Minimum Surface Energy Conditions

2T /r and RT nPi/ nP ) = IVa/r which leads to lnP,/lni = WilRT(r). These equations illustrate the observation that droplet and phase separated systems will naturally tend to form a curved surface to minimize the free energy. 

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Figure 10.1 Colloidal dispersions are Inherently unstable systems and in the long run the attractive forces will dominate and the colloidal system will destabilize. However, colloid stability depends on the attractive van der Waals and the repulsive electrical or steric (polymeric) forces. The repulsive forces stabilize a dispersion if they are larger than the van der Waals (vdW) ones (and the total potential is larger than the "natural" kinetic energy of the particles). Surfaces are Inherently unstable and the van der Waals forces "take the system" back to its stable (minimum surface area) condition and contribute to instability (aggregation)...

Hence, the minimum total surface energy condition at equilibrium can be written as... 

The basis for the determination of an upper bound on the apparent Young s modulus is the principle of minimum potential energy which can be stated as Let the displacements be specified over the surface of the body except where the corresponding traction is 2ero. Let e, Tjy, be any compatible state of strain that satisfies the specified displacement boundary conditions, l.e., an admissible-strain tieldr Let U be the strain energy of the strain state TetcTby use of the stress-strain relations... 

Proteins are dynamic molecules with respect to structure. The preferred folded structure for a given set of environmental conditions is that which has the minimum free energy. The driving force to assume a given folded structure is a thermodynamic force. In aqueous systems, the hydrophobic side-chains will endeavour to orient away from the surrounding water and towards the core of the molecule. However, for high surface activity, it is essential that the protein molecule should unfold and orient its hydrophobic side-chains towards the oil phase. A lack of hydrophilic residues usually does not restrict protein functionality at interfaces. Thus, flexible proteins can create a highly hydrated, mobile layer to stabilize an emulsion particle. 

Hence, when the control parameter / increases to such a value that the ratio l/R = X exceeds the critical value 1.3255, then the boundary conditions (3.2c) will not be met and the function (3.2b) ceases to be a solution for the minimum surface (minimum of the potential energy). The critical value of a is a = / /1.8102. 

The pore size distribution can be obtained from capillary pressure measurements or mercury porosimetry. The capillary pressure is related to the specific free energies of the interface between fluids and between the fluid and the capillary wall. At mechanical equilibrium, the surface free energy between the fluids is a minimum. The equilibrium condition is expressed by the Laplace equation ... 

Under conditions of very slow growth, the ultimate crystal shape is determined strictly by thermodynamics. Under such conditions, the faces appearing on the crystal correspond to the smallest convex polyhedron having minimum surface free energy. Gibbs... 

This equation is equivalent to the statement that the surface tension is exactly equal to the Helmholz free energy per unit area. Thus, we may make use of the thermodynamic equilibrium statement that isolated systems tend toward the condition of lowest free energy to show that the stable state of a system is the one with minimum free energy, including the contribution of the surface free energy. 

In so far as a condition of minimum potential energy is sought, massive phases are favoured, and small particles or droplets tend to fuse together into larger ones. H, for example, a drop of a hquid A is surrounded by another liquid -B, the condition that it should not dissolve is that the attractions between the molecules of A should outweigh those between molecules of A and molecules of B. This being so, molecules in the surface layer of A are subjected to a pull into the interior of the drop. The greatest response to this pull is made when the surface between the two liquids assumes its minimum area. 

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