Critical temperature and concentration versus molecular weight in polymer blends: Conformations, fluctuations, and the Ginzburg criterion

Abstract

We use the field-theoretical methods to derive the self-consistent one-loop equations (Hartree approximation) for the location of the critical point in binary mixtures of polymers. The small parameter in the loop expansion is the inverse of the square root of the number of segments (monomers) in a polymer chain 1/√N. Both the ideal chain conformation and fluctuation corrections to the Flory–Huggins mean field result are taken into account. For symmetric mixtures, the critical temperature Tc is shown to deviate from its mean field value (∼N) as √N, while the critical concentration remains unchanged in comparison to its mean field value φ̄c=1/2. Since the fluctuations tend to disorder the system, the real value of the critical temperature is lower than its mean field value. For asymmetric A,B mixture (NA≳NB), the critical concentration of A monomers φ̄c is shown to be larger than its mean field value. In the limit of NA→∞, the critical temperature attains its mean field value, even for finite NB. However, we estimate that for NA=NB∼104, the correction to the Flory–Huggins parameter at the critical point may still be of the order of 10%, although it depends on the details of the system. The explicit formulas for Tc and φ̄c as functions of NA and NB are given. The role of the upper wave vector cutoff in these formulas is emphasized and its proper estimate is given. The loop expansion, viewed as an expansion in a small parameter, is correct as long as both NA and NB are much larger than unity and breaks down when any of these quantities becomes of the order of unity. The calculation of the Ginzburg region in the temperature-concentration plane is also given. It is based on the analysis of the scattering intensity. The comparison with the earlier estimates of the Ginzburg criterion (for temperature) is made.