Critical Casimir forces between defects in the 2D Ising model

Abstract

An exact statistical mechanical derivation is given of the critical Casimir interactions between two defects in a planar lattice-gas Ising model. Each defect is a group of nearest-neighbor spins with modified coupling constants. Such a system can be regarded as a model of a binary liquid mixture with the molecules confined to a membrane and the defects mimicking protein inclusions embedded into the membrane. As suggested by recent experiments, certain cellular membranes appear to be tuned to the proximity of a critical demixing point belonging to the two-dimensional Ising universality class. Therefore one can expect the emergence of critical Casimir forces between membrane inclusions. These forces are governed by universal scaling functions, which we derive for simple defects. We prove that the scaling law appearing at criticality is the same for all types of defects considered here.

Numerically Exact Approach to Ultra-Small Particles in Near-Critical Mixtures

“…The transfer matrix method, allows to study the impact of temperature on the ensemble average distance in any pair of solute particles. In general, the method is applicable whenever the partition function can be expressed as a product of matrices [32]:…”

Dynamics and steady states of a tracer particle in a confined critical fluid

“…[5] for a review). Except in d = 2 spatial dimensions, where exact methods exist [6][7][8][9], the interaction between two colloidal particles or between a colloid and a solid wall is typically analyzed separately in the near-distance (Derjaguin) and the far-distance limit. In the former, an approximate parallel plate geometry is realized, allowing one to invoke known results for the CCF in a thin film [2,[10][11][12].…”

Critical Casimir effects in 2D Ising model with curved defect lines

“…Casimir effect emerges in various fields of physics. For example, critical Casimir forces in a biophysics application are calculated analytically in [9]. Membranes of living cells exhibit properties similar to those of the critical Ising model.…”

“…It should be noted that in our work we modify all the coupling constants around a selected spin. The Free energy of two interacting defects is evaluated in [9] in thermodynamic limit. One of the results, important for our research, is the attraction of identical defects, which is mentioned in the paper considered and known from an exact solution for the planar Ising model on the square lattice [10].…”