Polymer-mediated entropic forces between scale-free objects

Abstract

The number of configurations of a polymer is reduced in the presence of a barrier or an obstacle. The resulting loss of entropy adds a repulsive component to other forces generated by interaction potentials. When the obstructions are scale invariant shapes (such as cones, wedges, lines or planes) the only relevant length scales are the polymer size R0 and characteristic separations, severely constraining the functional form of entropic forces. Specifically, we consider a polymer (single strand or star) attached to the tip of a cone, at a separation h from a surface (or another cone). At close proximity, such that h ≪ R0, separation is the only remaining relevant scale and the entropic force must take the form F = A kBT /h. The amplitude A is universal, and can be related to exponents η governing the anomalous scaling of polymer correlations in the presence of obstacles. We use analytical, numerical and ǫ-expansion techniques to compute the exponent η for a polymer attached to the tip of the cone (with or without an additional plate or cone) for ideal and selfavoiding polymers. The entropic force is of the order of 0.1pN at 0.1µm for a single polymer, and can be increased for a star polymer.

Diffusion in the presence of scale-free absorbing boundaries

“…In simple geometries [such as circular cones (in d = 3) or wedges (in d = 2)] this exponent is known analytically [8,10,14,15]. (In some cases, η is known even for polymers in good solvents [22], where it is found by studying numerically SAWs in d = 3 for flat surfaces [19,20,25] and for circular cones or cone-plane geometries [14,15]. )…”

“…Thus solution of the diffusion problem provides a handle on counting the configurations of ideal polymers. The latter determines the free energy of the polymers and can be used to find forces between the polymers and confining surfaces [14,15].…”

“…(However, in a typical experimental situation the other end of the polymer is also attached to the stage.) The loss of polymer entropy leads to entropic force between the two objects [14,15]:…”

“…(4) over the separation h between a microscopic distance a and the maximal interaction distance ∼ aN ν fixes the value of the prefactor [14,15]:…”

“…In general, the diffusion problem can be related to statistical mechanics of ideal polymers in which self-interactions can be neglected [11][12][13]. In the case of SF surfaces the solutions of diffusion equations can be used to infer the prefactor in force-distance relation characterizing interactions between ideal polymers and the surfaces [10,14,15]. Section III explains the relation between the diffusion and ideal polymer problems, as well as discusses some general features of polymers near surfaces.…”