MULTI-DIMEKSIONAL GLASSTRANSITION
June, 1963
ture the curve for n = 15 gave the best fit, but to agree satisfactorily wiih the experimental data the curve for n = 15 would have to be shifted about 0.25 of a log unit toward lower pressures. It was of interest to compare the fall-off curves pre-. dicted a t 175.4' with the experjmental data since no further adjustment of parameters is possible in such calculations. The Kassel curves for the quantum form with s = 13 and the classical form with s = 10 were calculated a t 175.4' as in the case of the curves a t 150.4'. These curves are compared with the experimental results in Fig. 5. Over the limited range of the available data both curves predict approximately the shift in the fall-off region along the pressure axis which is caused by a change in temperature. It is hoped that it will be possible to study the isomerization of cyclobutene a t pressures below 0.0365 mm. in a larger reaction vessel, thereby overlapping the lowest pressure range a t 150' shown in Fig. 2 and extending the range of available data a t 175'. As has been shown in Table 11, the change in the falloff behavior of the isomerization of cyclobutene with temperature results in a decrease in activation energy from a value of 32.7 kcal./niole a t the high pressure
1333
limit (and a t 5 mm.) to a value of 30.7 kcal./mole a t 0.2 mm. and 30.2 koal./niole at 0.055 mm. Both the Kassel and Slater thleories predict a pressure dependence of the activatilon energy in the fall-off region. On the basis of the simplified form of the Slater theory2I with n = 15 the predicted decrease in activation eiiergy from the high pressure limit to the 0.055 mm. region is 2.6 kcal./mole; with the Kassel theory the decrease in activation eiiergy ranges from about 1.7 to 2.4 kcal./ mole depending on whether the fall-off behavior is estimated from the'quantum form with s = 13 or the classical version with s = 10. In view of the considerable effect of the possible experimental error upon the difference in two large quantities ( E , and Eo.o55), satisfactory agreement seems to exist between the experimental aiid predicted decreases in activation energy with decreasing pressure according to either the Slater or the Kassel theory. Acknowledgment.--The authors wish to thank Dr. E. W. Schlag and Dr. David Wilson for their computer programs and Mr. Carl Whiteman, Jr., for his assistance in connection with the least squares calculations and the computer operations.
THE &IULTI-DI&IEYSIONAL GLASS TRAKSITION BY ADI EISEKBERG Contribution No. 1515 f r o m the Department of Chemistry, University of California, Los Angeles, California Received January 2.4,196s The glass transition in polymeric materials is discussed as a phenomenon which can occur upon a change in any of the variables affecting its free volume, Le., temperature, pressure, diluent concentration, and molecular weight. Depending on which one of these variables is being changed, we can regard the glass transition as a glass transition temperature T,, a glass transition pressure P,, a glass transition concentration C,, and, in a formal sense, also a glass transition molecular weight Mg. A phenomenologically derived equation is presented giving the position of the glass transition in a four-dimensional space of T , P, C , and M , which enables one to calculate any one of these glass transitions from a knowledge of the three other variables.
1. Introduction The glass transition is normally regarded as a phenomenon which occurs when a glass former is heated or cooled through a specific temperature range. Since the glass transition depends very strongly on the free volume (which, while lacking an operational definjtion, has been very helpful in the formulation of phenomenological theories of the glassy state), it is normally determined by following some property dependent on the free voluine, for instance the specific volume, as a function of temperature, i.e., in one dimension. Since the expansion coefficients of the liquid and the glass are approximately linear except in the transition region, the intersectjon of the extrapolation of the linear portions of those lines through the transition region is called the glass transj tion temperature, Tg. However, the glass transition can be obtained by varying any of the other parameters which influence the free volume of a homopolymer, notably the pressure, molecular weight, and concentration of diluent. It has been shown1 that rates also have a very pronounced influence on the position of the glass transition, but these will be neglected in this discussion, and constant rates assumed throughout. Thus, the glass transition is clearly a multi-di( 1 ) A. ,J. K o % a c b , J . Polgmer Scz., SO, 131 (1958).
mensioiial phenomenon, the dimensions being temperature T , pressure P, molecular weight M , and conceiitration of diluent C. If one wishes to observe the transition rather than merely describe its location within these four dimeiisioiis, one must add a fifth, say the specific volume V , which would show some type of a discontinuity a t the transition. The first part of this work will present a discussion of sections through this five-dimensional space keeping all but two of the variables constant, L e . , varying only the volume and one of the others, T , P, I&?, or C. Subsequently, a simple, phenomenologically derived equation will be presented, which correlates the location of the transition obtained by varying any one of the above parameters with all tlie others, Le., a correlation of T,, P,, Xg, and C,. 11. The Individual Glass Transitions A. The Glass Tramsition Temperature, T,.-Since a change in temperai,ure is the most obvious way of bringing about a glass transition, the glass transition temperature has been investigated extensively, and will not be described here further, except to mention (2) W.Kauamann, Chem. IZev., 45, 219 (1948). (3) H. A Stuart, "Die Physik der Hoohpolyn~eren," Vol. 111, Spiingei, 1 B L 5 , Chagters 10 a n d 11.
ADI EISENBERG
1334
that a volume-temperature plot shows a change in slope at the glass transition temperature. B. The Glass Transition Concentration, C,.-Lewis and Tobin4 have recently investigated the dynamicmechanical behavior of a series of polymers as a fuiiction of diluent concentration a t room temperature, and a t a certain concentration (depending on the polymer-diluent system), they observed an inflection point in the modulus-temperature curve and a maximum in the damping-temperature curve, ie., behavior characteristic of a glassy transition. The concentration of polymer a t that point mas called the glass transition concentration, Cg. It is quite obvious that C, would vary with temperature, although no data were given. Glass transition temperatures as a function of concentration have, of course, been determined for a number of polymer-diluent systems and are discussed in the literature.j6 It is obvious that any plot of T , us. C can also be regarded as a plot of C, us. T , and the glass traiisitioii concentration for any temperature obtained from such a graph. Various expressions correlating T , and C have also been g i ~ e n . ~ , ~ C. The Glass Transition Molecular Weight, M,.Because of the difference in free volume associated with chain ends and chain middles, a method of changing
.860L ,858
,848 M
Fig. 1.-V~ir3
1's.
'X
io-3
UNITS.
%! f o r poly-(methyl
0,glass:
e, liquid.
Volume, arbitrary units. f1.02
Temp., "C. J
Pressure, bars. Fig. 2.-Pressure-volume-temperature (4) A. F. Lewis a n d R I .
for selenium.*
C. Tobin, Tra?is. Soc. Rheol., 6, 27 (1962).
( 5 ) E'. N. Kelley a n d F. Bueche. J . Polymer Sei., 5 0 , 549 (1961). (6) I€. A. Stuart. ref. 3, Vol. IV. Ch. 9. (7) R. B. Beevers and E. F. T. White, Tians. Faradag SOC.,56, 744 (1960). ( 8 ) G. Tarnman and TT. Jellinghaus, Ann. Phgs., [ 5 ] 2, 264 (1929).
Vol. 67
the free volume of a homopolymer at constant temperature (at atmospheric pressure and in the absence of diluent) is to change the degree of polymerization. This cannot, of course, be done continuously, but present techniques of polymerization permit excellent control of niolecular weights and distributions so that this may be regarded as more than just a formal may of changing the free volume. If the specific volume of a polymer is investigated isothernially as a function of molecular weight in the temperature region of the glass transition, one would expect to encounter a discontinuity a t the niolecular weight for which this teniperature is the T,. While no direct data are given in the literature for this type of experiment, a plot of the data of Beevers and White' for methyl methacrylate gives the density of the material a t any temperature, in the range investigated, for a number of molecular weights. A plot of specific volume us. niolecular weight, given in Fig. 1for 375"K., shows a definite discontinuity a t a molecular weight of ea. 15,000 which can be taken as the A I g a t that temperature. Similar to the results on concentration, a plot of T, us. 171 (or, preferably, 1," since this is linear) gives simultaneously M , us. T . The formulas of Beevers and White' give for 375°K. a value of the molecular weight of (1.8 0.3) X lo4; close to the above graphic estimate. The concept of an M,, therefore, seems quite ?yell founded on experimental fact. D. The Glass Transition Pressure, P,.-An early systematic investigation of vitrification under pressure mas undertaken by Tamman and Jellinghauss on three glass-formers, selenium, salicin, and colophony. Since subsequent investigatiolls on other materials do not differ in their conclusions from those of the above work (with one exception which will be discussed below), those of importance to this discussion will be reviewed briefly a t this point. Assuming the validity of the free-volume concept of the glass transition, the effect of pressure on T , can be predicted qualitatively. If hydrostatic pressure simply "squeezes out" free volume in a liquid in addition to conipressing the material itself, vitrification will take place as soon as the free volume of the glass transition is reached. Therefore, the effect of hydrostatic pressure is to raise the glass transition temperature. The simultaneous effect of temperature and pressure can perhaps be visualized best in a three-dimensional plot of V-T-P for amorphous selenium constructed from the data of Tamman and shown in Fig. 2. This plot illustrates the fact that while the volume-temperature curves a t any pressure are linear, the volume-pressure curves a t constant temperature are not, and also that the curvatures of the ( b l i / b P )curves ~ are different for the glass and the liquid, as was found by Tamman. To illustrate this last point, two volume-pressure curves are shown in Fig. 3, one for the liquid a t 70" and the other for the glass a t loo. A volume-pressure plot a t 40" is shown in Fig. 4, which shows the intersection of the glass-like curve a t high pressures with that of the liquid-like curve a t low pressures to lie a t ca. 1100 atm. This would be the transition pressure a t the teniperature. -4much more precise estimate of the transition pressure a t that or any other temperature within the region investigated can be obtaiiied by a projectioli on the P-T plane of the line of intersection of the glass and
MULTI-DIMENSIOSAL GLASSTRANSITIOPI'
June, 1963
I
3
0
1332
1
Phenol phi ha1ein ( l o )
P, BARS x IO-2.
Fig. S.--Volume
m. pressure for selenium8: 0,i o " , liquid; 0 , IOo, glass.
1.021
PRESSURE, B A R S .
Pig. 5.--T, us. I' or P , 2's. T for various materials.
P, BARS x
Fig. $.-Volume
10-2.
us. pressure for selenium*; 40".
liquid planes. This projection, shown in Fig. 5 together with those for other materials to be discussed later, gives, as in the other cases, both the variation of T , with P and of P, with T . It is noteworthy that of all the materials investigated, only in the case of selenium is the P-5" graph non-linear. It has been shown beforeg that amorphous selenium is a mixture of Ses and of polymeric selenium, and since the compressibilities of these two species are, most probably, not identical, it is not surprising that the (bT,/dP)c,l~fplot is curved. Vitrification under pressure was investigated by Shishkinl" for phenolphthalein, rosin, a phenol-formaldehyde resin, Bz03, poly-(methyl methacrylbte), and styrene. The last material was also studied by X a t suoka and Rfaxwell.ll O'Reilly'* studied the variation of l", with P for polyvinyl acetate, and it was he who defined the glass transition pressure as that pressure at which molecular rearrangements can 110 longer follow the applied pressure; at that point, the polymer exhibits a glass-like compressibility. The (9) A. EiseTberg a n d A . 5'. Tobolsky J . Polumer &z., 46, 19 (1960). (IO) A?. I Smshkin, Lh Telchn. Fzz., 26, 188 (1955), Fzz Tuerd. Tela, 2,
380 (1960). (11) S Matsuoka and B. Alaxaell, J . Polymer Scz., 32, 131 (1948). (12) J. AI. Q'Reilir, zhid., 67, 429 (1962).
(bT/bP)AM, c = plots for some characteristic materials which have been studied and the results on which are reported in the literature or can be estimated therefrom are given in Fig. 5 . They all show an unambiguous rise of T , with P. The only exception to this phenomenon was found by Weir, who, in an investigation of volume-temperature curves of sulfur-vulcanized rubber at various pressures, found no change in T , with pressure. No explanation seems to be available for this unique behavior. In view of the results on all the other materials, however, the glass transition pressure is certainly an experimental fad,. 111. Relation between T,, Cg, M g , and P, While it is impossible t o show graphically the relation between these four variables, an analytical expression can be derived easily. The influence of molecular weight on T , has been investigated by several authors' I 4 - l 6 and an equation developed which describes this influence in the high molecular weight region. This equation is
T g := T,"(Q) - A / M g (1) where Tgm(Q) is the glass transition temperature of a polymer of infinite molecular weight at zero pressure, A is a constant equal to - [ b T g / d ( l / ~ l ~ ) Jand p , ~M, g ( Q ie ) the molecular weight, the subscript g having been added to indicate that this equation expresses the simultaneouz variation of T , with 114 and Illgwith T . Since, as shown in Fig. 5 , all materials consisting of only one species show a linear dependence of T , on P (and vice versa), we can introduce the effect of pressure on T , by expanding equation 1 to (13) (14) (15) (16)
C. E. Weir, J . Res. .Vatl Bur. Std., 60, 311 (1953). T. G Fox a n d P. J. rloly, J . A p p l . Phys , 2 1 , 581 (1950). K. Uberreiter a n d G Kanlp, J . Coll. Set., 7, 569 (1952). T. G Fox and P. J. Floiy, J . Polymer Scz., 14, 315 (1984).
1336
DOUGLAS W. NCKEE
where D is a constant equal to (aT,/dP)hf and is assumed to be independent of M . It should be nieiitioned that Ferry” proposed the relationship (bT/aP)f = & / q for the change of T, with P at constant free volume, where pf is the “free volume compressibility,” approximately equal to PI - pg, and cyf is the “free volume expansion coefficient,” approximately equal to or1 - a,, the subscripts 1 and g referring to liquid and glass, respectively. A similar relation was also proposed by Shishkinj10 and, in a somewhat more complicated form, by Singh and Kolle.ls Finally, the behavior of plasticized polymers can be approximated by6 (3)
where C1 and C2 are the weight fractions of diluent and (17) J. D. Ferry and R. 4.Stratton, KoZZozd Z., 171, 107 (1960). (18) 13. Singh and A. W. Nolle, J . A p p l . Phys., 30, 337 (1969).
Vol. 67
polymer, respectively. Recalling that the diluent has only one molecular weight, so that we can rewrite equation 2 for diluent as Tgi
=
Tgi(0)
+ Dip
( 2 4
we can combine all equations to yield 1/Tg
=
(1 - Cg)/[Tg1(0)
+ D1PgI +
Cg/[Tg2”(0) - L4/Mg
+ DnPJ
(4)
where C, is the weight fraction of polymer. Thus, if me and Tgl(0), which are constants know 8 , D1, Tgpm(0), for the materials involved, we can correlate the simultaneous variation of all four variables influencing the glass transition and calculate any one from a knowledge of three of t’hefour. Acknowledgment.-It is a great pleasure to acknowledge the benefit of stimulating discussions with professors R. L. Scott and &I.. E. Baur in the course of this work.
CATALYTIC ACTIVITY XKD S I S T E R I S G OF PLATISURI BLACK. 11. DEMETHANATION A S D HYDROGENOLYSIS OF CYCLOPROPANE1 BY DOUGLAS W. IICKEE General Electrzc Research Laboratory, Schenectady, New York Received January 26, 1963 The kinetics of the demethanation (cracking) and hydrogenolysis of cyclopropane on unsupported platinum black have been studied between 100 and 250” by a static method. Thespecific catalytic activity of the metal decreased rapidly during sintering due to the elimination of active sites. Both reactions were found t o occur simultaneously, the products of the catalytic dissociation of cyclopropane being methane, ethane, propane, and a surface residue of average composition CHI.1. The apparent activation energy for demethanation on sintered platinum black was 14 kcal./mole but this reaction was strongly inhibited by the presence of hydrogen. Hydrogenolysis showed an activation energy of about 8.0 kcal./mole and was approximately zero order in both reactants a t low hydrogen concentrations but a large excess of hydrogen tended to retard the formation of propane. Possible mechanisms for these reactions are discussed.
Introduction Gas phase hydrocarbon reactions have rarely been studied on unsupported metal catalysts owing to the difficulty of preparing and maintaining a metal surface of sufficiently high area for appreciable activit’y. Metal blacks, however, although sintering readily when freshly reduced, provide materials of high surface area and avoid complications such as the uncertain influence of a catalyst support and unknown and possible metastable surface morphology of evaporated metal films. It’ is of interest to compare the behavior of metal catalysts in different states so that the role of these factors can be assessed. The kinetics of propane cracking on platinum black have been discussed in part I2of this series and the influence of sintering on the activity of the metal was investigated. This paper describes reactions involving dissociation of cyclopropane on a similar catalyst. Several recent st’udies3 have been concerned wit’h the hydrogenolysis of cyclopropane and its derivatives on supported nickel and platinum catalysts, and it is generally agreed that in this reaction cyclopropane (1) This work was made possible by the support of t h e Advanced Research Projects Agency (Order No. 247-61), through t h e United States Army Engineer Research and Development Laboratories, F o r t Belvoir, Virginia, under Contract No. DA-44-009-ENG-4853. ( 2 ) D. W. hIcKee, J . Phys. Chem., 67, 841 (1963). (3) For a review of t h e recent literature see, e.g., G. C. Bond, “Catalysis b y Metals,” Academic Press, New York, N. Y . , 1962, pp. 270-276.
appears to have properties intermediate between those of olefins and alkanes. These previous investigations have however mostly overlooked the simultaneous occurrence of catalytic cracking (demethanation), the products of which are likely to confuse the kinetic picture for the hydrogenolysis reaction. In addition, although it is agreed that cyclopropane chemisorbs readily on metal surfaces at elevated temperatures, neither the mode of attachment nor the mechanism of the subsequent cleavage has been established. The present investigation represents a further attack on these problems in the case where the hydrocarbon is present both alone and pre-mixed with hydrogen over the catalyst. Experimental Materials.-The platinum black used in this work was the same as in the previous investigation. The unreduced metal had a B.E.T. nitrogen surface area of 19.5 m,2/g., but this was reduced to about 12 m.a/g. after careful reduction with pure hy-drogen. Most of the measurements were carried out using sintering platinum black obtained by reduction with hydrogen a t 180” for over 2 hr. This material had a surface area of 5-6 m.2/g. and little further sintering occurred on further reduction and evacuation. Samples of platinum black weighing approximately 0.8 g. were used. The cyclopropane was obtained from the Matheson Co. Analysis showed this material t o be better than 99.87, pure, the main impurities being ethane and propylene. The gas was
