Machine Learning of a Density Functional for Anisotropic Patchy Particles.

Anisotropic patchy particles have become an archetypical statistical model system for associating fluids. Here, we formulate an approach to the Kern-Frenkel model via the classical density functional theory to describe the positionally and orientationally resolved equilibrium density distributions in flat wall geometries. The density functional is split into a reference part for the orientationally averaged density and an orientational part in mean-field approximation. To bring the orientational part into a kernel form suitable for machine learning (ML) techniques, an expansion into orientational invariants and the proper incorporation of single-particle symmetries are formulated. The mean-field kernel is constructed via ML on the basis of hard wall simulation data. The results are compared to the well-known random-phase approximation, which strongly underestimates the orientational correlations close to the wall. Successes and shortcomings of the mean-field treatment of the orientational part are highlighted and perspectives are given for attaining a full-density functional via ML.

Effects of flexibility in coarse-grained models for bovine serum albumin and immunoglobulin G.

We construct a coarse-grained, structure-based, low-resolution, 6-bead flexible model of bovine serum albumin (BSA, PDB: 4F5S), which is a popular example of a globular protein in biophysical research. The model is obtained via direct Boltzmann inversion using all-atom simulations of a single molecule, and its particular form is selected from a large pool of 6-bead coarse-grained models using two suitable metrics that quantify the agreement in the distribution of collective coordinates between all-atom and coarse-grained Brownian dynamics simulations of solutions in the dilute limit. For immunoglobulin G (IgG), a similar structure-based 12-bead model has been introduced in the literature [Chaudhri et al., J. Phys. Chem. B 116, 8045 (2012)] and is employed here to compare findings for the compact BSA molecule and the more anisotropic IgG molecule. We define several modified coarse-grained models of BSA and IgG, which differ in their internal constraints and thus account for a variation of flexibility. We study denser solutions of the coarse-grained models with purely repulsive molecules (achievable by suitable salt conditions) and address the effect of packing and flexibility on dynamic and static behavior. Translational and rotational self-diffusivity is enhanced for more elastic models. Finally, we discuss a number of effective sphere sizes for the BSA molecule, which can be defined from its static and dynamic properties. Here, it is found that the effective sphere diameters lie between 4.9 and 6.1 nm, corresponding to a relative spread of about ±10% around a mean of 5.5 nm.

Approaching the hard sphere limit in colloids suitable for confocal microscopy - the end of a decade lasting quest.

PMMA-PHSA particles serve as the hard sphere model system since the 1980s. We investigate the fluid structure of fluorescent ones in three different solvents by laser scanning confocal microscopy: a decalin-tetrachloroethylene (TCE)-mixture and a decalin-cyclohexylbromide (CHB)-mixture with and without tetrabutylammoniumbromide (TBAB). The experimental 3D radial distribution functions are modeled by analytical theory and computer simulations taking polydispersity and the experimental position uncertainty into account. The quantitative comparison between experiment and simulation/theory establishes hard sphere like behavior for particles in decalin-TCE for a wide range of particle packing fractions. To the best of our knowledge, we present the first experimental dataset of a fluid structure that agrees convincingly with Percus-Yevick over a wide concentration range. Furthermore, charged sphere behavior is confirmed both for the decalin-CHB and the decalin-CHB-TBAB solvents, and it is demonstrated that a finite particle concentration reduces screening in the decalin-CHB-TBAB system compared to the bulk solvent.

Short-Time Transport Properties of Bidisperse Suspensions of Immunoglobulins and Serum Albumins Consistent with a Colloid Physics Picture.

The crowded environment of biological systems such as the interior of living cells is occupied by macromolecules with a broad size distribution. This situation of polydispersity might influence the dependence of the diffusive dynamics of a given tracer macromolecule in a monodisperse solution on its hydrodynamic size and on the volume fraction. The resulting size dependence of diffusive transport crucially influences the function of a living cell. Here, we investigate a simplified model system consisting of two constituents in aqueous solution, namely, of the proteins bovine serum albumin (BSA) and bovine polyclonal gamma-globulin (Ig), systematically depending on the total volume fraction and ratio of these constituents. From high-resolution quasi-elastic neutron spectroscopy, the separate apparent short-time diffusion coefficients for BSA and Ig in the mixture are extracted, which show substantial deviations from the diffusion coefficients measured in monodisperse solutions at the same total volume fraction. These deviations can be modeled quantitatively using results from the short-time rotational and translational diffusion in a two-component hard sphere system with two distinct, effective hydrodynamic radii. Thus, we find that a simple colloid picture well describes short-time diffusion in binary mixtures as a function of the mixing ratio and the total volume fraction. Notably, the self-diffusion of the smaller protein BSA in the mixture is faster than the diffusion in a pure BSA solution, whereas the self-diffusion of Ig in the mixture is slower than in the pure Ig solution.

Elasticity in crystals with a high density of local defects: Insights from ultra-soft colloids.

In complex crystals close to melting or at finite temperatures, different types of defects are ubiquitous and their role becomes relevant in the mechanical response of these solids. Conventional elasticity theory fails to provide a microscopic basis to include and account for the motion of point defects in an otherwise ordered crystalline structure. We study the elastic properties of a point-defect rich crystal within a first principles theoretical framework derived from the microscopic equations of motion. This framework allows us to make specific predictions pertaining to the mechanical properties that we can validate through deformation experiments performed in molecular dynamics simulations.

Reconsidering power functional theory.

The original derivation of power functional theory [M. Schmidt and J. M. Brader, J. Chem. Phys. 138, 214101 (2013)] is reworked in some detail with a view to clarifying and simplifying the logic and making explicit the various functional dependencies. We note various issues with the original development and suggest a modification that allows us to avoid them. In the process, we also suggest an alternative interpretation of our results, which bears surprising similarities to classical density functional theory.

Protein Short-Time Diffusion in a Naturally Crowded Environment.

The interior of living cells is a dense and polydisperse suspension of macromolecules. Such a complex system challenges an understanding in terms of colloidal suspensions. As a fundamental test we employ neutron spectroscopy to measure the diffusion of tracer proteins (immunoglobulins) in a cell-like environment (cell lysate) with explicit control over crowding conditions. In combination with Stokesian dynamics simulation, we address protein diffusion on nanosecond time scales where hydrodynamic interactions dominate over negligible protein collisions. We successfully link the experimental results on these complex, flexible molecules with coarse-grained simulations providing a consistent understanding by colloid theories. Both experiments and simulations show that tracers in polydisperse solutions close to the effective particle radius Reff = ⟨ Ri3⟩1/3 diffuse approximately as if the suspension was monodisperse. The simulations further show that macromolecules of sizes R > Reff ( R < Reff) are slowed more (less) effectively even at nanosecond time scales, which is highly relevant for a quantitative understanding of cellular processes.

Bulk structural information from density functionals for patchy particles.

We investigate bulk structural properties of tetravalent associating particles within the framework of classical density functional theory, building upon Wertheim's thermodynamic perturbation theory. To this end, we calculate density profiles within an effective test-particle geometry and compare to radial distribution functions obtained from computer simulations. We demonstrate that a modified version of the functional proposed by Yu and Wu [J. Chem. Phys. 116, 7094 (2002)] based on fundamental measure theory for hard spheres produces accurate results, although the functional does not satisfy the exactly known low-density limit. In addition, at low temperatures where particles start to form an amorphous tetrahedral network, quantitative differences between simulations and theory emerge due to the absence of geometrical information regarding the patch arrangement in the latter. Indeed, here we find that the theory fits better to simulations of the floating-bond model [E. Zaccarelli et al., J. Chem. Phys. 127, 174501 (2007)], which exhibits a weaker tetrahedral order due to more flexible bonds between particles. We also demonstrate that another common density functional approach by Segura et al. [Mol. Phys. 90, 759 (1997)] fails to capture fundamental structural properties.

Phase diagrams and crystal-fluid surface tensions in additive and nonadditive two-dimensional binary hard-disk mixtures.

Using density functionals from fundamental measure theory, phase diagrams and crystal-fluid surface tensions in additive and nonadditive (Asakura-Oosawa model) two-dimensional binary hard-disk mixtures are determined for the whole range of size ratios q=smalldiameter/largediameter, assuming random disorder (lattice points or interstitial occupied by large or small disks at random) in the crystal phase. The fluid-crystal transitions are first order due to the assumption of a periodic unit cell in the density-functional calculations. Qualitatively, the shape of the phase diagrams is similar to the case of three-dimensional hard-sphere mixtures. For the nonadditive case, a broadening of the fluid-crystal coexistence region is found for small q, whereas for large q a vapor-fluid transition intervenes. In the additive case, we find a sequence of spindle-type, azeotropic, and eutectic phase diagrams upon lowering q from 1 to 0.6. The transition from azeotropic to eutectic is different from the three-dimensional case. Surface tensions in general become smaller (up to a factor 2) upon addition of a second species and they are rather small. The minimization of the functionals proceeds without restrictions and optimized graphics card routines are used.

A dynamic DFT approach to generalized diffusion equations in a system with long-ranged and hydrodynamic interactions.

We build on an existing approximation scheme to the Smoluchowski equation in order to derive a dynamic density functional theory (DDFT) including two-body hydrodynamic interactions. A generalized diffusion equation and a wavenumber-dependent diffusion coefficient D(k) are derived by linearization in the density fluctuations. The result is applied to a colloidal monolayer at a fluid interface, having bulk-like hydrodynamic interactions and/or interacting via long-ranged capillary forces. In these cases, D(k) shows characteristic singularities as [Formula: see text]. The consequences of these singularities are studied by means of analytical perturbation theory, numerical solution of DDFT and simulations for an explicit example: the capillary collapse of a finite, disk-like distribution of particles. There is in general a good agreement between DDFT and simulations if the initial density distributions for the theoretical prediction correspond to the actual initial configurations of simulations, rather than to an average over them. Otherwise, discrepancies arise that are discussed in detail.

A fundamental measure density functional for fluid and crystal phases of the Asakura-Oosawa model.

We investigate a density functional for the Asakura-Oosawa model of colloid-polymer mixtures, describing both fluid and crystal phases. It is derived by linearizing the two-component fundamental-measure hard sphere tensor functional in the second (polymer) component. We discuss the formulation of an effective density functional for colloids only. For small polymer-colloid size ratios the effective, polymer-induced potential between colloids is short-range attractive and of two-body form but we show that the effective density functional is not equivalent to standard mean-field approaches where attractions are taken into account by terms second order in the colloid density. We calculate numerically free energies and phase diagrams in good agreement with available simulations, furthermore we discuss the colloid and polymer distributions in the crystal and determine equilibrium vacancy concentrations. Numerical results reveal a fairly strong sensitivity to the specific type of underlying fundamental measure hard sphere functional which could aid further development of fundamental measure theory.

Fundamental measure theory for the inhomogeneous hard-sphere system based on Santos' consistent free energy.

Based on Santos' general solution for the scaled-particle differential equation [Phys. Rev. E 86, 040102(R) (2012)], we construct a free-energy functional for the hard-sphere system. The functional is obtained by a suitable generalization and extension of the set of scaled-particle variables using the weighted densities from Rosenfeld's fundamental measure theory for the hard-sphere mixture [Phys. Rev. Lett. 63, 980 (1989)]. While our general result applies to the hard-sphere mixture, we specify remaining degrees of freedom by requiring the functional to comply with known properties of the pure hard-sphere system. Both for mixtures and pure systems, the functional can be systematically extended following the lines of our derivation. We test the resulting functionals regarding their behavior upon dimensional reduction of the fluid as well as their ability to accurately describe the hard-sphere crystal and the liquid-solid transition.

3D hydrodynamic interactions lead to divergences in 2D diffusion.

We investigate the influence of 3D hydrodynamic interactions on confined colloidal suspensions, where only the colloids are restricted to one or two dimensions. In the absence of static interactions among the colloids, i.e., an ideal gas of colloidal particles with a finite hydrodynamic radius, we find a divergent collective diffusion coefficient. The origin of the divergence is traced back to the dimensional mismatch of 3D hydrodynamic interactions and the colloidal particles moving only in 1D or 2D. Our results from theory are confirmed by Stokesian dynamics simulations and supported by light scattering observational data for particles at a fluid interface.

Solid-liquid surface tensions of critical nuclei and nucleation barriers from a phase-field-crystal study of a model binary alloy using finite system sizes.

Phase-field-crystal (PFC) modeling has emerged as a computationally efficient tool to address crystal growth phenomena on atomistic length and diffusive time scales. We use a two-dimensional phase-field-crystal model for a binary system based on Elder et al. [Phys. Rev. B 75, 064107 (2007)] to study critical nuclei and their liquid-solid phase boundaries, in particular the nucleus size dependence of the liquid-solid interface tension as well as of the nucleation barrier. Critical nuclei are stabilized in finite systems of various sizes, however, the extracted interface tension as function of the nucleus radius r is independent of system size. We suggest a phenomenological expression to describe the dependence of the extracted interface tension on the nucleus radius r for the liquid-solid system. Moreover, the numerical PFC results show that this dependency can not be fully described by the nonclassical Tolman formula.

Multiple reentrant glass transitions in confined hard-sphere glasses.

Glass-forming liquids exhibit a rich phenomenology upon confinement. This is often related to the effects arising from wall-fluid interactions. Here we focus on the interesting limit where the separation of the confining walls becomes of the order of a few particle diameters. For a moderately polydisperse, densely packed hard-sphere fluid confined between two smooth hard walls, we show via event-driven molecular dynamics simulations the emergence of a multiple reentrant glass transition scenario upon a variation of the wall separation. Using thermodynamic relations, this reentrant phenomenon is shown to persist also under constant chemical potential. This allows straightforward experimental investigation and opens the way to a variety of applications in micro- and nanotechnology, where channel dimensions are comparable to the size of the contained particles. The results are in line with theoretical predictions obtained by a combination of density functional theory and the mode-coupling theory of the glass transition.

Strong effect of weak charging in suspensions of anisotropic colloids.

Suspensions of hard colloidal particles frequently serve as model systems in studies on fundamental aspects of phase transitions. But often colloidal particles that are considered as "hard" are in fact weakly charged. If the colloids are spherical, weak charging has only a weak effect on the structural properties of the suspension, which can be easily corrected for. However, this does not hold for anisotropic particles. We introduce a model for the interaction potential between charged ellipsoids of revolution (spheroids) based on the Derjaguin approximation of Debye-Hückel theory and present a computer simulation study on aspects of the system's structural properties and phase behaviour. In line with previous experimental observations, we find that even a weak surface charge has a strong impact on the correlation functions. A likewise strong impact is seen on the phase behaviour, in particular, we find stable cubatic order in suspensions of oblate ellipsoids.

Protein cluster formation in aqueous solution in the presence of multivalent metal ions--a light scattering study.

The formation of protein clusters as precursors for crystallization and phase separation is of fundamental and practical interest in protein science. Using multivalent ions, the strengths of both long-range Coulomb repulsion and short-range attraction can be tuned in protein solutions, representing a well-controlled model system to study static and dynamic properties of clustering during the transition from a charge-stabilized to an aggregate regime. Here, we study compressibility, diffusion, and size of solutes by means of static (SLS) and dynamic light scattering (DLS) in solutions of bovine serum albumin (BSA) and YCl3. For this and comparable systems, an increasing screening and ultimately inversion of the protein surface charge induce a rich phase behavior including reentrant condensation, liquid-liquid phase separation and crystallization, which puts the cluster formation in the context of precursor formation and nucleation of liquid and crystalline phases. We find that, approaching the turbid aggregate regime with increasing salt concentration cs, the diffusion coefficients decrease and the scattered intensity increases by orders of magnitude, evidencing increasing correlation lengths likely associated with clustering. The combination of static and dynamic observations suggests the formation of BSA clusters with a size on the order of 100 nm. The global thermodynamic state seems to be stable over at least several hours. Surprisingly, results on collective diffusion and inverse compressibility from different protein concentrations can be rescaled into master curves as a function of cs/c*, where c* is the critical salt concentration of the transition to the turbid aggregate regime.

Connectivity percolation in suspensions of hard platelets.

We present a study on connectivity percolation in suspensions of hard platelets by means of Monte Carlo simulation. We interpret our results using a contact-volume argument based on an effective single-particle cell model. It is commonly assumed that the percolation threshold of anisotropic objects scales as their inverse aspect ratio. While this rule has been shown to hold for rodlike particles, we find that for hard platelike particles the percolation threshold is nonmonotonic in the aspect ratio. It exhibits a shallow minimum at intermediate aspect ratios and then saturates to a constant value. This effect is caused by the isotropic-nematic transition preempting the percolation transition. Hence the common strategy to use highly anisotropic, conductive particles as fillers in composite materials in order to produce conduction at low filler concentration is expected to fail for platelike fillers such as graphene and graphite nanoplatelets.

Communication: Fundamental measure theory for hard disks: fluid and solid.

Two-dimensional hard-particle systems are rather easy to simulate but surprisingly difficult to treat by theory. Despite their importance from both theoretical and experimental points of view, theoretical approaches are usually qualitative or at best semi-quantitative. Here, we present a density functional theory based on the ideas of fundamental measure theory for two-dimensional hard-disk mixtures, which allows for the first time an accurate description of the structure of the dense fluid and the equation of state for the solid phase within the framework of density functional theory. The properties of the solid phase are obtained by freely minimizing the functional.