NOTES
949
oc =
(AM)-’;
CY0
= (XoM)-‘
(5)
with X and Xo corresponding to w and wo. Van Holde and Baldwin assigned the value of 0.001 for 6, but it seems to us that this value is too severe a criterion for the attainment of a sedimentation equilibrium state. Probably, 6 = 0.01 would be sufficient for most practical purposes. I n passing, it may be noted that eq 1 reduces to the result of Van Holde and Baldwin when wg = 0. With 6 = 0.01, Ill = M W= 5600, and D = 1.53 X cmz/sec, together with the proper experimental values for other parameters, we have calculated from eq 1 the values of t for the seven combinations of initial and final speeds which appear in Table I. These show a fairly satisfactory agreement between observed and calculated values.
Acknowledgment. This study was supported by Grant GB 3321, Sational Science Foundation, and was carried out in the laboratory of Dr. J. W. Williams.
mechanical, and nrnr experiments usually give points on the map that lie on common loci. The slopes of these lines are related to activation parameters. The fact of the correlation indicates that the same molecular motions underlie the various experiments. A review’ of the literature also shows that the intensities of corresponding relaxation effects are not always correlated. For example, an intense dielectric loss peak may correspond to a relatively weak mechanical loss peak, or vice versa. To understand the intensities we must consider individually the manner in which the various experiments couple to molecular motions. Relaxation theory, in general, is not well enough worked out to allow us to hope for quantitative prediction of the effects but it is possible, in many cases, to evaluate certain molecular parameters. These parameters can be judged reasonable or unreasonable by comparison with similar parameters deduced for nonpolymeric substances with similar molecular groupings. The intensityzaof a dielectric loss peak can be shown to bezb
The Interpretation of Secondary Transitions in Polymers
by David W. IIcCall Bell Telephone Laboratories, Incorporated, M u r r a y Hill,S e w Jersey (Recebed October 11, 1965)
The study of “secondary transitions” in polymers has received a great deal of attention in the past 10 years or so. Most of the work has gone into the experimental characterization of these relaxations. The three principal methods employed can be classified as dielectric, mechanical, and nmr, even though the technique within a given category may vary widely. Certain qualitative aspects of the subject can be regarded as firmly established. Molecular motion is certainly the underlying phenomenon. In this paper me use the expression “secondary transition” to distinguish processes that can be associated with small, definite molecular entities from processes that involve more general motions. The former might be exemplified by the rotation of a side group about a chemical bond while the latter will include glass transitions and, perhaps, certain other processes, e.g., the CY transition in polyethylene. A review’ of t,he existing literature reveals that the positions of the transitions on a frequency-temperature map are well correlated. That is, the dielectric,
where N is the number of electric dipoles/cm3, pz is the mean-square effectivezb dipole moment, and eB and E , are the low- and high-frequency dielectric constants. Thus, N p 2 is the molecular parameter deduced from dielectric loss intensities observed for polymers. In nmr experiments the depth of a 2’’ minimum can be shown to be proportional to a sum, Zrij-O, over internuclear vectors. 3 , 4 For an assumed molecular reorientation this sum can be calculated or, a t least, estimated. Kuclear magnetic resonance TZ, resonance width, or “second moment” measurements can be interpreted in terms of similar sums. Interpretations of nmr data in terms of internuclear distances and reasonable molecular motions are usually satisfying. However, there are several pitfalls and the interpretations may include complicating factors. For example, a small number of rotating groups can dominate TI through the process of spin diffusion.5 Also, the severe (1) D. W. McCall, unpublished; presented at the Polymer Research Group Meeting, Moretonhampstead, England, April 1964. (2) (a) The intensity is taken t o be the area under an e” os. log Y plot, where e” is the dielectric loss and Y is the frequency. (b) C. J. F. Bottcher, “Theory of Electric Polarisation,” Elsevier, Amsterdam, 1952. (3) N. Bloembergen, E. M. Purcell, and R. T’. Pound, Phys. Rea., 73, 679 (1948). (4) C. P. Slichter, “Principles of Magnetic Resonance,” Harper and Row, New York, N. Y., 1963. ( 5 ) D. W. McCall and D. C. Douglass, Polymer, 4, 433 (1963).
Volume 70,Number 3 March 1966
NOTES
950
limitation on frequency coverage makes intensity considerations more difficult when reorientation times are broadly distributed.6 Ifolecular mechanisms for mechanical relaxation in polymers are not so well established for secondary transitions although many workers have contributed in important ways to the proper appreciation of the factors involved. On the other hand, a very successful theory has been developed for the interpretation of sonic absorption in liquid^.^ We propose that this model is appropriate for the interpretation of many secondary transitions in mechanical loss studies of polymers. If a polymer has a side group that can reorient between two sites the mechanical loss can be shown to have a maximum given by7 tan 6,
=
(RT/2)(AHo/RT)2(M~2~2/Cp2) X exp( - AHo/RT)exp(AXo/R)
(1)
where cr is the expansion coefficient, C, the molar heat capacity, c is the speed of sound, and AH" and AS" are the enthalpy and entropy difference between the two sites. Thus, AH" can be determined by analysis of the temperature dependence of the mechanical loss intensity. (Equation 1 is a simplified form valid when A H " / R T is greater than 2 or 3. More general forms are given by Lamb.7 A plot of T X tan ,6 us. 1 / T will often yield a straight line with slope proportional to AH".) Let us now consider the /3 transition of poly(methy1 methacrylate) in the context of the foregoing discussion. This transition is evident in dielectric, nmr, and mechanical results and the frequency-temperature map shows excellent correlation between the various experiments, The resultant activation energy is about 20 kcal/mole. Analysis of the dielectric loss intensitp yields an effective dipole moment that increases from about 1.4 D. a t room temperature to 1.7 D. a t 130". This compares favorably with -1.9 D. found for ethyl acetate in ~ o l u t i o n . ~Nuclear magnetic resonance TI results make it clear that both the ester and main chain methyls are rotating rapidly at temperatures well below the p region.'O The depth of the TI minimum corresponding to the p transition is consistent with rotation of the ester side group but only an approximate analysis has been made. Analysis of the mechanical loss yields, for the enthalpy difference between sites, AH" g 3.4 kcal/ mole. This is close to the value AH" 3 kcal/mole found for ethyl acetate liquid by ultrasonic relaxation' We might suggest, on the basis of this comparison, that the energy difference has an intramolecular origin. The theory of Lamb7 reveals that the activation energy
-
T h e Journal
of
Physical Chemistry
measured is the lower of the two; ie., the barrier is -20 kcal/mole in one direction and -23 kcal/mole in the other. AS" must be 3 or 4 eu. These considerations make it quite obvious that the /3 transition in polymethyl methacrylate corresponds to a reorientation of the ester side group. In addition, parameters characterizing hindrances to thiq motion have been evaluated. I t will be of interest to see how widely detailed interpretations such as these can be applied. In any case, relaxation intensities are a relatively untapped resource. Intensity analyses will lead to a more secure understanding of the mechanisms by which electric, magnetic, and elastic energies are converted to heat and clearer pictures of molecular motion. Acknowledgment. It is a pleasure to acknowledge the assistance of W. P. Slichter in the development of this material. Helpful discussions were provided by D. C. Douglass, A. A. Bondi, and S. Natuoka. (6) T . 11.Connor, Trans. Faraday Soc., 60, 1574 (1964). (7) J. Lamb in "Physical Acoustics," 11, A, W. P . Mason, Ed., Academic Press, New York, N. Y., 1965, p 203. This chapter by Dr. Lamb contains a beautifully clear exposition of the theory and experimental documentation of its successes in liquids. (8) W. Reddish, Pure A p p l . Chem., 5 , 723 (1962). (9) C . P. Smyth, "Dielectric Behavior and Structure,'' McGrawHill Book Co., Inc., New York, N. Y., 1955. (10) J. G. Powles, B. I. Hunt, and D. J. H. Sandiford, Polymer, 5, 505 (1964). (11) G. W. Becker, Kolloid-Z., 140, 1 (1955). (12) J. Heijboer, ibid., 148, 36 (1956).
Free Energy of Formation of LizTe at 798°K
by an Electromotive Force Method' by M. S. Foster and C. C. Liu Chemical Engineering Division, Argonne National Laboratory, Argonne, Illinois (Received October $1, 1966)
The thermodynamic properties of the binary lithium-tellurium system have been studied using electromotive force measurements of a concentration cell without transference. Very little information pertaining to this system was found in the literature. The lattice constant of LizTe was reported by Zintl, Harder, and Danth.4 A semiconductor character was predicted for LizTe by Mooser and P e a r ~ o n . ~ (1) Work performed under the auspices of the U. S. Atomic Energy Commission. (2) E. Zintl, A. Harder, and B. Danth, 2. Elektrochem., 40, 588 (1934).
