Soft-sphere Liquid Research Articles

Structural relaxation, dynamical arrest, and aging in soft-sphere liquids.

We investigate the structural relaxation of a soft-sphere liquid quenched isochorically (ϕ = 0.7) and instantaneously to different temperatures Tf above and below the glass transition. For this, we combine extensive Brownian dynamics simulations and theoretical calculations based on the non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory. The response of the liquid to a quench generally consists of a sub-linear increase of the α-relaxation time with system's age. Approaching the ideal glass-transition temperature from above (Tf > Ta), sub-aging appears as a transient process describing a broad equilibration crossover for quenches to nearly arrested states. This allows us to empirically determine an equilibration timescale teq(Tf) that becomes increasingly longer as Tf approaches Ta. For quenches inside the glass (Tf ≤ Ta), the growth rate of the structural relaxation time becomes progressively larger as Tf decreases and, unlike the equilibration scenario, τα remains evolving within the whole observation time-window. These features are consistently found in theory and simulations with remarkable semi-quantitative agreement and coincide with those revealed in a previous and complementary study [P. Mendoza-Méndez et al., Phys. Rev. 96, 022608 (2017)] that considered a sequence of quenches with fixed final temperature Tf = 0 but increasing ϕ toward the hard-sphere dynamical arrest volume fraction ϕHS a=0.582. The NE-SCGLE analysis, however, unveils various fundamental aspects of the glass transition, involving the abrupt passage from the ordinary equilibration scenario to the persistent aging effects that are characteristic of glass-forming liquids. The theory also explains that, within the time window of any experimental observation, this can only be observed as a continuous crossover.

A classical density functional theory model for fragility in the hard-sphere limit

Abstract We study, using the classical density functional theory (DFT), the fragility and short-time elastic constants of a soft-sphere liquid. For the amorphous state, the order parameter is the inhomogeneous density function $\rho({\bf r})$ which is described in terms of Gaussian density profiles centered on a set random lattice points $\{{\bf R}_i\}$. The latter is characterized in terms of the Bernel pair function $g_\mathrm{B}(r)$. Based on the Adam–Gibbs-type relation between the $\alpha$ relaxation time $\tau_\alpha$ and the configurational entropy $\mathcal{S}_{\rm c}$, a thermodynamic fragility $m_\mathrm{T}$ for the liquid is defined. The concentration or average density of the liquid is treated as the control parameter here instead of temperature. The configurational entropy of the liquid is calculated using the DFT model. Variations in the short-range structure of the amorphous state are made with different choices for the value of $g_\mathrm{B}(r)$ at short distances, and its implications on the correlation between fragility $m_\mathrm{T}$ and the softness index $n$ are studied. The dependence of Poisson’s ratio $\nu$ on the softness index $n$ of the interaction potential is also obtained from the density dependence of the metastable state free energy. The correlation between $m_\mathrm{T}$ and $\nu$ follows.

Dynamic equivalences in the hard-sphere dynamic universality class

We perform systematic simulation experiments on model systems with soft-sphere repulsive interactions to test the predicted dynamic equivalence between soft-sphere liquids with similar static structure. For this we compare the simulated dynamics (mean squared displacement, intermediate scattering function, α-relaxation time, etc.) of different soft-sphere systems, between them and with the hard-sphere liquid. We then show that the referred dynamic equivalence does not depend on the (Newtonian or Brownian) nature of the microscopic laws of motion of the constituent particles, and hence, applies independently to colloidal and to atomic simple liquids. Finally, we verify another more recently proposed dynamic equivalence, this time between the long-time dynamics of an atomic liquid and its corresponding Brownian fluid (i.e., the Brownian system with the same interaction potential).

Equilibration and aging of dense soft-sphere glass-forming liquids

The recently developed nonequilibrium extension of the self-consistent generalized Langevin equation theory of irreversible relaxation [Ramírez-González and Medina-Noyola, Phys. Rev. E 82, 061503 (2010); Ramírez-González and Medina-Noyola, Phys. Rev. E 82, 061504 (2010)] is applied to the description of the irreversible process of equilibration and aging of a glass-forming soft-sphere liquid that follows a sudden temperature quench, within the constraint that the local mean particle density remains uniform and constant. For these particular conditions, this theory describes the nonequilibrium evolution of the static structure factor S(k;t) and of the dynamic properties, such as the self-intermediate scattering function F(S)(k,τ;t), where τ is the correlation delay time and t is the evolution or waiting time after the quench. Specific predictions are presented for the deepest quench (to zero temperature). The predicted evolution of the α-relaxation time τ(α)(t) as a function of t allows us to define the equilibration time t(eq)(φ), as the time after which τ(α)(t) has attained its equilibrium value τ(α)(eq)(φ). It is predicted that both, t(eq)(φ) and τ(α)(eq)(φ), diverge as φ→φ((a)), where φ((a)) is the hard-sphere dynamic-arrest volume fraction φ((a))(≈0.582), thus suggesting that the measurement of equilibrium properties at and above φ((a)) is experimentally impossible. The theory also predicts that for fixed finite waiting times t, the plot of τ(α)(t;φ) as a function of φexhibits two regimes, corresponding to samples that have fully equilibrated within this waiting time (φ≤φ((c))(t)), and to samples for which equilibration is not yet complete (φ≥φ((c))(t)). The crossover volume fraction φ((c))(t) increases with t but saturates to the value φ((a)).

Density-Temperature-Softness Scaling of the Dynamics of Glass-Forming Soft-Sphere Liquids

We employ the principle of dynamic equivalence between soft-sphere and hard-sphere fluids [Phys. Rev. E 68, 011405 (2003)] to describe the interplay of the effects of varying the density n, the temperature T, and the softness (characterized by a softness parameter ν(-1)) on the dynamics of glass-forming soft-sphere liquids in terms of simple scaling rules. The main prediction is the existence of a dynamic universality class associated with the hard-sphere fluid, constituted by the soft-sphere systems whose dynamic parameters depend on n, T, and ν only through the reduced density n*≡nσ(HS)(T*,ν). A number of scaling properties observed in recent experiments and simulations involving glass-forming fluids with repulsive short-range interactions are found to be a direct manifestation of this general dynamic equivalence principle.

Dissociative melting of ice VII at high pressure

We have used x-ray diffraction to determine the structure factor of water along its melting line to a static pressure of 57 GPa (570 kbar) and a temperature of more than 1500 K, conditions which correspond to the lower mantle of the Earth, and the interiors of Neptune and Uranus up to a depth of 7000 km. We have also performed corresponding first principles and classical molecular dynamics simulations. Above a pressure of 4 GPa the O-O structure factor is found to be very close to that of a simple soft sphere liquid, thus permitting us to determine the density of liquid water near the melting line. By comparing these results with the density of ice, also determined in this study, we find that the enthalpy of fusion (DeltaH(f)) increases enormously along the melting line, reaching approximately 120 kJ/mole at 40 GPa (compared to 6 kJ/mole at 0 GPa), thus revealing significant molecular dissociation of water upon melting. We speculate that an extended two-phase region could occur in planetary processes involving the adiabatic compression of water.

Model for dynamics of inhomogeneous and bulk fluids

An accurate model for the density of states (DOS) for strongly inhomogeneous and bulk fluids has been proposed based on gamma distributions. The contribution to the density of states from the collective dynamics is modeled as an incomplete gamma distribution and the high frequency region is obtained from the solution of the memory equation using a sech memory kernel. Using only the frequency moments as input, the model parameters for the collective dynamics are obtained by matching moments of the resulting distribution. The model results in an analytical expression for the self-diffusivity of the fluid. We present results for soft sphere fluids confined in slit-shaped pores as well as bulk soft sphere liquids. Comparisons of the DOS, velocity autocorrelation functions, and memory kernels with molecular dynamics simulations reveal that the model predicts features in the DOS over the entire frequency range and is able to capture changes in the DOS as a function of fluid density and temperature. As a result the predicted VACFs, memory kernels, and self-diffusivities are accurately predicted over a wide range of conditions. Since the frequency moments for bulk liquids can be obtained from pair correlation functions, our method provides a direct route from fluid structure to dynamics. For fluids confined in slit-shaped pores, where the frequency moments are obtained from molecular dynamics simulations, the predicted self-diffusivities capture the resulting oscillations due to variations in the solvation pressure, and in the case of smooth walled pores, the predictions are superior to those obtained using kinetic theory.

Relaxation and Short Time Dynamics of Bulk Liquids and Fluids Confined in Spherical Cavities and Slit Pores

The density of states for bulk and confined fluids have been modeled using a recently proposed gamma distribution (Krishnan, S. H.; Ayappa, K. G. J. Chem. Phys. 2004, 121, 3197). The gamma distribution results in a closed form analytical expression for the velocity autocorrelation function and the relaxation time of the fluid. The two parameters of the gamma distribution are related analytically to the second and fourth frequency moments of the fluid using short time expansions. The predictions by the proposed gamma model are compared with the velocity autocorrelation functions obtained using the theory of instantaneous normal modes (INMs) and from molecular dynamics simulations. The model is applied to a bulk soft sphere liquid and fluids confined in a spherical cavity and slit-shaped pores. The gamma model is able to capture the resulting changes in relaxation time due to changes in density and temperature extremely well for both the bulk liquid and confined inhomogeneous fluid situations. In all cases, the predictions by the gamma model are superior to those obtained from the INM theory. In the case of the fluid confined in a slit pore, the loadings were obtained from a grand canonical Monte Carlo simulation where the pore is equilibrated with a bulk fluid. This is similar to a confinement situation in a surface force apparatus. The predicted relaxation times vs pore widths from the gamma model are seen to accurately capture the oscillations due to formation and disruption of layers within the slit pore.

Biased Sampling of Nonequilibrium Trajectories: Can Fast Switching Simulations Outperform Conventional Free Energy Calculation Methods?

We have investigated the maximum computational efficiency of reversible work calculations that change control parameters in a finite amount of time. Because relevant nonequilibrium averages are slow to converge, a bias on the sampling of trajectories can be beneficial. Such a bias, however, can also be employed in conventional methods for computing reversible work, such as thermodynamic integration or umbrella sampling. We present numerical results for a simple one-dimensional model and for a Widom insertion in a soft sphere liquid, indicating that, with an appropriately chosen bias, conventional methods are in fact more efficient. We describe an analogy between nonequilibrium dynamics and mappings between equilibrium ensembles, which suggests that the practical inferiority of fast switching is quite general. Finally, we discuss the relevance of adiabatic invariants in slowly driven Hamiltonian systems for the application of Jarzynski's theorem.

Vapor-liquid coexistence in many-body dissipative particle dynamics.

Many-body dissipative particle dynamics is constructed to exhibit vapor-liquid coexistence, with a sharp interface, and a vapor phase of vanishingly small density. The application to fluid mechanics problems involving free surfaces is illustrated by simulation of a pendant drop. The model is an unusual example of a soft-sphere liquid with a potential energy built out of local-density-dependent one-particle self-energies.

INTERMOLECULAR MOMENTUM TRANSFER IN A SIMPLE LIQUID AND ITS CONTRIBUTION TO SHEAR VISCOSITY

Momentum transfer in a sheared liquid is analyzed with a focus on the intermolecular momentum transfer (IMT). The IMT, which determines shear viscosity of liquids, is examined based on the result of molecular dynamics (MD) simulations for a simple liquid modeled by the Lennard-Jones (12-6) potential. Distortion of liquid structure, which is characterized by spherical symmetry breaking in the pair distribution of molecules, is reproduced by the MD simulation and its effect on the characteristics of the IMT, which includes a negative momentum transfer in a certain range of intermolecular distance, is discussed. The appearance of negative IMT depends on characteristics of the intermolecular potential, and it is predicted that the IMT will show a different response for repulsive soft-sphere liquids.

Non-Newtonian flow between concentric cylinders calculated from thermophysical properties obtained from simulations

A study of the Weissenberg effect (rod climbing in a stirred system) based on nonequilibrium molecular dynamics (NEMD) is reported. Simulation results from a soft-sphere fluid are used to obtain a self-consistent free-surface profile of the fluid of finite compressibility undergoing Couette flow between concentric cylinders. A numerical procedure is then applied to calculate the height profile for a hypothetical fluid with thermophysical properties of the soft-sphere liquid and of a dense colloidal suspension. The height profile calculated is identified with shear thickening and the forms of the viscometric functions. The maximum climb occurs between the cylinders rather than at the inner cylinder.

Dynamic cross correlation in isotopic two-component liquids: Molecular-dynamics calculation results compared with predictions of kinetic theory.

For thermodynamically ideal two-component systems, as for example isotopic systems, the mass effect on the mutual diffusion coefficient is negligibly small for mass ratios of order 2. For higher mass ratios a significant influence on the diffusion coefficient may be expected. We present here accurate molecular-dynamics calculation results for the hard soft sphere and the Lennard-Jones liquids, which indicate quantitatively the mass-ratio effect on the mutual diffusion coefficient. The predictions of the Enskog kinetic theory are found to give the correct trend and order of magnitude of these effects but overestimate them at all the mass ratios studied. Both systems, the hard soft sphere liquid (the softness of the interaction is so small that it can be regarded as a hard one) and the Lennard-Jones liquid, show appreciable mass-ratio dependence of the diffusion coefficient only for ratios larger than 10, in agreement with our previous numerical work [Schoen and Hoheisel, Mol. Phys. 52, 1029 (1984); 53, 1367 (1984); 57, 65 (1986)].

A modified hypernetted-chain integral equation for the supercooled liquid of inverse power potentials

Using the integral equation of the theory of liquids, we discuss structural properties of highly supercooled soft-sphere liquids. Two different thermodynamic self-consistent (TC) integral equations, i.e., the Rogers-Young equation and the reference hypernetted-chain equation, have been tested for highly supercooled liquids. According to these results, we propose a new modified hypernetted-chain integral equation for highly supercooled liquids (MHNCS). The bridge function of the MHNCS equation is approximated by an appropriate interpolation of the bridge function of the Percus-Yevick hard-sphere model and that of the leading elementary diagram. We have obtained solutions of the MHNCS equation for sixth and twelveth inverse power potentials (soft-sphere model). Below the freezing point, the PDF calculated from the MHNCS approximation is found to show a splitting of the second peak, which is compatible with the results of computer simulations, while the TC equations show no such a splitting.

Properties of a soft-sphere liquid from non-Newtonian molecular dynamics

A soft-sphere, inverse-12 liquid is simulated in both the isokinetic-isochoric and the isokinetic-isobaric ensemble using nonequilibrium molecular dynamics. The simulation for the isobaric ensemble is discussed in detail. The non-Newtonian characteristics of the liquid are clearly demonstrated; namely, the shear-rate-dependent pressure and density (shear dilatancy), the shear-rate-dependent shear viscosity (shear thinning), and evidence of normal pressure differences. For the first time, it is clearly shown that a significant component of isobaric shear thinning is due to shear dilatancy. The isochoric and isobaric results are checked for consistency. Simple empirical relations for the equation of state and transport properties of the fluid are presented.

Prediction of shear viscosity and non-Newtonian behavior in the soft-sphere liquid

We show that a shear rate-dependent viscosity coefficient, normal pressure differences, and shear dilatancy can be predicted in a soft-sphere liquid given only the equilibrium radial distribution function and a relaxation time. Calculations are made using the relaxation-time theory of Hess and Hanley, and the results are compared with simulation data from nonequilibrium molecular dynamics.

The static structure factor of a soft sphere liquid under shear

An expansion for the structure factor of a simple liquid undergoing planar Couette flow is extended to tensor rank four. The coefficients are evaluated by the methods of nonequilibrium molecular dynamics (NEMD) for a soft sphere fluid at 7/8 of its freezing density. Intensity plots are constructed and it is shown that the fourth rank contribution to the structure factor is significant for a reduced shear rate of one.

Dynamic Slowing-Down and Nonexponential Decay of the Density Correlation Function in Soft-Sphere Supercooled Liquids

Molecular dynamics simulations were carried out for supercooled soft-sphere liquids. The computed density correlation function shows a clear nonexponential decay in intermediate times, as the liquid-glass transition is approached. The overall time dependence of the correlation function can be fitted by a stretched function of the type: exp [-( t / t 0 ) α ], where the exponent α varies from 1, in stable liquids, down to about 0.5, in the vicinity of the liquid-glass transition. The average correlation time =(1/α) G (1/α) t 0 ( G ( x ) being the gamma function) can be reasonably fitted by a scaling law of the type: \({\langle}T{\rangle}{\propto}(\varGamma_{\text{g}}-\varGamma)^{-\theta}\), where \(\varGamma\) is a coupling constant, the suffix g denotes the glass transition, and the exponent θ varies from about 2 to 5, depending on wave number k . The wave number dependence of the correlation function is discussed.

Now-Newtonian flow between concentric cylinders and the effects of finite compressibility

Previous studies of the flow of a model soft-sphere liquid between rotating vertical concentric cylinders have predicted an enhanced depression of the free surface at the inner cylinder and the necessity and importance of accounting for finite compressibility. In those studies the rheological properties of the liquids were taken directly from computer simulations, whereas in the present work the liquid properties are altered in a controlled manner and the fluid dynamics problems is again solved numerically and self-consistently with the original boundary conditions. Specific alterations include the removal of all non-Newtonian properties, the change in sign of a generalized viscosity to create a rodclimbing or Weissenberg effect, and the removal of shear dilatancy or increase in pressure with shear. Our conclusion is that nonzero compressibility needs to be taken into account only in the presence of shear dilatancy.

Non-Newtonian flow of a model liquid between concentric cylinders

Computer simulations of fluids out of equilibrium indicate that even the simplest fluid is in principle non-Newtonian. In particular, the simulations can provide explicitly the pressure tensor as a function of shear rate at a given temperature and density. In this paper the steady state flow of a model soft sphere liquid between rotating vertical concentric cylinders is discussed from a microscopic standpoint, given the coefficients that characterize the pressure tensor. The equations of motion are solved numerically. It is found that the normal pressure differences lead to an enhanced depression of the free surface at the inner cylinder, in contrast to a climbing (Weissenberg effect) which is usually regarded as the consequence of such differences. Reasons for the behavior observed for the soft sphere system are discussed. A consequence of the analysis is that a unique and self-consistent solution of the equations of motion is obtained only if the effects of finite compressibility are included.