Semi-Infinite Mixture Approach

The relation for the equilibrium contact angle was given almost two centuries ago by Young [152]  

Here subscripts s as well as 1 and 2 refer to the surface and the phases ( ) and (( 2, respectively, and y refers to interfacial (surface) tension between the i-th and 

Another insight into the wetting phenomena, alternative to that yielded by the contact angle 0, has been given by the profile (] (z) (composition vs depth z) [8,53,61,153,158]. The surface of a two component liquid mixture which favors one of the components will be enriched in that component, A say. When the region far from the surface (bulk region) is occupied by the B-rich phase the surface concentration s (( ls or 2S in Fig. 14) is higher than this binodal value 

The blend composition at the surface )s differs from the bulk concentration (fo,. Similar to the wetting layer (for 4 00=(])1), a surface excess layer is present for the mixture in the one-phase regime and it is defined as the difference between the real profile close to the surface (f (z) and the flat profile kept at bulk concentration (j). It is described by the surface excess z  

Here z(( )00) is the distance from the surface (at depth z=0) to the plateau in composition. The surface enriched/depleted in blend component A is characterized by positive/negative z (see Fig. 15). Relatively large correlation lengths for polymer mixtures (see Sects. 2.1 and 2.2.2) lead to the surface profiles ( )(z) of sufficient spatial extent that may be easily traced by current depth profiling techniques [29]. Surface enrichment has been observed at a free surface [164,165] and at a substrate [92] as well as at an interface between binary blend and a homopolymer [166]. 

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Section 3.1 considers the segregation from binary polymer blends towards external interface of a thin film described in a semi-infinite mixture approach. We relate the segregation with wetting phenomena. The role of a vapor and a gas in a classic formulation of this problem is played by two coexisting polymer phases. 

Most of the experiments reported so far have been performed for blends confined in a thin film geometry, but analyzed with a semi- infinite mixture approach. This is legitimate for thicker layers (Sect. 2.2.4) but it is not proper for thinner films, where both external interfaces of the film have to be considered... 

Since experiments on polymer mixtures in thin film geometry are often carried out in the intention to examine wetting behavior [69,71,81-83], we discuss here also the behavior of < )s as function of (]). In the semi-infinite case, we have a logarithmic divergence when phase coexistence is approached, (cf. Fig. 6b)... 

The structure of the interface formed by coexisting phases is well described by the Cahn-Hilliard approach [53] (developed in a slightly different context by Landau and Lifshitz [54]) extended to incompressible binary polymer mixtures by several authors [4,49,55,56]. The central point of this approach is the free energy functional definition that describes two semi-infinite polymer phases <]), and 2 separated by a planar interface (at depth z=0) and the composition ( )(z) across this interface. The relevant functional Fb for the free energy of mixing per site volume Q (taken as equal to the average segmental volume V of both blend components) and the area A of the interface is expressed by... 

Real polymer mixtures studied here do not form semi-infinite systems but rather they are confined in thin layers bounded by two surfaces. For relatively thick films (for a critical thickness evaluation see Sect. 3.2) the equilibrium profile ( >(z) of the whole film is described separately for each of the two surfaces allowing their independent characterization. This is based on the assumption that the profile ( >(z), describing the segregation to the respective surface, is in equilibrium with the plateau value of < > in the region adjacent to this surface. This approach was justified by theoretical [180, 181] and experimental [182] works on the dynamics of surface segregation, and is used here to focus on phenomena occurring near a single surface. 

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