percent of polymer formed to 5× 10-3 times the irradiation time. For the purpose of these investigations this approximation is valid and sufficient. The ratio of polymerized monomer units to the number of perfluorovinyl groups in the polymer was obtained by a Beer's law computation from data obtained from infrared spectra. This type of computation proved very useful in earlier work [2, 3]. With perfluorohexadiene and perfluorooctadiene polymers these values are thought to have an uncertainty factor of about 2. With perfluoro1.6-heptadiene polymer its crystallinity makes the accuracy of the CF2n-2/CF CF ratio rather indeterminate.

When the kinetic chain length is much greater than the CnF2n-2/CF CF2 ratio, this ratio should be about equal to the ratio of cyclization to noncyclization, but when the kinetic chain length and the aforementioned ratio differ by a factor less than 10, it is probable that an appreciable fraction of the observed double bonds may represent chain ends. These considerations give rise to the concept that the ratio of cyclization to noncyclization is about 20 for the octadiene, 200 for the hexadiene and a still larger but not precisely established value for the heptadiene. For each polymer, similar values of CnF2n-2/CF=CF2 were obtained under different polymerization conditions, suggesting that this ratio is not a strong function of either temperature or pressure.

Gelation occurs when a pendant vinyl group becomes incorporated in another chain, thus effecting a cross link between the two chains; thus, it may be used as a measure of the ratio of cyclization versus noncyclization. It is determined by extracting the polymer couple with boiling hexafluorobenzene and weighing both the soluble and insoluble fractions. The percent gelation is then determined by dividing the insoluble fraction by the weight of the total polymer sample. Polymer solution theory predicts that gelation will occur when the fraction of polymer formed is equal to the product of the ratio of cyclization to noncyclization and the reciprocal of the primary weight average degree of polymerization [18]. For cases where no gelation occurred a lower limit for the aforementioned ratio can be calculated by using 2y as the weight average degree of polymerization and multiplying this value by the observed conversion. Cases in which gel formed may be used in the same way to compute an upper limit for the ratio. In this manner, R, the ratio of cyclization to noncyclization, has been computed for perfluoro-1,5-hexadiene, perfluoro-1,6-heptadiene, and perfluoro-1.7-octadiene. The values are 230 < R <10, 230 < R, and 6<R < 320, respectively.

In cases where the polymer yield is fairly low, an additional check is possible since in such cases 2y is a good estimate of the weight average degree of polymerization. For a most probable distribution the sol fraction may be used to compute the number of cross links per weight average molecule [19]. In this work computations of this sort tended to confirm values estimated through use of Beer's law computa

tions.

Systematic studies with these monomers were hampered by sensitivity to impurities. The indicated results with perfluoro-1,7-octadiene were obtained with monomer which had not been purified on a prep scale vapor phase chromatograph. Polymer samples accordingly, gave values of y below those of other materials studied. Other samples in all cases were prepared from monomer which had been carefully chromatographed on a long column. The double bond migration which takes place so readily with perfluoro1.4-pentadiene [3] takes place less readily with the six, seven, and eight carbon fluorodienes. Whenever recovered monomer was used without chromatographic purification, lower rates were obtained. This diminution in rate was attributed to the presence of internal olefins resulting from double bond migration. Polymerization rates with pure monomer were about 200 to 500 times those obtained with perfluoro-1heptene in the same conditions [20]. It is uncertain whether this is due to a difference in monomer rate constants, difference in monomer purity, or to some effect of gelation on rate where dienes are concerned.

Attempts were made to avoid gelation by polymerization in solvents such as perfluorocyclobutane. The intention here was to take advantage of the hypothesis that the cyclization rate should be independent of monomer concentration while propagation should be directly related to monomer concentration so that dilution should increase the ratio of cyclization to noncyclization. Calculations using spectra of these polymers gave values of CnF2n-2/CF=CF2 of 103 or more. These spectra also showed intense absorption attributable to acid fluorides and internal double bonds. This is not easily explained since such absorptions from highly purified monomer. A comprehensive study were not observed with polymers prepared in bulk of the polymerization of these materials will be the subject of another communication.

4. References

[1] Fearn. J. E., and Wall, L. A., SPE Trans. 3, 231 (1963). [2] Fearn. J. E., Brown, D. W., and Wall, L. A., J. Polymer Sci. 4 Part A, 131-140 (1966).

[3] Brown. D. W., Fearn. J. E., and Lowry, R. E., J. Polymer Sci. 3 Part A, 1641-1660 (1965).

[4] Park, J. D., and Lacher, J. R., W.A.D.C. Tech. Report 56–590, Part 1, 21-22 (1957).

[5] Butler. G. B., and Agnello. R. J., J. Am. Chem. Soc. 79, 3128 (1957).

[6] Miller, W. T., private communication.

[7] Hauptschein, M., and Fainberg, A. H., J. Am. Chem. Soc. 83, 2495 (1961).

[8] Knunyants. I. L., Chai-yuan. Li, and Shokina, V. V., Doblady. Akad. Nauk. SSSR 136, 610–12 (1961).

[9] Haszeldine, R. N., J. Chem. Soc. 4291-4305 (1955).

[10]

[11]

[12]

[13]

Knunyants. I. L., Chai-yuan. Li, and Shokina. V. V., Izvestia

Akad. Nauk. SSSR, Otdelenie Khimicheskikh Nauk. 10, 1462-68 (1961).

Knunyants. I. L., Chai-yuan. Li, and Shokina. V. V., Izvestia

Akad. Nauk. SSSR, Otdelenie Khimicheskikh Nauk 10. 1910-11 (1961).

Miller, William T., U.S. Patent 2.668,182 (1954).

Hauptschein, M., and Braid, M., J. Am. Chem. Soc. 83, 2500

(1961).

where q=h/2u, (h is Planck's constant divided by 2. μ is the reduced mass of the molecule), and the B are the quantum corrections of order p. Kihara, Midzuno and Shizume [2] give expressions for these terms up through third order as integrals over functions of the intermolecular potential and its derivatives.

This expansion of B will not be applicable in cases where the intermolecular potential function or its derivatives contain discontinuities, as, for example, the hard sphere or square well cases. It has also been shown [3] that the exchange term of the second virial coefficient cannot be represented by such an expansion, so eq (1) will not be adequate for calculating B at temperatures so low that spin and statistical effects are important. In practice, however, these temperatures will usually be below the range in which eq (1) converges.

At very low temperatures the second virial coefficient may be accurately calculated by means of the phase shift formulation, and we can examine the range of validity of eq (1) by comparing results obtained by both methods. In the case of He1, for example, it has been shown [3] that eq (1) with three quantum cor

Figures in brackets indicate the literature references at the end of this paper.

B*():

S

-U*T*

e

(U*')2R2dR

8

These tables were calculated by Simpson numerical integration of eqs (4) from a value of R=R, below which the exponential becomes negligibly small. The contribution to the classical term is then simply R while this part of the integral gives a vanishingly small contribution to the quantum corrections and derivatives. The numerical integration is carried out from Rs to R10, and the contributions from the large R region are obtained by expanding the exponential and integrating analytically a sufficient number of terms to give the desired numerical accuracy. The resulting values were tested for accuracy by decreasing the interval size used in the integration until no changes resulted. For the (6-6) and (9-6) the value 1/128 was sufficient, for (12-6) through (24-6), 1/256 was used. and for the (36-6), 1/320 was required. Higher order integration procedures than Simpson were also tried and found to be unnecessary. The numbers given in the tables are believed to be correct to within 1 in the last digit given.

B() and its temperature derivatives could be checked against previous calculations for several of the potentials. For the (9-6) Epstein and Hibbert [5] give values for a number of T's ranging from 0.5 to 100 calculated by means of the series expansions (eq 6). Bird and Spotz [6] give results obtained in the same way for the (12-6) for T*'s covering our complete range, and Michels [7] also gives (12-6) results at low T* values (below 10), including the first two quantum corrections, obtained in terms of confluent hypergeometric functions. In all cases our results were in complete agreement to all figures available.

For the (18-6) Saxena and Joshi [8] calculated values for B*() and its first derivative from the series. but in terms of T, = (8/3√3)T*, so comparisons are somewhat difficult. Interpolating in our tables for a few values over their range (T*, = 0.5 to 400) indicates approximate agreement. Values of B* for a large number of values of m have been given by Sze and Hsu [9] but the formula they evaluated is incorrect except for m = 12. For this case results are also in agreement. Spot checks of values for the other potentials were made by comparison with values calculated by means of eqs (6).

The T intervals have been chosen such that fourpoint Lagrangian interpolation will suffice to give

values correct to within 5 in the sixth digit given throughout the table, with the exception of some T*'s < 1, and some higher order correction terms which have very small values and are given to fewer digits. We have included only a few small values of T* as eq (1) is convergent here only for very small values of 1*. i.e. nearly classical systems.

In considering the behavior of B* with m, it is useful to refer to figure 1 which shows the reduced potential

it is apparent that a nonvanishing well appears at a finite r only if m>n. As m→n, U(r) becomes purely attractive or repulsive depending on the relative values of X and u. and it vanishes identically when they are equal. If we now change over into a representation in terms of (€), the magnitude of the potential at the location where its derivative is zero, and (σ), the location of the zero of the potential, instead of A and μ, we obtain

function for selected values of m including the limiting cases m=6 and m=∞. One notes that as m increases, the well becomes narrower and the strength of the well can thus be considered as weaker. As a consequence, the reduced Boyle temperature (where the repulsive effects cancel the attractive effects and B* becomes zero) becomes lower as m increases. (It can also be noted from eq (4) that the first quantum correction is always positive and thus the more quantum mechanical the system, the more the Boyle temperature is lowered from the classical value.)

At higher T's the repulsive effect, which predominates, is eventually dimmed somewhat by core penetration, as was pointed out by Lennard-Jones, and B* passes through a maximum and then falls off as the system sees a smaller and smaller effective core size. This decrease is smaller the larger the m value, and the drop-off becomes less and less, until for m = ∞, no maximum occurs. The temperature at which the maximum occurs (Tmax) is the result of a competition between the rise in B* due to the loss of attractive effects from the declining strength of the well, and the drop in B* due to decreasing effective core size. Thus T* as a function of m passes through a minimum for m near 18.

max

Examination of the limiting cases is of interest to illustrate the caution which must be exercised in inferring the behavior of B in dimensioned units with m from that of B*.

which, when n=6, becomes eq (5).

Now, however, because of the variables chosen. eq (10) will always contain a well at a finite R. If m<n, it is apparent from eq (9) that the coefficient changes sign and thus the larger exponent is always on the repulsive term. When m→ n, eq (10) approaches the limiting form

which has a zero at R= 1 and a minimum at R= e1/". For this limiting case it must be noted that the variables σ and are no longer appropriate quantities for parameterizing the potential function as, when m→n, σ becomes infinite or indeterminate and e is zero. Equation (11) cannot distinguish between the vanishing and purely attractive or repulsive limits of U having lost the information about λ and μ.

When B*(T*) is calculated for the potential given by eq (11), it has the usual form, beginning at negative values for low T* and becoming positive at higher T's. Because of the loss of physical information in the potential function, however, these all correspond to 70 and indeterminate B's. Thus these values. which we include in our tables, should be regarded as useful only as limiting values for interpolating with respect to m and one must keep in mind the peculiar nature of the limit.

When m→ ∞, eq (10) becomes