To say nothing about the truly atomistic models, every flexible macromolecule can be universally presented as consisting of z freely jointed rigid segments (Kuhn segments, see Section 1.1) – this is considered as a microscopic ap-proach. So, the system of entangled macromolecules can be imagined as con-sisting of nz interacting segments, every z of them being connected in chain, and the basis heuristic model, kinetics of which has to be investigated, is a system of interacting rigid segments connected in chains. In other words, it is a system of interacting Kuhn-Kramers chains. The system is dense, the interactions are strong, and it seems to be a rather complex problem, which has not solved yet,1 so that one has to look for more coarse approximations to describe dynamics of this system. One can use the coarse-grained Gaus-sian model for every macromolecule (bead-and-spring model, see Section 1.2) to describe the behaviour of the system. This is a heuristic model easier to consider. To formulate an equation for the large-scale stochastic dynamics of the entangled system as dynamics of interacting chains of Brownian parti-cles, one can consider a system of interacting segments (atoms) and follow Zwanzig-Mori method (Zwanzig 1961; Mori 1965), described, for example, in monographs (Hansen and McDonald 1986; Boon and Yip 1980). There is no available solution of the problem; nevertheless, one can easily imagine a general form of the anticipated results.
3.1.1 Dynamics of Entangled Course-Grained Chains
When one is interested in slow modes of motion of the system, each macro-molecule of the system can be schematically described in a coarse-grained way as consisting of N + 1 linearly-coupled Brownian particles, and we shall be able to look at the system as a suspension of n(N + 1) interacting Brownian particles. An anticipated result for dynamic equation of the chains in equi-librium situation can be presented as a system of stochastic non-Markovian equations
md2raα dt2 =−
∞
0
Baα,bβ(s) ˙rbβ(t− s) ds − ∂U
∂raα + φaα(t),
a = 1, 2, . . . , α = 0, 1, 2, . . . , N, (3.1) where raα is a co-ordinate of a particle (where a is the label of the macro-molecule to which the Brownian particle belongs, and α is the label of the particle in the macromolecule), m is the mass of a Brownian particle associ-ated with a section of the macromolecule of length M/(1 + N ). The potential U (raα) depicts interaction of a particle raα with particles of its own and
1 Curtiss and Bird (1981a and 1981b) have posed and considered such a problem.
3.1 Admitted Approximations in the Many-Chain Problem 39
the other macromolecules. The integral term on the right is the friction force (external and internal resistance), determined through the memory matrix Baα,bβ(s), by all the Brownian particles in the system. The properly derived memory function must be expressed in terms of interaction between segments and includes relaxation time which, generally speaking, cannot be neglected.
One can think that this situation, described by equations (3.1), can be visualised as a picture of interacting (and connected in chains) Brownian par-ticles suspended in anisotropic viscoelastic ‘segment liquid’. Introduction of macroscopic concepts is unavoidable consequence of transition from micro-scopic to mesomicro-scopic approach, or better to say, from the micromicro-scopic model of interacting Kuhn-Kramers chains to mesoscopic model of interacting chains of Brownian particles.
Up to now no specific results for memory function and effective potential in equations (3.1) are available,2so that, to simplify the system (3.1), one has to make some suggestions, two of which are intensively exploited.
Course-Grained Interacting Chains in Non-Relaxing Medium The situation looks simpler, if one assumes that relaxation times of the sur-rounding can be neglected, and one obtains for the collective motion of the entire set of macromolecules, considered as a set of Brownian particles, a system of stochastic Markovian equations
md2raα
dt2 =−ζ ˙raα− 2μT Aαγraγ− ∂U
∂raα + φaα(t),
a = 1, 2, . . . , α = 0, 1, 2, . . . , N, (3.2) In other words, it is assumed here that the particles are surrounded by a isotropic viscous (not viscoelastic) liquid, and ζ is a friction coefficient of the particle in viscous liquid. The second term represents the elastic force due to the nearest Brownian particles along the chain, and the third term is the direct short-ranged interaction (excluded volume effects, see Section 1.5) be-tween all the Brownian particles. The last term represents the random thermal force defined through multiple interparticle interactions. The hydrodynamic interaction and intramolecular friction forces (internal viscosity or kinetic stiff-ness), which arise when the macromolecular coil is deformed (see Sections 2.2 and 2.4), are omitted here.
When this approximation is valid? The empirical estimation of the relax-ation time of the medium shows that, for the systems of short macromolecules (M < 2Me), the relaxation time of the medium indeed can be neglected, so that the approximation is valid for these systems. For the systems of longer
2 The methods used by Karl Freed with associates (Chang and Freed 1993; Tang et al. 1995;
Guenza and Freed 1996; Kostov and Freed 1997) for calculation time-correlation functions of single macromolecules can be apparently useful in a more complicated case of many interacting macromolecules.
40 3 Dynamics of a Macromolecule in an Entangled System
macromolecules (M > 2Me), when the entanglements exist, there seems to be no evidence that the relaxation times are equal to zero.
Course-Grained Non-Interacting Chains in Relaxing Medium As an alternative to the approximation of non-relaxing medium (equations (3.2)), one can suppose that the effect of the direct interactions between Brownian particles is less than the effect of the effective relaxing medium.
It is supported by the fact, that the number density of the Brownian particles in the coarse-grained approximation is much less than the number density of segments, so that the Brownian particles make up a weakly interacting system. If the effect of direct interactions of coarse-grained chains with each other can be neglected at all, the system (3.1) appears to be a collection of n independent equations, everyone of which describes effective dynamics of a chain of fictional Brownian particles. It is an amazing possibility: the system of entangled interacting macromolecules can be considered as a collection of non-interacting chains of Brownian particles suspended in a liquid, which is made up of interacting segments. The chains of Brownian particles appear to behave independently, though the system is closely packed. This is something of a paradox which, nevertheless, is confirmed in the following chapters.
Thus, one can choose from the two possibilities to simplify the system (3.1).
We are convinced, that the approximation of independent chains appears to be a very good initial approximation. The situation appears to be similar to a situation in dilute solutions discussed in the previous chapter. However, in contrast to the case of dilute solutions, the correlation times of the surrounding medium cannot be neglected for entangled systems. The initial phase of the theory might be found to be rather formal but the justification for every the-ory regarding physics eventually rests on the agreement between deductions made from it and experiments, and on the simplicity and consistency of the formalism. Comparison with experiment will be discussed in Chapters 5, 6, 9 and 10.
3.1.2 Dynamics of a Probe Macromolecule
The behaviour of a single macromolecule appears to be crucial in the dis-cussion of properties of entangled polymers. It is not difficult to imagine the result of eliminating all co-ordinates, apart of the course-grained co-ordinates of the only chosen single chain in the system of entangled macromolecules, as schematised in Fig. 5. Whatever the way one chooses, to start from dy-namics of interacting rigid Kuhn segments or from dydy-namics of interacting Brownian particles (equations (3.1) or (3.2)), an anticipated result in linear approximation and for the equilibrium situation can be written as
md2rα dt2 =−
∞
0
Γαγ(s) ˙rγ(t− s)ds − 2μT Aαγrγ(t) + φα(t),
α = 0, 1, 2, . . . , N, (3.3)
3.1 Admitted Approximations in the Many-Chain Problem 41
Figure 5. Macromolecule in an entangled system.
Entangled macromolecules, connected with weak van der Waals forces, make up a very concentrated polymer solution or a polymer melt. Mesoscopic approach considers the coarse-grained dynamics of a single macromolecule. The surrounding macromolecules are considered a reacting medium.
where the memory tensor function Γαγ(s) is connected with the correlation function of the random force φα(t)
φα(t)φγ(t) = 6T Γαγ(t− t).
In terms of our previous discussion, the result for the memory function Γαγ(s) has to retain traces from, generally speaking, two consequent steps: the tran-sition from the microscopic picture of interacting segments to a picture of a system of interacting coarse-grained chains (equation (3.1) or (3.2)) and transition from one of these systems to the single-chain equation (3.3).
The results for the memory function Γαγ(s) are available for the case, when the system (3.2) is chosen as a starting point of derivation of a dy-namic equation for a single chain in the system of entangled macromolecules (Schweizer 1989a, 1989b; Vilgis and Genz 1994; Guenza 1999; Rostiashvili et al. 1999; Fatkullin et al. 2000). It means that the segment carrier liquid is assumed viscous. Schweizer (1989a, 1989b) employed the Mori-Zwanzig pro-jector operator techniques to the problem. Another method of derivation of the same equation was presented by Vilgis and Genz (1994) and Rostiashvili et al. (1999). After having succeeded in eliminating all variables from the set of stochastic equations, other than those that refer to the chosen macro-molecule, and some approximations, the scholars came to equations (3.3) and evaluated, which is the most essential, the memory function Γαγ(s) through the intermolecular correlation functions and structural dynamic factor of the system of interacting Brownian particles. Their results allow us to estimate the contribution into the memory function from the second step of derivation and allow us to judge about importance of this contribution – the analysis that yet has to be done.
42 3 Dynamics of a Macromolecule in an Entangled System