829
STYRENE-METHYL METHACRYLATE RANDOM COPOLYMERS
Dilute Solution Properties of Styrene-Methyl Methacrylate Random Copolymers by Tadao Kotaka, Yoji Murakami, and Hiroshi Inagaki Institute for Chemical Research, Kyoto University, Takatsuki, Osaka, Japan
(Received June SO, 1967)
The results of an extensive study on dilute solution properties of three series of styrenemethyl methacrylate random copolymers are described. The copolymers are prepared by a free radical polymerization method (conversion = less than 10%) and subsequently fractionated by use of a butanone-diisopropyl ether system. The characterization of the fractions is made by combustion analysis, by osmometry, and by light-scattering methods. The 8 temperatures for the copolymers are determined as the temperatures at which the osmotic second virial coefficient A z = 0 in cyclohexanol and 2-ethoxyethanol. LMeasurements of intrinsic viscosity, [TI, under the 8 conditions show that [?le is proportional to M”* ( M , the molecular weight) in the copolymer solutions. Measurements of [TI are also made in toluene, 1-chloro-n-butane, and diethyl malonate, which have typically different solvent powers toward the parent homopolymers. Allthe [TI, A z ,and i W data are examined on the basis of the current excluded volume theories. Then the short-range and the longrange interaction parameters, A and B, are estimated for each copolymer-solvent pair, and their dependences on the monomer composition are examined in relation with those of the parent homopolymers. It is found that the values of A 2 = ((S2)o/M),with (Xz)o being the unperturbed mean-square molecular radius, are larger than the simple composition averages of those of the parent homopolymers. The parameter B shows the presence of repulsive interactions between the unlike monomer units.
Introduction Recent advances in the studies on the properties of dilute polymer solutions have made it clear that the conformational and thermodynamic properties of flexible macromolecules may be described essentially by two independent They are the short-range and the long-range interaction parameters, and are related, respectively, to the unperturbed average dimension4 and to the excluded-volume effect’ of a given polymer in a given environment. The adequacy of the basic assumptions of the two-parameter statisticalmechanical theories has been substantiated by experim e n t ~ . ~ - ~ + ’Of importance is the fact that these two parameters can be determined, in principle, consistently from adequate experimental data such as the molecular weight dependences of second virial coefficient,els of mean-square statistical radiu~,~,O of intrinsic v i s ~ o s i t y , ~ J ,etc. ~ - ~ ~I n fact, several graphical procedures for making such analyses have been proposed so far, still leaving certain points of dispute among them though. I n view of the current status of the two-parameter theories, it would be an interesting problem to extend the theories to examine the dilute solution properties of copolymers. Difficulties in copolymer studies would arise from an obvious fact that there are (even for a simple binary copolymer) a t least three factors: the molecular weight, the monomer composition, and the mode of monomer arrangements, for specifying a
copolymer sample. I n addition, the heterogeneities with respect to these three factors would further complicate the problem. Therefore, any copolymer materials should first be well characterized in these respects.12 For our study, we selected copolymers of styrene (ST) and methyl methacrylate (MMA) for several reasons. First of all, the properties of the parent homopolymers, polystyrene (PST) and poly(methy1 methacrylate) (PMMA), have been most extensively studied (1) P. J. Flory, “Principles of Polymer Chemistry,” Cornell University Press, Ithaca, N. Y.,1953. (2) W.H.Stockmayer, Makromol. Chem., 35, 54 (1960). (3) See, for example, M. Kurata and W. H. Stockmayer, Fortschr. Hochpolym. Forsch., 3, 196 (1963). (4) See, for example, 31. V. Volkenstein, “Configurational Statistics of Polymer Chain,” Moscow and Leningrad, Academy of Sciences, U. 8.8.R., 1959; T.M. Birshtein and 0. B. Ptitsyn, “Conformations of Macromolecules,” Nauk, Moscow, 1964. ( 5 ) H. Inagaki, H. Suzuki, M. Fujii, and T. Matsuo, J . Phys. Chem., 70, 1718 (1966). (6) (a) G.C. Berry, J . Polymer Sci., B4, 161 (1966);(b)G.C. Berry, J . Chem. Phys., 44, 4650 (1966). (7) G.C.Berry, ibid., 46, 1338 (1967). IS! M. Kurata, M. Fukatsu, H. Sotobayashi, and H. Yamakawa, zbzd., 41, 139 (1964). (9) W. H. Stockmayer and M. Fixman, J . Polymer Sci., C1, 137 (1963). (10) H.Inagaki, H.Suzuki, and M. Kurata, ibid., C15, 409 (1966). (11) J. M.G.Cowie, Polymer, 7, 487 (1966). (12) See, for example, H. Benoit, Ber. Bunsenges. Physik. Chem., 70, 286 (1966). Volume 7.2, Number S
March 1968
T. KOTAKA,Y. MURAKAMI, AND H. INAGAKI
830 in various solvents. Secondly, by the benefit of accumulated knowledge on the copolymerization kinetics of ST-NIMA random copolymers with a prescribed composition but with a negligible composition heterogeneity can be prepared by the free radical polymerization method. Thirdly, block copolymers with well characterized architecture can also be prepared by anionic polymerization methods,14 and thus the comparison of the ST-RUMA random and block copolymers would become possible in due course. I n this report, we deal with the properties of the STMMA random copolymers, i.e., those prepared by the free radical polymerization method. The studies of a similar nature have already been reported in several articles, notably by Stockmayer, Moore, Fixman, and Epstein15 and also by Utiyama.le The present work supplements the previous works of these authors in greater detail.
Experimental Section Materials. Commercial
styrene monomer was washed several times with 10% NaOH solution and followed by repeated water washing to remove the alkali. The monomer was dried over CaC12, filtered, and stored over CaHz in a sealed vessel at about 4'. Just prior to use the monomer was distilled under a reduced Nz atmosphere (12-14 mm pressure), and the middle fraction distilling at 38-40" was collected. Methyl methacrylate monomer was treated in the same way, except the middle fraction distilling at 40-41" under 82-83 mm of an Nz atmosphere was collected. Benzoylperoxide (BPO) was purified by recrystallizing it in methanol from chloroform solution; it was dried in vacuo at room temperature. All the solvents used for physicochemical measurements were carefully purified according to the standard procedures appropriate to each. Other solvents and nonsolvents used in large quantities for fractionation and subsequent purification were all purified by batch distillation. Preparation and Fractionation of Copolymers. Five copolymer samples of (three) different compositions were prepared in bulk at 60" by using BPO as an initiator. I n each polymerization, the reaction was terminated by cooling the mixture rapidly to room temperature and pouring it into an excess amount of methanol. The whole polymers obtained were purified by twofold precipitation from tetrahydrofuran (THF) or butanone solution into methanol, followed by vacuum drying at about 60". RIMA-rich copolymers often yielded somewhat sticky aggregates, even after vacuum drying. Therefore, they were further treated by redissolving them in benzene and subsequent freeze drying. The composition of each sample was determined from the carbon content by repeating the combustion analysis several times. Table I summarizes the copolymerization data. Tht Journal of Physical Chemistry
Table I: Polymerization Data of ST-MMA Random Copolymers
Sample code
S T in feed, mol %
Amount of BPO, mol %
Time, min
Conversion, wt %
S T in products, mol %
SM3 SM5 SM5' SM7 SM7'
21.5 55.8 55.8 75.1 75.1
0.0413 0.0444 0.84 0.046 0.90
265 270 90 285 85
7.0 5.1 9.5 6.5 10.0
29.6 55.9 53.6 71.2 70.1
Each whole polymer, coded as shown in Table I, was subjected to successive fractionating precipitation by using butanone and diisopropyl ether (DIPE) as solvent and pre~ipitant,'~respectively, repeating the following procedures. The precipitant was added dropwise to the mixture a t 30.0" with vigorous stirring, until the solution became sufficient cloudy. The temperature was raised (usually up to 50 55') until the precipitates were redissolved. Then the temperature was gradually lowered to 30.0" with constant stirring overnight to ensure the equilibrium between phases. Finally the mixture was kept standing still until the phase separation was completed, and the precipitates were recovered. The conditions for the fractionation are listed in Table 11. All the fractions were again
-
Table I1 : Conditions for the Copolymer Fractionation Amount of polymer,
Polymer concn, g / l W ml, and y values, vol fractiona Initial Final
Number Total of frac- recovery, tions wt 9%
Code
B
SM3
54.0
c = 1.41 y = 0.48
0.085 0.74
12
95.5
SM5
41.0
c = 1.35
0.11 0.72
12
93.8
0.51
c = 1.39 y = 0.68
...
6
...
0.83 98.4
y =
SM5'
35.0
SM7
51.0
c = 1.09 y = 0.57
0.082 0.77
12
SM7'
37.0
c = 1.36 y = 0.70
...
6
0.84
a y = volume fraction of the nonsolvent. For the first fractions, the data show the condition under which the initial precipitation took place. The last fraction was recovered by adding large amount of methanol to the final solution.
(13) See, for example, G. E. Ham, Ed., "Copolymerization," Interscience Publishers, Inc., New York, N. Y.,1964. (14) See, for example, M. Szwarc, and A. Rembaum, J . Polymer Sci., 22, 189 (1956); M. Baer, {bid., A2, 417 (1964); D. DeckerFreyss, P. Rempp, and H. Benoit, ibid., B2, 217 (1964). (15) W.H. Stockmayer, L. D. Moore, Jr., M. Fixman, and B. N. Epstein, {bid., 16, 517 (1955). (16) H. Utiyama, Dissertation, Kyoto University, Kyoto, Japan, 1963.
STYRENE-METHYL
METHACRYLATE
RANDOM COPOLYMERS
purified and their compositions were determined by the same procedures as for the whole polymers. Some of the fractions were used for the later measurements. Our laboratory stock of the parent homopolymers PST and PA4MA were used, whenever necessary, for the sake of comparison. Osmotic-Pressure Measurements. A Mechrolab highspeed membrane osmometer Model 502 with a variable temperature controller (Mechrolab, Mount>ain View, Calif .) was employed. Membranes used were ultracellafilter grade feinst or allerfeinst. The latter were used for low molecular weight samples. The membranes were conditioned through 1-propanol to each relevant solvent before use. The data were obtained at 30.0" in either or both toluene and l-chloron-butane (1-CB) solutions. The measurements were usually made at five or six different concentrations and the results were analyzed according to the equation' (T/C)"* = (RT/M,)'/'[l
+ (AzMJ2)C ]
where T is the osmotic pressure, C is the polymer concentration, M, is the number average molecular weight, A2 is the second virial coefficient, and R and T are the gas constant and the absolute temperature, respectively. Values obtained for M , and AZ are summarized in Table 111. Determination of 0 Temperatures. I n order to find pure 0 solvents for ST-ATMA copolymers, a variety of solvents were first examined by a simple cloud point test. For the test we used three random copolymer fractions with nearly equal M , (one from each of the three series), and also PST and PMMA samples. I n each case, a mixture of lc5mg of polymer and 3 ml of solvent was subjected to the cloud point test, within the temperature range from about 95" to room temperature. Several solvents were found to be promising.17 Two solvents, cyclohexano118 (CHL) and 2-ethoxyethanol" (2-EE), were chosen and the 0 temperatures were determined by osmotic-pressure measurements as the teniperatures a t which the osmotic second virial coefficient, A 2 , vanishes.' Values obtained for 0 and temperature coefficient of Az at 0 , (bA2/bT)e,are given in Table IV. Rejractive-Index Increment and Light-Scattering Measurements. Specific refractive-index increments of PST, PMMA, and some of the copolymer fractions were measured with 436-mw wavelength light in toluene, in dioxane, and in butanone at 30.0" by a Shimadzu (Debye type) differential refractometer equipped with a thermostated cell (Shimadzu-Seisakusho, Kyoto, Japan). Light-scattering measurements were made by a Shimadzu (Brice type) light-scattering photometer equipped with a thermostated jacket.6 A cylindrical cell was used throughout the work. Scattering intensities were usually measured at 11 different angles, ranging from 30 to 150", withvertically polarized light of 436-mp
831
Table 111: Results of Fractionation and Osmometric Data" 104~------.
A_.-.-
Code
SM3-1 -5 -6
-7 -8 -9 -10 -11 (0.29 f 0.01)' SM5-1 -2 -5 -6
-8
10-4~,
59.2 44.0 38.7 35.4 27.6 18.1 10.0 4.67 50.0 46.0 39.7 35.0 26.4 22.6 18.5 9.66 6.89 4.80 3.42
-9 -10 -11 SM5'-2 -4 -5 (0.56 f: 0.01)" SM7-2 43.2 -3 34.2 -6 27.4 -8 21.0 -9 18.1 -10 14.5 -1 1 6.67 SM7'-2 6.21 -4 4.73 -5 4.00 (0.70 i 0.lO)c
M,/M,b
1.78
...
TOL
I-CB
2.6 2.8
1.5 1.3 1.5 1.4 2.3 1.7 2.1
...
1.75
...
... 1.52 1.50 .
.
I
... 1.62 1.53 .
I
.
... ... 1.48 , . .
1.41 1.40
... 1.26 1.29
... 1.29
... 1.26
...
...
...
1.24
3.0 3.2 3.7 4.3 3.7 3.3 3.1 3.5 3.6 3.7 3.6 3.9 4.1
...
... .
.
I
2.9 3.1 4.0 3.5
...
4.7 4.1
...
... ...
...
2.2 2.1 2.2 2.2 2.3 2.6 2.7 3.4 4.6 4.6 5.6 2.1 2.6 3.0 3.0 3.3 3.2 5.8 3.0 4.1 5.2
a Units: average ST content = mole fraction, and Az = ml mol/gz. Abbreviations: TOL = toluene and 1-CB = 1chloro-n-butane, all measured a t 30.0'. For M , data, see The numbers in parentheses are the average Tables V and VI. ST content.
wavelength (for both incident and scattered lights). The calibration procedures of the apparatus were described e1sewhe1-e.~ For some of the fractions, measurements were carried out in three solvents, butanone, dioxane, and toluene, a t 30.0 f 0.1," and for some others only in butanone at 30.0 f: 0.1". All the test solutions were prepared by keeping them at about 60" in sealed glass tubes for at least 2 days, with occasional shaking. The solutions were filtered through Grade m ultracellafilter directly into the light-scattering cell. It has been known that the specific refractive index increment, v of a binary copolymer can be usually ex(17) For a preliminary report, see T. Kotaka, H. Ohnuma, and Y . Murakami, J . Phys. Chem., 70,4099 (1966). (18) D.Froelich and H. Benoit, MakromoE. Chem., 92, 224 (1966), reported values of 8 in CHL are 77.6' for PMMMA, 83.6' for PST, 68.6' for azeotropic ST-MMA random copolymer, and 81.6O for
equimolar (PST-PMMA) type block copolymer.
Volume 72, Number.3 March 1968
T. KOTAKA, Y. MURAKAMI, AND H. INAGAKI
832
Table IV : e Temperatures and ( bAz/b2')e for ST-MMA Random Copolymers ST Code
content, mole fraction
PMMA 14Mb SM3-7 SM5-6 SM7-3 PST 16Hb
0.00 0.285 0.552 0.694 1.00
F-0, 'C and 1O6(L4~/dT)e"Cyclohexanol
2-Ethoxyethanol
79.4(2.5) 68.2(2.3) 61.3(1.3) 63.0 ( 1 . 3 ) 81.8(1.7)
39.0(0.80) 40.0(0.46) 58.4(0.52) 72.8 (0.70) Insoluble
a Values in parentheses. Prepared by anionic polymerization method by using Na-biphenyl as an initiator. Values of M, are 7.21 X lo4 for PMMA 14M and 20.6 x lo4 for PST 16H.
pressed with a high precision as to be dependent linearly on the composition and independent of molecular weight, 1w5 Y
=
XYl
+ (1 - z)
e+o
lim Kc/Re
l/Mapp
+ 2(A2'),,, c + . . .
(l/Mapp) X
=
i
where fi is the weight fraction of all the components with zf,regardless of their molecular weights. Thus, measurements of Mapp in at least three solvents should allow one to determine M,, P , and Q independently. If the composition heterogeneity is very small, measurements, even in a single solvent with large v values, would allow one to determine M , with a reasonable accuracy. The data were analyzed according to the above equations; the results are given in Tables V and VI.
v2
where x is the weight fraction of monomer species I, and the subscripts 1 and 2 denote the quantities characteristic of the parent homopolymers I and I1 (in this study, PST and PR/IMA, respectively). The reduced scattering intensity, Re, at an scattering angle 8 for a copolymer solution can be expressed, following the procedure of Zimm,lQas lim Kc/RB =
and Q represent the heterogeneity in composition; P relates to the composition variation with molecular weight, and Q to its broadness. For azeotropic copolymers, or more generally, for low-conversion copolymers of any composition, it is very likely that the composition variation with molecular weight is negligible and the broadness is also The assumptions lead to: P = 0, Q = M , fi(xi - x)~,
Table V : Summary of Light Scattering Map, Data"
Code
SM3-1 -6 -9 SM5-2 -5 -10 SM7-2 -3
-8
0-0 a
,------10-4M~pp----MEK DOX TOL
111.1 69.0 27.0 74.6 61.7 26.6 57.1 48.3 28.2
112.9 71.4 28.2 87.7 67.1 28.6 62.5 52.6 30.8
10-4~,"
163.9 83.3
105.8 67.8 27.6 74.5 60.8 27.7 54.4 44.0 27.0
...
100.0 74.1 27.0 74.5 62.5 35.7
Q/M,
0.06 0.03 0.00 0.13 0.08 0.00 0.23 0.24 0.17
Abbreviations: MEK = butanone, DOX = dioxane, and = toluene, all measured a t 30.0'.
TOL
where K = ( 4 n 2 / h o 4 N , ) ( n o is ~ ) the 2 well known lightscattering factor; A 0 is the wavelength of light in vacuo; nois the refractive index of the solvent; and N , is the Avogadro number. In case of homopolymers, M,,,, (A2'),,,, and (Sf),,, are, respectively, equal to the weight-average molecular weight M,; the lightscattering second virial coefficient, A2'; and the xaverage mean-square statistical radius, (S2)z,of the polymer chainsa20 These quantities for copolymers are apparent values and different from those of homoif there is the slightest heterogeneity in the composition of the copolymer. As far as the assumption of the linear dependence of v on z is valid, M,,, can be expressed as12v16 Mapp
=
where p
+ 2 [ ( ~ -1 v z ) / ~ I P C
M,
= CYdMi(2i i
- $1, Q
[(vi
-
vz)/vI2
= CYdMl(Xl
-
%P/C
(In q r ) / c
=
= [71
+ k'[qI2 c + 0 ( c 2 )
[a1 - (1/2 -
k')[qI2
c
+ 0(c2)
Q XY,
where y t is the weight fraction of co&ponent i whose molecular weight is M , and the composition is xi; x = is the average composition of the sample (in i
weight fraction of component I). The parameters p The Journal of Physical Chemistry
Viscosity Measurements. The measurements were made by using three Ubbelohde dilution viscometers with flow times (for toluene at 30.0") more than 200 sec/ml. Neither the kinetic-energy correction nor the non-Newtonian correction were found to be necessary. The temperature was kept within h0.02" at each desired value. The intrinsic viscosity, [ q ] , was determined by using two types of viscosity-concentration plot, i.e., q s p / c and (In q r ) / c us. c, and by extrapolating them so as to yield a common intercept at c = 0
(19) B. H.Zimm, J . Chem. Phys., 16, 1093, 1099 (1948). (20) H. C. Brinkman and J. 3. Hermans, ibid., 17, 574 (1949); J. G. Kirkwood and R. J . Goldberg, ibid., 18, 54 (1950); W. H. Stockmayer, ibid., 18, 58 (1950). (21) W. Bushuk and H. Benoit, Compt. Rend., 244, 3167 (1958); Can. J . Chem., 36, 1616 (1958); H. Benoit and C. Wippler, J . Chim. Phvs.. 57, 524 (1960). (22) W. H.Stockmayer, J. Chem. Phvs., 13, 199 (1945).
STYRENE-METHYL METHACRYLATE RANDOM COPOLYMERS
833
Table VI : Summary of Viscometric Data of ST-MMA Copolymers in Three Good Solvents and in Two
SM3-1 -5 -6 -7
-8 -9 -10 -11
105.8 (77.4) 67.8 (60.6) (44.7) 27.6 15.0 (7.05)
2.01 1.62 1.43 1.32 0.996 0,768 0.507 0.299
1.42 1.15 1.09 0.974 0.774 0.585 0.379 0.242
1.15 0.946 0.865 0.830 0.666 0.518 0.360
...
0.628 (6.10)
...
-8 -9 -10 -11 SM5'-2 -4 -5
(81.0) 74.5 60.8 (53.5) (39.6) (33.4) 27.7 (13.7) 9.71 6.76 (4.79)
1.96 1.82 1.61 1.44 1.13 1.05 0.877 0.557 0.434 0.336 0.262
1.15 1.05 0.984 0.880 0.740 0.657 0.589 0.387 0.314 0.261 0.193
1.36 1.29 1.12 1.05 0.846 0.755 0.663
0.508 (6.55)
0.340 (6.48) 0.264 (6.83)
0.341 (6.50) 0.270 (6.98)
...
...
54.4 44.0 (35.2) 27.0 (23.0) 18.3 (8.40) (7.70) (5.86) 4.96
1.68 1.47 1.17 0.949 0.890 0.707 0.481 0.386 0.312 0.276
0.875 0.770 0.647 0.555 0.517 0,441 0,287 0.261 0.221 0.206
1.17 1.03 0.845 0.726 0.611 0.556
...
0.310 0.263 0.237
=
60.0'
... 0.682 (7.92) 0.622 (7.99)
.*.
...
T
-
64.0'
0.670 (7.44)
... ...
...
0.550 ( 7 , 5 4 )
...
0.443 (7.68)
...
...
0.408 (7.76)
...
...
0.221 ( 7 , 1 6 )
0.201(7.76)
...
...
...
T
SM7-2 -3 -6 -8 -9 -10 -11 SM7'-2 -4 -5
...
...
... 0.335 0.275 0.220
0.636 (6.18) 0.574 (6.54)
0.512 (6.24)
T
SM5-1 -2 -5 -6
e Solvents
-
72.0°
... ... T = 64.0'
0.580 (7.88)
0.578 (7.84)
0.401(7.75)
0.380 (7.35)
0.332 (7.78)
0.320 (7.50)
...
...
0.171 (7.72)
0.166(7.47)
... ...
...
... ...
... ... ...
... ...
Abbreviations: TOL = toluene, DEM = diethylmalonate, a Values in parentheses are those estimated from M , data; see text. Abbreviations: 2-EE = 2-ethoxyethanol and CHL = cyclohexanol; values and 1-CB = 1-chloro-n-butane, all measured at 30.0'. in parentheses are 104[7]/MdI2.
where qBP is the specific viscosity, qr is the relative viscosity, and k' is the Huggins constant. Values obtained for [ q ] in various solvents are summarized in Table VI.
Results Fractionation and Osmometric Data. Table 111 summarizes the results of fractionation and osmometric data. It is found from the results of elementary analysis that the composition fluctuation from one fraction to the other in each series is less than A l % . We assign the values (in mole fraction) 0.29 0.01 for the fractions from SM3, 0.56 cfi: 0.01 for those from SA45 and SM5', and 0.70 cfi: 0.01 for those from SM7 and SM7', as the average ST contents. For some of the fractions, the values of M,/M, are also listed in Table 111. The fractionation procedure developed by Stockmayer, et aZ.,15 appears to be successful in fractionating
ST-MMA copolymers, according only to molecular weight but not to average composition. However, in view of achieving an efficient fractionation by molecular weight, the butanone-DIPE system appears to be less efficient for MMA-rich copolymers. As shown in Table 111, the values of Mw/Mn are usually larger in SM3 fractions and smaller in SM7 fractions. A similar result was reported by Utiyama,lB who found a poor resolution of PMMA by molecular weights. I n this respect, use of some other system might be desirable for MMA-rich copolymers. 8 Temperatures. Table IV summarizes the results of 8 temperature determination in CHL and in 2-EE for the three copolymer fractions and two homopolymer ~ a m p l e s , ' ~and J ~ Figure 1 shows plots of 8 us. ST content m in the copolymers. It is interesting to note that the 8 temperatures in CHL of the copolymers are lower than either of those of PST and PMMA and have a minimum at nearly equimolar composition, Volume 7.2, Number 3 March 196
T. KOTAKA, Y. MURAKAMI, AND H. INAGAKI
834
100
0.20 90
E
-.
80
0.1 5
0
d
E
Y
a 0.10
0.05
30 0
0.4 0.6 0.8 1.0 ST-CONTENT rn (PST)
0.2
(PMMA)
Figure 1. The e temperatures of ST-MMA copolymers in cyclohexanol (CHL) and in 2-ethoxyethanol (2-EE) as functions of S T content (mole yo)in the copolymers. The points represent data for the random copolymers and the homopolymers. Bold dashed curves represent those for block copolymers.'?
while in 2-EE, which is a nonsolvent for PST, the 0 temperatures appear to have a shallow minimum at MAA-rich composition and increase rapidly with increasing ST content. Figure 1 also shows our previous data of 0 for ST-MMA block copolymer^'^ in these solvents. The composition dependences of 0 for the block copolymers are quite different from those of the random copolymers. Detailed discussion on the behavior of the block copolymers will be reported in a later publication. Specific Refractive-Index Increment and Light Scattering. Figure 2 shows the values of v as functions of ST content x. The results were found to be expressed with good approximations by the relations (for 436-mp light) : v = 0.232~ 0.133 (1 - x) in butanone, 30.0'; v = 0.1842 0.071(1 - x) in dioxane, 30.0'; 0.004(1 - x) in toluene, 30.0'. and v = 0.1132 The fluctuation in the values of v from one fraction to the other was negligible as anticipated; the result corroborates that of the elementary analysis. Table V summarizes the light-scattering data obtained in three solvents for some of the fractions from each series. Since we already found that the composition of each fraction in each series was virtually independent of molecular weight, we calculated M,v and Q by employing the assumption P = 0. We found that (i) the values of Q / M , were reasonably small as anticipated, and (ii) the values of Ma,, obtained in butanone, in which v is the largest among the three solvents, were close to the values of M,. Therefore, subsequent measurements were carried out only in butanone at 30.0'. The values of M,/Mn in each series derived from these data varied systematically with increasing M n ,and we found a correlation between M , and Mn
+ +
+
The Journal of Physical Chemistry
0 0 (PMMA)
0.2 0.4
0.6 0.8 1.0 ST CONTENT x (PST)
Figure 2. The specific refractive-index increments, Y, of ST-MMA copolymers as functions of ST content (wt %) in three solvents, i.e., in P-butanone, p-dioxane, and toluene, in order from top to bottom.
Figure 3. Intrinsic viscosity-molecular weight relationship for ST-MMA copolymers SM5 and 5' in toluene, 0; in DEM, a; in 1-CB, @; all a t 30.0°; in CHL a t 64.0°,0 ; and in 2-EE a t 60.0", e.
from plots of M,/Mn vs. M,. Then, we drew a smooth curve fitting the plots, and, by using this as the calibration curve we estimated M , of other fractions from the Mn data. The values of M , thus obtained are listed in Table VI, in which the values in parentheses are those estimated from the M , data. Intrinsic Viscosity. Table VI lists [ q ] data obtained in the two 0 solvents and in toluene, DEM, and 1-CB. By way of illustration, Figure 3 shows plots of log [ q ] us. log M , for SM5 and SM5' in these solvents. The plot for each solvent is fitted fairly well by a straight line over the range of M , examined. All these data are summarized in the form of the Mark-HouwinkSakurada relation, [v]= K'M,". The values of K' and a are listed in Table VII. It is interesting to note that
STYRENE-METHYL
METHACRYLATE RANDOM COPOLYMERS
the [ r ] ] vs. 1%' relations of the three copolymers are almost identical in toluene, which is a good solvent toward both of PST and PhIIRIA. On the other hand, in 1-CB, which is a 0 solvent for PMMA (0 = 35.4°),23the exponent a increases with increasing ST content. I n DEM, which is a 8 solvent for PST (0 = 35.9°),24,26 the tendency is apparently reversed. I n the 8 solvents the exponents a are close to 0.5, as is always the case for any homopolymer-8 solvent system.
Table VI1 : List of the Mark-Houwink-Sakurada Constants, [v] = K'MWaof ST-MMA Copolymers in Various Solvents Code
(ST,mol fraction)
Solvent
SM3 (0.29)
TOL DEM 1-CB 2-EE 40.0' CHL 68.0"
I . 14 1.57 2.65 9.73 9.73
0.70 0.66 0.60 0.47 0.47
SM5 (0.56)
TOL DEM 1-CB 2-EE 60.0' CHL 64.0'
1.32 2.49 2.49 6.85 7.00
0.71 0.62 0.63 0.51 0.51
TOL DEM 1-CB 2-EE 72.0' CHL 64.0'
0.832 3.13 1.76 7.16 7.16
0.75 0.60 0.67 0.51 0.51
SM7 (0.70)
104~'
a
a Measured a t 30.0', unless otherwise specified; for abbreviations, see Table VI.
Discussion It is well established that for solutions of flexible chain (homo)polymers the effects of long-range interactions (the excluded volume effects) between nonbonded sequents-i.e., those belonging to different polymer chains and those remotely separated by many other segments along the same polymer chain-should vanish under a special condition of solvent and temperature known as the Flory 0 condition.' Consequently, under the 0 condition, Az becomes zero and the polymer chains assume random flight configurations, the unperturbed dimensions being determined solely by the short-range interactions. The deviations of polymer dimensions and other properties, such as [ r ] ] , due to the excluded volume effects are usually expressed in terms of expansion factors such as a = ((X2)/(s~)o)1'a aq
= ([?lI/[?lle)l'a
with (Sz)o being the unperturbed mean-square statistical radius. The expansion factors are then ex-
835 pressed as functions of an excluded volume parameter z defined as x = ( B / A 3 )M'12
(1)
A 2 = (S2)o/M= az/61n,
(2)
with
B = (1/4a) "z(@/m,2) (3) Here, A and B are the parameters representing the short-range and the long-range interactions, reepectively; 1%' is the molecular weight of the polymer; u2 and m, are the mean-square length and the molar weight of a segmenl of the statistical model; and p is the binary cluster integral between any pair of the segments and vanishes at T = 0. As far as one assumes that the intra- and the interchain long-range interactions are identical in nature, the Az is also expressed in terms of the parameters as defined by eq 1-3.1-3,26 I n case of extending such concepts to the studies on copolymer solutions, one ought to ask, first of all, whether the concepts of the 8 conditions may stand valid for copolymer solutions as well, and then to ask whether it is possible to apply the procedures proposed for homopolymer solutions to deduce the parameters A and B of copo!ymer solutions. The results of our preliminary study17 showed that the answers to the above questions are both affirmative, at least for solution of random copolymers, although the evidence is somewhat indirect. Thus, in this section we shall attempt to apply the proposed procedures to the present copolymer data. Intrinsic Viscosity at 8 Conditions. At the 8 con, dition where A2 = 0, the intrinsic viscosity, [ ~ ] emay be written in the form
(4) where KO = +of((Sz)o/M)a~z = aOfA3;eofis the viscosity constantz7 a t the 8 condition. Thus one would expect that [r]le is proportional to M'l2 so far as the statistics of random flight prevails at the 8 condition, and, hence, the KO may be regarded as constant. It is seen in Table VI that [r]]e/&fN1/2 is constant, or nearly constant, in each system, satisfying the relation given in eq 4. Figure 4 shows plots of [r]]e/M,1/2vs. MN'l', from which values of KOmay be readily estimated. Use of eq 4 to deduce (X2)o (or A z, requires the knowledge of a,,'. As far as we are aware, such knowledge (23) G. V. Schulz and R. Kirste, 2. Physik. Chem., 30, 171 (1961). (24) T. A. Orofino and J. W. Mickey, Jr., J . Chem. PhyE., 38, 2512 (1963); T. A. Orofino, ibid., 45, 4310 (1966). (25) G. V. Sohuls and H. Baumann, Makromol. Chem., 60, 120 (1963). (26) B. H. Zimm, J. Chem. Phys., 14, 164 (1946); B. H. Zimm, W. H. Stockmayer, and M. Fixman, ibid., 21, 1716 (1953). (27) P. J. Flory and T. G Fox, J . Am. Chem. Soc., 73, 1904 (1951); see also ref 1, Chapter 14. Volume 79, Numbw 3 March 1968
T. KOTAKA, Y. NURAKAMI, AND H. INAGAKI
836
Table VIII: Values of the Parameters, KO,A*, and for ST-MMA Random Copolymers IO'Ko, m
dl/g
PMMA
0.00 0.29 0.546 0.56 0.70 1.00
5 . 0 zt 0 . 5 6.6f0.2 7.5 7.5i0.2 7.7+0.2 8.0zt0.5
SM3 SMA" SM5,5' SM7,7' PST
20
IO'SA2, om2
Code
15
a
et
5.42 zt 0.30 6.52f0.15 7.12 7.12i0.15 7.24i00.15 7.45zt0.30
u
C
1.87 i 0.07 2.05 f 0 . 0 2 2.15 2.15zt0.02 2.17=tOo.02 2.22 f 0 . 0 5
Data for ST-MMA azeotropic copolymer by Stockmayer,
~1.16
10
0
5
1
0
0
5
1
0
1O2Md"
Figure 4. Plots of [q]/MW'/'US. Mw'/' for ST-MMA copolymers (a) SM3; (b) SM5 and 5'; and (c) SM7 and 7' in various solvents (for sample codes, see Table V; solvents are identified by the symbols as in Figure 3); and for (d) ST-MMA azeotropic copolymers in butanone a t 25.0°, @;I6 PST in 1-CB at 40.8', @;le and in cyclohexane a t 34.5", 0;s PMMA in D E M a t 30.0", 0; and in I-CB a t 40.8', @.la The dashed lines show the values of aqa= 1.8 for the copolymer data.
in order (i) to estimate KOfrom the good solvent data to see if there is any specific solvent effect on KO, and then (ii) to estimate B for each copolymer-solvent pair. The simplest and the easiest to manipulate among those methods is that due to Stockmayer and Fixman,g which is written as [q]/&fl"
=
KO+ 1.55@o'Bf%f1''
(7)
The numerical constant, 1.55, is so chosen as to attain agreement with first-order perturbation t h e ~ r y . ~ ~ J ' The plots of [ q ] / M w ' / ' os. MW'" for the present is not yet available for the ST-MMA copolymer-8 and some other relevant data15!l6are shown in Figures solvent systems or any other copolymer systems. 4a-d. It is to be noted that [q]/Mwl"is not a linear However, in view of the evidence that the a' constant function of Mw'/', particularly for toluene data. The of the copolymers in good solvents is in agreement with use of eq 7 should be limited to rather poor solvent that for homopolymer-good solvent ~ y s t e m s ,it~ ~ ~data; ~ ~ otherwise it would result in an overestimate of KO. would not be unreasonable to employ the value of a0' Nevertheless, it is seen that the plots for each copolymer currently established for homopolymer-8 solvent syscan be reasonably well extrapolated to M,.,'/' = 0, tems. Thus, by employing the value of 10-21Cbo' = so as to yield a common value of KO. 39.4 ( [ q ] ,dl/g) given by Pyun and Fixmaq28we calcuUtiyama16 and Inagaki, Suzulii, and Kurata'o suglate A 2 (= (Sz)o/M,) for the three copolymer samples. gested a viscosity equation based on the Ptitsyn equaWe also calculate the steric factor 6 ,which is a measure tiona2 for a2 vs. x combined with a semiempirical of the hindrance to internal rotations about the skeletal r e l a t i ~ nbetween ~ ~ ~ ~a ~and a,,,such as aV3= a'/'. The link and is defined as equation due to Inagaki, et a2.,lo reads u = (A2/AfZ)'/2 ( [q]/M1/2)4/3 = 3.68 with Ai2 being the mean-square statistical radius of a [l 9.36(B/A3) M'/']l"~81/4.68 (8) All the values hypothetical freely rotating hai in.^,^ The authors further suggested that for very good solvent obtained are summarized in Table VIII. systems the unity in the right-hand side of eq 8 can be Analysis of [ q ]Data in Good Solvents. The intrinsic viscosity in good solvents may be written in the Flory(28) C. W. Pyun and M. Fixman, J . Chem. Phys., 42, 3838 (1965). Fox viscosity equationz7
+
+
tZ9
(29) For vinyl polymers, 108Ar = 1.256/MU'/a,with Mu being the molar weight of the repeating unit. For the ST-,MAMA copolymers with ST mole fraction m, we use Mu = 104m lOO(1 - m ) . or by the more explanatory form (30) H. Yamakawa and M. Kurata, J . Phys. SOC.Japan, 13, 94 (1958); M. Kurata and H. Yamakawa, J . Chem. Phys., 29, 311 (1958). [ q ] = ~ ~ ~ ' /=*~ a ~ ~~ ' /3' ( @ / + ~ ' ) a(6) 3 (31) M. Fixman, ibid., 23, 1656 (1955); 36, 3123 (1962). Therefore, eq 6 combined with an appropriate ex(32) 0. B. Ptitsyn, Vysokomol. Soedin., 3, 1673 (1961); the equation is known to fit second-order perturbation theory. pression for aq3us. x or for a3 and (@'/ao')vs. z can be (33) It is interesting to note that recent calculation by Fixman84 is expected to allow an estimate of A and B from [ q ] well approximated by the relation aqs = ( ~ 2 . 6 2over a considerably us. M data for a given solvent. We test the present large span of z . data by a few of the currently proposed m e t h ~ d s ~ J ~(34) ' ~ M. Fixman, J . Chem. Phys., 45, 785, 793 (1966).
[q] =
a'((sz)/M)8/* M1I2
The Journal of Physical Chemistry
(5)
+
837
STYRENE-METHYL METHACRYLATE RANDOM COPOLYMERS
,
neglected with respect to 9.362, and plots of ( [q]/M1/2)4/6 os. M’18 can be extrapolated linearly to yield KO. The authors emphasize that the linear extrapolation should be applied only to good solvent data for which ag3 > 2.2, and, hence, as such is complementary to eq 7. The present data are examined also by eq 8. We find that KO for toluene data are in good agreement with those determined in the 8 solvents. The plots for DEM and 1-CB solutions, both of which are poorer solvents than toluene, show nonlinear dependence on MW’/’, as anticipated. Berry has proposed still another viscosity equation on the basis of his experimental data on the correlations of A2, (S2),[ q ] , and M for solutions of monodisperse polystyrene in decalin and in t o l ~ e n e . ~He ? ~suggested that nonlinear functions of [q]/M’/*vs. MI/’, often encountered in good solvent systems, may be approximately split into two linear relations within the range of [q]/MI/’ =
+ 0.60@0‘BM1/2
KO
1.40Ko
+ 0.30@o’’B~’’*
I
I
I
I
I
f
-I
ZC
5 4
3 2 0
/
I
l
l
l
1
2
3
4
5
I
5
I
l
7
8
Figure 5. Plots of ( [q]/M$/’)’/’ US. M,/[q] for ( a ) SM3; (b) SM5 and 5‘; and ( c ) SM7 and 7’ in various solvents, as identified by the symbols as in Figure 3.
(10)
for 1.8 < aV8< 4 av3specified as above.
These equations retain the form of eq 7, but the constants are replaced by empirical ones. Therefore, so far as the estimates of K Oare concerned, the conclusions are the same to those of eq 7. However this is obviously not true for the estimates of B. The author further suggested that these two equations may also be approximated by a single relation’ ( [~]/M”*)”’ = Kol”{ 1
I
lo+ M w l [?I
(9)
for 1 < Av3 < 1.6 [ q ] / k f ’ / z=
I
I
+ 0.42@ofB(M/[~])} (11)
The present data are also tested by eq 11; such plots are shown in Figure 5. Again we find that the plots for each copolymer can be extrapolated linearly to M w / [ q ] = 0 to give the values of KO,’/*which are in practical agreement with each other and with those obtained in the e solvents, except for a few occasions. Such an exception is, for example, seen for the system SM7, 7’ in toluene, in which an extrapolation placing emphasis on all the experimental points (cf. the dashed line in Figure 5c) would result in an unusually small value KO1/’. Although the origin of such a descrepancy is not clear, it appears that this does not reflect the specific solvent effect, but rather implies an inadequate use of eq 11 (e.g., the range of M is not adequate). As we have shown above, one can expect that the use of the different methods to complement each other would minimize the uncertainties in the estimate of KO. Then we may conclude that the values of KO (and presumably A ) for the copolymers are not greatly influenced by the specific solvent effect and may be regarded as constant within the accuracy of, say, f 10%. Next we turn to the problem of estimating the values of B from [r]os. M data. Obviously any of the vis-
cosity equations, such as eq 7-11, can be expected to afford the estimates of B , but the results are never in agreement between those obtained by the different methods unlike for the estimates of KO. For example, if the results of eq 7 and 9 are compared, the latter equation predicts the values of B to be larger, by a factor of 2.58, than the former equation does for the same data. The situation is the same for other equations. Needless to say, it is not possible to judge from [ q ] vs. M data alone as to which one of these different methods would yield most appropriate estimates of B. The question shall be discussed in the next section in relation with the analysis of A2 vs. M data. Analysis of A2 Data. It is well known that A2 in the vicinity of 8 may be written as A2
+ O(B)]
=
4na’*N,B[1
B
=
(12)
with
~
~ - (e / 1q
(13)
where Bo is a parameter proportional to Flory’s entropy of mixing parameter.‘ Hence, one can estimate values of B in the vicinity of 8 by using eq 12 combined with the temperature coefficient of A2 at T = e.
(dAz/dT)T=e = (4n”*N./e) Bo The data in cyclohexanol are analyzed by these equations; values of B are given in Table IX, where T is arbitrarily taken as 80.0’. It has been shown that the interaction parameters for good solvent systems can be deduced from A , vs. M data, just as from [ q ] us. M data, by using an appropriate closed expression for AZas a function of A , B , Volume 72, Number 3 March 1968
838
T. KOTAKA, Y. MURAKAMI, AND H. INAGAKI
Table IX : Values of Long-Range Interaction Parameters B from [q] and A2 Data
Code
PMMA
SM3
SM5, 5'
SM7, 7'
PST
so1venta
----lOaOB-----, From 1711 (eq 11)
TOL DEM 1-CB CHL (SO')
60.5' 21.1
TOL DEM 1-CB CHL (SO')
82.3 50.0 21.5
TOL DEM 1-CB CHL (SO')
97.0 25.0 43.0
TOL DEM 1-CB CHL (SO')
120 24.0 55.6
TOL DEM 1-CB CHL (SO')
107. 5c 0 38. Od
0
...
...
...
...
...
From Aab 100
...
-
...
0 0.7
$-
I
\ f 50
120 i 10
+5 01
...
60 17
* 10
...
70 A 10 16 110"
... ...
-2.3
(15)
The function %?(a), representing the extent of mutual penetration of approaching polymer coils18is known to be a function which in general increases monotonically from zero (at 1 = 0, the 0 condition) and reaches an asymptotic limit as 1 increases. A number of approximate expressionsa6for *(I) have been proposed so far, and any of the appropriate combinations of *(I) and a3 can be expected to afford estimates of A and B from A z os. M data in good solvents. Each of these equations predicts more or less different dependence of A2M'IZ on x , and thus the application of the different equations to a given set of A2 vs. M data obviously yields different estimates of @ / A 3 ) value; the situation is the same as we are confronted with the analysis of the aV3 vs. x relations. Again it is not possible to judge from A2 vs. M data alone which one of them is the most appropriate. As far as one assumes that both the intrachain and the interchain long-range interactions are identical in nature, one would expect that the values of B derived from different sources, [v] and Az data, should be in The Journal of Physical Chemistry
I
0
1 .o
((71/(?Ie)
140 i 10
%?(I)((S2)/M)a'a = %?(I)A3a3 (14) z/a3
p
140 A 10
and Such expressions may be most conveniently cast into the forma5
I =
-
E
... 27 20
a For abbreviations, see Table VI. Data for CHL are treated by eq 12 and 13 taking arbitrarily T = SO0; others are treated by the correlations of A2 vs. M proposed by Berry.6 Data in ref 6 and 7. Data in ref 16.
AzM'19
-
2.0
3.0
-1
Figure 6. Plots of AzM,/[q] us. [ r ] / [ ~ ] e - 1 for ST-MMA copolymers in toluene and in 1-CB (solvents are identified by the symbols as in Figure 3, and the polymers by pip up for SM3, pip right for SM5 and 5', and pip down for SM7 and 7'); and for PST in toluenea17(A),PMMA in 1-CB2a(O),and also in other good solvents23 (+), Curves I and I1 represent theoretical correlations predicted by the Fixman-type approximations and by the Ptitsyn-type approximations,8 and curve I11 represents empirical correlation due to Berrys,' (see text).
agreement with each other (of course, if the particular combination of the theories on [v] and A2 employed for the analysis is adequate). The correlation between [ r ]and AZcan be most conveniently examined by the familiar dimensionless ratio of A 2 M / [ r ] = [\~(z)/G,'] (a3/aV3 as) a function of, say, ( [r]/[vle) - 1 = av3 1. Figure 6 shows such plots for the present copolymer data together with relevant homopolymer data. Figure 6 also shows two theoretical curves (referred to as curve I and II), which are constructed by combining the [ r ] equations (7 and 8) with two of the variation theories on A Z both due to Kurata, et al. [eq 74-76, and eq 10 and 101 in ref 81, respectively. Thus the first combination, curve I, is essentially related to the Fixman variation theory31 of the excluded effect, and the second one, curve 11, to the Ptitsyn variation theory.32 Figure 6 also shows a curve due to Berry's empirical correlat i o n ~ ~(referred ,' to as curve 111),which is constructed by combining Berry's A2 equation [e.g., eq 1 and 2 in ref 6a, or eq 7, 13, and 20 in ref 6b] and the [ r ]equation (11). Certain interesting features emerge from the figure. First of all, it is seen that the experimental points show (35) In eq 15, the expansion factor, a, for an isolated chain should be replaced by an expansion factor, a2, for a chain in a bimolecular cluster. They are often approximated as az = a. For detailed discussion, see ref 3, 6, and 8. (36) We do not list all the references here. Summaries and detailed discussion on them are found in several articles. See, for example, ref 2,3,6, and 8.
STYRENE-METHYL METHACRYLATE RANDOM COPOLYNERS a fairly definite tendency, in spite of the fact that they include both the copolymer and homopolymer data from different sources. Thus the behavior of the homopolymers and the random copolymers are indistinguishable in this respect. Curve I appears to predict correct behavior in the vicinity of the origin and a subsequent rapid approach to an asymptotic limit. However, the theoretical limit of A 2 M / [ r ]= 60 is far too small in comparison with the observed value of about 100-140. Consequently, the analysis based on eq 7 and the corresponding A 2 equation [e.g., eq 88 in ref 81, both of which are essentially based on the Fixman type approximation for the excluded volume eff e ~ t , would ~l result in the inconsistency between the viscometric and osmometric B values, namely the former is smaller than the latter by a factor of 2, a t least. Curve I1 also appears to predict correct initial behavior and subsequent increase of A d f / [ q ] up to about 120, which is about a right order of magnit~de.~’ However, a rather significant discrepancy between the theory and the experiments is found in the range of intermediate values of aq3,namely, the experimental values appear to approach to the asymptotic limit more rapidly than does the theoretical curve 11. Hence it would be expected that the analysis based on eq 8 and the corresponding A2 equation [e.g., such as eq 10 and 101 in ref 81 would result in a larger descrepancy between the viscometric and the osmometric values of B for moderately good solvent systems, rather than for very good solvent systems. I n fact, from the analyses of the present data by these equations, we find that for 1-CB solutions the values of B by the A2 method are about twice as large as those by the [q] method, and for toluene solutions, the former are about 20-50% larger than the latter. The inconsistencies found in the above two cases appears to be too large to be disregarded simply by attributing them to the usual poor experimental accuracy of Az data. This sort of inconsistency has already been reported for several homopolymer systems5-8 and is apparently real in the copolymer systems. To explain these inconsistencies between the theories and the experiments, a refinement of the theories of the excluded volume effect appears to be necessary. More recent show that the functions a(z) and a,,(x),which are consistent with the perturbation t h e o r i e ~ ~ in o , ~the ~ vicinity of z = 0, increase more slowly in the range of large 2 than the behavior predicted, for example, by the Ptitsyn equation or by eq 8. Finally, curve I11 appears to represent most closely the general tendency of the observed behavior of A 2 M / [ q ] us. ( [ q ] / [ q I e ) - 1, besides in the vicinity of the origin.40 This apparent success may not be surprising, since curve I11 plotted in this form is nothing else but Berry’s experimental data themselves. Then perhaps
839
the difference found between the asymptotic values of A2M/ [ q ] in Berry’s data (about 100) and in the present data (about 130-140) should deserve more careful examination. The difference between them would be expected t o result in the descrepancy in the estimates of B by the two methods using Berry’s equations by a factor of 20-40%,. A most likely reason one would think of is the difference in the polydispersity of the samples. I t has been known that the heterogeneity correction usually requires the reduction of (Po’ by a which increases the limit of factor of a few to A 2 M / [ q ]by the same factor. The fact that the present A , data are obtained by osmometry could be another reason; usually the osmometric Az are slightly larger than the light-scattering A2’, particularly for polydisperse samples. Anyway, in view of achieving consistent estimates of the values of B from both [ q ] and A z data, the use of eq 11 and the A2 equations proposed by Berry [e.g., eq 1 and 2 in ref 6a] would be expected to give most satisfactory results. The results of the estimates of B for the present systems are given in Table IX. The Short-Range and the Long-Range Interaction Parameters as Functions of Monomer Composition. Figure 7a shows plots of the steric factor, u2, us. ST content m taken from the data in Table VIII. Preproposed a simple relation viously Stockmayer, et for the values of A 2 of a binary copolymer as
A2 =
X(A2)I
+ (1 - X)(A2)2
(16)
where the subscripts 1 and 2 again denote the quantities characteristic t o the parent homopolymers (in this case PST and PMMA, respectively). Equation 16 can be readily rewritten as u2
= m(u2)1
+ (1 - m)(u”z
(16’) Apparently the experimental values are slightly larger than those expected from (16’). The characteristic ratio of the ST-MNA copolymer continuously increases with increasing ST content. Brant, Miller, and Flory4I reported that for the random coil configurations of copolypeptides, for example, of glycinealanine copolymer, introduction of minor proportion of glycine residues into poly-L-alanine brings about a (37) The subsequent slow but unlimited increase of curve I1 is apparently the result of the assumption that q 8 = a6/z in eq 8, which would be valid in the range of initial to intermediate values of a,,*,but would lose its significance in the range of very large aqs. The present data do not cover such a range of large 0 1 ~ 8 ,hence we disregard this behavior of curve 11. (38) Z. Alexandrowicz, IUPAC Symposium on Macromolecular Chemistry, Tokyo-Kyoto, 1966, preprint VI-20. (39) H. Yamakawa, private communication. (40) The large initial slope of curve I11 is apparently the result of the use of eq 11 for constructing the curve. The [VI, eq 11, is supposed to be good only for data confined in good solvents and with large M . (41)D. A. Brant, W. G. Miller, and P. J. Flory, J. Mol. Biol., 23, 47 (1967); W.G. AMiller,D. A. Brant, and P. J. Flory, ibid., 23, 67 (1967). Volume 72, Number 8 March 1968
T. KOTAKA, Y. MURAKAMI, AND H. INAGAKI
840
m W 0
0 c
0
0.2 0.4 0.6 0.8 ST-CONTENT rn
1.0
Figure 7. (a) Plots of the steric factor u2 us. ST content m; the dashed line represents the plots due to eq 16'; (b) plots of the long-range interaction parameter B us. m in various solvents as identified by the symbols as in Figure 3. Solid curves represent the plots due to eq 17'.
disproportionately large decrease in the average chain dimensions. Apparently introduction of ST units into PMMA chains results in an opposite effect, ie., a disproportionately large increase in the average dimensions. UtiyamalB also reported the similar tendency for the ST-NIXA random copolymers, and even the appearance of a maximum in the A 2 vs. m relation. This result might presumably be due to overestimation of KO (and hence of A 2 ) ,which were obtained from [ q ]data in a good solvent, but not by [ q l e data as in the present study. Obviously, the unperturbed dimensions of copolymers are a function of the monomer composition, the sequence length distribution, and the stereochemical ~onfigurations.4~~~ Miller, Brant, and Flory41 have treated the effects of these factors theoretically for random-coil copolypeptides, in which the choice of approximate representation of the dependence of the conformation energy on bond rotation angles is essential. For ST-MMA copolymers, it has been suggested that the analysis of the sequence length distribution and the stereochemical configurations is partly possible from the knowledge of the monomer reactivity ratiosla and nmr spectra.42However, the knowledge of the conformational energy functions is not yet available, and the analysis of the type such as suggested by Miller, et u Z . , ~ ~ is not applicable for the ST-MMA copolymers. Previously UtiyamalB suggested, by avoiding the estimation of the conformational energy functions, that the unperThe Journal of Physical Chemistry
turbed dimensions of copolymer chains may be described by six different parameters, each ascribed to one of the six different types of triadic sequences (e.g., 1-1-1, 1-1-11, 1-11-1, etc.). More recently, Nomura and I w a ~ h i d osuggested ~~ that the unperturbed dimensions may be described by three parameters each ascribed to one of the diadic sequences, and hence the deviations of u2 from those given by eq 16', ie., the extra shortrange interactions, are assumed to be proportional to the population of diadic sequence of unlike monomers. These types of analyses, however, have little theoretical foundations and it is difficult to evaluate the significance of the parameters derived by those procedures. The quantitative analyses on the dependence of the unperturbed dimensions of copolymers on the molecular factors, such as mentioned above, are subject to further investigation on both experimental and theoretical aspects. We now turn to the question on the long-range interaction parameters. As we have discussed in the previous section, the proposed theories for estimating B appear to be unsatisfactory and, hence, the values of B listed in Table IX would have only limited significance, Nevertheless, the data (or any other set of data estimated by different methods) distinctly show the presence of extra long-range interactions and their dependences on composition might have some qualitative significance. Figure 7 shows plots of B vs. m taken from Table IX. I n Figure 7, the maxima are seen at nearly equimolar composition in toluene and in CHL solutions, and in other solvents, 1-CB and DER4, the maximum positions shift to the good solvent sides, namely, in 1-CB to the ST-rich side and in DEM to the MMA-rich side. Stockmayer, et uZ.,16 and UtiyamalB suggested that the parameter B for a random (binary) copolymer may be written as a quadratic function of composition as
B = m2B1
+ (1 - m)2 Bz + 2m(l - m) Biz
(17)
where BIZis a parameter representing the (long-range) interactions between unlike monomers. As a zerothorder approximation, the extra interaction term, ABl2 = BIZ- (B1 Bz)/2, is supposed to be dependent on the solvent only through its molal volume and is characteristic to the type of monomer specie^.'^^'^ We calculated the values of Bl2 from the data such as listed in Table IX and obtained 10aoB12= 125 for toluene solutions, 60 for DEM solutions, 69 for 1-CB solutions, and 40 for CHL (80') solutions. From these values, the extra interaction parameters AB12 are calculated as 10a0AB12= 41 in toluene, 50 in DEM, 50 in 1-CB, and
+
(42) See, for example, U. Johnsen, Ber. Bunsenges. Physik. Chem., 70, 320 (1966).
(43) H. Nomura and T. Iwachido, IUPAC Symposium on Macromolecular Chemistry, Tokyo-Kyoto, 1966, preprint VI-52.
DIFFUSEDOUBLE LAYEROF WEAKELECTROLYTES
841
40 in CHL. The vadues of B for the ST-MRTA copolymers in various solvents may be approximated as
satisfactory. The positive value of ABl2 implies the existence of repulsive interactions between ST and h4h’IA units. The result is reasonably understood by the fact that PST and PMMA are incompatible with each other.16
B
=
mBl
+ (1 - m) Bz + 2m(l - VL) 10-30(45
* 5)
(17’)
The solid curves in Figure 7b show the relations given by eq 17’ with the values of B1 and Bz,such as shown in Table IX. It appears that the crudest assumption in the form of eq 17 for the long-range interaction parameter B for the random copolymer is reasonably
Acknowledgment. We wish to thank Messers H. Suzuki and H. Ohnuma for their competent help in carrying out the experiments. We are indebted to Dr. H.Utiyama for disclosing to us the valuable results of his unpublished experiments. Y. M . wishes to thank Teijin and Co.for a fellowship grant.
Diffuse Double Layer of Weak Electrolytes with Field Dissociation by Gilles G. Susbielles and Paul Delahay Department of Chemistry, New York Uni,uersitg, New York, New York lOOOS
(Received July 10, 1967)
The differential capacity of the diffuse double layer and the difference of potential across the diffuse double layer are computed as functions of the charge density on the metal for a metal in contact with a weak electrolyte. Field dissociation is considered without restriction on the field, and the field dependence of the dielectric constant in the diffuse double layer is introduced. The solution is obtained by numerical analysis after transformation into an initial value problem. Initial values are obtained by a perturbation treatment similar to but not identical with that reported by Bass. Application is made to 0.1 M acetic acid in water and to 0.5 M potassium acetate in glacial acetic acid. Departure from the Gouy-Chapman theory, as a result of field dissociation, is rather small for aqueous solution but significant for glacial acetic acid (20-30% change of differential capacity of diffuse double layer). The effect of dissociation of a weak electrolyte on its diffuse double layer at an interface with a metal was recently treated by Bass.’ This problem was also attacked by Sanfeld and Steichen-Sanfeld12by means of a local thermodynamic method,2 with the aim of calculating the dissociation constant in the double layer for a small charge density. The Bass treatment is applicable only for a small charge density on the electrode, as it uses a small perturbation method. This author calculated that the integral capacity of the diffuse double layer at the point of zero charge is changed only by 0.03 pF cm-2 as a result of the field dissociation for 1 M acetic acid in water. The field effect should increase with the charge density. It is determined here by numerical analysis, without the limitation imposed by a small perturbation method for a 1-1 weak electrolyte. The Onsager equation for field dissociation is introduced in its Bessel function form, and variation of the dielectric constant in the diffuse double layer is taken into account. Statement of Problem. The problem is clearly stated by Bass’ and is formulated by the following system of
differential equations (his eq 14) written in this author’ notations (see Appendix) -d(p -d(p
+ n)/df + f(p - n) - ads/@ - n)/dE + f(p + n ) - bds/df d2s/df2 = R(xs
= 0
(1)
= 0
(2)
- np) - n)
(3)
df/df = (‘/z)(P (4) The boundary conditions are discussed by Bass and are for f --+ m : n = 1, p = 1,f = 0, s = co( a ) / c ( m), and x = &/e( a) = e( a ) / c o ( m ) . The last two conditions are not independent ones if one selects the first three conditions as independent ones. Moreover, the flux of neutral species must be equal to zero for f = 0 and f --+ 00. The last condition for f 03 is not an independent one. The problem is to calculate the differential capacity of the diffuse double layer and the difference of poten(1) L. Bass, Trans. Faraday Soc., 62, 1900 (1966). (2) A.
Sanfeld and A. Steichen-Sanfeld, ibid., 62, 1907 (1966). Volume 72,Number 9 March 1968
