Entropic Depletion Forces Research Articles

Microbial biofilms are shaped by the constant dialogue between biological and physical forces in the extracellular matrix.

Microbial biofilms are shaped by the constant dialogue between biological and physical forces in the extracellular matrix.

Depletion force in the infinite-dilution limit in a solvent of nonadditive hard spheres.

The mutual entropic depletion force felt by two solute "big" hard spheres immersed in a binary mixture solvent of nonadditive "small" hard spheres is calculated as a function of the surface-to-surface distance by means of canonical Monte Carlo simulations and through a recently proposed rational-function approximation [R. Fantoni and A. Santos, Phys. Rev. E 84, 041201 (2011)]. Four representative scenarios are investigated: symmetric solute particles and the limit where one of the two solute spheres becomes a planar hard wall, in both cases with symmetric and asymmetric solvents. In all cases, the influence on the depletion force due to the nonadditivity in the solvent is determined in the mixed state. Comparison between results from the theoretical approximation and from the simulation shows a good agreement for surface-to-surface distances greater than the smallest solvent diameter.

Effective pair potentials between nanoparticles induced by single monomers and polymer chains

Using molecular dynamics (MD) simulations and density functional theory (DFT) calculations, we systematically study the effective pair potential between two particles induced by unconnected monomers and by polymers at various polymer concentrations (above the overlap), particle sizes, and polymer–particle interactions. In the case of athermal interactions, we verify that the entropic depletion forces between two nanoparticles inside a solvent of unconnected monomers oscillate in accordance with the radial distribution of monomers around one nanoparticle, and that the strength of polymer-induced entropic depletion forces rises linearly with the increase of nanoparticle size. These results are quite consistent with previously obtained experimental and theoretical results. When introducing attractive interactions between nanoparticles and polymers, the adsorption of polymer segments on the surface of each nanoparticle induces repulsive forces between the nanoparticles which can eliminate the depletion attractions. Enhancing the attraction between monomers and nanoparticles leads to the formation of thin polymer-layers on the surfaces of nanoparticles. As a consequence, the depletion attraction reappears at a somewhat increased particle distance. The observed phenomena become increasingly pronounced at higher polymer concentrations. Throughout this work we systematically compare computer simulation results with predictions from density functional theory and show that the data obtained with both approaches are quite consistent with each other.

Packing of the polynucleosome chain in interphase chromosomes: Evidence for a contribution of crowding and entropic forces

In the crowded intranuclear environment, entropic depletion forces between macromolecules are expected to be strong. A review of simulations of linear polymers leads to several predictions about probable conformations of a polynucleosome chain in these conditions. These include a globular conformation, variable compaction due to different local rigidity or curvature of the mosaic of isochores, satellite sequences, and nucleosomes containing different histone variants, and the possibility that chromosomes represent separate phases like those seen in heterogeneous particle mixtures by experiment and simulation. Experimental results which show that macromolecular crowding alone, in the absence of exogenous cations, can stabilise interphase chromosomes and cause self-association of polynucleosome chains are presented. Together, these considerations suggest that macromolecular crowding and entropic forces are major factors in packing polynucleosome chains in vivo.

Monte Carlo simulation for the potential of mean force between ionic colloids in solutions of asymmetric salts

A new technique for Monte Carlo sampling of the hard-sphere collision force has been applied to study the interaction between a pair of spherical macroions in primitive-model electrolyte solutions with valences 1:2, 2:1, and 2:2. Macroions of the same charge can attract each other in the presence of divalent counterions, in analogy with earlier observations for planar and cylindrical geometries. The attraction is most significant at intermediate counterion concentrations. In contrast to the entropic depletion force between neutral particles, attraction between macroions is of energetic origin. The entropic contribution to the potential of mean force is generally repulsive at conditions corresponding to aqueous colloids with or without salt. For systems with divalent counterions, the potentials of mean force predicted by mean-field approximations like the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory or the Sogami–Ise (SI) theory are qualitatively different from those observed in the simulations. However, for systems with monovalent counterions, predictions of DLVO theory are in fair agreement with simulation results.

Depletion forces in hard-sphere colloids

A system of monosized hard spheres is studied to elucidate the nature of entropic depletion forces. Our calculations include effective forces between two spheres, a hard sphere and a wall, and the behavior near a step edge and a corner. Qualitative differences between our results and those of the Asakura-Oosawa theory are found. We also demonstrate the nonadditivity of such entropic forces in a simple example.

Depletion forces in fluids

We investigate the entropic depletion force that arises between two big hard spheres of radius ${R}_{b},$ mimicking colloidal particles, immersed in a fluid of small hard spheres of radius ${R}_{s}.$ Within the framework of the Derjaguin approximation, which becomes exact as ${s=R}_{s}{/R}_{b}\ensuremath{\rightarrow}0$, we examine an exact expression for the depletion force and the corresponding potential for the range $0lhl{2R}_{s},$ where $h$ is the separation between the big spheres. These expressions, which depend only on the bulk pressure and the corresponding planar wall-fluid interfacial tension, are valid for all fluid number densities ${\ensuremath{\rho}}_{s}.$ In the limit ${\ensuremath{\rho}}_{s}\ensuremath{\rightarrow}0$ we recover the results of earlier low density theories. Comparison with recent computer simulations shows that the Derjaguin approximation is not reliable for $s=0.1$ and packing fractions ${\ensuremath{\eta}}_{s}=4\ensuremath{\pi}{\ensuremath{\rho}}_{s}{R}_{s}^{3}/3\ensuremath{\gtrsim}0.3$. We propose two new approximations, one based on treating the fluid as if it were confined to a wedge and the other based on the limit ${s=R}_{s}{/R}_{b}\ensuremath{\rightarrow}1$. Both improve upon the Derjaguin approximation for $s=0.1$ and high packing fractions. We discuss the extent to which our results remain valid for more general fluids, e.g., nonadsorbing polymers near colloidal particles, and their implications for fluid-fluid phase separation in a binary hard-sphere mixture.

Theory of the depletion force due to rodlike polymers

The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. In this paper, we consider a system of two parallel plates suspended in a semidilute solution of long thin rods of length L and diameter D. By numerically solving an integral equation, which is exact in the “Onsager limit” (D≪L), we obtain the depletion force between the plates. The second integral of this determines (via the Derjaguin approximation) the depletion potential between two large hard spheres of radius R, immersed in a solution of hard rods (satisfying D≪L≪R). The results for this potential are compared with our previous second order perturbation treatment [Y. Mao, M. E. Cates, and H. N. W. Lekkerkerker, Phys. Rev. Lett. 75, 4548 (1995)], as well as with newly computed third order perturbation results. There is good agreement at low and intermediate densities (which validates our numerical procedures for the integral equation) but the gradual failure of the perturbative treatments is revealed as the density increases. From the numerical results, we conclude that for typical colloidal sphere/rod mixtures, the repulsive barrier in the depletion interaction is likely to be less than the thermal energy kBT throughout the semidilute concentration range of the rods. This reverses our previous conclusion, based on second order perturbation theory, that this barrier could easily lead to kinetic stabilization of the mixture. However, it is possible that terms of order D/L (neglected here) could further modify the results. Within the present theory, the main modification to the well-known (attractive) first order interaction, as the density is raised, is by changing the depth and range of the primary attractive well.

Depletion effects in binary hard-sphere fluids

We report a molecular dynamics computation of the entropic depletion force induced between two large spheres (colloidal particles) immersed in a fluid of small spheres. The effective pair potential obtained by numerical integration of the force is used in a Monte Carlo study of the phase behaviour of the binary mixture. The simulation results are compared with the relevant theoretical predictions that follow from various integral equations for liquid mixtures. The simulations provide evidence for a spinodal instability in a liquid mixture of hard spheres with a size ratio of 0.1.

Depletion stabilization by semidilute rods.

The entropic depletion force in colloids arises when large particles are placed in a solution of smaller ones and sterically constrained to avoid them. We calculate the interaction between two hard spheres (of radius $R$) in a semidilute solution of hard rods of length $L$ and diameter $D$ ($D\ensuremath{\ll}L\ensuremath{\ll}R$) to second order in rod concentration. In addition to the well-known attractive force for separations less than $L$, we find a repulsive force between the spheres at larger separations. For semidilute rods, the resulting barrier can be large compared to ${k}_{B}T$, permitting kinetic stabilization.

Depletion Force in Polydisperse Systems

The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. We calculate the interaction between two flat plates or large hard spheres (of radius R) in a solution of small and polydisperse hard spheres (of average diameter $\bar{\sigma} \ll R$) to second order in small sphere concentration. We find that the effect of polydispersity is to generally smoothen out the depletion force while maintaining the recently predicted repulsive barrier; but for high polydispersities, the repulsion could be eventually wiped out.

Depletion force in colloidal systems

The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. We calculate the interaction between large spheres (of radius R) in a dilute solution of mutually avoiding small spheres (of diameter σ ⪡ R and volume fraction φ), to third order in φ. In addition to the well-known attractive force for 0 < h < σ, we find a repulsive barrier at larger separations, and beyond that a secondary minimum. Except for unusually large size ratios (perhaps abetted by relatively high density φ), these features of the interaction potential are too small, compared to k B T, for kinetic stabilization (arising from the barrier) or flocculation into the secondary minimum, to be widespread, although such effects are possible in principle. For feasible size ratios, the same features should have observable consequences for the radial distribution function of the large spheres. Such effects can be viewed as precursors, at low density, of liquidlike structuring (solvation forces) expected at higher φ. Our third order calculation gives satisfactory agreement with a recent computer simulation at moderate density and size ratio (2 R/ σ = 10; φ = π/15).