Fluctuation corrections

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

The results of models that include tube length fluctuation modes [Fig. 9.23(b)] are in much better agreement with the experimentally measured loss modulus G" (ic) of monodisperse melts than the prediction of the Doi Edwards reptation model [Eq. (9.83)]. Tube length fluctuation corrections predict that the loss peak broadens with decreasing molar mass because the fraction of the stress released by fluctuations is larger for shorter chains. 

Fig. 42. Theoretical phase diagram for diblock copolymers in the weak segregation limit The left side shows the mean field result of LeiUer [43], the right side the theory of Fredrickson and Jfelfand [58] which includes fluctuation corrections, for an effective degree of polymerization N = 104. LAM, Hex, BCC denote the various mesophases lamellar, hexagonal (Le., cylindrical morphology) and body-centered cubic (i.e., spherical micellar morphology). From Bates and Fredrickson (39)...

Another approach [64,338] combines the Hartree fluctuation corrections of the Fredrickson-Helfand theory [58] with contributions from multiple harmonics in the concentration expansion, chosen compatible with the considered... 

Table 8 Calculated direct Jd(K) and super-exchange Jse(K) coupling constants, fluctuation-corrected mean field Neel temperature, (K) and Bloch constant in MnO...

Because fluctuations become large at the critical point, the simple, mean-field theory used here breaks down. Large fluctuations mean that the approximation, JijSiSj -> JijSi( ), used to simplify the partition function, Eq. 0-65), is no longer valid since the local value of the concentration is no longer approximately given by the average concentration. Even if these fluctuations are included as corrections to the mean-field approximation, the theory becomes quantitatively inaccurate near the critical point. A detailed theoretical treatment of these critical phenomena is outside the scope of this book (see for example Ref. 24). However, analysis of both simple mean-field theories plus their fluctuation corrections includes most of the important physics and provides a guide to when one must include more sophisticated treatments very close to the critical point. 

Experiments of Roe et al. [26] and Hashimoto et al. [27] demonstrated that scattering experiments on disordered block copolymers may also be used to determine x I parameters, using the RPA theory of Leibler [9]. In a subsequent paper, Fredrickson and Helfand showed that fluctuation corrections to the RPA are important in block copolymer melts [28]. When available, x parameters obtained from block copolymer melts are reported after fluctuation corrections have been incorporated, k values obtained from block copolymers are often [29,30] but not always [31] larger than those obtained in homopolymer blends. 

Notice that the force does not vanish for 7 0, which is the hallmark of a buckling transition. Hu et al. [116] also estimate the fluctuation correction on this result and find it to be very small they apply this method to four different membrane models, ranging from strongly coarse grained to essentially atomistic, and argue that it is reliable and efficient. 

The R-MPY/HTA predictions for the symmetric thread blend contain local fluctuation corrections associated with the reference blend correlations. For example, the renormalization ratio in the N oo limit is [68] ... 

As a consequence, the structure factor no longer diverges at the MST, but reaches a finite value, leading to a first-order phase transition also for the symmetric diblock copolymer. Moreover, there is a finite composition region where a direct transition between disordered and lamellar phase is predicted and the fluctuation effects disappear for infinite large N. However, it has been noted that the fluctuation corrections are only valid for N > 10 (90). 

Figure 5 shows the structure factor from both mean-field theory (Leibler) and the fluctuation correction (Fredrickson-Helfand) for a symmetric diblock... 

Fig. 6. Phase diagram of a diblock copolymer according to Leibler s theory (left) and including fluctuation corrections according to Fredrickson and Helfand (right). From Ref. 91. Copyright (1990) American Institute of Physics.

The fluctuation corrections to the free energy of the solution of weakly charged polyelectrolytes with / 1 can be neglected and the terms in the brackets on the rhs of eqn [175] are sufficient to determine the stability region of the polyelectrolyte solution. The spinodal of the polyelectrolyte solution is given by the following equation ... 

Since PV is of order N, the correction term in (68) is of relative order. In the generalized canonical ensemble, fluctuation averages are related to thermodynamic functions, as in (48) and (49). Such results, together with the fluctuation correction equation (64), can be used to evaluate fluctuation averages in the MD ensemble. Two results of interest are. 

Within the compressibility route to the thermodynamics, the above two closures yield predictions that miss entirely the blend composition-dependent fluctuation corrections obtained numerically in the preceding subsec-... 

Analytic expressions for F are available for Gaussian chain models. " The self-consistent fluctuation correction arises from the (finite size) coupling of local, thermally driven changes of gAa(0 with the microdomain scale concentration fluctuations. Such a coupling is mediated by the chain and block connectivity constraints, and vanishes if the HTA becomes exact, that is, This condition can be achieved... 

Beside this fluctuation corrected mean field theory, an exact model has been developed by Pommier and Prost [105] in the limit of an infinite number of components n... 

A further consequence results when we note that x should simply be proportional to e = tAs/ksT for this model one then predicts that should simply decrease linearly with the product eN, and then determines the microphase separation transition temperature. In this section we have redefined the meaning of e, in order to have a notation consistent with the original literature.Figure 7.37 demonstrates that there is pronounced curvature on a plot of NS n q ) versus eN, rather than a linear decrease. This behavior is reminiscent of both experimental findings and theoretical predictions due to Fredrickson and Helfand, who have taken fluctuation corrections to the Leibler theory... 

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