Surface moment

Below, we first introduce the most general mechanical description of the surface moments (torques) exerted on the boundary between two fluid phases. Then, we consider the thermodynamics of a curved interface (membrane) in terms of the work of flexural deformation. Next, we specify the bending rheology by means of the model of Helfiich [202]. Finally, we review the available expressions for the contributions of the electrostatic, steric, and van der Waals interactions to the interfacial bending moment and curvature elastic moduli. These expressions relate the interfacial flexural properties to the properties of the adsorbed surfactant molecules. 

Figure 11 Components of the tensors (a) of the surface stresses, g , and (b) of the surface moments (torques), M.

For a complete mechanical description of the surface, one needs to specify expressions for the stresses and moments. This is usually done by postulating some constitutive relations between stress and strain, which pertain to a particular model for the rheological behavior of the interface. An example is the Scriven s constitutive relation [140] for a see Eq. (81). In Section IV.C, we discuss a constitutive relation for the tensor of the surface moments, M. Before that, we consider the thermodynamics of the curved interfaces. 

To derive a dynamic theory one can of course extend the above formulation of equilibrium theory employing generalized body and surface forces as in the initial derivation [7,15]. Here, however, we prefer a different approach [46], which, besides providing an alternative, is more direct in that it follows traditional continuum mechanics more closely, although introducing body and surface moments usually excluded, as well as a new kinematic variable to describe alignment of the anisotropic axis. 

Here x represents the position vector, K is the external body moment per unit volume, and t is any surface moment per unit area. In this equation, the velocity v is subject to the constraint... 

We shall always assume isothermal conditions and therefore ignore thermal effects. In these circumstances, as in any classically based continuum theory, conservation laws for mass, linear momentum and angular momentum must hold. The balance law for linear momentum, given below, is basically similar to that for an isotropic fluid, except that the resulting stress tensor (to be derived later) need not be symmetric. The balance law for angular momentum is also suitably augmented to include explicit external body and surface moments. 

If u denotes the outward unit normal to the surface 5, then the usual tetrahedron argument [161, p.l29] shows that the surface force U and surface moment U are expressible in terms of the stress tensor tij and couple stress tensor lij respectively, through the relations... 

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