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Register a new userDerivation of the Derjaguin approximation for the case of inhomogeneous solvents
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The Derjaguin approximation (DA) relates the force between curved surfaces to the interaction free energy between parallel planes. It is typically derived by considering the direct interaction between the bodies involved, thus treating the effect of an intervening solvent implicitly by a rescaling of the corresponding Hamaker constant. Here, we provide a generalization of DA to the case of a molecular medium between the bodies, as is the case in most applications. The derivation is based on an explicit statistical-mechanical treatment of the contribution to the interaction force from a molecular solvent using a general expression for intermolecular and molecule-surface interactions. Starting from an exact expression for the force, DA is arrived at by a series of well-defined approximations. Our results show that DA remains valid in a molecular solvent as long as (i) the surface-molecule interactions are of much shorter range than the radius R of the sphere and (ii) the density correlation length in the solvent is smaller than R. We then extend our analysis to the case where a phase transition occurs between the surfaces, which cannot easily be covered using a statistical-mechanical formalism due to the discontinuous change in the density of the medium. Instead using a continuum thermodynamic description, we show that this phase transformation induces an attractive force between the bodies, and that the force between curved surfaces can be related to the free energy in the corresponding planar case, in accordance with DA.
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The self-consistent field theory (SCFT) is a powerful framework for the study of the phase behavior and structural properties of many-body systems. In particular, polymeric SCFT has been successfully applied to inhomogeneous polymeric systems such as polymer blends and block copolymer melts. The polymeric SCFT is commonly derived using field-theoretical techniques. Here we provide an alternative derivation of the SCFT equations and SCFT free energy functional using a variational principle. Numerical methods of solving the SCFT equations and applications of the SCFT are also briefly introduced.
Responsive particles, such as biomacromolecules or hydrogels, display a broad and polymodal distribution of conformations and have thus the ability to change their properties (e.g, size, shape, charge density, etc.) substantially in response to external fields or to their local environment (e.g., mediated by cosolutes or pH). Here, we discuss the basic statistical mechanics for a model of responsive colloids (RCs) by introducing an additional property degree of freedom as a collective variable in a formal coarse-graining procedure. The latter leads to an additional one-body term in the coarse-grained (CG) free energy, defining a single-particle property distribution for an individual polydisperse RC. We argue that in the equilibrium thermodynamic limit such a CG system of RCs behaves like a conventional polydisperse system of non-responsive particles. We then illustrate the action of external fields, which impose local (position-dependent) property distributions leading to non-trivial effects on the spatial one-body property and density profiles, even for an ideal (non-interacting) gas of RCs. We finally apply density functional theory in the local density approximation (LDA-DFT) to discuss the effects of particle interactions for specific examples of i) a suspension of RCs in an external field linear in both position and property, ii) a suspension of RCs with highly localized properties (sizes) confined between two walls, and iii) a two-component suspension where an inhomogeneously distributed (non-responsive) cosolute component, as found, e.g., in the studies of osmolyte- or salt-induced collapse/swelling transitions of thermosensitive polymers, modifies the local properties and density of the RC liquid.
Three one-body profiles that correspond to local fluctuations in energy, in entropy, and in particle number are used to describe the equilibrium properties of inhomogeneous classical many-body systems. Local fluctuations are obtained from thermodynamic differentiation of the density profile or equivalently from average microscopic covariances. The fluctuation profiles follow from functional generators and they satisfy Ornstein-Zernike relations. Computer simulations reveal markedly different fluctuations in confined fluids with Lennard-Jones, hard sphere, and Gaussian core interactions.
We evaluate in this work the hydrodynamic transport coefficients of a granular binary mixture in $d$ dimensions. In order to eliminate the observed disagreement (for strong dissipation) between computer simulations and previously calculated theoretical transport coefficients for a monocomponent gas, we obtain explicit expressions of the seven Navier-Stokes transport coefficients with the use of a new Sonine approach in the Chapman-Enskog theory. Our new approach consists in replacing, where appropriate in the Chapman-Enskog procedure, the Maxwell-Boltzmann distribution weight function (used in the standard first Sonine approximation) by the homogeneous cooling state distribution for each species. The rationale for doing this lies in the fact that, as it is well known, the non-Maxwellian contributions to the distribution function of the granular mixture become more important in the range of strong dissipation we are interested in. The form of the transport coefficients is quite common in both standard and modified Sonine approximations, the distinction appearing in the explicit form of the different collision frequencies associated with the transport coefficients. Additionally, we numerically solve by means of the direct simulation Monte Carlo method the inelastic Boltzmann equation to get the diffusion and the shear viscosity coefficients for two and three dimensions. As in the case of a monocomponent gas, the modified Sonine approximation improves the estimates of the standard one, showing again the reliability of this method at strong values of dissipation.
The effective pair potentials between different kinds of dendrimers in solution can be well approximated by appropriate Gaussian functions. We find that in binary dendrimer mixtures the range and strength of the effective interactions depend strongly upon the specific dendrimer architecture. We consider two different types of dendrimer mixtures, employing the Gaussian effective pair potentials, to determine the bulk fluid structure and phase behavior. Using a simple mean field density functional theory (DFT) we find good agreement between theory and simulation results for the bulk fluid structure. Depending on the mixture, we find bulk fluid-fluid phase separation (macro-phase separation) or micro-phase separation, i.e., a transition to a state characterized by undamped periodic concentration fluctuations. We also determine the inhomogeneous fluid structure for confinement in spherical cavities. Again, we find good agreement between the DFT and simulation results. For the dendrimer mixture exhibiting micro-phase separation, we observe rather striking pattern formation under confinement.
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