Phase transition in tensionless surfaces

Abstract

We study the critical behavior of the Laplacian roughening model, which describes the growth of tensionless surfaces. This type of growth phenomena is very important, for instance, in biological membranes and in molecular beam epitaxy. We summarize previous analytical and numerical results and point out their contradictions and differences, thus making clear the context of our work. Our contribution, achieved through large-scale numerical simulations, is the confirmation that the model exhibits a single continuous phase transition: the transition takes place between a continuum massless (i.e., with infinite correlation length) bilaplacian behavior and a massive propagator (finite correlation length).

Introduction

Tensionless surfaces are of great importance in different fields and applications, ranging from biology, where lipid membranes are tensionless for practical purposes [1], to nanotechnology, where molecular beam epitaxy and related techniques used to grow devices are carried out in conditions for which surface tension is negligible [2]. Therefore, from the viewpoint of these and other applications, understanding the different regimes in which tensionless surfaces can grow is a very relevant issue. In this context, in this paper we study the critical behavior of a model which describes tensionless surfaces. In spite of its simplicity, the model has been the subject of a number of studies [3], [4], [5], [6], [7], [8], which have led to disparate and often contradictory predictions of its behavior. Our aim in this work is to carry out large-scale numerical simulations that can clarify the actual scenario in which the transition (or transitions) takes place. Indeed, one of the most intriguing previous results is the conjecture [3] that the model undergoes two different phase transitions between the so-called liquid, hexatic, and solid phases (names borrowed from studies of two-dimensional melting, but standard in this context). As we will see below, our numerical results do not support this picture and allow us to claim that there is no hexatic phase in the model. Furthermore, we find that the transition is continuous, contrary to earlier simulations [5] which reported a first-order transition.

In order to present clearly our conclusions, the outline of the paper is as follows. In the next two subsections we motivate the definition of the model and recall briefly examples of its applications. We then (Section 2) establish our notation and define the observables we study. Section 3 is devoted to summarizing all available analytical and numerical results to date. Section 4 reports the outcome of our numerical simulation program. Finally, we recapitulate on our work and its relation to previous ones in the conclusions, in Section 5.

Generally speaking (see, e.g., [2] for a detailed introduction to the subject), the free energy of a surface (F=μN−pV+σA) can be written as follows:

$$F=(\frac{\mu }{v}-p)\int \int \mathrm{d}x\mathrm{d}yh(x,y)+\int \int \mathrm{d}x\mathrm{d}y\varphi (h_x,h_y)\text{,}$$

where μ is the chemical potential, ν=V/N is the specific volume, h(x, y) is the height of the surface above point (x, y), p is the pressure, and $\varphi (h_x,h_y)=\sigma (h_x,h_y)\sqrt{1+h_x^2+h_y^2}$, with σ being the surface tension and A being the area. Working at constant pressure, we can study the in influence of either the chemical potential or the surface tension in the way the surface grows.

First, the main effect of the chemical potential is to induce diffusion on the surface from regions with higher chemical potential to regions with lower chemical potential. A conserved mass current (the number of particles is constant) is generated that is restricted to the surface, reads

$$j=-D_{\text{s}}\nabla \mu \text{,}$$

and drives dynamics for the surface height, in the form of a continuity equation

$$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}=-\nabla ·j\text{.}$$

Combining both expressions, and having in mind that, to linear order, the chemical potential (that is proportional to the mean surface curvature) is related to the surface height by μ ∝−∇²h, we arrive at

$$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}=D_{\text{s}}{\nabla }^2\mu =-\kappa {\left({\nabla }^2\right)}^{2}h,\text{with}\kappa >0\text{.}$$

Second, let us consider the other ingredient, surface tension. Assuming that the chemical potential of the vapor is uniform, μ₀, there will be evaporation in places where μ>μ₀. We can thus write the variation of the height with time as a linear function of the chemical potential difference:

$$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}\propto -(\mu -{\mu }_0)\text{,}$$

and so

$$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}=\nu {\nabla }^{2}h,\text{with}\nu >0\text{.}$$

If we now include all the above factors in our description, i.e., we take into account evaporation, surface diffusion and thermal fluctuations, we can write the following Langevin equation for the surface height

$$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}=\nu {\nabla }^{2}h-\kappa {\left({\nabla }^2\right)}^{2}h+\eta (\mathbf{r},t)\text{,}$$

where η(r, t) is a Gaussian white noise. This is the generalized, continuum Laplacian roughening model.¹ From this general expression, by letting ν=0, we are left with a model for continuum tensionless surfaces, henceforth referred to as the bilaplacian model.

One example of a tensionless surface is provided by a lipid membrane [1]. The fluid of lipids which conforms the membrane is above its melting point and hence lipids can move inside the membrane. Recently Gov and Safran [9] have proposed the following equation to model the action of the elastic cytoskeleton on the fluid membrane:

$${\kappa \left({\nabla }^2\right)}^2\chi \left(\mathbf{r}\right)+V\left(\mathbf{r}\right)\chi \left(\mathbf{r}\right)=0\text{,}$$

where χ(r)=〈h(r)〉 (i.e., we are in a mean field approximation), and V(r) represents the pinning effect of the cytoskeleton on the surface. Eq. (2) is the mean field analog of Eq. (1) for ν=0, in the presence of an external potential.

A second instance of tensionless surfaces arises in the growth of thin films by molecular beam epitaxy (MBE) [2], [6]. MBE is a process of high technological interest, in which atoms or molecules are deposited on the surface of the growing sample in conditions such that evaporation is largely suppressed. This suppression leads in turn, following the reasoning of the above subsection, to the continuum bilaplacian model [Eq. (1) with ν=0], referred to in this context as the linear MBE equation.

Finally, the third example is two-dimensional melting. Nelson [3] proposed the Laplacian roughening model in order to describe this phenomenon. We specify details of this model below. We refer the reader to the original work for details on how two-dimensional melting occurs within such approach.