Shear-stress Autocorrelation Function Research Articles

Invariant dynamics in a united-atom model of an ionic liquid.

We study a united-atom model of the ionic liquid 1-butyl-1-methylpyrrolidinium bis(trifluoromethyl)sulfonylamide to determine to what extent there exist curves in the phase diagram along which the microscopic dynamics are invariant when expressed in dimensionless, or reduced, form. The initial identification of these curves, termed isodynes, is made by noting that contours of reduced shear viscosity and reduced self-diffusion coefficient coincide to a good approximation. Choosing specifically the contours of reduced viscosity as nominal isodynes, further simulations were carried out for state points on these, and other aspects of dynamics were investigated to study their degree of invariance. These include the mean-squared displacement, shear-stress autocorrelation function, and various rotational correlation functions. These were invariant to a good approximation, with the main exception being rotations of the anion about its long axis. The dynamical features that are invariant have in common that they are aspects that would be relevant for a coarse-grained description of the system; specifically, removing the most microscopic degrees of freedom in principle leads to a simplification of the potential energy landscape, which allows for the existence of isodynes.

Dynamical behavior and transport coefficients of the pseudo hard-sphere fluid

In this work, we employ a recent approach to characterize the hard-sphere (HS) fluid by means of a continuous interaction potential, commonly referred to as pseudo hard-sphere potential, in order to determine HS transport coefficients as a function of the volume fraction for the three-dimensional mono disperse fluid. Using equilibrium molecular dynamics simulations, we determine time-dependent velocity, shear stress, and energy flux autocorrelation functions in order to use them within the Green–Kubo framework to compute the self-diffusion, shear viscosity, and thermal conductivity coefficients, respectively. Results are discussed as a function of the volume fraction and were compared to theoretical and simulations results previously reported by other authors. The main purpose of this work is twofold: first, testing the continuous approach of the HS fluid for the computation of dynamic properties and second, performing a systematic determination of aforementioned transport coefficients to analyze them as a function of fluid volume fraction. Furthermore, our results are used to provide a practical correction to the Chapman–Enskog equations for the HS self-diffusion, shear viscosity, and thermal conductivity predictions in a wide range of volume fractions.

Generalized hydrodynamics of the Lennard-Jones liquid in view of hidden scale invariance.

In recent years lines along which structure and dynamics are invariant to a good approximation, so-called isomorphs, have been identified in the thermodynamic phase diagrams of several model liquids and solids. This paper reports computer simulation data of the transverse and longitudinal collective dynamics at different length scales along an isomorph of the Lennard-Jones system. Our findings are compared to corresponding results along an isotherm and an isochore. Confirming the theoretical prediction, the reduced-unit dynamics of the transverse momentum density is invariant to a good approximation along the isomorph on all time and length scales. Likewise, the wave-vector dependent shear-stress autocorrelation function is found to be isomorph invariant (with minor deviations at very short times). A similar invariance is not seen along the isotherm or the isochore. Using a spatially nonlocal hydrodynamic model for the transverse momentum-density time-autocorrelation function, the macroscopic shear viscosity and its wave dependence are determined, demonstrating that the shear viscosity is isomorphic invariant on all length scales studied. This analysis implies the existence of a length scale that is isomorph invariant in reduced units, i.e., which characterizes each isomorph. The transverse sound-wave velocity, the Maxwell relaxation time, and the rigidity shear modulus are also isomorph invariant. In contrast to the isomorph invariance of all aspects of the transverse dynamics, the reduced-unit dynamics of the mass density is not invariant on length scales longer than the interparticle distance. By fitting to a generalized hydrodynamic model, we extract values for the wave-vector-dependent thermal diffusion coefficient, sound attenuation coefficient, and adiabatic sound velocity. The isomorph variation of these quantities in reduced units on long length scales can be eliminated by scaling with the density-scaling exponent, a fundamental quantity in the isomorph theory framework; this is an empirical observation that remains to be explained theoretically.

Dynamics of changes in stress autocorrelation functions of aluminum melt during ultrafast cooling

Abstract We present the results of the study of changes in liquid properties during ultrafast cooling. The molecular dynamics (MD) method is used, with aluminum melt as an example. Based on the changes in shear stress autocorrelation functions (SACF) with temperature in an ensemble of MD trajectories, we develop three approaches to the study of melt changes. In the first one, we investigate the asymptotic behavior of SACFs and a sharp increase in the melt viscosity, which is a conventional criterion. In the second approach, we present direct evidence of the transition of a metastable melt to a non-equilibrium state. In the third one, we show the appearance of transverse oscillations in a film of the melt. The most important observation is that all three phenomena occur in the same temperature range. On the basis of the current and the previous work, we conclude that there is a gap between the temperature of the splitting of the second peak of the pair correlation function and the temperature of the transition to a solid-like amorphous state. The splitting of the second peak occurs at a significantly higher temperature and this phenomenon is discussed.

Dynamics and Viscoelasticity of Disordered Melts of Symmetric Diblock Copolymers

Simulations of coarse-grained models are used to study relationships among chain motion, composition fluctuations, and stress relaxation in unentangled melts of symmetric diblock copolymers. Measurements of the dynamic structure factor S(q,t) are reported as a function of wavenumber q, time t, and χN, where χ is the Flory–Huggins interaction parameter and N is degree of polymerization. The function S(q,t) is found to be a nearly exponential function of time, S(q,t) ∝ e–t/τ(q), for wavenumbers similar to or less than the wavenumber q* at which the static structure factor S(q) ≡ S(q,t=0) is maximum. The relationship between the decay time τ(q) and S(q) is used to define an effective wavenumber-dependent diffusivity D(q) for fluctuations of wavenumber q. The function D(q) is shown to change very little with changes in χN and to be a monotonically decreasing function of the nondimensional wavenumber qRg, where Rg is polymer radius of gyration. The linear shear stress relaxation modulus G(t) is inferred from measurements of the shear stress autocorrelation function. At low values of χN, far from the order–disorder transition (ODT), the modulus G(t) agrees with predictions of the Rouse model. Near the ODT, G(t) develops an additional slowly decaying feature arising from slow decay of composition fluctuations with q ∼ q*. The behavior of G(t) near the ODT is predicted nearly quantitatively by a modified version of the model of Fredrickson and Larson (FL), in which the prediction of the FL theory for the slowly decaying component is added to the prediction of the Rouse theory for contributions arising from single-chain relaxation, using the independently measured behavior of S(q,t) as an input to the theory.

Density-dependent finite system-size effects in equilibrium molecular dynamics estimation of shear viscosity: Hydrodynamic and configurational study.

We study the intrinsic nature of the finite system-size effect in estimating shear viscosity of dilute and dense fluids within the framework of the Green-Kubo approach. From extensive molecular dynamics simulations, we observe that the size effect on shear viscosity is characterized by an oscillatory behavior with respect to system size L at high density and by a scaling behavior with an L-1 correction term at low density. Analysis of the potential contribution in the shear-stress autocorrelation function reveals that the former is configurational and is attributed to the inaccurate description of the long-range spatial correlations in finite systems. Observation of the long-time inverse-power decay in the kinetic contribution confirms its hydrodynamic nature. The L-1 correction term of shear viscosity is explained by the sensitive change in the long-time tail obtained from a finite system.

Rheology of symmetric diblock copolymers

Using large-scale simulations with the graphics processing unit (GPU) accelerated Highly Optimized Object-oriented Many-particle Dynamics (HOOMD)-blue software we investigate the anisotropic, dynamic-mechanical properties of lamellae-forming diblock copolymer melts. Both, the shear-stress autocorrelation function in equilibrium as well as stress-controlled simulations of shear flow are considered. The latter are implemented in (HOOMD)-blue via reverse nonequilibrium molecular dynamics simulation (RNEMDS).We find that copolymer lamellae that are perpendicularly oriented to the shear flow have similar rheological properties than unstructured homopolymer melts. Copolymer lamellae with the metastable parallel and the unstable transverse orientation exhibit similar dynamic-mechanical behavior in the linear regime. Most notably, they are characterized by a high dissipation.(RNEMDS) enables the study of oscillatory shear flow as well as the investigation of nonlinear rheology. Our simulation results for a soft, coarse-grained model at very high frequencies indicate that the finite speed of stress propagation has to be accounted for in the quantitative analysis, and we devise an analytic prediction for the nonlinear velocity and spatially varying stress profiles as a function of the loss and storage moduli.

Shear modulus of two-dimensional Yukawa or dusty-plasma solids obtained from the viscoelasticity in the liquid state.

Langevin dynamical simulations of two-dimensional (2D) Yukawa liquids are performed to investigate the shear modulus of 2D solid dusty plasmas. Using the known transverse sound speeds, we obtain a theoretical expression of the shear modulus of 2D Yukawa crystals as a function of the screening parameter κ, which can be used as the candidate of their shear modulus. The shear relaxation modulus G(t) of 2D Yukawa liquids is calculated from the shear stress autocorrelation function, consisting of the kinetic, potential, and cross portions. Due to their viscoelasticity, 2D Yukawa liquids exhibit the typical elastic property when the time duration is much less than the Maxwell relaxation time. As a result, the infinite frequency shear modulus G_{∞}, i.e., the shear relaxation modulus G(t) when t=0, of a 2D Yukawa liquid should be related to the shear modulus of the corresponding quenched 2D Yukawa solid (with the same κ value), with all particles suddenly frozen at their locations of the liquid state. It is found that the potential portion of the infinite frequency shear modulus for 2D Yukawa liquids at any temperature well agrees with the shear modulus of 2D Yukawa crystals with the same κ obtained from the transverse sound speeds. Thus, we find that the shear modulus of 2D Yukawa solids can be obtained from the motion of individual particles of the corresponding Yukawa liquids using their viscoelastic property.

Nature of intrinsic uncertainties in equilibrium molecular dynamics estimation of shear viscosity for simple and complex fluids.

We study two types of intrinsic uncertainties, statistical errors and system size effects, in estimating shear viscosity via equilibrium molecular dynamics simulations, and compare them with the corresponding uncertainties in evaluating the self-diffusion coefficient. Uncertainty quantification formulas for the statistical errors in the shear-stress autocorrelation function and shear viscosity are obtained under the assumption that shear stress follows a Gaussian process. Analyses of simulation results for simple and complex fluids reveal that the Gaussianity is more pronounced in the shear-stress process (related to shear viscosity estimation) compared with the velocity process of an individual molecule (related to self-diffusion coefficient). At relatively high densities corresponding to a liquid state, we observe that the shear viscosity exhibits complex size-dependent behavior unless the system is larger than a certain length scale, and beyond which, reliable shear viscosity values are obtained without any noticeable scaling behavior with respect to the system size. We verify that this size-dependent behavior is configurational and relate the characteristic length scale to the shear-stress correlation length.

Elastic moduli of a Brownian colloidal glass former

The static, dynamic and flow-dependent shear moduli of a binary mixture of Brownian hard disks are studied by an event-driven molecular dynamics simulation. Thereby, the emergence of rigidity close to the glass transition encoded in the static shear modulus is accessed by three methods. Results from shear stress auto-correlation functions, elastic dispersion relations, and the elastic response to strain deformations upon the start-up of shear flow are compared. This enables one to sample the time-dependent shear modulus consistently over several decades in time. By that a very precise specification of the glass transition point and of is feasible. Predictions by mode coupling theory of a finite shear modulus at the glass transition, of α-scaling in fluid states close to the transition, and of shear induced decay in yielding glass states are tested and broadly verified.

Viscosity of two-dimensional strongly coupled dusty plasma modified by a perpendicular magnetic field.

Transport properties of two-dimensional (2D) strongly coupled dusty plasmas have been investigated in detail, but never for viscosity with a strong perpendicular magnetic field; here, we examine this scenario using Langevin dynamics simulations of 2D liquids with a binary Yukawa interparticle interaction. The shear viscosity η of 2D liquid dusty plasma is estimated from the simulation data using the Green-Kubo relation, which is the integration of the shear stress autocorrelation function. It is found that, when a perpendicular magnetic field is applied, the shear viscosity of 2D liquid dusty plasma is modified substantially. When the magnetic field is increased, its viscosity increases at low temperatures, while at high temperatures its viscosity diminishes. It is determined that these different variational trends of η arise from the different behaviors of the kinetic and potential parts of the shear stress under external magnetic fields.

Numerical determination of shear stress relaxation modulus of polymer glasses.

Focusing on simulated polymer glasses well below the glass transition, we confirm the validity and the efficiency of the recently proposed simple-average expression [Formula: see text] for the computational determination of the shear stress relaxation modulus G(t). Here, [Formula: see text] characterizes the affine shear transformation of the system at t = 0 and h(t) the mean-square displacement of the instantaneous shear stress as a function of time t. This relation is seen to be particulary useful for systems with quenched or sluggish transient shear stresses which necessarily arise below the glass transition. The commonly accepted relation [Formula: see text] using the shear stress auto-correlation function c(t) becomes incorrect in this limit.

Shear-stress fluctuations in self-assembled transient elastic networks.

Focusing on shear-stress fluctuations, we investigate numerically a simple generic model for self-assembled transient networks formed by repulsive beads reversibly bridged by ideal springs. With Δt being the sampling time and t_{☆}(f)∼1/f the Maxwell relaxation time (set by the spring recombination frequency f), the dimensionless parameter Δx=Δt/t_{☆}(f) is systematically scanned from the liquid limit (Δx≫1) to the solid limit (Δx≪1) where the network topology is quenched and an ensemble average over m-independent configurations is required. Generalizing previous work on permanent networks, it is shown that the shear-stress relaxation modulus G(t) may be efficiently determined for all Δx using the simple-average expression G(t)=μ_{A}-h(t) with μ_{A}=G(0) characterizing the canonical-affine shear transformation of the system at t=0 and h(t) the (rescaled) mean-square displacement of the instantaneous shear stress as a function of time t. This relation is compared to the standard expression G(t)=c[over ̃](t) using the (rescaled) shear-stress autocorrelation function c[over ̃](t). Lower bounds for the m configurations required by both relations are given.

Simple average expression for shear-stress relaxation modulus

Focusing on isotropic elastic networks we propose a simple-average expression G(t)=μ_{A}-h(t) for the computational determination of the shear-stress relaxation modulus G(t) of a classical elastic solid or fluid. Here, μ_{A}=G(0) characterizes the shear transformation of the system at t=0 and h(t) the (rescaled) mean-square displacement of the instantaneous shear stress τ[over ̂](t) as a function of time t. We discuss sampling time and ensemble effects and emphasize possible pitfalls of alternative expressions using the shear-stress autocorrelation function. We argue finally that our key relation may be readily adapted for more general linear response functions.

Shear viscosity for dense plasmas by equilibrium molecular dynamics in asymmetric Yukawa ionic mixtures.

We present molecular dynamics (MD) calculations of shear viscosity for asymmetric mixed plasma for thermodynamic conditions relevant to astrophysical and inertial confinement fusion plasmas. Specifically, we consider mixtures of deuterium and argon at temperatures of 100-500 eV and a number density of 10^{25} ions/cc. The motion of 30,000-120,000 ions is simulated in which the ions interact via the Yukawa (screened Coulomb) potential. The electric field of the electrons is included in this effective interaction; the electrons are not simulated explicitly. Shear viscosity is calculated using the Green-Kubo approach with an integral of the shear stress autocorrelation function, a quantity calculated in the equilibrium MD simulations. We systematically study different mixtures through a series of simulations with increasing fraction of the minority high-Z element (Ar) in the D-Ar plasma mixture. In the more weakly coupled plasmas, at 500 eV and low Ar fractions, results from MD compare very well with Chapman-Enskog kinetic results. In the more strongly coupled plasmas, the kinetic theory does not agree well with the MD results. We develop a simple model that interpolates between classical kinetic theories at weak coupling and the Murillo Yukawa viscosity model at higher coupling. This hybrid kinetics-MD viscosity model agrees well with the MD results over the conditions simulated, ranging from moderately weakly coupled to moderately strongly coupled asymmetric plasma mixtures.

Fluctuation-dissipation relation between shear stress relaxation modulus and shear stress autocorrelation function revisited

The shear stress relaxation modulus G(t) may be determined from the shear stress after switching on a tiny step strain γ or by inverse Fourier transformation of the storage modulus G′(ω) or the loss modulus G″(ω) obtained in a standard oscillatory shear experiment at angular frequency ω. It is widely assumed that G(t) is equivalent in general to the equilibrium stress autocorrelation function which may be readily computed in computer simulations (β being the inverse temperature and V the volume). Focusing on isotropic solids formed by permanent spring networks we show theoretically by means of the fluctuation-dissipation theorem and computationally by molecular dynamics simulation that in general G(t) = Geq + C(t) for t > 0 with Geq being the static equilibrium shear modulus. A similar relation holds for G′(ω). G(t) and C(t) must thus become different for a solid body and it is impossible to obtain Geq directly from C(t).

Shear-stress relaxation and ensemble transformation of shear-stress autocorrelation functions.

We revisit the relation between the shear-stress relaxation modulus G(t), computed at finite shear strain 0<γ≪1, and the shear-stress autocorrelation functions C(t)|(γ) and C(t)|(τ) computed, respectively, at imposed strain γ and mean stress τ. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general G(t)=C(t)|(τ)=C(t)|(γ)+G(eq) for t>0 with G(eq) being the static equilibrium shear modulus. G(t) and C(t)|(γ) thus must become different for solids and it is impossible to obtain G(eq) alone from C(t)|(γ) as often assumed. We comment briefly on self-assembled transient networks where G(eq)(f) must vanish for a finite scission-recombination frequency f. We argue that G(t)=C(t)|(τ)=C(t)|(γ) should reveal an intermediate plateau set by the shear modulus G(eq)(f=0) of the quenched network.

Long-range spatial correlations of particle displacements and the emergence of elasticity.

We examine correlations of transverse particle displacements and their relationship to the shear modulus of a glass and the viscosity of a fluid. To this end we use computer simulations to calculate a correlation function of the displacements, S_{4}(q;t), which is similar to functions used to study heterogeneous dynamics in glass-forming fluids. We show that in the glass the shear modulus can be obtained from the long-time, small-q limit of S_{4}(q;t). By using scaling arguments, we argue that a four-point correlation length ξ_{4}(t) grows linearly in time in a glass and grows as sqrt[t] at long times in a fluid, and we verify these results by analyzing S_{4}(q;t) obtained from simulations. For a viscoelastic fluid, the simulation results suggest that the crossover to the long-time sqrt[t] growth of ξ_{4}(t) occurs at a characteristic decay time of the shear stress autocorrelation function. Using this observation, we show that the amplitude of the long-time sqrt[t] growth is proportional to sqrt[η] where η is the viscosity of the fluid.

Calculation of the second self-diffusion and viscosity virial coefficients of Lennard-Jones fluid by equilibrium molecular dynamics simulations

The second self-diffusion and viscosity virial coefficients of the Lennard-Jones (LJ) fluid were calculated by a detailed evaluation of the velocity and shear-stress autocorrelation functions using equilibrium molecular dynamics simulations at low and moderate densities. Accurate calculation of these coefficients requires corresponding transport coefficient values with low degrees of uncertainty. These were obtained via very long simulations by increasing the number of particles and by using the knowledge of correlation functions in the Green-Kubo method in conjunction with their corresponding generalized Einstein relations. The values of the self-diffusion and shear viscosity coefficients have been evaluated for systems with reduced densities between 0.0005 and 0.05 and reduced temperatures from 0.7 to 30.0. This provides a new insight into the transport coefficients beyond what can be offered by the Rainwater-Friend theory, which has not been developed for the self-diffusion coefficient.

An extended analysis of the viscosity kernel for monatomic and diatomic fluids

We present an extended analysis of the wavevector dependent shear viscosity of monatomicand diatomic (liquid chlorine) fluids over a wide range of wavevectors and for a variety ofstate points. The analysis is based on equilibrium molecular dynamics simulations, whichinvolve the evaluation of transverse momentum density and shear stress autocorrelationfunctions. For liquid chlorine we present the results in both atomic and molecularformalisms. We find that the viscosity kernel of chlorine in the atomic representation isstatistically indistinguishable from that in the molecular representation. The results furthersuggest that the real space viscosity kernels of monatomic and diatomic fluidsdepend sensitively on the density, the potential energy function and the choice offitting function in reciprocal space. It is also shown that the reciprocal spaceshear viscosity data can be fitted to two different simple functional forms overthe entire density, temperature and wavevector range: a function composed ofn-Gaussian terms and a Lorentzian-type function. Overall, the real space viscosity kernel hasa width of 3–6 atomic diameters, which means that the generalized hydrodynamicconstitutive relation is required for fluids with strain rates that vary nonlinearly overdistances of the order of atomic dimensions.