Instabilities coarsening and chaos in driven systems : Growing interfaces and sheared wormlike micelles

Abstract

We now provide a summary and outlook of the various topics discussed in this thesis. This thesis deals with systems driven far from equilibrium. Such systems show many interesting phenomena, notably nonequilibrium phase transitions, coarsening, and the presence of instabilities that lead to pattern formation. The two physical systems studied in this thesis concern growing interfaces and sheared wormlike micelles. A particular instability present in a class of models of growing interfaces is seen to lead to mound formation with slope selection and associated power-law coarsening, while the flow instabilities in a wormlike micellar fluid lead to spatiotemporal chaos.

In Chapter 1, we provided a brief introduction to systems out of equilibrium. These systems often hold surprises; for example, in many cases they exhibit first-order transitions in one dimension with short-range interactions, which are not observed in their equilibrium counterparts. They also exhibit nonequilibrium steady states, characterized by a zero probability current but potentially having other nonzero currents and fluxes arising from the competition between the drive and various dissipative mechanisms. Such nonequilibrium steady states are approached in time via a coarsening process.

In situations where a system out of equilibrium relaxes to equilibrium, one can use energy or free energy arguments to determine how the characteristic length scale grows as a function of time. Such arguments, however, do not hold for systems driven far from equilibrium. An analysis of coarsening behavior in truly nonequilibrium situations remains an open problem. We therefore provided a brief introduction to coarsening phenomena and phase ordering kinetics, as well as an introduction to pattern formation in systems out of equilibrium. This discussion sets the stage for the two physical systems analyzed in this thesis.

The first system concerns growing interfaces, particularly the growth of thin films by vapor deposition. One of the most important issues in surface growth experiments today is designing a dynamically controlled deposition process such that interfaces can be obtained that are atomically smooth (thin-film epitaxy) or exhibit desired patterns with well-defined geometry (as in nanotechnology). To this end, it is essential to understand the instabilities that may arise in a growing interface and how they can be controlled. We thus give an overview of the phenomenology and methods of analysis of a growing interface in connection to vapor deposition in thin films.

The second driven system studied involves sheared wormlike micelles. We provided a survey of the present state of understanding regarding rheological chaos in complex fluids in general and wormlike micellar systems in particular.

In Chapter 2, we focused on the theoretical aspects of growing thin films by deposition of atoms on a flat substrate. Such a growth mechanism plays a crucial role in modern semiconductor technology, which relies on the ability to fabricate high-quality thin films. On the theoretical side, the kinetics of the growth process brings up many interesting questions about generic scale invariance and pattern formation in nonequilibrium systems. While there has been extensive research on kinetic roughening leading to self-affine interface profiles, recent experimental and theoretical interest has shifted to a different mode of surface growth involving the formation of “mounds,” which are pyramid-like or “wedding cake-like” structures. The precise experimental conditions that determine whether the growth morphology is kinetically rough or mound-dominated are presently unclear. However, many experiments show mound formation that coarsens (the typical lateral size of the mounds increases) with time. During this process, the typical slope of the sides of the pyramid-like mounds may or may not remain constant. If the slope remains constant in time, the system exhibits slope selection.

Traditionally, mound formation has been attributed to the Ehrlich-Schwoebel (ES) step-edge barrier, which hinders the downward motion of atoms across step edges. This diffusion bias makes it more likely for an atom on a terrace to attach to an ascending step rather than a descending one. This leads to an effective “uphill” surface current that destabilizes the interface, causing mound formation. If the ES part of the surface current ?? j has one or more stable zeros as a function of the slope ?? s, then the slope of the mounds is expected to stabilize at the corresponding values of ?? s at long times, and the system exhibits slope selection.

In this context, we studied a class of spatially discretized nonequilibrium conserved growth models (the conserved KPZ equation and its atomistic version) that show mound formation, slope selection, and power-law coarsening without an ES instability, using numerical integration and stochastic simulation. A nonlinear instability in these growth equations, in which the height or depth of spontaneously generated pillars or grooves grows indefinitely once it exceeds a certain threshold, had been previously found [1]. When this instability is tightly controlled by introducing a control function that adds an infinite number of higher-order gradient nonlinearities, one recovers dynamical scaling with exponents matching one-loop dynamical renormalization group calculations [1].

Following this, in Chapter 2, we carried out detailed numerical investigations on the long-time behavior of the system when the instability is loosely controlled in one-dimensional systems. In this regime, we observed mound formation, slope selection, and power-law coarsening. This is an example of pattern formation out of equilibrium driven by a nonlinear instability controlled by higher-order nonlinearities, unlike usual pattern-forming systems where a linear instability is controlled by higher-order nonlinearities. As a function of model parameters—the strength of the nonlinearity and the strength of the control—the system undergoes a first-order dynamical phase transition from a mounded slope-selected phase to a rough self-affine phase. We performed a linear stability analysis to find the “spinodal boundary” across which the mounded state is locally unstable. We also defined an order parameter to distinguish the two phases and carried out a finite-size scaling analysis. Our analysis of phase fluctuations of the order parameter establishes that the slope-selected mounded phase is a true phase and not a transient.

In the parameter region where the mounded phase is stable, the pyramids coarsen with time, with larger mounds growing at the expense of smaller ones. Power-law scaling is recovered in this regime.

In Chapter 3, we studied the coarsening behavior arising in these models. Since the system is truly out of equilibrium, the coarsening process should be analyzed based on the relative stability of structures under the assumed dynamics. Linear stability analysis shows that any structure different from a single pyramid has a positive eigenvalue, implying instability and evolution toward a single pyramid. During coarsening, the valleys between mounds fill, and the peaks approach each other. Once a critical peak separation is reached, the interface region “melts” and eventually becomes the top part of a mound.

The coarsening exponent for the atomistic model and the continuum equation differs: ?? = 1 / 2 n=1/2 for the atomistic model and ?? = 1 / 3 n=1/3 for the continuum model. This is intriguing because the atomistic model was constructed to belong to the same universality class as the continuum model. When the instability is tightly controlled, the growth, roughness, and dynamic exponents of the continuum equation match those of the atomistic model and the one-loop RG calculations.

We also constructed a reduced model of the coarsening phenomenon to explain this difference. Observing that the coarsening exponents of the noiseless and noisy continuum models and the noiseless discrete model are all ?? = 1 / 3 n=1/3, we conjecture that coarsening is driven by an effective attractive interaction between peaks of the form ? 1 / ?? 2 ?1/x 2 , where ?? x is the peak separation. Analysis of a two-mound structure in the atomistic model with noise shows that peak positions have a fluctuating component along with an overall drift, leading to coalescence. The reduced model represents this as a Brownian particle in an attractive force field with absorbing boundaries at the origin, focusing on the mean first passage time.

We also studied the continuum model with conserved noise. The nonlinear instability persists, but unlike the nonconserved model—which exhibits anomalous dynamical scaling with nearest-neighbor height differences growing indefinitely—the conserved noise model exhibits usual dynamical scaling, with roughness exponent less than unity. The nearest-neighbor height difference saturates for flat initial states. Consequently, the instability may not occur if the nonlinear coefficient is sufficiently small. However, it can be initiated by choosing initial states with sufficiently high (or deep) pillars or grooves. This leads to “nonergodicity”: in a parameter region, both kinetically rough and faceted phases are locally stable, and the initial configuration determines the steady-state behavior, implying that long-time morphology and dynamics are history-dependent—relevant for growth on patterned substrates.

In Chapter 4, we considered competing linear and nonlinear instabilities. In the absence of noise, the steady state depends on which instability dominates early. Nonlinear instabilities lead to mounds with well-defined slopes, while linear instabilities lead to mounds without slope selection. Noise removes sensitivity to initial conditions, and parameters such as nonlinearity strength and control strength determine steady-state morphology. Coarsening behavior differs: for mounded phases with slope selection, the coarsening exponent from width vs. time is ?? = 1 / 2 n=1/2. In mounded phases without slope selection, the maximum slope increases with time, and the coarsening exponent, obtained by subtracting the steepening exponent from the width vs. time exponent, is ?? ? 1 / 3 n?1/3 with steepening exponent ?? ? 0.17 ??0.17.

The second system studied is a micellar fluid under shear. Surfactant molecules are amphiphilic, having hydrophilic and hydrophobic groups, and self-assemble in aqueous solutions into structures like wormlike micelles, vesicles, and bilayers. Wormlike micelles act like “living polymers,” reassembling even after being broken by flow. Rheological behavior of such systems is of technological and theoretical interest. In the dilute regime, they exhibit phenomena such as shear thickening and low-dimensional chaos. Shear-thickening theory exists, but the chaotic behavior remains poorly understood.

Rheological measurements of shear stress under controlled strain rate show signatures of low-dimensional chaos [5]. “Rheochaos” refers to macroscopic chaos in viscoelastic materials at negligible Reynolds number, implying nonlinearity arises from material constitutive behavior rather than momentum advection. For aqueous CTAT micellar systems, the most plausible mechanism is shear-banding instability, where high-shear bands coexist with low-shear bands.

In Chapter 5, we investigated rheochaos in CTAT micellar systems under shear flow. We studied the Johnson-Segalman (JS) model, accounting for non-affine deformations, but neither this model nor its diffusive variant reproduced chaotic oscillations in shear stress and the first normal stress difference as observed experimentally. Numerical studies of the JS model with flow-concentration coupling also showed no chaos. We then analyzed the equations of motion for the alignment tensor of a nematogenic fluid, following Reinacker et al. [6], where temporal fluctuations of the director produce chaos. Dynamical invariants and surrogate data analysis confirmed low-dimensional chaos: for all shear rates where the nematic director fluctuates chaotically, the correlation dimension remains greater than 2 but below the embedding dimension.

Including spatial inhomogeneities allowed us to study shear banding and other flow-induced structures. Analysis shows dynamic shear banding over a range of parameters—proximity to the nematic-isotropic transition, tumbling parameter from Leslie-Ericksen theory, shape factor (molecular aspect ratio), and diffusivities. Flow data reveal the presence of spatiotemporal chaos [7].