L. A. HOLMES, K. NINOMIYA, AND J. D. FERRY
2714
The Steady-State Compliance of Dilute Polymer Solutions
by Larry A. Holmes, Kazuhiko Ninomiya,’ and John D. Ferry Department of Chemistry, (initersity of Wisconsin, iMadiaon, Wisconsin 53706
(Receiced September 20, 1966)
In extremely dilute polymer solutions, the steady-state compliance J , should be directly proportional to concentration; with increasing concentration, it should pass through a maximum and become inversely proportional to concentration. Experimental determinations from low-frequency dynamic mechanical measurements on solutions of a polystyrene (111 = 267,000) in two chlorinated biphenyls, over a wide range of concentrations, confirm the presence of a maximum. The concentration dependence of J , in the neighborhood of the maximum is calculated from the theories of Rouse and Zimm and a phenomenological theory of Sinomiya and Ferry. The latter conforms best to the shape observed experimentally; it predicts the maximum to be at a concentration of the order of the reciprocal of the intrinsic viscosity. With increasing concentration, the magnitude of J , or of the low-frequency storage modulus shifts gradually from the Zimm prediction to the Rouse prediction; from this behavior, the concentration dependence of the hydrodynamic interaction parameter h can be obtained, and it agrees with previous conclusions from the form of the frequency dependence of dynamic properties. Values of J , are calculated from the phenomenological theory and compared with experimental results on a number of solutions of polystyrene and polyisobutylene. Over a 200-fold range of J,, the calculated values agree fairly well but are somewhat too small, a discrepancy which can be partly attributed to a slight degree of molecular weight heterogeneity.
Introduction
J,
The steady-state compliance, J,, is an important property of :iviscoelastic material, being a measure of the stored energy in steady flow under small stresses. Its product with the shear stress is sometimes called the recoverable shear.* Besides its role in linear viscoelastic behavior, it has added importance through its close relation to normal stress p h e n ~ m e n a . ~In concentrated macromolecular solutions of undiluted polymers, J , can be obtained from measurements of Cree7 or indirectly from stress rela~ation,~ but these methods are not readily applicable to dilute solutions and the latter have therefore received little attention. However, quite dilute solutions can be subjected to sinusoidally varying deformations5 to measure the storage and loss components of the shear modulus, G’ and G”. At low frequencies, G” is proportional to the circular frequency w and G’ is proportional to w 2 . Thus, at very low frequencies, G‘ > 1. A dilute polymer solution might be expected to resemble a binary polymeric blend with one component of very low molecular weight, so that at extremely low concentrations J , would increase with concentration but would soon pass through a maximum. At moderate to high concentrations, J , would be expected to be inversely proportional to polymer concentration, l1 and this relation has been confirmed in the range from 50 to 100% p ~ l y m e r . ~We ~ ’ develop ~~ here for dilute solutions the predictions of molecular theories and of the phenomenological theory described above, and compare them with experimental data, mostly on polystyrene. Predictions of Molecular Theories. In the theories of Rouse,14 Zimni,15 and Tschoegl,16 the coefficient A is given by
A = (cRT/M)712S’ 71
= (7
- V177s)/(CRT/IJ/I)S
s’ = c ( 7 k / 7 1 ) ’ s = C(nC/d
(34 (3b) (34
(34
where c is the polymer concentration in grams per milliliter, M is the polymer molecular weight, rls is the solvent viscosity, and v1 is its volume fraction in the solution; 71 is the terminal relaxation t h e and 7 k are the
2715
other characteristic relaxation times specified by the respective theories. Although these theories are subject to modification to take into account a finite solute contribution to dynamic viscosity a t very high frequencie~,”-’~Peterlin’s recent formulation20 of this feature shows that a t vanishing frequency the original equations are unaffected. (Otherwise, they would fail to reduce properly to the respective treatments of Debye and Kirkwood-Riseman for steady-state flow.) Thus eq 3 may be combined with eq 1 to obtain after rearrangement
J,
=
(IW/cRT)(l - Ui/~r)2(S’/Sz)
(4) where qr, the relative viscosity, is q / v s . For the Rouse theory, S’/Sz = 0.400; for the Zimm, it is 0.206; for the intermediate cases of the Tschoegl theory, intermediate values may be obtained. At high concentrations, this reduces to J e = (M/cRT)(S’/S2),as commonly used with the expectation that the Rouse ~ low value of S’/Sz is a p p l i ~ a b l e . ~ ~At- ~extremely [qlc, [ q ] being the inconcentrations, where qr = 1 trinsic viscosity in milliliters per gram, it reduces to
+
J , = (Af/RT)[qI2c(S’/S2) (5) so that J , is directly proportional to c. For extremely low concentrations, the Zimni value of S‘/S2 should be applicable.’l Prediction of Phenomenological Theory. For a binary polymeric blend, the composition dependence of J , is givenloby
If component 1 is a solvent of low molecular weight, Je1 can be neglected, and N ~ >> I 1, so this becomes (8) K. E. Van Holde and J. W. Williams, J . Polymer Sci., 11, 243 (1953). (9) H . Leaderman, R. G. Smith, and L. C. Williams, ibid., 36, 233 (1959). (10) K. Ninomiya and J. D. Ferry, J . Colloid Sci., 18, 421 (1963)‘ (11) Reference 6, p 172. (12) K. Ninomiya, J. R. Richards, and J. D. Ferry, J . P h y s . Chem., 67,327 (1963). (13) T.oyanagi and J. D. Ferry, J . Colloid Sci., 21, 547 (1966). (14) P. E. Rouse, Jr., J . Chem. P h y s . , 21, 12i2 (1953). (15) B. H. Zimm, ibid., 24, 269 (1956). (16) N. W. Tschoegl, ibid., 39, 149 (1963); 40, 473 (1964). (17) J. Lamb and A. J. Matheson, Proc. R o y . SOC.(London), A281, 207 (1964). (18) W. Philippoff, Trans. SOC.Rheology, 8 , 117 (1964). (19) J. D. Ferry, L. A. Holmes, J. Lamb, and A J. Matheson, J . P h y s . Chem., 7 0 , 1685 (1966). (20) A. Peterlin, J . Polymer Sci., in press. (21) J. E. Frederick, N . W. Tschoegl, and J. D. Ferry, J . P h y s . Chem., 68, 1974 (1964).
V o l u m e YO, Number 9
September 1966
L. A. HOLMES,K. NINOMIYA, AND J. D. FERRY
2716
J e
= N212Je2v2/(1
+ v2N21)’
(7)
A t very low concentrations of component 2 (polymer), this reduces to J e
= N212Je2v2
(8)
There are several possible ways of estimating N21 for specific systems,10p22but we shall do so by identifying eq 8 and 5. We assume that for the undiluted polymer Jez= (&!t/p2RT)(S’/S2)~,where p2 is the polymer density and the subscript R refers to Rouse; that in eq 5 the Zimm value of S’/Sz is applicable; and we note that c = v2p2. It follows that N21
[ 7 1 P 2 ~ ( ~ ’ / ~ 2 ) ~ / ( ~ ’ / ~ 2 ) R ] 1 ’ a[ ? l P Z / d Z
(9)
By a different derivation, involving the concentration dependence of the viscosity,23Nzl can be identified with [ q ] p 2 2 / 1 - 2kf, where k‘ is the Huggins constant. This is equivalent to eq 9 if k‘ = 1/4, which is in the range observed for good solvents. The identification of N21 as closely related to [7] is of course consistent with its interpretationlo as the ratio of flow energy dissipations, per unit volume, of solute and solvent. It cannot be expected that Nc?l will remain constant over a wide range of concentrat i o n ~ , but ’ ~the ~ above ~ ~ ~ value ~ ~ deduced for extremely dilute solutions may be used tentatively in the moderately dilute range.
urements on solutions with lower viscosities, an alternative rod was fabricated which gives a gap of 0.0460 cm and A , = 22,540 cm. It was necessary to determine the latter by calibration with Aroclor of known viscosity rather than from the apparatus dimensions, since calculation from the latter becomes numerically uncertain when the gap is small. The constants6 B and M are also changed somewhat. (A wide-gap combination with A, = 454.7 is also available for solutions of much higher viscosity.)
Results The low-frequency behavior is fully described by the two quantities q and A , which are given in Table I together with values for three other solutions of the same polymer S-108 previously The steadystate compliance Jewas calculated from eq 1 and then reduced to 25” (if the measurement was made a t another temperature) by the equation J e (25”) = J,Tp/ Topo,where TO = 298°K and p and po are the solution densities a t T and T o ; this reduction is a minor correction. Log J, is listed in Table I and plotted against concentration in Figure 1. The units are cm2/dyne. The
Experimental Section Low-frequency measurements of the storage and loss shear moduli were made on 14 solutions of a polystyrene with sharp molecular weight distribution, S108, generously provided by Dr. H. W. hfcCormick of the Dow Chemical Co., ranging in concentration from 0.75 to 10 wt %. The weight-average molecular weight, M w , was 267,000; M,/M, = 1.08. The solvents were two chlorinated biphenyls, with the following viscosities (711 and densities ( p ) a t 25”: A1248, 7 = 2.62 poises, p = 1.452 g/ml; A1232, 7 = 0.143 poise, p = 1.269 g,’ml. Measurements were made on each solution a t a single temperature, chosen between 0 and 35” to achieve the appropriate region of reduced frequency where G‘ and G” are proportional to w 2 and w, respectively. The Birnboim-Ferry apparatus5 with modifications previously d e s ~ r i b e d was ~ ~ , used ~ ~ to cover the frequency range from 0.06 to 400 cps. Most measurements in recent years have been made with a combination of cell and driving rod with a gap of 0.0617 cm and a cell constant5 A, = 4846 cm. (Previously5 denoted A , this constant is designated A , here to avoid confusion with the coefficient used in eq 1.) For measThe Journal of Physical Chemistry
-4.75‘ 0
1
I
I
0.05
0.10
0.15
Concentration, q m l . Figure 1. Log J , for solutions of polystyrene S-108, plotted against concentration in g/ml (from Table I). Black circles, solutions in Aroolor 1248; open circles, in Aroclor 1232. Circles with pips from ref 24. Rouse and Zimm curves, from eq 4; N21 curve from eq T and 9.
(22) K. Ninomiya, J. D. Ferry, and Y . Oyanagi, J . P h y s . Chem.. 67,2297 (1963). (23) K. Ninomiya and J. D. Ferry, Abstracts, 12th Symposium on Polymers, Nagoya, Japan, 1963, p 178. (24) R. B. DeMallie, Jr., M. H. Birnboim, J. E. Frederick, N. W. Tschoegl, and J. D. Ferry, J. P h y s . Chem., 66, 536 (1962). (25) N. W. Tschoegl and J. D. Ferry, KoZZoid-Z., 189, 37 (1963).
2717
STEADY-STATE COMPLIANCE OF DILUTE POLYMER SOLUTIONS
Table I : Low-Frequency Parameters for Solutions of Polystyrene 5-108, M , = 267,000 c
Solvent
A1248
poises
Log A
Log J e red. to 26", cmz/dyne
25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 35.0 35.0
5.96 6.47 7.36 8.88 15.5 20.8 31.4 28.8 27.9 41.7
-2.64 -2.64 -2.41 -2.22 -1.70 -1.52 -1.18 -1.30 -1.38 -1.02
-4.19 -4.26 -4.15 -4.12 -4.08 -4.16 -4.18 -4.22 -4.26 -4.24
0.6 0.0 0.0 0.2 0.0 10.0 0.1
4.51 8.79 15.19 24.8 39.7 22.3 241
-2.80 -2.22 -1.83 -1.39 -1.01 -1.60 0.36
-4.14 -4.16 -4.23 -4.22 -4.25 -4.34 -4.44
102, g/ml
Temp,
1.13 1.44" 1.53 1.84 2.86" 3.60 4.31 4.30" 7.13 8.52 2.53 ..3.76 5.04 6.28 7.50 8.77 12.40
A1232
" From ref
x
OC
?I
h from data of
Figure 2
100
... 12 6.3 1.3 1.6 0.8 3 0 0 1.3 1.3 1.6 0 0 0
0
24; solvent viscosity 2.2 poises.
two solvents give closely similar results and the expected maximum is observed. The predictions of the Rouse and Zimm theories from eq 4 (using measured values of vr) and the phenomenological theory from eq 7 are also plotted in Figure 1. For the latter calculation, Jez was taken from a direct measurement from creep studies of the undiluted polymer a t elevated temperaturespZ6reduced to 25": log Je2 = -5.31. Also included is the curve for Je2/v2, often used as an estimate of J,, in very concentrated solution^.^^-'^ The theories predict the maximum a t about the observed location on the concentration scale. In magnitude, J , appears to be approaching the Zimm prediction at low concentrations but closely conforms to the Rouse prediction a t higher concentration; a similar transition from Zimni-like to Rouse-like behavior with increasing concentration is observed with respect to the frequency dependence of dynamic viscoelastic properties.21 In the neighborhood of the maximum, the concentration dependence is best described by the phenomenological theory.
Discussion Magnitude of Low-Frequency Ratio G ' / d as a Measure of Hydrodynamic Interaction. From eq 3, it is evident that A c / ( ~-
=
(M/RT)(S'/SZ)
& -5 8
0
I
I
I
1
,
I
002
004
006
008
010
012
Concentration. q / m l
-
Figure 2. Log ( G ' / W ~ ) ~ , O C / (v1q.)2 ~ plotted against Concentration, for data of Table I. Key t o points same as Figure 1.
seen to approach the Zimm value a t low concentrations and the Rouse value a t high concentrations. The quantity S'/Sz is denoted J.R by T s c h ~ e g l , ~ ~ who points out that it is a potential source of information for the hydrodynamic interaction parameter h which characterizes the transition from Zimm-like to Rouse-like behavior. For such an evaluation, we take Tschoegl's other paramete9 e as zero in the expectation that in all but the most dilute solutions the coil configurations will be gaussian. Values of h calculated in this manner from the data of Figure 2, using Tschoegl's tables16of S and S', are also listed in Table
(10)
which each theory specifies as a constant. This quantity is plotted against concentrat#ionin Figure 2 and is
(26) G. Yasuda, E. Maekawa, T. Homma, and K. Ninomiya, GomuKvolcaishi, 39, 177 (1966). (27) N. W. Tschoegl, J. Chem. Phys., 44, 2331 (1966).
Volume 70, Number 9 September 1966
L. A. HOLMES, K. NINOMIYA, AND J. D. FERRY
2718
Table 11: Observed and Calculated Values of Log J , in Dilute Polymer Solutions (Temperature 25", Units cml/dyne) Sample
Mol wt x 10-3
s-102
82
s-111
239
S-1163
400
MDP-1 MDP-2
1000 1700
Solvent
A1248
A1232 A1248 A1232
HM From eq 11.
840
Primol D
C,
[?I,
g/ml
ml/g
Polystyrene 0.0288 0.0571 0.0288 0.0571 0.0288 0.0568 0.0127 0.0072 0.0144 0.0286 0.0127 0.0253 Polyisobutylene 0.0088 0.0176
38.5 7gb 100 205 205 270
194
LOPJ e , obsd
Log J e ,
-4.94 -4.76 -4.08 -4.05 -3.61 -3.82 -2.98 -2.86 -2.82 -2.95 -2.66 -2.76
-5.05 -5.06 -4.29 -4.40 -4.00 -4.14 -3.27 -3.00 -3.06 -3.20 -2.96 -3.12
-3.08 -3.23
-3.33 -3.42
oalcdn
Interpolated.
I. They show an essentially monotonic decrease from very high values to zero with increasing concentration. This conclusion was reached previously21 from the form of the frequency dependence of dynamic properties without regard to their magnitudes ; the present deduction is entirely independent, being based solely on the magnitudes, and it avoids the as yet unexplained feature of the previous curve-matching procedure that the apparent molecular weights are always somewhat too large. It is, however, restricted to polymers of sharp molecular weight distribution; a similar treatment by Tamura, et ~ 1 (who . ~denote ~ the quantity S'/S2 as y ) , showed how polydispersity can mask the distinction between high and low h. Approximate Estimate of Jefrom Intrinsic Viscosity by Phenomenological Theory. Predictions of J , for any solution of a polymer homogeneous with respect to molecular weight can be made easily from the phenomenological theory, provided it can be assumed that J e z = ( M / p 2 R T ) ( S ' / S 2 ) ~The . only additional information needed is the intrinsic viscosity in the solvent concerned. In terms of c, the relation can be written
Values of log J , calculated in this manner are compared in Table I1 with measured values for a number of other solutions of polystyrenez1 and polyi~obutylene,~~ derived from previous measurements. The comparison is shown graphically in Figure 3 together with the data for the solutions of Table I. Over a 200-fold range The Journal of Physical Chemistry
log Je, observed
Figure 3. Log J , calculated from eq 11 plotted against observed values. Pip up, polystyrene 5-102; successive 45" clockwise denote samples 5-111, S-108, 5-1163, MDP-2, and MDP-1. Internal pip denotes data from Table I. Crossed circles: polyisobutylene.
(28) M. Tamura, M. Kurata, K. Osaki, and K. Tanakz, J. P h y s . Chem., 70,516 (1966). (29) N. W.Tschoegl and J. D. Ferry, ibid., 68,887 (1964).
THEINFRAREDABSORPTIONSPECTRUM OF
OXYGEN-18-LABELED
there is fair agreement, although most of the calculated values are somewhat too small, some by as much as a factor of 2. Some deviation in this direction is to be expected from a slight molecular weight heterogeneity, but it js doubtful whether more than a factor of about 1.3 can be attributed to this cause, The remainder can be attributed in a formal manner to an increase in NB1 with concentration which has been neglected. However, the very simple formulation of
GLYCINE
2719
eq 11 is evidently useful for making estimates of J, in moderately dilute solutions over a considerable range of concentrations and molecular weights. In particular, eq 2 and 9 can provide an estimate of the location of the maximum.
Acknowledgment. This work was supported in part by grants from the National Science Foundation and the U. S. Public Health Service.
The Infrared Absorption Spectrum of Oxygen-18-Labeled Glycine
by I. Laulieht, S. Pinchas, D. Samuel, and I. Wasserman The Weizmann Institute of Science, Rehovoth, Israel
(Received October 26, 1965)
The infrared absorption of powdered 75 atom yo I 8 0 &-glycine was measured in the 4000400-cm-I region in comparison with the absorption of the corresponding normal glycine. The assignment of the various glycine absorption bands is discussed in the light of the observed lSO isotopic shifts. It is shown that the 1334-1324-cm-l band attributed previously mainly to a CH2 wagging has a pronounced C-C stretching character. The 894cm-I band of normal glycine, which was assigned to its C-C stretching, is now shown to be due probably to the Con- scissoring. The 700-685-cni-' band is assigned to the COzrocking while that about 502 cm is attributed to an NH3+ group deformation.
The infrared absorption spectrum of glycine is of special interest since, obviously, its understanding is essential for the interpretation of th? more complex spectra of peptides and proteins. This spectrum is also of importance in connection with the application of analytical infrared spectrophotometric methods to the problem of the composition and structure of various substances built up from amino acids. The infrared absorption of and variously deuterated',$ glycine has therefore been investigated thoroughly along with the Raman spectrum of these substance^^,^-^ and the infrared spectrum of 15s glycine.' On basis of most of these measurements Suzuki, et o,Z.,~ calculated the potential energy distribution among the various elementary modes of vibration of glycine for each of its fundamental frequencies. The correctness of these calculations and the extent to which the respective
underlying assumptions hold were checked for those vibrations which affect mainly the hydrogen atoms or the nitrogen atom, by studying the effect of deuteration or l 5 h i labeling on the corresponding absorption frequencies of glycine. No such check, however, could be carried out up to now with respect to those vibrations which affect mainly the oxygen atoms. (1) S. Suzuki, T. Shimanouchi, and 31. Tsuboi, Spectrochim. Acta, 19, 1195 (1963). (2) M. Tsuboi, T . Onishi. I. Nakagawa, T . Shimanouehi, and S. Mizushima, ibid., 12, 253 (1958). (3) K. Nakamura, J. Chem. SOC.Japan, 8 0 , 118 (1959). (4) M. Takeda, R. E. S. Izvazzo, D. Garfinkel, I. H. Scheinbergy and J. T. Edsall, J. A m . Chem. SOC.,80, 3813 (1958). (5) S.A. S. Ghazanfar, D. V. Myers, and J. T. Edsall, ibid., 86, 3439 (1964). (6) K. Balasubramanian and R. S. Krishnan, Proc. Indian Bcad. Sci., A59, 115 (1964).
Volume 70, Number 2
September 1966
