Phase transitions transition between minimal surfaces

Study of transitions between minimal surfaces of soap films discussed in Chapter 2 is an excellent exercise to understand the basics of phase transitions in much more complicated systems. Consider, for example, the four-pin arrangement between two parallel plates shown in Figure 3.1. 

When the membrane is placed in liquid water, rearrangement and a phase-transition occur. This could be due to surface rearrangements wherein the fluorocarbon-rich skin of the membrane is repelled from the interface between the water and membrane. What this means is that in order to minimize the energy of the system the side chains and backbone of the polymer reorient so that the chains are now arranged at the membrane/ water interface. This hypothesis agrees with the data that show that the water contact angle on the membrane surface becomes more hydrophilic after the membrane is placed in liquid water [32]. The presence of liquid water also results in the removal of a vapor-liquid meniscus, which could also aid in the above rearrangements [28]. 

Next we consider the effect of the block copolymer composition /= NfJN on the ordered morphology. In the limit of very strong segregation, that is, zero interface width, the natural idea is to let the stable ordered phase correspond to the phase with the minimal interface surface. To illustrate this principle and to obtain a semiquantitative estimate of the values of/for which the transitions between the three classical stmctures occur, we consider an LxLxL volume of the self-assembled diblock copolymer system. The ordered states that will be compared are the lamellar phase, a square lattice of cylinders, and spheres on a simple cubic (SC) lattice. L is the periodicity length scale of the layers, the square, and the cubic lattice (Figure 19). The LxLxL volirme element contains one cylinder resp. one sphere. Volirme conservation (Figure 20), therefore, requires fL = 7tRcL = 4n/SRs, where Rc and Rs are the radii of the cylinder and the sphere, respectively. 

To take place, these surface reconstructions may require a certain temperature. One distinguishes two cases. In the first, the surface reconstruction appearing above the transition temperature remains stable all the way down to low temperature, whereas in the second, there is a reversible first-order phase transition between two surface terminations. In the first case, the surface needs to overcome an activation barrier to get out of its metastable state and achieve the reconstruction, which is the thermodynamic ground state down to low T. The thermal evolution is dominated by kinetic barriers, but not by minimization of the surface free energy. In the second case, the higher temperature phase is a thermal surface reconstruction, which appears when the surface is annealed beyond the transition temperature and disappears once it is cooled below. We discuss one example for the first case of an irreversible transition and two for the second case of thermal surface reconstructions, which represent true phase transitions. 

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