# Critical Casimir effect in three-dimensional Ising systems: Measurements on binary wetting films

## Overview

**Authors:** Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan

**Source:** Physical Review Letters, Vol.94, 135702, (2005)

**Soft Matter key words:** thermodynamic Casimir force, correlation length, thin films, wetting

## Abstract

In analogy to the quantum electrodynamics Casimir force, arising between conducting plates due to confinement of zero-point fluctuations of vacuum fields, a thermodynamics Casimir force has been introduced. The latter arises by confining a fluid with diverging bulk correlation lenght <math>\xi</math> to a finite dimension L. Authors of this paper set out to experimentally confirm theoretical predictions for this force, in binary thin wetting films close to liquid/vapor coexistence. They extract a Casimir amplitude <math>\Delta_{+-}</math> as well as a Casimir scaling function <math>\theta_{+-}</math> which, they find, depends monotonically on dimensionality.

## Soft Matter Snippet

The experimental setup consists of a <math>SiO_2/Si</math> substrate on which methylcyclohexane and perfluoromethylcyclohexane form a 3D Ising film by complete wetting. The two solvents de-mix at a bulk critical point (BPC) of <math>T = 46.2^{\circ} C.</math> The authors measure film thickness while varying temperature t, mole fraction <math>\phi</math> or chemical potential <math>\Delta \mu</math>. They illustrate this schematically on the phase diagram of fig.1. They chose to measure film thickness employing x-ray reflectivity, for which they use a fixed anode tube. The radiation is reflected off of a vertically oriented substrate in the horizontal scattering plane, at an incident angle <math>\alpha</math>, corresponding to a wave vector <math>q_z = \frac{4 \pi}{\lambda} sin(\alpha)</math> normal to the surface. The film thickness <math>L = <\frac{n \pi}{q_{z,n}}></math> is obtained via the interference fringes arising from the substrate/film and film/vapor interfaces.

Figure 2 contains some of the results. On fig 2a, the variation of film thickness is plotted as a function of temperature. Open symbols and closed correspond to data on cooling and heating of the film respectively. Film thickening, which signifies the presence of Casimir force, is observed at a critical <math>T_c</math> regardless of the direction of temperature variation. However, for <math>T< T_c</math>, a hysteresis is observed between cooling and heating. No hysteresis is present when <math>T>T_c</math>.

The same film thickening at <math>x_c</math> is observed when plotting volume fraction <math>x</math> as a function of film thickness. In this case no hysteresis is present. These data are deemed robust and are subsequently used to extract the Casimir amplitude <math>\Delta_{+-}</math> and the Casimir scaling function <math>\theta_{+-}</math>. The scaling function is plotted on figure 3, as a function of rising temperature and rising molar fraction.