EMERGENT THERMODYNAMICS IN A SYSTEM OF MACROSCOPIC, CHAOTIC
SURFACE WAVES
by
KYLE J. WELCH
A DISSERTATION
Presented to the Department of Physics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
September 2016
DISSERTATION APPROVAL PAGE
Student: Kyle J. Welch
Title: Emergent Thermodynamics in a System of Macroscopic, Chaotic Surface Waves
This dissertation has been accepted and approved in partial fulfillment of the requirements for the
Doctor of Philosophy degree in the Department of Physics by:
Raghuveer Parthasarathy Chair
Eric Corwin Advisor
Benjamin McMorran Core Member
Michael Kellman Institutional Representative
and
Scott L. Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate School.
Degree awarded September 2016
ii
c© 2016 Kyle J. Welch
This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs (United States) License.
iii
DISSERTATION ABSTRACT
Kyle J. Welch
Doctor of Philosophy
Department of Physics
September 2016
Title: Emergent Thermodynamics in a System of Macroscopic, Chaotic Surface Waves
The properties of conventional materials are inextricably linked with their molecular
composition; to make water flow like wine would require changing its molecular identity. To
circumvent this restriction, I have contstructed and characterized a two-dimensional metafluid,
so-called because its constitutive dynamics are derived not from atoms and molecules but from
macroscopic, chaotic surface waves excited on a vertically agitated fluid. Unlike in conventional
fluids, the viscosity and temperature of this metafluid are independantly tunable. Despite this
unconventional property, our system is surprisingly consistent with equilibrium thermodynamics,
despite being constructed from macroscopic, non-equilibrium elements. As a programmable
material, our metafluid represents a new platform on which to study complex phenomena such
as self-assembly and pattern formation. We demonstrate one such application in our study of
short-chain polymer analogs embedded in our system.
Chapter 2 has been published in Physical Review E [15]. The writing and analysis were
performed by me as primary author. Eric Corwin is listed as a coauthor as he advised this work.
The material in this chapter is also co-authored with Raghuveer Parthasarathy, for use of his
image analysis software and contributions to the early design and motivation of the experiment,
and Isaac Hastings-Hauss, and undergraduate who helped with construction of the system.
Chapter 4 has also been published in Physical Review E [16]. The writing and analysis were
performed by me as primary author. Eric Corwin is listed as a coauthor as he advised this
work. The material in this chapter is also coauthored with Clayton Kilmer, an undergraduate
who helped with data collection. At the time of this writing, chapter 3 has been accepted
to Proceedings of the National Academy of Sciences but is not yet in print. The writing and
analysis were performed by me as primary author. Eric Corwin is listed as a coauthor as he
iv
advised this work. The material in this chapter is also coauthored with Alex Liebman-Pela´ez,
an undergraduate who helped with construction of parts of the experimental system.
This dissertation includes previously published and unpublished co-authored material.
v
CURRICULUM VITAE
NAME OF AUTHOR: Kyle J. Welch
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Washington State University, Pullman, WA
DEGREES AWARDED:
Doctor of Philosophy, Physics, 2016, University of Oregon
Bachelor of Science, Physics, magna cum laude, 2011, Washington State University
Bachelor of Science, Neuroscience, magna cum laude, 2011, Washington State University
AREAS OF SPECIAL INTEREST:
Soft Condensed Matter, Non-equilibrium Physics, Biophysics
PROFESSIONAL EXPERIENCE:
Research Assistant, Physics, University of Oregon, 2012-2016
Teaching Assistant, Physics, University of Oregon, 2011-2012, 2015, 2016
Teaching Assistant, Chemistry, Washington State University, 2008-2011
GRANTS, AWARDS AND HONORS:
Weiser Senior Teaching Assistant Award, University of Oregon, 2015
Weiser First Year Teaching Assistant Award, University of Oregon, 2012
Regents Scholar, Washington State University, 2006-2011
PUBLICATIONS:
K. J. Welch, A. Liebman-Pelaez, E. I. Corwin. “Chaotic surface waves form a metafluid
that is Newtonian, thermal and entirely tunable.” Proceedings of the National Academy of
Science (accepted).
K. J. Welch, C. S. G. Kilmer, E. I. Corwin. “Atomistic study of macroscopic analogs to
short chain molecules.” Physical Review E, 91, 022603 (2015).
K. J. Welch, I. Hastings-Hauss, E. I. Corwin. “Ballistic and diffusive dynamics in a two-
dimensional ideal gas of macroscopic chaotic Faraday waves.” Physical Review E, 89,
042143 (2014).
vi
ACKNOWLEDGEMENTS
All of this work has been done under the guidance of my graduate advisor, Eric Corwin,
who has taught me a great deal about science, the academic process, and life in general. I would
also like to thank Raghu Parthasarathy for his guidance, mentoring and honest feedback on paper
manuscripts and presentations throughout my time as a graduate student. For helpful discussions,
I would like to thank John Royer and John Toner. Olivia Carey-De La Torre and Cody Hill
provided initial machine work on the large dish used for the work presented in chapters 3 and
4. This work was primarily funded by the NSF under Career Award No. DMR-1255370. ALP (co-
author on chapter 3) was supported under the Research Experience for Undergraduates program
sponsored by the UO Materials Science Institute and the NSF.
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. BALLISTIC AND DIFFUSIVE DYNAMICS IN A TWO-DIMENSIONAL IDEAL
GAS OF MACROSCOPIC CHAOTIC FARADAY WAVES . . . . . . . . . . . . . . . . 4
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
III. FLUIDS BY DESIGN: CHAOTIC SURFACE WAVES FORM A METAFLUID
THAT IS NEWTONIAN, THERMAL AND ENTIRELY TUNABLE . . . . . . . . . . . 15
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
IV. ATOMISTIC STUDY OF MACROSCOPIC ANALOGS TO SHORT CHAIN MOLECULES 26
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Data Analysis and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
viii
Chapter Page
V. DETAILED MATERIALS AND METHODS FOR CHAPTER 3 . . . . . . . . . . . . . 38
Shaker Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Rheometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Derivation of Torque Due to Magnets . . . . . . . . . . . . . . . . . . . . . . . . 39
Measurement of Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
VI. SUPPLEMENTARY MATERIALS FOR CHAPTER 4 . . . . . . . . . . . . . . . . . . . 44
Effective Potentials for Chain With α = pi/6 . . . . . . . . . . . . . . . . . . . . 44
Semiflexible Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Self avoiding walk simulation algorithm . . . . . . . . . . . . . . . . . . . . . . . 46
VII. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
ix
LIST OF FIGURES
Figure Page
1. Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2. Map of parameter space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. Ballistic and Diffusive Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4. Gas behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5. Experimental setup and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6. Tunable Newtonian Fluid Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
7. Einstein Relation and Green-Kubo Relation . . . . . . . . . . . . . . . . . . . . . . . . . 22
8. Phase Space and Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
9. Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
10. Pair Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
11. Effective Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
12. Master curves for the scaling of winding angle variance σ2φ . . . . . . . . . . . . . . . . . 34
13. Scaling Laws for End-to-End Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
14. Comparison of Hookean approximation and full form for torque exerted by drive
arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
15. Effective Potentials in Total Bond Winding Angle φi,j . . . . . . . . . . . . . . . . . . . 44
16. Effective potentials for interparticle distance ri,j+1 . . . . . . . . . . . . . . . . . . . . . 45
17. Extrapolated Effective Potentials for End-to-End distance of Long Chains . . . . . . . . 46
x
CHAPTER I
INTRODUCTION
Materials science is a lot like cooking. We do our best to understand the properties of the
ingredients, then manipulate and combine them to create new, hopefully useful (or delicious)
substances. However, in cooking we are ultimately limited by the properties of our ingredients;
no matter how hard we try we can’t make pot roast out of a watermelon. Materials science seeks
to overcome the limitation of its ingredients by not only understanding, but directly controlling
the properties of matter. To do so, however, one must surmount the nearly impossible task of
dynamically manipulating the molecular nature of the material, because like a cake, the properties
of a conventional material are ultimately defined by the properties of its ingredients. For example,
you can tell the difference between water and corn syrup simply by watching them flow: corn
syrup has a much higher viscosity than water. To make water flow like corn syrup, however, you
would need to change the mass and interaction forces of the water molecules, or, put more plainly,
you would need to change the very molecular identity of the water, thus transforming one fluid
into the other. Such alchemy is beyond the scope of modern science.
What’s more, in conventional materials, there exist strict (though often complicated)
relationships between material properties. In the case of fluids, for example, there are well-defined
laws that relate temperature and viscosity. This means that you cannot change the temperature
of a fluid without inducing a commensurate change in the viscosity, as is apparent when one
microwaves honey to make it easier to pour. This restriction represents a serious limitation toward
attaining full control of the properties of matter as it means that a given fluid can only explore
certain combinations of temperature and viscosity. Again, in order to break this limitation, one
would need to achieve independent control over the molecular properties of the system. Thus, the
challenge of finding or creating truly programmable matter remains an active and exciting area of
study.
Recent work is tackling this challenge by creating metamaterials: constructed materials
that reject the molecule as the fundamental unit and instead substitute some athermal,
macroscopic structural element, achieving novel optical [1–3], acoustic [4–6], mechanical, [7–
9] and fluid properties [10, 11] that would otherwise be impossible. Remarkable though this
1
progress is, it comes at a cost. These materials, by deriving their properties from macroscopic,
non-equilibrium or anisotropic elements, abandon the physics of equilibrium thermal systems.
This means sacrificing a large body of tools that can be used to characterize and understand their
behavior. It also leads to a loss of universality; each metamaterial is governed by different physics,
and each must be circumstantially re-characterized to establish their governing laws.
My research has instead taken a bottom-up approach to the creation of programmable
matter. Whereas the existing body of work surrounding metamaterials focuses on careful, top-
down design of constitutive structures, my work instead seeks first to create a meta-system that
obeys the barest criteria of equilibrium thermodynamics, then to study the emergent properties
of said system. In particular, I have created and characterized a metafluid whose properties
are derived from macroscopic chaotic waves excited on the surface of water, independent from
the water itself. These waves (known as chaotic Faraday waves [12, 13]) arise on the surface
of a fluid when it is vertically agitated above some critical amplitude. They provide isotropic,
“molecular” chaos (where the “molecules” here are the waves themselves) and thus satisfy the
basic requirements of kinetic theory as outlined by Boltzmann [14]. By embedding buoyant
tracers in the waves, I can measure the basic properties of the metafluid: temperature, diffusion
constant and viscous drag coefficient. These properties turn out to be identical in nature, but
completely divorced in origin from the analogous properties in conventional fluids. They are
associated with the chaos of the waves in the same way that the temperature of a gas is associated
with the random motion of its molecules, yet they differ only in scale from their conventional
counterparts. As a dramatic comparison, for example, calculations of the ambient thermal
energy associated with the chaos of the waves suggests that the metafluid has a characteristic
temperature one trillion times hotter than the sun’s surface. This may seem like a ridiculous and
specious comparison, but it is made more reasonable when you consider that the sun is largely
composed of hydrogen atoms with mass approximately 1.7 × 10−27kg, whereas the tracer particles
thermalized by the chaotic waves have a mass of a few hundred milligrams ( 10−4kg).
This system carries with it great utility. It constitutes a truly thermal environment while
also exhibiting its “molecular” dynamics at the easy-to-access laboratory scale. This means that
macroscopic analogs to complicated microscopic phenomena can be studied in atomistic fashion,
observing and building up the dynamics from the smallest subunit to the gross bulk behavior.
2
I have demonstrated one such application in my study of scaling laws in short-chain polymer
analogs thermalized by the chaotic surface waves.
In a more far-reaching sense, this research hints at a broader universality of
thermodynamics. That all that was required to reproduce equilibrium thermal physics was
isotropic chaos suggests that the tools of thermodynamics may be applied to any system that
satisfies the basic criteria of kinetic theory. This may include such systems as turbulent weather,
bird flocks or complex financial markets.
This work shows that chaotic surface waves form a metafluid that is wholly consistent with
equilibrium thermodynamics, despite being composed of macroscopic, non-equilibrium elements.
Further, the viscosity and temperature of this metafluid are independently tunable. Chapter 2
begins the characterization of this system by examining the motion of passive tracer particles
embedded in the chaotic waves, measuring temperature and diffusion constant and how they
depend on experimental parameters. Chapter 3 expands this characterization by introducing
active rheology to directly measure viscous drag and showing that it, along with temperature
and diffusion constant, obeys equilibrium thermodynamics. Chapter 3 also shows, in detail, the
dependence of temperature and viscous drag on experimental inputs, and demonstrates that they
can be tuned independently of one another. Chapter 4 then illustrates the utility of this system
as a novel platform on which complex phenomena may be studied, in this case, the structural
dynamics of short-chain polymers analogs.
Chapter 2 has been published in Physical Review E [15]. The writing and analysis were
performed by me as primary author. Eric Corwin is listed as a coauthor as he advised this work.
The material in this chapter is also co-authored with Raghuveer Parthasarathy, for use of his
image analysis software and contributions to the early design and motivation of the experiment,
and Isaac Hastings-Hauss, and undergraduate who helped with construction of the system.
Chapter 4 has also been published in Physical Review E [16]. The writing and analysis
were performed by me as primary author. Eric Corwin is listed as a coauthor as he advised this
work. The material in this chapter is also coauthored with Clayton Kilmer, an undergraduate who
helped with data collection.
At the time of this writing, chapter 3 has been accepted to Proceedings of the National
Academy of Sciences but is not yet in print. The writing and analysis were performed by me
3
as primary author. Eric Corwin is listed as a coauthor as he advised this work. The material in
this chapter is also coauthored with Alex Liebman-Pela´ez, an undergraduate who helped with
construction of parts of the experimental system.
4
CHAPTER II
BALLISTIC AND DIFFUSIVE DYNAMICS IN A TWO-DIMENSIONAL IDEAL GAS OF
MACROSCOPIC CHAOTIC FARADAY WAVES
Abstract
We have constructed a macroscopic driven system of chaotic Faraday waves whose
statistical mechanics, we find, are surprisingly simple, mimicking those of a thermal gas. We
use real-time tracking of a single floating probe, energy equipartition, and the Einstein relation
to define and measure a temperature and diffusion constant, and then self-consistently determine
a coefficient of viscous friction for a test particle in this thermal-like gas. Because of its simplicity,
this system can serve as a model for direct experimental investigation of non-equilibrium
statistical mechanics, much as the ideal gas epitomizes equilibrium statistical mechanics.
This work was originally published in Physical Review E [15]. The writing and analysis
were performed by me as primary author. Eric Corwin is listed as a coauthor as he advised
this work. The material in this chapter is also co-authored with Raghuveer Parthasarathy, for
use of his image analysis software and contributions to the early design and motivation of the
experiment, and Isaac Hastings-Hauss, and undergraduate who helped with construction of the
system.
Background
Classical kinetic theory requires molecular chaos and homogeneity [14, 17]. Atomic-
scale collisions provide the source of homogeneous random motion in thermal systems and
form the foundation of classical thermodynamics. In equilibrium thermal systems the crossover
from ballistic motion to diffusive motion is a fundamental link between microscopic statistical
mechanics and macroscopic thermodynamics [18]. In Einstein’s classic thought experiment of a
pollen grain in water a thorough study of the grain’s ballistic motion would require simultaneous
temporal resolution of ' 10µs and spatial resolution of ' 1 nm [19]. Therefore, this crossover
from ballistic to diffusive has only recently been experimentally demonstrated in the equilibrium
thermal systems of rarefied gases [20, 21] and liquids [22, 23].
5
By contrast, macroscopic systems allow studies of their constituent dynamics that are
impossible in the thermal world. Macroscopic systems dilate the characteristic length and time
scales but lose the stochastic excitations of thermal systems. This means that any of the random
motion necessary for mimicking equilibrium thermal behavior macroscopically must be produced
by some stochastic energy input. Because chaotic Faraday waves are very well understood and
characterized they present an ideal source of randomness. The Faraday instability is excited on
the surface of a fluid subject to vertical oscillations beyond some critical amplitude [12, 13, 24].
Above a second, higher, critical amplitude the surface waves transition from stable, ordered waves
to spatio-temporal chaos [25]. Surface flows have been measured using fluorescent dyes [26, 27]
and tracer particles considerably smaller than the characteristic wave size [26, 28–30]. A ballistic
to diffusive crossover has been observed in chaotic Faraday waves using “virtual tracer” data
drawn from particle image velocimetry measurements[31]. However, for real tracer measurements
to date, diffusive motion and fractional Brownian motion have been observed at long and at
relatively short time-scales, respectively [30, 32]; the full crossover from the ballistic regime to
the diffusive regime for real particles has not been demonstrated before our present work.
It remains unproven, therefore, whether a driven (and therefore non-equilibrium), athermal
system like chaotic Faraday waves still exhibit all characteristics of equilibrium statistical
mechanics, such as a ballistic-diffusive crossover and a well-defined temperature derived from
atomistic chaos, as it is in classical kinetic theory. A variety of attempts to define temperatures
have been proposed for non-equilibrium systems [33]. These have had limited success, describing
aspects of the systems’ dynamics only over narrow parameter ranges and for few measured
properties. A typical approach is to use the Einstein relation [34, 35] or fluctuation-dissipation
theorem [35] to define an effective temperature. While successful in producing a well-defined
temperature, these studies do not comment on whether or not their internally determined
quantities exhibit behavior consistent with classical kinetic theory. In the present study we achieve
random excitation and homogeneity by floating a particle large relative to the characteristic
length of the Faraday waves on a chaotic fluid surface (Figure 1). Despite the decidedly non-
equilibrium nature of chaotic Faraday waves, we show that they drive a buoyant tracer to undergo
fully ballistic (short times) and diffusive (long times) Brownian motion, a hallmark of isotropic
equilibrium statistical mechanics. This system admits of a well defined temperature, diffusion
6
constant, and drag coefficient consistent with the system being a nearly ideal gas of excitations.
We have created a macroscopic thermal-like system (which, if treated like a conventional thermal
system would be one billion times hotter than the sun’s corona).
Methods
We generate the Faraday waves in a circular aluminum dish brim-full with water (Figure
1). We use a circular dish to discourage ordering of the waves [36]. The inner portion, where the
water is held, has a radius of 9cm and a depth of 1.27cm. A gutter to collect spillover is carved
around this inner region. The dish is vertically agitated by a shaker (Vibration Test Systems
VTS-100) whose frequency f and peak-peak amplitude A we control independently using a
digital function generator (Stanford Research Systems Model DS335) and a voltage amplifier
(Behringer Europower EP4000). Mounted on the bottom of the dish is an accelerometer (CTC
AC244-2D/010) used to measure stroke amplitude and frequency. The dish is filled to brim-full
conditions [24] against a knife-edge to avoid pinning of the contact angle which would create a
“cold” zone near the edges. However, the fluid position remains pinned to the knife edge leading
to some damping of the waves near the edges. Our floating test particle is a 3D printed section of
cone (selectively laser sintered nylon, coated in black spray paint, density approximately .7g/cm3)
with top radius .75 cm, height .25 cm, side angle 45 degrees, and mass 216 mg. The size of the
particle is chosen such that it is always larger than the characteristic size of the Faraday waves
which are on the order of millimeters. The particle is colored black and the dish is spray painted
white for maximum contrast. The system is imaged from ' 1.3m above the surface of the water
with a digital camera (Pointgrey Flea3 FL3-U3-32S2M-CS with a Pentax C32500 KP lens) with a
2080x1552 CMOS sensor.
We know a priori that certain regions of our parameter space are inaccessible (Figure
2): below a critical amplitude line chaotic Faraday waves do not exist; above a sufficiently high
amplitude line the surface undergoes a topological transition and begins to splash, which typically
sinks the particle. The shape of these critical lines is non-trivially dependent on the properties of
the fluid (i.e. viscosity, capillary length, skin depth), the frequency and amplitude of oscillation
and the container aspect ratio [37]. The details of the transition from ordered to chaotic Faraday
waves have been the subject of a great deal of scientific interest [38, 39]. However for this work we
7
Electromechanical Shaker
Accel
FIGURE 1. Experimental Apparatus
(cartoon) Cross-sectional diagram of shaker assembly. The inner portion of the dish is filled with
water to brim-full conditions. The electromechanical shaker vertically agitates the dish, exciting
the Faraday instability on the surface of the water. A buoyant particle is buffeted by the surface
waves. An accelerometer is attached to the bottom of the dish to directly measure peak-peak
amplitude A. (photo) Photo of the experimental apparatus from above. A floating particle is
shown, along with a typical position trace for 7.5 seconds of data collection. The surface of the
water is illuminated from an angle to show the chaotic Faraday waves.
8
35 45 55 65 75 85 95 1050
0.1
0.2
0.3
0.4
f (Hz)
A
 (m
m
)
Non-Chaotic
Splashing
0.2x10-8 J
4x10-8 J
FIGURE 2. Map of parameter space.
The parameter space is formed by the two externally-controlled parameters, shaker frequency f
and peak-peak amplitude A. This map shows approximate loci between the non-chaotic,
accessible, and splashing regimes. Plotted within the accessible region are points in parameter
space where measurements were taken, color indicating temperature T . Color gradient goes from
blue (dark grey) for “cold” to red (light grey) for “hot.” Plotted behind these points is an
interpolated contour map of T .
9
are only concerning ourselves with systems in which spatio-temporal chaos is fully developed, but
the vertical oscillation is not so great that the fluid splashes.
The floating test particle is set into the dish near the center and tracking is initiated. We
track the particle with sub-pixel accuracy using a radial symmetry algorithm [40]. By restricting
our region of interest of the current frame to the area near where the particle was found in the
previous frame, we are able to track in real time at 60 frames per second with a signal to noise
ratio 200:1 and a sub-pixel real space resolution of < 10µm. The particle is tracked for 7.5
seconds, chosen to allow for maximum data collection while reducing the likelihood that the
particle will reach the edge of the dish, where the physics is fundamentally different. The process
is repeated for N = 10 or 100 trials.
We measure the mean square displacement 〈∆r2〉 and use it to characterize the particle’s
motion. If motion is ballistic then
〈∆r2〉 = 〈v2〉∆t2, (2.1)
where 〈v2〉 is the mean square ballistic velocity and ∆t is the lag time. If motion is diffusive, then
〈∆r2〉 = 4D∆t, (2.2)
where D is the diffusion constant. Figure 3 shows a representative mean square displacement
demonstrating ballistic motion (〈∆r2〉 ∼ ∆t2) for short times and diffusive (〈∆r2〉 ∼ ∆t) for long
times. We measure the coefficients 〈v2〉 and D by fitting equations 2.1 and 2.2 to the data in the
ballistic and diffusive regimes, respectively.
Results and Discussion
The insets of Figure 3a show the scaled distributions of displacement magnitude for various
lag times ∆t. For short times in the ballistic regime the distributions collapse when scaled by
a factor of ∆t, and for long times in the diffusive regime the distributions collapse when scaled
by a factor of ∆t1/2. In both regimes, the distributions are well described by a two-dimensional
Gaussian integrated over all directions uniformly. This random walk behavior is an emergent
result of allowing a particle to interact with a chaotic surface.
10
10−2 10−1 100 101
10−4
10−3
10−2
10−1
100
101
∆t (s)
∆r〈
2 〉
 (c
m
2 ) 0 1 2 3 4
0
0.5
1
∆r/∆t (cm/s)
pd
f
0 1 2 3
0
0.5
1
1.5
∆r/(∆t)1/2 (cm/s1/2)
pd
f
∆t = 0.033s
∆t = 0.083s
∆t = 0.167s
Radial
Gaussian
∆t = 1.667s
∆t = 2.500s
∆t = 3.333s
∆t = 5.000s
Radial
Gaussian
Slo
pe
 2
Slo
pe 
1
FIGURE 3. Ballistic and Diffusive Behavior
Representative mean squared displacement 〈∆r2〉 (f = 85Hz, A = 0.159mm). 〈∆r2〉 is plotted
against lag time ∆t. At short lag times the mean squared displacement shows a power law with
an exponent of 2, indicating ballistic motion. For longer lag times the curve turns over to a power
law with an exponent of 1, indicating diffusive motion. Inset in the upper left is the distribution of
displacement ∆r divided by a factor of lag time ∆t for lag times in the ballistic regime. Inset in
the lower right is the distribution of ∆r divided by ∆t1/2 for lag times in the diffusive regime.
Both distributions are shown to collapse onto a 2D Gaussian integrated over all directions (radial
Gaussian).
11
In thermal systems the quantities 〈v2〉 and D are related to temperature T and the
coefficient of viscous friction ζ. In contrast to studies such as [34, 35], we use the average kinetic
energy of the tracer as it undergoes ballistic motion to define an effective temperature. By
equipartition the temperature is proportional to the average kinetic energy per particle as
1
2
m〈v2〉 = T, (2.3)
where m is the mass of a particle (m =216mg) and kB the Boltzmann coefficient. We regard T
as an energy, equivalent to kBT in molecular systems. T , D and ζ are related by the Einstein
relation as
Dζ = T (2.4)
where D and ζ both depend on T . T may be treated with the same physics as a conventional
temperature, as opposed to an “effective temperature” such as those used to describe granular
and other athermal systems. The friction term ζ is primarily associated with the viscosity of the
chaotic Faraday waves as a 2D medium through which the floating particle is moving. Thus, by
measuring 〈v2〉 and D we can determine the intrinsic properties T and ζ of a 2D thermal-like
system.
Analysis of the data reveals the dependence of 〈v2〉 and D, and therefore T and ζ, on
our externally-controlled parameters A and f . Figure 4a demonstrates that temperature T is
proportional to shaker amplitude A, independent of frequency. This trend is also apparent in
the temperature contours shown in figure 2. Thus, driving amplitude is the external control
for temperature. Since the typical Faraday wavelength is primarily determined by the driving
frequency and T is independent of frequency, we may also deduce that T is independent of the
ratio of particle size to wavelength over the range of parameters for which we have conducted our
study. We find a calibration curve between driving amplitude and the temperature of our bath:
T ≈ (1.3× 10−7 J
mm
)A− 1.0× 10−8J. (2.5)
The linearity of this relationship is expected because the height of the chaotic Faraday waves
should be proportional to the driving amplitude, and the particle moves in response to the height
difference between the two largest waves in contact with it. However, there is a curious feature to
12
this fit in that T crosses zero at finite A. This is due to the fact that below a particular driving
amplitude the temperature is ill-defined because the Faraday waves are not chaotic. Thus, A must
reach a critical value before T can exist and begin to increase.
While the existence of ballistic and diffusive motion is strongly suggestive that the
system of driven Faraday waves can be considered as a gas at some temperature T , we can more
rigorously examine whether other general properties of thermal gasses, such as the temperature
dependance of their viscosity, manifest themselves in this driven system. Recalling now the
Einstein relation (Eq. 2.4) we examine the behavior of the viscous drag coefficient ζ. Stokes’
equation for incompressible fluids states that ζ ∝ η where η is the viscosity. Sutherland’s formula
for the viscosity of a nearly ideal 2D gas (in which particles have finite collision radii and soft,
long range interaction potentials) [41] further relates η to T as
η = λ′
T√
T + C ′
(2.6)
where λ′ and C ′ are constants of the gas. We can now solve for D as a function of T for an ideal
gas to find
D = a
√
T + b, (2.7)
where all of our constants have been absorbed into a and b. Figure 4b shows our data on T and
D to be well fit by this simple ideal gas model with constants a = .16 ± .01 m/kg1/2 and b =
(−1.87± 0.65)× 10−9 J. The parameter a relates diffusion to energy and the parameter b contains
information on the interactions within the gas.
As a relates diffusion to energy it has units of length/mass1/2. Plugging in the length and
mass scale of the particle we would predict from simple dimensional analysis that a should be on
the order of 0.75 cm/(0.2 g)1/2 ' 0.5 m/kg1/2, which is comparable to the fitted value of a.
Since b is negative there is a positive value of T at which the diffusion constant should go
to zero and thus the viscosity to infinity. This is a singular point that does not exist for real ideal
gases but is difficult to probe directly because the timescales for diffusion diverge as T → −b and,
more practically, it is near the transition from chaos to order in our system.
Above T = −b the viscosity and diffusion constant maintain functional behavior consistent
with that of a nearly ideal gas, suggesting that the chaotic Faraday waves should be considered a
13
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
3
T (10−8 J)
D
 (1
0−
5  
m
2 /s
)
0 0.1 0.2 0.3 0.4
0
1
2
3
4
A (mm)
T
 (1
0−
8  
J)
f = 65Hz
f = 75Hz
f = 85Hz
f = 95Hz
a
b
FIGURE 4. Gas behavior
(a) temperature T plotted against shaker peak-peak amplitude A for various shaker frequencies f ,
with linear fits shown for frequencies plotted in large color symbols (circles, upward triangles,
downward triangles, diamonds). Grey crosses are the remaining frequencies shown in figure 2,
ranging from 35Hz-105Hz, and the dashed line is a linear fit to all data points shown. (b)
Diffusion constant D plotted against temperature T . Data points are taken for data sets where
N = 100 trials. Shown in red is a fit to the data of Equation 2.7. Error bars shown in (a) and (b)
are 1σ confidence intervals resulting from fits used to determine D and T .
14
nearly ideal gas of random surface excitations. The sign of the constant b tells us the sign of the
interaction between the elements of the gas: positive for attractive, and negative for repulsive.
One would expect that the interactions in our system are repulsive, as gravity will cause waves to
repel one another over short distances.
Conclusion
We have demonstrated for the first time that chaotic Faraday waves can be described as
a thermal-like, nearly ideal gas of random surface excitations for which a thermal energy scale,
diffusion constant, and viscosity are well defined. We directly observe the particle to receive
normally distributed, isotropic kicks from the chaotic waves. The particle’s displacement scales
with lag time exactly as would be expected for a Brownian particle, and exhibits a clear transition
from ballistic to diffusive behavior, showing that we have produced a laboratory-scale Brownian
system by Aristotelian means. This thermal-like gas takes difficult-to-access time and length scales
and dilates them to easily studied levels. In so doing we have defined a macroscopic effective
temperature in an out-of-equilibrium system functionally identical to, but completely divorced
in origin from a microscopic thermal temperature. The present single-particle study outlines the
method by which one would define a temperature for a system of chaotic Faraday waves, and
future studies will examine the dynamics of the multi-particle case.
This experimentally tractable system will be useful for studying the statistical mechanics of
actively driven non-equilibrium systems in general, especially interparticle interactions and driven
assembly on 2D substrates, which is a subject of active scientific interest [42]. An outstanding
theoretical question for 2D systems is whether it is possible to create long range order from short-
range interactions [43–45]. Our system offers a fully 2D environment in which these questions can
be studied.
15
CHAPTER III
FLUIDS BY DESIGN: CHAOTIC SURFACE WAVES FORM A METAFLUID THAT IS
NEWTONIAN, THERMAL AND ENTIRELY TUNABLE
Abstract
In conventional fluids, viscosity depends on temperature according to a strict relationship.
To change this relationship one must change the molecular nature of the fluid. Here, we create
a metafluid whose properties are derived not from the properties of molecules, but rather from
chaotic waves excited on the surface of vertically-agitated water. By making direct rheological
measurements of the flow properties of our metafluid, we show that it has independently tunable
viscosity and temperature, a quality that no conventional fluid possesses. We go on to show
that the metafluid obeys the Einstein relation, which relates many-body response (viscosity) to
single-particle dynamics (diffusion) and is a fundamental result in equilibrium thermal systems.
Thus, our metafluid is wholly consistent with equilibrium thermal physics, despite being markedly
non-equilibrium. Taken together, our results demonstrate a new type of material that retains
equilibrium physics while simultaneously allowing for direct programmatic control over material
properties.
As of this writing, this work has been accepted to but not yet published at Proceedings of
the National Academy of Sciences. The writing and analysis were performed by me as primary
author. Eric Corwin is listed as a coauthor as he advised this work. The material in this chapter
is also coauthored with Alex Liebman-Pela´ez, an undergraduate who helped with construction of
parts of the experimental system.
Background
Materials science seeks not only to understand, but to control the properties of matter.
To do so, one must dynamically change the nature of conventional materials at the molecular
scale. Recent work circumvents this problem by rejecting the molecule as the fundamental
unit and substituting a macroscopic structural element. This approach has successfully created
metamaterials with novel optical [1–3], acoustic [4–6], mechanical, [7–9] and fluid properties
[10, 11] that would otherwise be impossible. However, this comes at a cost: by deriving their
16
(a) (b)
Shaker
θ
γR
m1
m2
h
(c)
0 5 10 15 20 25 30 35 40
-0.4
-0.2
0
0.2
0.4
t (s)
po
si
tio
n 
(r
ad
)(d)
10-2 10-1 100 101
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
∆t (s)
ro
to
r M
SD
 (r
ad
 2 )
(e)
Bal
listi
c
Dius
ive
0 10 20-0.2
0.0
0.2
0.4
0.6
0.8
1.0
C(
Δ
t)
/C
(0
)
∆t (s)
FIGURE 5. Experimental setup and Characterization
(a) The rheometer assembly. h is the vertical separation of the drive arm and test rotor. (b)
Top-down cartoon illustrating the coodinates of the system. m1 and m2 are the moments of the
drive arm and test rotor magnets, respectively, R is the radius of rotation of the magnets, θ is the
angular position of the test rotor and γ is the relative angle between the drive arm and test rotor.
(c) Photo of rheometer. (d) Example (Shaker: fs = 60 Hz, As = 0.064 mm, drive arm:
ω/2pi = 0.8 Hz) time series of drive arm (red, light) and rotor (blue, dark) positions. (e) Example
(Shaker: fs = 60 Hz, As = 0.064 mm) mean squared displacement (MSD). Our data (points)
shown with the prediction for a truly Brownian particle (line) [46, 47]. (inset) Velocity
autocorrelation function corresponding to the plotted MSD, with the prediction of the Langevin
equation plotted over the data.
properties from macroscopic, non-equilibrium, or anisotropic elements, these materials necessarily
abandon the physics of thermal systems. In this work, using a combination of active and passive
rheology we show that macroscopic chaotic surface waves on a vertically agitated fluid form a fully
thermal metafluid with dynamically tunable material properties. In contrast to a conventional
fluid in which viscosity and temperature are inextricably linked, we show that these quantities
are independently tunable in our system. We further demonstrate that by satisfying the barest
criteria of isotropy and steady-state chaos (as required by kinetic theory [14, 17]) we have
created a system that obeys the Einstein relation [19]. Thus, despite being macroscopic and non-
equilibrium, it is well-described by equilibrium thermal physics.
The “molecules” of our metafluid are chaotic Faraday waves [12, 13], generated in a water-
filled aluminum dish that is vertically oscillated with RMS amplitude As and at frequency fs
(Figs. 5(a) & (c), see chapter 5 for technical details). The waves uniformly cover the surface of the
17
water and only experience significant pinning at the boundary, far from where our measurements
are conducted. The chaos and wave density is holistically steady state, though the existence
of a particular wave is transient. Thus, “collisions” in our system refer to encounters between
a buoyant tracer and the ephemeral excitations of the chaotic wave environment. We have
previously shown chaotic Faraday waves to have a well defined temperature decoupled from the
bath temperature by observing the Brownian motion of a tracer particle [15]. We found this
temperature to be proportional to shaker amplitude.
Results and Discussion
Drag coefficent
Our rheometer consists of a driving arm that applies a sinusoidally varying torque τ(t) at
a variable frequency to a buoyant test rotor via a magnetic interaction (Figs. 5(a), (b) & (c), see
chapter 5 for technical details). We derive the exact form of this torque in chapter 5 and find that
it is well approximated as a Hookean interaction between the drive arm and the test rotor with
rotational spring constant kr ≈ 6 × 10−5 Nm (Fig. 14 in chapter 5). Sample time series for the
drive arm and test rotor positions are given in Fig. 5(d).
We program the driving arm’s position β to oscillate sinusoidally as
β(t) = β0 sin(ωt), (3.1)
where β0 is the oscillation amplitude and ω is the angular driving frequency. The resulting motion
of the test particle is described by the differential equation
Iθ¨ = τ(t)− ζr θ˙, (3.2)
where θ is the angular position of the test particle and I is the moment of inertia. Treating the
rotor as a damped driven harmonic oscillator, the solution to this differential equation is
θ(t) = θ0 sin(ωt+ φ), (3.3)
18
where the response frequency ω is identical to the driving frequency ω after a brief period as the
rotor relaxes into its steady oscillation. All data sets are collected after allowing for this transient
period. As derived in chapter 5, the driving amplitude β0, oscillation frequency ω, response
amplitude θ0 and phase lag φ are related to the drag coefficient ζr as
ζr =
krβ0
ωθ0
sinφ. (3.4)
The rotor’s response θ(t) is recorded using a contactless magnetic encoder, then the
response amplitude and phase lag are obtained by using a simple lock-in measurement.
In Fig. 6(a) we demonstrate that the metafluid behaves as a Newtonian fluid with a drag
coefficient that is independent of the rheometer drive frequency over an experimentally relevant
range of ω. The above equations also predict (as shown in chapter 5) a mechanical impedance of
Zm(ω) =
√(
ζr
I
)2
+
1
ω2
(
kr
I
− ω2
)2
. (3.5)
Figure 6(b) demonstrates that the measured impedance agrees with the expected form.
Fig. 6(c) shows the results of our active measurement of drag coefficient at values for
shaker amplitude and frequency covering the phase space of chaotic Faraday waves. We find that
the viscous drag coefficient imparted by the chaotic waves is proportional to the product Asfs.
We can make a heuristic argument to show that this relationship is reasonable. If we hold the
shaker amplitude As fixed and increase the shaker frequency fs we will decrease the characteristic
wavelength of the surface waves and thus decrease the characteristic time between collisions of the
waves and the rotor. If we look to the derivation of the Einstein relation we see that ζr should be
inversely proportional to this characteristic time [19] and thus find ζr ∝ fs. Likewise, if we hold
fs constant and increase As we will increase the energy in each surface wave, thus increasing the
momentum transfer effected by each collision and so increasing the drag coefficient.
The onset of chaos limits our measurement range in shaker amplitude and frequency space,
because there exist regions in this space for which the Faraday waves are not chaotic (or indeed,
do not even exist). Nonetheless, we measure the drag coefficient on calm water to be ζr = 1.9 ×
10−5 Js, slightly below the lowest drag coefficient that we are able to measure with chaotic waves.
19
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
T 
(1
0-
7  J
)
As (mm)
(d)(b)
0 20 40 60 80 100
0
1
2
3
4
ζ r 
(1
0-
5  J
s)
Asfs (mm•Hz)
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
ζ r 
(1
0-
5  J
s)
ω/2� (Hz)
(a)
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Z m
 (s
-1
)
(c)
ω/2� (Hz)
FIGURE 6. Tunable Newtonian Fluid Behavior
(a) Demonstration that measured drag coefficient ζr is independent of drive arm frequency ω at
As = 0.85 mm and fs = 50 Hz (squares), As = 1.2 mm and fs = 50 Hz (triangles), and
As = 2.4 mm and fs = 30 Hz (circles). Dotted lines are mean drag coefficient at each amplitude.
(b) Mechanical impedance Zm measured at different drive arm frequency ω. Marker shape
indicates the same values of As and fs as in (a). Solid lines are the expected Zm(ω) for a damped
driven harmonic oscillator. (c) Fitted calibration curve relating rotational drag coefficient to the
product of shaker amplitude As and shaker frequency fs. Best fit shown is
ζr ≈ (3.6× 10−7 Js/(mm Hz))Asfs + 9.4× 10−6Js. Marker color illustrates amplitude; dark to
light gradient corresponds to low to high amplitude variation. Marker shape indicates shaker
frequency: fs = 30 Hz (stars), fs = 40 Hz (inverted triangles), fs = 50 Hz (squares), fs = 60 Hz
(upright triangles), and fs = 70 Hz (diamonds). (d) Fitted calibration curve relating wave
temperature T to shaker amplitude As. Best fit shown is
T ≈ (1.8× 10−7 J/mm)As − 7.3× 10−8 J. Marker shape indicates shaker frequency, as in (c).
20
Temperature and diffusion constant
To make passive measurements, we remove the driving arm of our rheometer and thus allow
the test rotor to diffuse freely while tracking its position. From the position trace we calculate the
mean squared displacement (MSD) 〈(∆θ)2〉 and extract two quantities: a mean squared ballistic
angular velocity 〈ω2b 〉 and a rotational diffusion constant Dr. A sample MSD is shown in Fig.
5(e). Plotted over our data in Fig. 5(e) is the prediction by Ornstein and Fu¨rth for the MSD of a
truly Brownian particle [46, 47]. Set into Fig. 5(e) is the velocity autocorrelation corresponding to
the plotted MSD, shown with the prediction of the Langevin equation for a thermally diffusing
particle. The agreement of predicted thermal behavior with our data demonstrates that our
system has the same constituent dynamics as a thermal particle, albeit at much larger length
and time scales.
For sufficiently small lag times ∆t, the MSD appears ballistic: 〈(∆θ)2〉∆tb = 〈ω2b 〉∆t2b . At
longer times, the MSD appears diffusive: 〈(∆θ)2〉∆td = 2Dr∆td. To avoid the biases associated
with the traditional method of fitting the MSD to estimate 〈ω2b 〉 and Dr, we instead choose the
smallest lag times ∆tb and ∆td that are convincingly ballistic and diffusive [48], respectively, and
estimate 〈ω2b 〉 and Dr using
〈ω2b 〉 =
〈(∆θ)2〉∆tb − σ2
∆t2b
(3.6)
and
Dr =
〈(∆θ)2〉∆td − 2σ2
2∆td
(3.7)
where σ2 = 3.7 × 10−5 rad2 is the positional error associated with the discrete nature of the
magnetic rotary encoder output. Finally, given the hydrodynamic moment of inertia Ih for the
rotor, equipartition gives us the temperature as T = Ih〈ω2b 〉.
Fig. 6(d) confirms our previous finding that temperature is proportional to shaker
amplitude As [15]. We would expect the temperature of the system to be proportional to the
energy density in the surface waves, which is itself proportional to the amplitude squared as
T ∝ A2waves. We can understand the linear dependence of T on the shaker amplitude As
by considering the response properties of Faraday waves. In such systems, the characteristic
response amplitude is generally proportional to the square root of the driving amplitude [49],
so Awaves ∝ A1/2s which leads to T ∝ As.
21
The temperatures we measure here from rotational motion are consistent with the
temperature we measured in earlier work on translational motion [15], giving further credence to
the notion that our system demonstrates energy equipartition and is truly thermal. The measured
temperature is a remarkable 14 orders of magnitude larger than the water bath temperature,
demonstrating that the thermal behavior of our system is completely divorced from that of
ordinary matter.
Einstein relation and hydrodynamic moment of inertia
Thermal fluids satisfy the Einstein relation, which states that Drζr = T . The power
of the Einstein relation is that it connects two very different properties of fluids: rheology
(a measurement of correlated many-body behavior) and diffusion (a measurement of single-
particle dynamics). There is no a priori reason to expect the Einstein relation to hold in a
non-equilibrium complex fluid such as ours in which energy is constantly being injected into
the system. Nonetheless, Fig. 7(a) demonstrates that the Einstein relation is satisfied. That we
discover an intact Einstein relation further convinces us that our macroscopic metafluid is in fact
an equilibrium fluid. Additionally, Fig. 7(b) shows that the diffusion constant calculated from
the MSD is consistent with the diffusion constant as calculated using the Green-Kubo relations,
which relate the diffusion constant to the integral of the velocity autocorrelation function. This
consistency convinces us that Einstein relation in our system may be trusted over all timescales
available to our measurement. Taken together, these results suggest that the universality of
thermal systems should be extended to systems which satisfy broader criteria of steady-state
chaos and isotropy.
This notion of universality is supported by work done in other driven, dissipative systems.
For example, the work of Ojha et al. [50] demonstrated that a particle agitated by turbulent
airflow is identically thermal with a temperature completely divorced from the room temperature.
Here, we have used a new mechanism for thermalization yet nevertheless recover thermal
behavior, also with with a temperature completely divorced from that of the surroundings. Our
work further demonstrates that these macroscopic thermal systems can be tuned in ways that
microscopic systems cannot.
22
0
0
1
2
3
4
5
D
 r ζ
 r  
 (1
0 
-7
 J
)
1 2 3 4
<ωb 
2> (10 -2 s-2)
0 2 4 6 8 10 12 14 16 18
0
2
4
6
8
10
12
14
16
18
Dr, Green-Kubo (10
-3 rad2/s)
D
r, 
M
SD
 (1
0-
3  r
ad
2 /
s)
(a)
(b)
FIGURE 7. Einstein Relation and Green-Kubo Relation
(a) Demonstration of the Einstein relation in our system. The product Drζr is proportional to the
angular mean square ballistic velocity 〈ω2b 〉, which is in turn proportional to the effective
temperature of the waves T . The line shown is a fit to the data of the Einstein relation
Drζr = Ih〈ω2b 〉, with the the hydrodynamic moment of inertia Ih left as a free parameter,
determined by the fitting to be Ih = 1.2× 10−5 kg m2. The error bars shown are 1σ confidence
intervals determined from the calculation of Dr and 〈ω2b 〉. (b) Comparison of diffusion constant
measured from MSD (Dr,MSD) and using the Green-Kubo relations (Dr,Green−Kubo). The dashed
line has slope 1, indicating good agreement between the two methods of calculating Dr.
23
The slope of the fitted line relating 〈ω2b 〉 and Drζr in Fig. 7(a) is the hydrodynamic
moment of inertia Ih, as T = Ih〈ω2b 〉. This quantity is larger than the moment calculated directly
from the mass and geometry of the test rotor by nearly an order of magnitude due to the fact that
a volume of water is swept along with the rotor.
We have thus far shown that T ∝ As, ζr ∝ Asfs and Drζr = T . Taken together,
these results imply that Dr ∝ 1/fs. We can heuristically understand this relationship by
realizing that shaker frequency is inversely related to the characteristic wavelength of the chaotic
surface waves. As shaker frequency decreases, the characteristic wavelength increases. Since
the average distance between wave excitations is greater, the mean free path and mean free
time of a diffusing particle increase proportionally. The diffusion constant is proportional to
(mean free path)
2
/(mean free time). By this line of reasoning, Dr should be proportional to 1/fs.
Our results also imply that Dr should not depend strongly on shaker amplitude As. We can
understand this heuristically as well. As the shaker amplitude increases, so too does the amplitude
of the waves, as we outlined in our discussion of why T ∝ As. This increases the strength of the
kicks that set a particle in motion, but also increase the strength of the kicks that slow it down.
The increase in thermal energy is offset by the energy dissipated by the drag coefficient, which
also increases with shaker amplitude. Thus, it is reasonable that Dr not depend strongly on As.
Tunable thermal Newtonian metafluid
We have demonstrated three properties of this metafluid: 1) The metafluid has a well-
defined Newtonian viscous drag which is controlled by the product of shaker amplitude and
frequency. 2) The metafluid has a well-defined temperature which is controlled by shaker
amplitude. 3) The metafluid satisfies the Einstein relation. These results lead to the inevitable
conclusion that our system is in fact a thermal Newtonian metafluid with independently tunable
material properties as shown in Fig. 8. Any arbitrary state or path in Dr − ζr − T space that
satisfies the Einstein relation is translatable using our measured calibration curves into the fs−As
parameter space. An example Dr−ζr−T path is shown in red in Fig. 8(a), then shown translated
into a corresponding fs − As path in Fig. 8(b) using the Einstein relation and the calibration
curves shown in Figs. 6(c) and (c).
24
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
ζr (10
-5 Js)
D
r (
10
-2
 ra
d2
 s-
1 )
0.125
0.25
0.5
1.0
2.0
4.0
20 25 30 35 40 451.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
fs (Hz)
A
s (
m
m
)
(a) (b)
ide
al 
ga
s m
od
el
FIGURE 8. Phase Space and Parameter Space
(a) Phase space. Measured diffusion constant plotted against measured drag coefficient (circles).
Marker color indicates measured temperature, which is in reasonable agreement with the
background gradient that represents temperature as calculated from the Einstein relation,
T = Drζr. Example isotherms (black lines) are shown, labelled in units of 10
−7 J. Our system can
take any path along the surface defined by the Einstein relation, whereas a conventional fluid is
confined to a particular curve (ideal gas behavior shown as an example). An example path
through Dr − ζr − T phase space beginning at the star is plotted, and is then transformed into
fs −As parameter space (b) via the Einstein relation and the calibration curves shown in Fig 6(c)
and 2(d).
The surface defined by the Einstein relation is shown as the background gradient in Fig.
8(a), with color indicating temperature. Each trial measurement of our system is able to sit
anywhere along this surface as shown by the plotted points in Fig. 8(a). The color of the plotted
points indicates the measured temperature for that trial, and we see, as in Fig. 7(a) that our
system agrees with the Einstein relation.
Conventional fluids, by contrast, lack the freedom to move arbitrarily about this surface,
as they suffer from the additional constraint that viscous drag is a function of temperature. For
example, viscous drag in an ideal gas is related to temperature as ζ ∝ T 1/2 (and thus ζ ∝ Dr)
as shown in the representative ideal gas model curve in Fig. 8(a) (dashed line). To change the
constant of proportionality one would need to change the molecular composition of the gas or,
put more plainly, one would need to change the very identity of the gas. A conventional fluid is
therefore confined to a single curve through Dr − ζr − T space.
25
Conclusion
In this paper we have presented a fully tunable, thermal Newtonian metafluid. By
combining passive and active rheology we directly measure diffusion, temperature, and viscous
drag in a system of chaotic surface waves and show that they are independently manipulable. The
Einstein relation is satisfied even though our system is both macroscopic and non-equilibrium,
hinting that a broader universality exists in random systems: so long as steady-state, isotropic
chaos exists, the system may be mapped onto an equilibrium statistical mechanical system. This
reading of the data is supported by previous studies of thermal behavior in driven, dissipitive
systems with inherent randomness [50].
We have followed in the footsteps of previous metamaterial research [1–11], creating and
characterizing a novel material by rejecting the atom as the fundamental thermal unit and
substituting a macroscopic element, in our case, chaotic surface waves. Our system provides
a novel platform in which experiments may be performed, where the tools of thermodynamics
remain intact and the fluid properties can be controlled in real time. There are obvious
limitations, for example, the metafluid can’t flow through a pipe, it can’t be compressed nor is
there an easy way to define pressure, and it is confined to two dimensions. However, we have
already shown it to be useful in observing the atomistic dynamics of 2D polymer analogs [16],
and we believe that it will prove elucidating in many open questions of the thermal world, such as
self-assembly and 2D pattern formation [51–53]. Perhaps, emergent metafluids such as ours exist
throughout the macroscopic world, be they swarms, turbulent weather, financial markets or any
other system with intrinsic randomness.
26
CHAPTER IV
EXPERIMENTAL AND COMPUTATIONAL STUDY OF MACROSCOPIC ANALOGS TO
SHORT CHAIN MOLECULES: AN ATOMISTIC APPROACH
Abstract
We use a bath of chaotic surface waves in water to mechanically and macroscopically mimic
the thermal behavior of a short articulated chain with only nearest-neighbor interactions. The
chaotic waves provide isotropic and random agitation to which a temperature can be ascribed,
allowing the chain to passively explore its degrees of freedom in analogy to thermal motion.
We track the chain in real time and infer end-to-end potentials using Boltzmann statistics. We
extrapolate our results, by using Monte Carlo simulations of self-avoiding polymers, to lengths not
accessible in our system. In the long chain limit we demonstrate universal scaling of the statistical
parameters of all chains in agreement with well-known predictions for self-avoiding walks.
However, we find that the behavior of chains below a characteristic length scale is fundamentally
different. We find that short chains have much greater compressional stiffness than would be
expected. However, chains rapidly soften as length increases to meet with expected scalings.
This work was originally published in Physical Review E [16]. The writing and analysis
were performed by me as primary author. Eric Corwin is listed as a coauthor as he advised this
work. The material in this chapter is also coauthored with Clayton Kilmer, an undergraduate who
helped with data collection.
Background
Anyone who has ever put the wrong weight motor oil into a car engine can attest to the
fact that the length of a chain molecule largely determines its mechanics [54]. A low-weight oil
may be too thin and a high-weight oil too viscous for efficient operation. Biopolymers such as
DNA also evidence changing mechanical behavior with changing length. Recent work shows
that short strands are far more flexible than would be expected from simply scaling down the
behavior of long strands [55]. Understanding this scale dependence in polymers is crucial to
creating new materials, as is being done with polymer thin films [56, 57] and so-called “DNA
origami” [58, 59]. Polymer mechanics is well explored in the coarse-grained sense, with many
27
established methods for direct measurement [60–62] and simulation [63–66]. However, there is
scant experimental evidence directly relating whole polymer mechanical properties to behavior
at the single bond level. These questions have been addressed in macroscopic granular polymer
studies [67, 68] however such systems lack a thermodynamic temperature, muddying the link to
true thermal systems. Statistical physics predicts [69], and empirical studies confirm [70] that the
end-to-end potential of a sufficiently long polymer chain is harmonic, regardless of the microscopic
interactions. Similarly, universalities are predicted for the scaling of statistical parameters (i.e.,
variance and mean) of both bond winding angle [71] and linear dimension [72, 73] in a polymer
chain when considered as a self-avoiding walk (SAW). While this is a powerful and robust result
regarding the chain-scale mechanics, it does little to elucidate the behavior towards the monomer
scale, as it does not address short chains.
Materials and Methods
In this paper we build a macroscopic analog to polymer physics in a bottom-up fashion
which allows us to observe not only chain-scale but also true monomer-scale behavior. We find
that short polymer chains exhibit behavior fundamentally different than that predicted for their
long counterparts. We use chaotic Faraday waves to create a macroscopic thermal-like bath
with a well-defined temperature T , as we have previously described in reference [15]. The waves
are generated by vertically oscillating a circular aluminum dish with a diameter of 17.5 inches
and a depth of 0.5 inches (Fig 9a). The dish is mounted on a ceramic-coated aluminum shaft
(diameter 2 inches), which passes through a 2 inch inner diameter air bushing (New Way Air
Bearings). This assembly is leveled by a stage mounted to the outside of the bushing mounting
block. The ceramic coated shaft is connected by a 24 inch length of 1 inch T-slot framing to
an electromechanical shaker (Vibration Test Systems VTS-100) powered by a digital function
generator (Stanford Research Systems Model DS335) through a voltage amplifier (Behringer
Europower EP4000). An accelerometer (CTC AC244-2D/010) is mounted to the bottom of the
dish to measure stroke amplitude and frequency.
We 3D-print our macroscopic polymer analogs in a clear photopolymer resin using a
Formlabs Form1. Our macroscopic polymer consists of two basic elements: buoyant particles
and links. The particles are cylindrical boats (diameter 15mm, height 3mm) with centered masts
28
(a)
(d)
17.5 in
θ1
θ2
θ3
θ4
θ5 = 0
anchor
5
43
2
1
6
θi - θi+1 Є [ -α , α]
R
R
RR
R
α
α
(b)
R(c)
FIGURE 9. Experimental Design
(a) The shaker assembly. (b) 3D schematic of one for the buoyant particles in the chain, showing
the definition of restricting angle α. (c) A link showing fixed length R. (d) A chain,
demonstrating the measurement of link angle θi. Bulk rotations are eliminated from our
coordinate system by measuring all θi off of the link most proximal to the anchor. This allows us
to work exclusively in terms of the relative coordinates of members of the chain.
29
(diameter 2mm, height 7.5mm). The links are rods with loops at each end, separated by 22mm
on center, that slide over the masts to link the boats together in a chain (Fig. 9c). Bond angle
is restricted by stops on the gunwales of the boats (Fig. 9b) separated by an angular distance
α. This chain is attached to a brass post set into the center of the aluminum dish using a link
that is free to rotate about the anchor point. This prevents the chain from interacting with the
boundary of the dish, where the physics is fundamentally different, and allows us to gather data
for arbitrarily long amounts of time, in our case 8 hours. In this study we present data for: 1) a
chain with α = pi/4 consisting of the post plus five other particles, five links, and 2) a chain with
α = pi/6 consisting of the post plus eleven particles, and eleven links.
To facilitate automated tracking each boat has a circular well imprinted in the bottom with
a diameter 1/2 that of the boat itself, uniformly filled with black oil paint. The anchor post is
capped with a black tracking point. The positions of these black spots are tracked in real time at
30 frames per second (Pointgrey Flea3 USB camera with a Pentax C32500 KP lens, mounted 8
feet above the dish) using a radial symmetry algorithm [40]. To gather data on the dynamics of
the chain we vertically oscillate our system with a frequency of 40Hz and an amplitude of 0.18mm
peak-to-peak creating a temperature of about 1.3× 108 J. These parameters were chosen such that
the Faraday waves would be well into the chaotic regime without causing the water to splash
and capsize the boats. We are limited in the length scales that we may probe in our system.
Particles or chains smaller than the typical Faraday wavelength (on the order of millimeters)
will not receive thermal-like agitation. Chains longer than the diameter of our dish (44.5 cm) can
interact with the side walls. The dish size was chosen as to allow for the largest possible system
while still remaining within the driving capability of our shaker.
Data Analysis and Simulation
In Figure 10 we demonstrate that the instantaneous bond angle velocities induced by the
chaotic waves are uncorrelated in a chain of rigidly linked particles for times longer than the
ballistic timescale (about 0.3 s). The motion of a given bond angle is initially anticorrelated with
its immediate neighbor due to the buckling of a chain of rigid links. As a bond angle is kicked in
one direction, its neighbor must react in the opposite direction as a result of the constraints in the
system. More distant neighbors show weak correlations that die off quickly. The 10Hz oscillation
30
in correlation results from the beating between the dish oscillation frequency (40Hz) and the
imaging frequency (30Hz). The lack of correlation beyond that induced by buckling of the chain
allows us to rule out the possibility of non-thermal correlations between particles.
To simulate the behavior of chains longer than experimentally accessible we employ a
simple unbiased Monte Carlo algorithm to simulate each chain as a SAW. We begin by generating
a random walk with N bond angles where each step is of length R and taken in a direction drawn
from the distribution of bond angles
p(φi,i+1) ∝ e−Vφ(φi,i+1)/T (4.1)
created by a given potential Vφ. The random walk is then checked for overlaps of the boats
and/or the links. If there are any overlaps, the chain is thrown out. If there are no overlaps then
the chain is indeed a SAW and the end-to-end distance and winding angle are computed. This
process is repeated until the data for 106 valid SAWs of length N have been collected.
We follow the Flory convention [73], which utilizes bond lengths and bond angles to
describe a polymer in generalized nearest-neighbor coordinates (Fig. 9d). To coarsen beyond
nearest-neighbor coordinates, we represent the configuration of the chain in terms of radial
distance ri,j between particles i and j and a winding angle between bonds i and j defined as
φi,j =
j−1∑
m=i
(θm − θm+1), (4.2)
where θi is the angle of the ith bond with respect to the link nearest the anchor point. The use
of a bond winding angle ensures that the angles we deal with are cumulative quantities and not
bounded between ±pi. Note that, since nbonds = nparticles − 1, the winding angle φi,j corresponds
to the interparticle distance ri,j+1.
We use Boltzmann statistics to measure effective potentials for the winding angle Vφ and
the radial distance Vr. We relate the probability distributions of winding angle p(φi,j) and end-to-
end distance p(ri,j) to the underlying potentials scaled by the temperature T [15] as
p(φi,j) ∝ e−Vφ(φi,j)/T (4.3)
31
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Bo
nd
 A
ng
le
 V
el
oc
ity
 C
or
re
la
tio
n
Lag Time (s)
FIGURE 10. Pair Correlation Function
Average bond angle velocity autocorrelation shown in red (thickest curve) for a chain with
α = pi/6. Nearest, second nearest, and third nearest bond angle velocity neighbor correlations
shown in blue, purple, and magenta (second, third, and fourth thickest curves) respectively. The
nearest neighbor bond angle velocities are initially anticorrelated due to buckling in the
interconnected chain, i.e. as one bond angle is kicked its neighbor will react in the opposite
direction. All correlations die off by ∼ 0.3s of lag time. The 10Hz beat frequency results from the
non-commensurate dish oscillation frequency (40Hz) and imaging frequency (30Hz).
32
−1 0 10
5
10
−2 −1 0 1 20
5
10
−3 −2 −1 0 1 2 30
5
10
−4 −2 0 2 40
5
10
0 1 20
1
2
3
4
5
6
7
0 1 2 30
1
2
3
4
5
6
7
0 1 2 3 40
1
2
3
4
5
6
7
0 1 2 3 4 50
1
2
3
4
5
6
7
V φ
(φ
i,j
)/
τ
j = i + 1 j = i + 2 j = i + 3 j = i + 4
φi,j /α φi,j /α φi,j /α φi,j /α
ri,j+1/R ri,j+1/R ri,j+1/R ri,j+1/R
V r
(r
i,j
+1
)/
τ
ii
j
j
iji
j
−3 −1 1 3
i
j+1
i
j+1
i
j+1
i
j+1
Linear Regim
e
FIGURE 11. Effective Potentials
Effective potentials for total bond winding angle φi,j (top row) and interparticle distance ri,j+1
(bottom row) with example subchains shown as inset diagrams. Potentials are determined from
distributions using equations 4.3 and 4.4. The blue (dark) curves are calculated directly from
empirical data and the red (light) curves are calculated using the distribution for φi,i+1 to
generate self-avoiding walks.
and
p(ri,j) ∝ ri,je−Vr(ri,j)/T . (4.4)
We show the measured effective potentials averaged over all possible sub-chains of each
given length for a system with α = pi/4 as blue (dark) curves in Figure 11. We present similar
data for α = pi/6 in figures 15 and 16 in chapter 6. Note that the measured potentials for both
systems reach beyond ±α. A small amount of play was required to allow the links to slide without
binding, creating a broadening of the potential. Further, water bridges form where links contacts
stops creating a more complicated potential energy landscape near the boundaries. However, as
we show in this paper, these details are immaterial as the variance of the nearest-neighbor bond
angle distribution is all that matters in determining the physics of the system.
The winding angle potential Vφ is anharmonic for short chains but approaches a harmonic
potential with increasing chain length, as predicted by a simple application of the renormalization
group [69]. The effective potential for end-to-end distance Vr, however, does not approach a
harmonic within the length of chain empirically available to us for both the α = pi/4 and pi/6
33
data. This stands in contrast to the harmonic behavior expected for a freely-jointed chain, due
to the fact that our chain has restrictions on bond angles and is by its nature non-intersecting.
Our data are much better to match to the worm-like chain model, showing qualitatively similar
end-to-end distributions (and equivalent potentials) [74].
In order to understand the behavior of these potentials for long chains we turn to
simulations of our polymer. The red (light) curves in figure 11 show the effective potentials for
simulated chains of the same lengths as our experimental results. The simulated data agrees with
the empirical data, confirming the validity of our simulation approach.
We extrapolate to long chains in order to determine the scaling of the statistical parameters
of the distributions of end-to-end distance and end-to-end winding angle. End-to-end quantities
are denoted as coordinates without i, j subscripts. Figure 12 shows the scaling of the variance
of end-to-end winding angle σ2φ with N , the number of defined bond angles. We find that for
sufficiently short chains, σ2φ is linear in N . This is identical to the scaling expected for a simple
random walk because at small scales self-intersections are unlikely due to the restriction of bond
angles. For sufficiently long chains we find that σ2φ exhibits a logarithmic scaling, as predicted by
[71]. To determine the critical N at which chains transition between “short” and “long” we fit the
data to the piecewise function
σ2φ =

σ2φ,0N N < N∗
σ2φ,0N∗(ln(N/N∗) + 1) N > N∗
. (4.5)
where σ2φ,0 is the already known variance of the nearest-neighbor link angle. The fit yields
the value of N∗, which we find to be inversely proportional to σ2φ,0 (Figure 12 (inset)) with a
phenomenological prefactor of order 1 that is likely determined by the relative size of the particles
to the links. N∗ plays a similar role to a Kuhn length in a real polymer. We can collapse all of our
data onto a master curve when scaled so that
N → N
N∗
, σ2φ →
σ2φ
σ2φ,0N∗
. (4.6)
We use three different families of bond angle distributions: 1) The experimentally measured
distributions from our α = pi/4 and α = pi/6 systems, the corresponding potentials for which
34
10−2
10−1
100
1
10−2 100 102
0
1
2
3
4
5
10−2 10−1 100
100
101
102
N/N*
10110−110−3
N
*
σφ,0
2
σ φ
 /(
σ φ
,0
 N
*)
2
2
σ φ
 /(
σ φ
,0
 N
*)
2
2
σ φ
 =
 σ φ
,0
 N
2
2
N
*  ~ 1/σ
φ,0
2
σ φ
 =
 σ φ
,0
 N *
(ln
(N
/N
*
) +
 1
)
2
2
2
10
Experimental, 
uniform, 
bidisperse
FIGURE 12. Master curves for the scaling of winding angle variance σ2φ
The data are plotted in both log-log (top) and semilog (bottom) to illustrate power law and
logarithmic scaling regimes, respectively. The shape of the markers indicates the probability
distribution of nearest-neighbor link angles; stars for experimental, circles for uniform, and
diamonds for bidisperse. The marker color goes from light to dark as the variance of the
nearest-neighbor probability distribution increases. Equation (4.5) is plotted over the data. (inset)
The critical length N∗ is inversely proportional to σ2φ,0, the variance of the nearest-neighbor link
angle for a given chain.
35
are shown in the top left of Figure 11 and 15, respectively. 2) A uniform distribution of bond
angles between ±α. 3) A bidisperse distribution of bond angles with value of ±α. We present
simulational data for SAWs from uniform and bidisperse distributions tuned to have the same σ2φ,0
as the α = pi/6 empirical data and find that all three associated scalings have precisely the same
N∗. Thus we see that the full physics of the system is controlled by the variance of the nearest-
neighbor angle distribution.
Figure 13 shows the scaling of the squared mean 〈r〉2 and the variance σ2r as a function of
N . We see that all curves, for sufficiently large N , go to a power law consistent with the Flory
prediction [72, 73], where
〈r〉 ∼ Nν , σ2r ∼ N2ν , ν =
3
4
(4.7)
in two dimensions. We collapse all of our data onto a master curve for N >> N∗ by scaling
N → N/N∗ and
〈r〉2 → 〈r〉
2
(C1 + C2)N
3/2
∗
, σ2r →
σ2r
(C1 + C2)N
3/2
∗
(4.8)
where the constants C1 and C2 are fitted prefactors (figure 13). This scaling was selected such
that 〈r2〉 = 〈r〉2 + σ2r = 1 at N = N∗, preserving relative magnitude of both statistical parameters.
Our simulations show that the for lengths much longer than experimentally accessible the
radial potentials eventually become harmonic as expected (Figure 17 in chapter 6). However, for
short and intermediate lengths the curves show non-trivial behavior. The compression stiffness
of the polymer, which is related inversely to the variance in end-to-end distance, is far higher in
short chains than would be predicted by the Flory power law. As these chains grow, the stiffness
necessarily softens very rapidly in order to meet the long chain limit and match expectations
above the crossover from “short” to “long.”
Our system shows clear hallmarks of semiflexibility [74, 75], with the end-to-end potential
transitioning from a rigid state in which the equilibrium position is near full extension of the
chain, to a flexible state with an equilibrium position near zero (Figure 17 in chapter 6). We
see further evidence of semiflexibility in the existence of a linear regime in our experimentally
determined potentials (Figure 11, lower right). In the intermediate lengths between the two
states, the potential temporarily becomes wide and flat before becoming harmonic. This
intermediate-scale widening explains the rapid scaling of the variance at these lengths. This roll-
36
10−3 10−2 10−1 100 101 102
10−9
10−6
10−3
100
103
100 101 102 103
100
101
N/N*
N*
 C
1 +
 C
2 
Sc
al
ed
 σ
r  ,
 S
ca
le
d 
<r
>2
2
<r>
2 = C 1N
3/2
σ r =
 C 2 N
3/2
2
Experimental, 
uniform, 
bidisperse
FIGURE 13. Scaling Laws for End-to-End Distance
Scaling of variance in end-to-end distance σ2r (lower curve) and squared mean of end-to-end
distance 〈r〉2 (upper curve). The shape and color of the markers indicates the probability
distribution of nearest-neighbor bond angles as in figure 12. Both curves are scaled by a factor of
(C1 + C2)N
3/2
∗ , where N∗ is the same critical length from figure 12, and collapse to N3/2 power
laws for N >> N∗. (inset) The sum of scaling coefficients C1 and C2 plotted against critical
length N∗.
over phenomenon is a result of our bonds being treated as rigid rods. This assumption is valid
as considering only torsional degrees of freedom is a standard in the field, consistent with the
rotational isomeric state model of polymers [76, 77].
Conclusion
In this study we have presented a simple mechanical analog to short chain molecules.
We see that the effective interaction potential for winding angle quickly becomes harmonic as
chain length increases, despite the decidedly non-harmonic nearest-neighbor interaction. Though
the effective potential for end-to-end distance does not become harmonic over the length of
our empirical chain, we are able to extrapolate for considerably longer chains and show that
it eventually assumes the familiar Hookean form. We show that our system may be treated as
37
a self-avoiding walk with scalings that are consistent with [71] and [72, 73] in the long-chain
limit and that the full character of these scalings is determined by the width of the nearest-
neighbor bond angle distribution. We show that short- and intermediate-length chains exhibit
unconventional behavior, a result likely connected the relative rigidity of bond lengths. Ultimately,
we demonstrate that over the full range of lengths studied all of the physics is controlled by the
variance of the nearest neighbor bond-angle distribution.
Our simulation method accurately predicts the behavior of our mechanical system at the
short scale as well as the well-known behavior of polymers in the long-chain limit. Therefore,
our mechanical analog is consistent with polymer physics and may be used to speak to the
non-universal mechanics of short chain molecules. Our macroscopic, mechanical system is
uniquely suited to passively study these lengths as we can build the physics from the bottom
up all while operating at easily observable length and time scales. It is remarkable that such a
simple macroscopic analog can point to a better understanding of the constitutive mechanics of
microscopic polymers. Beyond the study of polymers, the relative link angles in our chain can
be mapped as spins onto a 1D lattice, so this experiment can be viewed as a simple mechanical
simulation of a 1D X-Y model with peculiar nearest neighbor interactions.
38
CHAPTER V
DETAILED MATERIALS AND METHODS FOR CHAPTER 3
Shaker Assembly
The waves are generated in a water-filled aluminum dish (diameter 17.5 inches, depth 0.5
inches) that is vertically oscillated with RMS amplitude As and at frequency fs (Fig. 5(a) in
chapter 3). The dish is mounted on a ceramic-coated aluminum shaft (diameter 2 inches), which
passes through a 2 inch inner diameter air bushing (New Way Air Bearings). This assembly
is leveled by a stage mounted to the outside of the bushing mounting block. The ceramic-
coated shaft is connected by a 24 inch length of 1 inch T-slot framing to an electromechanical
shaker (Vibration Test Systems VTS-100) controlled by a digital function generator (Stanford
Research Systems Model DS335) through a voltage amplifier (Behringer Europower EP4000).
Four springs connect the T-slot framing shaft to the external armature, bearing nearly all of the
static load. An accelerometer is mounted to the bottom of the dish to measure stroke amplitude
and frequency through a digital oscilloscope (Tektronix TDS 2012C).
Rheometer
The driving arm of the rheometer is laser-cut from acrylic and has attached to it two
neodymium magnets (K&J Magnetics, cylinder magnets; thickness 0.125 in, diameter 0.5 in,
moment m1 = 0.42 A m
2). Each magnet is positioned R = 3cm from the the center of the
arm with their north poles up. The driving arm sits at the end of a long aluminum shaft and is
rotated about its center by a stepper motor driven by an Arduino Uno. The long shaft ensures
that there is no effect due to magnetic interactions with the stepper motor. The test rotor consists
of two closed-cell nylon foam disk floats held at the same R = 3cm from the center as the driving
arm magnets by a laser cut piece of balsa wood. The wood piece has a hole in its center which,
when slipped over a delrin post set into the center of the dish, holds the test rotor coaxial with
the driving arm but free to rotate. In the center of each float disk is a neodymium magnet (K&J
Magnetics, cylinder magnets; thickness 0.0625 in, diameter 0.25 in, moment m2 = 0.053 A m
2)
with its north pole up. The vertical separation between the driving arm and the test rotor is
h = 6cm. A contactlesss magnetic encoder (Avago Technologies, part number AEAT-6600-T16) is
39
used to track the angular position of the test particle. To facilitate this a cylindrical, diametric
(polarization orthogonal to the axis of symmetry) magnet with an axial pass-through hole is
mounted in the center of the wooden connector. The magnetic encoder is mounted on top of
the center post and is recorded by a second Arduino Uno.
Derivation of Torque Due to Magnets
Here we derive, in exact form, the torque on the test rotor as a function of relative angle γ.
We begin by defining all salient parameters and variables. Figs. 5(a) and (b) illustrate the
relationship between the quantities.
– θ: Angular position of the test rotor
– γ: Relative angle between the drive arm and test rotor when viewed from above
– R: distance of the magnets from the axis of rotation
– ~m1 = m1zˆ and ~m2 = m2zˆ: magnetic moments of the upper and lower magnets, respectively
– ~r: separation of the upper magnet from the lower magnet
– a: lateral distance between the upper and lower magnets (defining the xˆ direction)
– h: vertical distance between the upper and lower magnets
The coordinates and dimensions of the system are related to one another as
r =
√
h2 + a2 =
√
h2 + 4R2 sin2
(γ
2
)
. (5.1)
Given that ~m1 = m1zˆ, ~m2 = m2zˆ and ~r · zˆ = h, the force between the upper and lower
magnets is
~F (~r, ~m1, ~m2) =
3µom1m2
4pir5
(
2hzˆ +
(
1− 5h
2
r2
)
~r
)
. (5.2)
The exerted in-plane torque is driven by the component of the force that is directed tangent to
the arc of rotation. To obtain this, we first find the xˆ component of ~F which lies along a. Noting
that ~r · xˆ = a, we can write down the xˆ component of ~F as
Fx = ~F · xˆ = 3µom1m2R
2pir5
(
1− 5h
2
r2
)
sin
(γ
2
)
. (5.3)
40
We calculate the component of the force tangent to the arc of rotation Fγ = Fx cos(
1
2γ) and thus
the torque τ = FγR using the identity sin(
1
2γ) cos(
1
2γ) =
1
2 sin γ to obtain
τ(γ) =
3µom1m2R
2
2pi
(
1
r5
− 5h
2
r7
)
sin γ. (5.4)
This is the exact form for the torque on the test particle as a function of relative angle γ, with an
extra factor of 2 to account for the fact that our system has two pairs of magnets. For simplicity,
we would like to approximate this as a Hookean torque. To do this, we make the approximation
that r ≈ h and sin γ ≈ γ. This is a small-angle approximation, and will break down as γ increases.
The Hookean torque we obtain is
τ(γ) = −krγ (5.5)
where
kr =
6µom1m2R
2
pih5
. (5.6)
For the particular parameters of our system, kr ≈ 6× 10−5 Nm. Fig. 14 shows the torque in
exact form and the linear approximation over a range of relevant angles.
Measurement of Drag Coefficient
We program the driving arm’s position β to oscillate sinusoidally as
β(t) = β0 sin(ωt), (5.7)
where β0 is the oscillation amplitude and ω is the angular driving frequency. Assuming a well
defined coefficient of drag, the resulting motion of the test particle is described by the differential
equation
Iθ¨ = τ(t)− ζr θ˙, (5.8)
where θ is the angular position of the test particle, I is the the moment of inertia, τ(t) is the
torque applied by the driving arm and ζr is the rotational drag coefficient. The applied torque τ
is only dependent on the relative angle γ(t) = θ(t) − β(t) between the driving arm and the test
particle, and so can be parameterized as τ(t) = τ(γ(t)). We derive the exact form of this torque
41
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-4
-3
-2
-1
0
1
2
3
4
to
rq
ue
 (1
0-
5  N
m
)
relative angle γ (rad)
FIGURE 14. Comparison of Hookean approximation and full form for torque exerted by drive
arm
The full form torque (blue) is well approximated by a Hookean torque with spring constant
kr ≈ 6× 10−5Nm (red) over a range of angles relevant to our active measurement of drag
coefficient.
42
above and find that it is well approximated by a Hookean spring:
τ(γ) = −krγ, (5.9)
with spring constant
kr =
6µ0m1m2R
2
pih5
≈ 6× 10−5 Nm. (5.10)
Here, m1 and m2 are the magnitudes of the magnetic moments of the driving arm and test
particle magnets, respectively. This approximation allows us to solve equation 5.8 exactly as a
damped driven harmonic oscillator to find
θ(t) = θ0 sin(ωt+ φ), (5.11)
where
θ0 = β0
kr
IωZm(ω)
, (5.12)
the mechanical impedance is
Zm(ω) =
√(
ζr
I
)2
+
1
ω2
(
kr
I
− ω2
)2
, (5.13)
and the phase lag is
φ(ω) = tan−1
(
ζr
I
ω
ω2 − krI
)
. (5.14)
We solve for the drag coefficient to find
ζr =
krβ0
ωθ0
sinφ. (5.15)
Thus, we are able to calculate the drag coefficient by measuring the amplitude and phase lag of
the test rotor’s response relative to the the driving signal. We obtain the response amplitude and
phase lag using a simple lock-in measurement:
θ0 =
(〈θ(t) cos(ωt)〉2 + 〈θ(t) sin(ωt)〉2)1/2 , (5.16)
φ = tan−1
( 〈θ(t) sin(ωt)〉
〈θ(t) cos(ωt)〉
)
, (5.17)
43
where the angle brackets indicate time averages.
44
CHAPTER VI
SUPPLEMENTARY MATERIALS FOR CHAPTER 4
Effective Potentials for Chain With α = pi/6
−2 0 2
0
2
4
6
8
10
−4 −2 0 2
0
2
4
6
8
10
−5 0 5
0
2
4
6
8
10
−5 0 5
0
2
4
6
8
10
−5 0 5
0
2
4
6
8
10
−10 −5 0 5
0
2
4
6
8
10
−10 −5 0 5
0
2
4
6
8
10
−10 −5 0 5
0
2
4
6
8
10
−10 −5 0 5
0
2
4
6
8
10
j = i + 1 j = i + 3j = i + 2
j = i + 4 j = i + 5 j = i + 6
j = i + 7 j = i + 8 j = i + 9
φi,j /α φi,j /α φi,j /α
φi,j /α φi,j /α φi,j /α
φi,j /α φi,j /α φi,j /α
V φ
(φ
i,j
)/
τ
V φ
(φ
i,j
)/
τ
V φ
(φ
i,j
)/
τ
FIGURE 15. Effective Potentials in Total Bond Winding Angle φi,j
Potentials are determined from distributions using equation 3. The blue curves are calculated
directly from empirical data and the red curves are calculated using the distribution for φi,i+1 to
generate self-avoiding walks.
45
0 1 2
0
2
4
6
0 1 2 3
0
2
4
6
0 2 4
0
2
4
6
0 2 4
0
2
4
6
0 2 4 6
0
2
4
6
0 2 4 6
0
2
4
6
0 2 4 6
0
2
4
6
0 2 4 6 8
0
2
4
6
0 2 4 6 8
0
2
4
6
j = i + 1 j = i + 3j = i + 2
j = i + 4 j = i + 5 j = i + 6
j = i + 7 j = i + 8 j = i + 9
ri,j+1/R
ri,j+1/R ri,j+1/R ri,j+1/R
ri,j+1/R
ri,j+1/R ri,j+1/R
ri,j+1/R ri,j+1/R
V r
(r
i,j
+1
)/
τ
V r
(r
i,j
+1
)/
τ
V r
(r
i,j
+1
)/
τ
FIGURE 16. Effective potentials for interparticle distance ri,j+1
Potentials are determined from distributions using equation 4. The blue curves are calculated
directly from empirical data and the red curves are calculated using the distribution for φi,i+1 to
generate self-avoiding walks.
46
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
r / ((N+1)R)
V r
(r
)/
τ
N = 10
N = 51
N = 500
FIGURE 17. Extrapolated Effective Potentials for End-to-End distance of Long Chains
Shown is the crossover from rigid behavior (anharmonic) at low N to flexible behavior (harmonic)
at high N .
Semiflexible Behavior
Self avoiding walk simulation algorithm
We create self avoiding random walks with fixed bond length 22m, particle diameter 15mm
and bond angles φi,i+1 drawn from a given predefined distribution P (φi,i+1). In order to build a
chain with N bond angles we employ the following algorithm:
Begin
Draw N bond angles from the distribution
For each particle
Calculate the (x,y) value
Test for intersection with all previous particles and links
EndFor
Repeat until a chain of N non-intersecting links is found
47
CHAPTER VII
CONCLUSION
This work establishes chaotic Faraday waves as a metafluid that a) is wholly consistent
with equilibrium thermal physics, b) has independently tunable temperature and viscosity and c)
provides a new platform on which complex, many-body phenomena may be studied. It hints at a
new universality: systems need only meet the basic criteria of having steady-state and isotropic
chaos to be governed by the fundamental physics of equilibrium thermodynamics. This work
opens up a new set of exciting questions and applications to be addressed in future work.
Chapter 2 established the system of chaotic surface waves as a thermal-like fluid
composed of macroscopic elements, based on passive rheology of buoyant tracers. The motion
of passive tracer particles exhibits a clear crossover from ballistic to diffusive motion. This
motion is functionally identical to Brownian motion, differing only in length and timescales.
This scale difference is the utility of the system; it dilates the constitutive dynamics to easily
observable levels, where complicated optics and imaging setups are not required to obtain
detailed quantitative data. I showed how, from passive tracer motion, one is able to measure
the temperature of the chaotic waves. I also determined how this temperature depends on the
external driving parameters, in particular, that it is directly proportional to the amplitude of
the vertical agitation generating the waves. Further, I showed that the inferred viscous drag
coefficient (calculated using the temperature and diffusion constant and by assuming the Einstein
relation in our system) is consistent with what would be expected in a nearly ideal gas.
Chapter 2 also begins the discussion of applying equilibrium thermal physics to ostensibly
athermal or non-equilibrium systems. The chaotic waves are far from what would be traditionally
considered thermal or equilibrium: they are macroscopic, and their dynamics do not depend on
the room temperature, making them athermal with respect to their surroundings. The waves are
also indisputably non-equilibrium, as their existence depends on the input of energy in the form of
vertical agitation. Nevertheless, they seem to satisfy the basic criteria of kinetic theory in having
steady-state, isotropic chaos. Chapter 2 showed that chaotic Faraday waves are, at the very least,
consistent with an equilibrium thermal fluid, and that they have a well-defined and easy-to-control
temperature. The results however lack a key piece: a direct measure of viscous drag.
48
Chapter 3 addresses this deficiency by introducing an active rheology of chaotic Faraday
waves. Here, by applying a sinusoidally varying torque to a test rotor, I demonstrated the
measurement of a viscous drag coefficient. This “closes the loop” left open by chapter 2. With
direct measurement of temperature, diffusion constant and drag coefficient in the chaotic
waves I showed that this system does in fact obey the Einstein relation, a fundamental result
of equilibrium thermodynamics. This validated our assumptions regarding the thermal-like
nature of chaotic Faraday waves. This now leads to broader implications: perhaps, so long
as the fundamental units of a system exhibit isotropic, steady-state chaos, the system can be
treated with the same physics as an equilibrium thermal system. This suggests that equilibrium
thermodynamics may be extended to certain classes of otherwise non-equilibrium or athermal
systems, or, put another way, that equilibrium thermodynamics may be more universal than
previously thought.
Chapter 3 also showed the systematic dependance of the viscous drag coefficient on input
parameters. The viscous drag imparted by the chaotic waves is proportional to the product
of the vertical driving amplitude and frequency. Considering again that the temperature of
the waves is proportional only to the vertical driving amplitude, this now shows that chaotic
Faraday waves form a fluid that has independently tunable viscosity and temperature. This is
a remarkable property because in conventional fluids, there exists a strict relationship between
viscosity and temperature. Here, no such restriction exists. Instead, this fluid need only obey
the Einstein relation but is otherwise free to move about the temperature-diffusion constant-
viscous drag coefficient phase space. Additionally, my analysis showed that the drag coefficient
was independent of particle velocity, meaning that, for the range of parameters relevant to my
experiment, the chaotic Faraday waves are a Newtonian fluid.
When taken together, chapters 2 and 3 demonstrate that chaotic Faraday waves may
be considered a thermal, Newtonian, tunable metafluid. I call this system a metafluid because
it is constructed using the same philosophy behind all metamaterials: replacing atoms and
molecules with larger, athermal (with respect to the room or bath temperature) elements as the
fundamental unit from which emergent material properties are derived. While there is still work
to be done in rigorously characterizing the chaotic Faraday waves as a thermal metafluid, we may
still use it as a novel platform on which to study analogs to complex microscopic phenomena.
49
Chapter 4 shows one such application. Here, I constructed buoyant articulated chains as
macroscopic analogs to short-chain polymers and placed them in the chaotic Faraday waves. This
demonstrated the utility of the waves as a source of thermal agitation. Whereas in a microscopic
system, a polymer is allowed to explore its configurational space through random collisions
with the surrounding atoms and molecules, instead my polymer analog is allowed to explore its
configurational space through collisions with macroscopic chaotic wave excitations. My analysis
shows that, strikingly, these macroscopic polymer analogs are still entirely consistent with the
well-studied physics of real polymers.
Again, the utility of the chaotic wave system is in its scale. Whereas monomer-level
behavior must be inferred from bulk behavior in molecular systems, here we may observe the
dynamics of the polymer from the monomer scale up to the chain scale. This suggests that this
system will be able to shed light on other microscopic phenomena that occur at length and/or
time scales too small to easily observe.
There are many other applications that may be pursued in the future. The most
tempting is the study of many-particle phenomena. For example, buoyant particles with similar
hydrophobicity tend to attract on the surface of a fluid via the popularly-termed “Cheerios effect,”
after the way Cheerios tend to clump together on the surface of milk [78]. This leads to passive
agregation of buoyant particles at the air-water interface. At several points during my graduate
career I have taken exploratory forays into many-particle problems. Though I have not achieved
any powerful quantitative results, I have made several qualitative observations that I believe point
to the potential for impactful science. For example, if many uniform particles are placed in the
chaotic Faraday waves, they will tend to anneal into crystal structures at low shaker amplitudes.
At higher amplitudes (and thus higher temperature of the chaotic waves), small crystalline
clusters will sublimate into a gas-like state of free particles. Thus, we may observe each individual
particle in a material as it undergoes a phase transition. A serious systematic difficulty arises for
large clusters, however, in that the dense packing of particles in the center of a cluster damps
out the surface waves, effectively lowering the internal temperature of the cluster to our system’s
absolute zero (the point at which there are no chaotic waves). Nevertheless, I believe that through
careful design of experiments the chaotic Faraday wave system will prove to be a powerful tool in
studying phase transitions.
50
Another future direction for this work is to examine self-assembly and pattern formation.
Self-assembly is an active and exciting area of research [51], with applications to both inorganic
nanotechnology [79] and biology [80, 81]. In fluid interface systems, the Cheerios effect can in fact
be tuned via particle hydrophobicity and shape. Theoretically, buoyant particles could be designed
to bind to or repel from one another with preferred orientations, self-assembling into larger, more
complex macrostructures. For example, the chains of particles described in chapter 4 could be
modified to be heterogenous, leading them to self-fold into particular structures. The Faraday
waves provide an element of thermal agitation that allow particles to passively explore their state
space and anneal into their optimal structure.
There are a number of directions future researchers may carry this work in addition to
the aforementioned applications. To begin with, there are deeper characterization yet to try in
the chaotic Faraday waves. My research has focused on the use of tracer particles. We find that
the tracers diffuse identically to thermal particles, but we do not say anything about the waves
themselves. One possible method to study any diffusive nature of the waves would be to perform
a scattering-style experiment, similar to those used to study diffusive dynamics in colloidal glasses
[82]. The method is relatively simple: image the waves at a frequency commensurate with that of
the shaker, and examine the spatial correlations of the waves for frames seperated by various lag
times. The decay constant of these correlations can be used to determine if the waves themselves
are diffusive or not.
We have demonstrated many canonical results of equilibrium thermodynamics with
this work, but there are many more out there. Do the laws of thermodynamics exist in our
system? For example, can we recover a zeroth law? To answer this question, one would need to
demonstrate energy transport between two coupled chaotic Faraday wave systems. This could
be accomplished by embedding test rotors in each of two systems and couple the rotors via
electromechanical means. If the two baths are held at constant but different temperatures, a
zeroth law of thermodynamics would result in a measurable energy flux travelling from the probe
in the hotter reservoir to the probe in the colder reservoir. A similar experiment has already
been performed in a granular gas system [83] and indeed confirmed that energy flows from hot
to cold in accordance with the notion of irreversibility, at a rate proportional to the temperature
difference between the baths. Not only does this previous work show a proof-of-concept for how
51
one might measure the thermodynamics of a macroscopic, non-equilibrium system, it also once
again points toward a broader universality of thermal physics.
I have helped design an experiment in our lab that is still in its infancy to probe this notion
of universality. I have proposed that we build a granular gas system and extract from it thermal
behavior in the same way we have in the Faraday waves. Granular gasses are not a new system for
study [84], but I have proposed a new form of agitation. Granular gasses are generally created
by shaking grains or blowing air through them, often from beneath. This is an efficient way
for achieving the desired gas-like phase, but it inherently introduces anisotropy, thus leading it
away from the traditional tools of thermal physics. The system I have proposed uses isotropic,
turbulent (chaotic) air input as a means of fluidization for a two-dimensional granular gas. The
goal of the project, though I will no longer be present to see it through, is to once again attempt
to recover the Einstein relation in the gas just as we did in the chaotic Faraday waves. If the same
physics emerges, we will have two examples of macroscopic, non-equilibrium systems that exhibit
behavior consistent with equilibrium thermal systems, despite satisfying only the basic criteria of
steady-state chaos and isotropy. A successful result will lend further support to the notion that
thermodynamics may be more universal than previously thought.
I began this dissertation by saying that materials science is a lot like cooking. We do
our best to understand the ingredients, then we mix them together to make new, interesting
things. The properties of the new things must necessarilly be determined by the properties of
the ingredients. When baking a cake, it used to be said that “there’s no substitute for an egg.”
We now know that to be false; all it took was experimentation. I believe my work parallels
this sentiment. Thermodynamics has been traditionally applied to microscopic systems, but
that tradition does not preclude a broader universality. By stripping the physics down to bare
fundamentals, we have found that macroscopic systems can reproduce equilibrium thermal
behavior, and in so doing we have found a substitute for the proverbial egg.
52
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