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We present a new Monte Carlo method to calculate Casimir forces acting on objects in a near-critical fluid, considering the two basic cases of a wall and a sphere embedded in a two-dimensional Ising medium. During the simulation, the objects are moved through the system with appropriate statistical weights, and consequently are attracted or repelled from the system boundaries depending on the boundary conditions. The distribution function of the object position is utilized to obtain the residual free energy, or Casimir potential, of the configuration as well as the corresponding Casimir force. The results are in perfect agreement with known exact results. The method can easily be generalized to more complicated geometries, to higher dimensions, and also to colloidal suspensions with many particles.
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Using general scaling arguments combined with mean-field theory we investigate the critical ($T simeq T_c$) and off-critical ($T e T_c$) behavior of the Casimir forces in fluid films of thickness $L$ governed by dispersion forces and exposed to long-ranged substrate potentials which are taken to be equal on both sides of the film. We study the resulting effective force acting on the confining substrates as a function of $T$ and of the chemical potential $mu$. We find that the total force is attractive both below and above $T_c$. If, however, the direct substrate-substrate contribution is subtracted, the force is repulsive everywhere except near the bulk critical point $(T_c,mu_c)$, where critical density fluctuations arise, or except at low temperatures and $(L/a) (betaDelta mu) =O(1)$, with $Delta mu=mu-mu_c <0$ and $a$ the characteristic distance between the molecules of the fluid, i.e., in the capillary condensation regime. While near the critical point the maximal amplitude of the attractive force if of order of $L^{-d}$ in the capillary condensation regime the force is much stronger with maximal amplitude decaying as $L^{-1}$. Essential deviations from the standard finite-size scaling behavior are observed within the finite-size critical region $L/xi=O(1)$ for films with thicknesses $L lesssim L_{rm crit}$, where $L_{rm crit}=xi_0^pm (16 |s|)^{ u/beta}$, with $ u$ and $beta$ as the standard bulk critical exponents and with $s=O(1)$ as the dimensionless parameter that characterizes the relative strength of the long-ranged tail of the substrate-fluid over the fluid-fluid interaction. We present the modified finite-size scaling pertinent for such a case and analyze in detail the finite-size behavior in this region.
The confinement of critical fluctuations in soft media induces critical Casimir forces acting on the confining surfaces. The temperature and geometry dependences of such forces are characterized by universal scaling functions. A novel approach is presented to determine them for films via Monte Carlo simulations of lattice models. The method is based on an integration scheme of free energy differences. Our results for the Ising and the XY universality class compare favourably with corresponding experimental results for wetting layers of classical binary liquid mixtures and of 4He, respectively.
We study critical Casimir forces (CCF) $f_{mathrm C}$ for films of thickness $L$ which in the three-dimensional bulk belong to the Ising universality class and which are exposed to random surface fields (RSF) on both surfaces. We consider the case that, in the absence of RSF, the surfaces of the film belong to the surface universality class of the so-called ordinary transition. We carry out a finite-size scaling analysis and show that for weak disorder CCF still exhibit scaling, acquiring a random field scaling variable $w$ which is zero for pure systems. We confirm these analytic predictions by MC simulations. Moreover, our MC data show that $f_{mathrm C}$ varies as $f_{mathrm C}(wto 0)-f_{mathrm C}(w=0)sim w^2$. Asymptotically, for large $L$, $w$ scales as $w sim L^{-0.26} to 0$ indicating that this type of disorder is an irrelevant perturbation of the ordinary surface universality class. However, for thin films such that $w simeq 1$, we find that the presence of RSF with vanishing mean value increases significantly the strength of CCF, as compared to systems without them, and shifts the extremum of the scaling function of $f_{mathrm C}$ towards lower temperatures. But $f_{mathrm C}$ remains attractive.
Monte Carlo simulations based on an integration scheme for free energy differences is used to compute critical Casimir forces for three-dimensional Ising films with various boundary fields. We study the scaling behavior of the critical Casimir force, including the scaling variable related to the boundary fields. Finite size corrections to scaling are taken into account. We pay special attention to that range of surface field strengths within which the force changes from repulsive to attractive upon increasing the temperature. Our data are compared with other results available in the literature.
We numerically test an experimentally realizable method for the extraction of the critical Casimir force based on its thermodynamic definition as the derivative of the excess free energy with respect to system size. Free energy differences are estimated for different system sizes by integrating the order parameter along an isotherm. The method could be developed for experiments on magnetic systems and could give access to the critical Casimir force for any universality class. By choosing an applied field that opposes magnetic ordering at the boundaries, the Casimir force is found to increase by an order of magnitude over zero-field results.
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