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E-International Scientific Research JournalISSN: 2094-1749 Volume: 2 Issue: 4, 2010for all. Therefore, the temperature has as effect to reduce the induced force, when onegoes away from the critical point.Seventhly, formula (19) tells us that the reduced force,, is a two-factor scalingfunction of the renormalized distance and the ratio . In addition, it is a universalfunction of these variables, independently on the chemical structure of polymers and colloids.Eighthly, at fixed distance r and very close to the critical pointforce obeys an exponential decay, that is, the inducedWe have used the standard mathematical formula [42]: . The above expression tells us that thetemperature has as effect to ”screen” the induced force at . The correlation lengththen plays the role of the standard Debye screening length [60], but the computed inducedforce is much smaller than the screened Coulombian one.Finally, we give an order of magnitude of the induced force. We consider thehydrogenated polyolefin-deuterated polyolefin mixture incorporating silicon particles ofdiameter, for which the polymerization degree and critical temperature werefound to be [61] : , . We take as values for the inter particle distanceand correlation length : , . For these values, we find that the magnitudeof the force is :. This force must be compared to that relative tothe considered polymer mixture confined to two parallel plates separated by the same distance. The magnitude of the latter is . Thus, the forcebetween parallel plates is two times more important than that between a pair of colloids. Also,the same force must be compared to the electromagnetic one, which is of the order ofdyne (for ).326
E-International Scientific Research JournalISSN: 2094-1749 Volume: 2 Issue: 4, 2010In Fig.1, we report the reduced induced forces,, versus the dimension lessdistance , for two temperatures and . In Fig.2, we superpose tworeduced induced forces relative to two temperatures and away from , asfunctions of the dimensionless distance. These curves reflect the above deductions.ConclusionsWe recall that the purpose of the present paper was the determination of the spatial variation ofthe induced force between colloidal particles immersed in a binary polymer mixture, attemperatures far from the critical one. The colloidal aggregation is due to a preferentialattraction of one polymer by the particles.All calculations were done in the framework of the standard -theory, where the fieldis simply the composition fluctuation (order parameter). Applying this theory and using thecumulant method, we first showed that the resulting pair-potential is the sum of twocontributions, which are proportional to bulk two-point correlation functions constructed withfields and , respectively. Second, for polymer mixtures of high molecular weight, these(mean-field) correlation functions are known exactly, and we found that the expected force canbe expressed in term of the modified Bessel function .When it is adequately reduced, the force becomes a universal two-factor scalingfunction of the dimensionless variables and , where r is the inter particle-distance,is the correlation length andis the renormalized diameter.We emphasize that the force expression we obtained can be extended to two particles ofdifferent radii and . For this case, we state that the new expression would be similar tothat relative to a mono disperse colloidal system, by making the following change of therenormalized diameter : .We note that the above calculations make sense only for relatively high separations,that is for, with ∆ the surface-to-surface distance between two spheres of the same327
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