Structural and Dynamical Coupling in Solvent-Free Polymer Brushes Elucidated by Molecular Dynamics Simulations.

We investigate the chain configuration and segmental dynamics in interacting solvent-free polymer brushes using molecular dynamics simulations. The brush systems are designed to mimic the interstitial space between a pair of neighboring polymer-grafted nanoparticles in solvent-free nanoparticle-organic hybrid materials. Each brush consists of uniformly grafted chains formed by a given number of monomer beads. In monodisperse systems, two opposing brushes have the same chain length and grafting density. In mixed conditions, we consider binary systems with two surfaces being separately grafted with polymers of distinct chain lengths at different grafting densities as well as bidisperse systems with polymers of two different lengths being tethered to the surfaces at a fixed grafting density. We demonstrate that the brush configuration and interpenetration are both governed by the need that monomer beads have to uniformly fill the space. For systems with longer chain lengths and/or higher grafting densities, the larger interwall separation yields more stretched brush conformations and reduced extents of interbrush mixing. As a result, the polymer configurational entropy is generally decreased and the segment-to-segment relaxation dynamics is slowed down accordingly. The grafting of chains at a high density not only makes the relaxation dynamics deviate from the standard Rouse prediction but also leads to distinct relaxation times for the free and tethered segments. The more slowly relaxing tethered segments play a more important role in determining the overall end-to-end fluctuations. Moreover, the two distinct relaxation processes are consistent with the two-stage decay in the Rouse mode fluctuation autocorrelation function. In the presence of brush bidispersity, the collaboration between polymers of different lengths is evidently observed in the brush profiles. The variations of the chain configuration for the two polymers are complementary, and the associated relaxation dynamics of the two species are significantly coupled.

Entropic Effects in Solvent-Free Bidisperse Polymer Brushes Investigated Using Density Functional Theories.

Solvent-free polymer-functionalized nanoparticles form a special type of colloid composed of inorganic cores self-suspended by their grafted coronas. In the absence of intervening solvent molecules, the fluidity of the system is provided by these tethered polymers as they fill the space. Here, we study the structure and interaction of neighboring polymer-grafted surfaces in the solvent-free condition using mean-field density functional theories. For opposing flat surfaces, the brush configuration and the associated energy landscape are semianalytically investigated given the incompressibility of the tethered entropic chains. The effect of brush polydispersity (including variations in both chain length and surface grafting density) is considered by two bidisperse models corresponding to different physical scenarios: one for opposing brushes uniformly mixed with two species at a fixed grafting density, and the other for opposing brushes with distinct chain lengths and grafting densities. The space-filling capabilities of the neighboring coronas differ not only by their ratio of radii of gyration for the composing polymers but also by their ratio of grafting densities. We show that the system energy depicts a steric repulsion as the brushes are compressed, which is typical for hairy particles in a solvent. However, as the interwall separation increases, the cooperative stretching of the chains leads to an entropic attraction between them, a unique characteristic of solventless systems. The corresponding brush profiles change from a bell-like shape to a more step-function-like feature as the interwall spacing increases significantly. The interwall separation associated with the overall free energy minimum therefore characterizes the favorable interparticle spacing for solvent-free polymer-functionalized particles. The limiting accessible parameter space of polymer sizes and grafting densities subjected to the space-filling constraint is comprehensively explored for representative interparticle spacing characterizing the compressed, relaxed, and stretched regimes for a given polymer species, respectively. Such information would be useful for guiding the design of experimental solvent-free polymer-functionalized nanoparticles.

Adhesion of a polymer-grafted nanoparticle to cells explored using generalized Langevin dynamics.

We model a polymer-grafted stealth nanoparticle (SNP) as a composite system consisting of a spherical core coated with a porous polymeric brush with end-ligands. Adjacent to target cells, the near-wall hydrodynamics, thermal fluctuations, and thermodynamic adhesive interactions simultaneously impact the transient motion of the SNP. Employing both the Langevin framework for the effective hard sphere dynamics and the coupled generalized Langevin framework for the nanoparticle-polymer dynamics, we comprehensively investigate the velocity and position temporal relaxations of the SNP in the absence and presence of end-to-end distance fluctuations for the tethered polymer. We demonstrate that polymer structural relaxations substantially impact the SNP adhesive dynamics, especially when the grafted polymer is more flexible. Moreover, a long-time tail with t-3/2 scaling due to polymer chain-length fluctuations is observed in the velocity autocorrelation for a bound SNP. Finally, the thermodynamic effects of membrane morphology on SNP adhesion are explored by modifying the membrane-mediated binding potential of mean force.

Microstructure of Flow-Driven Suspension of Hardspheres in Cylindrical Confinement: A Dynamical Density Functional Theory and Monte Carlo Study.

We have studied the microstructure of a flow-driven hardsphere suspension inside a cylinder using dynamical density functional theory and Monte Carlo simulations. In order to be representative of various physical conditions that may prevail in experiments, we investigate the problem using both the grand canonical (µVT) ensemble and the canonical (NVT) ensemble. In both ensembles, the hydrodynamic effect on the suspension mediated by the presence of the confining wall is implemented in a mean-field fashion by incorporating the thermodynamic work done by the inertial lift force on the particle given the average flow field. The predicted particle distribution in the µVT ensemble displays strong structural ordering at increasing flow rates due to the correspondingly higher particle concentrations inside the cylinder. In the NVT ensemble, for dilute suspensions we observe a peak in the distribution of density at a location similar to that of the Segré-Silberberg annulus, while for dense suspensions the competing effects of the inertial lift and the hardsphere interaction lead to the formation of several annuli.

Computational Models for Nanoscale Fluid Dynamics and Transport Inspired by Nonequilibrium Thermodynamics.

Traditionally, the numerical computation of particle motion in a fluid is resolved through computational fluid dynamics (CFD). However, resolving the motion of nanoparticles poses additional challenges due to the coupling between the Brownian and hydrodynamic forces. Here, we focus on the Brownian motion of a nanoparticle coupled to adhesive interactions and confining-wall-mediated hydrodynamic interactions. We discuss several techniques that are founded on the basis of combining CFD methods with the theory of nonequilibrium statistical mechanics in order to simultaneously conserve thermal equipartition and to show correct hydrodynamic correlations. These include the fluctuating hydrodynamics (FHD) method, the generalized Langevin method, the hybrid method, and the deterministic method. Through the examples discussed, we also show a top-down multiscale progression of temporal dynamics from the colloidal scales to the molecular scales, and the associated fluctuations, hydrodynamic correlations. While the motivation and the examples discussed here pertain to nanoscale fluid dynamics and mass transport, the methodologies presented are rather general and can be easily adopted to applications in convective heat transfer.

Dynamics and yielding of binary self-suspended nanoparticle fluids.

Yielding and flow transitions in bi-disperse suspensions of particles are studied using a model system comprised of self-suspended spherical nanoparticles. An important feature of the materials is that the nanoparticles are uniformly dispersed in the absence of a solvent. Addition of larger particles to a suspension of smaller ones is found to soften the suspensions, and in the limit of large size disparities, completely fluidizes the material. We show that these behaviors coincide with a speeding-up of de-correlation dynamics of all particles in the suspensions and are accompanied by a reduction in the energy dissipated at the yielding transition. We discuss our findings in terms of ligand-mediated jamming and un-jamming of hairy particle suspensions.

Structure factor of blends of solvent-free nanoparticle-organic hybrid materials: density-functional theory and small angle X-ray scattering.

We investigate the static structure factor S(q) of solvent-free nanoparticle-organic hybrid materials consisting of silica nanocores and space-filling polyethylene glycol coronas using a density-functional theory and small angle X-ray scattering measurements. The theory considers a bidisperse suspension of hard spheres with different radii and tethered bead-spring oligomers with different grafting densities to approximate the polydispersity effects in experiments. The experimental systems studied include pure samples with different silica core volume fractions and the associated mean corona grafting densities, and blends with different mixing ratios of the pure samples, in order to introduce varying polydispersity of corona grafting density. Our scattering experiments and theory show that, compared to the hard-sphere suspension with the same core volume fraction, S(q) for pure samples exhibit both substantially smaller values at small q and stronger particle correlations corresponding to a larger effective hard core at large q, indicating that the tethered incompressible oligomers enforce a more uniform particle distribution, and the densely grafted brush gives rise to an additional exclusionary effect between the nanoparticles. According to the theory, polydispersity in the oligomer grafting density controls the deviation of S(q) from the monodisperse system at smaller q, and the interplay of the enhanced effective core size and the entropic attraction among the particles is responsible for complex variations in the particle correlations at larger q. The successful comparison between the predictions and the measurements for the blends further suggests that S(q) can be used to assess the uniformity of grafting density in polymer-grafted nanoparticle materials.

Predicting the disorder-order transition of solvent-free nanoparticle-organic hybrid materials.

The transition from a disordered to a face-centered-cubic phase in solvent-free oligomer-tethered nanoparticles is predicted using a density-functional theory for model hard spheres with tethered bead-spring oligomers. The transition occurs without a difference of volume fraction for the two phases, and the phase boundary is influenced by the loss of oligomer configurational entropy relative to an ideal random system in one phase compared with the other. When the particles are localized in the ordered phase, the cooperation of the oligomers in filling the space is hindered. Therefore, shorter oligomers feel a stronger entropic penalty in the ordered solid and favor the disordered phase. Strikingly, we found that the solvent-free system has a later transition than hard spheres for all investigated ratios of oligomer radius of gyration to particle radius.

Connecting Structural Characteristics and Material Properties in Phase-Separating Polymer Solutions: Phase-Field Modeling and Physics-Informed Neural Networks.

The formed morphology during phase separation is crucial for determining the properties of the resulting product, e.g., a functional membrane. However, an accurate morphology prediction is challenging due to the inherent complexity of molecular interactions. In this study, the phase separation of a two-dimensional model polymer solution is investigated. The spinodal decomposition during the formation of polymer-rich domains is described by the Cahn-Hilliard equation incorporating the Flory-Huggins free energy description between the polymer and solvent. We circumvent the heavy burden of precise morphology prediction through two aspects. First, we systematically analyze the degree of impact of the parameters (initial polymer volume fraction, polymer mobility, degree of polymerization, surface tension parameter, and Flory-Huggins interaction parameter) in a phase-separating system on morphological evolution characterized by geometrical fingerprints to determine the most influential factor. The sensitivity analysis provides an estimate for the error tolerance of each parameter in determining the transition time, the spinodal decomposition length, and the domain growth rate. Secondly, we devise a set of physics-informed neural networks (PINN) comprising two coupled feedforward neural networks to represent the phase-field equations and inversely discover the value of the embedded parameter for a given morphological evolution. Among the five parameters considered, the polymer-solvent affinity is key in determining the phase transition time and the growth law of the polymer-rich domains. We demonstrate that the unknown parameter can be accurately determined by renormalizing the PINN-predicted parameter by the change of characteristic domain size in time. Our results suggest that certain degrees of error are tolerable and do not significantly affect the morphology properties during the domain growth. Moreover, reliable inverse prediction of the unknown parameter can be pursued by merely two separate snapshots during morphological evolution. The latter largely reduces the computational load in the standard data-driven predictive methods, and the approach may prove beneficial to the inverse design for specific needs.

Deep learning and inverse discovery of polymer self-consistent field theory inspired by physics-informed neural networks.

We devise a deep learning solver inspired by physics-informed neural networks (PINNs) to tackle the polymer self-consistent field theory (SCFT) equations for one-dimensional AB-diblock copolymers. The PINNs framework comprises two parallel feedforward neural networks that separately represent the segmental partition functions and self-consistent chemical potential fields. The two networks are coupled through a loss function incorporating the governing equation, initial and boundary conditions, and the incompressibility constraint. To avoid the metastable homogeneous solution, the network parameters are initialized based on known self-consistent fields obtained from the numerical pseudospectral method. For copolymers of length N at a given volume fraction of A block (f) and the reduced Flory-Huggins interaction parameter (χN), the minimization of the loss function leads to the converged network parameters that successfully capture the stable lamellar phase. The periodicity of the lamellar structure is correctly reproduced for the explored sets of [f,χN], irrespective of the presumed computational domain size for initialization. Moreover, the proposed PINNs are applicable to the inverse discovery of the interaction parameter and the embedded chemical potential fields for an observed structure. This capability of solving the inverse SCFT problem demonstrates the potential of using PINNs to accelerate the exploration of new polymeric materials.

Enthalpic Interactions and Solution Behaviors of Solvent-Free Polymer Brushes.

We performed molecular dynamics simulations to characterize the role of enthalpic interaction in impacting the static and dynamic properties of solvent-free polymer brushes. The intrinsic enthalpic interaction in the simulation was introduced using different attraction strengths between distinct species. Two model systems were considered: one consisting of binary brushes of two different polymer types and the other containing a mixture of homopolymer brushes and free molecules. In the first system, we observed that, when two originally incompatible polymers were grafted to opposing surfaces, the miscibility between them was significantly enhanced. A less favorable intrinsic enthalpic interaction in the brushes resulted in a more stretched chain configuration, a lower degree of inter-brush penetration, and faster segmental relaxation. In the second system, we characterized the solvent capacity of the homopolymer brushes from variations in the energy components of the system as a function of the number of free molecules. We determined that molecular absorption was driven by the release of the entropic frustration for the grafted chains in conjunction with the chemical affinity between the solutes and polymers. The solute distribution function within the inter-wall space showed that solute-polymer mixing in the middle of the gap occurred preferentially when the enthalpic interaction was more favorable. When this was not the case, absorption was predominantly localized near the grafting surface. From the mean square displacement of the solute, we found that the brush profiles restrained the molecular diffusion perpendicular to the grafting wall; the weaker the attraction from the brush, the higher the solute mobility.

Structure of solvent-free grafted nanoparticles: molecular dynamics and density-functional theory.

The structure of solvent-free oligomer-grafted nanoparticles has been investigated using molecular dynamics simulations and density-functional theory. At low temperatures and moderate to high oligomer lengths, the qualitative features of the core particle pair probability, structure factor, and the oligomer brush configuration obtained from the simulations can be explained by a density-functional theory that incorporates the configurational entropy of the space-filling oligomers. In particular, the structure factor at small wave numbers attains a value much smaller than the corresponding hard-sphere suspension, the first peak of the pair distribution function is enhanced due to entropic attractions among the particles, and the oligomer brush expands with decreasing particle volume fraction to fill the interstitial space. At higher temperatures, the simulations reveal effects that differ from the theory and are likely caused by steric repulsions of the expanded corona chains.

Structure of solvent-free nanoparticle-organic hybrid materials.

We derive the radial distribution function and the static structure factor for the particles in model nanoparticle-organic hybrid materials composed of nanoparticles and attached oligomeric chains in the absence of an intervening solvent. The assumption that the oligomers form an incompressible fluid of bead-chains attached to the particles that is at equilibrium for a given particle configuration allows us to apply a density functional theory for determining the equilibrium configuration of oligomers as well as the distribution function of the particles. A quasi-analytic solution is facilitated by a regular perturbation analysis valid when the oligomer radius of gyration R(g) is much greater than the particle radius a. The results show that the constraint that each particle carries its own share of the fluid attached to itself yields a static structure factor that approaches zero as the wavenumber approaches zero. This result indicates that each particle excludes exactly one other particle from its neighborhood.

Binding of a Brownian nanoparticle to a thermally fluctuating membrane surface.

We investigate the Brownian dynamics of a nanoparticle bound to a thermally undulating elastic membrane. The ligand-functionalized nanoparticle is assumed to interact monovalently with the receptor expressed on the membrane. In order to resolve the nanoparticle transient motion subject to the instantaneous membrane configuration in a consistent manner, we employ a set of coupled Langevin equations that simultaneously incorporate the hydrodynamic effects, ligand-receptor binding interaction, intramembrane elastic forces, and thermal fluctuations. We show that the presence of a deformable, elastic fluid membrane not only affects the dynamics of a bound nanoparticle but also alters the effective binding potential felt by the nanoparticle. In contrast to a nanoparticle bound to a flat surface, the oscillatory characteristics of the nanoparticle velocity autocorrelation function are suppressed and transition to an anticorrelated long-time tail. Moreover, the nanoparticle position fluctuation becomes more coherent with that of the membrane binding site, and the width of the distribution of the nanoparticle distance from the membrane decreases with increasing membrane bending rigidity. By introducing a locally harmonic, bistable potential as an effective potential for the ligand-receptor pair, the rate of nanoparticle transitioning between two bound states is facilitated by membrane undulations as a result of stronger positional variations associated with the nanoparticle.

Rheology of colloidal suspensions in confined flow: Treatment of hydrodynamic interactions in particle-based simulations inspired by dynamical density functional theory.

We investigate the microstructure and rheology of a hard-sphere suspension in a Newtonian fluid confined in a cylindrical channel and undergoing pressure-driven flow using Monte Carlo simulations. We develop a hydrodynamic framework inspired by dynamical density functional theory approaches in which the contributions due to various flow-induced hydrodynamic interactions (HI) are included in the form of thermodynamic work done by these HI-derived forces in displacing the hard spheres. Using this framework, we can self-consistently determine the effect of the local microstructure on the average flow field, and vice versa, and coevolve the inhomogeneous density distribution and the flattening velocity profile with increase in the density of suspended particles. Specifically, we explore the effect on the local microstructure due to the inclusion of forces arising from confinement-induced inertial effects, forces due to solvent-mediated interparticle interactions, and the dependence of the diffusivity on the local density. We examine the dependence of the apparent viscosity of the suspension on the volume fraction of hard spheres in the cylinder, the flow rate, and the diameter of the cylinder and investigate their effects on the local microstructure.

Approximate analytical expressions for the electrical potential between two planar, cylindrical, and spherical surfaces.

Approximate analytical expressions for the electrical potential of planar, cylindrical, and spherical surfaces are derived for the case in which the dispersion medium contains counterions only. On the basis of the results for single surfaces, those for two identical surfaces can be derived. The curvature effect of a surface on the electrical potential distribution can be neglected when the order of its radius exceeds approximately 100 times the thickness of the corresponding double layer. If this effect needs to be considered, it can be taken into account by multiplying a correction function by the electrical potential of a planar surface. The electrical potential at the center between two derived surfaces is readily applicable to the evaluation of the electrostatic force per unit area between two surfaces, or the osmotic pressure. For the same set of parameters, the magnitudes of the osmotic pressure for various types of surfaces rank as follows: planar surface > cylindrical surfaces > spherical surfaces.

Critical coagulation concentration of a salt-free colloidal dispersion.

Both exact and approximate analytical solutions of the Poisson-Boltzmann equation for two planar, parallel surfaces are derived for the case when a dispersion medium contains counterions only, and the results obtained are used to evaluate the critical coagulation concentration of a spherical dispersion. A correction factor, which is a function of the valence of counterions, the surface potential of a particle, and the potential on the midplane between two particles at the onset of coagulation, is derived to modify the classic Schulze-Hardy rule for the dependence of the critical coagulation concentration on the valence of counterions. The correction factor is found to increase with the increase in the valence of counterions and/or with the increase in the surface potential. However, it approaches a constant value of 0.8390 if the surface potential is sufficiently high.

Effect of wall-mediated hydrodynamic fluctuations on the kinetics of a Brownian nanoparticle.

The reactive flux formalism (Chandler 1978 J. Chem. Phys.68, 2959-2970. (doi:10.1063/1.436049)) and the subsequent development of methods such as transition path sampling have laid the foundation for explicitly quantifying the rate process in terms of microscopic simulations. However, explicit methods to account for how the hydrodynamic correlations impact the transient reaction rate are missing in the colloidal literature. We show that the composite generalized Langevin equation (Yu et al. 2015 Phys. Rev. E91, 052303. (doi:10.1103/PhysRevE.91.052303)) makes a significant step towards solving the coupled processes of molecular reactions and hydrodynamic relaxation by examining how the wall-mediated hydrodynamic memory impacts the two-stage temporal relaxation of the reaction rate for a nanoparticle transition between two bound states in the bulk, near-wall and lubrication regimes.

Nanoparticle stochastic motion in the inertial regime and hydrodynamic interactions close to a cylindrical wall.

We have carried out direct numerical simulations (DNS) of the fluctuating Navier-Stokes equation together with the particle equations governing the motion of a nanosized particle or nanoparticle (NP) in a cylindrical tube. The effects of the confining boundary, its curvature, particle size, and particle density variations have all been investigated. To reveal how the nature of the temporal correlations (hydrodynamic memory) in the inertial regime is altered by the full hydrodynamic interaction due to the confining boundaries, we have employed the Arbitrary Lagrangian-Eulerian (ALE) method to determine the dynamical relaxation of a spherical NP located at various positions in the medium over a wide span of time scales compared to the fluid viscous relaxation time τv = a2/v, where a is the spherical particle radius and v is the kinematic viscosity. The results show that, as compared to the behavior of a particle in regions away from the confining boundary, the velocity autocorrelation function (VACF) for a particle in the lubrication layer initially decays exponentially with a Stokes drag enhanced by a factor that is proportional to the ratio of the particle radius to the gap thickness between the particle and the wall. Independent of the particle location, beyond time scales greater than a2/v, the decay is always algebraic followed by a second exponential decay (attributed to the wall curvature) that is associated with a second time scale D2/v, where D is the vessel diameter.