Karl Weissenberg - The 80th Birthday
Celebration Essays
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The
Effect of Molecular Weight and Concentration of Polymers in Solutions on the
G. V. VINOGRADOV, A. Ya. MALKIN
The
effect of the appearance of normal stresses in shear flow, which was discovered
by Weissenberg’ and bears now his name, has dual meaning. First, this is a
mechanical phenomenon essential for the understanding and description of the
properties of viscoelastic media showing high elastic
strain, and for the solution of the various related problems. This aspect of
the Weissenberg effect has been discussed most thoroughly in the literature on rheology. Second, it has been found lately that the
measurement of the quantities describing the Weissenberg effect may be used as
the most sensitive characteristic of polymer systems, associated with such
specific features of composition as the molecular weight and molecular-weight
distribution of polymers, branching of macromolecules, concentration of a
polymer in solution and the mode of interaction of macromolecules between each
other and with the solvent. We shall consider experimental data to illustrate
this and to show exceptional possibilities which are opened up through the use
of the Weissenberg effect as a physicochemical characteristic of polymer
systems.
1.
THE RELATION BETWEEN
Let
us compare the set of indices of the flow properties of a series of polybutadienes having a narrow molecular-weight distribution
(the ratio of the weight-average to the number-average molecular weight not
exceeding 1.1). For definiteness we shall consider the quantities characterizing
the properties of polymers in the region of low shear rates ý in which the tangential
(shear) stresses t are proportional to ý in steady shear flow. In this sense the flow is Newtonian and the
corresponding region of low shear rates may be termed linear. In visco-elastic polymeric liquids, however, even at these low
shear rates there arise recoverable deformations, ge which
are superimposed on plastic flow. This leads in turn to the appearance of
normal stresses characterized by the primary difference of normal stresses sw.
In
the region of low ý values the
relations ge ~ t and sw ~ ý2 are fulfilled. Formally,
these relations are limiting for the case ý → 0, but practically there can
always be found such a region of low shear rates in which deviations from the
indicated relations do not fall outside the limits of possible experimental
errors.
The
quantitative indices of the flow properties of polymers at low shear rates are
the “initial” values of the coefficients of viscosity ho= t/ý of the
high-elasticity (or shear modulus) Go=t/ýe, and of the primary difference of
normal stresses zo=sw /2 ý2.
The
character of the molecular weight dependence of all these quantities is shown
in Fig. 1. The data of Fig. 1 refer to the region of sufficiently high
molecular weights, M, exceeding substantially the critical molecular weight Mcr at which macromolecules become capable of
forming a network of entanglements 2.
For polybutadienes Mcr ~ 5,600. In the region M > > Mcr the high-elastic modulus of monodisperse polymers is independent of the molecular
weight (though it is strongly dependent on polydispersity).
In this case the molecular-weight dependences of ho and zo are
satisfactorily described by power functions such as ho ~ Ma
and zo ~ Mb, where the values of the constants a and b are, respectively, a ~ 3.5-3.7 and b ~ 7.4-7.6.
In
the physical chemistry of polymers, the various measured parameters associated
with the molecular characteristics of a polymer are usually represented in the
form of power functions. Thus, wide use is made of the measurement of the
intrinsic viscosity [h] as a measure of molecular weight. The dependence of
[h] on M is also expressed by a power law, [h] ~ Mα, the
value of the exponent a ranging, as a general rule, from 0.5 to 1.
Fig. 1. The dependences of the coefficients ho, Go and zo on the molecular wieght of polybutadienes with narrow molecular-weight distributions (25ºC).
From
this it follows that, as compared with the indices of the flow properties ho and
especially zo, the quantity [h] is but very slightly sensitive to a change in
molecular weight. Indeed, when ordinary polymer-solvent systems are used, with
a twofold change in the molecular weight of a polymer the intrinsic viscosity
seldom changes by more than 50 per cent. At the same time the value of the
initial Newtonian viscosity changes by more than 10 times, while the value of
the primary coefficient of normal stresses zo
changes by more than 100 times. That zo changes incomparably
stronger than ho
depending on molecular weight, is also evidenced by the data of Fig. 1.
From this figure it is seen that in the range of molecular weights under study ho
changes approximately by 200 times and zo, does
so by more than 104 times. Thus, even if one takes into account the
low accuracy of the measurement of zo in
comparison to ho, these effects will still
remain incomparable. The measurement of zo may
therefore serve as a basis for the most sensitive physicochemical method of
determining the molecular weight of a polymer or of controlling its change in
the various technological processes.
2.
THE DEPENDENCE OF
A
change in the concentration of a polymer in the solution affects exceedingly
sharply all the indices of the flow properties. However, in this case too, just
as with the above-mentioned molecular weight dependences of the various
parameters, the change of the normal stress coefficient zo is
found to be the greatest compared with the change of the high-elastic modulus Go
and viscosity ho. This is clearly seen in
Fig. 2, in which a comparison is made of the concentration dependences of the
parameters Go, ho and zo for a
series of concentrated solutions of polybutadienes. In
this case, with a tenfold change in Go and a change of the viscosity
ho by 7
decimal orders, the value of the normal stress coefficient changes by a factor
of 1012. Therefore
the measurement of zo may serve as a basis for
a method of controlling the concentration of a polymer solution, a method
highly sensitive to the minimum variations in the content of the polymer in the
system.
In comparing the concentration dependences of the viscosity ho and the coefficient of the primary difference of normal stresses zo of polymers having different molecular weights the choice of the main parameter is of decisive importance. One may use, as such parameter, the product of the volume fraction of a polymer in solution by the intrinsic viscosity (c[h]). Here it is convenient to make use3, as a viscosity-determining quantity, of the dimensionless parameter hsp/(c[h]), where hsp = (ho - hs)/hs, is the specific viscosity of the solution and hs is the viscosity of the solvent. It should in this case be borne in mind that the factor (c[h]) is a measure of the volume filling of the system by a polymer. As seen from Fig. 3, the results of the experimental measurements of ho, and zo for solutions of polymers having different molecular weights can in fact be represented in the form of reduced dependences on the dimensionless parameter (c[h]). This allows one to assert that the structural phenomena responsible for the strong concentration dependences of the rheological parameters ho, and zo are common for all polymer solutions. From Fig. 3 one can clearly see the same general regularity, which was noted in considering the data in Fig. 2, namely, that zo changes incomparably more strongly than ho. Indeed, for the range of the values of (c[h]) given in Fig. 3, the reduced viscosity changes by a factor of up to 108 and the normal stress coefficient zo by more than 1013 times.
Fig. 2. The concentration dependences of the coefficients ho, Go and zo for solutions of polybutadienes having a molecular weight of 1.52x105 in methyl-napthalene (25ºC).
Another
important fact emerges from Fig. 3, which is common for the manifestations of
the specific features of the visco-elastic properties
of polymers. This is the existence of a critical concentration or a narrow
region of concentrations, in which the character (rate) of the concentration
dependences of ho and zo
undergoes a change. By analogy with the dependence of ho on
molecular weight, such concentration is called critical and its existence
should be attributed to the formation in the solution of a fluctuating network
of entanglements. The corresponding concentration is marked in Fig. 3 by
arrows. It lies in the region of (c[h]) values of the order of about 6.
The
passage through the critical concentration is associated not only with the
change of the character of the concentration dependences of the parameters ho and zo but
also with the radical change in the shape of the concentration dependence of
the parameter (ho2/zo) which has the dimensions
of the modulus and characterizes the rigidity of the system.
As will be shown in the next section, in the region of high concentrations, much higher than the critical, this quantity has just the same physical meaning as the high-elastic modulus of the solution.
Fig. 3. The dependences of the dimensionless viscosity (hsp/ch) (filled signs) and the primary normal stress coefficient (open signs) on the dimensionless concentration (ch.) for solutions of polystyrene in decalin (25ºC). The molecular weights of the polymer are; 1.5 x 105 (r); 2.4 x 105 (O); 3.6 x 105 (¨). The arrows indicate the region of the critical concentration.
The
dependence of (ho - hs)2/zo on the dimensionless concentration (c[h]) is shown in Fig. 4. A correction for the viscous
dissipation of the solvent, (hs), is immaterial at high
values of (c[h]), but it must be taken into account in the region
of low concentrations since the contribution of the polymer itself to the visco-elastic properties of the solution is of interest. It
is obvious that in the region of (c[h]) <6 the rigidity of the system falls with
increasing concentration, while after passing through the critical
concentration it increases with increasing content of polymer in solution. A
change in the concentration dependence of the quantity (ho - hs)2/zo in the region of the critical value of the parameter
(c[h]) is determined by the
fact that the contribution of single macromolecules to the resistance of the
system to deformation is replaced by the determinative effect of the network of
entanglements which displays high-elasticity (i.e., the ability to undergo high
macroscopic recoverable deformations). In the region of dilute solutions the
deformation of individual chains is hampered by their quick relaxation and the
system therefore exhibits high rigidity. As the concentration increases the
relaxation slows down and the rigidity of the system decreases. After the
formation of a network the decisive factor hindering the deformation of a
macromolecular chain is the interaction in the entanglement nodes, which is
intensified due to an increase in the number of nodes per each chain.
Therefore, with increasing concentration the rigidity of the system increases
in this region of concentrations. Thus, the dependence of (ho - hs)2/zo on (c[h]) becomes complicated (see Fig. 4), and to the
minimum of the value of the parameter (ho - hs)2/zo there
correspond the difference in the structure of the solutions in the pre-and
post-critical regions and the effect of the formation of a structural network
of fluctuating entanglements of macromolecules in the solution.
3.
CORRESPONDENCE BETWEEN VARIOUS CHARACTERISTICS OF THE VISCO-ELASTIC PROPERTIES
OF POLYMER SYSTEMS
Even
in his early works Weissenberg pointed to the existence of a connection between
the appearance of normal stresses in shear flow and the high-elasticity of a
flowing medium. This idea as applied to the various rheological
models gave rise 4-6 to a theoretical relation between the above rheological parameters. This relation has the following
form
zo = ho2/Go
which
is a direct corollary of the Lodge equation:
ge = sw/2t
Though
there is a difference of opinion in the literature 7 as to the exact
form of the correspondence between the parameters characterizing the rheological properties of a flowing visco-elastic
liquid the very existence of such a correspondence seems to be indisputable.
Fig . 4. The Parameter (ho - hs)2/zo versus the dimensionless concentration (cho) for solutions of polybutadienes in methylnapthalene. The molecular weights of the polymer are; 1.5 x 105 (r); 2.4 x 105 (O); 3.6 x 105 (¨).
We
have directly measured all the three quantities figuring in the above formula with
a view to checking the theoretical equations given above. These measurements
were carried out in the region of low shear rates in which, within the accuracy
of measurements (not lower than ±10 per cent for each parameter) t ~ ý, ýe ~ t and sw ~ ý2, i.e., the
values of the parameters ho, zo and Go
could be found directly, without resorting to extrapolation. The results of the
measurements are presented in Fig. 5, where the dotted line indicates
the region of permissible experimental deviations from the strict equality of
the high-elastic modulus Go and the parameter ho2/ zo. The measurements were
performed in the region of concentrated solutions where (c[h])>>6 and a three-dimensional network of
entanglements exists in the solution. Evidently, in the region of high
concentrations the above theoretical relations are fulfilled. This result is in
agreement with the conclusions made in a recent, very thorough experimental
investigation 8.
Fig. 5. The relation between the parameter (ho2/zo) and the high-elastic modulus Go. The data have been obtained in investigations of the flow properties of solutions of polybutadiences having different molecular weights in methylnapthalene. The dashed line indicates the limits of experimental errors for the theoretical equality ho2/zo = Go.
A
direct correlation between the normal stresses and the high-elasticity in shear
flow however exists only in the region of sufficiently concentrated solutions.
The point is that upon transition to the region of dilute solutions, when the
value of (c[h]) becomes less critical, the solution ceases to be
high-elastic, though it still remains visco-elastic.
To this there corresponds the left portion of the curve in Fig. 4, where the
parameter (ho2/ zo) falls with increasing concentration. This was
explained above as being a consequence of the deformability of single
macromolecules not bound into a network. But in the absence of a network no
high (finite) recoverable deformations can develop as well. Therefore the
concept of the high-elastic modulus becomes inapplicable and the question of
the correlation between the parameter (ho2/ zo) and the modulus loses its physical meaning. Indeed,
in the region of dilute solutions, which is described, for polybutadiene
solutions, by the inequality (c[h])<6, no elastic recoil has been detected after
the interruption of flow, that is, all the accumulated deformation is
irrecoverable. However, normal stresses develop in this region of
concentrations too, which is evidently due to the deformability and orientation
of individual macromolecular chains not bound into a single network, though
capable of storing elastic energy because of the deviations from equilibrium
conformations. It is this factor that leads to the appearance of normal
stresses in shear flow.
Thus,
normal stresses in shear flow are in all cases caused by the deformability of
macromolecular chains. Two cases are possible, however: this elastic
deformability gives rise to the local effect in dilute solutions or has
continuum character in concentrated solutions. In the first case the system has
visco-elastic properties, which does not cause
development of macroscopic recoverable deformations. In the second case, the
elasticity of individual chains, because of their being bound by fluctuating
entanglement nodes, leads to the appearance of the high-elasticity of the
solution as a whole, i.e., the polymer system becomes not only visco-elastic but also high-elastic.
4.
With
increasing shear rate the dependence of normal stresses on shear rate becomes
weaker than the quadratic. This can be expressed as the decrease of the normal
stress coefficient z
= sw/2ý2
with increasing ý, just as the
effective viscosity h
= t/ý falls with increasing shear rate. However, in this region of
shear rates too, which is often defined by the term “non-linear region” up to
rather high shear rates, the correspondence between normal and shear stresses
is retained in the form
sw ~ t2
or in
one of the following formulae.
This
means that in the far non-linear region the mode of the development of
anomalous viscosity is intimately connected with the magnitude of normal
stresses, which is illustrated in Fig. 6.
It
is only at high shear rates, several decimal orders away from the linear region, that the relations given above are no longer
fulfilled and the rate of increase of normal stresses is slowed down.
Deviations from the limiting values of ho and zo for
shear and normal stresses begin to take place in one and the same region of
shear rates, though, according to the above relations, the sensitivity of the
coefficients h and z to the transition to the non-linear region is different. It is essential, however, that in the
non-linear region normal stresses continue to reveal themselves as the effect
quadratic with respect to the shear stresses (but not to the shear rate!).
Upon
transition to the non-linear region the high-elastic modulus becomes dependent
on the shear rate, namely, G=t/ge
increases, as a rule, with increasing t. As can be easily seen, the result is that in the
non-linear region Lodge’s formula is no longer valid since at high ý values the relationship sw /2t = Go £ G(ý) is
fulfilled. Deviations from this formula become larger with increasing shear
rate and may exceed the decimal order in the region of strongly pronounced
non-linearity.
In
describing the visco-elastic properties of polymer
systems in the non-linear region very wide use is made of the dimensionless
parameter (ýqo) as
the argument of the corresponding functions.
Here,
by the quantity qo is understood a certain
characteristic relaxation time for a system, whose definition is not unique.
Attempts are often made to connect it with the various relations following from
various molecular models. As shown in a discussion9 of the viscous
properties of polymer solutions in the non-linear region, generalization of the
dependences found experimentally for different solutions can be made if by the
quantity qo is meant the relation qo = ho/Go
= zo./ho. The
dependence of qo on the concentration and
molecular weight of a polymer are found to be different from those following from the familiar molecular models due
mainly to the complexity of the relations between Go and these
parameters of the system.
Fig. 6. The relation between the normal and shear stresses over a wide range of shear rates for solutions of polybutadienes in cetane. The concentrations of the polymer are 9(u); 20(r); 51(s); 85(¨); 100(O).
Fig. 7. The character of variation of the normal stress coefficient in the non-linear region versus the dimensionless shear rate (ýqo) for solutions of polystyrene in decalin (25ºC). The concentrations of the polymer are (in percent): 50(r); 40(¨); 30(s); and 20(u).
The
use of the quantity qo = ho/Go
as the principal relaxational characteristic of the
system enables one to generalize experimental data on the dependence of the
normal stress coefficient on the shear rate z(ý) by
using the same general method as in the case of the dependence h(ý). An
example is given in Fig. 7.
Thus,
the various aspects of the rheological behaviour of
polymer systems in the non-linear region of their mechanical behaviour are
found to be closely interrelated and the manifestations of the non-linear
effects are determined by the value of the dimensionless parameter (ýqo) = ýho/Go
= ýzo/ho. In this sense the
quantities zo and ho are
the fundamental characteristic of the properties of the system not only in the
linear region of low shear rates but also in the region of non-linear
behaviour. It should be noted here that the quantities ho and zo are
determined by the relaxation spectrum of the system in the linear region and
are expressed as its first and second moments 5,6.
REFERENCES
1. Weissenberg, K. and R. O. Herzog, Kol.-Z., 46, 277 (1928); K. Weissenberg, Nature, 159, 310 (1947); Proc. 1st Intern. Congr. Rheol., 1,
29, 46 (1948). Proc.
Royal Soc.,
2. Porter, R. S. and J. F. Johnson, Chem. Revs., 66, 1 (1966).
3. Utracki,
L. and R. Simba, J. Polymer
Sd., A, 50,
1084 (1963).
4. Lodge, A. S., Elastic Liquids,
Acad. Press,
5. Coleman, B. D. and H. Markovitz, J.
Appl. Phys., 35,
1(1964); H. Markovitz, in Rheology,
ed. F. R. Eirich, Acad. Press,
6. Malkin,
A. Ya, Rheol.
Acta, 7, 335 (1966).
7. Philippoff,
W., Trans. Soc. Rheol., 1O,
p. 1, 1 (1966); W. Philippoff and R. A. Stratton, Trans. Soc. Rheol.,
10, p. 2, 467 (1966).
8. Stratton, R. A.
and A. F. Butcher, J. Polymer Sci., A-2, 9, 1703 (1971).
9. Vinogradov, G. V., A. Ya. Malkin, and G. V. Berezhnaya, Vysokomol. Soedin., 13A, 2993 (1971).
Preface / Acknowledgements / Biographical Notes
Weissenberg’s Influence on Crystallography
Karl Weissenberg and the Development of X-Ray Crystallography
The Isolation of, and the Initial Measurements of the Weissenberg Effect
The Role of Similitude in Continuum Mechanics
Elasticity in Incompressible Liquids
The Physical Meaning of Weissenberg's Hypothesis with Regard to the Second Normal-Stress Difference
A Study of Weissenberg's Holistic Approach to Biorheology
The Weissenberg Rheogoniometer Adapted for Biorheological Studies
Weissenberg’s Contributions to Rheology
The Early Development of the Rheogoniometer
Some of Weissenberg's More Important Contributions to Rheology: An Appreciation
Publications of Karl Weissenberg and Collaborators / List of Contributors