Continuous chain model

The incompressibihty constraint 5(A + < >B — 1) has been explicitly included in partition function (37), and the continuous chain model, Eq. (19), is being used. Q is the partition function of an independent single chain subjected to the fields Ua and Ub, and it can be evaluated exactly. One writes the partition function as Q = f drq(r, 1), where... 

In the continuous chain model, a large part of the deformation during extension of the fibre consists of the shear deformation as a result of which the chain orientation distribution contracts. This leads to the concave shape of the tensile curve, often found for polymer fibres. Therefore, this description of the extension of the fibre implies a strain hardening process. 

Chain stretching is governed by the covalent bonds in the chain and is therefore considered a purely elastic deformation, whereas the intermolecular secondary bonds govern the shear deformation. Hence, the time or frequency dependency of the tensile properties of a polymer fibre can be represented by introducing the time- or frequency-dependent internal shear modulus g(t) or g(v). According to the continuous chain model the fibre modulus is given by the formula... 

The continuous chain model includes a description of the yielding phenomenon that occurs in the tensile curve of polymer fibres between a strain of 0.005 and 0.025 [ 1 ]. Up to the yield point the fibre extension is practically elastic. For larger strains, the extension is composed of an elastic, viscoelastic and plastic contribution. The yield of the tensile curve is explained by a simple yield mechanism based on Schmid s law for shear deformation of the domains. This law states that, for an anisotropic material, plastic deformation starts at a critical value of the resolved shear stress, ry =/g, along a slip plane. It has been... 

Indeed, it has been observed that the onset of yielding of isotropic polymers is approximately constant, 0.02< [<0.025, which implies that 0.04shear yield strain, the plastic shear deformation of the domain satisfies a plastic shear law. For temperatures below the glass transition temperature, the continuous chain model enables the calculation of the tensile curve of a polymer fibre up to about 10% strain [6]. Figure 7 shows the observed stress-strain curves of PpPTA fibres with different moduli compared to the calculated curves. 

Fig. 7 Comparison of the observed tensile curves of PpPTA fibres with three different moduli with the curves calculated with the continuous chain model [6]...

In a further development of the continuous chain model it has been shown that the viscoelastic and plastic behaviour, as manifested by the yielding phenomenon, creep and stress relaxation, can be satisfactorily described by the Eyring reduced time (ERT) model [10]. Creep in polymer fibres is brought about by the time-dependent shear deformation, resulting in a mutual displacement of adjacent chains [7-10]. As will be shown in Sect. 4, this process can be described by activated shear transitions with a distribution of activation energies. The ERT model will be used to derive the relationship that describes the strength of a polymer fibre as a function of the time and the temperature. 

In view of the development of the continuous chain model for the tensile deformation of polymer fibres, we consider the assumptions on which the Coleman model is based as too simple. For example, we have shown that the resolved shear stress governs the tensile deformation of the fibre, and that the initial orientation distribution of the chains is the most important structural characteristic determining the tensile extension below the glass transition temperature. These elements have to be incorporated in a new model. 

Before we enter upon the discussion of the lifetime of a polymer fibre, a brief presentation is given of the creep theory of polymer fibres according to the continuous chain model [7-10]. The total fibre strain is given by the sum of the... 

The total contribution of the shear strain to the fibre strain is the sum of the purely or immediate elastic contribution involving the change in angle, A0e=0o- , occurring immediately upon loading of the fibre at f=0, and the time-dependent or viscoelastic and plastic contribution A0(f)=0(f)-0o [7-10]. According to the continuous chain model for the extension of polymer fibres, the time-dependent shear strain during creep can be written as... 

All this analysis will be based on the mathematically most simple spring and bead model. Standard two parameter theory is based on the continuous chain model, which can be derived from our model by a not quite trivial limiting process. The derivation stresses that, standard two parameter theory is expected to hold only close to the (9-temperature, In this context we also exhibit the relation of polymer theory to a special quantum held theory. 

Continuous Chain Model and Naive Two Parameter Theory... 

Even for d < 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d < 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

We must note also a second important restriction of the continuous chain model. As we will see. by construction it deals with infinitely long chains n — oo. infinitesimally close to the -point , 5C — 0. Thus naive two parameter theory is valid only very close to the -temperature. In later chapters we will see how further renormalization leads to a theory of excluded volume effects valid for all /%, > 0. 

Finally, for completeness in Appendix A 7.1 we consider the formal relation of the continuous chain model to a field theoretic Hamiltonian, used to describe critical phenomena in ferrornagnets. It was this relation discovered by de Genries [dG72] and extended by Des Cloizeaux [Clo75, which initiated the application of the renormalization group to polymer solutions and led to the embedding into the larger realm of critical phenomena. 

The bust line of Eq. (7.9) follows since dimensionless quantities can depend only on dimensionless combinations of their variables. As a result of this simple dimensional analysis in the continuous chain model we have found the two parameter theory instead of the three independent parameters ,3A.nt physical observables involve only the two combinations Rq, z. Perturbation theory now proceeds in powers of 2. Thus the continuous chain limit gives a precise meaning to the simple argument presented in Chap. 6. 

Outside that region the continuous chain model is no valid representation of a polymer chain. (To avoid confusion we should recall that (T -0)/0[Pg.109]

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. 

This transformation leaves both jRq and 2 = Pen 2 (Eq. (7.10)) invariant. It just expresses naive dimensional analysis in the continuous chain model. The power of the RG-approach lies in the fact that we can construct nontrivial realizations. These take into account more than just the leading n-dependence of each order of perturbation theory and therefore obey the condition of invariance of the macroscopic observables up to much smaller corrections. 

We finally note that this discussion gains additional importance with respect to the continuous chain limit. In Chap. we have shown that we can construct the continuous chain model only after an additive renormalization. which essentially extracts a one-body part from the two-body potential. If we... 

This is consistent with our previous discussion (see Chap, 7), where we stressed that the continuous chain model, underlying naive two-parameter theory, is a valid representation of a general polymer model only close to the -point. 

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