Monge representation

Figure B3.6.2. Local mterface position in a binary polymer blend. After averaging the interfacial profile over small lateral patches, the interface can be described by a single-valued function u r. (Monge representation). Thennal fluctuations of the local interface position are clearly visible. From Wemer et al [49].

One can regard the Hamiltonian (B3.6.26) above as a phenomenological expansion in temis of the two invariants Aiand//of the surface. To establish the coimection to the effective interface Hamiltonian (b3.6.16) it is instnictive to consider the limit of an almost flat interface. Then, the local interface position u can be expressed as a single-valued fiinction of the two lateral parameters n(r ). In this Monge representation the interface Hamiltonian can be written as... 

Equation (2) is identified as a second-order, nonlinear differential equation once, the curvature is expressed in terms of a shape function of the melt/crystal interface. The mean curvature for the Monge representation y = h(x,t) is... 

Within this continuum description, the membrane is conceived as a thin elastic sheet. If one assumes that the membrane fluctuations are small, its surface can be parameterized within a Monge representation via a height function, h(y,z), describing its position over some reference plane. Then, the Helfrich-Hamiltonian takes the form... 

Show that a surface of minimal area has zero curvature using the Monge representation. What are the relations between the two curvatures and what is the Gaussian curvature ... 

In the Monge representation of the surface, we have from Eq. (1.94) that, the total area A for a surface described by a height h x,y) above the xy plane is... 

In order to study the fluctuations of interfaces and surfaces, one first needs an expression for the free energy of an interface with a shape that is not necessarily flat. Phenomenologically, one can write that the free energy is the product of the interfacial tension, y, and the area, which in the Monge representation for the surface, z = h Xy y), reads ... 

Consider the fluctuations of a surface defined in the Monge representation as z = h(x, y). The area of the flat surface is denoted by A. For slowly varying fluctuations of this surface about a flat shape (h = ho, where is a constant) the additional surface free energy of the undulated interface over that of the flat one (AFg = — yA) is approximately... 

Here Tframe is the stress or frame tension, Ap is the projected area in the plane of applied stress, and we have omitted the membrane edge term. Let us cmisider a membrane with fixed total area A. In Monge representation, one has... 

The intrinsic structure of a liquid-vapor interface resembles the surface of a polymer liquid in contact with a nonattractive solid substrate at the pressure where the liquid coexists with its vapor. In the latter case, the system is in the vicinity of the drying transition and a layer of vapor intervenes between the substrate and the polymer liquid. There is, however, one important difference between the vapor-polymer interface and the behavior of a polymer at a solid substrate the local position of the interface can fluctuate. Let us first consider the case where the film is very thick and the solid substrate does not exert any influence on the free surface of the film in contact with its vapor. The fluctuations of the free surface are capillary waves. Neglecting bubbles or overhangs, one can use the position of the liquid-vapor interface, z = h x,y), as a function of the two lateral coordinates, x and y, parallel to the interface to describe the system configuration on a coarse scale. In this Monge representation, the free energy of the interface is given by the capillary-wave Hamiltonian " ... 

A mathematical surface is a function of two coordinates. In the so-called Monge representation, it is simply expressed by z = z x,y), with x, y, and z being Cartesian coordinates. A more general description of a surface would be r = rix,y) with f being an arbitrary position in three-dimensional space. The surfaces are assumed to be continuous and differentiable, which implies that they are free of breaks and abrupt bends. 

The solution minimizing the elastic energy with PBC on this cell is obtained using a boundary layer method. We shall use the Monge representation where the deviation of the membrane from the (x, y) plane is given by y). 

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