# Constitutive equations from molecular network theories for polymer solutions

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## Summary

In this mainly expository paper, constitutive equations based on the network models of*Yamamoto*,*Lodge*, and*Kaye* are re-derived in a common notation involving the use of base vectors embedded in the deforming macroscopic continuum. The derivations are thereby simplified in some respects and the differences of detail between the models are clarified. In*Lodges* theory, the sub-network superposition assumption is replaced by alternative assumptions concerning the creation and loss of network segments, and the theory is extended to non-*Gaussian* networks.*Kayes* theory is extended to allow for the presence of entanglement junctions of different complexities.

### Notation

*b*_{n}= 3/(2

*n l*^{2})*n*= number of freely-jointed links, each of length

*l*, in a strand*r*= time-average (or ensemble-average) end-to-end vector for a typical network segment;

*r*= magnitude of*r**A*=

*Helmholtz*free energy per unit volume of solution*k*=

*Boltzmanns*constant*T*= absolute temperature

*F(x, n, t) d*^{3}*x=F(x*^{1},*x*^{2},*x*^{3},*n, t) dx*^{1}*dx*^{2}*dx*^{3}= concentration at time

*t*of*n*-segments whose*r*-vectors lie in the range (*x*^{i}*u*_{i}, (*x*^{i}+*dx*^{i})*u*_{i})*u*_{1},*u*_{2},*u*_{3}= orthonormal base vectors, fixed in space;

*r*=*x*^{i}*u*_{i}*e*_{1},*e*_{2},*e*_{3}= linearly independent, time-dependent base vectors, embedded in the macroscopic continuum

*ξ*^{i}= convected components of

*r: r*=*ξ*^{i}*e*_{i}*γ*_{jj}=

*e*_{i}*· e*_{j}(scalar product)*c*_{n}= concentration of

*n*-segments*γ*= det [

*γ*_{ij}]*γ*^{ij}= element of matrix reciprocal to matrix

*N*^{*}_{0}= concentration of segments

*π*^{ij}= stress components referred to basis

*e*_{1},*e*_{2},*e*_{3}*II*^{ij}=

*π*^{ij}+*p γ*^{ij},*p*arbitrary*x*= a parameter (of values 1, 2,...) labeling segments according to the complexity of their junctions with the network

*τ*_{xn}^{−1}= constant probability per unit time that an (

*n, x*)-segment will leave the network*Ψ(ξ, n, x, t' ¦t) d*^{3}*ξ dt'*= concentration at time

*t*≥*t'*of (*ξ, n, x*)-segments which were created in the interval (*t', t' + dt'*)*L*_{xn}= rate of creation, per unit volume, of (

*n, x*)-segments*N*^{*}(*t− t') dt'*= concentration at time

*t≥t'*of segments which were created in the interval (*t', t'+dt'*)*CdV*= density of configurations available to a strand having one end fixed at a given point and the other end within a volume element

*dV*about a second given point*p*_{11}−*p*_{22},*p*_{22}−*p*_{33}= primary and secondary differences of normal cartesian stress components for a liquid in steady shear flow in which the velocity components are

*v*_{1}=*Gx*^{2},*v*_{2}=*v*_{3}=0*G*= shear rate

*Φ (r, n, T)*= contribution to

*A*from a typical (*r, n*)-segment at time*t**β (r, n)*= probability per unit time that a given (

*r, n*)-segment will leave the network*G(r, n, t) d*^{3}*x dt*= concentration of (

*r, n*)-segments created during (*t, t + dt*)*Ψ (ξ, n, t′¦t) d*^{3}*ξ dt′*= concentration at time

*t≥t′*of (*ξ, n*)-segments which were created during (*t′, t′ + dt′*)*r′*=

*r(t′) r″*=*r*(t″)*P*^{ij}= components of extra stress tensor referred to basis

*u*_{1},*u*_{2},*u*_{3}*c (t)*= a cartesian space tensor defined by

*r(t) =e(*) · r*(0)*h*=

*r*(0) in [7.22],*r*(t′) in [7.24]*e*^{+}= transpose of

*e**e*^{−1}= reciprocal of e

*g*=

*g[Q*_{i}*(t), Q*_{2}*(t)]*= stress-dependent probability per unit time at time*t*of the loss of any given network junction*Q*_{1}*(t), Q*_{2}*(t)*= functions of invariants of stress at time

*t*, defined by [8.11], [8.12]*g″xn ≡ g [Qi(t″), Q*_{2}(*t″), x,n]*= probability per unit time at time

*t″*of the loss of any given (*n, x*)-segment*N*^{*}_{xn}= concentration of (

*n, x*)-segments*N*_{0}= concentration of junctions*s*=

*N*^{*}_{0}/*N*_{0}, a number whose value is about 1 or 2

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