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E-International Scientific Research JournalISSN: 2094-1749 Volume: 2 Issue: 4, 2010Here, r denotes the distance from the centre of the sphere. In these relations, and are thescaling dimensions of fields and(5)With the space dimension d, and and are the standard critical exponents of Ising-likemagnetic systems [40,41]. Using an adequate special conformal transformation, it was found[18,19] that the profiles relative to the one-sphere problem may be related to those in a halfspacebywhere z is the perpendicular distance, and and are those amplitudes appearing in Eqs.(4a) and (4b). This means that, for the determination of the value of these amplitudes, it will besufficient to consider a polymer mixture occupying a semi-infinite space limited by a planesurface. On the other hand, and are the amplitudes, at small-distances compared to thebulk correlation length ξ, of the (connected) two-point correlation functions,(The subscripts b is for bulk and c for connected).We recall that, as demonstrated by de Gennes [54,57], the polymer mixtures can becorrectly described using mean-field theory. This is true only for those polymer mixtures ofhigh-molecular-weight. Indeed, the critical region around the critical point, where a non-320
E-International Scientific Research JournalISSN: 2094-1749 Volume: 2 Issue: 4, 2010classical behaviour is expected, is very narrow (of the order of, so-that the mean-fieldtheory can be applied. Therefore, close to, one observes trivial critical exponents, and inparticular,. The critical behaviour of a free binary polymer mixture emerges inthe limit (4 being the critical dimension of the system), and then, and. By a simple map between the sphere and a semi-infinite space, we obtainFor the remaining amplitudes, we simply sketch the resultsTo derive the second relation in Eq. (9), we have used the Wick theorem [40,41].Combining relations (3), (8) and (9), we find the desired expressionsThese relations show the explicit dependence of the surface microscopic parameters uponcolloids geometry, via their radius R, and chains characteristics, through the monomer size aand the polymerization degree N.We finish this section by the following useful remark. When one is concerned withphenomena occurring at low temperature , the field must be naturally written aswhere the new field is the fluctuation of the order parameter around its bulk value321
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