Static and dynamic critical phenomena
Although critical phenomena have widely been investigated in the literature, almost all these studies consider the case of energy conserving systems. In particular, –to the best of our knowledge- no study exists for the case of an isothermal liquid close to the critical point. There are, however, a number of modern applications where such a situation may be relevant. An example here is a monolayer (Langmuir) film floating at the interface between two immiscible fluids that may undergo a first order phase transition in terms of monolayer thickness. Due to efficient heat transfer between the film and the surrounding liquid medium, the temperature of the film can be kept constant so that the isothermal assumption becomes a good approximation. Typically, monolayer films have critical temperatures around room temperature and are not only used in industrial applications but are also relevant in biology. The behaviour of such a system at room temperature is thus close to that of a two dimensional isothermal liquid at criticality. In order to study this interesting topic, we employed a variant of the lattice Boltzmann method, which has been developed within our group in the preceding year (2011) and has been successfully applied to a number of interesting test cases.
We used this approach to study a two dimensional liquid-vapour system close to the critical point. As a first step, we have shown that the static (time-independent) properties and thermodynamic quantities agree with the predictions of the Ising universality class. To illustrate this point, figure 2 shows the static correlation function of density fluctuations versus wave vector as obtained from our simulations. It is seen that the expected power-law is well reproduced, except for the smallest wave numbers, where finite-size effects show up . Interestingly, we find that the dynamical critical behavior of an isothermal non-ideal fluid is different from the predictions of theories valid for conventional (energy-conserving) fluids. The underlying reason for this is the absence of a heat-diffusion mode in the dynamical equations, which would otherwise dominate the order-parameter transport. Instead, for an isothermal fluid, sound waves are the only possible relaxation channel. As we have shown through a perturbation theory analysis of the associated nonlinear Langevin equations, at the critical point, sound waves are over-damped and the order-parameter dynamics in fact reduces to a time-dependent Ginzburg-Landau model for a non-conserved order-parameter (so-called "model A" in the classification of Hohenberg and Halperlin). In agreement with simulations, our theory predicts the relaxation time and bulk viscosity to diverge by power laws when approaching the critical point in 2D.
Correlation function of equal time density fluctuations, C(k) versus wave vector k at the critical point for two different system sizes as indicated. The dashed curve represents the power law, expected in the limit of large system size.
Growth of the effective bulk viscosity (normalized to its bare value) in the critical regime, in dependence of the wave number.