Primitive chain dynamics

L fluctuates in time as the chain (or snake) moves. A full description of chain dynamics requires knowledge of the probability distribution of the primitive path lengths. This problem has been solved exactly by Helfand and Pearson in 1983 for a lattice model of a chain in a regular array of... 

Doi has proposed a theory, which will be discussed in detail in Chapter 12, describing A) as the relaxation of tension on the primitive chain. The theory predicts that the t, A) process is not observable in the linear region, which has been found to be in agreement with experiment. However, corresponding to the dynamics of A), there is a process... 

To sort out such a complicated dynamic situation, we first assume that the primitive chain is nailed down at some central point of the chain, i.e. the reptational motion is frozen only the contour length fluctuation is allowed. This is equivalent to setting rg —> oo while allowing the contour length fluctuation 5L(t) to occur with a finite characteristic relaxation time Tb- In this hypothetical situation, the portion of the tube that still possesses tube stress tt fa tb is reduced to a shorter length Lq, because of the fluctuation SL(t). Then, tt tube length that still possesses tube stress can be defined by... 

The dynamics of the primitive chain is characterized by the following assumptions. 

We now study the dynamics of the primitive chain and show that certain time correlation functions can be calculated by a straightforward method. For example, consider the time correlation function of the end-to-end vector P(r) lf(L, r) -lf(0, t). Figure 6.4 explains the prin-dple of calculating tlfis correlation function. At r = 0, the chain is trapped in a certain tube. As time passes, the primitive chain reptates and at a certain later time (Fig. 6Ad), the part of the chain CD remains in the original tube while the parts AC and DB are in a new tube. To calculate... 

Stochastic equation for reptation dynamics Although the above probabilistic description is quite useful in understanding the essence of reptation dynamics, it becomes progressively more difficult to proceed with the calculation for other types of time correlation function. For example, it is not easy to calculate the mean square displacement of a primitive chain segment (R(s, t)-R(s, 0)) ) by this method. In this section we shall describe a convenient method" for calculating general time correlation functions. 

First we derive a simple mathematical equation for reptation dynamics. Let A (0 be the distance that the primitive chain moves in a time interval between t and t + Ar, then... 

Statistical distribution of the contour length In the previous sections we regarded the primitive chain as an inexten-sible string of contour length L. In reality, the contour length of the primitive chain fluctuates with time, and the fluctuation sometimes plays an important role in various dynamical processes. 

We shall now derive the constitutive equation for reptation dynamics. To simplify the analysis, we assume that tiie contour length of the primitive chain remains at the equilibrium value L under macroscopic deformation (inextensible primitive chain). This assumption is valid if the characteristic magnitude of the velocity gradient is much less than 1/Tj, i.e. 

Figure 6.19 Explanation of the stress relaxation after a large step strain, (a) Before deformation the conformation of the primitive chain is in equilibrium (t = — 0). (b) Immediately after deformation, the primitive chain is in the affinely deformed conformation (t = + 0). (c) After time Tr, the primitive chain contracts along the tube and recovers the equilibrium contour length (t ir). (d) After the time Td, the primitive chain leaves the deformed tube by reptation t Td). The oblique lines indicate the deformed part of the tube. (Reproduced from Doi and Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1986)...

In the semidilute solution, the hydrodynamic interactions are shielded over the distance beyond the correlation length, just as the excluded volume is shielded. We can therefore approximate the dynamics of the test chain by a Rouse model, although the motion is constrained to the space within the tube. In the Rouse model, the chain as a whole receives the friction of N, where is the friction coefficient per bead. When the motion is limited to the curvilinear path of the primitive chain, the friction is the same. Because the test chain makes a Rouse motion within the tube, only the motion along the tube survives over time, leading to the translation of the primitive chain along its own contour. The one-dimensional diffusion coefficient for the motion of the primitive chain is called the curvilinear diffusion coefficient. It is therefore equal to Dq of the Rouse chain (Eq. 3.160) and given by... 

The test chain would follow the dynamics of the unrestricted Rouse chain if the entanglements were absent, as would the primitive chain at f > t. In Section 3.4.9, we considered the mean square displacement of monomers on the Rouse chain. We found that the dynamics is diffusional at r < and Tj < t, where % is the relaxation time of the Mth normal mode but not in between. When the motion of the Rouse chain is resnicted to the tube, the mean square displacement of monomers along the tube, ([ (t) - x(0)] ), will follow the same time dependence as the mean square displacement of the unrestricted Rouse chain in three dimensions. Thus, from Eqs. 3.240 and 3.243,... 

The process of disentangling, as it is envisaged in the reptation model, is sketched in Fig. 6.11. The motion of the primitive chain , the name given to the dynamic object associated with the primitive path, is described as a diffusion along its contour, that is to say, a reptation . The associated curvilinear diffusion coefficient can be derived from the Einstein relation, which holds generally, independent of the dimension or the topology. Denoting it D, we have... 

Doi and Edwards analysed the described disentangling process of the primitive chain in more detail. As in the case of the Rouse-motion, the dynamics of the disentangling process can also be represented as a superposition of independent modes. Again, only one time constant, the disentangling time Td, is included, and it sets the time scale for the complete process. In the Doi-Edwards treatment, ra is identified with the longest relaxation time. Calculations result in an expression for the time dependent shear modulus in the terminal flow region. It has the form... 

F., and Marrucd, G. (2004) Highly entangled polymer primitive chain network simulations based on dynamic tube dilation. /. Chem. Phys., 121 (24), 12650-12654. 

K. E. Evans and S. F. Edwards, Computer Simulation of the Dynamics of Highly Entangled Polymers. Part 2 — Static Properties of the Primitive Chain , J. Chem. Soc., Faraday Trans. 2,11,1981, pp 1913-1927. 

Let us now consider the dynamics of a primitive chain. Within the concept of reptation, in the short timescale the motion of the polymer can be regarded as wriggling around the primitive path. On a longer timescale, the conformation of the primitive path changes as the polymer moves, creating and destroying the ends of the primitive path. In the absence of an external potential, the time evolution (i.e., the dynamics) of the primitive... 

In this section, we present the molecular theory for the linear dynamic viscoelasticity of miscible polymer blends by Han and Kim (1989a, 1989b), which is based on the concept of the tube model presented in Chapter 4. Specifically, the reptation of two primitive chains of dissimilar chemical structures under an external potential will be considered, and the expressions for the linear viscoelastic properties of miscible polymer blends will be presented. We will first present the expressions for zero-shear viscosity ob. dynamic storage and loss moduli G co) and G " co), and steady-state compliance J° for binary miscible blends of monodisperse, entangled flexible homopolymers and then consider the effect of polydispersity. There are a few other molecular theories reported... 

Then the dynamics of the primitive chain 1, representing polymer 1, can be expressed in the form of the Smoluchowski equation (Riskin 1989) ... 

These authors studied how tube models with reptation dynamics could be turned into a full theory of viscoelasticity. To do this one needs to describe the dynamics of the primitive path. Let the primitive chain make one step in time At. Define a random variable, (t) which is + 1( — 1) if the primitive chain moves backward (forward). Let p be a random vector of length a which is the new position of one of the ends of the primitive chain after one move (see Figure 23). Since the primitive chain is assumed to be made of N points R, . Rjy, connected by bonds of constant length a, the Langevin equation of the primitive chain is given by... 

Attempts have been made to identify primitive motions from measurements of mechanical and dielectric relaxation (89) and to model the short time end of the relaxation spectrum (90). Methods have been developed recently for calculating the complete dynamical behavior of chains with idealized local structure (91,92). An apparent internal chain viscosity has been observed at high frequencies in dilute polymer solutions which is proportional to solvent viscosity (93) and which presumably appears when the external driving frequency is comparable to the frequency of the primitive rotations (94,95). The beginnings of an analysis of dynamics in the rotational isomeric model have been made (96). However, no general solution applicable for all frequency ranges has been found for chains with realistic local structure. 

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

In the course of investigation of polymer dynamics, all topological states of chains are equivalent due to the presence of their free ends and it is not necessary to construct the topological invariant. However, for every instantaneous chain configuration, its topological state is definite and is determined by the topological invariant - the primitive path. 

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