This paper is the fourth in a series devoted to identifying and explaining the properties of strongly correlating liquids, i.e., liquids where virial and potential energy correlate better than 90% in their thermal equilibrium fluctuations in the $NVT$ ensemble. For such liquids we here introduce the concept of “isomorphic” curves in the phase diagram. A number of thermodynamic, static, and dynamic isomorph invariants are identified. These include the excess entropy, the isochoric specific heat, reduced-unit static and dynamic correlation functions, as well as reduced-unit transport coefficients. The dynamic invariants apply for both Newtonian and Brownian dynamics. It is shown that after a jump between isomorphic state points the system is instantaneously in thermal equilibrium; consequences of this for generic aging experiments are discussed. Selected isomorph predictions are validated by computer simulations of the Kob–Andersen binary Lennard-Jones mixture, which is a strongly correlating liquid. The final section of the paper relates the isomorph concept to phenomenological melting rules, Rosenfeld’s excess entropy scaling, Young and Andersen’s approximate scaling principle, and the two-order parameter maps of Debenedetti and co-workers. This section also shows how the existence of isomorphs implies an “isomorph filter” for theories for the non-Arrhenius temperature dependence of viscous liquids’ relaxation time, and it explains isochronal superposition for strongly correlating viscous liquids.

We derive the stationary probability distribution for a non-equilibrium system composed by an arbitrary number of degrees of freedom that are subject to Gaussian colored noise and a conservative potential. This is based on a multidimensional version of the Unified Colored Noise Approximation. By comparing theory with numerical simulations we demonstrate that the theoretical probability density quantitatively describes the accumulation of active particles around repulsive obstacles. In particular, for two particles with repulsive interactions, the probability of close contact decreases when one of the two particle is pinned. Moreover, in the case of isotropic confining potentials, the radial density profile shows a non trivial scaling with radius. Finally we show that the theory well approximates the “pressure” generated by the active particles allowing to derive an equation of state for a system of non-interacting colored noise-driven particles.

We show that a number of model liquids at fixed volume exhibit strong correlations between equilibrium fluctuations of the configurational parts of (instantaneous) pressure and energy. We present detailed results for 13 systems, showing in which systems these correlations are significant. These include Lennard-Jones liquids (both single- and two-component) and several other simple liquids, neither hydrogen-bonding liquids such as methanol and water, nor the Dzugutov liquid, which has significant contributions to pressure at the second nearest neighbor distance. The pressure-energy correlations, which for the Lennard-Jones case are shown to also be present in the crystal and glass phases, reflect an effective inverse power-law potential dominating fluctuations, even at zero and slightly negative pressure. An exception to the inverse power-law explanation is a liquid with hard-sphere repulsion and a square-well attractive part, where a strong correlation is observed, but only after time averaging. The companion paper [N. P. Bailey et al., J. Chem. Phys. 129, 184508 (2008)] gives a thorough analysis of the correlations, with a focus on the Lennard-Jones liquid, and a discussion of some experimental and theoretical consequences.

We present a detailed analysis and discuss consequences of the strong correlations of the configurational parts of pressure and energy in their equilibrium fluctuations at fixed volume reported for simulations of several liquids in the previous paper [N. P. Bailey et al., J. Chem. Phys. 129, 184507 (2008)]. The analysis concentrates specifically on the single-component Lennard-Jones system. We demonstrate that the potential may be replaced, at fixed volume, by an effective power law but not simply because only short-distance encounters dominate the fluctuations. Indeed, contributions to the fluctuations are associated with the whole first peak of the radial distribution function, as we demonstrate by an eigenvector analysis of the spatially resolved covariance matrix. The reason the effective power law works so well depends crucially on going beyond single-pair effects and on the constraint of fixed volume. In particular, a better approximation to the potential includes a linear term, which contributes to the mean values of potential energy and virial, but little to their fluctuations, for density fluctuations which conserve volume. We also study in detail the zero temperature limit of the (classical) crystalline phase, where the correlation coefficient becomes very close, but not equal, to unity, in more than one dimension; in one dimension the limiting value is exactly unity. In the second half of the paper we consider four consequences of strong pressure-energy correlations: (1) analyzing experimental data for supercritical argon we find 96% correlation; (2) we discuss the particular significance acquired by the correlations for viscous van der Waals liquids approaching the glass transition: For strongly correlating viscous liquids knowledge of just one of the eight frequency-dependent thermoviscoelastic response functions basically implies knowledge of them all; (3) we reinterpret aging simulations of ortho-terphenyl carried out by Mossa et al. [Eur. Phys. J. B 30, 351 (2002)], showing their conclusions follow from the strongly correlating property; and (4) we briefly discuss the presence of the correlations (after appropriate time averaging) in model biomembranes, showing that significant correlations may be present even in quite complex systems.

Microgels are colloidal-scale particles individually made of cross-linked polymer networks that can swell and deswell in response to external stimuli, such as changes to temperature or pH. Despite a large amount of experimental activities on microgels, a proper theoretical description based on individual particle properties is still missing due to the complexity of the particles. To go one step further, here we propose a novel methodology to assemble realistic microgel particles in silico. We exploit the self-assembly of a binary mixture composed of tetravalent (cross-linkers) and bivalent (monomer beads) patchy particles under spherical confinement in order to produce fully bonded networks. The resulting structure is then used to generate the initial microgel configuration, which is subsequently simulated with a bead–spring model complemented by a temperature-induced hydrophobic attraction. To validate our assembly protocol, we focus on a small microgel test case and show that we can reproduce the experimental swelling curve by appropriately tuning the confining sphere radius, something that would not be possible with less sophisticated assembly methodologies, e.g., in the case of networks generated from an underlying crystal structure. We further investigate the structure (in reciprocal and real space) and the swelling curves of microgels as a function of temperature, finding that our results are well described by the widely used fuzzy sphere model. This is a first step toward a realistic modeling of microgel particles, which will pave the way for a careful assessment of their elastic properties and effective interactions.

This series of papers is devoted to identifying and explaining the properties of strongly correlating liquids, i.e., liquids with more than 90% correlation between their virial W and potential energy U fluctuations in the NVT ensemble. Paper IV [N. Gnan et al., J. Chem. Phys. 131, 234504 (2009) https://doi.org/10.1063/1.3265957.] showed that strongly correlating liquids have “isomorphs,” which are curves in the phase diagram along which structure, dynamics, and some thermodynamic properties are invariant in reduced units. In the present paper, using the fact that reduced-unit radial distribution functions are isomorph invariant, we derive an expression for the shapes of isomorphs in the WU phase diagram of generalized Lennard-Jones systems of one or more types of particles. The isomorph shape depends only on the Lennard-Jones exponents; thus all isomorphs of standard Lennard-Jones systems (with exponents 12 and 6) can be scaled onto a single curve. Two applications are given. One tests the prediction that the solid-liquid coexistence curve follows an isomorph by comparing to recent simulations by Ahmed and Sadus [J. Chem. Phys. 131, 174504 (2009)] https://doi.org/10.1063/1.3253686.. Excellent agreement is found on the liquid side of the coexistence curve, whereas the agreement is less convincing on the solid side. A second application is the derivation of an approximate equation of state for generalized Lennard-Jones systems by combining the isomorph theory with the Rosenfeld-Tarazona expression for the temperature dependence of the potential energy on isochores. It is shown that the new equation of state agrees well with simulations.