Discretization chain

In order to obtain solutions with the desired flow properties, shear-induced degradation should be avoided. From mechanical degradation experiments it has been shown that chain scission occurs when all coupling points are loose and the discrete chains are subjected to the velocity field. Simple considerations lead to the assumption that this is obtained when y) is equal to T sp(c-[r ]) (Fig. 18). The critical shear rate can then easily be evaluated [22]. 

The variable q, which is conjugate to the arc length variable s, labels the normal mode. If we work with the discretized chain model (/ variable, instead of s... 

While the number of differently shaped clusters and rings of group 15 elements is quite limited, compounds with values between 6.11 and 6.8 e/atom display an impressive variety of dissimilar discrete chains. In Table 8, chain-containing sohd state structures of group 15 elements are listed. 

The prenyl chain elongation catalyzed by prenyltransferases is quite unique because the reaction proceeds consecutively and terminates precisely at discrete chain lengths according to the specificities of the enzymes. The chain length of products varies so widely that it ranges from geraniol (CIO) to natural rubber (C > 5000). 

It has the effect of subtracting v from the state vector. Thus, starting from an initial state n°, both reactions together cause the state vector to move over a discrete chain of lattice points lying on a straight line between two boundaries of the physical octant. The accessible points are... 

The Edwards Hamiltonian is an appealing but most formal object. To mention a simple fact, shrinking to zero the segment size of the discrete chain model as done in the continuous chain limit, we in general get a continuous but not differentiable space curve. Strictly speaking the first part, of Vj, does not exist. Further serious mathematical problems are connected to the (5-function interaction. Hie question in which sense Ve is a mathematically well defined object beyond its formal perturbation expansion is ari interesting problem of mathematical physics. 

Universality and two-parameter scaling in the general case of finite excluded volume, Be comes about by the much more sophisticated mechanism of renormalization. As will be discussed in later chapters (see Chap. 11, in particular) both the discrete chain model and the continuous chain model can be mapped on the same renormalized theory. The renormalized results superficially look similar to expressions like Eq. (7.13), but the definition of the scaling variables iie, z is more com plica led. Indeed, it is in the definition of R ) and z in terms of the parameters of the original unrenormalized theory, that the difference in microstructure of the continuous or discrete chain models is absorbed. 

The cut off renders the theory finite but ruins the two parameter structure. Indeed, it introduces a new dimensionless variable o/R ) 1/rt, resulting in perturbative expressions which closely resemble those of the discrete chain... 

To show this we recall that the path integral is defined as continuous chain limit of the discrete chain model ... 

We discuss here the basic ideas of the renormalization group, using the discrete chain model. This is not the most elegant or powerful approach, and in Part Til of this book we will present a much more efficient scheme. However, the present approach is conceptually the simplest, and it allows us to explain all the relevant features dilatation symmetry and scaling, fixed points and universality, crossover. Furthermore, technical aspects like the e-expansion also come up. We are then prepared to discuss the Qualitative concept of scaling in its general form and to work out some consequences. 

Is all of our preceding analysis based on an artificial model lacking physical relevance This fortunately is not the case, but to show the validity of our model it needs a more careful analysis of the effects of additional interaction terms. This is the topic of the present chapter. A full analysis is prohibitively complicated and has not been given, neither within the context of polymer theory, nor within field theory. We therefore restrict our discussion to illustrating the basic ideas. As a first step we consider the discrete chain model, pointing out two features ... 

In Chap. 8 we have constructed the renormalization group, starting from bare perturbation theory for the discrete chain model. This expansion involves nonuniversal microstructure corrections which we will now absorb into renormalization factors, introduced via a redefinition of the interaction constant and the chain length, According to Eq. (11.1) we write... 

The theorem of renormalizability can be read in two ways. With the renormalized theory taken to be fixed, it implies the existence of a one-parameter class, parameterized by , of bare theories, all equivalent to the given renormalized theory and thus equivalent to each other. This aspect is related to universality a whole class of microscopic models yields the same scaling functions. In the next chapter we will use this aspect to get rid of the technical complications of the discrete chain model. We can however also interpret the theorem as establishing the existence of a one-parameter class of renormalized theories, all equivalent to a given bare theory. This class is parameterized by the length scale r or the scaling parameter... 

In this chapter we first show that the continuous chain model is renor-malizable by taking the naive continuous chain limit of the theorem of renor-malizability. We then argue that we can construct renormalization schemes for the continuous or the discrete chain models, equivalent in the sense that they yield the same renormalized theory (Sect. 12.1). In Sect. 12.2 we estab-... 

In Chap. 7 we have shown that the bare discrete chain or continuous chain models are naively equivalent only close to the 0-point. We thus might wonder whether the equivalence of the two models, shown above to one loop order, can hold generally. We thus have to show that starting from these different bare theories we nevertheless can construct identical renormalized theories. We consider the renormalized continuous chain limit (RCL), used in the theorem of renormalizability. 

Thus asymptotically the NCL and the RCL, if applied to the bare functions for d < 4, are identical. In other words, for fo < 1 the set of equivalent theories for i —> 0 reaches the 0-region, where the continuous chain model and the discrete chain model coincide. The same renormalized theory can therefore be constructed from both models. 

Concerning the first question we note that the result of any renormalization scheme based on the continuous chain model via a finite renormalization can be mapped on the renormalized theory derived from the discrete chain model, and vice versa. After renormalization the models are completely equivalent. 

We thus take the following attitude. For technical reasons we calculate the renormalized theory starting from the continuous chain model. By equivalence of the bare theories for fo < 1 we know that we can derive the same theory from the discrete chain model. Since the renormalized theory in the way we construct it should show no singularity at u, we can use it for u > u. This region however can be interpreted only in terms of the discrete chain model. 

Starting from the discrete chain model we carry through the following steps. 

We should not close this section without touching upon some delicate technical point. A priori even for a discrete chain the critical chemical potential g s, defined as chemical potential per segment of an infinitely long chain, does not exist outside the -expansion. We already encountered this problem in Sect. 7.2, where we found that rC proportional to p, for d < 4 in naive perturbation theory suffers from infrared singularities. That problem has been considered in the field theoretic framework. The results, expressed in polymer language, show that the u-expansion is not invalidated, as long as we consider quantities which do not involve explicitly. Almost all the quantities of interest to us are of this type. Only in the equation of state relating gp(n) and cp(n) does an explicit contribution g n occur. But even... 

Our discussion of solitons and breathers has been purely classical, since we implied the existence of excitations corresponding to the classical solutions (7.71) and (7.75). To see this correspondence it is natural to invoke the quasiclassical approximation. Dashen et al. [1975] quantized the sine-Gordon problem and showed that in the WKB approximation for a discrete chain with periodic boundary condition the soliton mass is renormalized to... 

Discrete chains of three or four linked SiCU tetrahedra are extremely rare, but they exist in aminoffite Ca3(BeOH)2[Si30io], kinoite Cu2Ca2[Si30io]-2H20, and vermilion Agio[Si40i3]. 

The fact that composite inulin extracts can be fractionated on a commercial scale into reasonably discrete chain length classes allows tailoring of the product to intended market uses. Hence, the chain length condition of the raw product is important in that it is relatively easy to depolymerize inulin to give shorter-chain-length fractions however, current in vivo attempts to create longer-chain-length polymers on an economically viable scale have not been optimistic. 

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