Morphologies in the limit

In the simplest case, = 0 and all sites are equivalent. Thus, as we saw in Section 4.4.3 [see Eq. (4.94)], a closed expression for the grand potential may be derived where the occupation number sq = 0,1 is discrete and doublevalued. From Eqs. (1.76a) and (4.94), we conclude that 

The next slightly more complicated situation is one in which the lattice fluid is confined in the 2-direction by two planar hard substrates represented by 

According to our modular approach, the lattice fluid confined by hard repulsive substrates may be viewed as a bulk system, in which serves to introduce surfaces. We can then express the grand-potential density as 

However, as. So = 0,1, no new morphologies arise. The onty effect of confinement by hard, repulsive substrates is an upward shift in the chemical potential at gas liquid coexistence. By solving the analog of Eq. (4.99) we obtain 

The upward shift in effected by hard, repulsive walls relative to the bulk value may be interpreted as drying. This term refers to the fact that a larger chemical potential is needed to initiate condensation of the confined gas relative to its bulk counterpart. This is because the effect of the substrates represented by Eq. (4.100) is to create an energetically less favorable situation by reducing the number of nearest-neighbor attractions from six (bulk) to five in the surface planes of the confined fluid. 

From Eq. (4.99), //, = -3fff is readily deduced. Thus, for /r gas is the thermodynamically stable bulk phase, whereas for p this is true for the liquid phase. 

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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. 

While thin polymer films may be very smooth and homogeneous, the chain conformation may be largely distorted due to the influence of the interfaces. Since the size of the polymer molecules is comparable to the film thickness those effects may play a significant role with ultra-thin polymer films. Several recent theoretical treatments are available [136-144,127,128] based on Monte Carlo [137-141,127, 128], molecular dynamics [142], variable density [143], cooperative motion [144], and bond fluctuation [136] model calculations. The distortion of the chain conformation near the interface, the segment orientation distribution, end distribution etc. are calculated as a function of film thickness and distance from the surface. In the limit of two-dimensional systems chains segregate and specific power laws are predicted [136, 137]. In 2D-blends of polymers a particular microdomain morphology may be expected [139]. Experiments on polymers in this area are presently, however, not available on a molecular level. Indications of order on an... 

In the limit of nonvanishing temperatures, simple analytic forms for the w s, such as the ones compiled in Table 4.1, do not exist. Hence we need to resort to a numerical scheme to solve the Euler-Lagrange equations [see Eq. (4.86)]. This can be accomplished by an approach detailed in Appendix D.2.1 that starts from the set of (exact) morphologies M compiled in Table 4.1 at T = 0 as starting solutions for a temperature T > 0. Once convergence has been attained, the algorithm yields new morphologies for this sufficiently... 

However, with incrccising polymer concentration, effective repulsion of segments of different macromolecules should result in a decrease in the molecular coil sizes. This does qualitatively follow from Equations 113 and 114. An increase in polymer concentration for a labelled (eg. deuterated) macroniolecule can be regarded m effective increase in the solvent molecular volume, which lead.s to a decrease in Cm in Equation 114 and, therefore, to a decrezise in a in Equation 113. In the limit W — 0, the labelled macroniolecule finds itself to be among similar ones (in the thermodynamic and morphological sense). In view of Equations 113 and 114, K/ — 0, Cm —> 0 and o 1. 

In Fig. 3 the binodal curves represent the equilibrium phase boundaries and determine the compositions of the phases in equilibrium at any temperature. The spinodals represent the boundaries between totally unstable (inside the spinodal) and metastable (between the spinodal and binodal) homogeneous phases if created. The curves in Fig. 3 represent the variations in composition of the points in Fig. 4 with temperature. Phases with compositions between the binodal and spinodal are metastable and nucleation is required to initiate phase separation. In contrast, compositions within the spinodal are totally unstable and will inevitably imdergo phase separation if there is sufficient mobility to permit the necessary molecular motions. The mechanisms by which phase separation occurs in those two regions are different and give rise to different morphologies in the phase-separated materials. Thus, the curves define phase behaviour at equilibriiun and also define the limits of operation of mechanisms of phase separation, when appropriate. 

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