The steady state properties of an interface in a stationary Couette flow are addressed within the framework of fluctuating hydrodynamics. Our study reveals that thermal fluctuations are driven out of equilibrium by an effective shear rate that differs from the applied one. In agreement with experiments, we find that the mean square displacement of the interface is strongly reduced by the flow. We also show that nonequilibrium fluctuations present a certain degree of universality in the sense that all features of the fluids can be factorized into a single control parameter. Finally, the results are discussed in the light of recent experimental and numerical studies.
We study numerically phase separation in a binary fluid subject to an applied shear flow in two dimensions, with full hydrodynamics. To do so, we introduce a mixed finite-differencing/spectral simulation technique, with a transformation to render trivial the implementation of Lees-Edwards sheared periodic boundary conditions. For systems with inertia, we reproduce the nonequilibrium steady states reported in a recent lattice Boltzmann study. The domain coarsening that would occur in zero shear is arrested by the applied shear flow, which restores a finite domain size set by the inverse shear r…
In modeling nonequilibrium systems one usually starts with a definition of the microscopic dynamics, e.g., in terms of transition rates, and then derives the resulting macroscopic behavior. We address the inverse question for a class of steady state systems, namely complex fluids under continuous shear flow: how does an externally imposed shear current affect the microscopic dynamics of the fluid? The answer can be formulated in the form of invariant quantities, exact relations for the transition rates in the nonequilibrium steady state, as discussed in a recent letter [A. Baule and R. M. L. E…
We show that the shear rate at a fixed shear stress in a micellar gel in a jammed state exhibits large fluctuations, showing positive and negative values, with the mean shear rate being positive. The resulting probability distribution functions (PDF's) of the global power flux to the system vary from Gaussian to non-Gaussian, depending on the driving stress and in all cases show similar symmetry properties as predicted by Gallavotti-Cohen steady state fluctuation relation. The fluctuation relation allows us to determine an effective temperature related to the structural constraints of the jamm…
The nonequilibrium dynamics of a binary Lennard-Jones mixture in a simple shear flow is investigated by means of molecular dynamics simulations. The range of temperature investigated covers both the liquid, supercooled and glassy states, while the shear rate covers both the linear and nonlinear regimes of rheology. The results can be interpreted in the context of a nonequilibrium, schematic mode-coupling theory developed recently, which makes the theory applicable to a wide range of soft glassy materials. The behavior of the viscosity is first investigated. In the nonlinear regime, strong shea…
We investigate the nonequilibrium behavior of dense, plastic-crystalline suspensions of mildly anisotropic colloidal hard dumbbells under the action of an oscillatory shear field by employing Brownian dynamics computer simulations. In particular, we extend previous investigations, where we uncovered novel nonequilibrium phase transitions, to other aspect ratios and to a larger nonequilibrium parameter space, that is, a wider range of strains and shear frequencies. We compare and discuss selected results in the context of novel scattering and rheological experiments. Both simulations and experi…
We develop a nonequilibrium mode-coupling theory for uniformly sheared systems starting from microscopic, thermostatted SLLOD equations of motion. Our theory aims at describing stationary-state properties including rheological ones of sheared systems, and this is accomplished via two steps. Firstly, a set of self-consistent equations is formulated based on the projection-operator formalism and on the mode-coupling approach for the transient density correlators which measure the correlations between the density fluctuations in the initial equilibrium state and the ones at later times after the …
The influence of nonequilibrium bulk conditions on the properties of the interfaces exhibited by a kinetic Ising--like model system with nonequilibrium steady states is studied. The system is maintained out of equilibrium by perturbing the familiar spin--flip dynamics at temperature T with completely--random flips; one may interpret these as ideally simulating some (dynamic) impurities. We find evidence that, in the present case, the nonequilibrium mechanism adds to the basic thermal one resulting on a renormalization of microscopic parameters such as the probability of interfacial broken bond…
A nonequilibrium molecular dynamics computer simulation on microsegregated solutions of symmetrical diblock copolymers is reported. As the polymer concentration increases, the system undergoes phase transitions in the following order: body centered cubic (BCC) micelles, hexagonal (HEX) cylinders, gyroid (GYR) bicontinuous networks, and lamellae (L), which are the same morphologies that have been reported for block copolymer melts. Structural classification is based on the patterns of the anisotropic static structure factor and characteristic 3-dimensional images. The systems in the BCC micella…
A growing interface subject to noise is described by the Kardar-Parisi-Zhang equation or, equivalently, the noisy Burgers equation. In one dimension this equation is analyzed by means of a weak noise canonical phase space approach applied to the associated Fokker-Planck equation. The growth morphology is characterized by a gas of nonlinear soliton modes with superimposed linear diffusive modes. We also discuss the ensuing scaling properties.
The common rules of classical nonequilibrium thermodynamics do not allow an Onsager coefficient like the viscosity to depend on the shear rate. Such a dependence is experimentally well documented, however. In this paper it is shown, using nonequilibrium thermodynamics alone, how it is possible that the effective viscosity coefficient of an isotropic colloidal system depends on both the shear rate and the frequency.
We have performed small-angle light-scattering measurements of the static structure factor of a critical binary mixture undergoing diffusive partial remixing. An uncommon scattering geometry integrates the structure factor over the sample thickness, allowing different regions of the concentration profile to be probed simultaneously. Our experiment shows the existence of interface capillary waves throughout the macroscopic evolution to an equilibrium interface, and allows to derive the time evolution of surface tension. Interfacial properties are shown to attain their equilibrium values quickly…
The dynamics of systems out of thermal equilibrium is usually treated on a case-by-case basis without knowledge of fundamental and universal principles. We address this problem for a class of driven steady states, namely those mechanically driven at the boundaries such as complex fluids under shear. From a nonequilibrium counterpart to detailed balance (NCDB) we derive a remarkably simple set of invariant quantities which remain unchanged when the system is driven. These new nonequilibrium relations are both exact and valid arbitrarily far from equilibrium. Furthermore, they enable the systema…
The lectures examine several problems related to non-equilibrium fluctuations of interfaces and flux lines. The first two introduce the phenomenology of depinning, with particular emphasis on interfaces and contact lines. The role of the anisotropy of the medium in producing different universality classes is elucidated. The last two lectures focus on the dynamics of lines, where transverse fluctuations are also important. We shall demonstrate how various non-linearities appear in the dynamics of driven flux lines. The universality classes of depinning, and also dynamic roughening, are illustra…
We compare nonlinear stresses and temperatures for adiabatic shear flows, using up to 262,144 particles, with those from corresponding homogeneous and inhomogeneous flows. Two varieties of kinetic temperature tensors are compared to the configurational temperatures. This comparison leads to an improved form for the local and instantaneous smooth-particle averaged stream velocity and to a recognition of rotational contributions to the configurational temperature.
By means of classical density functional theory and its dynamical extension, we consider a colloidal fluid with spherically-symmetric competing interactions, which are well known to exhibit a rich bulk phase behavior. This includes complex three-dimensional periodically ordered cluster phases such as lamellae, two-dimensional hexagonally packed cylinders, gyroid structures or spherical micelles. While the bulk phase behavior has been studied extensively in earlier work, in this paper we focus on such structures confined between planar repulsive walls under shear flow. For sufficiently high she…
We present a detailed analysis of the fluctuation dissipation theorem (FDT) close to the glass transition in colloidal suspensions under steady shear using mode coupling approximations. Starting point is the many-particle Smoluchowski equation. Under shear, detailed balance is broken and the response functions in the stationary state are smaller at long times than estimated from the equilibrium FDT. An asymptotically constant relation connects response and fluctuations during the shear driven decay, restoring the form of the FDT with, however, a ratio different from the equilibrium one. At sho…
Using large deviation theory and principles of stochastic optimal control, we show that rare molecular dynamics trajectories conditioned on assembling a specific target structure encode a set of interactions and external forces that lead to enhanced stability of that structure. Such a relationship can be formulated into a variational principle, for which we have developed an associated optimization algorithm and have used it to determine optimal forces for targeted self-assembly within nonequilibrium steady-states. We illustrate this perspective on inverse design in a model of colloidal cluste…
Rigid clusters in shear-thickening suspensions: a nonequilibrium critical transition
Levich Institute and Dept. of Chemical Engineering, CUNY City College of New York（纽约市立大学城市学院莱维希研究所及化学工程系）；Department of Chemical Engineering, Indian Institute of Technology (ISM), Dhanbad（印度理工学院（ISM）达纳巴德化学工程系）；Martin Fisher School of Physics, Brandeis University（布兰代斯大学马丁·费舍尔物理学院）
We study the effect of a uniform shear flow on an interface separating the two broken-symmetry ordered phases of a two-dimensional system with nonconserved scalar order parameter. The interface, initially flat and perpendicular to the flow, is distorted by the shear flow. We show that there is a critical shear rate, γ_c, proportional to 1/L^2, (where L is the system width perpendicular to the flow) below which the interface can sustain the shear. In this regime the countermotion of the interface under its curvature balances the shear flow, and the stretched interface stabilizes into a time-ind…
We derive the transient-time correlation function (TTCF) expression for the computation of phase variables of inhomogenous confined atomistic fluids undergoing boundary-driven planar shear (Couette) flow at constant pressure. Using nonequilibrium molecular dynamics simulations, we then apply the TTCF formalism to the computation of the shear stress and the slip velocity for atomistic fluids at realistic low shear rates, in systems under constant pressure and constant volume. We show that, compared to direct averaging of multiple trajectories, the TTCF method dramatically improves the accuracy …
String Model Predicts Relaxation in Sheared Glass-Forming Liquids
Changchun Institute of Applied Chemistry, Chinese Academy of Sciences（中国科学院长春应用化学研究所）；University of Science and Technology of China（中国科学技术大学）；Institute of Mechanics, Chinese Academy of Sciences（中国科学院力学研究所）；University of Chinese Academy of Sciences（中国科学院大学）；National Institute of Standards and Technology（美国国家标准与技术研究院）；Shanghai Jiao Tong University（上海交通大学）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）
We propose a simple dynamic model of polymers under shear with an anisotropic mobility tensor. We calculate the shear viscosity, the rheo-dielectric response function, and the parallel relaxation modulus under shear flow deduced from our model. We utilize recently developed linear response theories for nonequilibrium systems to calculate linear response functions. Our results are qualitatively consistent with experimental results. We show that our anisotropic mobility model can reproduce essential dynamical nature of polymers under shear qualitatively. We compare our model with other models or…
Coarsening systems under uniform shear display a long time regime characterized by the presence of highly stretched and thin domains. The question then arises whether thermal fluctuations may actually destroy this layered structure. To address this problem in the case of non-conserved dynamics we study an anisotropic version of the Burgers equation, constructed to describe thermal fluctuations of an interface in the presence of a uniform shear flow. As a result, we find that stretched domains are only marginally stable against thermal fluctuations in $d=2$, whereas they are stable in $d=3$.
We have carried out nonequilibrium molecular dynamics simulations of a system of crosslinked particles under shear flow conditions. As the fraction of crosslinks $p$ is increased the system approaches a gel point at which the shear viscosity $η$ and the first and second normal stress coefficients $Ψ_1$ and $Ψ_2$ diverge. All three quantities seem to diverge with a power law form: $η\sim ε^{-s}$, $Ψ_{1,2}\simε^{-λ}$ where $ε=p_c-p$ and $s\approx 0.7$ and $λ\approx 3.15$ in three dimensions and $s\approx 2.0$ and $λ\approx 6.0$ in two dimensions.
Starting from an ideal crystalline state, we numerically study a nonequilibrium dynamical order- disorder transition promoted by the application of a periodic shearing protocol at low temperatures in model systems in two and three dimensions. We observe a continuous (2D) and discontinuous (3D) dynamical transition from an ordered to a disordered steady state. Through the analysis of large-scale simulations, we show that the amorphization mechanism around the discontinuous transition is reminiscent of spinodal decomposition.
We consider the interplay of thermal fluctuations and shear on the surface of the domains in various systems coarsening under an imposed shear flow. These include systems with nonconserved and conserved dynamics, and a conserved order parameter advected by a fluid whose velocity field satisfies the Navier-Stokes equation. In each case the equation of motion for the interface height reduces to an anisotropic Burgers equation. The scaling exponents that describe the growth and coarsening of the interface are calculated exactly in any dimension in the case of conserved and nonconserved dynamics. …
We use a phase-separated driven two-dimensional Ising lattice gas to study fluid interfaces exposed to shear flow parallel to the interface. The interface is stabilized by two parallel walls with opposing surface fields and a driving field parallel to the walls is applied which (i) either acts locally at the walls or (ii) varies linearly with distance across the strip. Using computer simulations with Kawasaki dynamics, we find that the system reaches a steady state in which the magnetisation profile is the same as that in equilibrium, but with a rescaled length implying a reduction of the inte…
The influence of an applied shear on the planar crystal-melt interface is modelled by a nonlinear stochastic partial differential equation of the interface fluctuations. A feature of this theory is the asymmetric destruction of interface fluctuations due to advection of the crystal protrusions on the liquid side of the interface only. We show that this model is able to qualitatively reproduce the nonequilibrium coexistence line found in simulations. The impact of shear on spherical clusters is also addressed.
Using nonequilibrium molecular dynamics (NEMD) simulations, we study the properties of supercooled liquids of Si under shear at T=1060K over a range of densities encompassing the low-density liquid (LDL) and high-density liquid (HDL) forms. This enables us to generate nonequilibrium steady-states of the LDL and HDL polymorphs, that remain stabilized in their liquid forms for as long as the shear is applied. This is unlike the LDL and HDL forms at rest, which are metastable under those conditions and, when at rest, rapidly undergo a transition towards the crystal, i.e. the thermodynamically sta…
We investigate the influence of a shear flow on the process of nucleation. Mesoscopic nonequilibrium thermodynamics is used to derive the Fokker-Planck equation governing the evolution of the cluster size distribution in a metastable phase subjected to a stationary flow. The presence of the flow manifests itself in the expression for the effective diffusion coefficient of a cluster and introduces modifications in the nucleation rate. The implications of these results in condensation and polymer crystallization are discussed.
We compute rates and pathways for nucleation in a sheared two dimensional Ising model with Metropolis spin flip dynamics, using Forward Flux Sampling (FFS). We find a peak in the nucleation rate at intermediate shear rate. We analyse the origin of this peak using modified shear algorithms and committor analysis. We find that the peak arises from an interplay between three shear-mediated effects: shear-enhanced cluster growth, cluster coalescence and cluster breakup. Our results show that complex nucleation behaviour can be found even in a simple driven model system. This work also demonstrates…
The nonequilibrium dynamical behavior and structure formation of end-functionalized semiflexible polymer suspensions under flow are investigated by mesoscale hydrodynamic simulations. The hybrid simulation approach combines the multiparticle collision dynamics method for the fluid, which accounts for hydrodynamic interactions, with molecular dynamics simulations for the semiflexible polymers. In equilibrium, various kinds of scaffold-like network structures are observed, depending on polymer flexibility and end-attraction strength. We investigate the flow behavior of the polymer networks under…
We propose a model for the nonequilibrium enhancement of colloidal self-diffusion in an externally imposed shear flow in charged systems. The enhancement of diffusion is calculated in terms of the electrostatic, two--body interactions between the particles in the shear flow. In the high shear rate, low volume fraction limit in which our model is valid, we compare these calculations to the experiments of Qiu et al. [Phys. Rev. Lett. 61, 2554 (1988)].
We report on the spatio-temporal dynamics of the interface in shear-banding flow of a wormlike micellar system (cetyltrimethylammonium bromide and sodium nitrate in water) during a start-up experiment. Using the scattering properties of the induced structures, we demonstrate the existence of an instability of the interface between bands along the vorticity direction. Different regimes of spatio-temporal dynamics of the interface are indentified along the stress plateau. We build a model based on the flow symetry which qualitatively describes the observed patterns.
We study numerically the nonlinear dynamics of a shear banding interface in two dimensional planar shear flow, within the non-local Johnson Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for sufficiently small ratio of the interfacial width $\ell$ to cell length $L_x$. The instability saturates in finite amplitude interfacial fluctuations. For decreasing $\ell/L_x$ these undergo a non equilibrium transition from simple travelling interfacial waves with constant average wall stress, to pe…
We present a large-scale molecular dynamics study of nematic-paranematic interfaces under shear. We use a model of soft repulsive ellipsoidal particles with well-known equilibrium properties, and consider interfaces which are oriented normal to the direction of the shear gradient (common stress case). The director at the interface is oriented parallel to the interface (planar). A fixed average shear rate is imposed with Lees-Edwards boundary conditions, and the heat is dissipated with a profile-unbiased thermostat. First we study the properties of the interface at one particular shear rate in …
In multi-component soft matter, interface properties often play a key role in determining the properties of the overall system. The identification of the internal dynamic structures in non-equilibrium situations requires the interface rheology to be characterized. We have developed a method to quantify the rheological contribution of soft interfaces and evaluate the dynamic modulus of the interface. This method reveals that the dynamic shear responses of interfaces in bilayer systems comprising polypropylene and three different polyethylenes can be classified as having hardening and softening …
The presence of even small amount of surfactant at the particle-laden fluid interface subjected to shear makes surface flow incompressible if the shear rate is small enough [T. M. Fischer et al, J. Fluid Mech. 558, 451 (2006)]. In the present paper the effective surface shear viscosity of a flat, low-concentration, particle-laden incompressible interface separating two immiscible fluids is calculated. The resulting value is found to be 7.6% larger than the value obtained without account for surface incompressibility.
We report on a non trivial dynamics of the interface between shear bands following a start-up of flow in a semi-dilute wormlike micellar system investigated using a combination of mechanical and optical measurements. During the building of the banding structure, we observed the stages of formation, migration of the interface between bands and finally the destabilization of this interface along the vorticity axis. The mechanical signature of these processes has been indentified in the time series of the shear stress. The interface instability occurs all along the stress plateau, the asymptotic …
A longstanding goal of nonequilibrium statistical mechanics has been to extend the conceptual power of the Boltzmann distribution to driven systems. We report some new progress towards this goal. Instead of writing the nonequilibrium steady-state distribution in terms of perturbations around thermal equilibrium, we start from the linearized driven dynamics of observables about their stable fixed point, and expand in the strength of the nonlinearities encountered during typical fluctuations away from the fixed point. The first terms in this expansion retain the simplicity of known expansions ab…
Interfacial rheology is important for understanding properties such as Pickering emulsion or foam stability. Currently, the response is measured using a probe directly attached to the interface. This can both disturb the interface and is coupled to flow in the bulk phase, limiting its sensitivity. We have developed a contactless interfacial method to perform interfacial shear rheology on liquid/liquid interfaces with no tool attached directly to the interface. This is achieved by shearing one of the liquid phases and measuring the interfacial response via confocal microscopy. Using this method…
In this article, we describe the instability of a contact line under nonequilibrium conditions mainly based on the results of our recent studies. Two experimental examples are presented: the self-propelled motion of a liquid droplet and spontaneous dynamic pattern formation. For the self-propelled motion of a droplet, we introduce an experiment in which a droplet of aniline sitting on an aqueous layer moves spontaneously at an air-water interface. The spontaneous symmetry breaking of Marangoni-driven spreading causes regular motion. In a circular Petri dish, the droplet exhibits either beeline…
Wormlike micelle solutions are submitted to small-amplitude oscillatory shear superimposed to steady shear in the shear banding regime. By imposing a shear oscillation, the interface between high- and low-shear regions oscillates in time. A two-fluid semi-phenomenological model is proposed for superposition rheology in the shear banding regime, which allows us to extract a characteristic velocity for the interface dynamics from experiments involving only a standard rheometer. Estimates of the stress diffusion coefficient ${\cal D}$ can also be inferred from such superposition experiments. The …
The nonequilibrium hydrodynamic correlations of a Multiparticle-Collision-Dynamics (MPC) fluid in shear flow are studied by analytical calculations and simulations. The Navier-Stokes equations for a MPC fluid are linearized about the shear flow and the hydrodynamic modes are evaluated as an expansion in the momentum vector. The shear-rate dependence and anisotropy of the transverse and longitudinal velocity correlations are analyzed. We demonstrate that hydrodynamic correlations in shear flow are anisotropic, specifically, the two transverse modes are no longer identical. In addition, our simu…
The effective dilatational and shear viscosities of a particle-laden fluid interface are computed in the dilute limit under the assumption of an asymptotically vanishing viscosity ratio between both fluids. Spherical particles with a given contact angle of the fluid interface at the particle surface are considered. A planar fluid interface and a small Reynolds number are assumed. The theoretical analysis is based on a domain perturbation expansion in the deviation of the contact angle from $90^\circ$ up to the second order. The resulting effective dilatational viscosity shows a stronger depend…
Using a dynamical density functional theory we analyze the density profile of a colloidal liquid near a wall under shear flow. Due to the symmetries of the system considered, the naive application of dynamical density functional theory does not lead to a shear induced modification of the equilibrium density profile, which would be expected on physical grounds. By introducing a physically motivated dynamic mean field correction we incorporate the missing shear induced interparticle forces into the theory. We find that the shear flow tends to enhance the oscillations in the density profile of ha…
We study the linear stability of planar shear banded flow with respect to perturbations with wavevector in the plane of the banding interface, within the non local Johnson Segalman model. We find that perturbations grow in time, over a range of wavevectors, rendering the interface linearly unstable. Results for the unstable eigenfunction are used to discuss the nature of the instability. We also comment on the stability of phase separated domains to shear flow in model H.
