Handbook of Solvents - George Wypych - ChemTech - Ventech!

7.2 Bubbles dynamics and boiling 361
( 1) ( 1)
⎤
( )
⎡ τ
τ = τ + τ τ + λ⎢
−α τ ⋅ + ⋅τ
⎥ ηβ
⎣
⎦
=
( 1)
( 1) ( 2) ( 1)
D
( 2)
,
e e 2 e, τ = 2η( 1−
β)
e [7.2.26]
Dt
Parameter β governs the contribution of the Maxwell element to effective viscosity,
η,(Newtonian viscosity of the solution). Equation [7.2.26] is similar to the Oldroyd-type
equation [7.2.15] with the only difference that in the former the upper convective derivative
is used to account for nonlinear effects instead of partial derivative, ∂/ ∂t.
Phenomenological parameters appearing in theoretical models can be found from appropriate
rheological experiments. 6 Certain parameters, the most important being relaxation
times and viscosities, can be estimated from molecular theories. According to molecular
theory, each relaxation time λk is relative to mobility of some structural elements of a polymer.
Therefore, the system as a whole is characterized by the spectrum of relaxation times.
Relaxation phenomena, observed at a macroscopic level, owe their origin to the fact that response
of macromolecules and macromolecular blocks to different-in-rate external actions
is described by different parts of their relaxation spectrum. This response is significantly affected
by temperature - its increase “triggers” the motion of more and more complex elements
of the macromolecular hierarchy (groups of atoms, free segments, coupled segments,
etc).
The most studied relaxation processes from the point of view of molecular theories are
those governing relaxation function, G1(t), in equation [7.2.4]. According to the Rouse theory,
1 a macromolecule is modeled by a bead-spring chain. The beads are the centers of hydrodynamic
interaction of a molecule with a solvent while the springs model elastic linkage
between the beads. The polymer macromolecule is subdivided into a number of equal segments
(submolecules or subchains) within which the equilibrium is supposed to be
achieved; thus the model does not permit to describe small-scale motions that are smaller in
size than the statistical segment. Maximal relaxation time in a spectrum is expressed in
terms of macroscopic parameters of the system, which can be easily measured:
λ
11 2
( ηp − ηs)
6 M
=
π cR T
G
[7.2.27]
where:
M molecular mass of the polymer
c concentration of polymer in solution
RG universal gas constant
The other relaxation times are defined as λ1k = λ11/k 2 . In Rouse theory all the modules
G1k are assumed to be the same and equal to cRGT/M. In the Kirkwood-Riseman-Zimm (KRZ) model, unlike Rouse theory, the hydrodynamic
interaction between the segments of a macromolecular chain is accounted for. In the
limiting case of a tight macromolecular globe, the KRZ theory gives the expression for λ11 that is similar to [7.2.27]:
λ
11
( η − η )
0 422
=
cR T
. p s
G
M
[7.2.28]