I
SURESH N. CHINA1 Plastics Research Section, Samuel Feltman Ammunition Laboratories, PicatinnyArsenal, Dover, N. J.
Calculation of Molecular Weight and Dimensions of Polymers from Viscosity Interaction Parameter An empirical equation of practical value to industrial polymer chemists for roughly estimating molecular weights of linear, flexible chain, synthetic polymers from viscosity measurements alone
A L T H O U G H viscometric measurements can be used for determining molecular weight, it has first been necessary to establish a correlation between intrinsic viscosity and absolute molecular weight ( 76). It now appears that it may be possible to approximate both molecular weight and dimensions of polymers by viscosity
measurements without any previous correlations with the absolute methods. This would be especially useful with new synthetic polymers for which the constants X and a in the modified Staudinger equation, [17] = KM", are not available. The relationship proposed here combines into a single, more useful form
several approximate relationships previously derived. These are the SchulzHuggins (73, 75) relation, S = k'[17I2 for a polymer-homologous series in a single solvent; and the observation of Alfrey and coworkers ( Z ) , Eirich and Riseman (8), and Streeter and Boyer (77) that k ' [ 7 ] 2 is approximately constant for a single polymer in different
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SLOPE
Figure 1.
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S
Relationship between
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[(M[ ~ ] ) 1 / * ] and (slope = S)
Least squares line Zones of scatter VOL. 49, NO. 2
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FEBRUARY 1957
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solvents and that [ q ] varies with M according to a power not too different from unity. These relations are combined here i n a slightly different way and empirical constants are evaluated by treating a large body of the most recent data.
Since the evaluation of molecular weight is of primary interest, it is of value to know the exact functional relationship between ( M [ q ] ) 1 / 3and S. T h e method of least squares was used to evaluate the function of Equation 4. This resulted in the estimating equation ( M [ 7 ] ) ’ / 3=
Experimental Data
T o plot Figure 1 data were used which had been obtained in this laboratory, as well as those published by other investigators on fractionated poly(viny1 acetate) (5: 72) in acetone and methyl ethyl ketone; poly(methy1 methacrylate) (4) in methyl ethyl ketone; polyisobutylene (70) in benzene, ethylbenzene, diisobutylene, cyclohexane, toluene, and n-heptane; polystyrene ( 7 , 3, 7, 77) in cyclohexane, ethylcyclohexane, benzene. toluene, ethylene dichloride, and methyl ethyl ketone; and a 50:50 copolymer of poly(methy1 methacrylate)pol>-styrene (74) in methyl ethyl ketone, dioxane, chloroform, and nitroethane.
I n a n attempt to extend the usefulness of viscosity data, it is assumed that the slopes ( S ) of the plots of
17s. - c may
be
related to the shape of the dispersed macromolecules, Thus, there may exist a functional relationship benveen a configuration parameter such as root mean square end-to-end distance, (i2)lI2, and the viscosity interaction parameter, ($)-that is, the slope of the plot of 17.. - c. Mathematically this may be C
represented by the equation : (y2)1/2
=
f
(3)
(1)
According to a theor!- presented by Flory, the polymer molecule is stretched more or less, depending upon the swelling action of the solvent used. T h e frictional characteristics of polymer molecules in dilute solutions, as manifested in solution viscosities, apparently depend largely on the configuration and size of the polymer molecule. From a theoretical treatment (9). Flory derived a relation between intrinsic viscosity and configuration parameter in the equation: (p)3/2
[VI = @-
iM
where [ q ] is intrinsic viscosity in deciliters per gram, @ is a numerical constant, F2 is mean square distance between the ends of a polymer chain, and M is molecular weight. Since Q, is a numerical constant, Equation 2 may be written (P)l/z =
f (A4[~])”3
(3)
Therefore, there follows from Equations l and 3 a n expression
( M 1 ~ 1 ) ”=~f (S)
304
(4)
(5)
T h e plot of Equation 5 is shown i n Figure 1. Cubing both sides of Equation 5 results in the equation i M [ q ] = (107)3S
(6)
or (7)
where = 1.23 X 106 = numerical constant having the dimensions (gram)2 per (deciliter) (mole). O n the basis of the above discussions the following approximate relationship for intrinsic viscosity may be written by substituting the factor slope term.
Theoretical Treatment of Data
107S”3
*a
for the
This gives
T h e accuracy of Equation 5 was checked by subjecting the data to rigorous statistical analysis ( 6 ) . A measure of the degree of relationship or the correlation between variables independent of the units in which they were originally expressed, is the coefficient of correlation, ( p ) . Values for p may range from 0 to 1. T h e closer p is to 1 the better the relationship between the variables. A closely related measure is the coefficient of determination, ( p “ , which is the measure of the relative amount of variations which have been explained by the estimating of Equation
examining the data it was found that five points out of 110 lie outside the 957, limits. This is to be expected under the above assumption. From this it can be concluded that the above results came from the same normal bivariate population. Therefore, the proposed equation is satisfactory for roughly estimating molecular weight of linear flexible polymers from viscosity measurements i n good solvents. O u r data show that 48% of the calculated molecular weights are within = 1576 of experimental molecular weights, and 66% of the calculated molecular weights are within &257& of experimental molecular weights obtained by a n absolute method. Kote that the agreement between the calculated and experimental values of would be much better than that for calculated and experimental values of ~f since (?)1/2 values are proportional to the cube root of the product M [ q ] , as shown by Equation 2. T h e accuracy of this method can probablv be improved by further theoretical treatment of the data already available along with additional data on other polymers. Acknowledgment
T h e assistance of Alonzo Bulfinich, of the Samuel Feltrnan Ammunition Laboratories, Picatinny Arsenal, with the statistical work described is gratefully acknowledged. Literature Cited
Alfrey, T., Bartovics, A., Mark, H., J . Am. Chem. Soc. 64, 1357 (1942). Zbid., 65,2319 (1943). Bueche, A. M., Zbid., 71, 1452 (1949). Chinai, S. N., Matlack, J. D., Resnick, A . L., Samuels, R. J., J . Polymer Sci. 17. 391 (1955). 5. (5) Chinai,‘S. N., Scherer, I?. C., Levi, Values of p and p2 were calculated to be D. W., Zbid., 17, 117 (1955). (6) Croxton, F., Cowden, D., “Applied 0.957 and 0.917, respectively. A p2 of General Statistics,” pp. 451-69, 0.917 indicates that 91.77, of the variaPrentice-Hall, N e w York, 1955. tions in the logarithms of the ( M [ q ] ) 1 / 3 ( 7 ) Doty, P., Affens, W., Zimm, B., values with reference to the logarithms Trans. Faraday Soc. 42B, 66 (1946). of the S values have been explained. ( 8 ) Eirich, F., Riseman, J., J . Polymer Sci. 4, 417 (1949). A measure of variation which is not (9) Flory, P. J., “Principles of Polymer explained by the estimation of Equation Chemistrv.” Cornel1 Universitv 5 is the standard error of estimate. Press, Ithyca, N. Y . , 1953. T h e standard error of estimate is a Fox, T., Flory, P., J . Am. Chem. SOC. 73, 1909 (1951). general measure of the variation of the I b i d . , p. 1915. actual logarithms of ( M [ v ] ) ’ /from ~ the Howard, R., Ph.D., dissertation computed logarithms of ( M [ q ] ) 1 / 3 I. n Massachusetts Institute of Technology, Cambridge, Mass., 1952. this work the standard error of estimate Hueeins. M. L.. J . Am. Chem. SOC. was found to be 0.06195 unit of (~kf[q])’/~. 62: 2716 (1942). I n Figure 1, the solid line is the line of (14) Moore, L., Ph.D. dissertation, Massaregression; the first set of dotted lineschusetts Institute of Technology, one above and one below the solid Cambridge, Mass., 1951. (15) Schulz, G. V., Blaschke, F., J . prakt. line-represents 1 2 confidence interval Chem. 158, 130 (1941). and the second set of dotted lines repre(16) Staudinger, H., Heuer, W., Ber. 63, sents 221 confidence interval. If we 222 (1 930). assume that the data in Figure 1 are (17) Streeter, D., Boyer, R., IND.END. CHEM.43,1790 (1951). normally distributed, 95% of the results would lie within the range established RECEIVED for review December 9, 1955 ACCEPTED -4ugust 21, 1956 by the average r t 2 (0,06195). I n
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