Bare Surface Free Energy

A surface energy difference Afs and (a surface tension difference Ay) between two pure blend components may be evaluated based on a (-dfs/d( )s vs (] s relation calculated for measured segregation isotherm data  

The surface energy difference may also be expressed by a dimensionless parameter Xs defined per lattice site. With a lattice site size taken as 21/3 and the corresponding surface area A=Q2li occupied by one lattice site we may relate the surface energy difference parameter %s with Afs as (see Eq. 24)  

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Note (from the relations between Hx, Z, p and g) that a useful order of magnitude of p and g is order 1/N, because only then is Hx (a field normalized by temperature) of order unity this is already clear from Eqs. (7) and (9), because f(<()) is of order 1/N, and in order to have a non-trivial competition between bulk terms and the bare surface free energy the latter also should be small of order 1/N, for the treatment in Eqs. (24)-(37) to be valid. Since g is of order 1/N, we write it as g=g /N, g being a constant of order unity, and thus Z=-2 N/9g. Now the condition Z /D=1, where crossover from Eq. (36) to Eq. (37) occurs, implies a crossover at a large thickness Dcross... 

When we now consider a thin film of thickness D, Eq. (41) must be supplemented by boundary conditions of the same type as in the polymer blend case, Eqs. (7) and (10), i.e. we add a (bare) surface free energy contribution to the free energy that accounts for preferential attraction of one kind of monomers to the walls, missing neighbors in the pairwise interactions, and possible changes in the pairwise interactions near the surface. As in the blend case, this surface contribution is taken locally at the walls only and expanded to second order in the local order parameter /(z). Per unit area of the wall, this free energy is written as... 

Enthalpic Contributions to "Bare"Surface Free Energy fs... 

The Cahn analyses already performed for early surface enrichment observations [167,170] have shown that the yielded relation of the surface energy derivative (—dfs/dc]))s vs (]>, cannot be described by the linear form suggested by simple arguments presented above. While it can be argued that the more sophisticated enthalpy-based models (e.g., [185]) might eventually account for these discrepancies, the relation (—dfs/dc]))s has been interpreted with the Cohen and Muthuku-mar model [183] instead. This model considers entropic effects due to the restriction of the configuration of polymers in the vicinity of an external interface (surface) and finds the additional entropic contribution to the bare surface free energy fss to be equal to... 

Fig. 20.a Results of the Cahn construction performed for the segregation data [16] of Fig. 19. Composition derivatives of bare surface free energy (-dfs/d( ))s calculated for different temperatures (symbols A, , O, and for T=99,142,165, and 184 °C, respectively) are fitted well by dashed lines, generated by the function (pf+g /ll+Y s). The hatched area marks the surface energy difference -Afs. b Surface energy derivatives (—dfs/d( >)s (dashed lines) and trajectories -2kV< ) (solid lines) plotted for T=99 °C and 184 °C. For T= 184 °C the surface boundary condition (Eq. 26) is met at point at ( >s>( >2, indicating complete wetting regime. If (—dfs/d([ )s was independent of temperature (and equal to that found at 184 °C) then the boundary condition (O) at 99 °C would correspond to partial wetting (c >s<( >2). In practice, however, (—dfs/d([ )s varies with temperature and the real boundary condition at 99 °C ( ) indicates complete wetting again...

A conventional understanding of the surface segregation from polymer blends is that the surface should be enriched in the component with lower bare surface free energy fs, regardless of the value of bulk composition This is however true only when (-dfs/d(j))s does not change its sign when surface concentration is varied (see Fig. 14b). For such blends, surface enrichment in the same... 

Fig. 32a-c. A thin blend film bounded by antisymmetric surfaces exerting opposing fields [93] a Cahn construction with trajectories -2kV( ) (solid and dashed lines) plotted for Ap= 0 and (here) for c —The bare surface free energy derivatives (—dfsL/dcf))s and (+dfsR/d( >)s due to left (L) and right (R) surface are marked by dotted lines. Boundary conditions (Eq. 51) are met at points 1 and 2 b the profile 1-2 as determined by Cahn construction (a) for a rather thick (due to the limit (ft,-Hfi) hlm c the corresponding bilayer equilibrium morphology with the interface between phases and < >2 running parallel to both surfaces... 

Here the bare surface free energy / = — Hi quadratic function of the local surface concentration < >i, Hi g are phenomenological coefficients. This ansatz can be justified as a Taylor expansion in the case where 4>i 1 or where 1 — < )i 1, respectively [125], or alternatively by... 

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