Casimir versus Helmholtz forces in the Gaussian model: exact results for Dirichlet--Dirichlet, Neumann--Dirichlet, Neumann--Neumann, and periodic boundary conditions

Abstract

We present results and compare the behavior of two fluctuation-induced forces pertinent for their corresponding ensembles: the critical Casimir force in the grand canonical (fixed external field $h$) one and the critical Helmholtz force in the canonical (fixed average value of the order parameter $m$) one. We do so by deriving exact results for their behavior near the bulk critical point at $T=T_c$ in the three-dimensional Gaussian model. We consider Dirichlet-Dirichlet, Neumann-Dirichlet, Neumann-Neumann, and periodic boundary conditions. For every boundary condition examined, we confirm that both forces follow a finite-size scaling. We find that for Dirichlet-Dirichlet and Neumann-Dirichlet boundary conditions the Casimir and the Helmholtz force differ from each other. For Dirichlet-Dirichlet boundary conditions the Casimir force is always attractive, while the Helmholtz force can be both attractive and repulsive as a function of $T$ and $m$. For Neumann-Dirichlet boundary conditions the Casimir force changes sign from repulsive to attractive with increase of $h$, while the Helmholtz force stays always repulsive. Under periodic and Neumann-Neumann boundary conditions the Casimir force and the Helmholtz force coincide - the first does not depend on $h$, while the latter does not depend on $m$; they are always attractive.

Figures (11)

• Figure 1: The behavior of the scaling function of the Casimir force $X^{\rm (DD)}_{(\rm Casimir)}(x_t,x_h)$ under Dirichlet-Dirichlet boundary conditions. The force is always attractive. It is strongest for nonzero field near $T=T_c$.
• Figure 2: The behavior of the scaling function $X_\chi^{(\rm DD)}$.
• Figure 3: The behavior of Helmholtz force as a function of $x_m$ and $x_t$. We observe that, depending on the values of the scaling variable $x_t$ and $x_m$, it can be both attractive and repulsive.
• Figure 4: The behavior of the excess free energy and Casimir force under Neumann -- Dirichlet boundary conditions as a function of the temperature scaling variable $x_t$ (left panel). For several fixed values of the field scaling variable $x_h$ the behavior of the forces is also visualized (the right panel). We observe that for zero field the force is always repulsive. However, when $h\ne 0$ the force becomes attractive very close to $T_c$ and slightly repulsive for moderate values of $x_h$ and large values of $x_t$. For large values of $x_h$ the force is always attractive.
• Figure 5: The behavior of Helmholtz force as a function of $x_m$ and $x_t$. We observe that the force is always repulsive.