DILUTE SOLUTION THERMODYNAMICS,
3.8 INTRINSIC VISCOSITY
Both the colligative and the scattering methods result in absolute molecular weights; that is, the molecular weight can be calculated directly from first prin-ciples based on theory. Frequently these methods are slow, and sometimes expensive. In order to handle large numbers of samples, especially on a routine basis, rapid, inexpensive methods are required. This need is fulfilled by intrin-sic viscosity and by gel permeation chromatography. The latter is discussed in the next section.
Intrinsic viscosity measurements are carried out in dilute solution and result in the viscosity-average molecular weight; see Figure 3.4 and equation (3.34).
Consider such a dilute solution flowing down a capillary tube (Figure 3.12).
The flow rate, and hence the shear rate, is different depending on the distance from the edge of the capillary. The polymer molecule, although small, is of finite size and “sees” a different shear rate in different parts of its coil. This change in shear rate results in an increase in the frictional drag and rotational forces on the molecule, yielding the mechanism of viscosity increase by the polymer in the solution.
Figure 3.12 The effect of shear rates on polymer chain rotation. Hydrodynamic work is con-verted into heat, resulting in an increased solution viscosity.
3.8.1 Definition of Terms
Several terms need defining. The solvent viscosity is h0, usually expressed in poises, Stokes, or, more recently, Pascal seconds, Pa·s. The viscosity of the polymer solution is h. The relative viscosity is the ratio of the two,
(3.84)
where h0is the viscosity of the solvent.
Of course, the relative viscosity is a quantity larger than unity. The specific viscosity is the relative viscosity minus one:
(3.85) Usually hspis a quantity between 0.2 and 0.6 for the best results.
The specific viscosity, divided by the concentration and extrapolated to zero concentration, yields the intrinsic viscosity:
(3.86)
For dilute solutions, where the relative viscosity is just over unity, the follow-ing algebraic expansion is useful:
(3.87)
Then, dividing ln hrelby c and extrapolating to zero concentration also yields the intrinsic viscosity:
(3.88)
Note that the natural logarithm of hrel is divided by c in equation (3.88), not hrel itself. The term (ln hrel)/c is called the inherent viscosity. Also note that the intrinsic viscosity is written with h enclosed in brackets. This is not to be confused with the plain h, which is used to indicate solution or melt viscosities.
Two sets of units are in use for [h]. The “American” units are 100 cm3/g, whereas the “European” units are cm3/g. Of course, this results in a factor of 100 difference in the numerical result. Lately, the European units are becom-ing preferred.
ln h( rel) h È
ÎÍ
˘
˚˙ =[ ] c c 0=
lnh lnh h h
rel sp sp
sp2
= ( +1) @ - +
2 L
hsp h
c c È ÎÍ
˘
˚˙ =[ ]
=0
hsp=hrel- 1
h h
rel=h
0
3.8 INTRINSIC VISCOSITY 111
3.8.2 The Equivalent Sphere Model
In assuming a dilute dispersion of uniform, rigid, noninteracting spheres, Einstein (69,70) derived an equation expressing the increase in viscosity of the dispersion,
(3.89) where the quantity v2represents the volume fraction of spheres. The intrinsic viscosity of a dispersion of Einstein spheres is 2.5 for v2, or 0.025 for concen-tration in units of g/100 cm3.
Now consider a coiled polymer molecule as being impenetrable to solvent in the first approximation. A hydrodynamic sphere of equivalent radius Rewill be used to approximate the coil dimensions (see Figure 3.13). In shear flow, it exhibits a frictional coefficient of f0. Then according to Stokes law,
(3.90) where Re remains quantitatively undefined.
The Einstein viscosity relationship for spheres may be written
(3.91)
where n2/V is the number of molecules per unit volume. Of course, Ve = (4p/3)R3e. The quantity n2Ve/V is the volume fraction of equivalent spheres,
h h h- h
= = Ê
Ë ˆ
¯
0
0
2 5 2
sp . n
V Ve
f0 = p h6 1 0Re
h=h0(1 2 5+ . v2)
Figure 3.13 The equivalent sphere model.
yielding the familiar result that the viscosity of an assembly of spheres is inde-pendent of the size of the spheres, depending only on their volume fraction.
Writing
(3.92)
where c is the concentration and NAis Avogadro’s number,
(3.93) Note that
(3.94)
and
(3.95) where a is the expansion of the coil in a good solvent over that of a Flory q-solvent.
The quantity R2e0/M is roughly constant. The same constant appears in Brownian motion statistics, where time takes the place of the molecular weight. This expresses the distance traveled by the chain in a random walk as a function of molecular weight. According to Flory (17), the expansion of the coil increases with molecular weight for high molecular weights as M0.1, yielding
(3.96)
3.8.3 The Mark–Houwink–Sakurada Relationship
In the late 1930s and 1940s Mark, Houwink, and Sakurada arrived at an empir-ical relationship between the molecular weight and the intrinsic viscosity (71):
(3.97) where K and a are constants for a particular polymer–solvent pair at a par-ticular temperature. Equation (3.97) is known today as the Mark–Houwink–
Sakurada equation. This equation is in wide use today, being one of the most important relationships in polymer science and probably the single most
[ ]h = KMVa
h p
[ ]= Ê a
Ë ˆ 2 54 ¯
3
1 20 3 2
1 2 3
. N R
Me M
A
Re =Re0a V
M
R M
R
M M
e e e
= = Ê
Ë ˆ
¯ 4
3
4 3
1 3
1 2 3 2
p p 1 2
hsp h A
c
N V
c M È e
ÎÍ
˘
˚˙ =[ ]=
=0
2 5. n
V cN
M
2 = A
3.8 INTRINSIC VISCOSITY 113
important equation in the field. Values of K and a for selected polymers are given in Table 3.10 (72). It must be pointed out that since viscosity-average molecular weights are difficult to obtain directly, the weight-average molecu-lar weights of sharp fractions or narrow molecumolecu-lar weight distributions are usually substituted to determine K and a.
According to equation (3.96) the value of a is predicted to vary from 0.5 for a Flory q-solvent to about 0.8 in a thermodynamically good solvent. This corresponds to a increasing from a zero dependence on the molecular weight to a 0.1 power dependence. More generally, it should be pointed out that a varies from 0 to 2; see Table 3.11.
The quantity K is often given in terms of the universal constant F,
(3.98) where represents the mean square end-to-end distance of the unperturbed coil. If the number-average molecular weights are used, then F equals 2.5 ¥ 1021dl/mol · cm3. A theoretical value of 3.6 ¥ 1021dl/mol · cm3can be calculated
r02
K r
= ÊM ËÁ ˆ
¯˜ F 0
2 3 2
Table 3.10 Selected intrinsic viscosity–molecular weight relationship, [h] = KMav(77)
Polymer Solvent T(°C) K ¥ 103a ab
cis-Polybutadiene Benzene 30 33.7 0.715
it-Polypropylene 1-Chloronaphthalene 139 21.5 0.67
Poly(ethyl acrylate) Acetone 25 51 0.59
Poly(methyl methacrylate) Acetone 20 5.5 0.73
Poly(vinyl acetate) Benzene 30 22 0.65
Polystyrene Butanone 25 39 0.58
Polystyrene Cyclohexane (q-solvent) 34.5 84.6 0.50
Polytetrahydrofuran Toluene 28 25.1 0.78
Polytetrahydrofuran Ethyl acetate hexane 31.8 206 0.49 (q-solvent)
Cellulose trinitrate Acetone 25 6.93 0.91
Source: J. Brandrup and E. H. Immergut, eds., Polymer Handbook, 2nd ed., Wiley, New York, 1975, sec. IV.
aEuropean units, concentrations in g/ml. Units do not vary with a. Units of K are cm3·mol1/2/g3/2.
bThe quantity a, last column, is the exponent in equation (3.97).
Table 3.11 Values of the Mark–Houwink–Sakurada exponent a
a Interpretation
0 Spheres
0.5–0.8 Random coils
1.0 Stiff coils
2.0 Rods
from a study of the chain frictional coefficients (17).†For many theoretical pur-poses, it is convenient to express the Mark–Houwink–Sakurada equation in the form:
(3.99) If the intrinsic viscosity is determined in both a Flory q-solvent and a
“good” solvent, the expansion of the coil may be estimated. From equation (3.95),
(3.100) Values of a vary from unity in Flory q-solvents to about 2 or 3, increasing with molecular weight.
3.8.4 Intrinsic Viscosity Experiments
In most experiments, dilute solutions of about 1% polymer are made up. The quantity hrel should be about 1.6 for the highest concentration used. The most frequently used instrument is the Ubbelhode viscometer, which equalizes the pressure above and below the capillary.
Several concentrations are run and plotted according to Figure 3.14. Two practical points must be noted:
h hq a [ ] [ ] = 3
h a a
[ ]= Ê ËÁ ˆ
¯˜ =
F r
M0 M M
2 3 2
1 2 3 K 1 2 3
3.8 INTRINSIC VISCOSITY 115
Figure 3.14 Schematic of a plot of hsp/c and ln hrel/c versus c, and extrapolation to zero con-centration to determine [h].
†A widely used older value of F is 2.1 ¥ 1021dl/mol · cm3.
1. Both lines must extrapolate to the same intercept at zero concentration.
2. The sum of the slopes of the two curves is related through the Huggins (73) equation,
(3.101) and the Kraemer (74) equation,
(3.102) Algebraically
(3.103) If either of these requirements is not met, molecular aggregation, ionic effects, or other problems may be indicated. For many polymer–solvent systems, k¢ is about 0.35, and k≤ is about 0.15, although significant variation is possible.
Noting the negative sign in equation (3.102), k≤ is actually a negative number.
The molecular weight is usually determined through light-scattering, as indicated previously. In order to determine the constants K and a in the Mark–Houwink–Sakurada equation, a double logarithmic plot of molecular weight versus intrinsic viscosity is prepared (see Figure 3.15) (75). Results of this type of experiment were used in compiling Table 3.11.
¢ + ¢¢ = k k 0 5. lnhrel h h
c =[ ]- ¢¢k [ ]2c
hsp h h
c =[ ]+ ¢k[ ]2c
Figure 3.15 Double logarithnmic plots of [h] versus MW for anionically synthesized poly-styrenes, which were then fractionated leading to values of MW/Mn of less than 1.06. Filled circles in benzene, half-filled circles in toluene, and open circles in dichloroethylene, all at 30°C (75). The arrows indicate the axes to be used. Units for [h] in 100 ml/g.
While use of the viscosity-average molecular weight of a polymer in cali-brating K and a in equation (3.97), MVvalues usually are not initially known.
The calibration problem may be alleviated by one or more of the following methods:
1. Use of fractionated polymers.
2. Use of polymers prepared with narrow molecular weight distributions, such as via anionic polymerization.
3. Use of weight-average molecular weight of the polymer, since it is closer than the number-average molecular weight.
3.8.5 Example Calculation Involving Intrinsic Viscosity
Say we are interested in a fast, approximate molecular weight of a polystyrene sample. We dissolve 0.10 g of the polymer in 100 ml of butanone and measure the flow times at 25°C in an Ubbelhode capillary viscometer. The results are
Pure butanone 110 s
0.10% Polystyrene solution 140 s
Starting with equation (3.84), and nothing that the flow time is proportional to the viscosity,
As an approximation, assume that the concentration is near zero, and the [h]
= 2.73 ¥ 102ml/g, equation (3.86). This obviates the extrapolation in Figure 3.14 that is required for more accurate results. Using the Mark–Houwink–
Sakurada relation, equation (3.97) and Table 3.11, we have
Note that the units of K are irregular, depending on the value of a. Usually the units of K are omitted from tables.