Molecular Weight Determination of Polystyrene Standards by Vapor Pressure Osmometry Alfred H. Wachter and Wilhelm Simon Department of Organic Chemistry, Swiss Federal Institute of Technology (ETH), 8006 Zurich, Switzerland Number average molecular weights Mnof polystyrene standards in the range 600 to 400,000 have been measured in benzene, carbon tetrachloride, and cyclohexane with a standard deviation of 1 to 12% by using an extended range vapor pressure osmometer. The values found for M, above 8,000 are lower than the nominal values obtained by membrane osmometry but are identical for different solvents. Determinations on polymer mixtures are consistent with the values calculated from these results. A plot of l/Mapp VS. concentration follows a three term virial expansion. Neither the Flory-Huggins nor the Flory-Krigbaum equations give an adequate fit over the concentration range studied. To reduce the standard deviation of the extrapolated M, the following procedures have been applied to allow a straight line extrapolation: measurement in a &solvent where the second virial coefficient equals 0, third term correction, and application of Maron's equation in a concentration range below 3 grams/100 cma.
BECAUSE of the speed, small sample requirement, and the possibility of using a wide choice of solvents and temperatures, vapor pressure osmometry ( I ) has become popular for the determination of number average molecular weight below about 20,000 ( 2 ) . The ability of vapor pressure osmometers to reproducibly detect temperature differences between a drop of solvent and a drop of solution, which are ideally linearly dependent on the concentration of the solution, sets the upper molecular weight limit. Although it has been said that polymers of 40,000 molecular weight have been determined with this method (3),membrane osmometry is generally used to determine number average molecular weights above 20,000, especially above 50,000. Membrane osmometry however suffers from the limitation of permeability of the membranes for low molecular weight compounds so that true number average molecular weights below 50,000 can only be obtained by a very careful selection of the membrane material ( 4 , 5). The disturbing effect of this permeability increases at a given number average value with the broadness of the molecular weight distribution. To evaluate number average molecular weights of macromolecules extrapolation procedures have to be used. The general virial expansion as given by McMillan and Mayer (6) is l/Mapp = A1
+ A2.c + A ~ * C+' . . . . A1
= 1/Mz
(1) (2)
where
(1) W. Simon and C. Tomlinson, Chimia, 14, 301 (1960). (2) A. Adicoff and W. J. Murbach, ANAL. CHEM., 39,302 (1967). (3) M. J. R. Cantow, R. S. Porter, and J. F. Johnson, J . Polym. Sci., Part A , 2 2547 (1964). (4) G. Mueh, Makromol. Chem., 77, 64 (1964). (5) H. T. Hookway and R. Townsend, J. Chem. SOC.,1952, 3190. (6) W. G. McMillan and J. E. Mayer, J. Chem. Phys., 13, 276 (1945).
90
ANALYTICAL CHEMISTRY
Maw
= apparent molecular weight calculated as-
surning ideal conditions as given in the experimental section (grams/mole) A1, A2, and Al = first, second, and third virial coefficient = sample concentration (grams/cm 9 C M2 = number average molecular weight of solute (grams/mole) The extrapolated value of M2 in the plot of l/Mappus. c increases in reliability when one decreases the lower limit of c and extends the sample concentration range. The application of vapor pressure osmometry in the measurement of very low solute concentrations has become possible with an instrument recently described (7). It allows the detection of temperature differences between a drop of solution and a drop of solvent with a standard deviation of 3.5 . 10-8 "C (7). For a lO-4M solution in carbon tetrachloride (30 "C) this corresponds to a standard deviation of 12x. In the present paper the application of this instrument in the range of molecular weights up to 400,000 is discussed, thereby extending the usable range of vapor pressure osmometers by at least one order of magnitude. EXPERIMENTAL
Although the extent of the influence of the drop size on the signal measured remains an open question (2, 8, 9), a close control of it seems of utmost importance if high reproducibility is to be obtained oriand if measurements have to be done on highly diluted samples. An extremely reproducible effective drop size can be simply obtained with an inverted arrangement of the sensors (8). In this design solvent and solution are added to platinum gauzes mounted on top of the thermistors through capillaries fixed on the lid of the cell. Because a highly reproducible amount of solvent and solution is retained by the platinum gauze and any surplus liquid flows down along the thermistor stem, there is no need for a direct visual observation of the cell interior (8). Details of the cell construction as well as of the readout of the signal using a Wheatstone bridge with dc amplifier, digital voltmeter, and printer are given elsewhere (7). A power dissipation of 2 . 10-j W per thermistor has been used throughout. The temperature of the cell was kept within +0.0015 OC over 12 hours by means of a system using water circulating thermostats [see (7) for details]. Solvents. Benzene, carbon tetrachloride, and cyclohexane (analytical grade, E. Merck AG, Darmstadt, Germany) were distilled over Linde molecular sieves (type 4A, l/s-inch pellets) immediately before use. Solutes. Dibenzyl disulfide and sulfonal (Organic Analytical Standards, The British Drug Houses Ltd., Poole, England) were taken as calibration standards and used as received. Polystyrene standards (Pressure Chemical Co., Pittsburgh, Apparatus.
(7) R. E. Dohner, A. H. Wachter, and W . Simon, Helo. Chim. Acta, 50,2193 (1967). (8) W. Simon, J. T. Clerc, and R. E. Dohner, Microchem. J . , 10, 495 (1966). (9) A. C. Meeks and I. J. Goldfarb, ANAL.CHEM., 39,908 (1967).
Table I. Coefficients of Virial Expansion (Equation 1) of Polystyrene Standards at 30 ' C Batch no. AI f s (AJa mole . gram-' A2 i s ( A @ mole . gram-2 cm3 A3 f s (A3)"mole . gram-3 cma
Solvent Benzene
16a 15a
I2a Ila 8a
2a 7a
4a la 3a
Carbon tetrachloride
Mixt ureb 8a 2a
la a
*
(1.73 0.01) (8.25 f 0.03) (5.77 i.0.02) (3.15 f 0.01) (1.14 f 0.01) (6.11 i 0.12) (2.36 i 0.26) (1.39 A 0.12) (0.91 f 0.23) (0.25 f 0.15) (6.06 0.12) (1.15 f 0.01) (6.19 i 0.03) (1.01 f 0.19)
*
. 10-3 . 10-4 . 10-4 . 10-4 . 10-5 . . 10-5 . loT5 . 10-5 . 10-4 *
e
*
.
*
.
(2 8 ) (2 i 3 ) (1 3 (1.2 0.2) (1.0 0.1) (8.5 h 0.4) (6.5 zk 0.6) (6.7 i 0.3) (6.6 f 0.6) (6.8 i 0.3) (7.6 i 0.6) (9.1 =t0.7) (9.3 + 0.1) (5.0 & 0.4)
* *
10-6
*
e
... ... ... ...
10-3 10-3 10-3 10-8
. . 10-3 . lo-* . . 10-4 . . . . 10-4 .
(5.4* 1.4). (6.4 =t0.8) . (7.5 1.1) . (6.1 i 0.4) . (5.6 i. 1.3) * (6.0 f 0.5) . (6.8 i: 1 . 8 ) .
10-3 10-3
( 5 . 2 f 0.5)
10-8
*
... ...
*
10-3 10-8 10-3 10-3
standard error of estimate. Consisting of batch 4a (54.01 wt %) and 8a. s (An)
=
Pa.)- a,/Mn < 1.10 for the batches I I a , I2a, 1.50, and Ida, _ MJM, < 1.06 for the batches l a , 2a, 3a, 4a, 7a, and 8a were used without further purification except for drying at 80 "Cand lop3mm Hg over 24 hours. Series of solutions of different concentrations were prepared by diluting stock solutions with appropriate amounts of solvent. The weight frsctions, w y ,of polymer in these solutions obtained by this procedure were converted into the concentration c (grams/cm3)by multiplying wzby the density of the solution calculated according to the equation
where
deoin, dl = density of solution and solvent, respectively (grams/cm3) = apparent specific volume of solute (cm3/gram) u2* The value of uz* (for w z = 0.01, T = 25 "C, M, = a) was obtained from data given by Schulz and Hoffmann ( I O ) and corrected for temperature dependence yielding the values 0.9198, 0.9110, and 0.9337 for benzene (30 "C), carbon tetrachloride (30 "C), and cyclohexane (34.4 "C), respectively. Molecular weight influence on VZ*, which becomes noticeable below = 10,000, was taken into consideration using the nominal M, values in the relation given in ( I O ) . The extrapolation in wP and T was checked against suitable data given by Streeter and Boyer (11) and McIntyre et al. (12). The resulting values of daolnwere found to be within 0.05 % throughout the whole concentration range used. The density dP of the polymer in solution was taken as l / u z *
a,
(IO). Procedure. The temperature of the cell was normally 30.0 "C except for the measurements performed in cyclohexane where it was 34.4 "C. After addition of the solvent and solution to the first and second thermistors, respectively, the bridge unbalance signal reached a steady state in three to five minutes and remained so for at least 15 minutes. Throughout this time values were printed out at minute intervals. Those from the 6th to the 11th minute were averaged and the mean of three such measurements, taken relative to the corresponding mean obtained by adding solvent (10) G. V. Schulz and M. Hoffmann, Makromol. Chem., 23, 220 (1957). (11) D. J. Streeter and R. F. Boyer, Ind. Eng. Chem., 43, 1790 (1951). (12) D. McIntyre, A. Wims, L. C. Williams, and L. Mandelkern, J . Phys. Chem., 66, 1932 (1962).
on both thermistors, was then calculated. For low molecular weight compounds this value A U is proportional to the molar sample concentration rn (mole/liter) : AU=k.rn
(4)
where AU
=
k
=
unbalance signal of the Wheatstone bridge caused by the temperature difference between the thermistors (Volt) a calibration constant (Volt liter . mole-').
For dibenzyl disulfide and sulfonal as calibration standards, a constant value for k was obtained (standard deviation, 0.7%, degrees of freedom, 30) over the concentration range 10-4 to lO-*M. With constant instrumental parameters such as temperature and bridge current the calibration constant measured at rn = 2 10-3M could be reproduced with a standard deviation of 0 . 6 z over a period of several weeks. The signals A U measured for the polystyrene solutions have been converted to Ma,, using
-
RESULTS AND DISCUSSION
Figures 1 and 2 show plots of l/Mappus. c for the 10 Pressure Chemical Co. polystyrene batches studied. The lines correspond to a least square fit. For batches I l a , 12a, 15a, and 16a, a linear, and for l a , 2a, 3a, 4a, 7a, and 8a, a quadratic regression have been calculated with the quadratic term significant for the latter batches only. The virial coefficients obtained by this least square approach are given in Table I. AP is in good agreement with literature data (13). The low concentration limits so far reported in use in vapor pressure osmometry are at least ten times higher than the ones used in our work (3, 14, 15). It is obvious from Figure 2 that a linear extrapolation over the concentration range generally used must lead to meaningless number average molecular weights. A linear extrapolation for batch 7a by using such a concentration range (3-7 grams/100 cm3) leads to a value of R, = 76,800 as compared with 42,400 with a quadratic (13) H. Sotobayashi and K. Ueberreiter, 2.Elektrochem., 66, 838 (1962). (14) R. K. Sharma and A. F. Sirianni, Can. J. Chem., 45, 1069 (1967). (15) J . Van Dam, Rec. Trau. Chim., 83, 129 (1964). VOL. 41, NO. 1 , JANUARY 1969
91
I8
16 10
s 2
2e
05
-
I
12
y1 u
=
o
IO
CONCENTRATION
Figure 1. Plots of l/Mappus. concentration Low molecular weight polystyrene standards in benzene (30 "C) 6
least square fit over the range 0.7-7 grams/100 cm3. For batch 3a the corresponding values are minus 110,000 and plus 404,000, respectively (see dashed line in Figure 2). A glance at Figures 1 and 2 shows that the significanceof R,,obtained by linear extrapolation increases with decreasing solute concentration. With the range 0.1-2.5 grams/100 cm3 accessible with the instrument described (7) and linear extrapolation R,,becomes 8,840 for batch 8a compared with 8,810 in the quadratic extrapolation in the range 0.1-5 grams/100 cm3. Unfortunately the use of a virial expansion with three terms leads to a high standard deviation of the intercept A , as compared with a linear fit (Table I). For this reason theories are usually applied which reduce the system to two parameters. Evaluation of the virial coefficients in terms of these theories gave the Flory-Huggins interaction parameters I.( (16) and the factors g of the Flory-Krigbaum treatment (17) presented in Table 11. Whereas the parameter I.( remains fairly constant for > 20,000, the third term in this treatment, is smaller than the corresponding A3 term in the virial expansion (see below). This prevents the Flory-Huggins equation from adequately following the curvature of the experimental points. From column 3 in Table I1 it is evident that the factor g is not constant for the polystyrene-benzene system studied, as is required by the theory, but depends on the molecular weight of the sample studied. These findings coincide with those obtained in membrane osmometry. As pointed out by McLeod and McIntosh (18) neither the FloryHuggins equation (16) nor the Flory-Krigbaum relation
a,,
an
(16) M. L. Huggins, Znd. Eng. Chem., 35,216 (1943). (17) W. R. Krigbaum and P. J. Flory, J. Amer. Chem. Soc., 75, 1775 (1953). (18) L. A. McLeod and R. McIntosh, Can. J . Chem., 29, 1104 (1951).
Table 11. Comparison of the Parameters p, g , and (PO for Different Polystyrenes in Benzene (30 "C) pa (Flory3 (Flory(PO - QO) Batch no. Huggins) Krigbaum) (Maron) 8a 2a
0.391 0.409 ?a 0.431 4a 0.429 la 0.430 3a 0.428 1 a Calculated from I.( = 2 b
92
Calculated from g
0.59 0.53 0.42 0.19 0.12 0.03
- A z . V1o.dza.
Aa.Ai As2 *
= __
ANALYTICAL CHEMISTRY
0.376 0.378 0.394 0.401 0.409 0.417
UO)
I
2 1 1 0
CONCENTRATION
Figure 2. Plots of l/Mappus. concentration High molecular weight polystyrene standards in benzene (30 "C) (17) allow a satisfactory fit if a larger concentration range is used. In a similar way and supported by theoretical reasons, Stockmayer and Casassa (19) affirm that whenever possible the virial expansion should be preferred except when the concentration range is too narrow. In accordance with the results obtained by Bawn, Freeman, and Kamaliddin (20), Table I suggests that the third virial coefficient is constant over a broad molecular weight range for the compounds studied. Subtracting a term, including the mean of the third virial coefficients, from experimental l/Mappvalues leads to a linear concentration dependence of this corrected l/Mapp(see Figure 3). Compared with A 3 given by the Flory-Huggins theory . cm6 a mean value of (16)as 2.88 . mole . 6.13 * 10-3 mole . gram-3 . cm6 was found in this procedure. The third column of Table I11 lists the number average molecular weights obtained in this way. According to literature data (21), cyclohexane at 34.3" to 34.5 "C should behave as a &solvent for polystyrene and should, therefore, give values of M,,, independent of the concentration. Figure 3 shows that this holds to a high degree for the four batches studied. Furthermore, Figure 3 demonstrates a perfect agreement of the extrapolated values obtained using benzene with the third term correction described and the O-solvent. The resulting molecular weights are given in column 5 of Table 111. The linear extrapolation procedures illustrated by Figure 3 (third term correction, O-solvent) are, in spite of their efficiency, limited in their routine application. The third term correction can be applied only if a relatively large number of homologs with constant third virial coefficients are studied. (19) W. H. Stockmayer and E. F. Casassa, J. Chem. Phys., 20, 1560 (1952). (20) C. E. H. Bawn, R. F. J. Freeman, and A. R. Kamaliddin, Trans. Faraday Soc., 46, 862 (1950). (21) W. R. Krigbaum, J. Amer. Chem. SOC.,76, 3758 (1954).
5 $ IE W
z
or0
ii
I
,
1
2
,
I
L
LL 0
,
,
1
6
,
Ib
8
1
:D-'mol./ct
Y = !+J$.c
2 u
Figure 4. Evaluation of experimental data obtained in benzene (30 "C) using Maron's method
P5
w u
a
o
I
.Z
3
L
5
6
g/ioo cm3
Upper concentration limit for linear extrapolation, 3 grams/100 ,ma
CONCENTRATION
Figure 3. Evaluation of high molecular weight polystyrene standards with (a) third term correction (benzene 30 "C)-.+.(b) 8-solvent (cyclohexane 34.4 "C) +The application of 6-solvents, on the other hand, is limited by solubility problems and for each given compound such a solvent has to be found. Because of the high sensitivity of the instrument described the use of solvent mixtures is not advisable. A generally applicable approach has been described by Maron (22, 2 3 ) which, for small sample concentrations, can be presented as follows:
q =
[VIP - 4/d2 (crn3/gram>
(7)
where [q] = intrinsic viscosity (cm3/gram)
In using Equation 7, [q] was calculated by means of the Mark-Houwink relationship, Equation 8, taking the constants given by Altares, Wyman, and Allen ( 2 4 ) and inserting nominal R, values [?I = 8.5
. 10-3
.
~
3
7
(8)
5
The validity of Equation 6 is demonstrated in Figure 4. It shows that a linear fit is adequate for the low concentration range (c < 3 gram/100 cm3, solid lines) accessible with the instrument described (7). This range is inaccessible with most of the vapor pressure osmometers in use at present. The parameters ( p o - ao) corresponding to the slope of the solid lines in Figure 4 are given in Table I1 and are in agreement with literature data (23). Similarly, the values calculated from the intercept of these lines agree with the other experimental values (Table 111). The standard deviations given for the number average molecular weights in Table 111 and IV include the uncertainty of the calibration factor. The R, values of the low molecular compounds listed in Table IV are in good agreement with the nominal values obtained by vapor pressure osmometry
a,
where VlO (po
= molar volume
of solvent (cm3/mole)
- m) = constant for a given molecular weight and at small concentrations deducible from viscosimetric data as given in (23) which may be transformed to
= a parameter
4
(24) T. Altares, D. P. Wyman, and V. R. Allen, ibid., Part A , 2 4533 (1964).
(22) S. H. Maron, J. Polym. Sci., 38, 329 (1959). (23) S. H. Maron and N. Nakajima, ibid., 42, 327 (1960),
a,,of Polystyrene Standards with Their Standard Deviation
Table 111. Nominal and Experimental
Batch no.
Nominal
8a 2a 7a 4a la
3a
(Mvis.0:
il?, determined by MO
Third term correction (Benzene, 30 "C)
Experimental R, Maron's theory" (Benzene, 30 "C)
8,800 f 1% 16,400 f 1 % 43,900 f 2 % 72,100 f 2 % 107,300 i 4 % 380,000 f 9 %
8,820 f 1% 16,500f 1 % 44,300 f 3% 72,500 f 3% 105,400 rt 5 % 320,000f 12%
10,900 f 5 % 19,800 f 5 % 50,100 rt 5 % 97,600 rt 5 % 160,000 f 3%)* 392,000 f 5 %
See references (22) and (23).
* Determinations show a volatile content of 3.0 f 0.3 wt
&solvent (Cyclohexane, 34.4 "C 8,910 f 1 % 16,400 f 1% 42,700 2%
...
107,200 & 3 %
...
% which cannot be reduced by heat and vacuum in the ordinary sense (Data sheet
No. 100,Pressure Chemical Co., Pittsburgh, Pa.)
VOL. 41, NO. 1, JANUARY 1969
93
mi,
Table IV. N o m i ~ and l Experimental of Polystyrene Standards with Their Standard Deviation M, obtained by VPO in benzene by linear extrapolation Nominal Experimental Batch no. 16u 15a
I2a Ila
5241585 A 92711060 =k 1710 ==! 2930 =k
7% 7 z 7% 7%
578 A 1 % 1212 f 1 % 1730 zk 1 % 3180 f 1 %
(VPO) (column 2). There are however considerable deviations in the values M n above 8,000. This is mainly due to the fact that those nominal values have been determined by membrane osmometry (MO), which is known to show some discrimination in regard to the low molecular weight components (5, 25). In agreement with this behavior, the values obtained by vapor pressure osmometry are somewhat lower than the ones determined by membrane osmometry (Table 111). The volatility of any low molecular weight contaminants which might be present is not expected to be comparable to that of the solvent for such components tend to spread out through the cell and do not enhance the measured (25) J. B. Donnet, B. Roth and G. Meyerhoff, ibid., 27, 591 (1958).
signal but tend to increase the equilibration time. No such effect was observed. A further check of the consistency of the results is the study of polymer mixtures. A mixture of batch 4a (54.01 wt %) and 8a should give a of 16,800 in perfect agreement with 2x. Assuming the the experimental value of 16,500 nominal values, the of this mixture would have been 19,900.
a,, a,
CONCLUSIONS
In contrast to suggested limitations in the measurement of polymer solutions arising from the overlap of polymer molecules and/or surface films (3, 26), vapor pressure osmometers can be used to determine true number average molecular weights of polystyrene up to at least 400,000. At present this extended range vapor pressure osmometry is the only method to measure true M, values over the whole range from above 400,000 down to about 100. This lower limit is dictated by the volatility of the solute only ( I , 27). RECEIVED for review June 5 , 1968. Accepted September 3, 1968. The present work has been supported by the Schweizerischer Nationalfonds zur Foerderung der wissenschaftlichen Forschung (Research Projekt Nr. 4312). (26) W. Scholtan and S. Y. Lie, Makromol. Chem., 108, 104 (1967)~ (27) W. I. Higuchi, M. A. Schwartz, E. G. Rippie, and T. Higuchi, J . Phys. Chem., 63, 996 (1959).
Column Efficiency and Electrolyte Effects of Inorganic Salts in Aqueous Gel Chromatography P. A. Neddermeyer’ and L. B. Rogers Department of Chemistry, Purdue University, Lafayette, Ind. 47907
A number of polyphosphate and metaphosphate salts of sodium were eluted f r o m columns containing BioGel P-2, Sephadex G-10, 6-25, and 6-50 with electrolyte solutions. Elution orders for the linear polyphosphate, Na5P3010, and the cyclic metaphosphates, Na4P4OI2, and Na3P309,were i n the order Na4P4011< Na6P3Ol0 < Na3P309,as expected from a steric exclusion mechanism. The peaks for Na16P14043 and Na9P,022 were broad, confirming the reported heterogeneity of these preparations. Elution of a sample salt different f r o m the eluent salt often resulted in two positive peaks, one caused by the sample salt and the other by eluent salt. The peak for the eluent saltwas attributed to a Donnan “diffusion effect” and its size depended directly upon the sample concentration and the composition of both the sample and eluent. Zone broadening was investigated and compared to theory where possible. Although peak width increased only with the square root of column length, the separation of NasP3010,Na4P207,and Na2HP04 did not appear to improve on going f r o m a 63-cm to a 500-cm column. Decreasing the flow rate decreased the plate height to values equal to the particle diameter of the swollen gel, and increased peak skewing for salts eluted with water f r o m Sephadex gels. Plate heights also decreased by as much as 50% upon increasing the temperature f r o m 22.0 O C to 50.0 O C .
Present address, Research Laboratories, Eastman Kodak Co., Rochester, N. Y. 14650 94
ANALYTICAL CHEMISTRY
GELCHROMATOGRAPHY has become a very useful method for separating and characterizing organic and biological materials (1-5). To date, comparatively little effort has been made to apply the advantages of this technique to the fractionation of inorganic species, especially polymers. Saunders and Pecsok (6) reported on the elution of aqueous samples of strong electrolyte salts from tightly cross-linked polyacrylamide gel. Although separations did occur, they were the result of an adsorption rather than a steric exclusion mechanism. In an earlier study, we reported (7) that for aqueous salt samples eluted from the polydextran gel, Sephadex, a Donnan saltexclusion effect prevented penetration of the gel to the full extent allowed by the steric exclusion mechanism itself. The presence of background electrolyte minimized the interfering Donnan mechanism and allowed elutions to proceed in accordance with the steric exclusion mechanism. Ohashi, Yoza, and Ueno (8) similarly noted an elution in the order (1) J. G. Hendrickson and J. C. Moore, J . Polymer Sci. Part A-1, 4, 167 (1966). (2) P. Flodin, Anal. Chim. Acta, 38,89 (1967). (3) B. Gelotte, J . Chromatog., 3, 330 (1960). (4) J. G. Hendrickson, ANAL.CHEM., 40,49 (1968). (5) J. R. Whitaker, ibid., 35, 1950 (1963). (6) D. Saunders and R. L. Pecsok, ibid., 40,44 (1968). (7) P. A. Neddermeyer and L. B. Rogers, ibid., 40,755 (1968). (8) S. Ohashi, N. Yoza and J. Ueno, J . Chromatog., 24,300 (1966).
