Equations for the Non-Equilibrium Moments

Local Anisotropy

The situation is getting considerably complicated, if one takes local anisotropy into account. The mobility of a particle along the axis of a macromolecule is considered to be bigger than that in the perpendicular direction, so that the entire macromolecule can move easier along its contour. Introduction of the local anisotropy of mobility in Chapter 3 (equations (3.12) and (3.13)) allows us to specify the extra forces of external and internal resistance and define the quantities (7.5) as

where a_e and a_i are measures of local anisotropy. Every internal particle of the chain can be ascribed by the direction vector

e^α= r^α+1− r^α−1

|r^α+1− r^α−1|, α = 1, . . . , N− 1,

while the zeroth and the last particles having no direction, so that there are N− 1 vectors for a chain. It is convenient formally to consider the product of components of vectors e⁰ and e^N to be defined as

e⁰_ie⁰_j = e^N_i e^N_j =1 3δij.

While introducing of the global anisotropy, the equation for the macro-molecular dynamics remains linear in co-ordinates and velocities, the intro-duction of the local anisotropy makes it non-linear in co-ordinates. Both global and local anisotropy are needed to describe the non-linear effects of the relax-ation phenomena in the mesoscopic approximrelax-ation.

7.2 Equations for the Non-Equilibrium Moments

The dynamics of the macromolecule in the form of a set of differential equa-tions of the first order is convenient for derivation of relaxation equaequa-tions (Volkov and Vinogradov 1984, 1985; Volkov 1990). As a starting point, we use equations (7.6) and consider m = 0 in this system, so that the second equation allows us to define

140 7 Equations of Relaxation

ϕ^α_k = ζ(ψ^α_k − νkjρ^α_j) + 2μT λαρ^α_k + ¯ξ_k^α (7.13) and to rewrite the system of equations (7.6) as

d

dtρ^α_i = ψ_i^α, d

dtψ^α_i = 0, τ d

dtϕ^α_i = τ ωilϕ^α_l − ζ(ψi^α− νijρ^α_j)− 2μT λαρ^α_i

− ζB^ανij (ψ_j^ν− νjlρ^ν_l)− ζGij^αν(ψ_j^ν− ωjlρ^ν_l) + ¯ξ^α_i + σ^α_i. These equations allow one to find relaxation equation for different mo-ments, if the quantities Bij^αν and Gij^αν are given. We consider here that the quantities are independent on the co-ordinates; the more complicated case, when these quantities depend on the co-ordinates of particles, is left for other researchers. It is convenient in this section to omit the mode label and write down the above equations in the form

dρ_i dt = ψi, d

dtψ_i= 0, (7.14)

dϕi

dt = κijρj+ λijψj+ ωijϕj+1

τ( ¯ξ_i^α+ σ^α_i), where

κ_ij = ζ τ



− 1

2τ^Rδ_ij+ ν_ij+ B_ilν_lj+ E_ilω_lj

 , λij=−ζ

τ (δij+ Bij+ Eij) , (7.15)

Bij = Bβij, Eij = Eij, τ^R= ζ 4μT λ.

Further on we shall use the following symbols for the moments of the considered variables

r_ik=ρiρ_k, n_ik=ρiϕ_k, y_ik=ρiψ_k, m_ik=ψiϕ_k, z_ik=ψiψ_k, l_ik=ϕiϕ_k.

The moments can be found as solutions of the set of equations, which are followed by set (7.14)

7.2 Equations for the Non-Equilibrium Moments 141

drik

dt = yik+ yki, dyik

dt = zik, dz_ik

dt = 0, dnik

dt = m_ik+ κ_kjr_ji+ λ_kjy_ij+ ω_kjn_ij+1

τρi( ¯ξ_i^α+ σ_i^α), dmik

dt = κkjyji+ λkjzji+ ωkjmij+1

τψi( ¯ξ_i^α+ σ^α_i), dlik

dt = κijnjk+ κkjnji+ λijmjk+ λkjmji+ ωijljk+ ωkjlji

+1

τϕi( ¯ξ_i^α+ σ_i^α(t)) + ϕk( ¯ξ^α_i + σ^α_i).

To determine the average quantities, which contain the random forces in the above-written equations, one can consider the equilibrium situation. The unknown terms can be evaluated through the equilibrium values of moments, which can be used to rewrite the equations for the moments. Finally, the set of relaxation equations has the form of the above-written equations, where the terms with random forces and the terms containing the product of velocity gradient and an equilibrium moment are omitted. Instead of moments, the differences in the moments and their equilibrium values, such as r_ik− r⁰_ik instead of rik, for instance, ought to be written, so that one has

drik

dt = yik+ yki, dyik

dt = zik, dz_ik

dt = 0, dnik

dt = mik− m⁰ik+ κkjrji− κ⁰kjr⁰_ji+ λkjyij+ ωkjnij, dmik

dt = κkjyji+ ωkjmij, dl_ik

dt = κijnjk− κ⁰ijn⁰_jk+ κkjnji− κ⁰kjn⁰_ji+ λijmjk+ λkjmji

+ ω_ijl_jk+ ω_kjl_ji

(7.16)

where κ⁰_ij=−^ζ_τ2τ¹Rδij is the value of κij given by (7.15) at zero velocity gradients. The fact that some of the equilibrium moments are equal to zero, as shown below, has already been taken into account in equations (7.16).

The system of equations (7.16) determines the unknown quantities as func-tions of time and the parameters of the problem: ζ, B, E, and the parameters of anisotropy. To find a solution, one uses the equilibrium values of moments, three of which are found in Section 4.1.2

142 7 Equations of Relaxation

r_ik⁰ = 1

2μλδik, z_ik⁰ = T

mδik, y_ik⁰ = 0. (7.17) The others are contained in the following relations, which are consequences of relation (7.13)

nik= n⁰_ik+ ζ(yik− νkjrji) + ζ

2τ^R(rik− r⁰ik), m_ik= m⁰_ik+ ζ(z_ik− νkjy_ji) + ζ

2τ^Ry_ki, (7.18)

lik= l⁰_ik+ ζ(mki− νkjnji) + ζ

2τ^R(nki− n⁰ki).

One can note that, with help of the one of the above relations, the fourth equation in the set (7.16) can be written as

dnik

dt = ζzik− ζνkjyji+ ζ

2τ^Ryki+ κkjrji− κ⁰kjr_ji⁰ + λkjyij+ ωkjnij. From the other side, after having differentiated the first of equations (7.18), one has

dn_ik

dt = ζ(z_ik− νkj(y_ji+ y_ij)) + ζ

2τ^R(y_ik+ y_ki).

These equations are followed by the relation  τ

2τ^Rδkj+ δkj+ Bkj+ Ekj

yij

=− 1

2τ^R(rik− r⁰ik) +  τ

2τ^Rδkl+ δkl+ Bkl+ Ekl

ωljrji+ Bklγljrji

+ τ νkjyij+ τ ωkjyji+τ

ζωkjn⁰_ji+ τ

2τ^Rωkjr_ji⁰, (7.19) where terms containing velocity gradients in the second power are already excluded.

The equation (7.19) has to be considered as an equation for the quantity yik. When the velocity gradients are absent,

yik=− 1

2τ_♦(rik− r⁰ik), τ_♦ =τ

2+ τ^R(1 + B + E) . (7.20) The symbol_♦is used here to show the place of label of relaxation times which are identical to relaxation times (4.26). Then, one can see that the last four terms in (7.19) can be neglected and the last equation allows us to find the relation

y_ik=− 1

2τ_♦(r_ij− rij⁰)b_jk+ ω_kjr_ji+Bτ^R

τ_♦ r_ijγ_jlc_lk, (7.21) where the following notations are used

In document CHEMICALPHYSICS 95 (pagina 152-156)