Cautionary note on extrapolation methods for determining unperturbed coil dimensions

Notes and Communications Cautionary Note on Extrapolation Methodg f o r Determining Unperturbed Coil Dimensions DURING the course of some light scattering experiments on polyi.sobutene (PIB) fractions, covering a 15-fold range of molecular weight, in cyclohexane it became necessary to try and estimate the unperturbed dimensions of these polymer fractions. A graph of log (S~>z versus log (M>w is shown in Figure 1, and the (SZ>/(M> relationship derived from this is (S~>z= 0"046 (M> 1"16 wFor other purposes, measurements were carried out on six of the PIB fractions at a number of temperatures (the number ranging from five to eight) in two different solvents and so it was possible to analyse the results for errors. The mean fractional error in (M> was found to be 0.031. The error in (SZ> must be at least as great and we would suggest that it is more likely to be about twice as great although we cannot estimate it from our data. In order to estimate the unperturbed dimensions, the extrapolation methods of Kurata and Stockmayef (K-S) and of Ptitsyn 2 were tried. The plots of w

versus

(M) w (S~>+z and

w

CM~,,/s versus - - - - w

(the latter is equivalent to Ptitsyn's plot of R2/(DP) versus (DP)+, where DP is the degree of polymerization) were both curved over the whole range. Over no range did th~ curves appear linear and at low values of M they tended towards the origin (Figure 2), especially if extra values were taken from the line of Figure 1. This behaviour made'it impossible to carry out any linear extrapolation to find {(S~>/(M>}0 and caused us to examine the theory behind the extrapolation methods. Both methods rely on the use of a closed expression for the expansion factor a2=(S~)/($2>o of the form a"-og"=KM+, ( n = 3 , m = l , in K-S, n =3, m = 0 in Ptitsyn). However, it is now generally accepted ~-8 and can be seen from Figure 1, that for a linear polymer in solution ( S 2>=kM 1+~ where E varies from 0, in a 0-solvent, to 0.25 or thereabouts, and is 0.16 in the case of PIB in cyclohexane. In the light of this we can reconsider the nature of the extrapolation plots. The initial K-S plot amounts to a graph of kM ~ versus M~l-~/2x k-+ and will, in general, be a curve passing through the origin. This is precisely what we observe for PIB. For any extrapolation of any part of the PIB curve, estimation of g ( a ) = {8a'~/(3a2+ 1)'~/~} from the intercept gives values ranging from about 1 up to 1.4 as M covers its whole range. Use of these values to adjust the curve only serves to increase the curvature and does not improve the' chances of extrapolation. 585

NOTES AND COMMUNICATIONS 3.5

/ ~,3"C .go

Figure 1

c~

2'5

5-5

I 6"0

I 6'5

I 7.0

tog <~w

0.6

0'5

Figure 2--Abscissa scales: 0 upper, A lower

0"4

0-3

I

1000 50

•

I

2000 100

I

3000 150

I

4200000

Similar considerations apply to the Ptitsyn plots of /DP versus Dpt. This is the same as
NOTES AND COMMUNICATIONS is examined; and if the experiments only cover a small range and the plots appear linear, the intercept on the /(M> axis will depend on the particular range of molecular weights covered. It would thus seem that the extrapolation methods discussed above will not, in general, work and indeed it is noticeable that many of the plots of the K-S or Ptitsyn types in the literature are in fact rather poor straight lines. However, it may be that in some cases meaningful extrapolatiorts can be made and for some systems the methods appear to work reasonably well s. In all cases when E=0 the analyses obviously become meaningful but unnecessary. A. J. HYDE and A. G. TANNER

Department of Pure and Applied Chemistry, Strathclyde University, Glasgow, C.1. (Received December 1967) REFERENCES KUR^T^, M. and STOCKMAYER,W. H. Fortschr. Hochpolym. Forsch. 1963, 3, 196 2PTITSYN,O. B. Vysokomol. Soedineniya, 1961, 3, 1673 3 EDWARDS,S. F. Proc. phys. Soc. 1965, 85, 613 4 PISHER, M. E. and HILLY,B. J. 1. chem. Phys. 1961, 34, 1253 FISrIER,M. E. and SYKES,M. F. Phys. Rev. 1959, 114, 45 6 DOMB,C., GILLIS,J. and WILMERS, G. Proc. phys. Soc. 1965, 85, 685 7 SCHULTZ,A. R. 1. Amer. chem. Soc. 1954, 76, 3422 8HYDE,A. J. Polymer, Lond. 1966, 7, 459 a ALLEN,G. Personal communication

S e c o n d Virial Coefficients in T e r n a r y S y s t e m s IN A recent study 1 of preferential adsorption in solgent (1), non-solven,t (2), polystyrene (3) systems, values of the second virial coefficient (A2) were obtained and compared with three theoretical treatmen.ts, which attempt to formulate the molecular weight dependence ~-~ of A2. It was concluded that, under the experimental conditions used, the approach which best described the behaviour of the second virial coefficient was that of Casassa and Markovitz ~, where the general equation describing the molecular weight dependence of A2 is A s = 6, (1 - O/T) (-~1 v,) × F (X)

(1)

and the function F ( X ) , as given by Casassa and Markovitz, has the form F ( X ) = (1 / 1.093X) [1 - exp ( - 1-093X)]

(2)

in which X is defined as X = 4C,,,¢, (1 - O/T) M+ x a -3

(3)

Here qJ, is the entropy of mixing parameter, ~ the partial specific volume of the polymer, V~ the molar volume of solvent, T and 0 the temperature and 587