H. INAGAKI, H. SUZUKI,RII. FUJII,AND T. MATSUO
1718
Note on Experimental Tests of Theories for the Excluded Volume
Effect in Polymer Coils
by Hiroshi Inagaki, Hidematsu Suzuki, Makoto Fujii, and Togoro Matsuo Institute for Chemical Research, Kyoto University, Kyoto, J a p a n
(Received August 97,1966)
Results are reported for the molecular weight determination of polystyrene fractions in butanone by light scattering and by intrinsic viscosity measurements, [ q ] ,for dilute transdecalin solutions of these fractions at various temperatures ranging from 18.7 to 100". The 8 temperature for the system polystyrene-trans-decalin was determined by light scattering to be 23.8". Using these data and applying the perturbation theory of [ q ] developed by Kurata and Yamakawa, parameters A and B, which describe the shortrange and long-range interactions between segments, respectively, have been evaluated. Good agreement in A was found between the values obtained from [ q ] and from meansquare statistical radii (s?) at 0. Significant inconsistencies were found between the values for B from [ q ] and those from the second virial coefficient A?. Values of the excluded volume variable, z, for several sets of molecular weight M and temperature T have been calculated using A and B which were obtained from [ q ] alone. These values of z were applied to Berry's data for (9)measured as a function of M and T for the present system. According to our analytical result, the experimental relation between (s?) and z does not seem to agree with any of the existing theories. Satisfactory agreement with equations of the Fixman type is found only if it is assumed that B depends on T in a manner different from the usual approximation B = Bo(1 - 8 / T ) . The Stockmayer-Fixman equation is used to explain this relation and its applicability to well-behaved solvent systems is also discussed in the light of results of the present study.
Introduction The configurations of flexible chain polymers in solution deviate from the statistics of random flight because of the long-range interactions between nonadjacent segments of a polymer chain. The effects of such interactions are usually expressed in terms of the linear expansion factor a which is defined as the ratio of the root-mean-square statistical radii of a polymer chain in the presence and absence of these interactions. Since the first attempt by Flory' to relate a to molecular constants of polymers and solvents, great efforts have been devoted to test his theory both theoretically and e ~ p e r i m e n t a l l y . ~ After ~ ~ a number of attempts to refine the defects in the Flory theory, very similar equations of closed form for a have been proposed by F i ~ m a nKurata, ,~ Stockmayer, and Roigj5 and Ptitsyn.6 All of these equations are represented in terms of a single variable z given by The Journal of Physical Chemistry
2 =
BA-3M'l'
(1)
where
A 2 = ( s ~ ) ~ /=Ma2/6nz, B
E (1/4~a'2)(~/2n1s2)
Here (s2), and a are the unperturbed mean-square statistical radius and the effective length of a segment, respectively, t3 is the binary cluster integral, M is the solute molecular weight, and m, is the molar weight of (1) P. J. Flory, J . Chem. Phys., 17, 303 (1949). (2) P. J. Flory, "Principles of Polymer Chemistry," Cornell University Press, Ithara, N. Y., 1953. (3) M. Kurata and W.H. Stockmayer, AdDan. Polymer Sei., 3, 196 (1963). (4) M. Fixman, J . Chem. Phys., 23, 1656 (1955); 36, 3123 (1962). ( 5 ) M. Kurata, W. H. Stockmayer, and A. Roig, ibid., 33, 151 (1960). (6) 0. B. Ptitsyn, Vysokomolekul. Soedin., 3, 1673 (1961).
THEORIES FOR
THE
1719
EXCLUDED VOLUME EFFECT IN POLYMER COILS
segment. Obviously the variable z involves two independent paramet,ers, A , which describes the short-range interactions between adjacent segments, and B, which describes the long-range interactions between nonadjacent segments. Thus the theories available to date are generally called two-parameter theories. To test the validity of the two-parameter theory experimentally, parameters A and B for a given system must be first determined. To estimate parameter B, Schulz and his associates have recently proposed a method that makes use of temperature derivatives of the osmotic second virial coefficient A z a t the 0 point,' i.e., (bAz/bT)e. The measurement of Az, however, often suffers greatly from experimental errors, especially if the measurement is made for high molecular weight fractions and/or a t a temperature below the 8 point. More recently, a theory has been established by Kurata, et U Z . , ~ which allows one to determine B from the slope of a plot of AZM112 us. M"'. They emphasize, however, that there is always a significant difference between B values derived from two different sources, A z and the limiting viscosity number [7].9 Thus the determination of these parameters still remains an important subject for testing these theories, as has recently been pointed out by Berry and Fox.'O In this communication the parameters A and B are evaluated using viscosity data of the polystyrenetrans-decalin system, in which the solute molecule reaches the 6 condition a t 23-24'. The perturbation theory of [7] developed by Kurata and Yamakawa" is employed for this evaluation. The excluded volume variable z is then calculated for various sets of molecular weights and temperatures. It is then applied to analyze the experimental results of Berry,12,13who recently measured the dependence of the mean-square statistical radius (s2)on molecular weight M and temperature T for the same system. According to our analytical results, the experimental relation between (s2) and z does not seem to be in agreement with any of the theories presented to date. A good agreement between experiment and theory is, however, obtained by abandoning the usual assumption, viz., p = po(l 8 / T ) , where Po is a constant independent of T and 8 is the 8 temperature. Thus the behavior exhibited by a3(2)is fairly well described in terms of the family of equation^^-^ represented by Fixman's formula4 a3 =
1
+ 1.912
(2)
On the basis of this result, the validity of the Stockmayer-Fixman equationg proposed to estimate unperturbed dimensions from viscosity data obtained in well-behaved solvents, is also discussed.
Experimental Section Polymer Samples. Unfractionated polystyrenes of different molecular weights were used. The samples were prepared by thermal bulk polymerization at different temperatures. In each polymerization the conversion of monomer to polymer was kept below ca. 10% to avoid possible chain-branching reactions. Table I lists polymerization conditions, degrees of conversion, and viscosity-average molecular weights M , of unfractionated polymers computed by log M , = (log 17.73 2.013)/0.74 established by Ewart in benzene solvent.14
+
Table I : Bulk Polymerization Data of Polystyrene Samples Polymer sample
J P a
Polymerization temp,
OC
Time, hr
95 68
6 48
Conversion,
M"
x
70
lo-@
11 9
0.47 1.40
According to t h e Ewart equation (see text).
Each polymer was fractionated into ten to twelve fractions by usual fractionational precipitation with a benzene-n-butyl alcohol system a t 30'. Seven samples from these fractions were used for subsequent measurements. Each sample was redissolved in benzene and precipitated with methanol before drying to constant weight under vacuum over P205a t 50". In addition to these fractions, two low molecular weight fractions obtained in a previous studyi5 in this laboratory (referred to as G-1 and G-2) have also been used. Solvents. The limiting viscosity numbers of polystyrene solutions are different in cis- and trans-decalin. In the present work, pure trans-decalin (K & I< Laboratories and Eastman Kodak Co.) was used. Its purity (7) G. V. Schulz, A. Haug, and R. Kirste, 2.Physik. Chem. (Frankfurt), 38, 1 (1963); G. V. Schulz and H. Baumann, Mukromol. Chem., 6 5 , 101 (1963). (8) M. Kurata, N.Fukatsu, H. Sotobayashi, and H. Yamakawa, J . Chem. Phys., 41, 139 (1964). (9) W. H. Stockmayer and M.Fixman, J . Polymer Sci., C1, 137 (1963). (10) G. C. Berry and T. G Fox, J . Am. Chem. Soc., 86, 3540 (1964). (11) M. Kurata and H. Yamakawa, J . Chem. Phys., 29, 311 (1958); Makromol. Chem., 34, 139 (1959). (12) G. C. Berry, Preprints, Division of Polymer Chemistry, 145th National Meeting of the American Chemical Society, New York, N. Y., Sept 1963, Vol. 4, p 141. (13) G. C. Berry, private communication to Dr. &I. Kurata of this institute. (14) R. H. Ewart, Abstracts, 111th National Meeting of the American Chemical Society, Atlantic City, N. J., April 1947. (15) H. Inagaki and S. Kawai, Mukromol. Chem., 79, 42 (1964).
Volume 70, Xumber 6 J u n e 1966
H. INAGAKI, H. SUZUKI, M. FUJII,AND T. MATSUO
1720
was checked by determining the characteristic infrared absorption band of trans-decalin a t 10.8 p16 and by gas chromatography. Thus the content of the trans form was determined to be over 99.7%. The other physical constants were also measured as follows: the density and refractive index at 20.0" were 0.8700 and 1.4674, respectively, which agree fairly well with those reported by other authors." Before use, the solvent was further purified by vacuum distillation. All other solvents used, benzene, butanone, and cyclohexane, were carefully purified and dried in accordance with standard procedures. Light Scattering. Scattered light intensities were determined with a modified photometer of the Brice type (a Shimadzu light scattering photometer'*) equipped with a constant-temperature jacket.19 Using a cylindrical cell, the angular variation of the scattered intensity was determined at eleven different angles ranging from 30 to 150". To calibrate the photometer, the absolute scattering data of Carr and Zimm obtained for benzene with natural light were used (the reduced scattered intensity for natural light U , = 48.5 X cm-' and the corresponding depolarization pu = 0.41 at 25" and for the wavelength 436 mp).20 Throughout the present study the incident light was vertically polarized to avoid optical and electronic disturbances caused by the photomultiplier used. Only the vertical component of scattered light was received with an analyzer attached t o the front of the photomultiplier. Thus ? wasI converted , into V , by
V,
=
UU(2 -
PU)/O
+
PU)
which is necessary under our experimental conditions. The above equation yielded 54.7 X 10-6 cm-' for V,. Then the calibration constant (by is given by 4,
=
54.7 X 10-6(Gg0/Go)-1D-'
where Go and GgO are the galvanometer readings at 0 and go", respectively, and D is the transmittance of a neutral filter inserted between the light source and the solution When the refractive index of solvent used (denoted by n') differs from that of benzene at 25" (denoted by n), further correction for (by was made by multiplying it by (n'/n)2.22 Molecular weight determinations of all the samples were carried out in butanone. To measure some of the thermodynamic properties, such as A2 and the 8 point for some of the samples, trans-decalin was used as solvent. In both of these cases solutions of desired concentrations were prepared by heating to 80" in sealed glass tubes for at least 1 day, followed by filtration with no. 4 sintered glass plates. The butanone solution was placed directly into the cell through grade m UltraThe Journal of Physical Chemistry
cellafilter, while the decalin solution was centrifuged with a Hitachi preparative ultracentrifuge Model 55P at a rotor speed between 16,000 and 20,000 rpm for about 2 hr at 30". Ultracentrifugation was necessary because of the high viscosity of decalin. All measurements using butanone were carried out at 20" and those with trans-decalin at a variety of temperatures ranging from 18 to 76". The specific refractive index increment (dnldc) of the butanone solution at 20" was 0.211 ml/g. It was determined with a differential refractometer of the Debye type2a at 436 mp. For the decalin solution the empirical rule of Dale and Gladstone2*was used to calculate the values of (dnldc) for measurements at different temperatures. Dale and Gladstone's rule can be expressed by dnldc = Rz - (nl - l)/& where R is a temperature-independent constant, n is the refractive index, d is the density, and the subscripts 1 and 2 refer to the solvent and polymer, respectively. According to Outer, Carr, and Zimm,25the following values were assumed: Rz = 0.581 and d2 = 1.05 a t 20" and dz = 1.02 a t 67". Viscosity Measurements. The viscosity measurements were carried out using three dilution viscometers of the Ubbelohde type, hereafter designated no. 1, no. 2, and no. 3. Viscometer no. 1 was specially designed to suppress the average shear stress below 10 dynes cmW2in its capillary for trans-decalin in a temperature range from 18 to 60". Using similar viscometers, the average shear stresses in no. 2 and no. 3 were kept lower than 8 dynes cm-2 for cyclohexane at ca. 35" and for trans-decalin in a temperature range from 80 to loo", respectively. Due to large efflux times of these viscometers, no need for the kinetic energy correction has become apparent. In each determination the temperature of a water or silicone oil (16)Report of the American Petroleum Institute, Research Project
No. 44. (17)W. F. Seyer and R. D . Walker, J . Am. Chem. Soc., 60, 2125 (1938); A. Weissberger and E. S. Proskauer, "Organic Solvents," 2nd ed, Interscience Publishers, Inc., New York, N.Y., 1955. (18) H.Inagaki and T. Oyama, J . Chem. Soc. Japan, 78,676 (1957). Chem., 55, 150 (1962). (19) T.Matsuo and H. Inagaki, MU~TOWZ. (20) C. I. Carr and B. H. Zimm, J . Chem. Phys., 18, 1616 (1950). (21) B. A. Brice, M. Halwer, and R. Speiser, J. O p t . SOC.A m . , 40, 768 (1950). (22) J. J. Hermans and S.Levinson, ibid., 41,460 (1951). (23) P. W. Debye, J . A p p l . Phys., 15, 338 (1944). (24) D.Dale and F. Gladstone, Phil. Trans., 148, 887 (1858); 153, 317 (1864). (25) P. Outer, C . I. Carr, and B. H. Zimm, J . C h m . Phys., 18, 830 (1950).
THEORIES FOR
THE
EXCLUDED VOLUME EFFECT IN POLYMER COILS
bath was kept within *0.02" of each desired temperature. The viscometer was placed vertically in the constant temperature bath by means of a special device.
Results Molecular Weight Detemzination. The molecular weight of each sample was determined by light scattering in butanone solvent. Required values of Kc/Ro, where K is the well-known light scattering factor, c iS the polymer concentration, and Ro is the reduced intensity of the scattered light at zero angle, were obtained by the usual extrapolation of the data on the Zimm plot.26 The values obtained for the weightaverage molecular weight M, and the second virial coefficient Az are given in Table 11. The molecular weight of fraction G-2 was found to be 14.3 X lo* by the Archibald ultracentrifugation procedure* in butanone at 30". This is in satisfactory agreement with the value found for this sample by light scattering (see Table 11). Because of the complexities of the
Table II : Light Scattering Data in Butanone at 20' and Viscosity Data in Cyclohexane at 34.5' ~ - 1 n butsnone at ZOOFraction
M n x lo-'
G-1 G-2 5-4 5-7 J-10 J-11 P-4 P-7 P-8
5.8 14.3 22.2 31.4 51.3 72.5 112 139 200
AI
x lo',
8 Temperature Determination in trans-Decalin. To test the two-parameter theory, the 8 temperature for a given solution system must be known. Consequently, the scattered light was determined once more for G-2, 5-4, J-10, and P-4in trans-decalin solutions at various temperatures ranging from 22 to 30". The second *a1 coefficient A2 at each temperature was computed by plotting Kc/Ro us. c. Subsequently, A2 was plotted against T to find out, by interpolating, the temperature at which A2 vanishes. Examples of these plots are shown in Figures 1 and 2. Details of the data obtained are given in Table I11 together with the values of ( ~ 2 ) ~ .
'4-
I
I
P4
I
I
1
2
I
3
I
I
I
3
4
-I
I
1O.c in glml Figure 1. Light scattering plots of various temperatures near 8 for P-4 fraction in truns-decalin. The temperature of measurement which corresponds to each line is given.
-In cyclohexane at 34.S0[.?It
cgs
ml/g
2.18 2.16 1.61 1.66 1.50 1.44 1.48 1.51 1.38
...
...
32.2 40.0 47.0 61 .O 72.7 91.0 101.5 116.4
0.496 0.524 0.566 0.739 0.724 0.628 0.83 0.81
kt4
Huggins' constant.
Archibald procedure, this was not used further. However, the agreement between the values found by these two methods may be regarded to indicate the reliability of the results obtained by the present light scattering measurement in butanone. In fact, the values of [q] of all fractions determined in cyclohexane at the corresponding 8 point 34.5" were ascertained to be exactly proportional to the square roots of molecular weights obtained in butanone. The plot of [qle vs. M'I' gives the equation
[?le
1721
(ml/g) = 8.50 X 10-*M,"'
This equation agrees well with that reported recently by Altares, et al."
24
27
3q
21
T in"C Figure 2. Plots of A Sagainst T for 5-4, J-10, and P-4 fractions. The same plot for G-2 is omitted (see Table 111).
As can be seen from Figure 2, the 8 temperature observed for each fraction may be regarded to be independent of the solute molecular weight in the present study. Thus the 8 point for this system is at 23.8'. This finding, however, differs somewhat from that reported by Berry,'* who has found a 8 temperature between 21 and 22" for the same system. At present, we see no reason why such a discrepancy should arise. (26) B. H.Zi", J . C h m . Phys., 16, 1099 (1948). (27) H. Wits, H. Inagaki, T.Kotaka, and H. Utiyama, J . Phys. Chem., 66,4 (1962). (28) T.Altarea, Jr., D.P.Wyman, and V. R. Allen, J. Polynet Scd., A2,4533 (1964).
Volume 70, Number 6 Juw 1066
H. INAGAKI, H. SUZUKI,M. FUJII,AND T. MATSUO
1722
Table I11 : Light Scattering Data in trans-Decalin a t Various Temperatures
18.0" 20.0" 30.0' 60.0' 76.0"
-0.750 -0.353 1.50 3.70 6.20
22.3' 24.0' 25.9' 28.9'
P-4
J-10
5-4
G-2
Measuring temperature -0.160 0.014 0.250
and A Z X lo4 22. 0' -0.125 24.0" 0.045 26.9" 0.286
22.0" 24.0" 27.0" 29.6'
0.450
-0.150 0.012 0.143 0.212
9 temperature
23.8"
23.8'
1.43
1.14
23.7"
23.8"
1.11
0.69
(bA&T)e X 10' (s2) X 1012 and A* X 1018' at 24.0'
...
(7 * 4)
1.5
3.8
(6.8)
(7.6)
8.6
(7.7)
' Values in parentheses.
5.01 ' I ' I !' I ' 1 Figure 2 indicates also that the values of ( b A ~ l b T ) ~ are different for different molecular weights, showing a distinct decreasing tendency with M . The same trend has already been observed by Schulz, et al., for the 45 system polydimethylsiloxane and bromocyclohexane,7 but no theoretical interpretation has yet been offered. Limiting Viscosity Numbers in the Vicinity of the 8 Point. Table IV summerizes experimental results of m [v]of the fractions in trans-decalin at different temperaE4D c . tures. Figure 3 shows semilogarithmic plots of [ q ] os. T. Except for the fraction P-8 having the highest molecular weight, all of the data points can be expressed by straight lines as shown in Figure 3. Thus 3.5 the temperature derivative of In [v] at the 8 point can be precisely evaluated. The values of (b In [ v ] / b T ) e obtained (see Appendix) are given in the last row of Table IV. The [qle values are determined by interpolating each straight line shown in Figure 3 to the 8 point. These values are included in Table IV. Evaluation of the Variable z. In principle, it is impossible to estimate both parameters A and B from viscosity data alone, unless we first assume that a valid equation exists that describes [q I near the 8 point as a function of M and T. Thus the parameters, hence with the variable z, will now be evaluated using the theory of Kurata and Yamakawa" and the present viscosity [7]e = 6"'%A8M"' data and will be applied for the analysis of Berry's where a,,is the linear expansion factor for the hyresults of (s2).la drodynamic radius8 and % is the universal constant of According to perturbation calculations on the basis Floryaoa t the 8 point, which has been calculated to be of the Kirkwood-Riseman mode1,29 the limiting viscosity number [v] near the 8 point has been formulated by Kurata and Yamakawa asaJ1 I
8
"
-
-
-
[SI
=
le^^
+ 1.552)
= [~le(l
The Journal of Phvsical Chemistry
(3)
(29) J. G. Kirkwood and J. Riaeman, J . Chem. Phye., 16,565 (1948). (30) P. J. Flow and T.G Fox, J. Am. Chem. Soc., 73, 1904 (1951).
THEORIES FOR
THE
EXCLUDED VOLUMEEFFECT IN POLYMER COILS
1723
2.87 X cgs.al,aa From eq 3, parameter A can b e evaluated knowing the relation of [Q Je to M'/' and the value of @o, which has been established both theoretically and experimentally, since no hydrodynamic draining effect exists at the 9 point. Thus using eq 3, A a has been computed to be 7.1 X 10-l8 cma. This value is in good agreement with that obtained by Berry (7.0 X 10-I8) by direct measurements of (s2) (see also the last row of Table III).aa To estimate parameter B we assume that the binary cluster integral p is temperature dependentl~2
P
=
- 9/T)
PO(1
and hence
B = Bo(1
- 8/T)
(4)
with
Bo =
(l/q~"p)
@0/2m,')
Then eq 3 can be rewritten [ Q ] / [ T ] ~=
1
+ 1.55A-a&(l
- 8/T)M"'
(5)
Equation 5 shows that the plot of [7]/[71e against (1 - 6/T)M'/*should give a straight line with a slope equal to 1.55A-3B0 passing through unity on the ordinate. Such a plot for the present viscosity data is shown in Figure 4. The data in the range of -0.15 < z < 0.15 can be well approximated by a straight line
Figure 4. Plots of according to eq 5. (31)
[ql/[qle
against (1
- e/T)M'/*
P. L. Auer and C. G. Gardner, J . Chem. Phya.,
23, 1645, 1540
(1955).
(32) B.H. Zi", ibdd., 24,209 (1956). (33) This experimental result should be compared to the theoretical result of Hearst, who has given 2.2 X 10sa for L ( J . Chem. Phya., 40, 1600 (1984)).
Volume 70, Number 6 June 1966
H. INAGAKI, H. SUZUKI, M. FUJII,AND T. MATSUO
1724
3.0
whose slope is 10.0 X using the least-squares method. This value is divided by 1.55 to yield the equation z = 6.45 X lO-’(l
- 8/T)M”’
(6)
2.5
It should be mentioned that the coefficient 1.55 in the final factor of eq 3 is only an approximation because of the mathematical treatment introduced for calculating the frictional resistance force exerted on fluid by segments.ll Thus this number may later have to be changed a bit. On the other hand, Berry has measured the temperature dependence of A2 for the same system and gave 10.2 X for BOA-’. This value significantly differs from that evaluated from the viscosity data. The difference can be attributed to different Bo values because our A value and that found by Berry are in agreement. However, the average value of (dA2/bT)e calculated from Berry’s original data and that estimated for our sample P-4 are in fairly good agreement, i e . , 0.76 X and 0.69 X respectively. The molecular weights of Berry’s samples and our P-4 sample were in the same range. Similar discrepancies in other systems have been pointed out by Kurata, et a1.* The origin of this discrepancy in BO is still open for discussion. Comparison with Theory. Berry’s data of (9) measured as a function of M and T will be analyzed with z expressed by eq 6. Figure 5 shows a plot of a3 against z together with two dashed lines (I) and (11) obtained according to eq 2 and Flory’s equation’J CY‘
-d
=
1.282
(7)
respectively. The initial slope of the experimental curve estimated within the range of z < 1 is 1.61. According to Figure 5, Flory’s &type equation appears to provide a better fit within the range of z < 2 than a3-type equations do. However, when the FloryFox-Schaefgen plot based on the combination of eq 7 and Flory-Fox’s viscosity equationm is applied to relations between [?] and M obtained in well-behaved solvents, it leads to a serious underestimation of the unperturbed dimension.’ In contrast to the above fact, the similar plots based on the &type equation always give reasonable unperturbed dimensions. I n the next section this inconsistency will be discussed and an assumption for p will be introduced.
Discussion Before discussing the inconsistence between the experimental relation of a’ to z and the existing theories, let us examine the approximation made for the temperature dependence of B, i.e., eq 4. We apply the The J w r d of Physical Chemiatqi
0
x
2.0
1.5
1.0
0
1.o
2.0 z=&&(l- W T ) M’”
3.0
Figure 5. Experimental relation between (Y’ obtained from Berry’s data of (st) and z expressed by eq 6. Dotted curves I and I1 are drawn according to eq 2 and 7, respectively. Full line indicates the initial slope of experimental curve (for details, see text).
Stockmayer-Fixman equationBto our viscosity data, which is essentially equivalent to eq 2 and which d e scribes the isothermal dependence of [ q ] on M . With the present notations this is rewritten
+
[q]/M’/’ = 6 a / ’ @ ~ a 22.7@0l3M”’
(8)
Figure 6 shows plots of eq 8 with values of [?] determined in trans-decalin for various temperatures ranging from 18.7 to 100” and M determined by light scattering in butanone. When the temperature is close to the 8 point (
