Handbook of Solvents - George Wypych - ChemTech - Ventech!

358 Semyon Levitsky, Zinoviy Shulman
where:
Gik partial modules, corresponding to λik δ(λ - λik) Dirac delta function
In this case the integration over the spectrum in Equation [7.2.8] is replaced by summation
over all relaxation times, λik. For polymeric solutions, as distinct to melts, it is convenient to introduce in the
right-hand sides of equations [7.2.4] additional terms, 2ηssij and 3ρ0ηvekk, which represent
contributions of shear, ηs, and bulk, ηv, viscosities of the solvent. The result has the form
t ∞
⎛ t − t′
⎞
τij = 2∫∫F() λ ⎜−
⎟ sij () t′ dλdt′ 1 exp + 2ηssij
[7.2.10]
⎝ λ ⎠
−∞ 0
t
t ∞
t t
p = p − G ekk() t′ dt′− ⎛ − ′ ⎞
0 20∫ ∫ ∫ F2()
λ exp⎜− ⎟ ekk() t′ dλdt′+ G30θ+
⎝ λ ⎠
−∞
−∞ 0
() ′
∞
⎛ − ′ ⎞
+ ∫ ∫ () ⎜−
⎟ ′ − =−
⎝ ⎠ ′
F
t
t t ∂θ t
3 λ exp dλdt ρ0ηvekk, p 1/ 3σ
kk [7.2.11]
λ ∂t
−∞ 0
Equilibrium values of bulk and shear viscosity, η b and η p, can be expressed in terms of
the relaxation spectra, F 1 and F 2, as: 1
∞
∞
η − η = λF λ dλ, η − η = λF λ dλ
[7.2.12]
() ∫ ()
p s ∫ 1 b v
2
0
0
In the special case when relaxation spectrum, F 1(λ), contains only one relaxation time,
λ 11, equation [7.2.10] yields
τ
t
t − t′
= G −
λ
⎛ ⎞
2 11
2
−∞ ⎝ 11 ⎠
∫ exp ⎜
⎟
s () t′ dt′ + ηssij,
G11
= ( − )
ij ij
η η λ
p s / 11 [7.2.13]
At η s = 0 the integral equation [7.2.13] is equivalent to the linear differential Maxwell
equation
τ λ τ η s
+ � = 2
[7.2.14]
ij 11 ij p ij
Setting η s = λ 2η 0/λ 1, λ 1 = λ 11, η p = η 0, where λ 2 is the retardation time, one can rearrange
equation [7.2.14] to receive the linear Oldroyd equation 3
( s s )
τ + λ �τ = 2η
+ λ � , λ ≥λ
[7.2.15]
ij 1 ij 0 ij ij ij 1 2
Thus, the Oldroyd model represents a special case of the general hereditary model
[7.2.10] with appropriate choice of parameters. Usually the maximum relaxation time in the
spectrum is taken for λ 1 in equation [7.2.15] and therefore it can be used for quantitative description
and estimates of relaxation effects in non-steady flows of polymeric systems.
Equation [7.2.11], written for quasi-equilibrium process, helps to clarify the meaning
of the modules G 20 and G 30. In this case it gives