Chemically striped substrates

Because we are concerned in this tutorial with the effects of chemical heterogeneity at the nanoscale on the behavior of the confined film, we expect the details of the atomic structure not to matter greatly for our purpose. Therefore, we adopt a continuum representation of the interaction of a film molecule with the substrate, which we obtain by averaging the film substrate interaction potential over positions of substrate atoms in the x-y plane. The resulting continuum potential can be expressed as 

tia = 2/ is the areal density of the (100) plane of the fee lattice. The position of a film molecule is denoted by r, and r = (x, y, z = Sz/2 vi 5t) represents the position of a substrate atom, where refers to the lower k = 1), to the upper (k — 2) substrate, and 6i is the spacing between successive crystallographic planes in the z direction. We note that, because all features of the substrate at the atomic scale have been washed out in 1 1, our continuum model cannot account properly for solid fonnation, which, as mentioned briefly at the beginning of Section 5.4, is strongly influenced by the atomic structure of the substrate. 

The remaining integration over x cein also be carried out analytically (see, for example. No. 244 in Ref. 141). A tiresome computation yields 

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If the external potential is noneonstant aeross the. t-j/ plane but varies periodically along the x-axis, say. as the one describing the chemically striped substrate surfaces depicted in Fig. 5.7, the symmetry of the fluid is reduced even further. This causes the equation of state to depend on even more parameters as in the previously discvvssed case. This can be realized from Eqs. (1.66) and (168), which permit us to derive yet another Gibbs-Duhem cqiiation, namely... 

For the chemically striped substrate, the external field representing the composite solid material can be cast as... 

As a first illustration we consider the model discussed in Section 1.3.3, namely a fluid of simple molecules confined between chemically striped solid surfaces (see Fig. 5.2). As before in Section 5.4 we treat the confined fluid as a thermodynamically open system. Hence, equilibrium states correspond to minima of the grand potential 11 introduced in Eqs. (1.66) and (1.67). The fluid fluid interaction is described by the intermolecular potential ug (r) introduced in Eq. (5.38) where the associated shifted-force potential is introduced in Eq. (5.39). The fluid substrate interaction is described by 1 1 (x, z) in the continuum representation [see Eq. (5.68)], where x replaces x because of the misaligmncnt of the sul)stratcs relative to each other [see Eq. (5.103)]. 

In 2000, Pereira et performed the earliest 3D simulation of a block copolymer in the presence of combined chemical and topographical patterns. A symmetric AB diblock was simulated over a substrate patterned with wide chemical stripes, where the film thickness was either uniform or increased over the stripe. The lamellae parallel to the substrate exhibited a kink that allowed one microdomain to be continuous across the thicker region. 

The above considerations may be extended to the situation of primary in-tert t here The fltiid is constrained in one dimension z) by plane-parallel substrates that are chemically decorated with weakly and strongly adsorbing stripes. These stripes alternate periodically in the x-direction, so that the external potential depends only on Xj and Zj (see Eq. (4.52), Fig. 4.8). Thus, for a given value of Xj and z the occupation numbers do not vary with the V-coordinate of the lattice site. That is, by symmetry all densities along lines parallel with the j/-axis are equivalent. Thus, using Eq. (4.86) we can write for a particular site i... 

Fig. 11 Tapping mode AFM image of polymerized vesicle stripes deposited on glass substrates with pCP. Reprinted with permission from [94]. Copyright 2005, American Chemical Society...

Fig. 20 Multilayer deposition on a hydrophilically/hydrophobically patterned gold substrate. Upper AFM images height (left) and friction force (right) images of patterned methyl- and hydroxyl- alkanethiol self-assembled monolayers. Adsorption of poly(ferrocenylsilane) polyions (5/7, 12 bUayers) occms selectively on the broad methyl-terminated stripes (lower AFM images). Reprinted with permission from [94]. Copyright 2002, American Chemical Society...

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