## Geographic Tongue as a Reaction-Diffusion System

### Margaret K. McGuire, Chase A. Fuller, John F. Lindner, Niklas Manz

*Chaos: An Interdisciplinary Journal of Nonlinear Science*, volume 31, pages 033118(1-9) (2021)

Geographic tongue or benign migratory glossitis is a condition of unknown cause characterized by chronic lesions that slowly migrate across the surface of the tongue. The condition’s characteristic wavefronts suggest it can be modeled as a reaction-diffusion system. Here we present a model for geographic tongue pattern evolution using reaction-diffusion equations applied to portions of spheroids and paraboloids that approximate a tongue shape. We demonstrate that the observed patterns of geographic tongue lesions can be explained by propagating reaction-diffusion waves on these variably curved surfaces.

## Disruption and recovery of reaction-diffusion wavefronts interacting with concave, fractal, and soft obstacles

### Yang F. Yu, Chase A. Fuller, Margaret K. McGuire, Rebecca Glaser, Nathaniel J. Smith, Niklas Manz, John F. Lindner

*Physica A*, volume 565, pages 125536(1-15) (1 March 2021)

In a recent publication (Smith et al., 2019), we documented the distinct recovery of reaction-diffusion wavefronts disrupted by hard convex obstacles. Here, we extend that work to include concave, spiral, fractal, random, and soft obstacles. Curvature dependent wavefront velocities ultimately restore the wavefronts, with perturbations that decay as power-law functions of time. But concave, spiral, and fractal obstacles can sustain wavefronts locally for long times. Softobstacles with variable diffusivity, either intrinsically or due to light sensitivity, can enforce one-way propagation and,appropriately configured, can locally and indefinitely sustain incident wavefronts, creating clocks or repeaters, beating hearts for these excitable systems.

## Modeling and measuring the absorption-induced expansion of swellable organically modified silica

### Paul A. Bonvallet, John F. Lindner, Brian Corbin, Duncan Crow, Ryan Reffner, Theresa Albon Riedl, Susan Y. Lehman, Paul Edmiston

*AIP Advances*, volume 10, pages 065136(1-13) (June 2020)

We present a theoretical framework that describes the force generated by the expansion of swellable organically modified silica (SOMS) upon exposure to organic solvent. The total swelling force, produced from the differential contributions of localized swelling domains, is related logarithmically to the amount of material confined to a rigid space. The model is further parameterized according to the physical dimensions of that space and the intrinsic swellability of SOMS. This mathematical representation is validated experimentally with a piston force sensor apparatus, which shows that the solvent-induced force and pressure exerted by SOMS increases logarithmically with the amount of material that is present. Comparison with theory implies that the commercially available varieties of SOMS CyclaSorb™ and Osorb™ have Young’s expansion moduli Y_{c} ∼0.8 MPa and Y_{o}∼0.7 MPa respectively, which succinctly quantifies their relative behavior. The theoretical model and experimental technique should be widely applicable to other swellable and stimuli-responsive materials.

## The Twists and Turns of a Dynamic, Nonlinear Career Path

### Niklas Manz

#### Career Path Story in *ACS Graduate & Postdoctoral Chemist Magazine* (15 December 2019)

What is life if we don’t even make an effort to follow our dreams?

At a glance, the career path of someone you admire can look like a series of simple steps that led to a dream job—the person got a degree, got experience, got awards, got advancements, and found a perfect balance between career and life. Sounds simple, right? Such a snapshot perspective misses the truth of many career paths that are nonlinear, full of surprising twists and turns, and as distinctive as the values, personality, and strengths of a particular individual. No one can give you a perfect map toward your own career success. The most that can be offered are some pathfinding wisdom and helpful resources to assist you in making the most of your unique journey. We hope the adventures that Dr. Niklas Manz shares here will give you helpful tips and encouragement in figuring out your own career path, no matter where in the world of science it’s headed.

## Patterns and Humans

### Niklas Manz, Flavio H. Fenton

#### in Kinko Tsuji and Stefan C. Müller (Eds.), *Spirals and Vortices*, Springer International Publishing, pages 217-224 (5 April 2019)

The appearance of spiral structures on the human tongue (geographic tongue) and skin (pathological rashes) are described. Affected and often migrating areas on the tongue can be bistable, patch-like fronts or waves. Pathological rashes with an expanding region of redness is often a result of an autoimmune disease in which the human immune system becomes hyperactive and attacks healthy tissues. In addition, we show a rotating spiral created by humans, which has been described as a reaction-diffusion waves in an excitable medium.

## Disruption and Recovery of Reaction-Diffusion Wavefronts Colliding with Obstacles

### Nathaniel J. Smith, Rebecca Glaser, Vincent W. H. Hui, John F. Lindner, Niklas Manz

*Physica A*, volume 517, pages 307-320 (1 March 2019)

We study the damage to and restoration of planar reaction-diffusion wavefronts colliding with convex obstacles in narrow two-dimensional channels using finite-difference numerical integration of the Tyson-Fife reduction of the Oregonator model of the Belousov-Zhabotinsky reaction. We characterize the obstacles’ effects on the wavefront shape by plotting wavefront delay versus time. Due to the curvature dependent wavefront velocities, the initial planar wavefront (or iso-concentration line) is restored after a relaxation period that can be characterized by a power-law. We find that recovery times are insensitive to obstacle concatenation or to the upstream obstacle shape but are sensitive to the downstream shape, with a vertical back side causing the longest disruption. Delays vary cyclically with obstacle orientations. The relaxation power-laws confirm that larger obstacles produce larger wavefront delays and longer recovery times, and for a given area larger obstacle width-to-length ratios produce longer delays. Possible applications include elucidating the effect of inhomogeneities on wavefront recovery in cardiac tissue.

## Hannay’s Hoop Beyond Asymptotics

### Hwan Bae, Norah Ali, John F. Lindner

*Chaos: An Interdisciplinary Journal of Nonlinear Science*, volume 28, pages 083107(1-6) (2018)

Certain systems do not completely return to themselves when a subsystem moves through a closed circuit in physical or parameter space. A geometric phase, known classically as Hannay’s angle and quantum mechanically as Berry’s phase, quantifies such anholonomy. We study the classical example of a bead sliding frictionlessly on a slowly rotating hoop. We elucidate how forces in the inertial frame and pseudo-forces in the rotating frame shift the bead. We then computationally generalize the effect to arbitrary — not necessarily adiabatic — motions. We thereby extend the study of this classical geometric phase from theory to experiment via computation, as we realize the dynamics with a simple apparatus of wet ice cylinders sliding on a polished metal plate in 3D printed plastic channels.

## A wind-powered one-way bistable medium with parity effects

### Tessa Rosenberger, Graham Schattgen, Matthew King-Smith, Prakrit Shrestha, Katsuo J. Maxted, John F. Lindner

*Chaos: An Interdisciplinary Journal of Nonlinear Science*, volume 27, 023114(1-5) (2017)

We describe the design, construction, and dynamics of low-cost mechanical arrays of 3D-printed bistable elements whose shapes interact with wind to couple them one-way. Unlike earlier hydromechanical unidirectional arrays, our aeromechanical one-way arrays are simpler, easier to study, and exhibit a broader range of phe- nomena. Solitary waves or solitons propagate in one direction at speeds proportional to wind speeds. Periodic boundaries enable solitons to annihilate in pairs in arrays with an even number of elements. Solitons propagate indefinitely in odd arrays that frustrate pairing. Large noise spontaneously creates soliton-antisoliton pairs. Soliton annihilation times increase quadratically with initial separations, as expected for random-walk models of soliton collisions.

## Heat capacity anomaly in a self-aggregating system: Triblock copolymer 17R4 in water

### L.V. Dumancas, D.E. Simpson, D.T. Jacobs

*Journal of Chemical Physics*, volume 142, 174902 (2015)

The reverse Pluronic, triblock copolymer 17R4 is formed from poly(propylene oxide) (PPO) and poly(ethylene oxide) (PEO): PPO_{14} – PEO_{24} – PPO_{14}, where the number of monomers in each block is denoted by the subscripts. In water, 17R4 has a micellization line marking the transition from a unimer network to self-aggregated spherical micelles which is quite near a cloud point curve above which the system separates into copolymer-rich and copolymer-poor liquid phases. The phase separation has an Ising-like, lower consolute critical point with a well-determined critical temperature and composition.We have measured the heat capacity as a function of temperature using an adiabatic calorimeter for three compositions: (1) the critical composition where the anomaly at the critical point is analyzed, (2) a composition much less than the critical composition with a much smaller spike when the cloud point curve is crossed, and (3) a composition near where the micellization line intersects the cloud point curve that only shows micellization. For the critical composition, the heat capacity anomaly very near the critical point is observed for the first time in a Pluronic/water system and is described well as a second-order phase transition resulting from the copolymer-water interaction. For all compositions, the onset of micellization is clear, but the formation of micelles occurs over a broad range of temperatures and never becomes complete because micelles form differently in each phase above the cloud point curve. The integrated heat capacity gives an enthalpy that is smaller than the standard state enthalpy of micellization given by a van’t Hoff plot, a typical result for Pluronic systems.

## Broad Absorption Line Variability in Radio-Loud Quasars

### C. A. Welling, B. P. Miller, W. N. Brandt, D. M. Capellupo, R. R. Gibson

*Monthly Notices of the Royal Astronomical Society*, volume 440, pages 2474-2497 (2014)

We investigate C IV broad absorption line (BAL) variability within a sample of 46 radio-loud quasars (RLQs), selected from SDSS/FIRST data to include both core-dominated (39) and lobe-dominated (7) objects. The sample consists primarily of high-ionization BAL quasars, and a substantial fraction have large BAL velocities or equivalent widths; their radio luminosities and radio-loudness values span ~2.5 orders of magnitude. We have obtained 34 new Hobby-Eberly Telescope (HET) spectra of 28 BAL RLQs to compare to earlier SDSS data, and we also incorporate archival coverage (primarily dual-epoch SDSS) for a total set of 78 pairs of equivalent width measurements for 46 BAL RLQs, probing rest-frame timescales of ~80-6000 d (median 500 d). In general, only modest changes in the depths of segments of absorption troughs are observed, akin to those seen in prior studies of BAL RQQs. Also similar to previous findings for RQQs, the RLQs studied here are more likely to display BAL variability on longer rest-frame timescales. However, typical values of |Delta_EW| and |Delta_EW|/ ⟨ EW ⟩ are about 40±20% lower for BAL RLQs when compared with those of a timescale-matched sample of BAL RQQs. Optical continuum variability is of similar amplitude in BAL RLQs and BAL RQQs; for both RLQs and RQQs, continuum variability tends to be stronger on longer timescales. BAL variability in RLQs does not obviously depend upon their radio luminosities or radio-loudness values, but we do find tentative evidence for greater fractional BAL variability within lobe-dominated RLQs. Enhanced BAL variability within more edge-on (lobe-dominated) RLQs supports some geometrical dependence to the outflow structure.

## Avalanches on a conical bead pile: scaling with tuning parameters

### S. Y. Lehman, Elizabeth Baker, Howard A. Henry, Andrew J. Kindschuh, Larry C. Markley, Megan B. Browning, Mary E. Mills, R. Michael Winters IV, D. T. Jacobs

*Granular Matter*, volume 14, pages 553-561 (2012)

Uniform spherical beads were used to explore the behavior of a granular system near its critical angle of repose on a conical bead pile. We found two tuning parameters that could take the system to a critical point where a simple power-law described the avalanche size distribution as predicted by self-organized criticality, which proposed that complex dynamical systems self-organize to a critical point without need for tuning. Our distributions were well described by a simple power-law with the power τ = 1.5 when dropping beads slowly onto the apex of a bead pile from a small height. However, we could also move the system from the critical point using either of two tuning parameters: the height from which the beads fell onto the top of the pile or the region over which the beads struck the pile. As the drop height increased, the system did not reach the critical point yet the resulting distributions were independent of the bead mass, coefficient of friction, or coefficient of restitution. All our apex-dropping distributions for any type of bead (glass, stainless steel, zirconium) showed universality by scaling onto a common curve with τ = 1.5 and σ = 1.0, where 1/σ is the power of the tuning parameter. From independent calculations using the moments of the distribution, we find values for τ = 1.6 ± 1.0 and σ = 0.91 ± 0.15. When beads were dropped across the surface of the pile instead of solely on the apex, then the system also moved from the critical point and again the avalanche size distributions fell on a common curve when scaled similarly using the same values of τ and σ. We also observed that an hcp structure on the base of the pile caused an emergent structure in the pile that had six faces with some fcc or hcp structure.

## What the oblique parameters S, T, and U and their extensions reveal about the 2HDM: a numerical analysis

### Gerhardt Funk, Deva O’Neil, and R. Michael Winters

*International Journal of Modern Physics A*, volume 27, 1250021 [21 pages] (2012)

The oblique parameters S, T and U and their higher-order extensions (V, W and X) are observables that combine electroweak precision data to quantify deviation from the Standard Model. These parameters were calculated at one loop in the basis-independent CP-violating Two-Higgs Doublet Model (2HDM). The scalar parameter space of the 2HDM was randomly sampled within limits imposed by unitarity and found to produce values of the oblique parameters within experimental bounds, with the exception of T. The experimental limits on T were used to predict information about the mass of the charged Higgs boson and the difference in mass between the charged Higgs boson and the heaviest neutral Higgs boson. In particular, it was found that the 2HDM predicts -600 GeV < m_{H}^{±} – m_{3} < 100 GeV, with values of m_{H}^{±} > 250 GeV being preferred. The mass scale of the new physics (M_{NP}) produced by random sampling was consistently fairly high, with the average of the scalar masses falling between 400 and 800 GeV for Y_{2} = m_{W}^{2}, although the model can be tuned to produce a light neutral Higgs mass (~120 GeV). Hence, the values produced for V, W and X fell well within 0.01 of zero, confirming the robustness of the linear expansion approximation. Taking the CP-conserving limit of the model was found to not significantly affect the values generated for the oblique parameters.

## Refractive Index of Liquid D_{2}O for Visible Wavelengths

### Hosanna Odhner and D.T. Jacobs

*Journal of Chemical and Engineering Data* volume 57, pages 166-168 (2012)

The index of refraction for D_{2}O at common wavelengths was measured for several temperatures at atmospheric pressure. While heavy water’s refractive index was precisely measured decades ago using the transition lines of elements, those wavelengths are seldom used now that inexpensive lasers provide a range of available wavelengths. We review those measurements, note some inconsistencies between research groups, and fit the best of the literature data to a simple equation that allows an easy calculation for the refractive index of D_{2}O with an accuracy of ±0.0002 at any visible wavelength and between (278 and 359) K. To verify the equation, we then compare the calculated refractive index to our measured values for three He-Ne laser wavelengths (543.5, 594.1, and 632.8) nm over a temperature range from (288 to 338) K and find good agreement.

## Micellization and Phase Separation for Triblock Copolymer 17R4 in H_{2}O and in D_{2}O

### Alison Huff, Kelly Patton, Hosanna Odhner, Donald T. Jacobs, Bryna C. CLover, Sandra C. Greer

*Langmuir,* volume 27(5), pages 1707-1712 (2011)

The reverse Pluronic, triblock copolymer 17R4 is formed from poly(propylene oxide) (PPO) and poly(ethylene oxide) (PEO): PPO_{14}-PEO_{24}-PPO_{14}, where the subscripts denote the number of monomers in each block. In water, 17R4 shows both a transition to aggregated micellar species at lower temperatures and a separation into copolymer-rich and copolymer-poor liquid phases at higher temperatures. For 17R4 in H_{2}O and in D_{2}O, we have determined (1) the phase boundaries corresponding to the micellization line, (2) the cloud point curves marking the onset of phase separation at various compositions, and (3) the coexistence curves for the phase separation (the compositions of coexisting phases). In both H_{2}O and in D_{2}O, 17R4 exhibits coexistence curves with lower consolute temperatures and compositions that differ from the minima in the cloud point curves; we take this as an indication of the polydispersity of the micellar species. The coexistence curves for compositions near the critical composition are described well by an Ising model. For 17R4 in both H_{2}O and D_{2}O, the critical composition is 0.22 ± 0.01 in volume fraction. The critical temperatures differ: 44.8 degrees C in H_{2}O and 43.6 degrees C in D_{2}O. The cloud point curve for the 17R4/D_{2}O is as much as 9 degrees C lower than in H_{2}O.

## Electronic and mechanical realizations of one-way coupling in one and two dimensions

### Barbara J. Breen, Aaron B. Doud, Jamie R. Grimm, Andrew H. Tanasse, Stuart J. Tanasse, John F. Lindner, Katsuo J. Maxted

*Physical Review E,* volume 83, pages 037601(1-4) (2011)

One-way or unidirectional coupling is a striking example of how topological considerations — the parity of an array of multistable elements combined with periodic boundary conditions — can qualitatively influence dynamics. Here we introduce a simple electronic model of one-way coupling in one and two dimensions and experimentally compare it to an improved mechanical model and an ideal mathematical model. In two dimensions, computation and experiment reveal richer one-way coupling phenomenology: in media where two-way coupling would dissipate all excitations, one-way coupling enables soliton-like waves to propagate in different directions with different speeds.

## Induced smectic phases in phase diagrams of binary nematic liquid crystal mixtures

### Tsang-Min Huang, Kathleen McCreary, Shila Garg, Thein Kyu

*Journal of Chemical Physics,* volume 134, 124508 (2011)

To elucidate induced smectic A and smectic B phases in binary nematic liquid crystal mixtures, a generalized thermodynamic model has been developed in the framework of a combined Flory-Huggins free energy for isotropic mixing, Maier-Saupe free energy for orientational ordering, McMillan free energy for smectic ordering, Chandrasekhar-Clark free energy for hexagonal ordering, and phase field free energy for crystal solidification. Although nematic constituents have no smectic phase, the complexation between these constituent liquid crystal molecules in their mixture resulted in a more stable ordered phase such as smectic A or B phases. Various phase transitions of crystal-smectic, smectic-nematic, and nematic-isotropic phases have been determined by minimizing the above combined free energies with respect to each order parameter of these mesophases. By changing the strengths of anisotropic interaction and hexagonal interaction parameters, the present model captures the induced smectic A or smectic B phases of the binary nematic mixtures. Of particular importance is the fact that the calculated phase diagrams show remarkable agreement with the experimental phase diagrams of binary nematic liquid crystal mixtures involving induced smectic A orinduced smectic B phase.

## Tracking Stars, Sun, and Moon to Connect with the Universe

### Todd C. McAlpine, Corwin Atwood-Stone, Travis Brown, John F. Lindner

*American Journal of Physics,* volume 78, pages 1128-1131 (November 2010)

We describe the theory, design, and construction of simple electromechanical devices that automatically and continually track celestial objects. As Earth rotates and revolves, a star tracker always points at a star or other object fixed to the celestial sphere, such as the center of the Milky Way galaxy. A planet tracker can fixate on any celestial object, including planets, the Sun, or the Moon. A sidereal clock mechanism drives the star tracker, and software which encoding astronomical algorithms controls an inexpensive robot that drives the planet tracker. The star tracker acts like a gyroscope, rigidly oriented in space, despite Earth’s motion. Both trackers indicate the passing of time just like clocks and calendars. The resulting lecture, hallway, or museum displays promote awareness of and excitement about our place in the universe.

## Order and chaos in the rotation and revolution of a line segment and a point mass

### John F. Lindner, Jacob Lynn, Frank W. King, Amanda Logue

*Physical Review E,* volume 81, pages 036208(1-10) (2010)

We study the classical dynamics of two bodies, a massive line segment or slash (/) and a massive point or dot (.), interacting gravitationally. For this slashdot (/.) body problem, we derive algebraic expressions for the force and torque on the slash, which greatly facilitate analysis. The diverse dynamics include a stable synchronous orbit, generic chaotic orbits, sequences of unstable periodic orbits, spin stabilized orbits, and spin-orbit coupling that can unbind the slash and dot. The extension of the slash provides an extra degree of freedom that enables the interplay between rotation and revolution. In this way, the slashdot body problem exhibits some of the richness of the three body problem with only two bodies and serves as a valuable prototype for more realistic systems. Applications include the dynamics of asteroid-moonlet pairs and asteroid rotation and escape rates.

## Investigating the inner structure of irregular β-lactoglobulin spherulites

### Kristin R. Domike, Eric Hardin, Doug N. Armstead, Athene M. Donald

*The European Physical Journal E,* volume 29, pages 173-182 (2009)

When β-lactoglobulin in low pH aqueous solutions is exposed to high temperature for extended time, spherulites composed of amyloid fibrils of the β-lactoglobulin protein form. Many of these spherulites have fibrils that radiate out from a centre and, under crossed polarisers, exhibit a symmetric Maltese Cross structure. However, a significant fraction (50 of 101 observed spherulites) of β-lactoglobulin spherulites formed under these conditions demonstrate various forms of irregularity in apparent structure. The irregularities of spherulites structures were qualitatively investigated by comparing optical microscopy images observed under crossed polarisers to computationally produced images of various internal structures. In this way, inner spherulite structures are inferred from microscopy images. Modeled structures that were found to produce computed images similar to some of the experimentally viewed images include fibrils curving as they radiate from a single nucleation point; multiple spherulites nucleating in close proximity to one another; and fibrils curving in opposite directions above and below a single nucleation point.

## Experimental observation of soliton propagation and annihilation in a hydromechanical array of one-way coupled oscillators

### John F. Lindner, Kelly M. Patton, Patrick M. Odenthal, James C. Gallagher, Barbara J. Breen

*Physical Review E,* volume 78, pages 066604(1-5) (2008)

We have experimentally realized unidirectional or one-way coupling in a mechanical array by powering the coupling with flowing water. In cyclic arrays with an even number of elements, soliton-like waves spontaneously form but eventually annihilate in pairs, leaving a spatially alternating static attractor. In cyclic arrays with an odd number of elements, this alternating attractor is topologically impossible, and a single soliton always remains to propagate indefinitely. Our experiments with 14 and 15-element arrays highlight the dynamical importance of both noise and disorder and are further elucidated by our computer simulations.

## Invitation to embarrassingly parallel computing

### Barbara J. Breen, Christine E. Weidert, John F. Lindner, Lisa May Walker, Kasey Kelly, Evan Heidtmann

*American Journal of Physics,* volume 76, pages 347-352 (April/May 2008)

A surprising number of physics problems are well suited to “embarrassingly parallel” computations that do not require complicated software algorithms or specialized hardware. As faculty and students at small institutions, we are readily incorporating parallel computing in diverse levels of our curricula, and we are embracing the opportunity to utilize high performance computing to attack contemporary research problems in summer research, senior theses, and course work. This article describes how we do this in three significant examples: spatiotemporal patterns of one-way coupled oscillators, ray-tracing in curved spacetime, and solar escape as a three-body problem.

## Precession and Chaos in the Classical Two-Body Problem in a Spherical Universe

### John F. Lindner, Martha I. Roseberry, Daniel E. Shai, Nicholas J. Harmon, Katherine D. Olaksen

*International Journal of Bifurcation and Chaos*, volume 18, No. 2, 455-464 (February 2008)

We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe’s radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.

## Universality in 8-arm star polystyrene and methylcyclohexane mixtures near the critical point

### D.T. Jacobs, Clinton I. Braganza, Andy P. Brinck, Adam B. Cohen, Mark A. Lightfoot, Christopher J. Locke, Sarah J. Suddendorf, Henry R. Timmers, Angela L. Triplett, Nithya L. Venkataraman, and Mark T. Wellons

*Journal of Chemical Physics* **127**, 124905 (2007)

Measurements of the coexistence curve and turbidity were made on different molecular mass samples of the branched polymer-solvent system 8-arm star polystyrene in methylcyclohexane near its critical point. We confirmed that these systems belong in the Ising universality class. The location of the critical temperature and composition as well as the correlation length, susceptibility, and coexistence curve amplitudes were found to depend on molecular mass and the degree of branching. The coexistence curve diameter had an asymmetry that followed a “complete scaling” approach. All the coexistence curve data could be scaled onto a common curve with one adjustable parameter. We found the coexistence curve amplitude to be about 12% larger for branched than linear polystyrenes of the same molecular mass in either solvent cyclohexane or methylcyclohexane. The twoscale- factor universality ratio R was found to be independent of molecular mass or degree of branching.

## Heat capacity of the liquid-liquid mixture nitrobenzene and dodecane near the critical point

### Nathan J. Utt, S.Y. Lehman, and D.T. Jacobs

*Journal of Chemical Physics* **127**, 104505 (2007)

The heat capacity of the liquid-liquid mixture nitrobenzene-dodecane has been measured for the first time near its upper critical consolute point using an adiabatic calorimeter. The theoretical expression for the heat capacity near the critical point was applied to our combined data runs. The critical exponent α was determined to be 0.124±0.006, which was consistent with theoretical predictions. When α was fixed at its theoretical value of 0.11, our value for the amplitude ratio A^{+}/A^{–} = 0.58±0.02 was consistent with experimental determinations and theoretical predictions. However, the two-scale-factor universality ratio X, now consistent among experiments and theories with a value between 0.019-0.020, was violated in this system when using a previously published value for the correlation length.

## Self-erasing perturbations of Abelian sandpiles

### Jeffrey R. Moffitt, Patrick Macdonald, John F. Lindner

*Physical Review E,* volume 70, 016203(1-6) (2004)

We investigate generalized seeding of the attracting states of Abelian sandpile automata and find there exists a class of global perturbations of such automata that are completely removed by the natural local dynamics. We derive a general form for such *self-erasing perturbations* and demonstrate that they can be highly nontrivial. This phenomenon provides a new conceptual framework for studying such automata and suggests possible applications for data protection and encryption.

## Dielectric Properties of a Nematic Binary Mixture

### Shila Garg and Tom Spears (The College of Wooster)

*Molecular Crystals Liquid Crystals,* volume 409, 335-342 (2004)

In this paper, a phase diagram is developed for the molar mixtures of nematic liquid crystals of 5CB and MBBA. In order to understand the interaction of the two systems, dielectric permittivities ε_{||} and ε_{⊥} were measured for mixtures of various concentrations. The usual assumption is that in the absence of chemical reactions the bulk physical properties add up as a weighted sum of the individual properties. Our dielectric permittivity data clearly show a correlation to the phase diagram and the existence of the induced phase. In order to understand the interactions from a fundamental level, we modeled the 5CB and MBBA molecules using a Silicon Graphics O2 workstation running the software Spartan 5.1. Different electrical surfaces were calculated for a geometrically optimized molecule. Our investigations support the idea of strong charge interactions between the nematic systems.

## Turbidity determination of the critical exponent η in the liquid-liquid mixture of methanol and cyclohexane

### Amy Lytle and D. T. Jacobs

*Journal of Chemical Physics* **120**, 5709-5716 (22 March 2004)

The turbidity of the liquid-liquid mixture methanol-cyclohexane has been measured very near its critical point and used to test competing theoretical predictions and to determine the critical correlation-correction exponent η. By measuring the ratio of the transmitted to incident light intensities over five decades in reduced temperature, we are able to determine that Ferrell’s theoretical prediction for the turbidity explains the data with the correlation length amplitude ξ_{0}=0.330±0.003 nm and critical exponents η=0.041±0.005 and ν=0.632±0.002. These values are consistent with the values measured before for ξ_{0} in this system and with the exponents predicted by theory. The data allow five different theoretical expressions to be tested and to select two as begin equivalent when very close to the critical point.

## The Flux Creep Automaton

### John F. Lindner, Scott B. Hughes, David J. Miller, Bradley C. Thomas, Kurt Wiesenfeld

*International Journal of Bifurcation and Chaos, *volume 14, pages 1155-1175 (March 2004)

We study a cellular automaton derived from the phenomenon of magnetic flux creep in two-dimensional granular superconductors. We model the superconductor as an array of Josephson junctions evolving according to a set of coupled ordinary differential equations. In the limit of slowly increasing magnetic field, we reduce these equations to a simple cellular automaton. The resulting discrete dynamics, a stylized version of the continuous dynamics of the differential equations, is equivalent to the dynamics of a gradient sand pile automaton. We study the dynamics as we vary the symmetry of the underlying lattice and the shape of its boundary. We find that the “simplest” realization of the automaton, on a square lattice with commensurate boundaries, results in especially simple dynamics, while “generic” realizations exhibit more complicated dynamics characterized by statistics with broad distributions, even in the absence of noise or disorder.

## Electric-Field Induced Nucleation and Growth of Focal Conic and Stripe Domains in Smectic A

### Shila Garg, Kirstin Purdy, Erica Bramley (The College of Wooster)

I. I. Smalyukh, and O. D. Lavrentovich (Chemical Physics Interdisciplinary Program and Liquid Crystal Institute, Kent State University)

*Liquid Crystals* **30**, 1377 (2003)

We study the electric field-induced first-order transition from a homeotropic smectic A structure into a polydomain texture that occurs through nucleation of toric focal conic domains (TFCDs). The process involves two steps: first nucleation of TFCDs of a size larger than a critical radius a*, and then a steady growth of TFCD to a secondary critical radius a** when surface anchoring effects become dominant and cause a transition from a circular TFCD to an elongated stripe domain (SD). Studies are performed for pure smectic A materials and for smectic A doped with kunipia nanoparticles. Non-destructive 3D imaging with the fluorescence confocal polarizing microscopy (FCPM) shows that the field-induced TFCDs can nucleate in the smectic A bulk. Clay particles reduce the energy barrier for nucleation as they distort the smectic A layers and thus increase the ground state energy. Simple elastic models of TFCD and SD allow us to describe the qualitative features of the observed phenomena.

## Optimal Exit: Solar Escape as a Restricted 3-body Problem

### Nicholas Johann Harmon, Christine Leidel, John F. Lindner

*American Journal of Physics,* volume 71, pages 871-877 (2003)

We analyze solar escape as a special case of the restricted three-body problem. We systematically vary the parameters of our model solar system to show how optimal launch angle and minimum escape speed depend on the mass and size of Earth. In some cases, it is best to launch near the direction of Earth’s motion, but slightly outward; in other cases, it is best to launch near the perpendicular to Earth’s motion, but inward, toward Sun (so as to obtain a solar gravity assist). Between direct escapes for high launch speeds and trapped trajectories for low launch speeds is an irregular band of chaotic orbits that reveals something of the true complexity of solar escape and the three-body problem.

## Self-Organized Criticality in a Bead Pile

### Rachel M. Costello, K.L. Cruz, Christie Egnatuk, D.T. Jacobs, Matthew C. Krivos, Tim Sir Louis, Rebecca J. Urban, Hanna Wagner

*Physical Review E* **67**, 041304 (2003)

Self-organized criticality has been proposed to explain complex dynamical systems near their critical points. This experiment examined a monodisperse conical bead pile and how the distribution of avalanches is affected by the pattern of beads glued on a base, by the size or shape of the base, and by the height at which each bead was dropped onto the pile. By measuring the number of avalanches of a given size that occurred during the experiment, the resulting distribution could be compared to a power law description. When the beads were dropped from a small height, all of the data were consistent with a simple power-law of exponent 1.5, which is the mean-field model value. The data showed that neither the bead pattern on the base nor the base size or shape significantly affected the power-law behavior. This is the first time that the mean-field exponent has been observed in a granular pile. However, when the bead is dropped from different heights then the power-law description breaks down and a power-law times an exponential is more appropriate. We found a scaling relationship in the distribution of avalanches for different heights and relate our data to an energy dissipation model. We both confirm self-organized criticality and observe deviations from it.

## Phase Transitions in a nematic binary mixture

### A. Christine Rauch, Shila Garg, D. T. Jacobs

*Journal of Chemical Physics* **116**, 2213-228 (1 February 2002)

Our objective was to study mixtures of nematic liquid crystals with dissimilar dielectric anisotropies but similar phase properties. Using light scattering and microscopy, we have established the phase boundaries and transition widths of mixtures of 4′-*n*-pentyl-4-cyanobiphenyl and 4′-methoxybenzylidene-4-butylaniline. In addition to the isotropic-nematic transition, there is a second induced phase for certain concentrations, which we conclude is an induced smectic B phase. Recent theoretical works provide a model for nematic to induced smectic A transition by combining Flory-Huggins and Maier-Saupe-McMillan theories. From our phase transition data and the application of the above theoretical framework, we conclude that there is a possibility of strong interaction between the two mesogens that produces the smectic B phase.

## Monostable Array Enhanced Stochastic Resonance

### John F. Lindner, Barbara J. Breen, Meghan E. Wills, Adi R. Bulsara, William L. Ditto

*Physical Review E,*volume 63, 051107 (2001)

We present a simple nonlinear system that exhibits *multiple* distinct stochastic resonances. By adjusting the noise and coupling of an array of underdamped, monostable oscillators, we modify the array’s natural frequencies so that the spectral response of a typical oscillator in an array of *N* oscillators exhibits *N* – 1 different stochastic resonances. Such families of resonances may elucidate and facilitate a variety of noise-mediated cooperative phenomena, such as Noise Enhanced Propagation, in a broad class of similar nonlinear systems.

## Heat Capacity and Turbidity Near the Critical Point of Succinonitrile-Water

### A.W. Nowicki, M. Ghosh, S. M. McClellan, and D.T. Jacobs

*Journal of Chemical Physics* **114**, 4625 (2001)

Both the heat capacity and the turbidity of the liquid-liquid mixture succinonitrile-water near its upper critical consolute point were measured and two amplitude relations were tested. Using an adiabatic calorimeter to measure the heat capacity and the transmitted light intensity to determine the turbidity, precise and reproducible data determined the critical exponents α, η, and γ, consistent with theoretical predictions. The correlation length ξ_{o} = 1.68±0.004 nm was determined from the turbidity experiment while the heat capacity amplitudes were A_{+} = 0.0543±0.0004 J/(cm^{3}K) in the one-phase region and A_{–} = 0.1013±0.0004 J/(cm^{3}K) in the two-phase region. The amplitude ratio A_{+}/A_{–} = 0.536±0.005 was consistent with other experimental determinations in liquid-liquid mixtures or liquid-vapor systems, and with recent theoretical predictions. The two-scale-factor universality ratio Χ, now consistent among experiments and theories with a value between 0.017 and 0.020, was determined to be 0.0187±0.0013.

## Heat capacity of the liquid-liquid mixture per-fluoroheptane and 2,2,4-trimethylpentance near the critical point

### E.R. Oby, D.T. Jacobs

*Journal of Chemical Physics* **114**, 4918 (2001)

The heat capacity of the liquid-liquid mixture perfluoroheptane and 2,2,4-trimethylpentane (also known as iso-octane) has been measured for the first time near its upper critical consolute point using an adiabatic calorimeter. The theoretical expression for the heat capacity near the critical point was applied to our combined data runs. The critical exponent α was determined to be 0.106±0.026, which agreed with theoretical predictions. When α was fixed at its theoretical value of 0.11, our value for the amplutde ratio A_{+}/A_{–} = 0.59±0.05 was consistent with experimental determinations and theoretical predictions. However, the two-scale-factor universality ratio χ, now consistent among experiments and theories with a value between 0.019 and 0.020, was violated in this sytem when using the published value for the correlation length.

## Turbidity in a Near-Critical, Liquid-Liquid System: A Precise, Automated Experiment

### D.T. Jacobs, S.M.Y. Lau, A. Mukherjee, and C.A. Williams

*International J. of Thermophys.* **20**, 877 (1999)

A ground based (1-g) experiment is in progress that measures the turbidity of the density-matched, binary fluid mixture methanol-cyclohexane extremely close to its liquid-liquid critical point. By covering the range of reduced temperatures t = (T-T_{c}) / T_{c} from 10^{-8} to 10^{-2}, the turbidity measurements should allow the Green-Fisher critical exponent eta to be determined. This paper reports measurements showing ±0.1 percent precision of the transmitted and reference intensities, and ±4 μK temperature control near the critical temperature of 320 K. Preliminary turbidity data show a non-zero eta consistent with theoretical predictions. No experiment has precisely determined a value of the critical exponent η, yet its value is significant to theorists in critical phenomena. Relatively simple critical phenomena, as in the liquid-liquid system studied here, serve as model systems for more complex behavior near a critical point.

## Electric Field Induced Transient Effects in a Nematic Liquid Crystal in the Presence of a Stabilizing Magnetic Field

### Shila Garg, Christopher Ditchman, Nathan Schiffrik, U.D. Kini

*Molecular Crystals Liquid Crystals* **328**, 581 (1999)

A homeotropically aligned nematic liquid crystal with positive dielectric and diamagnetic anisotropies is subjected to a destabilizing AC electric field * E* in the bend geometry in the presence of a stabilizing magnetic field

*. When the applied voltage*

**B***V*is gradually increased at a given frequency, the distortion that results above a threshold

*V*is spatially periodic with the wavevector depending on the electric frequency

_{th}*f*. Sudden application of a voltage step,

*V*, higher than

_{s}*V*causes a temporal evolution of the director field, which finally attains the homogeneously distorted

_{th}*H*state; the nature of temporal evolution depends on V

_{D}_{s}. If V

_{s}is slightly higher than

*V*, the transient deformation is periodic and the wavevector of periodicity depends on

_{th}*f*. When

*V*is high enough, the transition to

_{s}*H*occurs via a turbulent state.

_{D}## Taming Chaos with Disorder in a Pendulum Array

### Woodrow L. Shew, Hanna A. Coy, John F. Lindner

*American Journal of Physics,* volume 67, pages 709-711 (August 1999)

We designed and constructed an array of ten forced damped nonlinear pendulums. We drove the pivot of the pendulums in a circle and torsionally coupled them with springs. We analyzed the motion using digitized videotape. The behavior of the real array closely mirrored the behavior of its computer simulation. For a homogeneous array of identical pendulums, the spatiotemporal dynamics was chaotic; for a heterogeneous array of nonidentical pendulums, the spatiotemporal dynamics was periodic. Such temporally fixed but spatially varying chaos control has been called “disorder taming chaos”.

## Electric field induced walls in the bend geometry of a nematic liquid crystal

### Shila Garg, Erica Bramley, U.D. Kini

*Molecular Crystals Liquid Crystals* **325**, 209 (1998)

A homeotropically aligned nematic is subjected to the action of an ac electric field applied in the sample plane. With progressively increasing electric voltage, walls move away from the electrodes, approach each other and merge. A subsequent decrease of voltage to zero causes the reverse process to occur except for hysteresis. The hysteresis width is employed to estimate the adhesion surface energy density of the walls; the surface energy density is of the same order as the anisotropy in surface tension of nematics. The wall thickness diminishes with increasing voltage. This shows that the observed walls are similar to those produced by magnetic fields. The walls exhibit curvature in the sample plane, the undulation in a wall being regular at sufficiently elevated frequencies. The walls are decorated along their length by a zigzag defect pattern which is being reported in the bend Freedericksz geometry for the first time. Some of the observations are explained qualitatively.

## Noise Enhanced Propagation

### John F. Lindner, Sridhar Chandramouli, Adi R. Bulsara, Markus Löcher, William L. Ditto

*Physical Review Letters,* volume 81, pages 5048-5051 (7 December 1998)

We use noise to extend signal propagation in one and two-dimensional arrays of two-way coupled bistable oscillators. In a numerical model, we sinusoidally force one end of a chain of noisy oscillators. We record a signal-to-noise ratio at each oscillator. We demonstrate that moderate noise significantly extends the propagation of the sinusoidal input. Both the optimal noise and the maximum propagation length scale like the square root of the coupling. We obtain similar results with two-dimensional arrays. The simplicity of the model suggests the generality of the phenomenon.

## Heat capacity anomaly near the critical point of aniline-cyclohexane

### Paul F. Rebillot and D. T. Jacobs

*Journal of Chemical Physics,* **109**, 4009 (1998)

The heat capacity of the liquid-liquid mixture aniline-cyclohexane has been measured for the first time near its upper critical consolute point using an adiabatic calorimeter. Two data runs provide heat capacity data that are fitted by equations with background terms and a critical term. The critical exponent alpha was determined to be 0.104±0.011, consistent with theoretical predictions. When alpha was fixed at its theoretical value of 0.11 to determine the critical amplitudes A_{+} and A_{–}, our value for the amplitude ratio A_{+}/A_{–} = 0.50±0.03 was consistent with most experimental determinations in liquid-liquid mixtures, but was slightly larger than either theoretical predictions or recent experimental values in liquid-vapor systems. The two-scale-factor universality ratio χ, now consistent among experiments and theories with a value between 0.019 and 0.020, is consistent in this system using one published value for the correlation length, but not with another.

## Frequency Variation of Periodic Distortion Thresholds in a Nematic

### Shila Garg, Steven Wild, Ben Zurn, Salman Saeed, U. D. Kini

*Liquid Crystals,* **24**, 501 (1998)

Investigations are reported on the electric field induced orientational transitions in the bend Freedericksz geometry under the action of a stabilizing magnetic field. When the magnetic field is strong enough, the deformation above electric threshold is periodic with the periodicity disappearing at a higher voltage. The alignment does not remain homeotropic below threshold and the sample exhibits pretransitional biaxiality. Every transition is discontinuous and accompanied by hysteresis. A form of scaling appears to hold for all the observed thresholds. The thresholds and the direction of the wavevector are frequency dependent showing that the instability mechanism involves electrical conductivity.

## Optimal disorders for taming spatiotemporal chaos

### John F. Lindner, Bryan S. Prusha, Kimberly Elizabeth Clay

*Physics Letters A,* volume 231, pages 164-172 (7 July1997)

We study a coupled array of torqued damped nonlinear pendulums. Disordering this system can convert chaotic spatiotemporal evolution into periodic motion. Here, in numerical experiments, we elucidate and quantify this phenomenon. For each of several types of disorder, we find an optimal magnitude of disorder which minimizes the system’s largest Lyapunov exponent.

## Frequency dependent threshold in bend geometry of 5CB

### Shila Garg, Salman Saeed, Steven Wild, Erica Bramley, U. D. Kini

*Molecular Crystals Liquid Crystals,* **302**, 379 (1997)

We report the effect of electric frequency on deformation threshold in the bend Freedericksz geometry in the presence of a stabilizing magnetic field applied normal to the plates with the destabilizing electric field impressed parallel to the sample. In general, the observed threshold and deformation above it are strongly dependent on frequency and magnetic strength associated with a pretransitional field-induced biaxiality. The periodic deformation observed under a strong magnetic field has wavevector along the electric field at low frequencies. Above a cut-off frequency, the direction of the wavevector becomes normal to the electric field. Hysterisis is present between increase and decrease of voltage. At DC or low frequency excitation, there is clear evidence of hydrodynamic flow which can become turbulent for some values of parameters.

## Heat capacity anomaly near the lower critical consolute point of triethylamine-water

### Anne C. Flewelling, Rohan J. DeFonseka, Nikfar Khaleeli, J. Partee, D. T. Jacobs

*Journal of Chemical Physics,* **104** , 8048 (1996)

The heat capacity of the binary liquid mixture triethylamine-water has been measured near its lower critical consolute point using a scanning, adiabatic calorimeter. Two data runs are analyzed to provide heat capacity and enthalpy data that are fitted by equations with background terms and a critical term that includes correction to scaling. The critical exponent a was determined to be 0.107±0.006, consistent with theoretical predictions. When a was fixed at 0.11 to determine various amplitudes consistently, our values of A_{+} and A_{–} agreed with a previous heat capacity measurement, but the value of A_{+} was inconsistent with values determined by density or refractive index measurements. While our value for the amplitude ratio A_{+}/A_{–} = 0.56±0.02 was consistent with other recent experimental determinations in binary liquid mixtures, it was slightly larger than either theoretical predictions or recent experimental values in liquid-vapor systems. The correction to scaling amplitude ratio D_{+}/D_{–} = 0.5±0.1 was half of that predicted. As a result of several more precise theoretical calculations and experimental determinations, the two-scale-factor universality ratio χ, which we found to be 0.019±0.003, now is consistent among experiments and theories. A new “universal” amplitude ratio involving the amplitudes for the specific heat was tested. Our determination of = -0.5±0.1 and = -1.1±0.1 is smaller in magnitude than predicted and is the first such determination in a binary fluid mixture.