Liquid State Integral Equation Theories Research Articles

Theoretical analysis of the structure, thermodynamics, and shear elasticity of deeply metastable hard sphere fluids.

The structure, thermodynamics, and slow activated dynamics of the equilibrated metastable regime of glass-forming fluids remain a poorly understood problem of high theoretical and experimental interest. We apply a highly accurate microscopic equilibrium liquid state integral equation theory, in conjunction with naïve mode coupling theory of particle localization, to study in a unified manner the structural correlations, thermodynamic properties, and dynamic elastic shear modulus in deeply metastable hard sphere fluids. Distinctive behaviors are predicted including divergent inverse critical power laws for the contact value of the pair correlation function, pressure, and inverse dimensionless compressibility, and a splitting of the second peak and large suppression of interstitial configurations of the pair correlation function. The dynamic elastic modulus is predicted to exhibit two distinct exponential growth regimes with packing fraction that have strongly different slopes. These thermodynamic, structural, and elastic modulus results are consistent with simulations and experiments. Perhaps most unexpectedly, connections between the amplitude of long wavelength density fluctuations, dimensionless compressibility, local structure, and the dynamic elastic shear modulus have been theoretically elucidated. These connections are more broadly relevant to understanding the slow activated relaxation and mechanical response of colloidal suspensions in the ultradense metastable region and deeply supercooled thermal liquids in equilibrium.

Liquid state theory study of the phase behavior and macromolecular scale structure of model biomolecular condensates.

Biomolecular condensates can form through the liquid-liquid phase separation (LLPS) of proteins and RNAs in cells. However, other states of organization, including mesostructured network microstructures and physical gels, have been observed, the physical mechanism of which are not well understood. We use the Polymer Reference Interaction Site Model liquid state integral equation theory to study the equilibrium behavior of (generally aperiodic in sequence) biomolecular condensates based on a minimal sticker-spacer associating polymer model. The role of polymer packing fraction, sequence, and the strength and range of intermolecular interactions on macromolecular scale spatial organization and phase behavior is studied for typical sticker-spacer sequences. In addition to the prediction of conventional LLPS, a sequence-dependent strongly fluctuating polymeric microemulsion homogeneous state is predicted at high enough concentrations beyond the so-called Lifshitz-like point, which we suggest can be relevant to the dense phase of microstructured biomolecular condensates. New connections between local clustering and the formation of mesoscopic microdomains, the influence of attraction range, compressibility, and the role of spatial correlations across scales, are established. Our results are also germane to understanding the polymer physics of dense solutions of nonperiodic and unique sequence synthetic copolymers and provide a foundation to create new theories for how polymer diffusion and viscosity are modified in globally isotropic and homogeneous dense polymeric microemulsions.

A comparative study of aqueous DMSO mixtures by computer simulations and integral equation theories

ABSTRACTSeveral computer simulation studies of aqueous-dimethylsulfoxide with different force field models, and conducted by different authors, point out to an anomalous depressing of second and third neighbour correlations of the water–water radial distribution functions. This seemingly universal feature can be interpreted as the formation of linear water clusters. We test here the ability of liquid state integral equation theories to reproduce this feature. It is found that the incorporation of the water bridge diagram function is required to reproduce this feature. These theories are generally unable to properly reproduce atom–atom distribution functions. However, the near-ideal Kirkwood–Buff integrals are relatively well reproduced. We compute the X-ray scattering function and compare with available experimental results, with the particular focus to explain why these data do not reproduce the cluster pre-peak observed in the water–water structure factor.

Calculation of Ion-Dependent RNA Folding Free Energy using Coarse-Grained Simulation

Magnesium ions (Mg2+) are fundamentally important in RNA biology, playing indispensable roles in RNA structure, folding and function. In order to develop a model that treats monovalent ions implicitly while retaining explicit divalent cations we developed a theory based on liquid state integral equation theory that captures both inner shell and outer shell bindings of Mg to phosphate groups. Coupling this approach with our thermodynamically consistent RNA model reproduces Mg2+-RNA free energies for several RNA molecules, ranging from a small pseudoknot to the aptamer domain of adenine riboswitch. The model not only works for the RNA folded state, but also can be used to give accurate Mg2+-RNA thermodynamics for intermediate and unfolded states. In addition, we provide here one of the first attempts to elucidate the full three dimensional distribution of Mg2+ ions around RNA. Comparing Mg2+ with Ca2+ , it is revealed that magnesium binds using inner shell and outer shell equally well while calcium prefers to bind directly with phosphate groups due to the ease of dehydration of the lower charge density cation. Our model provides a comprehensive treatment of Mg2+-RNA interaction in terms of both energetic and structural features, allowing us to study ion-dependent RNA folding.

Designing molecular complexes using free-energy derivatives from liquid-state integral equation theory

Complex formation between molecules in solution is the key process by which molecular interactions are translated into functional systems. These processes are governed by the binding or free energy of association which depends on both direct molecular interactions and the solvation contribution. A design goal frequently addressed in pharmaceutical sciences is the optimization of chemical properties of the complex partners in the sense of minimizing their binding free energy with respect to a change in chemical structure. Here, we demonstrate that liquid-state theory in the form of the solute–solute equation of the reference interaction site model provides all necessary information for such a task with high efficiency. In particular, computing derivatives of the potential of mean force (PMF), which defines the free-energy surface of complex formation, with respect to potential parameters can be viewed as a means to define a direction in chemical space toward better binders. We illustrate the methodology in the benchmark case of alkali ion binding to the crown ether 18-crown-6 in aqueous solution. In order to examine the validity of the underlying solute–solute theory, we first compare PMFs computed by different approaches, including explicit free-energy molecular dynamics simulations as a reference. Predictions of an optimally binding ion radius based on free-energy derivatives are then shown to yield consistent results for different ion parameter sets and to compare well with earlier, orders-of-magnitude more costly explicit simulation results. This proof-of-principle study, therefore, demonstrates the potential of liquid-state theory for molecular design problems.

Mixture models with weak micro-heterogeneity

Weakly micro-heterogeneous binary systems are modelled as a mixture of a simple Lennard-Jones liquid and water-like models with reduced partial charges, hence named ‘weak-waters’. Such mixtures do not have the statistic problems of the usual micro-heterogeneous mixtures, and they allow to test how well liquid state integral equation theories are able to describe locally heterogeneous structures. It is found that such theories describe quite well the short-range structure, but fail to reproduce the heterogeneous structure in the nanometer range. In particular, it is found that the theories are not able to distinguish between micro-segregation and concentration fluctuations, a flaw apparent through the small wave vector behaviour of the structure factors. This finding helps provide further insight into both the nature of the micro-heterogeneity of aqueous mixtures in general, and the routes to improve the ability of integral equation theories to describe such mixtures.

Packing correlations, collective scattering and compressibility of fractal-like aggregates in polymer nanocomposites and suspensions

We employ a small scale Monte Carlo method plus microscopic liquid state integral equation theory to study two models of fractal-like aggregates (single and quenched disordered averaged) as a function of concentration. The models employed are relevant to non-compact carbon black or silica aggregates in polymer nanocomposites and suspensions. When the primary nanoparticles interact as hard spheres the resulting center-of-mass level effective pair potential has a distinctive soft repulsion form which is sensitive to whether geometric disorder at the single aggregate level is included or not. The aggregate fluids show a considerable amount of interpenetration and complex non-hard-sphere packing features in their pair correlation functions. The latter are also different, both qualitatively and quantitatively, for the single versus average aggregate systems. Calculations of the collective static structure factor and compressibility (or inverse bulk modulus) are in qualitative agreement with a recent scattering experiment on suspensions of fumed silica hard fractals. Use of the theoretical structural information in a microscopic dynamical theory of kinetic arrest suggests the presence of multiple length scales and disordered local packing frustrates the emergence of glassy dynamics in these aggregate fluids even at high volume fractions.

Closure-based perturbative density-functional theory of hard-sphere freezing: Properties of the bridge functional

The study of freezing using perturbative classical density-functional theory is revisited, using a bridge functional approach to resum all terms beyond second order in the free energy expansion. More precisely, the first-order direct correlation function of the solid phase is written as a functional expansion about the homogeneous liquid phase, and the sum of all higher-order terms is represented as a functional of the second-order term. Information about the shape and uniqueness of this bridge functional for the case of hard spheres is obtained via an inversion procedure that employs Monte Carlo fluid-solid coexistence data from the literature. The parametric plots obtained from the inversion procedure show very little scatter in certain regions, suggesting a unique functional dependence, but large scatter in other regions. The scatter is related to the anisotropy of the solid lattice at the particle scale. Interestingly, the thermodynamic properties of the phase transition are quite insensitive to the regions where the scatter is large, and several simple closures (i.e., analytical forms of the bridge function) reproduce exactly the liquid-solid coexistence densities and Lindemann parameter from simulation. The form of these closures is significantly different from the usual closures employed in liquid-state integral equation theory.

Effective interactions in polydisperse colloidal suspensions investigated using Ornstein–Zernike integral equations

We present a mean of calculating the effective interactions in polydisperse colloidal suspension from liquid state integral equation theory. The method is based on Lado’s expansion of correlation functions in a suitable set of orthogonal polynomials. The outlined approach is subsequently used to investigate the effects of polydispersity on the effective potentials for model systems with attractive and repulsive bare interactions. The dominant effect of polydispersity of the smaller species is to weaken the effective potentials between big colloidal particles. This can be exploited as another way of tuning the interactions in colloidal suspensions to match the desired properties.

Phase diagram of a model anticlustering binary mixture in two dimensions: A semi-grand-canonical Monte Carlo study.

The temperature-density phase diagram of a model binary mixture in two dimensions is investigated using a semi-grand-canonical Monte Carlo simulation scheme which allows for exchange between the two species while keeping the total number of atoms fixed. The gas-liquid and the gas-solid regions of the phase diagram are mapped out using the efficient block analysis method incorporating finite-size scaling of the various coexisting densities. An ordered square lattice structure is seen to be stable at low temperatures. Interesting short-range ordering phenomena resulting in a ``disorder line'' in the fluid phase are also analyzed and compared with results from liquid-state integral equation theories.

Liquid-state integral equations at high density: On the mathematical origin of infinite-range oscillatory solutions

Analytic asymptotic analysis and finite element numerical procedures are used to elucidate the mathematical reasons for the appearance of infinite-range oscillatory solutions to certain integral equation theories of wall–fluid interfacial structure and liquid state radial distribution functions. The results contribute to two issues of recent debate: (i) what physical significance (if any) can be attributed to the apparent ‘‘solidlike’’ structure that is often (but not always) seen in high density solutions to liquid state integral equation theories and (ii) is the same mathematical structure present in density functional theories (i.e., in the presence of a variational condition arising from a free energy functional)?