Reptation-Tube Model

, (3.38)

which can be looked upon as the equation for the random force ˜φ^γ_i for the given random quantity σ^α_i. Then, if the relation

σi^γ(t)σ^μ_j(t^) = 2ζ(B Hij^γμ+ E G^γμ_ij ) δ(t− t^) (3.39) is satisfied, the random force correlator satisfies the following relation

 ˜φ^γi(t) ˜φ^μ_j(t^) = T ζ τ

B H_ij^γμ+ E G^γμ_ij  exp



−t− t^ τ



. (3.40)

This relation, in line with relation

 ¯φ^γ_i(t) ¯φ^μ_j(t^) = 2T ζδγμδ_ijδ(t− t^), (3.41) return us to the random-force correlation function (3.36).

The set of stochastic equations given by (3.37) is equivalent (in the linear case) to equations (3.11) with the memory functions defined in Section 3.3, but, in contrast to equations (3.11), set (3.37) is written as a set of Markov stochastic equations. This enables us to determine the variables that describe the collective motion of the set of macromolecules. In this particular approx-imation, the interaction between neighbouring macromolecules ensures that the phase variables of the elementary motion are co-ordinates, velocities, and some other vector variables – the extra forces. This set of phase variables de-scribes the dynamics of the entire set of entangled macromolecules. Note that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approxi-mations of the memory functions. We are considering the case with a single relaxation time, but generalisation for a case with a few relaxation times is possible.

3.5 Reptation-Tube Model

The system of dynamic equations (3.37) for a chain of Brownian particles with local anisotropy of mobility appears to be rather complicated for direct anal-ysis, and one ought to use numerical methods, described in the next Section,

3.5 Reptation-Tube Model 57

to be convinced that equations (3.37) really describe the observed effects. Not to explore non-linear equations, one can exaggerate anisotropy of mobility, assuming that unbounded lateral motion of particles is completely suppressed due to the presence of many neighbouring coils. By this way, one comes to a very elegant linear model of reptating macromolecule proposed by Doi and Edwards (1978) (see also Doi and Edwards 1986).

Following Doi and Edwards (1978), we shall consider a bead-spring model consisting of Z = M/Me subchains and assume that the distance between adjacent particles along the chain is constant and equal to a certain interme-diate length ξ, which is considered to be the radius of ‘a tube’, so that the number of particles is not arbitrary, but satisfies the condition

Zξ²=R². (3.42)

The states of the macromolecule will be considered in points of time in a time interval Δt, so that the stochastic motion of Brownian particles of the chain can be described by the equation for the particle co-ordinates

r⁰(t + Δt) = 1 + φ(t)

2 r¹(t) +1− φ(t)

2 [r⁰(t) + v(t)], r^ν(t + Δt) = 1 + φ(t)

2 r^ν+1(t) +1− φ(t)

2 r^ν−1(t), ν = 1, 2, . . . , Z− 1,(3.43) r^Z(t + Δt) = 1 + φ(t)

2 [r^Z(t) + v(t)] +1− φ(t)

2 r^Z−1(t)

where φ(t) is a random quantity, which takes the values +1 or−1, and v(t) is a vector of constant length ξ and random direction, so that

φ(t)φ(u) = δtu, φ(t) = 0,

v(t)v(u) = δtuξ², v(t) = 0. (3.44) The set of equations (3.43) describes the stochastic motion of a chain along its contour. The “head” and the “tail” particles of the chain can choose ran-dom directions. Any other particle follows the neighbouring particles in front or behind. The smaller the time interval Δt is the quicker moves the chain.

Clearly, the time interval cannot be an arbitrary quantity and is specified by the requirement that the squared displacement of the entire chain by diffusion for the interval Δt is equal to ξ², so that

ξ²= 2D0Δt = 2T

ζZΔt (3.45)

where D₀ = T /(ζZ) is the diffusion coefficient of the macromolecule in a monomeric viscous liquid (see Section 5.1.1 for explanation). Note that we follow the original Doi-Edwards model in which diffusion of the chain is con-sidered to be one-dimensional.

58 3 Dynamics of a Macromolecule in an Entangled System

The model described by equations (3.42)–(3.45) is valid for equilibrium situations. For chain in a flow, one ought to define displacements of the par-ticles under flow and to consider the average values (3.44) to depend on the velocity gradient (Doi and Edwards 1986). McLeish and Milner (1999) consid-ered mechanism of reptation motion of branched macromolecules of different architecture.

It is convenient to rewrite equations (3.43) in more compact form, taking also definition of Δt into account,

r^α(t + Δt)− r^α(t)

Δt = − T

ζξ²Z³Aαγr^γ(t)+σ^α(t), α = 0, 1, 2, . . . , Z, (3.46) where the stochastic force is defined as

σ^α(t) = 1 2φ(t)×

⎧⎪

⎪⎨

⎪⎪

⎩

r¹(t)− r⁰(t) + v(t), α = 0,

r^α+1(t)− r^α−1(t), α = 1, 2, . . . , N− 1, r^Z−1(t)− r^Z(t) + v(t), α = Z.

(3.47)

To obtain relation (3.46), one has to take into account that motion of the particles of the chain ought to be considered to be coherent. Now, it is not difficult to pass from equation (3.46) to the normal-mode equation

dρ^α

dt = −π²T α²

ζξ²Z³ ρ^α+ Qγασ^γ(t), α = 0, 1, 2, . . . , Z. (3.48) These equations describe the reptation normal relaxation modes, which can be compared with the Rouse modes of the chain in a viscous liquid, described by equation (2.29). In contrast to equation (2.29) the stochastic forces (3.47) depend on the co-ordinates of particles, equation (3.48) describes anisotropic motion of beads along the contour of a macromolecule.

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length ξ emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful.