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STATISTICAL MECHANICS
OF NANOPARTICLE SUSPENSIONS AND GRANULAR MATERIALS
A dissertation submitted
to Kent State University in partial
fulfillment of the requirements for the
degree of Doctor of Philosophy
by
Lena M. Lopatina
August, 2011
Dissertation written by
Lena M. Lopatina
Bachelor of Science, Taras Shevchenko National University of Kyiv, 2004
Master of Science, Taras Shevchenko National University of Kyiv, 2005
Ph.D., Kent State University, 2011
Approved by
Dr. Jonathan Selinger
, Chair, Doctoral Dissertation Committee
Dr. Robin Selinger
, Members, Doctoral Dissertation Committee
Dr. Cynthia Olson Reichhardt
Dr. John West
Dr. David Allender
Dr. Alexander Seed
Accepted by
Dr. Liang–Chy Chien
, Director, Chem. Phys. Interdisciplinary Prog.
Dr. Timothy Moerland
, Dean, College of Arts and Sciences
ii
TABLE OF CONTENTS
LIST OF FIGURES AND TABLES . . . . . . . . . . . . . . . . . . . . . . . .
v
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
CHAPTER 1. Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
CHAPTER 2. Thermal conductivity and particle agglomeration in nanofluids .
10
1.1
2.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2
Theoretical model and discussion . . . . . . . . . . . . . . . . . . . .
14
2.3
Shape effects: Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.4
Shape effects: Dendrites . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.5
Effect of surface thermal resistance . . . . . . . . . . . . . . . . . . .
25
2.6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.8
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
CHAPTER 3. Theory of Ferroelectric Nanoparticles in Nematic Liquid Crystals. Landau-like Approach . . . . . . . . . . . . . . . . . . . . . .
34
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.2
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.3
Effect of Ionic Impurities . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.4
Kerr Effect
43
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
3.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.6
46
CHAPTER 4. Theory of Ferroelectric Nanoparticles in Nematic Liquid Crystals. Maier-Saupe-like Approach . . . . . . . . . . . . . . . . . . .
47
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.2
Overview of Maier-Saupe Theory . . . . . . . . . . . . . . . . . . . .
49
4.3
Transition Temperature . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.4
Kerr effect
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
4.6
65
CHAPTER 5. Jamming in Granular Polymers . . . . . . . . . . . . . . . . . .
66
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5.2
Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.3
Classical Bidisperse System . . . . . . . . . . . . . . . . . . . . . . .
70
5.4
Granular Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5.4.1
Effect of length and shape . . . . . . . . . . . . . . . . . . . .
73
5.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.6
78
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
iv
LIST OF FIGURES AND TABLES
Figure 2.1. Illustration of the temperature distribution problem for a sphere in
a fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Figure 2.2. Schematic representation of nanoparticle agglomerates . . . . . .
18
Figure 2.3. Illustration of the temperature distribution problem for an ellipse
in a fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Figure 2.4. Nanoparticle shape effect on the thermal conductivity of nanofluids 21
Figure 2.5. Illustration of the lattice geometry for theoretical study of the
dendritic agglomeration effect. . . . . . . . . . . . . . . . . . . . . . . . .
22
Figure 2.6. Random distribution of the dendritic particles with 1% concentration 23
Figure 2.7. Calculated profile of the heat flux in the system . . . . . . . . . .
24
Figure 2.8. Calculated thermal conductivity enhancement vs concentration of
dendritic nanoparticles compared to circular nanoparticles within effective
medium theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Figure 2.9. Thermal conductivity of suspensions prepared from 11-, 20-, and
40-nm nominal size alumina nanoparticles in (a) water and (b) ethylene
glycol as a base fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
Figure 3.1. Nanoparticles surrounded by liquid crystal . . . . . . . . . . . . .
36
Figure 3.2. Isotropic-nematic transition temperature as a function of ion concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Figure 3.3. Field-induced order parameter for several values of applied electric
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
Figure 4.1. Schematic illustration of ferroelectric nanoparticles suspended in a
liquid crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Figure 4.2. Four regimes of the Kerr effect . . . . . . . . . . . . . . . . . . . .
63
Figure 5.1. Granular configurations . . . . . . . . . . . . . . . . . . . . . . .
68
Figure 5.2. Pressure vs. density curves for bidisperse system and granular
polymers systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Figure 5.3. Scaling of pressure and contact number parameters vs. density . .
71
Figure 5.4. Contact number parameter and bulk pressure tensors vs. density
72
v
Figure 5.5. Plot of shear velocity of grains adjacent to the stationary wall and
the corresponding pressure in the packing vs. density . . . . . . . . . . .
73
Figure 5.6. The jamming threshold versus chain length . . . . . . . . . . . . .
74
Figure 5.7. P vs φ for a chain
. . . . . . . . . . . . . . . . . . . . . . . . . .
76
Table 2.1. Comparison of experimental heat transfer enhancements in alumina
nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Table 2.2. Surface thermal conductance in alumina nanofluids . . . . . . . . .
32
33
vi
ACKNOWLEDGMENTS
Most of all I would like to thank Dr. Jonathan Selinger for the years of leading me
towards my degree. Over this time he has shared have learned tremendous amount
of knowledge with me not only in scientific area but in the all aspects of life. Also,
I would like to thank Dr. Robin Selinger for all the academic and personal guidance
and advice over those years.
I would like to thank Dr. Cynthia Olson Reichhardt and Dr. Charles Reichhardt
for an amazing opportunity to work under their supervision in Los Alamos National
Laboratory. This was an huge step in my academic development, but also an unique
experience of every day life in the governmental laboratory compared to the life in
the university.
Also, I would like to thank Dr. David Allender, Dr. John West, Dr. Alexander Seed for taking their time read my dissertation, offer their advice, and serve as
committee members for my dissertation defence.
I would like to extend special thank to Dr. Yuri Reznikov and Dr. Mitya Reznikov
who have recruited me to join Chemical Physics Interdisciplinary Program, and have
been excellent friends and great to work with collaborators over the years.
I would like to acknowledge all the Liquid Crystal Institute, Chemical Physics
Interdisciplinary Program students, researchers and faculty for the useful discussions,
support, and great time of working together. I would like to acknowledge all the
researchers from other science departments of Kent State University for sharing their
knowledge and useful discussions and collaborations.
vii
Obtaining this degree would have never been possible without Liquid Crystal
Institute administrative and technical staff: Lynn Fagan, Dawn Miller, Janet Cash,
Mary Ann Kopcak, Betty Hilgert, Brenda Decker, and James Francl.
I would like to acknowledge Kent State University for supporting Chemical Physics
Interdisciplinary Program program, and also I would like to extend my thank to
the Institute for Complex Adaptive Matter (ICAM) for their efforts and financial
support in developing young generation of scientist. Also, I would like to thank
Zonta International for awarding me Amelia Earhart fellowship.
I thank to all my family and friends for ongoing support over the years.
I would like to finish by acknowledging the financial support which allowed me
to work on my projects. In particular I should mention the Office of Naval Research
Grant No. N000140610029, National Science Foundation Grant (DMR-0605889),
LANL Contract No. DE-AC52-06NA25396, Zonta International (Amelia Earhart
fellowship), and Kent State University (University Fellowship).
viii
CHAPTER 1
INTRODUCTION
1.1
Introduction
This work is motivated by both the desire to advance basic science and the need
to engineer advanced materials.
In this dissertation I present three projects united by the idea of using statistical
mechanics to study systems in which the main component is an ensemble of particles.
Each project is a study of a distribution of particles either in an interacting ensemble
by itself or in a host medium (such as a fluid or liquid crystal). I analyze connections
between the internal properties of individual particles and the resulting macroscopic
properties of the material with many particles.
Shape is one of the most important properties of the particles. Spherical particles
are one of the most common and most studied objects — presumably because the
first methods of manufacturing colloids led to spherical objects. However, even if
all particles were manufactured as perfect spheres, interparticle interactions would
still cause them to stick together and form asymmetrical shapes. In other cases,
researchers combine spherical particles together by adjusting their interaction with
each other to form aggregates with predefined shapes of interest. Modern techniques
of chemical engineering provide many ways to produce nonspherical colloids, such as
ellipsoidal or banana-shaped particles. Consequently, there is a need in soft matter
physics for a theoretical description of the behavior of nonspherical objects. In this
dissertation I model the effects of the shape of the objects on material properties.
1
2
Apart from the shapes of the particles, another way to produce asymmetrical
properties is by incorporation of internal characteristics that break the symmetry, for
example, permanent dipole moment. In this dissertation I study the effects of the
internal asymmetrical properties of the particles on the global material properties.
The first project is a study of thermal conductivity in suspensions of nanoparticles in base fluids, called nanofluids. In such systems, the shapes of the nanoparticle
agglomerates and the distribution of those shapes with respect to heat flow is a key
factor controlling the thermal conductivity of the nanofluid. In the second project,
the system of interest is a liquid crystal with a suspension of ferroelectric particles.
Although the particles of interest are considered to be spherical, their permanent
dipole moments play the role of the symmetry breaker in this case. By analogy with
the nanofluid system where the shape of the nanoparticles is a symmetry breaker, in
the liquid crystal the distribution of the permanent dipole moments of the nanoparticles determines the properties of the resulting composite material. The third project
is a study of the jamming transition in granular materials consisting of nonspherical
objects. In this project I study granular materials made of spherical particles connected together forming chains. I investigate whether the jamming of chains is similar
to or different from the jamming of spherical objects.
Nanoparticle suspensions are of interest in the first two projects in this dissertation. One of the reasons soft matter scientists are so interested in nanoparticle inclusions is their size. At the length scale of a few nanometers, nanoparticles
are essentially intermediate between colloidal inclusions (at the micron scale) and
molecular-size additives at the nanometer scale. This feature provides new opportunities for designing new materials with enhanced properties, not by synthesizing them
3
in the chemical laboratory but by adding molecular-size additives into existing materials. Such an approach is especially beneficial for industrial applications, because
this scenario would not require dramatic changes in the production lines. Nanoparticles also come in a great variety of shapes, ranging from simple spheres or elongated
objects with highly variable aspect ratio to complicated geometrical shapes such as
spirals, stars, or boomerangs. The possibility of producing molecular-size additives in
a variety of shapes and sizes opens new horizons for engineering new materials with
enhanced properties.
The small size of nanoparticles provides both technological advantages and challenges. A major advantage of nanoparticles, compared with micron-scale colloids, is
that they do not disturb the alignment of liquid crystals. This feature makes nanoparticles ideal as dopants for liquid crystals, because preserving liquid crystalline order is
essential. On the other hand, the nanometer scale of the particles is a serious challenge
for particle characterization and processing. It is especially difficult to characterize
nanoparticles by themselves, or to characterize their distribution in the host material, since they cannot be seen under a standard optical microscope and one must
find other characterization methods.
1. Thermal Conductivity and Particle Agglomeration in Nanofluids.
One of the first projects in my graduate research was studying thermal conductivity and particle agglomeration in nanofluids. By the year 2005 many experimental groups had reported an anomalously enhanced thermal conductivity in
liquid suspensions of nanoparticles; also there were a few theoretical reports investigating that phenomenon. Since conventional effective medium theory was
not able to explain such anomalous enhancement in thermal conductivity of
4
nanofluids, there was a need for a new theory to describe this phenomenon.
Despite the importance of this effect for heat transfer applications, at that time
there was no agreement about the underlying mechanism, or even about the
experimentally observed magnitude of the enhancement.
To address those issues, in 2005 we started a collaboration with groups in the
Chemistry Department and the Physics Department of Kent State University.
That collaboration allowed our team to perform in-depth combined experimental (performed by our collaborators) and theoretical study (performed by my
advisor and me) of heat conduction and particle agglomeration in nanofluids.
The theoretical part of this study is presented in detail in Chapter 2 of this
dissertation.
On the experimental side, nanofluids of alumina particles in water and ethylene glycol were characterized by a variety of methods. One of the main issues
was to build a device to measure the coefficient of thermal conductivity for
fluids of variable densities and optical properties, such as ethylene glycol and
water, that would eliminate as many extraneous effects as possible, such as
convection. Apart from thermal conductivity measurements, we used viscosity
measurements, dynamic light scattering, and other techniques to fully explore
the nanofluid system and its properties. The results showed that the particles
were agglomerated, with an agglomeration state that evolved in time, and unfortunately, that the thermal conductivity enhancement was within the range
predicted by effective medium theory.
In our theoretical study, we developed a model for heat conduction through a
fluid containing nanoparticles and agglomerates of various geometries. Along
5
with experimental findings, theoretical calculations showed that effective medium
theory for nonspherical agglomerates was sufficient to describe the experimental
results. We showed that elongated and dendritic structures are more efficient
in enhancing the thermal conductivity than compact spherical structures of the
same volume fraction, and that surface (Kapitza) resistance was the major factor resulting in the lower than effective medium conductivities measured in our
experiments.
Together, those experimental and theoretical results proved that the geometry, agglomeration state, and surface resistance of nanoparticles were the main
variables controlling thermal conductivity enhancement in nanofluids. We concluded that suspending nanoparticles in base fluid does not result in an anomalous thermal conductivity enhancement. In year 2008 most of the experimental
and theoretical researchers in the area of thermal conductivity of nanofluids
united in the conclusion that nanofluids did not have anomalously enhanced
thermal conductivity and were not a promising material for applications that
demanded high thermal conductivities.
2. Theory of Ferroelectric Nanoparticles in Nematic Liquid Crystals: Landau and
Maier-Saupe Type Approaches.
The second project is a study of the effects of ferroelectric nanoparticles on liquid
crystal materials. Similarly to the thermal conductivity project, we studied the
effects of nanoparticles on the macroscopic properties of the composite material.
However, in this case the nanoparticles are not metal (which was important
for thermal conduction), but are made of a ferroelectric material, whose main
6
characteristic is sensitivity to the electric field. The other major difference is
that nanoparticles are suspended in a nematic liquid crystal that has much more
complicated interactions with inclusions compared to a simple fluid.
Liquid crystals have high sensitivity to external conditions and influences such
as temperature, light, and electric and magnetic fields, which makes them perfect candidates for various applications. Liquid crystals are used in modern
thermometers, molecular and biological sensors, optical recording, signal transfer in telecommunications and computing, and in display technologies. In watch,
laptop, computer and TV screens, liquid crystal materials are the main component of the display matrix because of their ability to respond quickly to applied
electric fields. Due to numerous applications of liquid crystal materials there is
a great demand for improving their properties. Optimizing liquid crystal properties for improved and new applications is the leading goal in liquid crystal
research.
Experiments report that doping a liquid crystal with ferroelectric nanoparticles increases the nematic-isotropic transition temperature — one of the main
characteristics of nematic liquid crystals. Ferroelectric materials are highly sensitive to electric field, and consequently the main characteristic of nanoparticles
made of ferroelectric material is their intrinsic permanent dipole moments. One
would expect that inclusion of ferroelectric nanoparticles in the liquid crystal matrix would highly increase the sensitivity of the mixture to the applied
electric field, but it was unexpected to discover an increase in the nematic
to isotropic transition temperature. We want to understand why ferroelectric
nanoparticles exhibit this effect and how we can enhance it further. We develop
7
a Landau-like theory for the statistical mechanics of ferroelectric nanoparticles in liquid crystals to understand the enhancement of the nematic-isotropic
transition temperature in that system. Previous theoretical research assumed
the main mechanism of interaction in such a system to be the interaction between nanoparticle dipole moments and the induced polarization of liquid crystal molecules. As a new approach, we develop a model taking into account
more “direct” interaction between the liquid crystal molecules and the electric
field produced by permanent dipole moments of nanoparticles. The other major
difference is that we consider the statistical distribution of the orientations of
nanoparticle dipoles, whereas all previous models assumed uniform orientation
of nanoparticle dipoles. Our model predicts the enhancements of liquid-crystal
properties such as the nematic-isotropic temperature transition and sensitivity
to the applied electric field, and is in good agreement with experiments. These
predictions apply even when electrostatic interactions are partially screened by
moderate concentrations of ions.
The Landau-like model has one important limitation: Like all Landau theories,
it involves an expansion of the free energy in powers of the order parameters, and
hence it overestimates the order parameters that occur in the low-temperature
phase. For that reason, we expand the parameter range in which our prediction
will be valid by developing a Maier-Saupe-type model, which explicitly shows
the low-temperature saturation of the order parameters. This model reduces
to the Landau theory in the limit of high temperature or weak coupling, but
shows different behavior in the opposite limit. We compare these calculations
with experimental results on ferroelectric nanoparticles in liquid crystals.
8
3. Jamming in Granular Polymers.
In the summer of 2009 I worked with Cynthia Olson Reichhardt and Charles
Reichhardt as an intern at the Los Alamos National Laboratory (LANL), doing
theoretical research on jamming in granular materials.
Apart from its importance for fundamental properties of materials science, the
physics of granular materials has great economic impact. Granular and fluid
flows are part of daily processes in food and drug industries, cosmetics, textiles,
construction, oil, and plastics. NASA reports that these materials and processes account for 4.9% of the gross domestic product (GDP) and 31% of the
manufacturing output of the U.S. alone (∼ $850 Billion/year)1 . Unfortunately,
industrial facilities typically operate well below design efficiency, resulting in to
great overhead in expenses and frequent catastrophic failures.
Granular materials are different from ordinary solid materials. Unlike a regular
solid, where an applied force propagates in the direction of application, in granular materials an applied force can propagate in any direction depending on the
internal structure of the material, and force is redistributed in a way that may
result in points with localized high stress (pressure).
One of the fascinating topics of research on the granular materials is the jamming transition. Jamming is a development of resistance to shear. We come
across jamming transitions in granular materials on an everyday basis. For
example, jamming occurs when our vitamins get stuck in the bottle neck, or
when we have to keep shaking the box while pouring the cereal into the bowl
for breakfast, and it also occurs for those of us who have an “advantage” of
1
http://gltrs.grc.nasa.gov/reports/2003/CR-2003-212618.pdf
9
living in big cities - traffic jams.
One important question in research on jamming in granular materials is whether
the static and dynamic properties of systems approaching the jamming transition are universal. I have modeled the jamming transition of particles of different
shapes, such as banana-shaped particles, or flexible chains of spherical particles,
and compared those to the classical and well understood bidisperse system —
a mixture of perfectly spherical particles of two different sizes. Because there
was an experimental study on flexible chains of spherical particles (also called
granular polymers) I have concentrated on that particular shape.
Using computer simulations we have examined the jamming transition in a twodimensional granular polymer system. One of the characteristics of the jamming
transition is the density at which the system starts to resist shear - called the
jamming density, or “Point J”. We learned that the jamming density decreases
with increasing length of the granular chain due to the formation of loop structures. This conclusion was in excellent agreement with experiments. We learned
that a jamming density can be further reduced in mixtures of granular chains
and granular rings, also as observed in experiment. We showed that the nature of jamming in granular polymer systems has pronounced differences from
the jamming behavior observed for polydisperse two-dimensional disk systems
at Point J. This result indicates that there is more than one type of jamming
transition.
CHAPTER 2
Thermal conductivity and particle agglomeration in nanofluids
2.1
Introduction
Over the past 15 years, many experimental studies have reported anomalous en-
hancement in the thermal conductivity of nanoparticle suspensions in liquids (known
as nanofluids) compared to the same liquids without nanoparticles. Even though
there had been earlier reports of such an effect [1], research in this area acquired a
major thrust only after publications from a group at Argonne National Laboratory,
who studied water- and oil-based nanofluids containing copper oxide nanoparticles,
and found a striking 60% enhancement in thermal conductivity for only a 5% volume
fraction of nanoparticles [2]. Since then, there have been similar reports of anomalous
enhancement of the thermal conductivity of various nanofluids, using nanoparticles of
oxides as well as of metals and carbon (for reviews, see Refs. [3–10]). Such enhancement of heat transport offers important benefits for numerous applications which rely
on liquid coolants for carrying heat away from electronics or machinery.
Despite numerous studies stimulated by the fundamental and practical importance
of this subject, it has proven rather difficult to establish either the magnitude or the
mechanism of the thermal conductivity enhancement in nanofluids when following the
early reports. Indeed, one of the review articles has commented that “experimental
values on the thermal conductivity of nanofluids published in the literature show an
astonishing spectrum of results” [8]. Published results show enhancement in the thermal conductivity ranging from anomalously large values – i.e., much greater than the
10
11
prediction of the classical Maxwell-Garnett effective medium theory – to values that
are similar to or even less than the prediction of effective medium theory. Remarkably
these discrepancies occur even for the same base fluid and the same nominal size and
composition of the nanoparticles.
In addition to this range of experimental results, there is also a wide range of theoretical approaches for modeling thermal transport in nanofluids [11]. Some researchers
have used variations of effective medium theory, involving nonspherical shapes [12–16]
or a layer of ordered fluid around the nanoparticles [14, 17–21]. Other studies have
considered thermal transport by the motion of nanoparticles, or convective thermal
transport due to fluid flow entrained by nanoparticle motion [22–26]. The situation
in the field has been described as “investigations of the properties of nanofluids have
reached the awkward situation of having a greater number of competing theoretical
models than systematic experimental results” [27].
The need for continuing studies to characterize individual nanofluid systems in
greater depth, and to identify and correlate factors underlying their thermal conductivity, is therefore clear. During the period of 2005 to 2008 we had a collaboration
with two experimental groups in Kent State University: the group of Samuel Sprunt,
Department of Physics, and Yuriy V. Tolmachev, Department of Chemistry. We
worked together on a combined, in-depth experimental and theoretical study of the
thermal conductivity of alumina Al2 O3 nanofluids, one of the most commonly studied
nanofluids yet still a controversial system (see Table 2.1) [1, 2, 28–39]. In this chapter,
we present the theoretical part of this joint experimental and theoretical study, and
we compare our developed theory and the experimental results obtained by our collaborators, the Yuriy V. Tolmachev and Samuel Sprunt groups at Kent State University.
12
The results of this collaborative in-depth study were published in [40].
To summarize the experimental results, nanofluids of Al2 O3 are prepared in water
and ethylene glycol, and characterized with an unprecedentedly broad array of techniques, including thermal conductivity, viscosity and zeta-potential measurements,
dynamic light scattering, and powder x-ray diffraction. The main experimental results are as follows:
(i) Our collaborators’ experiments do not reproduce the anomalously high enhancements of thermal conductivity and the temperature dependence of the enhancement reported by other groups; our results are closer to the predictions of effective
medium theory. This discrepancy may be associated with the differences in the shape
and size of the agglomerates as well as with the differences in particle-liquid and
particle-particle heat transfer resistances on the surface and within agglomerates,
respectively.
(ii) In the alumina nanofluids there is a significant nanoparticle agglomeration, as
shown by the dynamic light scattering results as well as by viscosity measurements.
Moreover, the agglomeration state of the particles evolves as a function of time as the
nanofluid ages (a process that is distinct from agglomerate sedimentation, which is
carefully controlled in our experiments). Variations in the agglomeration state may
well explain the variations in reported thermal conductivity in previous studies, which
generally have not considered this phenomenon.
(iii) The crystal structure of the Al2 O3 nanoparticles, obtained from commercial
sources, varies significantly with nominal particle size, even from the same supplier,
which complicates comparison of measured nanofluid properties. Furthermore, our
collaborators find that the properties of nanofluids correlate better with the crystallite
13
size (obtained from x-ray diffraction) than with nominal particle size (obtained from
surface area measurements via gas sorption).
On the theoretical side, we estimate the rate of thermal transport through particle
motion, and compare it with thermal transport due to heat diffusion through a static
composite of particles and fluid. In the latter case, the particles may be spheres
or nonspherical agglomerates, and they may have a finite surface thermal (Kapitza)
resistance. The main theoretical results are as follows:
(i) Our calculations show that nanoparticle motion does not make a substantial
contribution to thermal transport, compared with the diffusion of heat through static
composites of nanoparticles and fluid.
(ii) However, the geometry of nanoparticles and particle agglomerates has a very
important effect. Classical effective medium theory only applies to systems of spherical particles, and must be modified if the particles are elongated, or if they form
agglomerates that are either elongated or dendritic (fractal). Earlier studies have
considered certain elongated or fractal shapes, and have found a greater enhancement than for spheres. Here, we consider other shapes and confirm that a modified
effective medium theory gives a greater enhancement in these cases. Furthermore,
we show that the modified theory can account for previously published reports of
anomalous enhancement in thermal conductivity.
(iii) Elongated and dendritic geometries can only explain thermal conductivity
enhancements that are greater than effective medium theory for spheres. To explain
thermal conductivity enhancements less than effective medium theory for spheres,
such as those observed in the experiments performed by our collaborators in Kent
14
State University, we must consider thermal resistance at the nanoparticle-fluid interface. We show explicitly that those experimental results are consistent with an
effective medium theory that includes interfacial thermal resistance.
Taken together, these experimental and theoretical results imply a unified picture
of thermal conduction through nanofluids. In this picture, heat is transported diffusively through the composite, slowly through fluid and rapidly through particles
and aggregates. For increasing the thermal conductivity of the composite one should
consider (i) formation of extended particles or aggregates, (ii) enhancement of the
orientational order of the particles or aggregates, and (iii) reduction of the surface
resistance at the particle-liquid and particle-particle interfaces.
2.2
Theoretical model and discussion
Experiments performed by our collaborators at Kent State University have shown
that the enhancement in thermal conductivity of water and ethylene glycol-based
Al2 O3 nanofluids, in which there is a substantial degree of particle agglomeration, is
below the level expected from classical effective medium theory [40]. This finding,
together with the disagreement between those results and results from a number of
previous reports on similar systems (which present anomalously high values of thermal
conductivity knf ), warrants a critical assessment of possible factors responsible for
thermal conductivity enhancement.
To develop a theory for thermal conduction in nanofluids, the first essential issue
is to identify the primary mechanism for heat transport. As noted in the Introduction 2.1, some studies have modeled heat transport using versions of effective medium
theory. In these models, nanoparticles are assumed to be stationary or slowly moving, with the heat diffusing through the “effective medium” composed of particles
15
and fluid. Because the thermal conductivity of solids is usually much greater than
that of liquids, the particle-liquid-particle pathways can lead to faster heat conduction through the medium below the percolation threshold. In addition, some studies
modeled heat transport based on the motion of nanoparticles. The particle motion
may also entrain the motion of the fluid, which will carry even more heat. This heat
transport may provide an alternative mechanism for the enhancement of thermal
conductivity of nanofluids [22, 23, 41–43].
Of course, both of these mechanisms may contribute to the thermal conductivity
enhancement in nanofluids; the question is their relative magnitudes. To estimate the
order of magnitude for the enhancement in effective medium theory, we can use the
classical prediction for highly conducting spherical particles, i.e.,
knf
= 1 + 3φ.
k0
(2.1)
With the thermal conductivity of water k0 = 0.6 W m−1 K−1 , and the typical nanoparticle volume fraction ρ = 0.05, this equation gives the enhancement knf −k0 = 0.09 W
m−1 K−1 . To estimate the order of magnitude for thermal transport through nanoparticle motion, we can calculate ∆kparticle = DCparticle c, where D is the diffusivity of
the particles, Cparticle is the heat capacity of each particle, and c is the number of
particles per unit volume of nanofluid. Equivalently, this estimate can be rewritten
as kparticle = DCV φ, where CV is the specific heat per volume of solid particle. The
diffusivity is given by the Stokes-Einstein relation as D = kB T /6πηR. Using the
thermal energy kB T = 4 × 10−21 J at room temperature, the viscosity of the water
η = 10−3 Pa s, and the radius R = 10 nm, we obtain D = 2 × 10−11 m2 / s. The
16
highest specific heat capacity of the aluminum oxide polymorphs is CV = 3 × 106 J
m−3 K−1 (for the α phase, also known as corundum). Combining these values gives
∆kparticle = 3 × 10−6 W m−1 K−1 for the same volume fraction 0.05. Of course, this
is just a rough estimate of the thermal conductivity enhancement associated with
particle motion, and it does not include the heat transported by entrained fluid motion. Still, one must note that this value is four orders of magnitude smaller than the
thermal conductivity enhancement expected from effective medium theory. Thus it
seems unlikely that particle motion contributes significantly to the thermal conductivity enhancement in nanofluids. Rather, the enhancement must be understood by
regarding the particles as effectively fixed (moving slower that heat diffusion), with
heat diffusing through and around them.
In order to model the diffusion of heat through the suspension, we must consider a
range of geometries. For that reason, we briefly review the classical Maxwell-Garnett
theory for spherical particles, and then discuss how the predictions are modified by
nonspherical geometries.
T0 (r ) k0
T1 (r )
k1
∇T
FIG. 2.1. Illustration of the temperature distribution problem for a sphere in a
fluid used for theoretical modeling of the nanoparticle shape effect on the thermal
conductivity of nanofluids within effective medium theory.
The classical Maxwell-Garnett theory considers a system that consists of the base
17
fluid with the thermal conductivity k0 and one spherical nanoparticle with the thermal conductivity k1 , as shown in Fig. 2.1. When a thermal gradient is imposed on
the system, the temperature distribution in the fluid and in the spherical particle is
described by the functions T0 (r) and T1 (r), respectively. In steady state, the temperature profiles obey Laplace’s equation,
∇2 T0 = 0, ∇2 T1 = 0
(2.2)
with the boundary conditions for the temperatures
T0 = T1 ,
(2.3)
and the normal derivatives at the interfaces between the two media
k0
∂T0
∂T0
= k1
.
∂n
∂n
(2.4)
The first of the boundary conditions Eq. (2.3) implies that there is no resistance to
heat transfer at the fluid-particle interface, and the second Eq. (2.4) implies that the
heat current is continuous across the interface. Solving these equations and averaging
the results over a random distribution of particles, we obtain the effective thermal
conductivity of the nanofluid correct to first order in φ [44].
3(k1 − k0 )
knf
=1+
φ,
k0
k1 + 2k0
(2.5)
where φ is the particle volume fraction. If the particles are much more conducting than
18
the base fluid, i.e., k1 À k0 , this result reduces to Eq. (2.1). The result corresponding
to Eq. (2.5) for a circular nanoparticle in a two-dimensional nanofluid is
knf
2(k1 − k0 )
=1+
φ ≈ 1 + 2φ (in two dimentions),
k0
k1 + k0
(2.6)
which does not apply to the three-dimensional (3D) experiments but is useful for
theoretical comparisons.
One should notice that the slope of 3 in Eq. (2.1) is specific for spherical particles.
However, the nanoparticles studied experimentally are not necessarily spherical. Furthermore, our experimental results show that the particles in nanofluids agglomerate
substantially. Even if the nanoparticles are spherical when initially prepared, the
agglomerates will generally not be spherical. Thus it is essential to determine how
nonspherical geometries change the prediction for the thermal conductivity enhancement.
FIG. 2.2. Schematic representation of nanoparticle agglomerates.
Figure 2.2 shows a schematic illustration of the geometry of nanoparticle agglomerates. From this picture we can see that the agglomerates have two important
geometrical features – they are elongated and they have a dendritic or fractal structure. Nanoparticle agglomerates will generally be elongated, merely because of the
19
statistics of random clustering. For example, random-walk polymers typically have
an aspect ratio of 3.4 : 1.6 : 1, compared with the spherical shape 1 : 1 : 1, as discussed in Ref. [45]. Elongated objects can transfer heat faster along the long axis.
The dendritic or fractal structure of nanoparticle agglomerates is another important
consideration, as has recently been pointed out by Prasher et al. [46]. Many types
of formation conditions – such as diffusion-limited aggregation – lead to dendritic or
fractal agglomerates, with complex rarified geometries of armlike dendrites separated
by fluid interstices. Such structures can transport heat over long distances, characterized by a large radius of gyration. In that way, the nanofluids may act as if they
had an effective volume fraction of nanoparticles that is much greater than the actual
volume fraction. We can model each of these geometrical features separately.
2.3
Shape effects: Ellipses
In earlier research, other investigators have considered the effects of certain elon-
gated shapes on the thermal conductivity of nanofluids. For example, Hamilton and
Crosser considered cylindrical geometries [12], and Nan et al. considered ellipsoids
and other 3D shapes [47]. Here, to see the effect of elongation in its simplest form,
we consider 2D ellipses-either aligned or randomly oriented – and compare the results
with the 2D prediction for circular disks.
The calculation for elliptical nanoparticles is analogous to the calculation above
for circular nanoparticles, but with the geometry shown in Fig. 2.3. The temperature
profiles again obey Laplace’s equation Eq. (2.2), with the boundary conditions in
Eq. (2.3) and Eq. (2.4) now applied to the boundary of an ellipse. Solving these
equations for an ellipse with axes a along the gradient and b normal to the gradient
20
T0 (r )
k0
T1 (r ) k1
∇T
T1 (r )
k1
FIG. 2.3. Illustration of the temperature distribution problem for an ellipse in a
fluid used for theoretical modeling of the nanoparticle shape effect on the thermal
conductivity of nanofluids within effective medium theory.
gives the thermal conductivity enhancement:
knf
(a + b)(k1 − k0 )
=1+
φ,
k0
bk1 + ak0
(2.7)
which is a generalization of Eq. (2.6) and is correct to first order in φ [44]. Reversing
a and b gives the result for ellipsoids with long axes perpendicular to the temperature
gradient. Equation (2.7) can be compared with calculations for three-dimensional
ellipsoids by other methods [12, 47].
This expression shows explicitly that the thermal conductivity of the system depends on the orientation of the elliptical particles with respect to the temperature
gradient. As shown in Fig. 2.4 for ellipses with a : b = 5 : 1, particles with long
axes aligned parallel to the temperature gradient produce thermal conductivity enhancement much higher than predicted for circles. On the other hand, particles that
are aligned perpendicular to the temperature gradient produce lower enhancement
than predicted for circles. If we average over all possible orientations, representing an isotropic distribution of elliptical particles, we obtain the intermediate case
21
1.7
1
2
3
4
1.6
knf / k0
1.5
(d)
1.4
+
1.3
1.2
1.1
1.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
P article area fractio n φ
FIG. 2.4. Nanoparticle shape effect on the thermal conductivity of nanofluids. Averaged solution for different volume fractions of (1) ellipses oriented parallel to the
temperature gradient; (2) ellipses oriented perpendicular to the temperature gradient;
(3) isotropic distribution of the ellipse orientations; (4) effective medium theory for
circles. The aspect ratio of the ellipses is 5 : 1.
also shown in Fig. 2.4. Note that this random distribution of orientations produces
a greater enhancement than predicted for circles: particle elongation enhances the
thermal conductivity even if the particles are not aligned.
2.4
Shape effects: Dendrites
Apart from elongation, a further geometrical issue is how the effective thermal
conductivity is affected by a dendritic (fractal) shape of the nanoparticle agglomerates. This is an important question, because agglomerating particles commonly form
such structures. In such structures there are extended dendritic arms of highly conducting solid particles, separated by interstitial regions of the less conducting fluid. In
the steady state, the fluid regions between the arms will have approximately the same
temperature as the solid arms themselves. Hence the whole complex of nanoparticles
plus interstitial fluid will function as a single effective particle from the perspective of
enhancing the thermal transport. The volume taken up by such an effective particle
22
can be much greater than the volume of the constituent nanoparticles themselves.
We would expect the thermal conductivity enhancement to depend on the effective
volume fraction of such agglomerates, rather than on the actual volume fraction of
the particles. Thus thermal conductivity enhancement should be significantly greater
for dendritic or fractal agglomerates than for isolated nanoparticles or compact agglomerates.
In a recent paper, Prasher et al. modeled the thermal conductivity of a nanofluid
composed of fractal aggregates [15]. In this study, they used a specific model of the
agglomeration process based on the model of cluster-cluster agglomeration, which
gives rarified agglomerates with a fractal dimension df = 1.8. They calculated the
thermal conductivity enhancement associated with such agglomerates, and showed
that it is much larger than that for well-dispersed particles.
T1
T2
FIG. 2.5. Illustration of the lattice geometry for theoretical study of the dendritic
agglomeration effect.
Here we would like to assess how general this result is. We would like to determine
whether it depends on the specific model of cluster-cluster agglomeration, and indeed
whether it depends on having agglomerates that obey fractal scale invariance, or if
it is a general feature of disordered dendritic structures. For that reason, we calculate the thermal conductivity enhancement for simpler random structures, which
23
are constructed by self-avoiding random walks of eight steps on a square lattice, as
shown in Fig. 2.5. These structures are not truly fractal, but they have disordered
shapes and dendritic arms, and hence can serve as models for experimental random
agglomerates.
FIG. 2.6. Illustration of one of the random distribution of the dendritic particles with
1% concentration.
As an analytic solution of Laplace’s equation in the presence of disordered particles
is not feasible, we use a numerical approach to determine the temperature profile on
a discretized lattice representing the nanofluid. For a disordered system, Laplace’s
equation for the temperature profile takes the form
∇[k(r)∇T (r)] = 0,
(2.8)
where k(r) is the position-dependent thermal conductivity. We solve this equation on
a 2D square lattice, where the temperature is defined on the lattice sites and thermal
conductivities are defined on the bonds, as shown in Fig. 2.5. The temperature is
fixed on two sides of the sample, creating a temperature gradient. Periodic boundary
24
conditions are enforced on the other two sides. To decrease computation time we
use the alternating-direction implicit method [48]. For any random distribution of
nanoparticle clusters, such as shown in Fig. 2.6, we solve Eq. (2.8) numerically to
obtain the temperature profile. From this temperature profile, we obtain the heat
flux and hence the average thermal conductivity.
FIG. 2.7. Calculated profile of the heat flux for the same distribution as in Fig. 2.6.
Darker to lighter color change corresponds to increase in the heat flux.
The results can be analyzed in two ways. In Fig. 2.7, we show a visualization
of the heat current through the sample, for a specific realization of the nanoparticle
cluster distribution. This picture shows explicitly that the random clusters provide
highly conducting paths for the heat. In the steady state, the system takes advantage
of these paths by concentrating the heat current in the clusters, thereby enhancing
the overall heat transport. In Fig. 2.8 we summarize numerical results for the effective
thermal conductivity of the nanofluid with random-walk shaped agglomerates. For
every particle concentration three independent realizations of shapes and positions of
cluster were tried with no significant effect on the calculated thermal conductivities.
25
The thermal conductivity enhancement varies linearly with the occupied lattice fraction, and with a higher slope than predicted from the analytic solution for circular
nanoparticles (2.6), shown by the solid line. Thus dendritic structures provide another
mechanism for enhancing the thermal conductivity beyond the classical prediction.
1.12
knf / k0
1.10
1.08
1.06
1.04
1.02
1.00
0.00
0.01
0.02
0.03
0.04
Occupied lattice fraction φ
FIG. 2.8. Calculated thermal conductivity enhancement vs concentration of dendritic nanoparticles (dashed line with symbols) compared to circular nanoparticles
(solid line) within effective medium theory. Three data points for each concentration correspond to three independent random distributions of particle shapes and
positions.
2.5
Effect of surface thermal resistance
So far we have studied suspensions of elliptical nanoparticles (or elliptically shaped
aggregates), as well as random dendritic or fractal nanoparticle aggregates. All of
these cases show a higher effective thermal conductivity than predicted for spherical or circular nanoparticles dispersed in the base fluid. These theoretical results
are consistent with experimental results of many research groups, as outlined in Table 2.1, which often show a thermal conductivity enhancement beyond the classical
prediction. However, experimental results of our collaborators from Kent State University [40], show a thermal conductivity enhancement that is somewhat lower than
26
0.75
20 nm Al2O3
1.25
40 nm Al2O3
40nm
1.20
0.70
11nm 1.15
20nm 1.10
0.65
knf /k0
knf (W/mK)
1.30
Effective Medium Theory
knf/k0=1+3ϕ
11 nm Al2O3
1.05
1.00
0.60
(a) water
0.95
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
Al2O3 volume fraction, φ
Effective Medium Theory
knf/k0=1+3ϕ
11 nm Al2O3
0.32
20 nm Al2O3
40 nm Al2O3
1.25
0.30
1.20
1.15
0.28
knf /k0
knf (W/mK)
1.30
1.10
1.05
0.26
(b) ethylene glycol
1.00
0.24
0.00
0.02
0.04
0.06
0.08
0.10
Al2O3 volume fraction φ
FIG. 2.9. Thermal conductivity at 23◦ C of suspensions prepared from 11-, 20-, and 40nm nominal size alumina nanoparticles in (a) water and (b) ethylene glycol as a base
fluid. The right axes show the enhancement effect relative to thermal conductivity of
the base liquid. The dotted lines indicate predictions of the effective medium theory
for spherical particles with infinite heat conductivity and φ ¿ 1. Error bars indicate
standard deviation over ten consequent measurements.
the classical prediction (Fig. 2.9). To explain this smaller enhancement in the thermal conductivity, we cannot rely on geometrical effects. Rather we must consider the
surface thermal resistance between the base fluid and the nanoparticles, as has been
done in Refs. [22, 47, 49, 50], and compare the results with experimental data.
To see the effect of surface thermal resistance, we can perform a calculation for a
spherical nanoparticle with a resistive interface by analogy with our calculation above
for a perfectly conducting interface. The system still obeys Laplace’s equation (2.2)
27
for the temperature, and the boundary condition (2.4) for the temperature gradients
still applies. However, the boundary condition (2.3) for the temperatures is now
changed to
T0 − T1 =
k0 ∂T0
,
β ∂~n
(2.9)
where β is the surface thermal conductance, the inverse of the surface thermal resistance. When β → ∞, this boundary condition reduces to the equality of temperatures
across a perfectly conducting interface Eq. (2.3). By solving Laplace equation with
this new boundary condition, and averaging the results over a uniform distribution
of 3D spherical nanoparticles, we obtain a prediction for the thermal conductivity
enhancement:
knf
3(k1 − k0 )
=1+
φ,
k0
k1 + 2k0 + 2k0 k1 /(Rβ)
(2.10)
consistent with Refs. [22, 47, 49, 50] in the limit of small volume fraction. Note that
this prediction depends on the radius R of the nanoparticles, unlike all the predictions
above for nanoparticles with no thermal resistance, which depend only on the volume
fraction.
We fit the experimental data obtained by our collaborators in Kent State University (Fig. 2.9) to this prediction, in order to extract the composite parameter Rβ
that enters the equation. The results of this fitting are shown in Table 2.2. Note
that in the case of aqueous nanofluids the fitted parameter Rβ does not follow a
consistent trend with the nominal particle size (first column in Table 2.2); however, it
does increase monotonically with the crystallite sizes determined from powder x-ray
difraction (XRD) [40] (second column in Table 2.2). Moreover, the values of β for all
particle sizes are about the same if crystallite size, Dvol , is used as 2R, i.e., β = 5×108
28
W/ K m2 for an alumina-water interface. This value is at the high end of the values
reported for metal nanoparticle-water interfaces [51]. Thus the simple classical model
incorporating surface resistance reasonably accounts for the relatively low thermal
conductivity enhancement in our experiments.
2.6
Discussion
At this point we can ask why the thermal conductivity enhancement in alumina-
water nanofluids correlates with crystallite particle size but not with surface-area
averaged particle size nor with the size of agglomerates. More specifically, we need
to explain why the least agglomerated 40-nm alumina nanofluids show the highest
heat transport and the lowest viscosity enhancement [40], whereas 20-nm nanofluids
show the lowest heat transport and the highest viscosity enhancement, while 11-nm
nanofluids exhibit an intermediate behavior.
Despite the variety of techniques employed [40], our experimental data [40] are
still insufficient to provide a definite answer. Nevertheless, having identified the main
factors in heat transfer enhancement in nanofluids – the particle shape and the surface
resistance – we can suggest the following scenario.
The 40-nm particles, due to the presence of acidic δ phase, are highly charged
and do not undergo significant agglomeration in solution, as confirmed by viscosity,
dynamic light scattering (DLS), and the aging experiments [40]. This leads to the
behavior closest to the ideal spherical particle case given by Eq.(2.1). The slightly
smaller than ideal slope can be accounted for by the finite surface heat-transfer resistance (high β ⇒ low 1/β). Note that an earlier work reported identical heat transfer
enhancement for α- and γ-alumina particles of the same specific surface area [30].
On the other hand, the more agglomerated 11-nm and 20-nm alumina nanofluids
29
are expected to produce a higher enhancement than nanofluids with spherical particles. The viscosity and the aging experiments data [40] suggest that the 20-nm
nanofluids are the most agglomerated. This may be due to the presence of smaller
alumina crystallites in this sample (as indicated by XRD [40]), which makes it more
prone to agglomeration via the dissolution-precipitation mechanism. The question
now becomes why the more agglomerated nanofluids show a weaker heat transport
enhancement than the less agglomerated nanofluids, contrary to the theoretical considerations given above. It is possible that the heat-transfer resistance between the
crystallites (and therefore between nanoparticles) within the agglomerate plays a critical role here. If this particle-particle resistance is large, it effectively eliminates the
enhancement due to agglomeration and leads to the situation described by Eq.(2.10)
with R being the radius of the crystallites in the agglomerate. This hypothesis is
supported by the correlation of the heat transfer enhancement with the crystallite
size and the consistency of β under this assumption shown in Table 2.2, as well as
the data on the thermal conductivity of the nanoparticle powder [40].
One further issue, which also requires an explanation, is why there is no particlesize effect in ethylene glycol nanofluids, while there is a strong particle size effect
in aqueous nanofluids. Agglomeration is not likely to account for this difference,
because the state of agglomeration in ethylene glycol and aqueous nanofluids is quite
similar [40]. However, we can explain all the experimental data in terms of the
base fluid thermal conductivity k1 and the interfacial thermal resistance β −1 . The
base fluid thermal conductivity of ethylene glycol is about 2.4 times lower than that
of water. Furthermore, we can hypothesize that the interfacial thermal resistance
β −1 of nanoparticles in ethylene glycol is lower than that of the same nanoparticles
30
in water. For those two reasons, the 2k1/(Rβ) term in Eq.(2.10) would be much
smaller in ethylene glycol nanofluids, and hence there would be no particle size effect.
A lower interfacial thermal resistance also results in a higher slope of the thermal
conductivity enhancement in ethylene glycol suspensions compared to water, as found
in the experiments (see Table 2.2). Unfortunately, the interfacial thermal resistance
in the nanofluids studied in this work cannot be directly measured, and the proposed
explanation still requires an independent confirmation. We also cannot give a simple
account for a smaller interfacial thermal resistance in ethylene glycol compared to
water, as such quantities do not always follow a simplified phonon spectra mismatch
model, and they usually cannot be predicted based on rule-of-thumb arguments [52].
2.7
Conclusions
In this chapter, we have presented a theoretical study of thermal conduction in
nanofluids. In this work, we have assessed possible mechanisms for thermal conductivity enhancement in nanofluids. By estimating characteristic magnitudes, we find
that the contribution associated with nanoparticle motion is much smaller than the
contribution associated with heat diffusion through the effective medium of particles
and fluid, with the particles providing a path for rapid heat conduction. Furthermore, the geometry of nanoparticles and agglomerates plays a very important role in
determining the thermal conductivity enhancement in effective medium theory. The
prediction for compact spherical particles is the “worst case” for thermal conductivity
– the enhancement is greater for extended elliptical particles, even randomly oriented
ellipses, and the enhancement is also greater for dendritic or fractal aggregates. Thus
there is no need to invoke theoretical mechanisms beyond effective medium theory
to explain the anomalously high enhancement reported by other investigators; it is
31
sufficient to consider effective medium theory for appropriate geometries and thereby
to take into account higher “effective” volume fractions.
For improving thermal conductivity at fixed volume fraction, nanofluids should
have extended particles or agglomerates, which can transport heat rapidly over significant distances within a sample. Ideally, these particles or agglomerates should be
oriented with their long axes along the thermal gradient, in order to provide conducting paths in the optimum direction, as shown by our calculation for oriented
ellipses. However, a related and equally important factor – which apparently plays
the greater role in the alumina system studied here – is the heat transfer resistance
at the particle-liquid interface, as well as interfacial resistance between the particles
in the agglomerates, which must obviously be minimized. In our view, these factors
should be the primary focus for the continued development of nanofluids for thermal
management applications.
2.8
Acknowledgments
This work was done in collaboration with Elena V. Timofeeva, Alexei N. Gavrilov,
James M. McCloskey, and Yuriy V. Tolmachev from the Department of Chemistry,
Kent State University, and Samuel Sprunt from the Department of Physics, Kent
State University.
This research was supported by the Office of Naval Research Grant No. N000140610029,
the Ohio Board of Regents Research Challenge, and the National Science Foundation
Research Experiences for Undergraduates Program. We would like to thank M. S.
Spector for suggesting this project and for many helpful discussions.
32
Table 2.1. Comparison of experimental heat transfer enhancements in alumina
nanofluids reported in earlier literature and in the present study.
Base
Fluid
Nominal
Al2O3
Particle
Size
Voluem
Fraction
Thermal
conductivity
enhancement
Enhancement
slope
and temperature
Masuda et al. [1]
(transient hot wire)
water
13 nm
4.3%
33
7.7
Eastman et al. [2]
water
33 nm
4.3%
9%
2.1
Lee et al. [28]
water
EG
38 nm
5%
5%
12%
17%
2.4 at 25◦ C
3.4 at 25◦ C
Wang et al. [29]
(parallel plates)
water
EG
pump oil
engine oil
28 nm
5%
5%
5%
5%
14%
26%
12%
26%
2.8
5.2
2.4
5.2
Xie et al. [30, 31]
water
EG
GLY
pump oil
60.4 nm
5%
5%
5%
5%
22%
29%
27%
38%
4.4 at 25◦ C
5.8 at 25◦ C
5.4 at 25◦ C
7.6 at 25◦ C
Das et al. [32]
(temperature oscillation)
water
38 nm
4%
8%
25%
2.0 at 21◦ C
6.25 at 51◦ C
Putra et al. [33]
(steady-state parallel
plates with convection)
water
131 nm
4%
25%
6.3 at 50◦ C
Wen and Ding [34]
water
27 – 56 nm
1.6%
10%
6.3 at 22◦ C
Nara et al. [35]
(temperature oscillation)
water
EG
PG
40 nm
0.5%
34%
5%
0%
68 at 85◦ C
10 at 85◦ C
0 at 85◦ C
Chon et al. [36]
water
13 nm
50 nm
182 nm
1%
4%
1%
15%
30%
5%
15 at 60◦ C
7.5 at 70◦ C
5 at 60◦ C
Li and Peterson [37]
(steady-state parallel plates)
water
36 nm
47 nm
6%
6%
28%
26%
4.6 at 36◦ C
4.3 at 36◦ C
Krishnamurthy et al. [38]
(unspecified, possibly
like Ref. [35])
water
20 nm
1%
16%
16 at room temperature
Zhang et al. [39]
water
20 nm
5%
15%
3
water
11 nm
20 nm
40 nm
all sizes
5%
5%
5%
5%
8%
7%
10%
13%
1.6
1.3
2.0
2.6 all at 10 – 60◦ C
Researcher/
Reference
(Method)
Our results [40]
EG
33
Table 2.2. Surface thermal conductance in alumina nanofluids.
Nominal
Particle Size
Crystallite
size (from XRD) nm
Slope
in Fig. 2.9
Fitted
Rβ (W/m K)
40 nm in water
20 nm in water
11 nm in water
In ethylene glycol
12.5
5.3
5.6
2.0
1.3
1.6
2.6
3.7
1.1
1.7
4.5
β × 108 W/m2 K
5.8
4.0
6.2
CHAPTER 3
Theory of Ferroelectric Nanoparticles in Nematic Liquid Crystals.
Landau-like Approach
3.1
Introduction
In recent years, many experiments have found that colloidal particles in nematic
liquid crystals exhibit remarkable new types of physical phenomena. If the particles
are micron-scale, they induce an elastic distortion of the liquid-crystal director. This
elastic distortion leads to an effective interaction between particles, and offers the
possibility of organizing a periodic array of particles, with possible photonic applications [53–58]. If the particles are 10–100 nm in diameter, they are too small to distort
the liquid-crystal director, and hence the system enters another range of behavior.
Experiments have shown that low concentrations of ferroelectric nanoparticles can
greatly enhance the physical properties of nematic liquid crystals [59–66]. In particular, Sn2 P2 S6 or BaTiO3 nanoparticles at low concentration (<1%) increase the orientational order parameter of the host liquid crystal, and increase the isotropic-nematic
transition temperature by about 5 K. The nanoparticles also decrease the switching
voltage for the Fredericksz transition. These experimental results are important for
fundamental nanoscience, because they show that nanoparticles can couple to the orientational order of a macroscopic medium. They are also important for technological
applications, because they provide a new opportunity to tune the properties of liquid
crystals without additional chemical synthesis.
34
35
A key question in this field is how to understand and control the properties of liquid crystals doped with ferroelectric nanoparticles, and to make further progress with
these materials, it is essential to develop a theory for the interaction between liquid
crystals and ferroelectric nanoparticles. In previous theoretical research, Reshetnyak
et al. have developed a theoretical approach based on electrostatics [64, 65, 67]. In this
theory, the key issue is how an ensemble of nanoparticles with aligned dipole moments
can polarize the liquid-crystal molecules, hence increasing the intermolecular interaction. This electrostatic effect enhances the isotropic-nematic transition temperature
and reduces the Frederiks transition voltage. In related research, Pereira et al. have
performed molecular dynamics simulations of ferroelectric nanoparticles immersed in
a nematic liquid crystal [68]. These simulations also assume that the nanoparticles
are aligned, and they also find a substantial enhancement of liquid-crystal order.
In this chapter, we propose a new theory for the statistical mechanics of ferroelectric nanoparticles in liquid crystals, which is based specifically on the orientational
distribution of the nanoparticle dipole moments. This distribution is characterized by
an orientational order parameter, which interacts with the orientational order of the
liquid crystals and stabilizes the nematic phase. We estimate the coupling strength
and calculate the resulting enhancement in TNI , in good agreement with experiments.
This enhancement occurs even when electrostatic interactions are partially screened
by moderate concentrations of ions in the liquid crystal. In addition, we predict the
response of the isotropic phase to an applied electric field, known as the Kerr effect,
and show that it is greatly enhanced by the presence of nanoparticles. This work is
published [72].
36
FIG. 3.1. Nanoparticles surrounded by liquid crystal. (a) Particle with no electric
dipole moment, in the isotropic phase. (b) Ferroelectric particle with electric dipole
moment, which produces an electric field that interacts with the orientational order
of the liquid crystal.
3.2
Theory
To begin the calculation, consider a spherical nanoparticle with radius R and
electrostatic dipole moment p, surrounded by a nematic liquid crystal, as shown in
Fig. 3.1. The electric field E generated by the nanoparticle interacts with the order
tensor QLC
αβ of the liquid crystal through the free energy
Fint
²0 ∆²
=−
3
ˆ
d3 rQLC
αβ (r)Eα (r)Eβ (r),
(3.1)
where ∆² is the dielectric anisotropy of the fully aligned liquid crystal. In all calculations in this chapter, we assume the dielectric anisotropy ∆² to be positive, meaning
that molecules of nematic liquid crystal prefer to align parallel to the electric field. In
3
1
the case of a fully ordered liquid crystal with QLC
αβ = 2 nα nβ − 2 δαβ , Eq. (3.1) reduces
to the well known expression for free energy of a liquid crystal in an electric field
37
Fint
²0 ∆²
=−
2
ˆ
d3 r(n · E)2 .
(3.2)
The electric field of the nanoparticle has the standard dipolar form
1
E(r) =
4π²0 ²
µ
3r(r · p)
p
− 3
5
r
r
¶
,
(3.3)
neglecting higher-order corrections due to the dielectric anisotropy of the liquid crystal. This is valid assumption since in the nematic phase the corrections are of the
order of nematic liquid crystal order parameter SLC , and after substituting this expression in the equation for free energy Eq. (3.1) we will obtain terms of higher order
in SLC that we are neglecting.
Near the nanoparticle, the electric field varies rapidly as a function of position.
However, the liquid-crystal order cannot follow that rapid variation, because it would
cost too much elastic energy. Hence, for sufficiently small nanoparticles, we can
assume that the order tensor QLC
αβ is uniform in space. In that case, we can integrate
the interaction free energy to obtain
Fint = −
∆²
QLC pα pβ .
180π²0 ²2 R3 αβ
(3.4)
Now consider a low concentration ρNP of nanoparticles dispersed in the liquid
crystal. The dipole moments of these nanoparticles will not all have the same orientation; rather there must be a distribution of orientations. As a result, the interaction
38
of liquid crystal and nanoparticles gives the free energy density per unit volume
Fint
∆²ρNP
=−
QLC hpα pβ i,
V
180π²0 ²2 R3 αβ
(3.5)
averaged over the distribution of nanoparticle orientations. This distribution can be
expressed in terms of a nanoparticle order tensor
QNP
αβ =
3 hpα pβ i 1
− δαβ ,
2 p2
2
(3.6)
analogous to the standard liquid-crystal order tensor. Hence, the interaction free
energy density becomes
Fint
∆²ρNP p2 LC NP
=−
Q Q ,
V
270π²0 ²2 R3 αβ αβ
(3.7)
which shows an explicit coupling between the order tensor of the liquid crystal and
the order tensor of the nanoparticles. If we make the reasonable assumption that
both of these tensors are aligned along the same axis, then this interaction reduces to
Fint
∆²ρNP p2
=−
SLC SNP ,
V
180π²0 ²2 R3
(3.8)
where SLC and SNP are the scalar order parameters of the liquid crystal and the
nanoparticles, respectively.
To model the statistical mechanics of nanoparticles dispersed in the liquid crystal,
39
we must expand the free energy in both order parameters SLC and SNP , which gives
F
V
=
a0LC (T − T ∗ ) 2
b 3
c 4
SLC − SLC
+ SLC
2
3
4
2
∆²ρNP p
aNP 2
SNP −
SLC SNP .
+
2
180π²0 ²2 R3
(3.9)
Here, the first three terms are the standard Landau-de Gennes free energy of a nematic
liquid crystal. The first-order isotropic-nematic transition of the pure liquid crystal
occurs at TNI = T ∗ + (2b2 )/(9a0LC c), and the leading coefficient in this expansion
can be estimated through Maier-Saupe theory as a0LC = 5kB ρLC , where ρLC is the
concentration of liquid-crystal molecules per volume [69]. The fourth term in the
free energy is the entropic cost of imposing orientational order on the nanoparticles.
By expanding the entropy in terms of the orientational distribution function of the
nanoparticles, we can estimate the coefficient as aNP = 5kB T ρNP . The final term is
the coupling between the liquid-crystal order and the nanoparticle order, calculated
above.
We minimize the free energy of Eq. (3.9) over the nanoparticle order parameter
to find the optimum value
SNP =
∆²p2
SLC .
900π²0 ²2 R3 kB T
(3.10)
This equation shows that the liquid crystal induces orientational order of the nanoparticles, with a nanoparticle order parameter proportional to the liquid-crystal order
parameter. Note that the induced order is independent of the nanoparticle concentration, which is reasonable because it arises from the interaction of individual
nanoparticles with the liquid crystal, not from interactions between nanoparticles.
40
We then substitute this expression back into the free energy to obtain
F
V
"
µ
¶2 #
2
2
a0LC
ρ
∆²p
2
=
T − T ∗ − 0 NP
SLC
2
aLC aNP 180π²0 ²2 R3
c 4
b 3
+ SLC
.
− SLC
3
4
(3.11)
2
has been shifted by the interaction with the
In this equation, the coefficient of SLC
nanoparticles. This shift increases the isotropic-nematic transition temperature by
∆TNI
¶2
∆²p2
180π²0 ²2 R3
µ
¶2
πφNP R3
2∆²P 2
=
.
3TNI ρLC 675kB ²0 ²2
ρ2
= 0 NP
aLC aNP
µ
(3.12)
The last expression has been simplified by writing p = ( 34 πR3 )P and ρNP = φNP /( 34 πR3 ),
where P is the polarization and φNP the volume fraction of the nanoparticles.
To estimate ∆TNI numerically, we use the following parameters appropriate for
Sn2 P2 S6 nanoparticles in the liquid crystal 5CB: φNP = 0.5%, R = 35 nm, TNI = 308
K, ρLC = 2.4 × 1027 m−3 , P = 0.04 Cm−2 , kB = 1.38 × 10−23 JK−1 , ²0 = 8.85 × 10−12
C2 N−1 m−2 , and ∆² ≈ ² ≈ 10 [70]. With those parameters, we obtain ∆TNI ≈ 5 K,
which is roughly consistent with the increase that is observed experimentally. Of
course, there is a substantial uncertainty in this estimate, because the parameters R
and P are not known very precisely in the experiments.
Note that our model predicts that the enhancement ∆TNI should be first-order in
volume fraction φNP , fourth-order in polarization P , and third-order in R. In particular, increasing R should increase ∆TNI as long as the nanoparticles are not large
enough to disrupt the liquid-crystal order. This prediction disagrees with Ref. [65],
41
which predicts that ∆TNI should be first-order in φNP , second-order in P , and independent of R.
3.3
Effect of Ionic Impurities
At this point, we must consider the effects of ionic impurities in the liquid crys-
tal. Any liquid crystal contains some concentration of free positive and negative ions,
which can redistribute in response to electric fields. One might worry that these ions
would screen the electric field of the nanoparticles, and hence prevent the enhancement of TNI . To address this issue, we start with well known solution of linearized
Poisson-Boltzmann equation for the single charge in the presence of ions. The electric
potential has form:
Φ(r) =
Q e−κ|r|
.
4π²0 ² |r|
(3.13)
In this expression, κ−1 is the Debye screening length given by
µ
κ
−1
=
²0 ²kB T
2nq 2
¶1/2
,
(3.14)
where n is the concentration and q the charge of the ions. We modify Eq. (3.13) for
the dipole, taking to account that dipole is just a combination of two opposite charges
separated by the distance δr:
Q
Φ(r) =
4π²0 ²
µ
e−κ|r−δr| e−κ|r|
−
|r − δr|
|r|
¶
.
(3.15)
42
Expanding Eq. (3.15) for small δr, and taking to account that dipole moment p =
Qδr, we obtain electric field potential for the dipole p in the presence of ions:
Φ(r) = −
p · r −κr
e
(1 + κr)
r3
(3.16)
Then, the electric field around a dipole in the presence of ions is
·
µ
¶
¸
e−κr
3r(r · p)
p
κ2 r(r · p)
E(r) =
(1 + κr)
− 3 +
.
4π²0 ²
r5
r
r3
(3.17)
With this expression for the field, we can repeat the calculation above for the enhancement in TNI , leading to
∆TNI
µ
¶2
2∆²P 2
πφNP R3
=
e−2κR
2
3TNI ρLC 675kB ²0 ²
¡
¢
× 1 + 2κR + κ2 R2 + κ3 R3 .
(3.18)
320
318
TNI (K)
316
314
312
310
308
18
10
19
10
20
10
21
10
22
10
23
10
24
10
-3
Ion concentration (m )
FIG. 3.2. Predicted isotropic-nematic transition temperature as a function of ion
concentration in a nanoparticle-doped liquid crystal, using numerical parameters presented after Eq. (3.12).
To interpret this result, note that the key parameter is κR, the ratio of the
43
nanoparticle radius to the Debye screening length. If the ion concentration is low, then
the screening length is large compared with the nanoparticle radius, and hence ∆TNI
is as large as in the unscreened case. However, if the ion concentration is sufficiently
high, then the screening length becomes comparable to the nanoparticle radius, and
hence the enhancement is screened away. For a specific example, Fig. 3.2 shows TNI
as a function of ion concentration n, using the numerical parameters discussed above.
In this example, the full unscreened enhancement persists up to n ≈ 1020 ions/m3 . It
then decays away as a function of ion concentration, and is virtually eliminated by
n ≈ 1023 ions/m3 .
Typical measurements of the ion concentration in 5CB show n ≈ 1020 ions/m3
and hence κ−1 ≈ 260 nm [71]. Because this screening length is much greater than
the nanoparticle radius, the enhancement should indeed be observable in realistic
experiments. Note that the ion concentration in a liquid crystal varies over several
orders of magnitude, depending on preparation conditions. Hence, we speculate that
variations in ion concentration may be one explanation for variations in published
experimental measurements of ∆TNI .
3.4
Kerr Effect
So far, we have modeled the spontaneous ordering of a nanoparticle-doped liquid
crystal. We can also use the same theoretical approach to predict how the system
responds to an applied electric field. For a specific example, we investigate the Kerr
effect, in which an applied electric field E induces orientational order in the isotropic
phase, slightly above the isotropic-nematic transition. In a pure liquid crystal, the
Kerr effect is a weak alignment proportional to E 2 . In a liquid crystal doped with
ferroelectric nanoparticles, we expect that an applied electric field will induce polar
44
order of the nanoparticles, proportional to E. This polar order will necessarily induce
nematic order of the nanoparticles, proportional to E 2 , which will in turn induce
nematic order of the liquid crystal, also proportional to E 2 . Hence, the nanoparticledoped liquid crystal should have an enhanced Kerr effect with the same symmetry as
the standard Kerr effect, but with a much larger magnitude.
To model the enhanced Kerr effect, we must generalize the Landau theory presented above in three ways.
First, we must introduce a polar order parameter
Mα = hpα i/p for the nanoparticles, as well as the nematic order parameters QNP
αβ
and QLC
αβ . Second, we must consider the energetic coupling of an applied electric
field to the order parameters. In the free energy density, an applied field couples
linearly to the polar order parameter of the nanoparticles through the interaction
−ρNP pEα Mα , and couples quadratically to the nematic order parameter of the liquid crystal through the interaction − 31 ²0 ∆²Eα Eβ QLC
αβ . Third, we must calculate the
entropy of a nanoparticle distribution characterized by both order parameters Mα
and QNP
αβ , following the method of Ref. [69]. Assuming that all order parameters are
aligned along the electric field direction, the free energy becomes
F
V
=
a0LC (T − T ∗ ) 2
b 3
c 4
²0 ∆² 2
SLC − SLC
+ SLC
−
E SLC
2
3
4
3
∆²ρNP p2
−
SLC SNP − ρNP pEM
180π²0 ²2 R3
µ
¶
3 2
5 2
2
+kB T ρNP
S + M − 3SNP M .
2 NP 2
(3.19)
We minimize this free energy over all three order parameters, M , SNP , and SLC . In
the high-temperature isotropic phase, in the limit of small electric field, the resulting
45
liquid-crystal order parameter is
SLC
"
µ
¶2 #
²0 ∆²E 2
φNP 4πP 2 R3
= 0
1+
.
3aLC (T − T ∗ − ∆TNI )
3
45²0 ²kB T
(3.20)
In this expression, note that the induced order parameter depends on electric field and
temperature exactly as in the standard Kerr effect, but the coefficient is increased by
the coupling with nanoparticles. In the square brackets, the first term of 1 indicates
the standard Kerr effect for pure liquid crystals, and the second term indicates the
relative enhancement due to nanoparticle doping.
For a specific numerical example, we use the same parameters presented after
Eq. (3.12). With these parameters, the relative enhancement in the Kerr effect is
extremely large, of order 107 . Figure 3.3 plots the predicted order parameter SLC
as a function of temperature for several values of the applied electric field, with and
without nanoparticles. This plot shows explicitly that the presence of nanoparticles
greatly enhances the sensitivity to applied electric fields in the isotropic phase, as well
as enhancing the isotropic-nematic transition temperature. This prediction should be
tested in future experiments, and should provide an opportunity to build liquid-crystal
devices that can operate at lower electric fields.
As a final point, we should mention one limitation of our model. Like all Landau
theories, our model involves an expansion of the free energy in powers of the order
parameters, and hence it overestimates the order parameters that occur in the lowtemperature phase. Future work may extend this model through asymptotic lowtemperature approximations to the free energy. Nevertheless, our Landau-like model
clearly shows the effects of nanoparticle doping at and above the isotropic-nematic
46
0 V/µm
20 V/µm
40 V/µm
60 V/µm
0 V/µm
0.02 V/µm
0.04 V/µm
0.06 V/µm
1.0
Order Parameter
0.8
0.6
0.4
0.2
0.0
290
300
310
320
330
340
350
Temperature (K)
FIG. 3.3. Prediction for field-induced order parameter SLC as a function of temperature for several values of applied electric field, with and without ferroelectric
nanoparticles, using numerical parameters presented after Eq. (3.12).
transition, in the regime where Landau theory is valid.
3.5
Conclusions
In conclusion, we have developed a theory for the statistical mechanics of ferroelec-
tric nanoparticles in nematic liquid crystals. This theory predicts the enhancement in
the isotropic-nematic transition temperature and in the response to an applied electric field, which can be tested experimentally. The work demonstrates the coupling
of nanoparticles with macroscopic orientational order, and provides an opportunity
to improve the properties of liquid crystals without chemical synthesis.
3.6
Acknowledgments
We would like to thank J. L. West, Y. Reznikov, and P. Bos for many helpful
discussions.
This work was supported by NSF Grant DMR-0605889.
CHAPTER 4
Theory of Ferroelectric Nanoparticles in Nematic Liquid Crystals.
Maier-Saupe-like Approach
4.1
Introduction
In chapter 3, we proposed a Landau-like theory for the statistical mechanics of
ferroelectric nanoparticles in liquid crystals. In that theory, we suppose that both the
liquid crystals and the nanoparticles have distributions of orientations, as illustrated
in Fig. 4.1. These distributions are characterized by two orientational order parameters, which interact with each other. Using a Landau theory, we showed that the
coupling stabilizes the nematic phase. By estimating the strength of the coupling, we
calculated the enhancement in the isotropic-nematic transition temperature. We also
predicted that the nanoparticles would greatly increase the Kerr effect, the response
of the isotropic phase to an applied electric field.
FIG. 4.1. Schematic illustration of ferroelectric nanoparticles suspended in a liquid
crystal. The electrostatic dipole moments of the nanoparticles have a distribution of
orientations.
47
48
Although the work in chapter 3 demonstrates an important physical mechanism,
we must acknowledge that it has one mathematical limitation: Like all Landau theories, it involves an expansion of the free energy in powers of the order parameters.
This expansion is valid when the order parameters are small, but it breaks down
when they become large. In particular, the theory allows the order parameters to
become larger than 1, which is clearly impossible. For ferroelectric nanoparticles in
a liquid crystal, the nanoparticle order parameter is not necessarily small, even near
the isotropic-nematic transition.
The purpose of the current chapter 4 is to generalize the previous theory by eliminating the assumption that the order parameters are small. For this generalization, we
now use a Maier-Saupe-type theory instead of a Landau theory. We still consider the
same physical concept of coupled orientational order parameters for the liquid crystals and the nanoparticles, and we still use the same energy of interaction between
them. However, we now use a more general expression for the entropy, not a power
series, which enforces the constraint that the order parameters cannot become larger
than 1. This change allows us to avoid the potential mathematical inconsistency of
Landau theory.
Like our previous calculation, the work presented here shows that doping liquid crystals with ferroelectric nanoparticles enhances the isotropic-nematic transition
temperature. In the limit of weak coupling between the nanoparticles and the liquid
crystal, the Maier-Saupe-type theory exactly reduces to the Landau theory. However,
in the case of strong coupling, the new theory predicts a smaller but still substantial
enhancement. Rough estimates suggest that the experimental system is in the limit
of strong coupling, so it is important to use this modified theory. Furthermore, the
49
work presented here also predicts the Kerr effect as a function of applied electric field.
In the limit of low electric field, the Maier-Saupe-type theory exactly reduces to the
Landau theory. However, for larger field, the nanoparticle order saturates and the
enhanced Kerr effect is cut off.
In this chapter 4 we present the formalism of Maier-Saupe theory, with interacting
orientational distributions for liquid-crystal molecules and nanoparticles; then we
apply this formalism to calculate the isotropic-nematic transition temperature, and
determine the enhancement due to nanoparticles. We calculate the Kerr effect of
induced orientational order under an applied electric field, and investigate how this
effect depends on the magnitude of the field.
4.2
Overview of Maier-Saupe Theory
In this section we introduce the free energy for a system of liquid-crystal molecules
with ferroelectric nanoparticles. To construct the free energy, we use the fundamental
equation of mean-field theory,
F = hHi + kB T hln
N
LC
Y
%i i,
(4.1)
i=1
where the first term is the energy, the second term is the entropic contribution to the
free energy, and the averages are taken over the single-particle distribution function
%, which is the same for all liquid crystal molecules in the mean field approximation.
Hence, the first step is to define the distribution functions for liquid-crystal molecules
and nanoparticles.
Liquid-crystal molecules are rod-shaped objects, with each molecule characterized
by the direction of its long axis m. In the nematic phase, these axes are preferentially
50
oriented along the average director n. Because the individual molecules are equally
likely to point along +n or −n, the single-molecule distribution function can be
written as
%LC (θ) = ´ 1
−1
exp(ULC P2 (cos θ))
d(cos θ) exp(ULC P2 (cos θ))
,
(4.2)
where θ is the angle between the molecular orientation m and the average director
n, and P2 is the second Legendre polynomial. The parameter ULC is a variational
parameter, which acts as an effective field on the molecular orientation. It is related
to the standard nematic order parameter SLC = hP2 (cos θ)i by
´1
−1
SLC =
d(cos θ)P2 (cos θ) exp(ULC P2 (cos θ))
´1
d(cos θ) exp(ULC P2 (cos θ))
−1
(4.3)
Note that ULC ranges from 0 to ∞, while SLC ranges from 0 to 1.
We can now calculate the free energy of a pure liquid-crystal system. Maier-Saupe
theory assumes that the interaction energy between neighboring molecules i and j is
proportional to −(mi · mj )2 . With the assumed distribution function, the average
interaction energy becomes
1
2
LC
,
Fenergetic
= hHi = − JNLC SLC
3
(4.4)
where NLC is the number of liquid-crystal molecules in the system, and J is an energetic parameter proportional to the interaction strength and the number of neighbors
per molecule. Furthermore, the entropic contribution to the free energy can be written
51
as
LC
Fentropic
= kB T NLC hln %LC i
h
= kB T NLC ULC SLC
h´
ii
1
− ln −1 d(cos θ) exp(ULC P2 (cos θ)) .
(4.5)
By combining these pieces, we obtain the total free energy of the liquid crystal,
F
NLC kB T
J
2
SLC
+ ULC SLC
3kB T
h´
i
1
− ln −1 d(cos θ) exp(ULC P2 (cos θ))
= −
(4.6)
The free energy of Eq. (4.6) is a function of the temperature T and the variational
parameter ULC , with SLC defined implicitly as a function of ULC through Eq. (4.3). By
minimizing the free energy over ULC for varying temperature, we can find the liquid
crystal has a first-order transition from the isotropic phase with ULC = SLC = 0 to
the nematic phase with
ULC = 1.95,
SLC = 0.429.
(4.7)
The numerical solution for the transition temperature in this pure liquid crystal is
TNI = 0.147
J
.
kB
(4.8)
Also, we can find an analytic solution for the limit of supercooling,
T∗ =
2J
J
= 0.133 .
15kB
kB
(4.9)
52
From experiments we know T ∗ and TNI for any particular liquid-crystal material, so
we can use Eq. (4.8) or (4.9) to determine J for that material,
J = 6.81kB TNI .
(4.10)
Once we add nanoparticles to the system, we get another distribution function
for the orientations of the nanoparticle dipole moments. By symmetry, we expect
that this distribution should be aligned along the same axis n as the liquid-crystal
distribution. However, the magnitude of the order may be different. Hence, we can
write the nanoparticle distribution function as
%NP (θ) = ´ 1
exp(UNP P2 (cos θ))
d(cos θ) exp(UNP P2 (cos θ))
−1
,
(4.11)
where UNP is a variational parameter for the nanoparticles. The orientational order
parameter SNP of the nanoparticles can be defined by analogy with the liquid-crystal
order parameter as
´1
SNP =
−1
d(cos θ)P2 (cos θ) exp(UNP P2 (cos θ))
.
´1
d(cos
θ)
exp(U
P
(cos
θ))
NP
2
−1
(4.12)
Just as in the liquid-crystal case, note that UNP ranges from 0 to ∞, while SNP ranges
from 0 to 1.
As we discussed in chapter 3, the ferroelectric nanoparticles create static electric
fields, which interact with the dielectric anisotropy of the liquid crystal. By averaging
53
the interaction energy over the distribution functions %LC and %NP , we obtain
Finteraction = −KNP NNP SLC SNP .
(4.13)
In this expression, NNP is the number of nanoparticles in the system, and KNP is an
energetic parameter representing the strength of the interaction. For an unscreened
electrostatic interaction, we derived
KNP =
4πε0 ∆εP 2 R3
ε0 ∆εp2
=
.
180π(ε0 ε)2 R3
405(ε0 ε)2
(4.14)
where p, P , and R are the dipole moment, polarization, and radius of a nanoparticle,
and ε and ∆ε are the dielectric constant and dielectric anisotropy of the bulk liquid
crystal (as we pointed out in chapter 3 all the calculations are done under assumption
of positive dielectric anisotropy of the liquid crystal). If the interaction is screened
by counterions, then KNP is somewhat reduced, but it is still substantial as long as
the Debye screening length is greater than the nanoparticle radius. Hence, orientational order of the liquid-crystal molecules tends to favor orientational order of the
nanoparticles, and vice versa.
Whenever there is an aligning effect, there must be an entropic cost. By analogy
with the entropic term for liquid-crystal molecules, the entropic penalty for aligning
the nanoparticles is
NP
Fentropic
= kB T NNP hln %NP i
h
= kB T NNP UNP SNP
h´
ii
1
− ln −1 d(cos θ) exp(UNP P2 (cos θ)) .
(4.15)
54
The total free energy for liquid-crystal molecules and nanoparticles is now the
combination of Eqs. (4.6), (4.13), and (4.15),
F
NLC kB T
= −
J
νKNP
2
SLC
−
SLC SNP
3kB T
kB T
+ULC SLC + νUNP SNP
h´
i
1
i
h´
1
−ν ln −1 d(cos θ) exp(UNP P2 (cos θ)) .
(4.16)
Note that we have normalized this free energy by the number of liquid-crystal molecules,
not by the number of nanoparticles. For that reason, all the nanoparticle terms in
Eq. (4.16) contain a factor of ν = NNP /NLC , the ratio of the number of nanoparticles
to the total number of liquid-crystal molecules.
To summarize, we have derived the free energy for the system of ferroelectric
nanoparticles suspended in a liquid crystal. The first term represents the aligning
energy favoring orientational order of the liquid crystal, while the second term describes the mutual aligning interaction between nanoparticle order and liquid-crystal
order. The last terms are entropic terms that give the free-energy penalty for any
liquid-crystal or nanoparticle order. The free energy is a function of two variational
parameters, ULC and UNP , and we formulate our problem as minimization over those
quantities. Once we find them, we can calculate the order parameters SLC and SNP
using Eqs. (4.3) and (4.12).
55
4.3
Transition Temperature
Experiments show a substantial increase in the isotropic-nematic transition tem-
perature for liquid crystals doped with ferroelectric nanoparticles. In order to understand this phenomenon and predict how to enhance it further, we investigate the
isotropic-nematic transition using the free energy of Eq. (4.16).
Two distinct limiting cases of this transition are possible. If the nanoparticle order
is small, then all of the integrals in Eq. (4.16) can be expanded in Taylor series for
small ULC and UNP . The expressions for the order parameters SLC and SNP from
Eqs. (4.3) and Eq. (4.12) can also be expanded in power series in ULC and UNP .
Hence, the free energy can be expressed as a series in ULC and UNP , or equivalently
as a series in SLC and SNP . After some algebraic transformations, we obtain
F
NLC kB T
µ
= const +
−
5
J
−
2 3kB T
¶
5 2
2
SLC
+ νSNP
2
νKNP
SNP SLC + . . . .
kB T
(4.17)
This expression is exactly the Landau free energy as a series in the order parameters,
as discussed in chapter 3. To find the isotropic-nematic transition, we first minimize
over SNP to obtain
SNP =
KNP
SLC .
5kB T
(4.18)
We then substitute this value into the free energy series to obtain
F
= const +
NLC kB T
µ
2
5
J
νKNP
−
−
2 3kB T
10(kB T )2
¶
2
+ ....
SLC
(4.19)
56
2
The change in the coefficient of SLC
shows that the isotropic-nematic transition tem-
perature is shifted upward by
∆TNI
2
νKNP
=
25kB2 TNI
(4.20a)
In the notation of the previous paper, this shift can be written as
∆TNI
πφNP R3
=
3TNI ρLC
µ
2∆εP 2
675kB ε0 ε2
¶2
,
(4.20b)
where ρLC is the number of liquid-crystal molecules per unit volume and φNP =
4
πR3 ρLC ν
3
is the volume fraction of nanoparticles.
Note that the power-series approximation works well as long as the energetic
parameter KNP is small compared with 5kB T . In that case the nanoparticle order
parameter SNP is small compared with SLC , which is approximately 0.429 just below
the isotropic-nematic transition. However, the approximation breaks down if KNP
becomes large compared with 5kB T , so that SNP is large compared with SLC . In the
latter case, the prediction for SNP would be greater than 0.429 on the nematic side
of the transition. It might even be greater than 1, which would be unphysical. This
unphysical prediction arises because the power-series expansion cannot take account
of the saturation of the order parameters at low temperatures. Hence, for large KNP
we must consider a different limiting case.
In the limit of large KNP , the nanoparticle order is large; i.e. the variational
parameter UNP approaches infinity and the order parameter SNP approaches 1. In
57
that case we can approximate Eq. (4.12) to obtain
SNP = 1 −
1
.
UNP
(4.21)
We can then put this approximation into the free energy of Eq. (4.16), expand the
nanoparticle entropic integral for large UNP , and minimize the resulting free energy
over UNP . This calculation gives
KNP
SLC ,
kB T
kB T
.
= 1−
KNP SLC
UNP =
(4.22a)
SNP
(4.22b)
Note that this calculation is self-consistent, showing large nanoparticle order when
KNP À kB T . Using Eqs. (4.22), we obtain the approximate free energy of the nematic
phase
F
NLC kB T
J
S 2 + ULC SLC
3kB T LC
h´
i
1
µ
¶
νKNP
3KNP SLC
−
SLC + ν ln
.
kB T
2kB T
= −
(4.23)
This free energy is equivalent to the classical Maier-Saupe free energy of Eq. (4.6),
except for the last two terms, which represent the energy and entropy of well-ordered
nanoparticles interacting with the liquid crystal. These terms are proportional to
the nanoparticle concentration ν = NNP /NLC , which is small. These terms shift the
nematic free energy, and hence shift the isotropic-nematic transition temperature. To
find the value of the shift, we must minimize the free energy.
58
To minimize the free energy, we use perturbation theory. For this calculation, we
define the parameters
0
ULC = ULC
+ ∆ULC ,
(4.24a)
0
+ ∆TNI ,
TNI = TNI
(4.24b)
0
0
and TNI
are the known results from the classical Maier-Saupe free energy,
where ULC
given in Eqs. (4.7) and (4.8), and ∆ULC and ∆TNI are perturbations due to the
addition of ferroelectric nanoparticles. For low nanoparticle concentrations, these
perturbations should both be of order ν. We now expand the free energy to lowest
order in these pertubations, minimize over ∆ULC , and solve for ∆TNI such that the
isotropic and nematic free energies are equal. The resulting shift in the transition
temperature is
∆TNI = 1.03
νKNP
φNP ∆εP 2
= 1.03
.
kB
135kB ρLC ε0 ε2
(4.25)
Comparing Eqs. (4.20) and (4.25), we can see that there are two regimes for
the shift in the transition temperature. For small interaction KNP (i.e. the Landau
2
regime), the shift ∆TNI increases as KNP
, but for large KNP , it increases more slowly
as KNP . In both cases it is proportional to the nanoparticle concentration ν. Equivalently, if we work at fixed nanoparticle volume fraction φNP , our theory predicts that
∆TNI will increase with the nanoparticle material polarization P 4 and radius R3 in
the weak-interaction regime, but it will only increase as P 2 and will be independent
of R in the strong-interaction regime. (It will be independent of R as long as the
particles are small enough so that they do not distort the liquid-crystal alignment.)
Our predictions for ∆TNI can be compared with the previous predictions of Li
59
et al. [65]. They calculated that ∆TNI should increase as the volume fraction φNP
and as the polarization P 2 , and should be independent of the radius R. These predictions for the scaling agree with our predictions for the strong-interaction regime
(although not for the weak-interaction regime). We believe that this agreement is
just a coincidence, because the theories are quite different. One way to see the difference is through the dependence on dielectric anisotropy ∆ε: Li et al. predict that
∆TNI should scale as (∆ε)2 , but we calculate that it should scale linearly with ∆ε
in the strong-interaction regime. This difference arises because the Li et al. model
considers one liquid-crystal molecule interacting through the dielectric anisotropy ∆ε
with one nanoparticle, which then interacts through ∆ε with another liquid-crystal
molecule, thus giving an effective liquid-crystal interaction proportional to (∆ε)2 . By
comparison, in the strong-interaction regime our model considers the direct influence
of well-ordered nanoparticles on the liquid crystal, and hence has only one power of
∆ε.
For a numerical estimate, we use typical experimental values of the parameters
φNP = 0.5%, P = 0.26 Cm−2 , R = 35 nm, ρLC = 2.4 × 1027 m−3 , kB = 1.38 × 10−23
JK−1 , ²0 = 8.85 × 10−12 C2 N−1 m−2 , and ∆² ≈ ² ≈ 10. Those parameters imply
ν = 1 × 10−8 , KNP = 1 × 10−15 J, and hence KNP /(kB T ) = 2 × 105 , so the system is
definitely in the strong-interaction regime. Our prediction for the shift in transition
temperature is then
∆TNI ≈ 1 K.
(4.26)
This value is consistent with the order of magnitude that is observed in experiments.
Note that in this prediction we are using the bulk polarization of the ferroelectric
material BaTiO3 , which is P = 0.26 Cm−2 . In this respect, our current estimate
60
is different from chapter 3, where we assumed P = 0.04 Cm−2 because of an understanding that the bulk polarization is reduced by surface effects in nanoparticles. The
issue of estimating the polarization of nanoparticles is subtle, as discussed in Ref. [67].
As a final point about the phase diagram, we should mention that the model defined by the free energy (4.16) can exhibit one additional phase, between isotropic and
nematic, which occurs if the parameter KNP is sufficiently large. In this intermediate
phase, the nanoparticles have substantial orientational order (with SNP comparable to
the Maier-Saupe order parameter of 0.429), but the liquid crystal has only very slight
orientational order (with SLC of order νKNP /(kB T )). For that reason, we might call
it a “semi-nematic” phase. It is a perturbation on the pure liquid crystal’s isotropic
phase, not on the nematic phase. The semi-nematic phase is probably an artifact of
the mean-field theory used here. It can only exist because the very slight order of the
liquid crystal mediates an aligning interaction between the nanoparticles. This slight
orientational order is unlikely to persist when one includes fluctuations in the liquid
crystal.
4.4
Kerr effect
Apart from the phase diagram, another important issue is the response of a liquid
crystal to an applied electric field. In the isotropic phase, an applied field E induces
orientational order proportional to E 2 , known as the Kerr effect. In most pure liquid
crystals, the Kerr effect is quite small, and can only be observed for very large fields.
However, in chapter 3, we predicted that ferroelectric nanoparticles can enhance the
Kerr effect by several orders of magnitude. We would like to assess how this prediction
is modified by the Maier-Saupe theory presented here.
In the presence of an electric field, ferroelectric nanoparticles will have polar order
61
along the field; i.e. the orientational distribution function will no longer have a symmetry between the directions +n and −n. Hence, we must change the nanoparticle
distribution of Eq. (4.11) to
NP P
eU1
%NP (θ) = ´ 1
−1
NP
1 (cos θ)+U2 P2 (cos θ)
NP P (cos θ)+U NP P (cos θ)
1
2
2
d(cos θ)eU1
.
(4.27)
Here, U1NP and U2NP are two variational parameters, which act as effective fields on
the polar and nematic order of the nanoparticle distribution function, as described
by the Legendre polynomials P1 (cos θ)) and P2 (cos θ)), respectively. They generate
polar and nematic order parameters, defined as
´1
MNP =
−1
´1
SNP =
−1
NP
NP
d(cos θ)P1 (cos θ)eU1 P1 (cos θ)+U2 P2 (cos θ)
,
´1
U1NP P1 (cos θ)+U2NP P2 (cos θ)
d(cos
θ)e
−1
(4.28a)
U1NP P1 (cos θ)+U2NP P2 (cos θ)
d(cos θ)P2 (cos θ)e
´1
NP
NP
d(cos θ)eU1 P1 (cos θ)+U2 P2 (cos θ)
−1
.
(4.28b)
We still assume that the liquid-crystal distribution function is purely nematic, not
polar, as given by Eq. (4.2).
The applied electric field E adds two contributions to the energy of the system,
field
=−
Fenergetic
ε0 ∆ε 2
E SLC NLC − pEMNP NNP .
3ρLC
(4.29)
Here, the first term is the interaction of the field with the dielectric anisotropy of
the liquid crystal, and the second term is the interaction with the dipole moments
of the nanoparticles. With these energetic terms, together with the entropy of the
62
distribution function, the free energy becomes
F
J
νKNP
2
=−
SLC
−
SLC SNP
NLC kB T
3kB T
kB T
ε0 ∆εE 2
νpE
SLC −
MNP
−
3kB T ρLC
kB T
+ULC SLC + νU1NP MNP + νU2NP SNP
i
h´
1
− ln −1 d(cos θ)eULC P2 (cos θ)
i
h´
NP
NP
1
−ν ln −1 d(cos θ)eU1 P1 (cos θ)+U2 P2 (cos θ) .
(4.30)
The next step is to minimize this free energy over all three variational parameters
ULC , U1NP , and U2NP . For this minimization, there are four distinct regimes of electric
field, as indicated in Fig. 4.2.
(a) If the field is sufficiently small, E . kB T /p, it induces only slight order in the
liquid-crystal and nanoparticle distributions. In that case, we can expand the free
energy as a power series in all the variational parameters. This expansion is exactly
the Landau theory presented in chapter 3. We can then minimize the free energy over
all the variational parameters to obtain
SLC
E2
=
∗
15kB (T − Tdoped
)
µ
ε0 ∆ε νKNP p2
+
ρLC
5(kB T )2
¶
,
(4.31)
∗
where Tdoped
= T ∗ +∆TNI is the limit of supercooling of the nanoparticle-doped liquid
crystal, combining Eqs. (4.9) and (4.20). In this expression, the first term is the
conventional Kerr effect without nanoparticles, and the second term is an additional
contribution due to the aligning effect of the nanoparticles. Note that both terms
are proportional to E 2 . With the numerical estimates presented above, the second
63
term is several orders of magnitude larger than the first, and hence the nanoparticles
greatly increase the Kerr effect in this regime.
Saturated LC order
Slc
1
0.01
SLC
Particles align LC
10-4
10-6
10-8
10
1000
105
107
109
E
1011
Electric field (V/m)
FIG. 4.2. Four regimes of the Kerr effect, derived from a numerical minimization of
Eq. (4.30) with typical experimental parameters. A log-log scale is used to show all
the regimes on a single plot.
(b) For larger field, in the regime kB T /p . E . [νKNP ρLC /(ε0 ∆ε)]1/2 , the
nanoparticle order parameters MNP and SNP saturate near the maximum value of
1. In that case, we can no longer expand the free energy as a power series in the
nanoparticle parameter, but we can still expand it in the liquid-crystal parameter.
Minimizing the free energy then gives
SLC
1
=
15kB (T − T ∗ )
µ
¶
ε0 ∆εE 2
+ 3νKNP .
ρLC
(4.32)
Once again, the first term is the conventional Kerr effect without nanoparticles, and
the second term is the additional contribution from the nanoparticles, but now the
second term is independent of electric field. The second term is still much larger than
64
the first, and hence the Kerr effect is approximately constant with respect to field in
this regime.
(c) For even larger field, [νKNP ρLC /(ε0 ∆ε)]1/2 . E . [kB (T − T ∗ )ρLC /(ε0 ∆ε)]1/2 ,
the order parameter SLC is still given by Eq. (4.32), but now the first term becomes
larger than the second. In this regime, SLC again increases as E 2 . It is similar to the
conventional liquid-crystal Kerr effect, but with an extra constant contribution from
the nanoparticles.
(d) For the largest field, [kB (T − T ∗ )ρLC /(ε0 ∆ε)]1/2 . E, the order parameter SLC
saturates at the maximum value of 1.
To get a full picture of the behavior through all these regimes, we minimize the
free energy of Eq. (4.30) numerically, using the typical experimental parameters listed
at the end of Sec. 4.3. The results of this calculation are shown by the black line in
Fig. 4.2. By comparison, the red line shows the limiting case of regime (a), and the
green line shows the approximation for regimes (b-d). We see that the numerical
solution overlaps the limiting cases and connects them.
Note that the low-field regime (a) is the regime where Landau theory is valid,
and it is where the nanoparticles give the greatest enhancement of the conventional
Kerr effect. However, this regime will be difficult to observe in experiments, because
the induced order parameter SLC is so small, on the order of 10−4 . Typical optical
experiments can only detect a birefringence corresponding to SLC on the order of
10−2 , which does not occur until regime (c), which is closer to the conventional Kerr
effect.
65
4.5
Conclusions
In chapter 3, we developed a Landau theory for the statistical mechanics of fer-
roelectric nanoparticles suspended in liquid crystals. This theory differs from other
models by considering the orientational distribution function of the nanoparticles as
well as the liquid crystal. It shows a coupling between the nanoparticle order and
the liquid crystal order, which leads to an increase in the isotropic-nematic transition temperature and in the Kerr effect. In chapter 4, we consider the same physical
concept, but we improve the mathematical treatment by using a Maier-Saupe-type
theory. This theory reduces to the previous Landau theory in the limit of weak interactions (for the isotropic-nematic transition) or weak electric fields (for the Kerr
effect). However, it changes the results in the opposite limit, when the order parameters begin to saturate. For that reason, the new theory should make more accurate
predictions for experiments.
In general, the concept of coupled orientational distribution functions should be
useful for many other systems beside ferroelectric nanoparticles in liquid crystals.
For example, it applies to any type of nonspherical colloidal particles, such as carbon nanotubes, in a liquid-crystal solvent. It also applies to two distinct species of
nonspherical colloids suspended in an isotropic solvent, which could have a coupled
ordering transition. Such systems would provide further opportunities to investigate
the theory presented here.
4.6
Acknowledgments
We would like to thank Y. Reznikov and J. L. West for many helpful discussions.
This work was supported by NSF Grant DMR-0605889.
CHAPTER 5
Jamming in Granular Polymers
5.1
Introduction
Jamming, or the development of a resistance to shear, is a phenomenon that occurs
when a disordered assembly of particles subjected to increasing density, load or other
perturbations exhibits a transition from a liquid-like state that can flow to a rigid state
that acts like a solid under compression. Tremendous recent growth in this field has
been driven by the prospect that jamming may be associated with universal properties
across a wide class of systems including granular media, foams, emulsions, colloids,
and glass forming materials [73]. One of the most accessible routes for exploring the
jamming transition is gradually increasing the density of a sample in the absence of
shear or temperature. Here, jamming occurs at a density termed “Point J.” [74].
Jamming transitions have been studied both for noncohesive granular media and
for cohesive and/or nonspherical granular materials [75–78]. There is considerable
evidence that for frictionless disordered disk assemblies, critical behavior occurs near
Point J, with both the pressure and the particle coordination number Z exhibiting
power law behavior as a function of packing density φ [74, 79–81]. Similar behavior
appears when the shear, external forcing, or temperature are finite, providing further
evidence that the jamming transition may indeed exhibit universal properties [82–88].
If such universal behavior holds for other systems that undergo jamming, it would
have profound implications for the understanding and control of disordered and glassy
systems.
66
67
The most widely studied two-dimensional (2D) jamming system contains bidisperse frictionless disks. When two sizes of disks with a radius ratio of 1 : 1.4 are
mixed in a 50:50 ratio, a jamming transition occurs at a density of φ = 0.84 [74, 81–
84, 87, 88]. To explore whether the jamming transition is universal in nature, it would
be ideal to have a system in which the jamming density φc could be tuned easily. Here
we propose that one model system which meets this criterion is an assembly of 2D
granular polymers. This model is motivated by experiments on granular polymers
or chains of the type used for lamp pulls, where various aspects of knot formation,
diffusion processes, and pattern formation have been explored [89–91]. We model the
chains as coupled harmonically repulsive disks similar to those studied in the polydisperse disk system, with a constraint on the minimum angle that can be spanned
by a string of three disks. Other workers have considered freely-jointed chains [92]
or chains of sticky spheres [93]. Although our model is 2D and neglects friction,
we show that it captures the same features found in recent three-dimensional (3D)
granular polymer compaction experiments [94, 95]. To study the jamming transition,
we construct pressure versus density curves by compressing the chains between two
walls, and compare our results to compression experiments on 2D polydisperse disks
[81, 89]. We show that the jamming density φc decreases with increasing chain length
and saturates at long chain lengths, in agreement with the experiments of Ref. [94].
The decrease of φc occurs due to the formation of rigid loops along the chains which
stabilize voids inside the packing. Unlike the bidisperse disk system where the pressure scales linearly with φ, in the chain system the pressure increases with a power
law or stretched exponential form as the jamming transition is approached. Jamming
of our frictionless granular chains shares several features with jamming of frictional
68
disks and could be distinct from the jamming transition for frictionless disks.
(b)
(c)
(a)
(d)
FIG. 5.1. Granular configurations in a portion of the sample. The x direction runs
vertically and the fixed wall is at the top of each panel. (a) An unjammed system of
N = 67 granular polymers with length Mb = 16. (b) The jammed configuration for
the same system contains voids which appear when the chains form ring structures.
(c) The jammed configuration for N = 67 loops of length Mb = 16. The jamming
density is lower than for systems of chains or individual disks. (d) The jammed
configuration at φ = 0.84 for a sample of N = 1500 bidisperse frictionless disks
contains no significant voids.
5.2
Simulation Model
We simulate a 2D system confined by two walls at x = 0 and x = L and with
periodic boundaries in the y direction. The wall at x = L is held fixed while the
position of the other wall is allowed to vary in order to change the density. The system
contains N chains or loops, each of which is composed of Mb individual disks that are
strung together by harmonic springs and that experience a constraining force which
69
limits the bending radius of the chain. In loops, the two ends of a chain are connected
together. The disk-disk interaction is modeled as a stiff harmonic repulsion, and the
motion of all disks is taken to be overdamped in order to represent the frictional force
between the disks and the underlying floor. A given disk i moves according to the
following equation of motion:
η
dRi
= Fidd + Fic + Ficc + Fiw .
dt
(5.1)
Here we take the damping constant η = 1. The disk-disk interaction potential is
P Mb
Fidd = N
j6=i kg (reff − rij )Θ(reff − rij )r̂ij where the spring constant kg = 300, rij =
|Ri − Rj |, r̂ij = (Ri − Rj )/Rij , Θ is the Heaviside step function, and reff = ri + rj ,
where ri(j) is the radius of disk i(j). For the chains and loops we set ri = 1; for a
bidisperse disk system we set ri = 1 for half of the disks and ri = 1.4 for the other
P
half of the disks. The chain interaction potential is Fic = k kg (reff − rik )r̂ik , and it
acts only between a disk and its immediate neighbors along the loop or chain. The
P
bending constraint potential Ficc = l kg (rstiff − ril )Θ(rstiff − ril )r̂il acts between disks
separated by one chain element, with rstiff = 2reff sin(θs /2) and θs = 0.82π unless
otherwise noted. Smaller values of θs produce more bendable chains. The disk-wall
interaction force Fiw is computed by placing a virtual disk at a position reflected across
the wall from the actual disk, and finding the resulting disk-disk force. To initialize
the system, we place the chains, rings, or individual disks in random non-overlapping
positions to form a low density unjammed phase, such as in Fig. 5.1(a). The x = L
wall is held fixed while the other wall is gradually moved from x = 0 to larger x in
small increments of δx. The waiting time between increments is taken long enough
70
so that the system has sufficient time to relax to a state where the velocities of all
particles are indistinguishable from zero.
We identify the jamming transition by measuring the total force exerted on the
P Mb i
fixed wall by the packing, P = N
Fw · x̂, and the average contact number Z =
i
P Mb
zi as a function of the total density of the system defined by the
(N Mb )−1 N
i
spacing between the two walls. To determine the contact number zi of an individual
grain in a chain, we first count the immediate neighbors of the grain along the chain,
and then add any other grains that are in contact with the individual grain. The force
P is proportional to a component of the pressure tensor. At the jamming transition,
the pressure in the system becomes finite [74, 79, 81], while below jamming P = 0.
5.3
Classical Bidisperse System
Previous simulations on 2D disordered disk packings have revealed the onset of a
finite pressure near φc = 0.84 which grows as P ∝ (φ − φc )ψ with ψ = αf − 1, where
αf is the exponent of the interparticle interaction potential [74, 79]. Theoretical work
on the jamming of 2D disks also predicts a power law scaling of the pressure versus
density [80], and indicates that the contact number Z should scale as Z ∝ (φ − φc )β ,
with β = 0.5. Experiments using a combination of shear and compression on the same
disk system found that after performing cycling to reduce the effect of friction, the
pressure and Z both scale with the density as power laws with ψ = 1.1 and β = 0.495
[81].
We first test our simulation geometry using the bidisperse individual disk system.
A configuration of N = 1500 disks appears in Fig. 5.1(d) just above the onset of a
finite pressure P at φ ≈ 0.84. In Fig. 5.2(a) we show that for the disk system above
jamming, P increases linearly with φ, consistent with a scaling exponent ψ = 1.0.
71
100
500
(a)
60
300
(b)
P
400
P
80
40
200
20
100
0
0.82
0.83
0.84
φ
0
0.85
0.65
0.7
φ
0.75
0.8
FIG. 5.2. (a) The pressure P vs φ for a bidisperse disk system. Above the jamming
transition at φ ≈ 0.84, P increases linearly with φ. (b) P vs φ for the granular
polymer system with chains of length Mb = 6 (H), 8 (4), 10 (¨), 16 (¤), and 24 (•).
As Mb increases, the onset of a finite value of P indicating jamming drops to lower
values of φ and the linear scaling of P with φ is lost.
This indicates that our compressional geometry captures the jamming behavior found
in other studies of bidisperse disks.
5.4
Granular Polymers
We use the same compression protocol to study the jamming behavior of granular
polymers, as illustrated in Fig. 5.1(a,b) for a system with N = 67 chains that are each
of length Mb = 16. In Fig. 5.2(b) we plot P versus φ for granular polymer chains of
lengths Mb = 6, 8, 10, 16, and 24. For the chains, the onset of finite P indicating the
100
(a)
(Z-Zc) Mb
6
P Mb
2.9
10
(b)
10
3
10
1
0
10
0.001
0.01
0.1
φ-φc
1
0.001
0.01
φ-φc
0.1
1
FIG. 5.3. (a) Scaling of pressure vs density, P Mb2.9 vs φ − φc close to the jamming
transition for chains of length Mb = 6 (H), 8 (4), 10 (¨), 16 (¤), and 24 (•). (b)
Scaling of (Z − Zc )Mb vs φ − φc for chains of length Mb = 6, 8, 10, 16, and 24, with
the same symbols as in panel (a).
72
beginning of jammed behavior occurs at a much lower density than for the bidisperse
disk system shown in Fig. 5.2(a), and as Mb increases, the jamming transition shifts
to even lower φ and the linear dependence of P on φ is lost. We illustrate scaling of
P near the jamming transition in Fig. 5.3(a) where we plot P Mb2.9 vs φ − φc . Here we
find ψ ≈ 3. The jammed state develops isotropic rigidity, as indicated by the plot of
the bulk pressure tensor components pxx and pyy in Fig. 5.4(b). To test whether the
packing also develops a finite response to shear at the jamming transition, we fix the
packing density and apply a shear to the system by applying a force Fshear = 0.04ŷ
to all particles that are in contact with the mobile wall. We measure the resulting
velocity Vshear = dRi /dt · ŷ of all particles that are in contact with the stationary wall
on the other side of the sample. In Fig. 5.5 we plot Vshear and P versus φ for samples
with chains of length Mb = 6, 8, 10, 16, and 24. In each case, a finite shear response
and a finite pressure P appear simultaneously at the jamming density. We define
the jamming threshold φc as the density at which P rises to a finite level. The same
threshold also appears as the sudden onset of an increase in Z, as seen in Fig. 5.4(a).
The scaling of (Z − Zc )Mb vs φ − φc appears in Fig. 5.3(b), where the exponent β
falls in the range β = 0.6 to 0.8.
4
(a)
(b)
200
Z
pxx, pyy
3.5
100
3
2.5
0.6
0.7
φ
0.8
0
0.7
φ
0.8
FIG. 5.4. (a) Contact number Z vs φ for chains of length Mb = 6 (H), 8 (4), 10 (¨),
16 (¤), and 24 (•). (b) Bulk pressure tensors pxx (°) and pyy (¨) vs φ for Mb = 16.
73
3
0.5
0.4
2
P
Vshear
0.3
0.2
1
0.1
0
0.55
0.6
0.65
φ
0.7
0.75
0
0.8
FIG. 5.5. The shear velocity Vshear of grains adjacent to the stationary wall vs φ
(black symbols) and the corresponding pressure P in the packing vs φ (red symbols)
for a sample in which a shear force is applied using the mobile wall with chain lengths
Mb = 6 (H), 8 (4), 10 (¨), 16 (¤), and 24 (•). The onset of a finite pressure and a
finite shear response occur at the same value of φ for each sample.
5.4.1
Effect of length and shape
In Fig. 5.6 we plot φc versus Mb for chains of two different stiffnesses: θs = 0.82π
and θs = 0.756π. In both cases, φc decreases monotonically with increasing Mb and
saturates for large Mb . Recent experiments on the packing of granular polymers
showed the same behavior: the final packing density decreased for increasing chain
length and saturated for very long chains [94]. This was attributed to the formation
of rigid semiloops which stabilized voids in the packing and decreased the jamming
density. Loops have also been observed in dense 3D packings of freely-jointed chains
[96]. Since the minimum area spanned by a semiloop increases with θs , the jamming
density should be lower for larger θs when larger voids are stabilized. Figure 5.1(b)
illustrates the voids that appear in our chain packings due to the formation of rigid
74
0.65
0.6
0π 0.4π 0.8π
θc
0.75
φc
0.8
0.7
0.6
φc
φc
0.8
0.55
0.65
0.6
0.5
(a)
10
20
30
Chain length Mb
40
0.45
0
(b)
0.2
0.4
0.6
0.8
1
Chain fraction Nc/N
FIG. 5.6. (a) The jamming threshold φc versus chain length Mb for chains with a
bending angle of θs = 0.82π (circles) and θs = 0.756π (squares). φc decreases with
increasing Mb and saturates at large Mb . Inset: φc vs θs for a system with Mb = 16.
(b) φc vs chain fraction Nc /N for a system with Mb = 16 and a mixture of loops and
chains. As the fraction of chains decreases, φc decreases.
semiloops. For comparison, the bidisperse disk system shown in Fig. 5.1(d) contains
no large voids. If the rattler disks in Fig. 5.1(d) were removed, the amount of void
space present would increase, but the chain system would still be able to stabilize a
larger amount of void space since the constraint of the chain backbone permits the
formation of larger arches around the voids than the arches that can be stabilized in
the bidisperse disk system. The semiloops increase in size for increasing θs and φc is
reduced at higher θs , as shown in Fig. 5.6(a). The plot of φc versus θs in the inset of
Fig. 5.6(a) for fixed Mb = 16 shows that φc monotonically decreases with increasing
θs . For perfectly flexible chains with θs = 0, we find φc ≈ 0.8. This is lower than
the density of a triangular lattice due to the trapping of voids within the packing by
the physical constraints of the bonding between chain elements. If the system were
annealed or shaken for a sufficiently long time, these voids could eventually be freed
and the perfectly flexible chains would form a perfect triangular lattice.
As a confirmation of the idea that the formation of rigid semiloops is the mechanism by which the jamming density is depressed, Ref. [94] includes experiments
75
performed on mixtures of granular polymers and granular loops with equal length
Mb . In this case, φc decreased linearly as the fraction of loops increased. We find the
same effect in our 2D system by varying the number of loops Nl and chains Nc in a
sample with fixed N = Nl + Nc and fixed Mb . In Fig. 5.6(b) we plot φc versus Nc /N ,
where Nc /N = 1 indicates a sample containing only chains and Nc /N = 0 is a sample
containing only loops. As Nc /N decreases, φc decreases. The jammed configuration
for a sample with Mb = 16 containing only loops, Nc /N = 0, appears in Fig. 5.1(c).
The number of voids present is much larger than in the Nc /N = 1 sample shown in
Fig. 5.1(b). Interestingly, the voids began to form a disordered triangular packing.
Our results indicate that the experimentally observed dependence of the jamming
density on chain length or fraction of loops in Ref. [94] is not caused by friction or other
possible spurious effects, but is instead a product of the geometrical configuration of
the chains and loops. The surprisingly good agreement between our 2D simulations
and the 3D experiments may be due to the fact that in each case, the semiloops
formed by the chains are 2D in nature. Additionally, in the experiment the container
used to hold the sample induced ordering of the chains and loops near the walls and
may have caused the system to act more two-dimensional. The fact that much of
the physics observed for the 3D system can be captured in 2D models means that
2D experiments, which are much easier to image than 3D experiments, could provide
many of the same insights for understanding jamming in a system where the jamming
density can be tuned easily.
For a packing of bidisperse harmonic disks, the pressure increases linearly or equivalently as a power law with ψ = 1 for increasing φ, as shown in Fig. 5.2(a) and found
in earlier works [74, 79, 81]. In our granular chain system, the interactions between
76
1000
400
100
P
P
600
P
800
200
400
0.7
10
φ
0.8
200
1
0
(a)
0.05
0.1
0.15
φ - 0.6
0.2
0.25
0
0.5
(b)
0.6
φ
0.7
0.8
FIG. 5.7. (a) P vs φ for a chain sample with Mb = 20 on a log-linear scale. Light
line: initial compression; heavy line: steady state reached after four compressions;
dashed line: an exponential fit P ∝ eφ−0.6 for φ > φc . (b) P vs φ for a system with
Mb = 38 for the initial compression (light line) and for the fourth compression (heavy
line). The response is hysteretic but the curves remain nonlinear on all compressions.
Inset: A portion of the P vs φ curve during the second compression cycle in the same
sample showing a sudden pressure drop associated with the collapse of an unstable
void.
the disks composing the chains are also harmonic, so it might be expected that the
pressure would also increase linearly with increasing φ. Instead, Fig. 5.2(b) shows
that the P vs φ curves for granular chains are highly nonlinear and can be fit to the
exponential form P ∝ eφ−0.6 for φ > φc , as illustrated in Fig. 5.7(a). The range of
the exponential behavior increases for increasing Mb .
The chain system exhibits a pronounced hysteresis effect that can seen by cycling
the mobile wall in and out to repeatedly compress and uncompress the packing.
Figs. 5.7(a) and (b) show a comparison of the responses during the first compression
and during the fourth compression. After four compressions the system does not
exhibit further hysteresis. In contrast, we find little or no hysteresis for the bidisperse
disk system. During the initial compression, the chain systems often exhibit sizable
fluctuations in P above the onset of jamming. Sudden drops in the pressure, such
as that shown in the inset of Fig. 5.7(b), occur due to the collapse of semiloops that
77
are larger than the minimum stable size. After all semiloops have reached a stable
size, we find no further hysteresis. Even after cycling to a steady state, the P vs
φ curves remain power law or stretched exponential in nature and do not become
linear. Simulations of compressed 2D frictional bidisperse disk systems show that
void structures can form during the initial compression but collapse during subsequent
cycles, allowing the sample to reach the same density as a frictionless disk sample
[97]. In the chain system, the void structures are associated with semiloops that have
formed in the chains and, unlike in the disk system, the voids can never be fully
collapsed by repeated cycling.
The fact that the granular polymers do not exhibit the same behavior at the
jamming transition as the bidisperse disk systems do at Point J provides additional
evidence that jamming does not occur with universal features in all systems, and that
the criticality found in the bidisperse disk systems may be associated with a special
type of jamming. Additional studies on a variety of different types of systems would
need to be performed to confirm whether the jamming behavior is indeed different
for each system or whether there is a small number of different classes of jamming
behaviors, with the granular polymer system and the bidisperse disk system falling
into separate classes.
5.5
Conclusions
In summary, we have introduced a numerical model of 2D granular polymers
that can be used to study the jamming transition. The onset of jamming occurs at
a density that decreases with increasing chain length and saturates for long chain
lengths. The decrease of the jamming density results from the formation of rigid
semiloops in the granular chains which permit stable voids to exist in the packing, in
78
excellent agreement with recent 3D experiments on granular polymers. For fixed chain
length, the jamming density decreases when the chains are made stiffer since the rigid
semiloops, and the voids stabilized by them, are larger. The jamming density can
also be further decreased by increasing the fraction of granular loops present in the
packing, which is also in agreement with experimental observations. The fact that our
2D simulations agree so well with the 3D experiments of Ref. [94] indicates that the
formation of semiloops in the chains is essentially a 2D phenomenon. In comparison
to bidisperse disk systems which show a linear increase in the pressure as a function of
density, characteristic of a critical phenomenon, in the granular polymer systems the
pressure increases as a power law or stretched exponential with density, suggesting
that the jamming transition in the granular chain system is different in nature from
jamming in the bidisperse disks and may be related to the type of jamming that
occurs for frictional grains.
5.6
Acknowledgments
We thank R. Ecke and R. Behringer for useful comments.
This work was carried out under the auspices of the NNSA of the U.S. DoE at
LANL under Contract No. DE-AC52-06NA25396.
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