Page 308 - Academic Press Encyclopedia of Physical Science and Technology 3rd Polymer

P1: GQT/MBQ P2: GPJ Final Pages
Encyclopedia of Physical Science and Technology EN014C-660 July 28, 2001 17:14

246 Rheology of Polymeric Liquids

exhibits purely an elastic effect (i.e., as a Hookean solid) of these variables (e.g., λ i and η i for the ith submolecule).
and the dashpot exhibits purely a viscous effect (i.e., as a If one assumes that the components of a stress are linearly
Newtonian fluid). related to the components of the rate of deformation, then
Therefore, the total strain of the “spring and dashpot” at the overall response of the N submolecules may be ex-
any time t is the sum of that due to the spring (reversible) pressed by
and that due to the dashpot (irreversible). Combining
N
Eqs. (33) and (34), we obtain σ(t) = 2 t
η 0 −(t−t )/λ i d(t ) dt . (41)

e
dσ −∞ i=1 λ i
σ + λ 1 = η 0 ˙γ (35)
dt Ifoneconsiderstherateofdeformationasthecauseandthe
stress as the resulting effect, then the observed “resulting
which is known as the one-dimensional Maxwell mechan-
effect” at the present time is due to the sequence of causes
ical model. Note that λ 1 = η 0 /G in Eq. (35) is a time con-
up to the present time t from the remote past.
stant,oftenreferredtoastherelaxationtime.Equation(35)
For large deformations, one can still write differential-
is capable of qualitatively explaining many well-known
type rheological equations of state with integral represen-
viscoelastic phenomena, such as stress relaxation follow-
tation, using appropriate transformations of the coordinate
ing a sudden change in deformation and elastic recovery
systems involved, as
following a sudden release of imposed stress.
If the partial derivative ∂/∂t, appearing in Eq. (35), is t

σ(t) = 2 m(t − t ) (t, t ) dt , (42)
replaced with the convected derivative ∂/∂t, we obtain
−∞
∂σ where m(t − t ) is referred to as the memory function and

σ + λ 1 = η 0 d. (36)
∂t (t, t ) is the rate-of-deformation tensor in a coordinate

2
This model predicts η = η 0 and N 1 = 2λ 1 ˙γ for steady- system rotating with a fluid element. The memory function

state shear flow. There are many different types of non- m(t − t ) for Eq. (40) is given by
linear rheological models suggested in the literature. One η 0 −(t−t )/λ 1

m(t − t ) = e . (43)
of the simplest modifications that can be introduced into λ 1
Eq. (36), in order to empirically correct the inherent de-
One can derive expressions for three material functions,
fects that the linear Maxwell model has in predicting
σ, N 1 , and N 2 , from Eq. (37), Eq. (40), or Eq. (42). Such
the rheological properties of viscoelastic fluids, would be
expressions are well documented in the literature.
to make the material constants become shear dependent.
There have been several attempts made to accomplish this,
and one such generalization can made as Molecular Approach to Constitutive Equations
∂σ The molecular approach is to relate the rheological be-
σ + λ(II) = 2η(II)d, (37)
∂t havior of polymeric liquids to their molecular parameters,
such as molecular weight, molecular weight distribution,
where II represents the second invariant of the rate-of-
and the extent of side chain branching. It is not difficult
deformation tensor d. For steady-state shear flow, the two
to surmise that the rheological properties of polymers are
parameters λ(II) and η(II) may be expressed by
greatly influenced by the molecular parameters. The pre-
η 0
, (38) dictions of the rheological properties of polymers on the
(1−n)/2

η( ˙γ ) =
1 + (η 1 ˙γ ) 2 basis of phenomenological theory is of little help either
to control the quality of polymers produced or to improve
λ 0
. (39)
(1−m)/2 the performance of polymers, unless the parameters ap-
λ( ˙γ ) =
1 + (λ 1 ˙γ ) 2
pearing in various continuum constitutive equations are
There are many other differential-type constitutive equa- related to molecular parameters.
tions that can be found in the literature. It is well established that the zero-shear viscosity (η 0 )
For infinitesimally small deformations, the integration of polymer is proportional to the molecular weight (M)
of Eq. (35) gives below a critical value M c , whereas above M c it increases
rapidly and becomes proportional to M 3.4 , i.e.,
t

η 0 −(t−t )/λ 1
σ(t) = 2 e d(t ) dt . (40) KM, for M ≤ M c ,
−∞ λ 1
η 0 = 3.4 (44)
KM , for M > M c .
Since polymeric materials consist of many segments of
different submolecules, the properties of a polymeric ma- The critical molecular weight M c is believed to correspond
terial may be thought of being given in terms of a spectrum to a value beyond which molecular entanglements (i.e.,