Thermal composition fluctuations

D. Schwahn, G. Meier, K. Mortensen, and S. Janssen (1994) On the N-scaling of the ginzburg number and the critical amplitudes in various compatible polymer blends. J. Phys. II (France) 4, pp. 837-848 H. Frielinghaus, D. Schwahn, L. Willner, and T. Springer (1997) Thermal composition fluctuations in binary homopolymer mixtures as a function of pressure and temperature. Physica B 241, pp. 1022-1024... 

The thermal composition fluctuations tend to stabilize the disordered phase, giving rise to a renormalized critical temperature. The relation between the mean-field value T and the real Tc is expressed by the Ginzburg rdation [8] ... 

Schwahn, D. and Mortensen, K. (2000) Thermal Composition Fluctuations in Polymer Blends studied with Small-Angle... 

For a multicomponent glass, thermal composition fluctuations exist at temperatures above the glass transition. These tend to be frozen in on cooling. The loss contribution due to scattering from such fluctuations can be described by terms proportional to... 

According to the fluctuation-dissipation theorem the equilibrium mean square deviation of thermal composition fluctuations (5 ) is related to the first derivative of the order parameter with respect to the chemical potential [3,4]. If the order parameter is defined as the composition

conjugate field is represented by the difference of the chemical potentials Afi(= - mb) so the degree of thermal fluctuations is related to dAfi idA/x = d/dA/x) which is equivalent to 1/9 AG, where AG represents the Gibbs free energy of mixing with its natural parameters temperature T, pressure P, and composition This shows how the thermodynamic parameters can be determined from measurements of thermal fluctuations. 

Monte Carlo simulations very early demonstrated the effect of thermal composition fluctuations in low molecular blends. Studies by Sariban et al. [16] exclusively found Ising critical behavior in blends of molar volume up to about 16000 cm /mol and no indications of a crossover to mean field behavior. Such a mean field crossover was later detected by Deutsch et aL [ 17] in blends with an order of magnitude larger chains. These results and the techniques of Monte Carlo simiflations have been extensively reviewed by Binder in [4]. 

From these introductory remarks one already gets the impression about the prominent and diversified role of thermal composition fluctuations on the properties of polymer blends, which will be detailed in the following sections mainly from an experimental point of view. 

The RPA is a mean field approximation that neglects contributions from thermal composition fluctuations and that assumes the chain conformations to be unperturbed Gaussian chains. The last assumption becomes visible from the Debye form factor in the first two terms, which for Vp, = are in accordance with Eq. 7, while the third term involves the FH interaction parameter. 

Fig. 4 Structure factor of binary blend dPB/PS in Zimm representation at different temperatures (top) and pressures (middle), and polymer concentrations (bottom), with the other parameters constant. From the fitted straight lines the susceptibility and correlation length is evaluated. The increase of scattering is caused by stronger thermal composition fluctuations when approaching the critical point...

Equation 12 can also be considered as an Ornstein-Zernicke equation describing the degree of thermal composition fluctuations of correlation length The correlation length is evaluated from V 2 S(0) and becomes infinite at the critical point as described by the scaling law The... 

Thermal fluctuations can be described within the Gaussian approximation at sufficiently high temperatures above the critical temperature. For these situations, the system fulfills the conditions of mean field approximation [9]. On the other hand, thermal composition fluctuations become strong near the critical temperature, leading to non-Hnear effects which asymptotically close to the critical temperature imply that the system obeys the universality class of 3D-Ising critical behavior. Thermal fluctuations are described by the Ginzburg-Landau Hamiltonian which is written as a fimctional of the spatially varying order parameter

[Pg.21]

Linear and non-linear effects of thermal composition fluctuations become visible in a scattering experiment. Within the mean field and Ising regimes the susceptibility S(0) and the correlation length are described by simple scaling laws as functions of the reduced temperature r according to Cr ) with the critical amplitudes C( o) and the critical exponents y(v). The critical exponents y(v) are known to be equal to y = 1 and 1.239 0.003 and V = 0.5 and 0.634 0.001 in the mean field and Ising cases, respectively [66]. The mean field case has already been discussed in the context with Eq. 11. 

Besides temperature and polymer composition, pressure is a third parameter needed to completely determine the thermodynamic equilibrium state of a binary mixture. So far, only a few systematic SANS studies exist for polymer blends in external pressure fields [34-41]. Those experiments were also performed in our laboratory for which a temperature-pressure cell was developed for in-situ investigations allowing pressure and temperature fields between 0.1 < P(MPa) < 200 and - 20 < T(°C) < 200, respectively. A temperature control better than 0.01 K allowed also precise exploration of thermal composition fluctuations near the critical point [34]. 

Also including contributions from thermal composition fluctuations the Clausius-Clapeyron equation for critical polymer blends can be expressed... 

Here we have the situation that one of the (A/B) monomers of the homopolymer and of the copolymer was always deuterated with the same relative amount of deuterium. Under such conditions the structure factor S(Q) measures thermal composition fluctuations with respect to the total monomer fraction, which corresponds to a scalar order parameter represented by the local concentration =

basic thermodynamic features of those systems near their binodal line are well described by the common Landau expansion of the free energy according to... 

Near the Lifshitz line thermal composition fluctuations are expected to become strong over a larger temperature range because of the reduced smTace energy (c2 a I2 = 0), leading to a lower threshold force for thermal fluctuations. On a more abstract level this effect can also be interpreted in terms of a larger upper critical dimension Du = 8 beyond which thermal fluctuations become irrelevant, and Gi is twice as large as for ordinary binary polymer blends [86]. 

Figure 25 displays the effects of thermal composition fluctuations on the inverse susceptibility S(0) for a (PEE PDMS) mixture (sample 10 in Table 2) versus 1/T for different diblock concentrations below the Lifshitz line [48]. The critical temperatures determined from S Ho) = 0 decrease with increasing diblock content in a similar way as shown for the (PB PS) blend (Fig. 23). The = 4.3% sample behaves as a pure blend At high temperatures S (0)...

In this article we considered thermal composition fluctuations in binary polymer blends under various conditions. Samples of critical composition were studied in temperature and pressure fields, with small additions of a non-selective solvent, or in mixtures with a symmetric diblock copolymer with the same monomers as the homopolymers. Blends of critical composition were chosen in order to follow the fluctuations up to the critical point which represents the stabiUty limit of miscibility. The strength of thermal fluctuations is estimated by the Ginzburg criterion which in the incompressible limit follows the universal scaling law l/U and predicts that binary polymer blends... 

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