Fixman theory

Figure 4.37. Comparison between the present theory (dashed), the Fixman theory (solid curves) and the exact numerical results by DeLacey and White (diamonds) Ka = 20, a = 200 nm, Ff /RT = 4. Left excess conductivity per unit volume fractions of the colloid right, excess dielectric loss.

When flexible coils are dissolved in good solvents, the theoretical formulation of kjy is more difficult. The Pyun-Fixman theory represented by Equation 69 indicates kt should increase monotonically from a value around 2.23, corresponding to theta solvents, to the hard sphere value 7.16, as the excluded volume effect increases. Yamakawa notes (38) that the Pyun-Fixman equation can be put in the approximate form ... 

Equation 73 satisfactorily explains frictional coeflBcient data for PS in a variety of solvents (87). The thermodynamic contributions to kjy are modelled quite well using the two-parameter theory of polymer solutions (87). Recent QLLS experiments on PS/tetrahydrofuran solutions are also in accord with the Pyun-Fixman theory (37,44), However, other studies of the frictional coeflBcient of poly-a-methylstyrene in trans-decalin and toluene indicate that the Pyun-Fixman theory but not the Yama-kawa theory fits the data at small excluded volumes (88). Neither theory works well at large excluded volumes (88). 

In Figure 2 is shown the calculated curve of the ratio [n]/ [n]o at various values of shear stress for the bead-spring (Fixman) theory with an = 1.00 and a, = 2.5. It is apparent that Fixman s theory without any adjustable parameters can explain the data at least qualitatively, for an = 2.5. This bead-spring model can give a reasonably accurate representation of the specific shear stress at which [n] first decreases, and the rate of decrease with shear stress is also predictable. 

The Fixman theory was quite heuristic and bears little resemblance to later versions of mode-mode coupling. The theory was based on macroscopic hydrodynamic considerations of energy dissipation in a sound wave. The theory makes beautiful physical sense, but does not concern itself with the classical linear formalism that we have just discussed. Thus, the Fixman theory allows no determination of why our assumptions about K Xt) break down. Understanding of this point was achieved via the more formal theories of Kawasaki and of Kadanof and Swift. ... 

Spectroscopic Methods, [Biological] Applications of Spectroscopy, EPR, Recent Advances in (Smaller). Spectroscopy, Infrared, Use in Biology (Lecomte). Spectroscopy of Transition-Group Complexes (Jorgensen) Statistical-Mechanical Theory of Transport Processes. X. The Heat of Transport in Binary Liquid Systems (Bearman, Kirkwood, Fixman). ... 

Bearman, Kirkwood, Fixman). Transport Processes, Formal Statistical Theory of 1 1... 

In both this theory and that presented by Casassa, the differences between linear and branched polymers are ultimately referrable to the fact that the branched polymers have not only a lower mean-square radius for a given MW, but also a higher mean segment density, and a considerably higher central segment density, for a given mean-square radius the importance of this was pointed out by Stockmayer and Fixman in 1953 (2). Otherwise, the two treatments give different reasons for the lower 0Al of branched polymers, for the expression (6.9), used by Casassa, implies that a and b of Eq. (6.12) are zero. 

This discussion highlights the difficulty of deciding at what separation A and B form an encounter pair and then whether this reacts or separates. Noyes [5] and Wilemski and Fixman [51] have taken the encounter distance to be that separation which, if reduced slightly, will lead to reaction. Where these authors disagree is that Noyes [5] only allows reaction to occur in a very narrow range of separation distances about R (which is the usual assumption) and Wilemski and Fixman [51] assume that any separation distance less than the encounter distance, R, can lead to reaction between A and B and that A and B can diffuse through each other till their centres of mass coincide (Chap. 9, Sect. 4). Neither assumption is good, but the differences in predicted rate coefficients are so small that an experimental test of these theories could not be definitive. 

The first reasonably successful theory of diffusion-limited chemical reactions which specifically endeavoured to develop a model that could described, in principle, the competitive effect was introduced by Wilemski and Fixman [51], They considered the fluorescence of a species A which can be quenched by natural decay (lifetime t) and by a quencher, Q, of concentration [Q]... 

In the previous chapter, the theory developed by Wilemski and Fixman [51] is discussed in some detail (see Chap. 9, Sect. 4). While there are a number of reservations about this approach to describing diffusion-limited reaction rates (see Chap. 9, Sect. 4.3), it is very useful analysis because it is capable of further refinement. A most interesting case in point is the variation analysis by Doi [485]. This section discusses his analysis in more detail. 

Williams begins with Fixman s equation (220) for the stress contributed by intermolecular forces in flexible chain systems. The theory assumes that the polymer concentration is high enough that intermolecular interactions control the stress. The shear stress contributed by polymer molecules in steady shear flow is expressed in the form... 

Fiir Knauelmolekiile im Theta-Zustand existieren je nach Theorie eine groBe Zahl numerischer Werte fiir Sie werden in der Regel auf den Fadenendenabstand bezogen. In neueren theoretischen Arbeiten werden dabei fur Werte von 2,8 1021 (Auer u. Gardner, 1955) 2,84 1021 (Zimm, 1956), 2,87 1021 (Kurata u. Yamakawa, 1958), 2,86-1021 (Eizner etal, 1963) und 2,68-1021 (Pyun u. Fixman, 1965) in 100 (mol Makromolekiil)-1 angegeben. In der Literatur wird jetzt meist ein Wert von 2,87 1021 [100 (mol Makromolekiil)-1] benutzt. 

Unfortunately, Fixman has not yet given a value for the reduced steady-state shear compliance. However, from a comparison of eqs. (3.60a), (3.64) and (3.66) the impression is obtained that the theory of Ptitsyn and Eizner overestimates the influence of the excluded volume on 0 and JeR. As will be shown in the experimental section of this chapter, this impression is supported by flow birefringence measurements on solutions in 0- solvents and in good solvents. 

Some additional remarks should perhaps be made with respect to Fixman s theory (107). This theory provides a straight foreward explanation of non-Newtonian intrinsic viscosity in terms of excluded volume. According to this theory, the behavior of a solution in an ideal solvent is almost Newtonian [cf. a similar result derived by Subirana (108)]. In fact, the influence of excluded volume on the viscosity must decrease with increasing shear rate due to an increasing probability for elongated... 

There have been several treatments to calculate correlation functions and the transport coefficients near the critical point (Fixman [35], Kawasaki [36] and Kadanoff and Swift [37]). All these treatments embody essentially the same physical ideas and contains the genesis of the modem mode coupling theory. Here we discuss the treatment of Kadanoff and Swift [37] because this is physically the most transparent one and seems to have influenced the latter development of the mode coupling theory in a more significant manner. 

Here Lw is the contour length of the chain. In an effort to generalize Eq. (14) for non-theta solutions, Reed et al. [46] have provided an ad hoc treatment by combining theories of Odijk [35], Odijk and Houwaart [36], Skolnick and Fixman [37], and Gupta and Forsman [49]. Here they append an additional contribution f30 to A arising from short-ranged non-electro-static interactions, so that f3 of Eq. (11) is given by... 

In the second place, the maximum in u(f) for xa S 3 is striking but theoretically well-established it is also predicted by Dukhin s, Fixman s and, less explicitly, by Wiersema et al. s theories. On first sight such a maximum is counterintuitive how can a particle move slower when its electroklnetlc charge is Increased It must mean that, upon increasing f, the driving force increases less than the braking forces. The former is proportional to the latter, force of sec. 4.3a (1), is proportional both to cr and to the electric field produced by the polarized double layer, which is also hence the brake scales as... 

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