The Rouse-Model

The dynamics of an isolated Kuhn segment chain in its bead-and-spring form is considered in a viscous medium without hydrodynamic backflow or excluded-volume effects. The treatment is based on the Langevin equation generalized for Brownian particles with internal degrees of freedom. A first, crude formalism of this sort was reported by Kargin and Slonimskii [45]. In- 

The effective intramolecular interactions between the segments are approximated by entropic harmonic interactions, reflecting the Gaussian character of the large-scale chain conformation. Intermolecular interactions (with the surrounding viscous medium) are taken into account by friction and stochastic forces acting on the segments. 

The entropic spring constant and the friction coefficient of a Kuhn segment are given by 

The boundary conditions reflecting the existence of only one neighbor segment at the chain ends are 

The first term on the right-hand side of Eq. 46 represents an intramolecular force whereas the second and third terms are forces of an intermolecular nature. 

The Rouse model starts from such a Gaussian chain representing a coarsegrained polymer model, where springs represent the entropic forces between hypothetic beads [6] (Fig. 3.1). 

We are interested in the motion of segments on a length scale r R , where Rl=N is the end-to-end distance of the chain. The segments are subject to an entropic force resulting from Eq. 3.4 (x-components)  

The boundary condition of force-free ends requires —— =0. The par- 

Here is the Rouse time - the longest time in the relaxation spectrum - and W is the elementary Rouse rate. The correlation function x(p,t) x p,0)) of the normal coordinates is finally obtained by  

Scattering experiments are sensitive to the mean-square segment correlation functions  

In the Rouse model, the excluded volume interaction and the hydrodynamic interaction are disregarded and the mobility tensor and the interaction potential are written as 

In this model the Langevin equation (4.2) becomes a linear equation for R . For internal beads (/i = 2, 3. iV - 1), 

The distribution of the random force/, is Gaussian, characterized by the moments given by eqns (3.40) and (3.41)  

As in the case of the Gaussian chain, the suffix n in the Rouse model can be regarded as a continuous variable. In the continuous limit, eqn (4.6) is rewritten as (see the transformation rule given in Table 2.1, Section 2.2) 

To rewrite eqn (4.7) in the continuous limit, we note that eqn (4.7) is included in the general equation (4.6) if the hypothetical beads Rq and are defined as 

Let us assume that when the polymer is disturbed due to being in a shear gradient there are only two major forces, the hydrodynamic force F and the restoring (or spring) force F( ) that the bead exerts on the liquid. The hydrodynamic force F( ) exerted on the liquid by the bead, the components of which are assumed to be proportional to the relative velocity of the bead and solvent through the fluid, is expressed by 

Equation (4.47) can be solved using the normal coordinates. Briefly stated, one ends up with solving the following differential equation for the plh mode of the normalized coordinates Qp  

Rouse (1953) has shown that the relaxation times (i.e., the time constants associated with the rate of stress dissipation after a given strain) can be obtained from 

Note that the largest or terminal relaxation time Tj for the Rouse chain (i.e., for p = m Eq. (4.52)) is given by 

Note further that the free-draining expression for the translational diffusion coefficient Dq (i.e., the diffusion coefficient of the center-of-mass for the Rouse chain) is given by (Doi and Edwards 1986) 

Fluctuations arise from all dynamical modes in the system however, different experiments associate them with different weights. Take, for example, the glass-rubber transition. The transition shows up in measurements of the dynamic compliance, the d3mamic modulus and the dielectric function. Even if the shapes E w) and differ from each other, the maxima of 

The fluctuation-dissipation theorem may be regarded as an interface between the microscopic and the macroscopic properties of a sample. It provides us with a prescription of how to proceed when these two are to be related. On the microscopic side, a theoretical analysis of dynamical models often enables us to calculate equilibrium correlation functions for properties of interest. The fluctuation-dissipation theorem then relates these correlation functions with the results of measurements of corresponding response functions. 

In the following, we will discuss some microscopic dynamical models. We begin with the Rouse model, which describes the dynamics of chains in a non-entangled polymer melt. The effects of entanglements on the motion can be accounted for by the reptation model, which we will treat subsequently. Then we shall be concerned with the motion of polymer chains in a solvent, when the hydrodynamic interaction between the segments of a chain plays a prominent role. At the end of this chapter, we briefly discuss the modes of motion in polyelectrolyte solutions, which are strongly affected by Coulomb forces. 

We shall derive this force in the next chapter when dealing with the elasticity of rubbers. If Gaussian properties are assumed the result, as given by Eq. (9.13), is as follows If a sequence within a chain that has a mean squared end-to-end distance (Ar ) is chosen and its endpoints are at a distance Ar, then the tensile force is 

The result implies that a sequence behaves like a spring, showing a linear dependence of the force on the extension. The force constant b is proportional to the absolute temperature T, as is characteristic for forces of entropic origin. Note furthermore that b decreases on increasing the size of the sequence. 

If a polymer molecule is stretched out by applying forces to the end groups and then the forces are removed it returns to the initial coiled conformation. The reason for this behavior has already been mentioned The transition back to an isotropic coil increases the number of available rotational isomeric states and thus the entropy. The recoiling effect can also be expressed in mechanistic terms, by stating that, if the two endgroups of a polymer chain are held fixed at a certain distance, a tensile force arises due to the net moment transfer onto the ends. If, rather than keeping hold of the endgroups, two arbitrary points within a polymer molecule are kept at constant positions, a tensile force arises as well. 

A polymer chain in a melt moves in the surroundings set up by the other chains and at first, this looks like a rather complicated situation. However, as it turns out, one can employ an approximate simple treatment. For particles of at least mesoscopic size, the various interactions with adjacent molecules may be represented in summary by one viscous force. This is well-known from treatments of the dynamics of a colloid in a solvent. There it can be assumed that if a colloid moves with a velocity u, the solvent molecules in contact with its surface create a force which is proportional to u and the solvent viscosity Vs 

Rouse devised a treatment of the dynamics of polymer chains in a melt which makes use of this notion of a viscous force and also takes account of the tensile forces arising in stretched parts of the chain. The procedure used 

The derivation of the constitutive equation for the dumbbell model can also be extended to allow multiple beads and springs. From the force-balance equations for such a model, one obtains (Bird et al. 1987b Larson 1988) 

Rouse (1953) transformed this matrix equation into a set of uncoupled equations—that is, into Ns independent equations for the normal modes -------------------------- 

The normal-mode decomposition permits one to break the stress into a sum of contributions. 

Each ji obeys an upper-convected Maxwell equation 

The longer relaxation times, for which i/Ng is small, are given approximately by 

As discussed in Chapter 1, a Gaussian chain is physically equivalent to a string of beads connected by harmonic springs with the elastic constant ikT/lP (Eq. (1.47) with 6 given by Eq. (1.44)). Here each bead is regarded as a Brownian particle in modeling the chain d3mamics. Such a model was first proposed by Rouse and has been the basis of molecular theories for the dynamics of polymeric liquids. 34 

When Eq. (3.28) is expressed in the form of Eq. (3.20), the mobility matrix L for a chain of N beads will have the dimension 3N x SN. 

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A polymer chain can be approximated by a set of balls connected by springs. The springs account for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

Fig. 13 shows this autocorrelation function where the time is scaled by mean square displacement of the center of mass of the chains normalized to Ree)- All these curves follow one common function. It also shows that for these melts (note that the chains are very short ) the interpretation of a chain dynamics within the Rouse model is perfectly suitable, since the time is just given within the Rouse scaling and then normalized by the typical extension of the chains [47]. 

Before turning to dynamics, we should hke to point out that, because no solvent is explicitly included, the Rouse model [37,38] (rather than the Zimm model [39]) results in the dilute limit, as there is no hydrodynamic interaction. The rate of reorientation of monomers per unit time is W, and the relaxation time of a chain scales as [26,38]... 

D. H. King, D. F. James. Analysis of the Rouse model in extensional flow. J Chem Phys 72 4749 754, 1983. 

In the Rouse model [45], the drag velocity is considered to be uniform on any bead and equal to the relative velocity of the center of mass of the bead-spring... 

Combining Eqs. (24) and (25) gives the longest relaxation time for the Rouse model as ... 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

In analogy with the Rouse model, the longest relaxation time (Xj) according to the Zimm model can again be put into a form which does not depend on N [44] ... 

An even more serious problem concerns the corresponding time scales on the most microscopic level, vibrations of bond lengths and bond angles have characteristic times of approx. rvib 10-13 s somewhat slower are the jumps over the barriers of the torsional potential (Fig. 1.3), which take place with a time constant of typically cj-1 10-11 s. On the semi-microscopic level, the time that a polymer coil needs to equilibrate its configuration is at least a factor of the order larger, where Np is the degree of polymerization, t = cj 1Np. This formula applies for the Rouse model [21,22], i. e., for non-... 

Within the Rouse model for polymer dynamics the viscosity of a melt can be calculated from the diffusion constant of the chains using the relation [22,29,30] ... 

This section presents results of the space-time analysis of the above-mentioned motional processes as obtained by the neutron spin echo technique. First, the entropically determined relaxation processes, as described by the Rouse model, will be discussed. We will then examine how topological restrictions are noticed if the chain length is increased. Subsequently, we address the dynamics of highly entangled systems and, finally, we consider the origin of the entanglements. 

In the case of coherent scattering, which observes the pair-correlation function, interference from scattering waves emanating from various segments complicates the scattering function. Here, we shall explicitly calculate S(Q,t) for the Rouse model for the limiting cases (1) QRe -4 1 and (2) QRe > 1 where R2 = /2N is the end-to-end distance of the polymer chain. 

For different momentum transfers the dynamic structure factors are predicted to collapse to one master curve, if they are represented as a function of the Rouse variable. This property is a consequence of the fact that the Rouse model does not contain any particular length scale. In addition, it should be mentioned that Z2/ or the equivalent quantity W/4 is the only adjustable parameter when Rouse dynamics are studied by NSE. 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

Fig. 6. Dynamic structure factor as observed from PI for different momentum transfers at 468 K. ( Q = 0.038 A"1 Q = 0.051 A-1 A Q = 0.064 A-1 O Q = 0.077 A"1 Q= 0.102 A-1 O Q = 0.128 A 1 Q = 0,153 A "" 11. The solid lines display fits with the Rouse model to the initial decay. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

Over the entire Q-range within experimental error the data points fall on the line and thus exhibit the predicted Q4 dependence. The insert in Fig. 7 demonstrates the scaling behavior of the experimental spectra which, according to the Rouse model, are required to collapse to one master curve if they are plotted in terms of the Rouse variable u = QV2 /wt. The solid line displays the result of a joint fit to the Rouse structure factor with the only parameter fit being the Rouse rate W 4. Excellent agreement with the theoretical prediction is observed. The resulting value is W/4 = 2.0 + 0.1 x 1013 A4s 1. 

In summary, the chain dynamics for short times, where entanglement effects do not yet play a role, are excellently described by the picture of Langevin dynamics with entropic restoring forces. The Rouse model quantitatively describes (1) the Q-dependence of the characteristic relaxation rate, (2) the spectral form of both the self- and the pair correlation, and (3) it establishes the correct relation to the macroscopic viscosity. 

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

Tube confinement leads to strong alterations of the mean square segment displacements as compared to the Rouse model. 

Like the dynamic structure factor for local reptation it develops a plateau region, the height of which depends on Qd. Figure 20 displays S(Q,t) as a function of the Rouse variable Q2/ 2X/Wt for different values of Qd. Clear deviations from the dynamic structure factor of the Rouse model can be seen even for Qd = 7. This aspect agrees well with computer simulations by Kremer et al. [54, 55] who found such deviations in the Q-regime 2.9 V Qd < 6.7. 

The long-range coupling via the flow field which only decreases with 1/r leads to a qualitatively different behavior from that of the Rouse model. Equation (75) is approximately solved by transformation to Rouse normal coordinates. Its solution [6,91] leads to the spectrum of relaxation rates... 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

The coherent structure factor of the Zimm model can be calculated [34] following the lines outlined in detail in connection with the Rouse model. As the incoherent structure factor (83), it is also a function of the scaling variable ( fz(Q)t)2/3 and has the form... 

Within the framework of the Rouse model the characteristic frequency for the center of mass diffusion follows the equation... 

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