CHAPTER
3.3 Dilute Solution Viscometry
3.3.3 Measurement of Intrinsic Viscosity
Laboratory devices are available to measure intrinsic viscosities without human intervention. These are useful where many measurements must be made. The basic principles involved are the same as those in the glass viscometers which have long been used for this determination. An example of the latter is the Ubbelohde suspended level viscometer shown in Fig. 3.7. In this viscometer a given volume of polymer solution with known concentration is delivered into bulb B through stem A. This solution is transferred into bulb C by applying a pressure on A with column D closed off. When the pressure is released, any excess solution drains back into bulb B and the end of the capillary remains free of liquid. The solution flows from C through the capillary and drains around the sides of the bulb E. The volume of fluid in B exerts no effect on the rate of flow through the capillary because there is no back pressure on the liquid emerging from the capillary. The flow timetis taken as the time for the solution meniscus to pass from marka to mark b in bulb C above the capillary. The solution in D
Table 3.1 MarkHouwinkSakurada Constantsa
Polymer Solvent
Temperature (C)
Molecular weight range of calibration samples (M31024)
K3103 (cm3g21) a
Polystyrene Toulene 25 1-160 17 0.73
Polystyrene Benzene 25 0.4-1 100 0.50
Polyisobutene Decalin 25 .500 22 0.70
Poly(methyl acrylate)
Acetone 25 28-160 5.5 0.77
Poly(methyl methacrylate)
Acetone 25 8-137 7.5 0.70
Poly(vinyl alcohol)
Water 30 1-80 43 0.64
Nylon-6 Trifluoroethanol 25 1.3-10 53.6 0.74
Cellulose triacetate
Acetone 20 2-14 2.38 1.0
aFrom Brandrup, J., Immergut, E. H., Grulke, E. A., Abe, A., Bloch, D. R., Eds.,“Polymer Handbook”, 4th ed., Wiley, New York, 2005
Capillary
Ubbelohde viscometer E B
A D
c
a b
FIGURE 3.7
Ubbelohde suspended level viscometer.
122 CHAPTER 3 Practical Aspects of Molecular Weight Measurements
can be diluted by adding more solvent through A. It is then raised up into C, as before, and a new flow is obtained.
The flow time is related to the viscosity η of the liquid by the HagenPoiseuille equation:
η5πPr4t=8Ql (3-87)
wherePis the pressure drop along the capillary which has lengthland radiusrfrom which a volumeQof liquid exits in timet. It is necessary to compare the flow behav- ior of pure solvent with that of solution of concentrationc. We will subscript the terms related to solvent behavior with zeros. The average hydrostatic heads,hand h0, are the same during solvent and solution flow in this apparatus, becausetis the time taken for the meniscus to pass between the same fiducial marksa andb. Then the mean pressures driving the solvent and solution arehρ0gandhρg, respectively, where g is the gravitational acceleration constant and ρ is a density (compare Eq. 3-28). For dilute solutionsρis very close toρ0and it follows fromEq. (3-87)that
η=η05t=t0 (3-88)
wheret0is flow time for the solvent andtthat for the solution. Thus, the ratio of viscosities needed inEq. (3-64)can be obtained from flow times without measur- ing absolute viscosities. The intrinsic viscosity [η] is defined in the above equa- tion as a limit at zero concentration. Theη/η0ratios which are actually measured are at finite concentrations, and there are a variety of ways to estimate [η] from these data. The variation in solution viscosity (η) with increasing concentration can be expressed as a power series in c. The equations usually used are the Huggins equation[9]:
1 c
η η0
21
51 c
t t0 2t
5½ η 1kH½η2c (3-89) and the Kraemer equation[10]:
c21lnðη=η0Þ5½η2k1½η2c (3-90)
EXAMPLE 3-2
The results of one lab group for the polystyrene/toluene solution at 25C were as follows:
Solution Flow times (s)
Pure toluene 72.9 73.6 73.8
Initial solution (0.7 g/100 mL toluene) 228.9 229.4 228.9 10 mL initial solution110 mL toluene 137.6 136.8 137.4 Using given data andKandavalues for the above solution fromTable 3.1, estimate the viscosity average molecular weight of the polymer.
Solution
By examining the Huggins equation, one finds that only two flow times corresponding to two concentrations,C1andC1/2, are needed to obtain [η]:
1 C
tt0
(x0, y0)
(x1/2, y1/2)
(x1, y1)
C1 = 2C1/2
C1/2 C1
C [η]
— 1
By collinearity,
y1/22y0
x1/22x05y12y0
x12x0
i:e:;
C11/ 2
t1/2
t0 21
2½ η C1/220 5
1 C1
t1
t021
2½ η C120 C1
C1/2 t1/2
t0 21
2C1½ η5C1/2 2C1
t1
t021
2C1/2½ η η
½ 5 2 C1/2
t1/22t0
t0
2 1 2C1/2
t12t0
t0
η
½ 52t114t1/223t0
C1t0
Taking the average flow times from experimental data, t0573:4 s; t15229:1s; t1/25137:3s
½η5 2229:114ð137:3Þ23ð73:4Þ 73:430:7 g=100 mL 5194 mL=g
FromTable 3.1,a50.73 andK50.017 at 25C.
i.e.,
19450:017ðMvÞ0:73 ln 194
0:017 0
@ 1
A50:73lnðMvÞ MvD361000
124 CHAPTER 3 Practical Aspects of Molecular Weight Measurements
It is easily shown that both equations should extrapolate to a common inter- cept equal to [η] and that kH1k1should equal 0.5. The usual calculation proce- dure involves a double extrapolation ofEqs. (3-89) and (3-90) on the same plot, as shown in Fig. 3.8. This data-handling method is generally satisfactory.
Sometimes experimental results do not conform to the above expectations. This is because the real relationships are actually of the form
c21ðη=η021Þ5½η1kH½η2c1k0H½η3c21? (3-91) and
c21lnðη=η0Þ5½η2k1½η2c2k01½η3c22? (3-92) and the preceding equations are truncated versions of these latter virial expres- sions in concentration. No two-parameter solution such asEq. (3-89) or (3-90) is universally valid, because it forces a real curvilinear relation into a rectilinear form. The power series expressions may be solved directly by nonlinear regres- sion analysis[11], but this is seldom necessary unless it is desired to obtain very accurate values of [η] and the slope constants kHandk1.
The term kHinEq. (3-89)is called “Huggins constant.” Its magnitude can be related to the breadth of the molecular weight distribution or branching of the sol- ute. Unfortunately, the range ofkHis not large (a typical value is 0.33) and it is not determined very accurately because Eq. (3-89) fits a chord to the curve of Eq. (3-91), and the slope of this chord is affected by the concentration range in which the curve is used.
A useful initial concentration for solution viscometry of most synthetic poly- mers is about 1 g/100 cm3 solvent. High-molecular-weight species may require lower concentrations to produce a linear plot ofc21(η/η021) againstc(Fig. 3.8),
C [η]
η η0
( — –1) C– 1
ηη0 n (—–) C– 1
FIGURE 3.8
Double extrapolation for graphical estimation of intrinsic viscosity.
which does not curve away from the c axis at the high concentrations. At very low concentrations, such plots may also curve upward. This effect is thought to be due to absorption of polymer on the capillary walls and can be eliminated by avoiding such high dilutions.
EXAMPLE 3-3
The tables below give the mean flow times (t) in a suspended-level viscometer recorded for solutions of two of five monodisperse samples of polystyrene at various concentrations (c) in cyclohexane at 34C. Under these conditions, the mean flow time (t0) for cyclohexane is 151.8 s.
Sample B
c,3103g/cm3 t, s
1.586 158.5
3.172 166.5
Sample E
c,3103g/cm3 t, s
1.040 176.1
2.080 209.20
Determine the intrinsic viscosities of these samples. The intrinsic viscosities of the other three polystyrene samples were evaluated under the same conditions and are given in the fol- lowing table together withMwvalues determined by light scattering.
Polystyrene sample Mw, g/mol [η], cm3/g
A 37,000 15.77
B 102,000
C 269,000 42.56
D 690,000 68.12
E 2,402,000
Using these data together with the calculated values of intrinsic viscosities for samples B and E, evaluate the constants of the MarkHouwinkSakurada equation for polystyrene in cyclohexane at 34C. What can you deduce about the conformation of the polystyrene chains under the conditions of the viscosity determinations?
Solution
Sample B:t05151:8 s; t15166:5 s; t1/25158:5 s
½η5 2166:514ð158:5Þ23ð151:8Þ 3:172310233151:8 525:13 cm3=g
126 CHAPTER 3 Practical Aspects of Molecular Weight Measurements
Sample E:t05151:8 s; t15209:2 s; t1/25176:1 s
½η52209:214ð176:1Þ23ð151:8Þ 2:08310233151:8 5126:05 cm3=g
A plot of ln [η] vs. lnMwill yield the MarkHouwink constants for the system of interest:
M lnM ln [η]
37,000 10.5 2.76
102,000 11.5 3.22
269,000 12.5 3.75
690,000 13.4 4.22
2,402,000 14.7 4.84
10.0
ln K = –2.5⇒ K = 0.08 cm3/g ln[η]
Slope = a≈ 4.5 – 2.5 14 – 10 5
4
3
2
1
0
11.0 12.0 13.0
ln M
14.0 15.0
= 0.5
The polystyrene chains are in their unperturbed state sincea50.5.