflow, possibly from the density difference between solidified
and liquid solvent, which contributes to whiskers moving into interdendritic space.
In the case of spherical particles, particles are either fully engulfed or pushed away.
However, partial engulfment—where only a portion of particles is engulfed while the
rest are pushed away—is common in freeze casting of high-aspect ratio particles.
These two parameters (𝜃 and 𝑏), introduced due to the geometry of high-aspect
ratio particle and non-planar freezing front, can explain partial engulfment in the
case of high-aspect ratio particles. For example, the high-aspect ratio particles can
rotate out of the way even if the freezing velocity is above the critical freezing front
velocity. The percentage of the engulfed whiskers is determined by the solidification
conditions and will be further discussed in the next section.
3.3.2.4

Freezing Simulations and Bridge density

The 2D solidification simulation based on the proposed model can reveal the effect
of intersecting angles and offset distances on the engulfment of SiC whiskers. The
lamellar spacing and the freezing front velocity were determined experimentally
(Appendix E) for 10, 20, 30 vol.% SiC whisker suspensions. The visualization of
the simulation is shown in Fig. 3.11(a). Blue lines represent freezing dendrites.
Whiskers are randomly distributed and randomly oriented at the start of the simulation (simulation details: whisker concentration, width (d), and length (L) are 2
vol.%, 2 𝜇m, and 30 𝜇m respectively. Freezing front velocity is 45 𝜇m/s). In this example (Fig. 3.11(a)), some whiskers never interacted with the freezing front, while
some were pushed aside/engulfed by the freezing front. This simulation shows both
alignment and partial engulfment of whiskers as observed in 2D in situ solidification
videos and 3D freeze-casting systems. With the simulation, the engulfed whisker
fraction is found to vary even for the same freezing conditions, i.e. same freezing
front velocity and lamellar spacing (Fig. 3.11(b)). This variance in the engulfed
whisker ratio was caused by the starting angles and positions of whiskers.
Previous particle engulfment models of planar freezing front and spherical particles
only considered the particle size and freezing front velocity. Here, the variance
shows that in addition to the particle size and freezing front velocity, the intersecting
angle and offset distance also play a major role in whether a high-aspect ratio particle
is engulfed.

48

Figure 3.11: (a) Visualization of simulation (SiC concentration = 2 vol.%, 𝑑 = 2
𝜇m, 𝐿 = 30 𝜇m, 𝑉 𝑓 𝑓 = 45 𝜇m/s). Blue lines represent freezing dendrites. Whiskers
are randomly distributed and orientated. (b) Probability of engulfment fraction from
the simulation for 10, 20, and 30 vol. % SiC solidification.

Experimental values of engulfed whisker percentages were calculated for each SiC
concentration from the bridge density of freeze-cast samples. As previously mentioned in Section 3.2.2.4, each simulated engulfed whisker fraction is averaged from
2000 repeated simulations for each solidification condition to gain an overall sense
of engulfment due to the variance in simulation results shown in Fig. 3.11(b). Both
experimental and simulated engulfed whisker fractions are reported in Table 3.3 and
are of the same order of magnitude. We hypothesize that the force of gravity, present
only in freeze-cast experiments, could explain some of the difference between the
simulated and the experimental result. Furthermore, the simulation is simplified
and does not include the particle-particle interactions and the interaction between a
single particle with multiple dendrites. With particle-particle interactions and the
interaction of single particles with multiple dendrites, we expect to see a higher
fraction of engulfed whiskers due to the interlocking of SiC whiskers in the former
case and a higher probability of having at least one dendrite meet engulfment criteria

49
with a whisker in the latter.
Table 3.3: Comparison of the engulfed whisker ratio between the freeze-cast samples
and freezing simulation
SiC whisker/all solid Experimental engulfed Simulated engulfed
(vol.%)
fraction (%)
fraction (%)
10.00
14.13 ± 2.48
10.07
20.00
12.68 ± 2.32
9.39
30.00
15.47 ± 4.18
9.54

The model further predicts a higher fraction of engulfment with higher freezing
front velocities (Fig. 3.12(a)) and longer whiskers (Fig. 3.12(b)). The higher
engulfment fraction is caused by a higher probability of engulfment for individual
particles. Both predictions are consistent with the results from previous spherical
particle engulfment models.

Figure 3.12: Averaged simulated engulfed whisker fraction as a function of (a)
Freezing front velocity and (b) whisker length.

These results show that the proposed model and the subsequent simulation are able to
reproduce results that are consistent with previous engulfment models, i.e. those that
include higher probabilities of engulfment with higher freezing front velocities and
large particles. Furthermore, the proposed model and the simulation can produce
results unachievable with the previous models, specifically for partial engulfment
and alignment effects common in freeze casting high-aspect ratio particles. A deeper
understanding of particle engulfment of high-aspect ratio particles gained from this
model will allow us to better design the processing of materials in solidification and
freeze casting.

50
3.4

Conclusions

High-permeability lamellar structures often fail in shear fracture at low stress, limiting their wider adoption as filter materials. To achieve better pore morphologies and
microstructures, we introduced interlamellar bridges by freeze casting high-aspect
ratio SiC whiskers with polysiloxane preceramic polymer. The compressive strength
shows a 3-fold increase with 30 vol.% whiskers while maintaining a high permeability in the range of 10−12 m2 . Two types of interlamellar bridges were identified:
one formed from particles trapped between secondary dendrite arms and the other
from particle engulfment of high-aspect ratio particles (SiC whiskers). The density
of interlamellar bridges increases linearly with the concentration of SiC whiskers.
To remedy the deficiency of current engulfment theories, we proposed a new model
to account for two new variables, the intersecting angle between a pair of the
interacting whisker and freezing dendrite and the offset between the center of a
whisker and that of the nearest dendrite, introduced due to the geometry of highaspect ratio particles and lamellar/dendritic freezing fronts.
The critical freezing front velocity was calculated from the model and the result was
in reasonable agreement with the experimental values. Moreover, a simulation based
on the proposed model successfully reproduced the whisker rotation, the sequential
alignment, and the partial engulfment of high-aspect ratio particles observed in the
experiments, while the engulfed whisker fractions from the simulation and those
from the experiment were in close range. In addition, the simulation showed higher
probabilities of engulfment with higher freezing front velocities and longer whiskers,
consistent with the previous engulfment models.
This chapter explored reinforced pore structures with engulfed whiskers with both
theory and experiment, providing an in-depth understanding of the freezing process
of high-aspect ratio particles, the resultant microstructures, the mechanical properties, and the fracture behavior of porous composites. This research can further
improve the prediction of both structure and resultant properties and result in more
accurate designs for porous composites with high-aspect ratio particles.

51
Chapter 4

CARBON NANOTUBE-REINFORCED COMPOSITES
Material in this chapter is reproduced in part from "Permeable carbon nanotubereinforced silicon oxycarbide via freeze casting with enhanced mechanical stability", C.T. Kuo, K.T. Faber; Journal of the European Ceramic Society, 40 (2020)
2470–2479. The work was done in collaboration with Katherine T. Faber. T. Kuo
fabricated and characterized CNT composites, and wrote the manuscript. K. Faber
supervised this work.
4.1

Introduction

Chapter 3 demonstrated that SiC whiskers as reinforcing fillers increase the compressive strength of lamellar structures dramatically at a small expense in permeability
by significantly changing the lamellar structure. Here we explore the possibility
of retaining the original freeze-cast pore structures with alternative reinforcement
methods. To accomplish this, the reinforcing fillers need to fulfill the following
requirements. First, the reinforcement must change the strength significantly with
small quantities so that there is little risk of altering the freezing dynamics, and
by consequence, the structure. The second requirement is for reinforcements to
be small enough, and of low enough density relative to the solution, to be pushed
by the freezing front and embedded in the walls without protrusion. Lastly, the
reinforcements have to be dispersible in the solution/suspension to achieve uniform
distribution in the final freeze-cast structures.
In light of these constraints, we chose multi-walled carbon nanotubes (MWCNTs)
to be the reinforcing fillers. MWCNTs can have strengths as high as 63 GPa [194]
and are known to increase the strength of a matrix as much as five times with as little
as 1 wt.% CNT [84]. It has been shown previously that the mechanical properties
(e.g. Young’s modulus, compressive strength, and toughness) are very sensitive to
the quality of dispersion of CNTs in the matrix; the better the dispersion, the stiffer,
stronger, and tougher the composite [9]. This gives MWCNTs an advantage over
single-walled carbon nanotubes (SWCNTs), since MWCNTs, unlike SWCNTs, are
less likely to form bundles, making uniform dispersion in the matrix more feasible
[34].

52
CNTs as reinforcements in polymeric matrices have been well studied [167] and
some research has explored ceramic matrices [3, 11, 163]. However, in the latter,
the high pressures and temperatures required to densify ceramic powders by hot
pressing might deteriorate CNTs, which limits their reinforcing effect [197]. To avoid
the damage caused by high temperature and pressure during hot pressing, several
groups have previously dispersed CNTs in preceramic polymers and pyrolyzed
the suspensions to produce CNT-ceramic composites [74, 85, 152, 199]. Using
this approach, CNTs are largely preserved after pyrolysis, resulting in a noticeable
reinforcement effect. Gutierrez et al. have reported a series of studies on freeze
casting of CNTs [1, 70], where the CNTs were used for their low electrical percolation
limit, high conductivity, and excellent biocompatibility, but not as reinforcements.
The work described here is the first where CNTs are used as the reinforcing fillers
in a freeze-casting system to create porous CNT-ceramic composites and composite
design in such systems is addressed. In this chapter, we will demonstrate both
the potential and challenges of incorporating CNT reinforcements in freeze-cast
systems. The processing and dispersion methods are described, as well as the
effect of reinforcements on the microstructure, electrical conductivity, permeability,
compressive strength, and toughness.
4.2
4.2.1

Experimental Methods
Materials and Processing

A commercially available polysiloxane preceramic polymer (Silresr MK Powder,
Wacker Chemie, CH3 -SiO1.5 , Munich, Germany) was dissolved in organic solvents
and freeze cast to produce porous silicon oxycarbide (SiOC). MWCNTs (purity:
>90 wt.% carbon nanotubes, outside diameter: 8-15 nm, inside diameter: 3-6 nm,
length: 30-50 𝜇m, density: 2.1 g/cm3 , US Research Nanomaterials, Inc., TX,
USA) were introduced to reinforce the porous SiOC matrix. After assessing several
combinations of dispersants and solvents, dimethyl carbonate (DMC, m.p.= 2-4℃
density: 1.073 g/cm3 , Sigma-Aldrich, MO, USA) and Hypermer KD1 (Croda, NJ,
USA) were determined to be the optimal solvent and dispersant, respectively, for
dispersing carbon nanotubes. Hypermer KD1 is a cationic dispersant, which has
been proven effective in dispersing graphene-based nanomaterials [115].
To prepare porous SiOC, MK was first dissolved in DMC. A cross-linking agent (N(2-Aminoethyl)-3-aminopropyltrimethoxysilane, Geniosil GF 91, Wacker Chemie,
Munich, Germany) was added into the MK-DMC solution and stirred for 3 min

53
before freezing. To prepare CNT-SiOC composites, KD1 was first dissolved in
DMC. MWCNTs were then added to the solution and sonicated for 60 min in Branson
3800 Ultrasonic Cleaner (Branson, OH, USA). After sonication, MK was added into
the CNT suspension and sonicated for another 60 min. In most composites prepared
via solution processing, sonication is one of the most effective ways to break up
CNT agglomerates and achieve uniform dispersion in the composites [34]. Geniosil
GF 91 was added into the MK-CNT-DMC suspension and stirred for 3 min before
freezing.
The suspension or the solution was then poured into a glass mold with an inner
diameter of 24 mm, and frozen at a cooling rate of 3℃ /min from 10℃ to -30℃Ȧfter
the completion of solidification, the body was dried in a lyophilizer (Virtis Wizard
2.0, SP Scientific, PA, USA) until all solvent was removed. The green sample was
then cured at 150℃ for 24 h to increase ceramic yield, and pyrolyzed at 1100℃ for
4 h with a 2℃ /min heating and cooling ramp rate under argon to obtain a CNTamorphous silicon oxycarbide (SiOC) composite. The pyrolyzed samples were
sectioned for SEM micrographs and imaged at their mid-section.
Eight concentrations of CNTs and the respective KD1 concentrations, shown in Table
4.1, were chosen to determine the conductivities of porous CNT-SiOC composites.
Four concentrations of CNTs in the pyrolyzed solid, 0.0, 1.3, 4.3, and 8.2 wt.%,
were used for all other measurements. To make comparisons easier with other
studies, units of mg/ml are used with CNT-preceramic polymer suspensions while
wt.% is used with pyrolyzed freeze-cast composites or dense disks for toughness
measurements. The units were converted assuming a total decomposition of KD1
and a ceramic yield of 75% for MK [126]. Pyrolyzed freeze-cast CNT-SiOC
composites with 1.3, 4.3, and 8.2 wt.% CNTs were made from suspensions with 3,
10, and 20 mg/ml CNTs. The volume percent (vol.%) MWCNTs in the solid scales
with the weight percent (wt.%) as follows: 1.5, 4.7, and 9.0 vol.% correspond to
1.3, 4.3, and 8.2 wt.%.

54
Table 4.1: Compositions of suspensions and pyrolyzed samples.
CNT/DMC
KD1 (g)
(mg/ml)
0.00
0.1
0.02
0.25
0.02
0.5
0.02
0.02
0.06
10
0.10
20
0.20

4.2.2

CNT (g)

DMC (g)

MK (g)

0.00
0.001
0.0025
0.005
0.01
0.03
0.1
0.2

10.73
10.73
10.73
10.73
10.73
10.73
10.73
10.73

3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00

CNT concentration
of pyrolyzed solid (wt.%)
0.00
0.04
0.11
0.22
0.44
1.32
4.26
8.16

X-ray Diffraction

Phase compositions of as-received CNTs and samples with 0, 4.26 wt.% and 8.16
wt.% CNTs were evaluated using X-ray diffraction (XRD). Scans were conducted
with Cu-K𝛼 radiation form 10° to 60° with a scan speed of 0.057°/s (PANalytical
X’Pert Pro). Background subtraction and peak fitting were performed to identify
phases.
4.2.3

Electrical Conductivity

Resistance measurements were performed at room temperature with a Keithley 614
Electrometer for five lower CNT concentrations (≤0.44 wt.%) and a Fluke 115
True-rms Multimeter for three higher CNT concentrations (≥1.32 wt.%). Because
freeze-cast and pyrolyzed samples have an uneven top surface due to the meniscus formed during freezing, samples were cut at 2 mm from the top surface to
produce two parallel surfaces at the ends. The two ends were coated with carbon
paste and sandwiched between two copper plates for resistance measurements. The
conductivity was calculated without any adjustment for porosity.
𝜎 = 𝜌 −1 =

𝑅𝐴

(4.1)

where 𝜎 is the conductivity, R is the measured resistance, 𝜌 is the resistivity, and L
and A are the length and the contact area of the sample.
4.2.4

Permeability

The permeability of the porous solids was assessed from the flow rate of water over
a series of pressure drops. The permeability test setup consisted of a water tank,
a pressure sensor (Omega PX409 pressure transducer, Omega Engineering, CT,

55
USA), a balance, and a voltage-controlled gear pump. (A schematic of the setup is
shown in Fig. 4.1(a).) Pressures from 6 to 40 kPa were applied.

Figure 4.1: (a) Schematic of the experimental setup for permeability measurement
[125]. (b) Schematic of the experimental setup for diametral compression (Brazilian
disk) test. Disks were placed between compression platens with an aligner before
the start of the compression testing to ensure the load was applied along the long
axis of pre-crack.

Cylindrical pyrolyzed samples were core-drilled to be 13 mm in diameter, and
the circumferences of the samples were enclosed by a low shrinkage mineral-filled
acrylic system (VariDur 3003, Buehler, IL, USA) to prevent the leakage from the circular side of the samples. The samples were then sectioned at 4 mm and 9 mm from
the sample base, and enclosed along a length of 5 mm tubing for the permeability
investigation. Three separate samples were tested for each CNT concentration, and
the Darcy-Forchheimer equation was used to determine the permeability constants
[78].
Δ𝑃
= 𝑣 + 𝑣2
(4.2)
𝑘1
𝑘2
where Δ𝑃 is the pressure drop across the sample, 𝐿 is the thickness of the sample,
𝜇, 𝜌, 𝑣 are, respectively, the dynamic viscosity, the density, and the flow velocity of
water, and 𝑘 1 , 𝑘 2 are the intrinsic and inertial permeability, respectively.

56
4.2.5

Mechanical Properties

4.2.5.1

Compressive Strength

Uniaxial compression tests were performed on an Instron 4204 universal testing
machine (Instron, MA, USA) in constant displacement mode at a displacement
rate of 0.1 mm/min. To prepare samples for compressive strength measurements,
cylindrical pyrolyzed samples were core-drilled to be 13 mm in diameter. The
measurements for compressive strength for brittle porous materials are susceptible to
a wide variance, arising from the crushing behavior due to high local contact stresses,
leading to inconsistent results. A study by Mehr showed that the incorporation of a
rigid interface layer could spread the applied stress more evenly and provide reliable
results [117]. Therefore, VariDur 3003 was used to cap both ends of the sample, and
a pair of spherical washers were placed between the sample and the compression
platens to ensure the compression load was applied along the axis of the sample (A
schematic of the setup can be found in [125]).
Three samples were tested for each CNT concentration. The height and diameter were recorded and used to calculate the engineering stress and strain. The
compressive strength was determined by the peak of the stress-strain curve.
4.2.5.2

Fracture Toughness

The diametral compression test was performed to determine the fracture toughness
of dense CNT-SiOC composites and assess the effect of CNTs on the CNT-SiOC
walls. To prepare the Brazilian disks, CNTs were mixed with MK powder via
rotation in a polypropylene bottle at 70 rpm for at least 12 h to obtain a uniform
mixture. The powder mixture was then sifted through a mesh with 500-𝜇m nominal
sieve opening. The powder mixture was poured into a mold with a diameter of 65
mm. Two pieces of Teflon-coated Mylar of 0.07-mm thickness and 10-mm length
were taped together and inserted into the powder mixture to create the pre-crack at
the center of the disk after pyrolysis.
The mold was heated at 180℃ for 48 h to cure the MK. Pure MK and powder
mixtures with 1.3, 4.3 wt.% CNTs produced smooth and dense solids after curing.
Mixtures with 8.2 wt.% CNTs were bubble-filled and did not produce a smooth
surface, and were therefore not included in the toughness testing. The cured samples
were then were core-drilled to be 29 mm in diameter. The circumferential side
and the edges were polished with 240, 400, 600, 800, and 1200 grit silicon carbide
sandpaper to obtain a uniform disk shape. The samples were pyrolyzed in an alumina

57
crucible filled with SiOC powders pyrolyzed from MK to achieve uniform heating.
The samples were heated to 1100℃ and held for 4 h with a 2℃ /min heating and
cooling ramp rate under argon. During pyrolysis, the Mylar decomposed, leaving
a pre-crack in the center of the disk. The pre-crack length, thickness, and diameter
were recorded for each sample. At least four samples for each CNT concentration
were tested.
Two pieces of rubber were placed between the sample and the compression platens
to avoid the crushing at contact points mentioned earlier. An aligner was used
to set up the disk perpendicular to the compression platens. Once the disk was
clamped between two compression platens with 1 N applied force, the aligner was
removed before the compression test. The setup is shown in Fig. 4.1(b). Uniaxial
compression tests were performed on an Instron 4204 universal testing machine in
constant displacement mode at a displacement rate of 0.5 mm/min.
The toughness was calculated following [7, 28]
𝜋𝑅𝑡
𝑁𝐼 = √ 𝐾𝐼
𝑃 𝜋𝑎

(4.3)

=𝑇1 𝐴1 (𝜃) + 𝑇2 𝐴2 (𝜃) + 𝑇3 𝐴3 (𝜃) + ...
where 𝑁 𝐼 is the normalized stress intensity, 2𝑎 is the length of the pre-crack, 𝑅 is
the radius and 𝑡 is the thickness of the Brazilian disk, 𝑃 is the applied load, 𝐾 𝐼 is
the stress intensity factor of mode I, and 𝐴𝑖 and 𝑇𝑖 s are, respectively, angular and
numerical constants. For 𝜃 = 0, 𝐴1 = 1 and 𝐴𝑖 = 0 if 𝑖 ≠ 1, the expression reduced
to
𝜋𝑎
𝑃 × 𝑇1
(4.4)
𝐾𝐼 =
𝜋𝑅𝑡
where 𝑇1 is a function of 𝑎/𝑅 and was evaluated/interpolated from the values
reported in [7] for each sample. The maximum load associated with crack extension
is where K 𝐼 = K 𝐼𝑐 .
4.3
4.3.1

Results and Discussion
Carbon Nanotube Dispersion in Solution and Freeze-cast Structures

Fig. 4.2(a, b) illustrate the effect of dispersants on the carbon nanotube dispersion.
The suspension without KD1 in Fig. 4.2(a) had a large cluster of interconnected
agglomerates. It is believed that although sonication broke up agglomerates, CNT
agglomerates re-formed without a dispersant present [107]. The suspension with
KD1 and sonication in Fig. 4.2(b) had no visible agglomerates and did not settle

58
within the duration of the freezing process. Sonication broke up the CNT agglomerates and the dispersant attached to the surfaces of CNTs, preventing them
from entangling with other CNTs again. Adding dispersants and sonicating are two
essential steps in successfully dispersing carbon nanotubes and forming a stable
dispersion to freeze.

Figure 4.2: Suspensions with 3 mg/ml CNTs and 300 mg/ml MK (a) without, and (b)
with KD1. Representative SEM micrographs of the resulting freeze-cast structures
with 1.3 wt.% CNTs (c) without and, (d) with KD1.

The effectiveness of the dispersion in suspension has a direct correlation with the
uniformity of CNT distribution in freeze-cast structures. Shown in Fig. 4.2(c) is an
SEM image of the freeze-cast structure made from the suspension without the dispersant, which had massive agglomerates containing concentrated clusters of CNTs.
Macro-pores of more than 100 𝜇m in diameter were created by air bubbles trapped
in the suspension due to the high viscosity from poorly-dispersed CNTs. Shown in
Fig. 4.2(d) is an SEM image of the freeze-cast structure made from the suspension
with the dispersant. It has no visible agglomerates and is similar to typical lamellar
freeze-cast structures made with DMC [125]. The comparison demonstrates that
the effectiveness of dispersion of CNTs is especially crucial for freeze casting to
produce CNT-SiOC composites. In the case of poor CNT dispersion, not only will
agglomerates be present, but equally important, the freezing process during freeze
casting will be disrupted. With large CNT agglomerates present, the freezing front

59
splits and engulfs CNTs instead of ejecting them to pore walls as it does with the
MK preceramic polymer.

Figure 4.3: Representative SEM micrographs of freeze-cast structures (a) pure
SiOC, (b) 1.3, (c) 4.3, (d) 8.2 wt.% CNT composites.

Shown in Fig. 4.3 are SEM images of four freeze-cast samples prepared with
both dispersant and sonication, with equal preceramic polymer contents, but with
increasing CNTs. With an increasing concentration of CNTs, the agglomerates
could not be prevented and protruded from the walls, making the surfaces of walls
rougher. If agglomerates were sufficiently large, they formed bridges of CNTs
and SiOC between walls as in Fig. 4.3(d). However, despite extra bridges seen
in freeze-cast samples with 8.2 wt.% CNTs, overall, the structure did not change
significantly compared to CNT-free compositions. Even the freeze-cast structure of
the highest CNT concentration (8.2 wt.%) has few small (<20 𝜇m) agglomerates, a
stark contrast to that made without dispersants in Fig. 4.2(a), having a large number
of >50 𝜇m agglomerates with just 1.3 wt.% CNTs.
A closer look at cross-sections of lamellar walls in Fig. 4.4 provides an added view
of CNT dispersion—this time within the SiOC walls. We observed localized CNT-

60
dense regions of 20 to 500 nm, noted by circles in Fig. 4.4(b) through (d), indicative
of sub-micron CNT agglomerates, which were reduced in size by several orders
of magnitude compared to those of as-received CNTs. As CNT concentration
increases, the number of agglomerates increases. No pores or delamination are
observed between CNTs and SiOC.

Figure 4.4: Representative SEM micrographs of freeze-cast walls of (a) pure SiOC,
(b) 1.3 (c) 4.3, (d) 8.2 wt.% CNT composites. The white dots, present in 1.3, 4.3,
8.2 wt.% samples, are carbon nanotubes. Sub-micron agglomerates are highlighted
with green circles.

X-ray diffraction patterns in Fig. 4.5(a) shows that pure SiOC pyrolyzed at 1100℃ is
structurally amorphous, with a broad peak centered at 23.4°. No distinct crystalline
peaks were observed, consistent with the literature [36, 72, 166]. As CNTs are
added, new peaks begin to emerge. With peak fitting software, Fityk [187], the
XRD pattern was resolved into several peaks, shown in Fig. 4.5(b). Peaks located
at 26.0° and 43.6° are attributed to MWCNT. The ratio of the height of the two
peaks (ratio ∼3) suggest carbons are in the form of MWCNT (ratio ∼5) rather than
graphite (ratio
20) [148]. The peaks from MWCNT confirmed the preservation

61
of CNTs in the SiOC matrix throughout pyrolysis at 1100℃.
The peak located at 35.6° is likely from SiC [79], formed from the redistribution
reactions between Si-O and Si-C bonds and subsequent phase separation of SiOC
glass usually at 1200-1400℃ [22, 132, 165]. We hypothesize that the extra carbon
introduced into the system might have facilitated the formation of SiC. Sujith et
al. [171] added graphene nanoplatelets to polysiloxane and also saw peaks of SiC
emerge at temperatures as low as 1000℃.

Figure 4.5: (a) XRD pattern of pure SiOC, 4.3, 8.2 wt.% CNT composites and
MWCNTs. (b) Background subtraction and peak fitting performed on XRD pattern
of 8.2 wt.% CNT composite to identify phases present.

4.3.2

Electrical Conductivity

Plotted in Fig. 4.6 are the conductivities of porous CNT-SiOC composites. Pure
SiOC freeze-cast structures are insulating with a conductivity value around 10−12
S·cm. Polymer-derived ceramics are typically insulating (𝜎<10−10 S·cm) if pyrolyzed at low temperatures (<600℃) and semiconducting if pyrolyzed at medium
temperatures (800∼1400℃) [36], significantly higher than conductivity values of
pure SiOC observed here. However, the free carbon concentration and its distri-

62
bution after pyrolysis play a major role in determining the electrical behavior of
polymer-derived ceramics [38]. The ceramics from polysiloxanes have lower carbon contents than those from polycarbosilanes. Among polysiloxanes, MK also
produces a much lower carbon content (∼8 wt.%) compared to those (∼21 wt.%)
from other polysiloxanes (Silres 602, Silres 30, REN 100, H 44, Silres 601, Silres
600 from Wacker Chemie, Munich, Germany) [154]. Therefore, it is expected that
ceramics produced from MK will have a lower conductivity than other polymerderived ceramics. Another factor for the low conductivity observed here is that all
the conductivity measurements in [36] were performed on dense ceramics while the
conductivity of porous freeze-cast structures was measured here. Conductivity is
known to decrease with increasing porosity [104].

Figure 4.6: Electrical conductivity, calculated without adjustment for porosity, as a
function of CNT concentration of freeze-cast composites. The resistance of one pure
SiOC sample was above the equipment limit (200 GW) of Keithley 614 Electrometer
and its conductivity was calculated using the limit. This data point was plotted with
a circle and a downward arrow.

More importantly, a transition from insulating to conductive behavior was observed

63
with increasing CNT concentration. With as little as 4.3 wt.% CNTs, the conductivity increases by ten orders of magnitude, an indication that the CNTs were largely
preserved after the pyrolysis. The critical CNT concentration of electrical percolation was in the range of 0.1 to 0.5 wt.%, well in the range of reported percolation
thresholds for nearly 150 CNT-polymer composites reviewed by Bauhofer et al [15].
The excluded volume theory is often used to estimate the electrical percolation limit
for non-spherical fillers in a composite. Two conductive fillers would be in contact
and increase the conductivity of the composite if the center of one enters a certain
surrounding volume of another. This specific volume is the excluded volume of the
filler (𝑉𝑒𝑥 ). The ratio of the excluded volume to the real volume (𝑉) of the filler
determines the percolation threshold, Φ𝑐 , and is roughly equal to the inverse of
twice the aspect ratio, D/2L [12].
For an aspect ratio ∼3000 (D = 12 nm and L = 40 𝜇m), the excluded volume
theory predicts a percolation limit of 0.017 vol.%, which is an order of magnitude
lower than that observed in the experiment. The discrepancy came from the fact
that the excluded volume theory is intended for straight and stiff rods. In addition,
the excluded volume theory fails to account for the dispersion and entanglement
of objects, which becomes critical for CNTs because of their large aspect ratio.
To address these deficiencies, Li et al. [106] introduced two parameters (𝜉, 𝜀) to
describe the state of the dispersion specifically for CNTs, where 𝜀 is a measure of
how tightly packed CNTs are in an agglomerate and 𝜉 accounts for the fraction of
agglomerates in the dispersion. The percolation limit can be calculated with the
following equation:
𝜉𝜀𝜋 27(1 − 𝜉)𝜋𝐷 2
(4.5)
Φ𝑐 =
4𝐿 2
For high aspect-ratio CNTs dispersed by ultrasonication for hours, Li et al. found
𝜉 and 𝜀 to range from 0.01 to 0.05. Based upon Eqn. 4.5, the percolation limit of
CNTs was determined to be 0.13 vol.% (D = 12 nm, L = 40 𝜇m and 𝜉 = 𝜀 = 0.05),
which closely agrees with the observed percolation limit.
4.3.3

Permeability

Pressure vs. flow rate plots for porous SiOC and CNT-SiOC composites are shown
in Fig. 4.7(a), and are largely linear. Inertial permeability (𝑘 2 ) in Eqn. 4.2 accounts
for any non-linearity in the pressure-versus-flow rate data, a common occurrence
when the fluid is a gas and the flow velocity is large. Since water was used as the
fluid of interest, the inertial permeability term was neglected.

64
Plotted in Fig. 4.7(b) are the experimental permeability and measured porosity
for porous CNT-SiOC composites as a function of CNT concentration. Average
permeability constants ranged from 7.9 to 14.9 × 10−12 m2 with no observable trend
with reinforcement fraction. Given the similarity in microstructure (Fig. 4.3), we
did not expect to see a significant change in permeability, which was confirmed.
Likewise, porosity, often related to permeability, demonstrated no noticeable change
with the addition of CNTs. This was expected since the variance in total solid loading
in suspensions was less than 1 vol.% (from 19.5 to 20.1 vol.% of the suspension).
It is reasonable to deduce that the addition of a small quantity of well-dispersed
CNTs neither changes the freeze-cast structures nor the fluid-dynamic properties
significantly.

Figure 4.7: (a) Pressure vs. flow velocity to establish permeability of pure SiOC and
CNT-SiOC composites. (b) Permeability (circles with lefthand axis) and porosity
(triangles with righthand axis) as a function of CNT concentration of freeze-cast
composites. Error bars represent ± one standard deviation.

4.3.4

Mechanical Properties

Shown in Fig. 4.8(a) are representative stress-strain curves of each CNT concentration tested under uniaxial compression. Typically, the fracture behavior for brittle
porous structures, as described by Gibson and Ashby, demonstrates a linear rise to
a peak followed by a lower stress plateau [65]. This is well-documented in porous
structures composed of pure SiOC. With increasing CNT concentration, the stress-

65
strain curves diverge more and more from the typical fracture behavior. Stress-strain
curves of 8.2 wt.% CNT composites sustain a rapidly increasing stress followed by a
higher plateau, while CNT concentrations of 1.3 wt.% and 4.3 wt.% show a transient
behavior between pure SiOC and 8.2 wt.%. In all cases, CNTs raise the peak stress
and plateau in the stress-strain curves.

Figure 4.8: (a) Representative stress-strain curves of freeze-cast samples. The first
load drop around 1 MPa in most samples is suspected to be from the realignment
of the washer. Examples of fractured samples after compression with (b) pure
SiOC and (c) 8.2 wt.% CNTs. The blue disk-like capping is a cured mineral-filled
acrylic applied to avoid crushing behavior common in compression tests of porous
ceramics.

The change in fracture behavior with the addition of CNTs can also be seen in

66
Fig. 4.8(b, c), macroscopic views of fractured samples of uniaxial compression.
In pure SiOC samples, the fracture occurs along lamellar walls, along directions
of maximum shear. The samples fail by bridge fracture and lamellar wall sliding,
resulting in a substantial load drop as observed early in the stress-strain curve. At
the largest CNT concentration tested (8.2 wt.%), fracture occurs perpendicular to
the applied force, exhibiting a rough fracture plane. Here samples fail by lamellar
wall bending and buckling. The failure arises cell-by-cell, which manifests as an
effective hardening accentuated with small load drops.
One likely explanation for this difference in fracture mode (shear vs. wall bending and buckling) with the addition of carbon nanotubes is bridging from CNT
agglomerates, as seen in Fig. 4.3(d).
For pure SiOC, DMC produces relatively smooth lamellae with few bridges inbetween (Fig. 4.9(a)). As the CNT concentration is increased, the agglomerates,
noted first with 4.3 wt.% CNTs, increase in size. If agglomerates are larger than the
lamellar spacing, we hypothesize that they form bridges between the lamellar walls,
consistent with the images in Fig. 4.9(b). The bridges between lamellar walls make
shear failure unfavorable, in contrast to that demonstrated in pure SiOC and SiOC
with low CNT concentrations, and ultimately lead to a higher plateau in stress-strain
curves.
The evidence of the structural difference contributing to the change in fracture mode
can also be observed in the fracture surfaces (Fig. 4.9(c, d)). For pure SiOC, the
fracture surface is a clean lamellar surface with little contact between one lamella
and another. Lamellae are able to slide over one another easily without breaking
lamellar planes. (Fig. 4.9(c)). On the other hand, the fracture surface of the 8.2 wt.%
CNT composite exhibits a broken surface of lamellae with bridging agglomerates.
The bridges prevent the sliding between lamellae (Fig. 4.9(d)).

67

Figure 4.9: SEM micrographs of freeze-cast and pyrolyzed walls of (a) pure SiOC
and (b) 8.2 wt.% CNTs. In pure SiOC, the lamellae are smooth with few features
and bridges. In the 8.2 wt.% composite, the lamellae are bridged with CNT agglomerates. SEM micrographs of fracture surfaces with (c) pure SiOC and (d) 8.2 wt.%
CNTs. The green arrows in (d) point to the agglomerate bridges between lamellae.

The strengthening effect of carbon nanotubes in freeze-cast structures is illustrated
in Fig. 4.10(a), which documents the compressive strength as determined by the
initial peak load. The strength increases with increasing concentration of CNTs
with the exception of SiOC with 8.2 wt.% CNTs. Unlike those for other CNT
concentrations, the peak stress can hardly be distinguished from the high plateau,
and is therefore not included here. With the exception of agglomeration effects in
the 8.2 wt.% CNT composite and discussed in [9], the strengthening enhancement

68
is consistent with short-fiber reinforcement models, where load can be transferred
between the CNT and SiOC matrix [34, 60].

Figure 4.10: (a) Compressive strength and fracture toughness as a function of CNT
concentration. (b) Compressive strength vs. permeability constant of freeze-cast
CNT-SiOC composites compared with that of pure SiOC both in this work and the
values reported by Naviroj [125]. Error bars represent ± one standard deviation.

To understand the reason for the increase in the peak load, we examined how CNTs
influence the fracture toughness of the SiOC-CNT walls. Shown in Fig. 4.10(a) are
values of fracture toughness for SiOC samples as a function of CNT concentration.
These values follow the same trend as the compressive strength, with 57% and 74%
increase for 1.3 wt.% and 4.3 wt.% CNTs, respectively. It is reasonable to argue the
freeze-cast walls with higher toughness can withstand a higher stress before fracture
as exemplified by an increase the peak load of freeze-cast CNT-SiOC composites.

69

Figure 4.11: Representative SEM micrographs of fracture surfaces of Brazilian disks
of (a) pure SiOC and (b) 4.3 wt.% CNTs. The insets in (b) shows a magnified image
of a fracture surface of 4.3 wt.% CNT composites. The rough patches, appearing in
4.3 wt.% composites, are clusters of CNT agglomerates.

Fracture surfaces of Brazilian disks of pure SiOC and with 4.3 wt.% CNTs are
compared in Fig. 4.11. Pure SiOC (Fig. 4.11 (a)) exhibits the typical glassy and
smooth surface of amorphous brittle materials while that of 4.3 wt.% CNT composite
(Fig. 4.11(b)) has rough patches where CNT agglomerates are located. The inset
shows a magnified image of Fig. 4.11(b). Interestingly, the CNT agglomerates in
the Brazilian disk are larger both in size and number than those in freeze-cast CNT
composites of the same CNT concentration. It is likely due to the difference in
fabrication methods used for freeze-cast samples and Brazilian disks. For Brazilian

70
disks, the CNT dispersion was achieved through mixing CNTs in powder form with
the preceramic polymer powders without sonication or the addition of dispersant.
As previously mentioned, two major challenges with CNT reinforcement of ceramics
are the damage done to CNTs by high temperature and pressure required to densify
ceramic powders, and difficulties to achieve homogeneous dispersion of CNT with
powder process. To prevent damage, some researchers report a low-temperature
fast-sintering technique, spark-plasma sintering, while others used preceramic polymers, which convert to bulk ceramics under ambient pressure and relatively low
temperature. In this work, we have addressed both issues by utilizing both preceramic polymer and sonication in solution, known to produce well-dispersed CNT
composites. We compared our results with CNT-ceramic composites sintered by
spark plasma sintering (Zhan et al. [197]) and those made from preceramic polymers
(Katsuda et al. [85]). Zhan et al reported 139% and 176% increase in toughness
for 5.7 and 10 vol.% (approximately 3.1 and 5.6 wt.%), respectively. Katsuda et al.
reported 49% and 72% increase in toughness for 1, 2 wt.%. Both results are on par
with the toughness increase we observed.
The reinforcing effect of CNTs on freeze-cast CNT-SiOC composites is demonstrated both in the fracture toughness and compression tests; CNTs in freeze-cast
walls result in tougher walls, and consequently increase the peak load of the freezecast structure. The agglomerate bridges of CNTs prevent shear failures by making
fracture along the lamellar walls unfavorable, leading to a higher plateau in stressstrain curves. The two factors, both introduced by the addition of CNTs, improve
the durability of freeze-cast CNT-SiOC composites.
Finally, Fig. 4.10(b) shows compressive strength versus permeability of freeze-cast
CNT-SiOC composites compared with that of freeze-cast pure SiOC both from this
work and values reported by Naviroj [125]. Freeze-cast pure SiOC samples in [125]
have similar values in compressive strength and permeability. The best CNT-SiOC
composite, however, achieved a markedly higher compressive strength of 3.3 MPa
and a similar permeability constant of 11.7 × 10−12 m2 . It is demonstrated here that
by introducing CNTs into the SiOC matrices as reinforcements, the compressive
strengths were doubled at virtually no cost to permeability, making this method
ideal for fabricating high load-bearing directional porous ceramics for transport and
filter applications.

71
4.4

Conclusions and Implications

Challenges of combining the high aspect ratio reinforcements with freeze-casting to
produce porous composites have been systematically addressed. To avoid altering
freezing dynamics, the concentrations of fillers were chosen to be less than 2 wt.%
of the suspensions. MWCNTs were selected as fillers so that the reinforcement was
sufficient even at low concentrations. The addition of dispersants (KD1) combined
with sonication was shown to greatly reduce CNT agglomerates, allowing for stable
suspensions. Not surprisingly, there was an increase in the size and number of CNT
agglomerates with increasing CNT concentration. However, it was shown that if
CNTs were well dispersed in suspension, the addition of CNTs would not change
the pore structures notably.
The electrical conductivity changed drastically by 10 orders of magnitude with the
addition of 8.2 wt.% CNTs, and the transition from insulating to conductive was
found to be around 0.1 to 0.5 wt.%. This significant increase in conductivity indicated the preservation of CNTs after pyrolysis. Permeability was largely unaffected
by the addition of CNTs and followed a similar trend with porosity as unreinforced
lamellar freeze-cast solids.
With the introduction of as little as 1.3 and 4.3 wt.% CNTs in porous SiOC ceramics,
the compressive strength increased by 88% and 118%, respectively. Two different
fracture behaviors were observed. Freeze-cast samples of pure SiOC fractured in
shear parallel to lamellar walls. The stress-strain response featured a peak stress
followed by a low-strength plateau. At the highest CNT concentration tested (8.2
wt.%), fracture perpendicular to the load direction and lamellar walls was favored.
In stress-strain curves, the peak became less pronounced and more difficult to
distinguish from the following high-strength plateau. In addition, the prevention
of shear failure by bridges formed from CNT agglomerates increases the stress at
which the plateau occurs.
Disks of SiOC and CNT-SiOC were made to investigate the effects of CNTs on the
walls. The fracture toughness, determined with diametral compression, increased
by 57% and 74 %, respectively, for 1.3 and 4.3 wt.% CNTs added. The toughness
follows a similar trend seen in compressive strength. The toughening of bulk disk
samples, analogous to the walls, increases the peak stress in the stress-strain curve
of the freeze-cast structure. The reinforcing effect of CNTs on the walls, along with
prevention of shear failure by CNT bridges, increases the compressive strength,
changes the fracture behavior and, overall, improves the load-bearing capabilities of

72
freeze-cast CNT-SiOC composites. The potential of this method has been demonstrated through measurements of permeability, conductivity, compressive strength,
and fracture toughness. CNTs as reinforcements have successfully improved the
mechanical properties of porous SiOC without affecting permeability, rendering
this technique powerful in fabricating robust porous ceramics.

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Chapter 5

HIERARCHICAL COMPOSITES VIA SELF-ASSEMBLY OF
BLOCK COPOLYMER
Material in this chapter is reproduced in part from "Hierarchical porous SiOC via
freeze casting and self-assembly of block copolymers", C.T. Kuo, L.M. Rueschhoff,
M.B. Dickerson, T.A. Patel, K.T. Faber; Scripta Materialia, 191 (2021) 204–209.
The work was done in collaboration with Lisa M. Rueschhoff, Matthew B. Dickerson, Tulsi A. Patel of the Air Force Research Laboratory, and Katherine T. Faber.
T. Kuo fabricated and characterized hierarchical ceramics using SEM, MIP, and
N2 adsorption isotherms, characterized weight loss, volume shrinkage, compressive
strengths, and permeabilities, conducted image analysis on X-ray computed tomography (XCT) datasets, and wrote the majority of the manuscript. L.M. Rueschhoff
characterized the BCP-PCP and its PDC system. T.A. Patel performed XCT. K.
Faber and M.B. Dickerson supervised this work.
5.1

Introduction

Chapters 3 and 4 explored the challenges and opportunities of filler-reinforced highpermeability pore structures by SiC whiskers and CNTs, respectively. Once a pore
structure is optimized for its mechanical and transport properties, its surface area
can be increased by creating a high-surface-area mesoporous coating on top of the
original pore structure, effectively constructing a hierarchical structure with large
pores for fluid transport and small pores which produce high surface area for capture
or catalytic reactions.
Procedures that utilize molecular templates such as surfactants or block copolymers
(BCPs) are advantageous for the creation of mesoporous ceramics [39, 96, 139, 158].
Rueschhoff et al. [147] deposited polymer films with a BCP and polycarbosilane
preceramic polymer (PCP) blend that formed interconnected spherical micelles.
These were pyrolyzed to produce mesoporous silicon carbide-based (SiC-based)
ceramic films, shown in Fig. 5.1(a). In this work, we adopt this BCP/PCP system
with freeze casting to fabricate a hierarchical structure.
While various synthesis methods of hierarchical structures and their respective surface areas have been discussed in literature [36, 195, 198], mechanical performance

74
is often overlooked and warrants consideration. Retaining the desired mechanical properties and permeability with added pore hierarchy is often a challenge to
practicability for applications such as filters, electrodes, and catalyst supports. In
this chapter, we address this challenge by coupling directional freeze casting of
a preceramic polymer as the macroporous template with a self-assembled block
copolymer-preceramic polymer blend to provide a conformal mesoporous coating
on the macropore walls. The processing methodology, and resultant microstructures, compressive strength, permeability and surface area of the hierarchical pore
structures are featured.
5.2
5.2.1

Experimental Methods
Materials and Processing

The macroporous scaffold for the hierarchical porous structure was prepared via directional freeze casting as described in [124]. Twenty wt.% polysiloxane preceramic
polymer (Silres® MK Powder, CH3 -SiO1 .5, Wacker Chemie, Munich, Germany)
was dissolved in dimethyl carbonate (DMC, Sigma-Aldrich, MO, US). Ten µl/g
(agent/solution) of cross-linking agent (N-(2-Aminoethyl)-3-aminopropyltrimethoxysilane,
Geniosil GF 91, Wacker Chemie, Munich, Germany) was added into the solution.
The solution was then poured into a glass mold with an inner diameter of 24 mm, and
quenched at -30℃. After solidification, the sample was dried in a lyophilizer (Virtis
Wizard 2.0, SP Scientific, PA, US) until the solvent was removed. The bottom one
millimeter of the sample was polished off with sandpaper to improve infiltration
before curing at 200℃in advance of infiltration with BCP/PCP gel.
The mesoporous coating was prepared via self-assembly of the BCP (KurarityTM
LA4285, PMMA-PnBA-PMMA, Kuraray America, Inc., Houston, TX, US). The
BCP and a polycarbosilane preceramic polymer (StarPCSTM SMP-10, Starfire
Systems Inc., Glenville, NY, US) were dissolved in 2-ethyl-1-hexanol (2EH, SigmaAldrich, MO, US) to form a BCP/PCP gel, a physical polymer blend with no
chemical interaction between the two polymers [147]. The freeze-cast scaffold was
infiltrated with the BCP/PCP solution at 80℃ using vacuum. After infiltration, the
sample was dried for at least 4 days, then cured and pyrolyzed at 230 and 900℃,
respectively. During pyrolysis, freeze-cast polysiloxane walls were converted into
amorphous SiO2 -rich silicon oxycarbide (SiOC)
[36, 125, 154], while the porous coating formed on the freeze-cast walls through
the removal of the fugitive BCP phase and the conversion of polycarbosilane into

75
amorphous SiOC (Table 5.1) [86]. Elemental analysis of the derived SiOC from
the BCP/PCP gel system has been reported in [21, 147]. Thermogravimetric (TGA)
analysis [147] indicates a general rule of mixtures for the ceramic yield of a similar BCP/PCP systems, demonstrating that the BCP addition does not change the
ceramic yield of the SMP-10. The synthesis process of these hierarchical samples (designated as Hierarchical20) is illustrated in Fig.5.1(b). Fourier-transform
infrared spectroscopy (FTIR) spectra of both MK and SMP-10 polymers and the
derived SiOC ceramics indicate complete conversion of the preceramic polymers to
ceramics after pyrolysis [125, 147].
Table 5.1: Chemical composition (in weight percent) of MK- and SMP-10-derived
ceramics. The elemental analyses were conducted with a thermobarometrical redox
analysis [154, 185] and by a carbon analyzer [86].
Precursor polymer SiO2 SiC
MK [154]
76.01 15.98 8.00
SMP-10 [86]
10.69 82.08 7.21

For comparison, the sample denoted as Freezecast20 had the same polymer concentration as Hierarchical20 in the freeze-casting stage. However, it was pyrolyzed
directly after freeze casting without infiltration of BCP/TCP. Three other freeze-cast
samples with varying polymer concentrations (Freezecast15, Freezecast25, Freezecast35) were produced to establish baseline compressive strength and permeability.
Polymer concentrations and average porosities ranging from 63.0 to 83.1% were
determined by Archimedes’ method (following ASTM C373) and are summarized
in Table 5.2.
Table 5.2: Average porosity after pyrolysis of all Freezecast and Hierarchical20
samples. Six samples were tested for each polymer concentration.
Freezecast15
Freezecast20
Freezecast25
Freezecast35
Hierarchical20

MK Polymer concentration (wt.%)
15
20
25
35
20

Average porosity (%)
83.1 ± 0.4
78.3 ± 0.6
73.1 ± 0.9
63.0 ± 0.8
72.8 ± 0.7

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5.2.2

Pore Structure

The pore structure was imaged using scanning electron microscopy (ZEISS 1550VP
FESEM, Carl Zeiss Microscopy GmbH, Jena, Germany) and X-ray computed tomography (X-ray CT, ZEISS Xradia 520 Versa, Carl Zeiss Microscopy GmbH,
Jena, Germany). For the latter, a 120 kV and 10 W X-ray source was used with
an exposure time of 25 sec. A total of 2401 projections were obtained about the
field of view from -180° to 180°, which were manually reconstructed in the Zeiss
Reconstructor program; beam hardening was corrected for and 2D reconstructed
slices of all three orthogonal planes were written through the Zeiss XMController
program. The resulting tomograms have a resolution of 1.52 µm/pixel and were
circles with a radius of 736 µm in one direction and 1482 × 1487 µm2 rectangles in
two directions.
The pore size distribution was determined with a mercury intrusion porosimeter
(MIP, AutoPore IV 9500, Micromeritics Instrument Corp., Norcross, GA, US).
Nitrogen adsorption data were collected with TriStar II 3020 analyzer (Micromeritics
Instrument Corp., Norcross, GA, US). The samples were ground into powders and
degassed at 300℃ under vacuum for 3 h prior to measurements. The BET surface
area [23] was determined under relative pressures of 0.02-0.15 with a correlation
coefficient of 0.999.
5.2.3

Permeability

Permeability was determined from the flow rate of water through cylindrical samples
∼4.5 mm tall and ∼9.5 mm wide over a series of pressure drops, using an experimental set-up described in [97]. Three samples were tested for each processing
condition. The permeability was calculated with Darcy-Forchheimer equation [78]:
Δ𝑃
= 𝑣 + 𝑣2
𝑘1
𝑘2

(5.1)

where ΔP is the pressure drop across the sample, L is the thickness of the sample, and
µ, 𝜌, v are the dynamic viscosity, density, and flow velocity of water, respectively.
The parameters 𝑘 1 and 𝑘 2 are the intrinsic and inertial permeabilities, respectively;
the latter describes non-linear flow. Given the linearity of the data collected, the
non-linear term was ignored.
5.2.4

Mechanical Properties

To determine the compressive strength and Young’s modulus, mechanical tests were
performed on cylindrical samples ∼6 mm tall and ∼9.5 mm in diameter using an

77
Instron 5960 Series Universal Testing System (Instron, MA, US). The compressive
load was applied parallel to the axis of freezing direction under a constant displacement rate of 0.05 mm/min. The top and bottom of the samples were capped
with a low-shrinkage acrylic system (VariDur 3003, Buehler, IL, US) to minimize
contact stresses. Details of the setup can be found in [97]. The peak load was used
to calculate compressive strength. Three samples were tested for each processing
condition.

78
5.3

Results and Discussion

Figure 5.1: (a) AFM phase view of as-deposited films (left) and SEM image of
pyrolyzed films (right) made with BCP/PCP gel. (b) Schematic of the synthesis
process for Hierarchical20. SEM images of pyrolyzed freeze-cast structures of (c)
Freezecast20 and (d) Hierarchical20, with (e) enlarged lamellar spacings and (f)
porous conformal coating on the freeze-cast pore wall of Hierarchical20.

79
Fig. 5.1 shows cross-section SEM images of pyrolyzed (c) Freezecast20 and (d)
Hierarchical20. Lamellar freeze-cast structures, typical of DMC, were seen in both
Freezecast20 and Hierarchical20. The freeze-cast structure remains largely intact
after infiltration and pyrolysis, except for the creation of selected large lamellar
spacings shown in Fig. 5.1(e). Fig. 5.1(f) reveals the mesoporous coating on the
surface of the freeze-cast pore walls, resulting from the infiltration of BCP/PCP
gel. There was no delamination between the coating and the scaffold during the
simultaneous conversion of polysiloxane and polycarbosilane into SiOC during
pyrolysis. This is in contrast to flaking that is seen in some other synthesis processes
for hierarchical structures [39].
This bimodal pore size distribution of the hierarchical pore structure was confirmed
with MIP in Fig. 5.2, with peaks around 30 nm and 20 µm for Hierarchical20
sample. Macropores from freeze casting are observed in both Hierarchical20 and
Freezecast20 albeit smaller in size and volume for Hierarchical20 due to the mesoporous coating on freeze-cast pore walls. The presence of 30-nm pores from BCP
templating was unique to Hierarchical20 and consistent with the nominal size gathered from image analysis reported in [21]. Of the 69.1% porosity in Hierarchical20,
2.3% were smaller than 1 µm.

80

Figure 5.2: Pore size distribution of Hierarchical20 and Freeezecast20 pore structure
from MIP. Smoothing of the data, indicated by solid lines, was performed by
averaging sets of five consecutive points minus outliers.

In Fig. 5.1(e), dilated lamellar spacings were estimated to be nearly ten times
larger than surrounding lamellar spacings. The enlarged spacings are likely a
result of the infiltration and subsequent drying of the BCP/PCP gel. To understand
how infiltration affected the freeze-cast structure quantitatively, volume and weight
changes after each processing step were recorded (Table 5.3).
Table 5.3: Volume and weight changes (in percent) after each processing step of
Freezecast20 and Hierarchical20. At least five samples are measured for each step.
Infiltration
Volume (%)
NA
Weight (%)
NA
Volume (%) +15.7 ± 2.4
Hierarchical20
Weight (%) +264.3 ± 5.2
Freezecast20

Solvent
evaporation
NA
NA
-8.0 ± 2.6
-61.0 ± 0.7

Pyrolysis

Total

-47.2 ± 1.4 -47.2 ± 1.4
-16.3 ± 0.5 -16.3 ± 0.5
-48.5 ± 0.7 -45.2 ± 0.6
-25.8 ± 1.0 +5.4 ± 0.6

During infiltration, BCP/PCP gel completely fills the freeze-cast pores and pushes
pore walls apart, resulting in an increase in volume. It has been reported that

81
during the drying stage of similar BCP gel systems, the capillary stresses from the
evaporation of solvent could cause warpage [162]. In the current work, the stresses
could pull some walls back while leaving others separated, generating a decrease in
volume. The infiltration and drying steps combined create a 6% and 42% increase in
volume and weight, respectively, before pyrolysis. Pyrolyzed Hierarchical20 shrank
less than Freezecast20 (45.2% vs. 47.2%) compared to the freeze-cast green body.
The difference in volume change was found to be statistically significant with a pvalue of 2.1×10−3 . The concentration of enlarged lamellar spacings was determined
from stereological arguments [44], but was found to be only 0.27% showing that
the freeze-cast structures were still largely preserved. The large increase in weight
for Hierarchical20 in contrast to Freezecast20 results in a lower porosity despite the
sporadic presence of large lamellar spacings.

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Figure 5.3: Stages of the lamellar spacing analysis from X-ray CT (A tomogram
of Hierarchical20 is used as an example) (a) Original 2D image and segmented
image (b) Euclidean distance map (c) 2D skeleton of lamellar pores (d) Graphic
representation of lamellar spacing measurements (e) Average lamellar spacings
from image analysis of 2D X-ray tomograms.

Image analysis was performed on about 300 X-ray tomograms (of 2401 collected)

83
for Hierarchical20 and Freezecast20 in three directions using MATLAB to assess
average interlamellar spacing. Three orthogonal directions were chosen to reflect
the overall structure, with one parallel to freezing direction, reflecting the samples’
anisotropy, and two directions perpendicular to the freezing direction and each other
to account for the effect of image sectioning on lamellar spacing.
Examples of images from the step-wise analysis are shown in Figs. 5.3(a) through
(d). Each of the images was cropped, thresholded, and cleaned to perform image
segmentation (Fig. 5.3(a)). The distance from any point to its nearest center of
lamellar wall was measured to construct the Euclidean distance map (Fig. 5.3(b)).
The lamellar pores were then skeletonized (Fig. 5.3(c)). The distances between
the pore centers and the centers of nearest walls, i.e. half of the lamellar spacing,
were assessed by multiplying the Euclidean distance map (Fig. 5.3(b)) by the pore
skeleton (Fig. 5.3(c)) and are mapped in Fig. 5.3(d). The average lamellar spacings
of three imaging directions are presented in Fig. 5.3(e). The bar chart shows
that, on average, the lamellar spacing increases with the infiltration. The increase
is attributed to selectively enlarged lamellar spacings (0.27%) noted in SEM (Fig.
5.1(f)) and in the volume difference (∼3%) between pyrolyzed Hierarchical20 and
Freezecast20 noted earlier.
From Fig. 5.3(e), the effect of image sectioning was also prominent, as evidenced by
the lamellar spacing of Hierarchical20 being much larger than that of Freezecast20 in
one perpendicular direction than the other. This is not surprising since lamellae are
randomly aligned in the directions perpendicular to the freezing direction. Lamellar
spacing would be the smallest if the image was sectioned perpendicular to lamellar
walls and the largest (theoretically infinite) if sectioned paralleled to the walls.
Therefore, the angle of image sectioning in the directions perpendicular to freezing
direction can have a large influence on average lamellar spacing. The Freezecast20
sample has a specific surface area of 0.36 m2 /g, typical for a macroporous material.
The hierarchical sample has a specific surface area of 5.23 m2 /g, which implies
that surface area of the conformal mesoporous coating is 25.78 ± 2.60 m2 /g, as
calculated from the weight and surface area difference between Hierarchical20 and
Freezecast20 samples. By depositing the porous coating on the pore walls, the
coating provides at least 70 times more surface area than the freeze-cast scaffold.
An even higher surface area could possibly be achieved by infiltrating a different
BCP that provides a higher surface area.

84

Figure 5.4: (a) Permeability as a function of open porosity and (b) compressive
strength and Young’s modulus as a function of relative density of all Freezecast and
Hierarchical20 samples. Error bars represent ±1 standard deviation. Dotted lines
and the dashed line are linear regressions on log-log plots of baseline materials,
included to guide the reader.

Fig. 5.4(a) shows permeability constants for all Freezecast samples compared to
Hierarchical20. As noted earlier, the porosity decreased with infiltration. According
to the baseline, a decrease in porosity should be accompanied by a decrease in
permeability. However, permeability of Hierarchical20 was nearly the same as that
of Freezecast20. One likely explanation for this unexpected behavior is the presence
of enlarged lamellar spacings (3.4% by volume) as shown in both Fig. 5.1(f) and
Fig. 5.3(e) which allowed more flow, and hence produced a higher permeability
than anticipated with the lower porosity.

Figure 5.5: Strain at maximum stress as a function of relative density of all Freezecast
and Hierarchical20 samples. Error bars represent ±1 standard deviation. Dotted
lines is a linear regressions of baseline materials, included to guide the reader.

85
The compressive strength and Young’s modulus for all Freezecast and Hierarchical20
samples are reported in Fig. 5.4(b) and follow the generally observed trend of
increasing strength and Young’s modulus with relative density [141]. Gibson and
Ashby [65] showed that compressive strength and Young’s modulus are powerlaw functions of relative density. In Fig. 5.4(b), Hierarchical20 is also shown
to be within a reasonable range of the same power-law relation established with
Freezecast samples and other studies of freeze-cast polysiloxanes-derived ceramics
[125]. Likewise, the strain at maximum stress for the Hierarchical20 (Fig. 5.5) is
also consistent with that of the Freezecast samples, accounting for density. This
agreement demonstrates that the incorporation of a highly-porous conformal coating
does not diminish the expected strengthening with increased density.

86

Figure 5.6: Representative stress–strain curves of (a) Freezecast20 and Hierarchical20 and (b) Freezecast15, Freezecast25, and Freezecast35. SEM micrographs of
fracture surfaces of (c) Freezecast15 and (d) Freezecast35. Examples of fractured
samples after compression of (e) Freezecast15 and (f) Freezecast35.

As previously mentioned in Sections 4.3.4 and 3.3.1.2, brittle lamellar structures
with insufficient interlamellar bridges, such as freeze-cast structures from DMC,
often fail in shear [111]. In a typical stress-strain curve representing shear failure of
a porous lamellar solid, the stress increases linearly to a peak value at which point
the interlamellar bridges fracture and lamellar walls slip over one another, resulting
in a substantial load drop at strains <3%. The peak is followed by a low stress plateau
during crack propagation, and finally a gradual stress increase due to the densification
of the fractured pore structure [65]. Fig. 5.6(a) shows that both Freezecast20 and
Hierarchical20 failed in shear failure, exhibiting the signature stress-strain response

87
of porous brittle materials. The stress-strain curves, independently of SEM images,
also demonstrate that the introduction of a mesoporous coating has not noticeably
changed the structure of the underling freeze-cast scaffold.
Interestingly, while Freezecast15, Freezecast25, Freezecast35 have fairly different
porosities (83.1-63.0%) and mechanical strengths (2.30-9.68 MPa), they all failed in
shear as evidenced by similar shapes of the stress-strain curves. Additional evidence
of the shear failure was observed in macroscopic (Fig. 5.6(c, d)) and microscopic
(Fig. 5.6(e, f)) fracture surfaces of uniaxial compression tests. Largely throughthickness intact lamellar walls can be observed in fracture samples of Freezecast15
(Fig. 5.6(c)) and Freezecast35 (Fig. 5.6(d)), indicating that the fracture occurred
along lamellar walls, along directions of maximum shear. Fracture surfaces of
Freezecast15 (Fig. 5.6(c)) and Freezecast35 (Fig. 5.6(d)) show clean cleavage
fracture between lamellar walls with few connections between one lamella and
another. Lamellae are able to slide over one another easily without breaking lamellar
planes. These findings imply that the pore structures of various porosities and their
resultant fracture behavior are similar despite having higher solids loading. Denser
freeze-cast pore structures do not automatically create noticeably more interlamellar
bridges than those with lower relative densities.
5.4

Conclusions and Implications

In summary, a bulk hierarchical ceramic structure was achieved via solution-based
freeze casting coupled with a conformal coating using self-assembled block copolymers. The hierarchical material demonstrates an increase in surface area and mechanical properties, while retaining permeability of the macroporous structure as
a result of the introduction of pore hierarchy. These changes in properties are expected to produce more robust filters, reactors, and electrodes with higher throughput
(permeability), higher reaction rate (surface area), and prolonged product life (compressive strength). Furthermore, this study opens new avenues to independently
optimize two pore length scales for creating hierarchical pores. In the freeze-casting
stage, pore alignment and permeability can be controlled by adopting a grain selector
[127] or by thermal gradient control [109].
For solution-based freeze casting using preceramic polymers, compositions are
restricted to silicon-based materials. However, provided good interfacial adhesion
exists, suspension freeze casting can potentially provide porous templates so that
a wide range of starting materials with bio-compatibility [102], ionic conductivity

88
[110], or shape-memory effect [196, 202] can be used. Conformal coatings with
self-assembled block copolymers are also tailorable. Pore size, connectivity and
surface area can all be adjusted by changing either the relative block size in the BCP
or homopolymer concentrations [83, 90].

89
Chapter 6

SUMMARY, CONCLUSIONS, AND FUTURE WORK
6.1

Summary and Conclusions

This work has successfully supplied strategies for creating optimized pore structures
that provide sufficient strength, permeability, and surface area under the framework
of freeze-casting based upon composite material principles of both particle reinforcement and hierarchical design.
To optimize the mechanical and transport properties, we first chose a highly permeable lamellar structure of SiOC ceramic fabricated from freeze casting with DMC
and polysiloxane. This high-permeability pore structure was reinforced by SiC
whiskers and CNTs. The two fillers provided two different reinforcement mechanisms of pore structures—structural reinforcement across pore walls discussed in
Chapter 3 and material reinforcement within the pore walls explored in Chapter 4.
Lamellar structures often fail in shear at low stresses. Therefore, in Chapter 3, we
utilized the phenomenon of particle engulfment to create SiC whisker interlamellar
bridges, which prevented shear failure and increased the compressive strength 3-fold
with 30 vol.% whiskers. Whiskers were present both across and inside pore walls,
providing reinforcements to the walls and the overall structure. The permeability
decreased slightly but maintained in the range of 10−12 m2 as the porosity of the
composites decreased.
In addition, we proposed a new model to remedy the deficiency of current engulfment
theories outlined in Section 3.1. The proposed model accounts for two new variables:
the intersecting angle between a freezing dendrite and a whisker, and the offset
between the center of a whisker and that of the nearest dendrite These was found to be
necessary due to the geometry of high-aspect ratio particles and non-planar freezing
fronts. We have implemented a simulation based on the proposed model and a 2D
freezing setup to observe solidification in situ. The engulfed whisker fraction from
the freeze-cast samples was in reasonable agreement with the simulated values. The
critical freezing front velocity from the 2D freezing setup agreed with the calculated
values from the model. The simulation also reproduced the whisker alignment and
the partial engulfment observed in the experiments and showed higher probabilities
of engulfment with higher freezing front velocities and longer whiskers, consistent

90
with previous engulfment models.
We also explored another route as an alternative reinforcing strategy in Chapter 4 for
situations where it might be crucial to maintain the original pore structure. CNTs
were used as reinforcement fillers and were pushed aside by the freezing front to
pore walls due to their small size. The compressive strength increased by 118%
with 4.3 wt.% CNTs, likely a result from the 74% increase in pore wall toughness.
The pore structure remained similar for low CNT concentration composites (1.3 and
4.3 wt.%). However, CNT agglomerates formed and were engulfed by the freezing
front creating interwall bridges as the concentration increased to 8.2 wt.%. The
shear failure was prevented by the interwall bridges as in the case of SiC whisker
discussed in Chapter 3. Additionally, CNTs were able to change the conductivity by
10 orders of magnitude with the addition of 8.2 wt.% of the reinforcement.
As demonstrated, pore structure can be tuned by the methods outlined in Chapters 3
and 4 to achieve desired mechanical and transport properties. In contrast, Chapter 5
outlined the method to increase the surface area of the pore structure by depositing
a high-surface-area conformal coating via self-assembly of BCP. The hierarchical
structure displayed bimodal pore size distribution with pores around 30 nm from
BCP self-assembly and 20 𝜇m from freeze casting. The surface area of the coating
is 70 times higher than that of the scaffold. The introduction of pore hierarchy also
resulted in an increase in mechanical properties, while retaining permeability of the
macroporous structure.
All the pore structures discussed in this thesis are lamellar structures made from
freeze casting with DMC as the solvent, and as such, can be readily compared. It is
useful in summary to examine how different composite strategies affect the fracture
behavior and the mechanical and transport properties of the various porous solids.
The compressive strength versus relative density of freeze-cast pure SiOC from all
three projects—the SiC-reinforced, the CNT-reinforced, and the BCP-hierarchical
porous solid—are shown in Fig. 6.1. Pure SiOC from two of the projects, the
SiC-reinforced and the BCP-hierarchical projects, are very similar. In fact, a linear
regression with R2 = 0.9311, plotted with dotted line, can be determined from
these two sets of data. Gibson and Ashby [65] have shown that the compressive
strength and relative density follows a power-law relation for brittle porous materials
as briefly discussed in Section 3.3.1.2:

91

Figure 6.1: Compressive strength as a function of relative density of freeze-cast
pure SiOC from the CNT-reinforced, the SiC-reinforced, and the BCP-hierarchical
porous solid projects.

= 𝐶 ( )𝑛
𝛼0
𝜌0

(6.1)

They found n to be 1, 23 , and 2 for honeycombs, open-cell, and closed-cell foams,
respectively. Seuba et al. [159] observed a contradictory result with n = 3 for freezecast ceramics that fail in bulking. The scaling factor, n, calculated from these two sets
of data in this work is 2.05, similar to that of freeze-cast dendritic structures reported
by Naviroj [125] despite having different morphologies. The possible reason for
the variation in the scaling factor, n, is that while freeze-cast cellular structures are
honeycombs, dendritic or lamellar structures possess a complex geometry that is
neither open-cell, close-cell, nor honeycomb-like.
Fig. 6.1 also shows that pure SiOC samples from the CNT-reinforced study have
lower strength than those from the other two projects. There are two contributing
factors to this result. First, porous SiOC samples from the CNT-reinforced project
were frozen with a -3°/min cooling rate while those from the SiC-reinforced and the

92

Figure 6.2: Compressive strength vs. permeability constant of CNT-reinforced,
SiC whisker-reinforced, and hierarchical composites, and corresponding fracture
surfaces and macro-images of fractured samples. The inset includes the stressstrain curves of freeze-cast pure SiOC of various densities from BCP-hierarchical
project for reference.

BCP-hierarchical projects were quenched at -35℃ and -30℃, respectively. This is
expected to result in higher freezing front velocities, finer freeze-cast structures and
ultimately higher mechanical strengths [159]. Second, porous SiOC samples from
the CNT-reinforced project are taller. For brittle materials, a size effect exists such
that samples with larger volumes are more likely to contain a larger flow, making
such samples fail at lower stresses [16, 149, 164].
The stress-strain curves, SEM images of the fracture surfaces, and macro images
of fractured samples of CNT-reinforced, SiC whisker-reinforced, and hierarchical

93
composites are shown in Fig. 6.2. The inset includes the stress-strain curves
of freeze-cast pure SiOC of various densities from BCP-hierarchical project for
reference. The composite with 30 vol.% SiC whisker has the highest strength
followed by Hierarchical20 and the 8.2 wt.% CNT composite, in the same order
as their relative density. Two different stress-strain responses are discernible in
the figure. Hierarchical composites exhibited the stress-strain behavior of freezecast pure SiOC lamellar structures with a linear increase to a peak followed by a
lower-stress plateau as previously shown in the inset and Sections 3.3.1.2, 4.3.4, and
5.3.
The shapes of these curves imply that the hierarchical sample failed by shear fracture
where it underwent catastrophic failure, as indicated by a large drop in stress.
The fracture surface and macroscopic view of the fractured sample of hierarchical
composites display large intact lamellae, from which other lamellae slip off after the
breaking of few interlamellar connections. This indicates that the introduction of
hierarchy to the structure did not change the connectivity of lamellae despite having
a higher relative density than those of CNT composites.
In contrast, the stress-strain curves of CNT and SiC composites had no large drops
in stress that indicate failure. Instead, a series of micro-fracture events beginning
at about 3% strain indicate that the shear failure (manifested as a large drop in
stress in a stress-strain curve) was likely prevented by the interlamellar bridges
formed by the engulfment of CNT agglomerates or SiC whiskers. The fracture
surfaces of CNT and SiC composites both exhibit a broken surface of lamellae
with intact interlamellar connections as the fracture propagates through lamellae.
These observations demonstrate that the fracture behavior is heavily influenced by
the density of interlamllar connections rather than the density of the porous ceramic
and that the fracture behavior is independent of relative density or strength.
Finally, Fig. 6.3 compares the compressive strength versus permeability constant of
CNT-reinforced, SiC whisker-reinforced, and hierarchical composites with freezecast pure SiOC all with lamellar microstructures. Also is included is data for
pure SiOC, both lamellar and dendritic structures, reported by Naviroj [125]. The
figure shows that CNTs and SiC whiskers have successfully reinforced the lamellar
structure at little expense of permeability. Interestingly, the hierarchical structure
is also in the same range as reinforced composites despite not being reinforced by
fillers. As discussed in 5.3, the compressive strength of the hierarchical structure is
similar to that of pure SiOC of the same relative density. However, the permeability

94

Figure 6.3: Compressive strength vs. permeability constant of CNT-reinforced, SiC
whisker-reinforced, and hierarchical composites compared with that of pure SiOC
reported by Naviroj [125].

is higher than that of pure SiOC of the same porosity due to the presence of enlarged
lamellar spacings. Although unintentional, this demonstrates another strategy to
create robust permeable pore structure—instead of reinforcing a highly permeable
structure with fillers, increasing the permeability of the structure with lower porosity.
These composites have widened the range of mechanical and transport properties and
pushed the inversely-related strength and permeability line to the top right corner,
where a robust and highly permeable pore structure for filtration and transport lies.
Strategies based upon the design of composites allow us go beyond the current limitation of materials. In this thesis, we have shown that with the provided strategies,
robust, highly-permeable and high surface area filtration materials are achievable.
The strategies are not limited to the above properties. Other combinations of desirable properties are also possible. For example, robust, bio-compatible CNT-chitosan
composites with high surface area are fabricated for bilirubin adsorption [131]. Flexible, thermally stable PVdF-HFP-MgAl2 O4 with good wettability by electrolytes are

95
used as battery separator [140]. A lead zirconate titanate-polydimethylsiloxane pressure and shear sensor is self-powered, flexible, and highly active [190] to list a few.
Multi-functionality devices can be fabricated with the strategies proposed in this
work.
On the scientific side, due to the interplay between each constituent component and
processing, the structure, and the resultant properties, the composites are inherently
complex. This complexity poses many interesting questions. Particle engulfment
discussed in the thesis is an example, where the PCP in suspension affects the shape
and the spacing of freezing dendrites, affecting the probability of engulfment for
whiskers. The engulfment probability of the whiskers, in turn, affects the mechanical
and transport properties of the PDC derived from the said PCP. In this work, we
have tackled a few of these questions. However, many are left for the future studies
in this burgeoning field of composites via freeze casting.
6.2

Suggestions for Future Work

A stable, well-dispersed CNT suspension is a requirement for CNTs to act as reinforcement fillers that only alter the pore wall chemistry without changing pore
structures. As shown in Chapter 4, with higher CNT concentrations, CNT agglomerates form and change the freezing dynamics and subsequently the pore structure.
CNT walls can be modified chemically to improve dispersion in a solvent and obtain
a stable dispersion in spite of high CNT concentrations. This modification of CNT
walls may also improve the adhesion between the matrix and CNTs, further improving its mechanical properties [167]. We have not seen engulfment of any individual
CNT. However, Gutiérrez has shown a pore structure with interwall bridges possibly
formed via the engulfment of individual CNTs [70].
There are a variety of future directions for particle engulfment theory and experiments. First, the model discussed in Chapter 3 is a simplified, bare-bones model
that includes only two major forces, the drag force from the fluid motion and the
repulsive force from the surface energy. As discussed in Section 2.1.4, other forces
(buoyancy, gravity, osmotic forces, and externally applied forces), factors, such as
particle roughness, the Gibbs-Thomson effect, and particle wettability, and particleparticle and multiple-dendrite interactions can be included. With the addition of
these parameters, it is likely that an analytical solution, such as that shown in Section
3.3.2.1 would not be possible. Numerical solution of Navier-Stokes equation will
have to be attempted instead. The engulfment model could be extended to work with

96
other high-aspect ratio particles and verified with other freezing casting systems.
On the experimental side, the 2D in-situ setup can be adapted to include control
of the thermal gradient. The possible minimal distance between a particle and
the freezing front, i.e. the gap of the liquid film, has been under debate and is
often used as a fitting parameter [6]. This is unlikely to be resolved by adopting a
confocal microscope since the gap is on the order of nanometer. However, a confocal
microscope can be used to construct 3d images of the dendrite tip shape and the
relative location of the center of a dendrite tip and a whisker [43]. Combined with
numerical solutions, a confocal microscope may allow us to model the solidification
process more accurately.
Chapter 5 shows the introduction of enlarged spacing in the hierarchical structures
possibly due to drying pressure of BCP gel. This phenomenon remains the major
challenge with thermoreversible gelcasting [162] and other processing methods that
utilize the self-assembly of BCP gels. For small-size samples that dry quickly,
freeze-drying is able to mitigate the deformation since the solvent stays in the solid
state during drying. However, commonly-used solvents for BCP gels, such as 2EH
and isopropanol, have low freezing points, which are difficult to maintain over a
long period of time required for the drying of larger samples. Alternatively, liquid
desiccants have been used to minimize warping by transferring the solvent from
the gel body to the desiccant bath [13]. Another logical next step is to construct a
device made from hierarchical structures via BCP self-assembly and freeze casting
and measure the reaction rate to ascertain if the higher surface area is reflected in the
reaction rate or whether the enlarged lamellar spacings diminish the reaction rate.
Last but not least, while compressive strength is often used to characterize the
mechanical properties of porous ceramics, burst test for pressure vessels may be
more suitable measurement for devices intended for filtration and transport. Maximum pressure before failure of filtration and transport devices should be tested and
compared with their compressive strength.

97

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Appendix A

STRESS-STRAIN CURVES OF SIC WHISKER COMPOSITES
AND PURE SIOC

Figure A.1: Stress-strain curves of freeze-cast pure SiOC ceramics (black lines) and
30 vol.% SiC composites (blue lines)

As described in Section 3.6, lamellar structures of pure SiOC fail in shear fracture
with a large drop in stress, followed by a gradual increase in stress. Most of the
pure SiOC curve shows this behavior with only one exception. We hypothesize the
natural variation in fracture strengths of ceramics and in pore alignment from freeze
casting could cause this anomaly. All stress-strain curves of 30 vol.% SiC whisker
composites have a series of micro-fracture events leading to a maximum in stress,
followed by a softening. The differences in the uniaxial compression of the 30 vol.%
SiC whisker composites and pure SiOC are shown to be repeatable and consistent.

121
Appendix B

SURFACE ENERGY DIFFERENCE AS THE ORIGIN OF THE
REPULSIVE FORCE
After engulfment, a new solid/whisker interface, designated WS, is created and the
interface of whisker/liquid (WL) is eliminated (Fig. B.1(a)). These interfaces are
described by 𝛾𝑊 𝐿 , the surface energy between liquid DMC and the whisker, and by
𝛾𝑊 𝑆 , the surface energy between solidified DMC and the whisker. The repulsive
force arises from the difference in energy between these two interfaces. A schematic
of a whisker engulfed by a dendrite is shown in Fig. B.1(b). The repulsive force 𝐹𝑟
is calculated from the free energy difference before and after the engulfment, Δ𝐺
[168].
2𝑑
(B.1)
Δ𝐺 = 𝐴Δ𝛾 = 2 × 𝑑 × Δ𝛾 = 𝐹𝑟 × (𝑑 + 2𝑎 0 )
Then it follows that
2
4𝑑
𝑎0
(𝛾𝑊 𝑆 − 𝛾𝑊 𝐿 )
(B.2)
𝐹𝑟 ≈
𝑎 (𝑎 0 + ℎ)
where 𝐴 is the contact surface area between the whisker and the dendrite, 𝑎 0 is the
atomic distance, and Δ𝛾 is the surface energy difference between the WS and WL
interfaces. The surface energy between solidified DMC and the whisker, 𝛾𝑊 𝑆 was
calculated following [168]
𝛾𝑊 𝑆 = 𝛾𝑆𝑉 + 𝛾𝑊𝑉 − 𝑊𝑎𝑑ℎ𝑒𝑠𝑖𝑜𝑛
where 𝛾𝑆𝑉 and 𝛾𝑊𝑉 are the surface energies of solidified DMC and whiskers,
respectively, and 𝑊𝑎𝑑ℎ𝑒𝑠𝑖𝑜𝑛 is the work of adhesion between DMC and whiskers.
The surface energy of solid DMC, 𝛾𝑆𝑉 was calculated from the heat of sublimation,
Δ𝐻𝑠𝑢𝑏𝑙𝑖𝑚𝑎𝑡𝑖𝑜𝑛 [122, 130] from the following:
𝛾𝑆𝑉 × 𝐴𝑠𝑜𝑙𝑣𝑒𝑛𝑡 = Δ𝐻𝑠𝑢𝑏𝑙𝑖𝑚𝑎𝑡𝑖𝑜𝑛
where 𝐴𝑠𝑜𝑙𝑣𝑒𝑛𝑡 is the area occupied by one mole of surface atoms of the solvent, and
Δ𝐻𝑠𝑢𝑏𝑙𝑖𝑚𝑎𝑡𝑖𝑜𝑛 is the sum of the heat of fusion (Δ𝐻 𝑓 𝑢𝑠𝑖𝑜𝑛 , 11.58 kJ/mol [50]) and heat
of evaporation, Δ𝐻𝑣𝑎 𝑝𝑜𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 , 36.4 kJ/mol [145]) following [122, 130]. Based upon
these values, 𝛾𝑆𝑉 was found to be 3.9 ×10−2 J/m2 From the droplet experiment, 𝛾𝑊 𝐿
and 𝑊𝑎𝑑ℎ𝑒𝑠𝑖𝑜𝑛 were determined to be 23.1×10−2 and 5.7×10−2 J/m2 , respectively.

122
Finally, the surface energy difference, 𝛾𝑊 𝑆 −𝛾𝑊 𝐿 , is 10.7 mJ/m2 , within a reasonable
range [138].

Figure B.1: A schematic of (a) the surface energy difference, and (b) the surface
area of the dendrite in contact with the whisker during the engulfment process.

123
Appendix C

CALCULATION OF TORQUES
C.1

Torque from Fluid Motion

The pressure difference in the gap of the left side of the whisker, 𝜕𝑃
𝜕𝑥 , is calculated
from the Navier-Stokes equations:
𝜕𝑃 12𝜂𝐿𝑉 𝑓 𝑓
𝜕𝑥
ℎ3
The pressure difference, 𝜕𝑃
𝜕𝑥 , is then integrated to produce the pressure between the
left end of the whisker and any point to the left of the middle of the whisker, Δ𝑃(𝑥):
∫ −𝐿 sin 𝜃 +𝑏𝐿
12𝜂𝐿𝑉 𝑓 𝑓
−𝐿 sin 𝜃
𝑑𝑠
+ 𝑏𝐿 − 𝑃(𝑥) =
Δ𝑃(𝑥) =
ℎ3

The torque from the left side of the whisker, 𝜏𝐿𝑒 𝑓 𝑡 , is calculated by multiplying the
force with the distance between a point to the middle of the whisker:
∫ −𝐿 sin 𝜃 +𝑏𝐿

∫ 0

𝜏𝐿𝑒 𝑓 𝑡 = 𝑑

−𝐿 sin 𝜃
+𝑏𝐿

12𝜂𝐿𝑉 𝑓 𝑓
𝑑𝑠𝑑𝑥
ℎ3

The same calculation is repeated for the right side of the whisker.
𝜕𝑃 −12𝜂𝐿𝑉 𝑓 𝑓
𝜕𝑥
ℎ3
∫ 𝐿 sin 𝜃 +𝑏𝐿
−12𝜂𝐿𝑉 𝑓 𝑓
𝐿 sin 𝜃
Δ𝑃(𝑥) =
+ 𝑏𝐿 − 𝑃(𝑥) =
𝑑𝑠
ℎ3

∫ 𝐿 sin 𝜃 +𝑏𝐿 ∫ 𝐿 sin 𝜃 +𝑏𝐿
−12𝜂𝐿𝑉 𝑓 𝑓
𝜏𝑅𝑖𝑔ℎ𝑡 = 𝑑
𝑑𝑠𝑑𝑥
ℎ3

124
The total torque, 𝜏𝑇 𝑜𝑡𝑎𝑙 , is the summation of the left and the right torque:
𝜏𝑇 𝑜𝑡𝑎𝑙 =

6𝑑𝜂𝐿𝑉 𝑓 𝑓 sin 𝜃
(𝑎 − cot 𝜃) 2
ln ℎ − ln ℎ − 𝑎𝑏𝐿 + 0.5𝐿(𝑎 sin 𝜃 − cos 𝜃 + 2𝑏 cot 𝜃)
cos 𝜃 − 𝑎 sin 𝜃
𝐿(sin 𝜃 − 2𝑏) [−8ℎ sin 𝜃 − 6𝐿 (sin 𝜃 − 2𝑏)(𝑎 sin 𝜃 − cos 𝜃)]
4[2ℎ sin 𝜃 + 𝐿(sin 𝜃 − 2𝑏)(𝑎 sin 𝜃 − cos 𝜃)] 2

6𝑑𝜂𝐿𝑉 𝑓 𝑓 sin 𝜃
(𝑎 + cot 𝜃) 2
1.5 + ln ℎ − ln ℎ + 𝑎𝑏𝐿 + 0.5𝐿 (𝑎 sin 𝜃 + cos 𝜃 + 2𝑏 cot 𝜃)
cos 𝜃 + 𝑎 sin 𝜃

2ℎ sin 𝜃 [ℎ sin 𝜃 + 𝐿(sin 𝜃 + 2𝑏)]
[2ℎ sin 𝜃 + 𝐿 (sin 𝜃 + 2𝑏)(𝑎 sin 𝜃 + cos 𝜃)] 2

2ℎ
𝐿 cos 𝜃 + 2(ℎ + 𝑎𝑏𝐿 + 𝑏𝐿 cot 𝜃 + 𝑎𝐿 sin 𝜃)(𝑎 sin 𝜃 + cos 𝜃)] 2 )
C.2

Torque from Frictional Force

The Reynolds number, 𝑅𝑒, of SiC whiskers in the freeze-cast and solidification is
estimated to be
0.1. For small Reynolds numbers (<0.1), the drag coefficient of a
cylinder, 𝐶𝑑 , approaches 60 [41]. The drag force arising from the frictional force is
calculated from the following:
𝐹𝑑 =

1 2
𝜌𝑣 𝐶𝑑 𝐴𝑤ℎ𝑖𝑠𝑘𝑒𝑟

where 𝑣 is the velocity at which whiskers rotate. 𝐴𝑤ℎ𝑖𝑠𝑘𝑒𝑟 is the cross-section area
of the whisker facing the fluid. The torque from the frictional force, 𝜏𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛 , is
calculated below and always works against the whisker motion.
𝜏𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛 = 𝐹𝑑 ×

×2

125
Appendix D

EXAMPLES OF ROTATION FROM THE FREEZING VIDEO

Figure D.1: Predicted rotations based on the model and examples of the rotation in
the solidification video. Each image sequence is arranged chronologically with (a)
original images and (b) images with green lines marking the whisker configuration.

Shown in Fig. D.1 are the predicted rotations based on the model and examples of
the rotation in the solidification video. The white fingers on the right are freezing
dendrites and they advanced to the left as time progressed. Each image sequence
is arranged chronologically. Fig. D.1(a) shows the images directly taken from the
video. In Fig. D.1(b), we marked the whiskers with green lines to highlight the
rotation of the whisker. The whisker shown in the top sequence had 𝜃 > 90° and
it rotated counterclockwise as the model predicted. The one shown in the bottom
sequence had 𝑏 < 0 and rotated clockwise, also in agreement with the model.

126
Appendix E

LAMELLAR SPACING AND FREEZING FRONT VELOCITY
During the freezing step of freeze casting, images of the freezing process were captured every 20s by a camera with an intervalometer. Image analysis was performed
using ImageJ [155] to determine the freezing front velocity. The samples were
quenched at -35℃. Fig. E.1(a) shows high starting freezing front velocities which
rapidly level off to a steady-state value from the bottom to the top of the sample. The
freezing front velocities were similar, regardless of the SiC whisker solids loading.
The freezing front velocities at 3 mm were used in the simulation to calculate the
engulfed whisker fraction.
To measure the lamellar spacing, the pyrolyzed samples were sectioned through
the radial center parallel to the freezing direction for longitudinal views of the
pore structures. Three SEM images were taken across the height of each sample.
Image analysis was performed using ImageJ to determine the lamellar spacing.
The lamellar spacings increased with the sample height, as expected from a lower
freezing front velocity (Fig. E.1(b)).
The relations between lamellar spacings and freezing front velocities were established through the sample height for 10, 20, 30 vol.% SiC whiskers in Fig. E.1(c).
Since we could not directly measure lamellar spacing as a function of freezing front
velocity, the relations were calculated from a best-fit power-law function of lamellar spacing and sample height, and a best-fit power-law function of freezing front
velocity and sample height.
The measured lamellar spacing and freezing front velocity from the 2D solidification setup agrees well with those from the freeze-cast samples, indicating the 2D
solidification setup is able to produce similar freezing conditions (freezing front
velocity and dendritic/lamellar spacing) to those of the freeze-cast samples.

127

Figure E.1: (a) Freezing front velocity and (b) lamellar spacing as functions of
sample height. (c) Lamellar spacing as a function of freezing front velocity