Des Cloizeaux theory

The theory based on this assumption is expected to give zeroth order predictions of global polymer properties, and is usually called the blob theory. However, this nomenclature does not seem relevant to the author (see Section 1.4). Hence, in this book, we call it the Weill-des Cloizeaux theory. 

The Weill-des Cloizeaux theory has acquired considerable popularity in recent years, because its basic assumption (eq 1.4) permits analytical calculation of various polymer properties in non-0 solvents. However, we should not expect it to be more than a semi-quantitative theory. For its improvement Francois et al. [3] proposed allowing y in eq 1.2 to vary with i — j according to either... 

The Weill-des Cloizeaux theory assumes that no expansion takes place within subchains shorter than aN. On this assumption we may consider a chain model which consists of Gaussian subcoils, each being made of Nc beads and interacting with one another. With such a subcoil viewed as a blob, this model is often referred to as the blob model. However, it is important to recognize that the Weill-des Cloizeaux theory concerns interactions between beads but not those between blobs. Therefore, contrary to many other authors, the author does not consider it relevant to call it the blob theory. 

When Nc is set equal to zero, the Weill-des Cloizeaux theory appears to reduce to the earlier e theory of Peterlin [7] and Ptitsyn and Eizner [8] (see... 

X. SUN, B. FARNOUX, G. JANNINK AND J. des CLOIZEAUX Theory of Dynamic Screening in Macromolecular Solutions 269... 

Freed KF (1987) Renormalization group theory of macromolecules. John Wiley, New York des Cloizeaux J, Jonnink G (1990) Polymers in solution Clarendon Oxford... 

Modern theories do not solve analytically the excluded volume problem, instead they pursue the same problem from different angles. This is because the excluded volume problem is placed as a category of typical many body problems [36], hence an intractable one. In spite of such a situation, there is an approach to persistently seek closed solutions of the excluded volume chain. Following this trend, some empirical formulations have been put forth for the limiting case of N—>°° [37-41]. The most successful one is that of the des Cloizeaux type equation, written by the form [41] ... 

In the subsequent 20 years (1960-80), the main principles of modern polymer physics were developed. These include the Edwards model of the polymer chain and its confining tube (Chapters 7 and 9), the modern view of semidilute solutions established by des Cloizeaux and de Gennes (Chapter 5), and the reptation theory of chain diffusion developed by de Gennes (Chapter 9) that led to the Doi-Edwards theory for the flow properties of polymer melts. 

In fact, the decisive experimental proof of the macromolecular hypothesis has been given by osmotic pressure measurements in very dilute solutions. A second fundamental contribution is more recent. It deals with the study of systems of strongly overlapping chains. The observations made by Noda and his collaborators (1980) definitely showed that, in this case, the osmotic pressure dependence of the solute concentration follows a universal law the crossover with the dilute state is also universal. These results gave an experimental proof of the principles which constitute the basis of the modern theory introduced by de Gennes and des Cloizeaux. 

This property of Edwards s approximation is also a result of the more exact theory established by des Cloizeaux, but only in the domain where the theory applies (see Chapter 13). It can be interpreted by observing that the excluded volume interactions are screened in the semi-dilute regime. This effect thus diminishes the probability of contact between segments. 

The introduction of many fugacities is a source of complications which one would like to avoid. Thus, in order to apply field theory and renormalization principles to polymer solutions, des Cloizeaux used a simpler ensemble2 admitting that for long polymers, this ensemble has the same scaling properties as the exact grand canonical ensemble. 

However, the principles and the techniques of renormalization theory are not directly related to the existence of fields. They apply whenever one deals with a critical system, i.e. whenever one has to describe large-scale phenomena which depend only globally on the chemical microstructure. Thus, because an ensemble of long polymers in a solution constitutes a critical system, renormalization principles and renormalization techniques must be directly applicable to their study. Actually, this idea appeared quite naturally. It led to the decimation method which has been described previously and which lacks efficiency. However, the same idea can be applied in a much better way. This direct renormalization method (des Cloizeaux 1980)37,38 consists in adapting to polymers methods which had been successful in field theory.39 In other words, the aim is to bypass the Laplace de Gennes transformation (see Chapter 11). This method applies to semi-dilute solutions as well as to dilute solutions. 

Finally, we touch upon des Cloizeaux s theory [53], which dealt with W,j(R) in the limit of R 0 for subchains in an infinitely long chain. It derived 5 = 2.273, 2.46, 2.71 for cases I, II, IB, respectively. The s value for case I is very close to, but those for cases II and HI are somewhat smaller than... 

This formula indicates that for 3-dimensional chains becomes a universal function of as in the two-parameter theory, in the Douglas-Freed scheme and under the condition (27rL/A) > 1. It predicts 0.269 for at the selfavoiding limit ( = oo). This limiting value is in accordance with 0.268 by Witten and Schafer [12] and 0.269 by des Cloizeaux [13] from different RG calculations. Though slightly laiger than the experimental estimates 0.22 — 0.25, it represents a remarkable success of the RG theory. 

To calculate k) theoretically we must determine rj(r) by solving the equations of motion for the chain. Akcasu and Gurol [41] in 1976 attacked this on the basis of the Kirkwood diffusion equation [42], and Akcasu et al. [43] presented a more general theory in a review article of 1980. In what follows, without going to mathematical details, we summarize some important results on for Gaussian chains. Benmouna and Akcasu [44] and Akcasu et al. [43] extended the calculation of to non-Gaussian chains by invoking the Weill-des Cloizeaux approximation, eq 1.4. However, as mentioned in Section 1, this approximation seems too crude to explore excluded-volume effects on Q, quantitatively. 

Using a very sophisticated method based on field theory, des Cloizeaux [3] showed that Ilr(c) in the semi-dilute regime can be expressed in the form... 

When the actual experimental temperature used is equal to 6, xi = 1/2, at which point all excess contributions to the solution thermodynamics disappear and the solution exhibits ideal behaviour since the second virial coefficient has a value of zero. At this point the excluded volume effects that cause an expansion of the polymer molecule are exactly balanced by the unfavourable polymer-solvent interactions and the molecule adopts imperturbed, random walk dimensions. The influence on polymer dimensions and the highly detailed theories of polymer configuration in relation to the excluded volume parameter are beyond the scope of this book but are extensively covered by Yamakawa (1971) and to some extent by des Cloizeaux and Jannink (1990). 

If the treatment was limited to these two determinants with a common set of 3sci3pci atomic orbitals (AO) taken from either the Cl- or Cl Hartree-Fock (HF) calculation, the treatment would be quite incorrect and would predict an erroneous curve crossing distance and avoidance. But one may use these two determinants to define a 2 x 2 model space and apply the theory of effective Hamiltonians, as suggested by Levy " (with a slightly non-orthodox definition of the effective Hamiltonian). One may use either the Bloch or des Cloizeaux definition of /f " as a 2 x 2 matrix, the eigenvalues of which are the exact adiabatic eigenvalues... 

Figure 4.1. Graphs of the Green function (a) and of the ordered Green function (6) for Lagrangian 4.2 8 graphs of polymer theory (c) (des Cloizeaux, 1975) [Reprinted with permission from Des Cloizeaux. J. de Phys. 36 (1975) 281-291. Copyright 1975 by EDP Sciences]...

For this behaviour of physical quantities, the statistical integrals must be renormalized (once again ), and des Cloizeaux assumes that only few renormalization f2ictors are enough for this. By analogy with field theory, these renormalization factors arc associated with the end effects, i.e. with the vertex insertions (see section 2.6). 

To compare experimental results with theory, first, the quantities 6, c, and h should be determined. The parameter b has been obtained from coil swelling data at a temperature above as against the theoretical curve of swelling (des Cloizeaux ct al., 1985)... 

Figure 5.50. X values normalized to I at Ti = 308K the crosses stand for experimental values, the straight lines stand for theory (Duplantier et al., 1986) [Reprinted with permission from I).Duplantier, O.Jannink, J. des Cloizeaux. I liys. Rev. lett. 56 (1986) 2080-2083. (Vpyriglit 1986 by the Anmerican Phy.sical Sticicty)...

A version of the direct renormalization method in the polymer theory was proposed by des Cloizeaux. 

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