Total free energy per unit area

We now determine the equilibrium value of the density on the surface and hence the equilibrium film thickness — again for the case where the film is uniform in the x direction. Using Eq. (4.45) for the local grand potential, W, and performing the integration in Eq. (4.49), we find that the total free energy per unit area is... 

The total free energy per unit area at the surface A is... 

The total free energy (per unit area) of the system is... 

We can balance these free-energy expressions and integrate over the width of the device d to obtain the total free energy per unit area ... 

The total free energy of the system is then made up of the molar free energy times the total number of moles of the liquid plus G, the surface free energy per unit area, times the total surface area. Thus... 

The total change in free energy per unit area in charging up the interface is obtained by integrating Equation 14... 

The total surface energy per unit area of liquids, U, is related to measurable surface tension or the specific free surface energy 7 by the thermodynamic equation... 

Let us denote by V the interaction potential for planar bilayers. Assuming that the fluctuations of adjacent bilayers are totally uncorrelated, and using locally the planar approximation, the free energy per unit area can be written in terms of the distance distribution function... 

After the solution of the Poisson—Boltzmann equation is obtained, the total double layer free energy per unit area is obtained by adding the electrostatic... 

The total electrostatic free energy per unit area, between the two parallel plates, due to the orientation of dipoles m z) in the local field E = m(z)/y is given by26... 

Consequently, the total electrostatic free energy per unit area (accounting for all the interactions between charges and dipoles) is provided by the expression... 

The free energy function provides a mesoscopic framework for understanding the terraced spreading phenomena. Since the surface energy is defined as the free energy per unit area, the total disjoining pressure (n) for these fluids can be derived from the experimental surface energy data by ... 

If fii is the fraction of the area of plate I covered by i and fnj is the corresponding quantity for polymer j on plate II, then fifnj is the fraction of unit area over which i on I is aligned with J on II. The total change in free energy per unit area for close approach is obtained by summing over all possible polymer interactions... 

The total interfacial free energy per unit area, consists of the sum of /o and the free energy per unit area that comes from the liquid-vapor interface. In equilibrium, one minimizes the total free energy subject to the conservation constraint — i.e., one works at fixed chemical potential. As explained in the discussion of the gas-fiquid interface in Chapter 2, the appropriate bulk free energy to minimize to find the interfacial profile is the grand potential per unit area, gs, which is written ... 

In order to obtain the values of the surface coupling energies from the temperature dependence of So, we have to calculate the value of So for a semi-infinite sample, where the interface is located in the xy plane at = 0. The total free energy per unit surface area is F = Fb + Fs, where Fb = /B(2 )d2 and Fs = w So + w2. After minimization with respect to So, the surface value of the order parameter is... 

The total free energy of the system is composed of the energy of the bulk liquid and the surface liquid. The latter is equal to the surface free energy per unit area (Gg) multiplied by the surface area as shown in Eq. (2.2). Combining Eqs. (2.1) and (2.2) results in Eq. (2.3), which illustrates that the free surface energy of a pure substance is equal to its surface tension. 

The total surface energy per unit area, however, may not be equal to the surface free energy alone. As heat must be supplied to the expanding surface to keep the temperature in the surface constant, a heat term has to be added to tlie surface free energy. The total energy at the surface per sq cm then will be equal to... 

Integration of Equation 12.12 is performed only for the part of the boundary that is located between a and ao phases. The second item in Equation 12.12 corresponds to the loss in the change of total free energy and to the formation of the new part of the boundary between a and P phases, which is of length h. Therefore, for a unit volume we get where y is the free energy per unit area of the a-P... 

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