Theory and Derivation of Basic Equations

Let us consider for simplicity a linear chain of amphiphilic molecules, because a switch to the two-dimensional case is self evident. Let Pi(t) be the probability of the occurrence of molecule i at the instant t in the overturned state then, the same probability at the time t + At will be equal to 

FIGURE 5.29 Polymeric substrate containing rotationally mobile amphiphilic chains in contact with air or water v — air, w — water s — polymeric substrate. 1 — amphiphihc chains in the normal state, 2 — amphiphihc chains in the overturned state. 

Let us consider Equation 5.191 in more detail. If expressions (5.189) and (5.190) involve both overtumings due to the thermal fluctuations and those caused by the interaction with the surrounding media, then, in contrast to these expressions, Equation 5.191 involves only the overtumings related to the interactions between the neighbors hence, random overtumings caused by thermal fluctuations should not be taken into account because they do not result in the transfer of the overturned state. This is why the unity is subtracted in Equation 5.191. 

Let a be the mean distance between molecules capable of overturning. Then, Equation 5.192 may be rewritten in the following form  

The latter part of Equation 5.193 is a discrete analogy of the second-order spatial derivative. Hence, using the continnons coordinate x = ia, we obtain the following second-order partial differential eqnation instead of Equation 5.193  

---