Long-wavelength fluctuations and dimensionality crossover in confined liquids*

The phase behavior of liquids confined in a slit geometry does not reveal a crossover from a three- to a two-dimensional behavior as the gap size decreases. The hexatic phase in two dimensions only occurs in liquids in a monolayer. In this work, the dimensionality crossover becomes apparent in the lateral size dependence of the relaxation dynamics of confined liquids. We develop a Debye model for the density of vibrational states of confined liquid systems and perform extensive numerical simulations. In these systems, the amplitude of vibrational motion is enhanced by Mermin-Wagner fluctuations or the Debye-Waller factor- by a quantity that scales as the inverse gap width and is proportional to the logarithm of the aspect ratio, a clear signature of two-dimensional behavior.  As the temperature or lateral system size increases, the crossover to a size-independent relaxation dynamics occurs when structural relaxation takes place before the vibrational modes with the longest wavelength develop. Our results show that confined systems exhibit a gradual dimensionality crossover controlled by the gap width and the temperature, which is appreciable when investigating the lateral size dependence of the dynamics. The physics of confined liquids are thus richer than previously realized.