3.3 Molecular Interpretation of the Dissipative Terms
3.3.2 External Friction
If we consider very slow motion of a macromolecular coil with constant veloc-ity, the force of internal resistance can be neglected and the resistance-drag coefficient for the external force can be written down as
ζB =
∞
0
β(s)ds, (3.18)
where the non-dimensional quantity B is a measure of the increase in the fric-tion coefficient, due to the fact that the particle is moving among neighbouring macromolecules, perturbing them. There is a slight difference in resistance, when the particle moves along the chain or in a perpendicular direction, but, in this subsection, the anisotropy of resistance will be neglected for simplicity.
Overlapping-Coils Friction
Let us imagine, following Pokrovskii and Pyshnograi (1988), the shear motion of the system as a motion of overlapping macromolecular coils, each of which is characterised by the function (1.24) of the mean number density of Brownian particles
3.3 Molecular Interpretation of the Dissipative Terms 49 where r is the distance from the mass centre of the macromolecule.
The motion of a Brownian particle of the chosen macromolecule agitates a volume with size ofS21/20 through its adjacent chain particles. This vol-ume is the bigger the longer the macromolecules are. Note once more that the agitation comes through the chain, not through viscous friction. In this situation, for a particle with radius a S21/20 , the average environment has to be considered as a non-local liquid for which the following stress tensor can be written
σij =−pδij+ 2
η(r− r)γik(r)dr.
If the influence function η(r) is known, the resistance-drag coefficient of a Brownian particle can be calculated (see Appendix E) as
ζB = 6πa
η(r)dr. (3.20)
To find the influence function η(s), we shall consider shear deformation of the system at velocity gradient γij, while two macromolecular coils, separated by a distance dj, move beside each other at velocity γijdj. We add to the sum the contributions of every coil, apart from the chosen one, and find the density distribution of the energy dissipation for the chosen coil. The proportionality coefficient depends only on the concentration of the Brownian particles, if an assumption is made that local dissipation is determined by relative velocities of macromolecular coils,
η(r)γijγij∼
a
ρ(r)ρ(r− da)daldakγilγjk. (3.21)
When the linear in velocity gradients approach is considered, the equilib-rium density distribution function (3.19) can be used. We turn the sum into the integral and, after calculating, obtain
η(r)γijγij ∼ nN2 Now the friction coefficient of the Brownian particle is calculated according to (3.20)
ζB∼ nN2. It means that
B∼ M2. (3.23)
50 3 Dynamics of a Macromolecule in an Entangled System
The approximation of overlapping coils is very rough, the real value of the in-dex in the dependence of the coefficient B on the length of the macromolecules apparently ought to be bigger than the estimated one, because one has to take into account extra motions of the macromolecule among the other chains in order to disentangle itself from its neighbours, as it was speculated by Bueche (1956).
Constraint-Release Estimate
The constraint-release theory, due to Graessley (1982), Klein (1986) and many others (see review by Watanabe 1999), studied the detailed mechanism of a large-scale lateral motion of a macromolecule in an entangled system, due to process of release of some constraints of the probe chain and jumps of some parts of the chain in lateral direction. The result of this consideration (specif-ically, the relaxation times of the macromolecule) is equivalent to the formal assumption that a particle of the chain is moving through the environment as a particle in viscous medium. The friction coefficient of the particle can be presented as a product of the friction coefficient in ‘monomer’ liquid, multi-plied by some measure of enhancement of the friction coefficient due to the neighbouring chains. This measure of enhancement corresponds to the above parameter B, so that, referring to the results of the calculations, one can say that the constraint-release mechanism determines the dependence of the coefficient B on the lengths of the neighbouring chains as
B∼ M3. (3.24)
Approximation of the Dependence
So, according to the alternative estimations (3.23) and (3.24), the coefficient of friction of a Brownian particle increases with increase in length of macro-molecules. One has to distinguish the probe macromolecule (with molecular weight or length M ) and the neighbouring macromolecules (with the length M0), even if all of them are equal. The derived estimates of the parameter B show that the parameter depends only on the length of neighbouring macro-molecules, so that the derived dependence of the enhancement coefficient on molecular weights of the macromolecules can be written as
B ∼ Z0δZ0, Z0=M0 Me
, Z = M Me
.
The length of macromolecules is measured in units of Me, Mebeing the length of a part of a macromolecule between ‘adjacent entanglements’). For the index δ, the above estimations give the values between 2 and 3.
It is convenient to have an approximate expression for the dependence of the parameter B on molecular length, and, accepting B = 1 at Z0 = 2, one can write the universal function
3.3 Molecular Interpretation of the Dissipative Terms 51
B≈
Z0
2
δ
. (3.25)
The formula contains only one index δ, which has been estimated theoreti-cally (δ = 2 or 3) and empiritheoreti-cally. In the last case, note, that in virtue of equation (6.52), the parameter B can be derived directly from measurements of coefficient of viscosity η
B = 6η π2nT τ∗,
where τ∗ is the characteristic Rouse relaxation time of the macromolecule in viscous ‘monomer’ liquid (see Section3.3.1). According to experiment, co-efficient of viscosity is proportional to M3.4, so that the reliable empirical estimation of index δ, is δ = 2.4. This value corresponds to the above theo-retical estimation of index.
3.3.3 Intramolecular Friction
If deformation of the system is fast enough (that is, before relaxation of chains can occurs), one expects that macromolecules deform affinely, i.e., for every particle ˙rαi = νijrαj, where νij is the velocity gradient, and rjαis the position in space of a particle α of a chain. Under given deformation, the external force (3.6) is equal to zero, while the intramolecular resistance force (3.7) is proportional to ˙riα− ωijrαj, where ωijis the vorticity, or γijrjα, where γijis the symmetric part of the velocity gradient, so that this force is a force of the in-tramolecular resistance due to the change in shape of the macromolecular coil (kinetic stiffness). As far as we consider the coarse-grained approximation, all the neighbouring chains, or, one can say, the particles of coarse-grained chains follow the deformation affinely, and there is no apparent cause for this force.
To explain the emerging of the force, we have to refer to more detailed model of macromolecule – to the chain of freely-jointed rigid segments. Apparently, small parts of macromolecules cannot follow the deformation affinely, seg-ments can only rotate, and an extra force is needed to change the direction of a segment in the case, when the segments of the other chains present around.
That is why we can say that the internal resistance force for a macromolecule in a polymer melt has to be attributed to the interaction with neighbouring chains, though in the coarse-grained approximation we forget about segments, and this force is characterised by only phenomenological coefficient of internal resistance, which, in the simplest case, can be denoted as
ζE =
∞
0
ϕ(s)ds. (3.26)
This quantity has value of zero for non-entangled systems and increase with increase in the length of macromolecules. As for external force, there is a slight difference in resistance, when the particle moves along the chain or in a perpendicular direction, but, in this subsection, this effect is neglected for simplicity.
52 3 Dynamics of a Macromolecule in an Entangled System
Approximation of Internal Resistance Matrix
The force of intramolecular resistance appears, when relative motion of the particles exists, so that one can write a general expression (which is identical to expression (2.21) for a chain in a dilute solution)
Gαγij =
γ=α
Cαγ(uαj − uγj)eαγj eαγi ,
where eαγj = (rαj − rγj)/|rα− rγ|. Matrix Cαγis symmetrical, the components of the matrix are non-negative and may depend on the distance between the particles. The diagonal components of the matrix are equal to zero. One can also reasonably assume that components of the matrix are equal to zero, if the difference between indexes |α − γ| is less than a certain value.
One can rewrite the matrix of internal resistance in the following form Gαγij = δαγ
β=α
Cαβeαβi eαβj − Cαγeαγi eαγj . (3.27)
This expression defines the general form of a matrix of internal friction, which allows the force to remain unchanged on the rotation of the macromolecular coil as a whole. The written matrix is symmetrical with respect to the up-per and lower indices and, in contrast to matrix Cαβ, has non-zero diagonal components, which are depicted by the first term in (3.27). In equilibrium sit-uations, after averaging over the orientation, matrix (3.27) can be presented as
Gαγij = Gαβδij, Gαβ=1 3
⎛
⎝δαγ
β=α
Cαβ− Cαγ
⎞
⎠ .
Since the components of matrix Cαβ are non-negative, the diagonal compo-nents of matrix Gαβ exceed the non-diagonal ones and can be considered to be approximately diagonal with respect to the indices α and β. The effect is very strong for long macromolecules and reduces to zero at M ≈ 2Me. As an initial approximation, to express the idea of severe confinement, one can assume that the intramolecular resistance force is determined equally by all the particles of the chain, so that the matrix is reduced to the already written matrix (3.10). It is easy to find, that in normal co-ordinates (1.13), matrix (3.10) has a diagonal form with the eigenvalues
ϕα=
0, α = 0,
1, α= 0. (3.28)
One can directly check that, if matrix (3.10) is modified, for example, zeros are placed on a few diagonals next to the main diagonal in the matrix, the transformed matrix retains approximately its diagonal form, while eigenval-ues are close to unity and decrease slightly when the index of an eigenvalue
3.3 Molecular Interpretation of the Dissipative Terms 53
increases. So, the effect of diagonals with zeros can be neglected indeed and the above matrix does approximate the situation for large-scale motions of the chain at N → ∞
Approximation of the Measure of Internal Resistance
For the systems of long macromolecules (strongly entangled systems), the requirements of universality and self-consistency allow us to write practically identical asymptotic relations (5.17) and (6.53) between the parameter χ, introduced in Section3.3.1, and the ratio E/B, which allows us to write for this case
E∼ M0δM.
For the weakly entangled systems, one can expect, that the ratio E/B, that is the parameter of ‘internal’ viscosity is small. It can be demonstrated in Sec-tion 4.2.3, that transiSec-tion point from weakly to strongly entangled systems occurs at E≈ B. To describe these facts, one can use any convenient approx-imate function for the measure of internal resistance, for example, the simple formula
E≈ B(Z0)12 (Z− 2)2
Z + 768 , Z0=M0
Me
, Z = M Me
. (3.29)