J. Phys. Chem. 1992, 96, 9551-9554
9551
Temperature Dependence of '*@XeNMR of Xenon in Polymers T.T.P.Cbeung* and P . J. Chu Phillips Research Center, Phillips Petroleum Company, Bartlesville. Oklahoma 74004 (Received: June I I , 1992; In Final Form: August 3, 1992) Temperature dependence of "Xe NMR of xenon dissolved in solid polymers is investigated in terms of the free volume model of polymers. It is shown that transitions in polymers can lead to discontinuities in the slope of the chemical shift as a function of the inverse temperature. In particular, a discontinuity in the slope of the chemical shift will develop at the glass transition and is directly related to the discontinuity in the thermal expansion coefficient of the polymer at the transition. Examples of transitions (discontinuities) observed in 12'Xe NMR are given for xenon trapped in isotactic poly(4-methyl- l-pentene) and isotactic polypropylene.
I. Introduction While 129Xenuclear magnetic resonance (NMR) of xenon adsorbed in crystalline materials such as zeolites has been extensively investigated' in recent years, there are comparatively few s t ~ d i e s ~on- ~xenon dissolved in polymers. Part of the difficulties in applying lz9Xe NMR to polymers stems from the complicated spatial structures in solid polymers involving various amorphous and crystalline phases where the xenon may be located. Some properties of polymers are sensitive to the thermal history of the polymers, which introduce variations in the 129XeNMR data, resulting in ambiguities in their interpretations. But most importantly, there is a lack of sound understanding of how the environment in solid polymers affects the '%e resonance. In this paper, we present a theoretical model of '29XeNMR of xenon in solid polymers based on the free volume concept.6 The goal is to investigate the relationships between the temperature dependence of 129XeNMR chemical shift and the transitions occurred in polymers. To provide a context for our discussion, we shall briefly describe the temperature dependence of lBXe NMR of xenon in isotactic poly(4-methyl-l-pentene)(i-PMP) and isotactic polypropylene (i-PP).
II. Examples: '%e NMR of Xenon in LPMP and i-PP 12%e chemical shifts of xenon dissolved in i-PMP and i-PP are shown in Figure 1 as a function of inverse temperature, TI.The shifts of i-PMP in a larger temperature range are given in Figure 2. Details of the properties of the polymers, experimental conditions, and results will be given elsewhere. Briefly, '29XeNMR spectra were obtained from the polymers under 2 atm of pressure of xenon. The polymers in the form of fluffs of the size 40 mesh and less were loosely packed in NMR tubes and outgassed for several hours in a vacuum less than 1 X lod Torr before the introduction of xenon. Only loosely packed samples were used in order to avoid contributionsfrom the interparticle space volume to the xenon chemical shift. Compression of a powder sample may create interparticle pores which have properties that mimic those of the intraparticle pores and would modify the xenon chemical shift.' The sample in the NMR coil was equilibrated for at least 15 min at each temperature before data acquisition began. 129XeNMR spectra of xenon in polymers are only slightly affected by the pressure of the xenon used. In some polymers, slight increases in the chemical shift with xenon pressure in the range of 0 . 5 4 atm are observed. However, the glass transition temperatures as determined by '29XeNMR (see below) remain unchanged. We believe that the increase of the chemical shift with xenon pressure depends on the solubility of the xenon in the polymers. In polymers with high permeability to xenon, xenonxenon interactions become important at high xenon presure, which lead to increases in the chemical shift. At 2 atm of xenon, effects of polymer plasticization due to xenon sorption are minimal since no variation in the glass transition temperature is observed by '%e NMR by either increasing or decreasing the xenon pressure. Isotactic poly(4-methyl-1-pent) is one of the most permeable polymers to gases. Puleo, Paul, and Wong* have found that 0022-3654/92/2096-955 1$03.OO/O
significant amounts of carbon dioxide and methane can diffuse into the crystalline phase, although the permeabilities of the crystalline phase to these gases are 25-3096 less than those of the amorphous phase. This is in contrast to other semicrystalline polymers like polyethylene and polypropylene in which permeabilities of these gases in the crystalline phase arc negligible. Considering similar dynamic radii between xenon and these gases, it is believed that xenon dissolves in both the amorphous and crystalline phases of i-PMP. The fact that we observe only a single resonance for the dissolved xenon indicates rapid movements of xenon atoms between different phases or that the chemical shifts of xenon in these two phases are very similar. At temperatures below 180 K, the 129XeNMR line does indicate the presence of a shoulder at the downfield side of the main resonance, indicating that the xenon atoms are probing different morphological domains in the polymers. For the present discussion, we consider only the averaged chemical shifts. In Figure 1, one observes that, for i-PMP, there is a discontinuity in the slope of the chemical shif-T1 w e at a temperature T,(i-PMP) = 323 K. We identify this discontinuity with the glass transition of the polymer since it falls within the temperature range where the glass transition of amorphous i-PMP occws.9 Dynamic mechanical analysis'O indicates another transition at 220 K,which may be due to the densification process by which excess free volume in the amorphous phase is reduced by better packing of amorphous chains and side groups. 129XeNMR results do not indicate any transition at that temperature but instead show evidence of three discontinuities (in the slope of the chemical shift-r' curve) at temperatures T, = 288 K, Tb = 233 K, and T, = 200 K. The latter two may be associated with the densification process. It is interesting to note that, in Figures 1 and 2, the slope decreases a c r w the discontinuitiesat T,(i-PMP), T,, and Tc as one goes from the high temperature side (smaller 1 / T ) of the discontinuity to the low temperature side (larger 1/7). A similar change in the slope has been observed by Stenglc: and Williamson' at the glass transition temperature of poly(ethy1 methacrylate). On the other hand, the change in the slope at Tb is the opposite. A discontinuity in the slope of the chemical shift-T' plot of isotactic polypropylene occurs at a temperature T,(i-PP) = 266 K, as shown in Figure 1. As in the case of i-PMP, the discontinuity is associated with the glass transition in the i-PP polymer," which takes place at about 270 K. Again, the slope decreases across T,(i-PP) as one goes from the high temperature side to the low temperature one. In contrast to i-PMP, the discontinuity in i-PP is more dramatic. We shall return to this point later in our discussion.
III. Free Volume Model of '%e NMR in Polymer Results shown above and from others suggest that discontinuities in the slope of the '%e NMR chemical shifts reflect the presence of transitions in the polymers. From studies of 129XeNMR of xenon trapped in porous crystalline materials, it is generally agreed that one of the important factors affecting the l%e chemical shift is the free volume in which the xenon atoms are trapped. In the Q 1992 American Chemical Society
9552 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 220t
2001 2.5
two basic assumptions: (1) the short-range potential -Ul(F-F,) vanishes when i - i,lies outside a volume Y around the ith site, and (2) there is no overlap in the potentials that originate from different sites. The fvst assumption implies that the chemical shift ul(F-i,)due to -Ul(F-F,) is also zero when 7- i,is outside the volume u around the ith site. With these two assumptions, one finds
t
2.7
2.9 3.1 1000/T (K-')
1
3.3
'd'iu(7) exp[BU(fl] = NJ"d3iul(W,) exp[BUl(F-i,)] (4)
3.5
235t
Cheung and Chu
thus
t
d[
'd3Fu(fl exp[BU(i)]l/dV = 0
I'd3? exp[BU(fl] = N~h3flexp[BUl(F-F,)]- 1) + V
(5) (6)
and thus 2101
3.5
4.0 1000/T (K-1)
3.0
d(~'d3iexp[BU(7)]~/dV =1
4.5
Figwe 1. I2%e NMR chemical shift of xenon in isotactic poly(4methyl-1-pentene) (I-PMP) and isotactic polypropylene (i-PP) as a function of the inverse temperature.
(7)
From eqs 1, 5, and 7, we obtain, at a constant pressure, d(U(fl)T/dB = Kl(7')
+ K2(7')
(8)
where
.
a 230 v
K1(T ) = -( dfl ) T[(d V/dB)/ VI t(7') = (dr3U(i))T - ( Q ( f l ) T ( U ( f l ) T
T,
(9) (10)
We have used the general relation
(af/aY)x = (af/aY)* + (af/az),@z/aY)X 200 1 2
3
4
5
6
to arrive at eqs 8-10. Here,f, x, y , and z represent respectively ( ~ ( 7 )the ) ~prea3ure, 8, and the freevolume K Kl(7j comes from the second term on the right side of the relation while K 2 ( Q originates from the first term. ((7') is defined as
7
100Ofl (K-')
Figwe 2. Iz9Xe NMR chemical shift of xenon in isotactic poly(4methyl-1-pentene)as a function of the inverse temperature in a larger temperature range.
followin& we describe a model of '%e NMR of xenon in polymers based on the free volume ~ n c e p t ~ Jof~ -polymers. '~ We show that the discontinuities are directly related to processes in polymers which result in abrupt changes in the free volume or in the slope of the free volume as a function of temperature. In this model, a dissolved xenon atom is confiied in an enclosure of free volume Vwithin the polymer where the xenon atom is free to move about. The xenon atom interacts with the polymer segments, which define the surface of the enclosure, via a short-range pairwise attractive potential -Ul(F-ii)such as the van der Waals interaction, where ?and Fi refer to the positions of the xenon atom and the ith "interaction site" on the surface of the enclosure. The potential -Ul(W,) gives rise to a 129Xechemical shift ul(F-i,). In the fast exchange approximation, the observed IsXe chemical shift is given by the canonical ensemble average (Q(fl)T
( 4 3 )= ~1'd3Wr3
exp[BU(fl1/~'d3i exp[BU(flI
(1)
where
J-
((7') = V/ 237 exp~BU(r31 51 (11) It equals unity when (1) k T is much larger than the maximum of -Ul(h*,), the high temperature approximation, or (2) V>>Nu. The ensemble averages on the right side of eq 10 are defined in the same way as that in eq 1. It is clear from eq 8 to aq 10 that a discontinuity in d(u(7)),/@ will occut at a temperature T, if there is a discontinuity in dV/@ or Vat the same temperature. Let T+ and T- be respectively the temperatures infinitesimally larger and smaller than the transition temperature Tt. We define the discontinuity in d( ~ ( 7 ) ) ~ / dbyB
d(m)T/dBlT+ - d(Q(fl)T/dBIT-
A
and consider the discontinuities in dV/dB and Vseparately. These two types of discontinuities are shown schematically in Figure 3. (In order to adhere to tradition, we sketch the free volume Vas a function of T instead of 1/T. The translation from the V-T plot to the V-T1 plot is straightforward. Note that dV/@ is given by -kT2(dV/dT).) Vis continuous but dV/d@is discontinuous at T, (Figure 34. Using the fact that K2(7') is continuous at Tt (since Vis continuous at T J , one obtains from eq 9 A = -A(",€(Tt)
N
u(fl =
Cul(+ii) i
(2)
N
U(3 = ZU,(+F,) i
(3)
with N being the total number of interaction sites on the surface of the enclosure and B = l/kT. The subscript Ton the left side of eq 1 emphasizes the T dependence of the ensemble averaging. To investigate transitions in polymers, we need to determine d((a(i)),)/dg. In order to simplify the calculations, we introduce
(12)
(13)
where A is the discontinuity of dV/dB at Tt A
l(dV/dB)lT+ - (dV/dB)l~J/v
(14)
Note that A and A have opposite signs and that, according to eq 14, A is simply the difference of the thermal empansion coefficient of the free volume above and below the transition temperature Tt multiplied by -kT$. If the thermal expansion coefficient of the polymer is governed by the thermal expansion of its free volume, that is, if (1/ V) dV/d@is approximately given by (1 / Vp)
The Journal of Physical Chemistry, Vo1. 96, No. 23, 1992 9553
129XeNMR of Xenon in Polymers
volume in the amorphous phase of the i-PP, while it probes the free volume in both the crystalline and amorphous phases of i-PMP. This difference in the xenon permeabilities may be the origin of the more dramatic change in the slope of the chemical shift-T' plot at the glass transition of i-PP in comparison to that of i-PMP observed in Figure 1. V changesfrom V to ( V - c) when T goesfrom T+ to T- while dV/dfl remains constant aboue and below T, (Figure 3b). With expansion of Kl(T-) in cq 9 and K2(T-) in eq 10 to the fmt order in c / V , eq 12 becomes A = ( ~ / V ) € ( T + ) ( ~ ( U ( ~ ) ) T -+ ((dr3 V ~ )W T , ) T-+ 2( dr3 ) T+€(T+)[(dV/dB) / VI IT*) (1 7)
r
Free Volume
V
/ Temperature T
Figure 3. (a) Continuous V-Tcurve but with a discontinuity in dV/dT (thus, also in dV/d@) at the temperature T,. (b) Discontinuous V-T curve at Tt but dV/dT (therefore, also dV/d@) remains constant above
and below Tt. TABLE I
isotactic poly(4-methyl-
cpcalculated from cq 16, K-'
1-pentene) 2.56 X la-'
The sign of A depends on that of 6 as well as that of the quantity in the { ) bracket. If the latter has a negative sign, a contraction in Vwill lead to an increase in the slope of the chemical shift-T1 curve. Q(TJ denotes the terms in the (1 bracket in eq 17. To determine the sign of Q( TJ requires the knowledge of U(7).For simplicity, we consider a square well potential in which Ul(%T,) equals a positive constant VI, when T - 7, is within the volume v and zero otherwise. Then Q(Tt) can be expressed as a product of two factors. The factor which determines the sign can be written as Nv[exP(ulo/kTt) + 11 - vll
isotactic poly(ethy1 polypropylene methacrylate) 7.17 X la-' 4.71 X lo-*'
reported in 3.16 X lo-' 4.43 X la-" 2.95 X lo4 literature: K-' OReferenct 3. bReference 17. 'There is a range of q, from 3.22 X 104/K to 6.45 X la-'/K, reported in literature for polypropylene. The value given in the table is the average of five different results. (p
dV /d& where Vp is the total volume of the polymer, then by debning cp cp = W p / d n l T + - (dVp/dT)IT3/V, one obtains A/ (.(? )T,I = €(Tt)kT?V
(15)
In the high temperature approximation, or if V >> Nv, €(TI) 1 and eq 15 reduces to A / ( u ( r 3 ) ~ ,= kTt%
-
(16)
The discontinuity in the slope of the chemical shift-T' plot at the transition temperature Tt is therefore given by the discontinuity in the thermal expansion coefficient of the polymer at the same temperature. It is known that changes in the free volume of polymers near the glass transition temperature follow that sketched in Figure 3a.15J6 Therefore, eq 13 or eq 16 can be used to describe the effect of glass transition on Iz9XeNMR. To test these equations, one may compare v obtained from eq 16 using lZ9XeNMR data in Figure 1 with those determined by other meth0ds.l' Results are summarized in Table I. We also include in the table those of poly(ethy1 methacrylate)? the 129XeNMR data of which are available. One sees that the values of cp obtained from '%e NMR agree, within a factor of 2, with those in literature. In view of the assumptions and approximations in arriving at eq 16, we consider that the agreement is reasonably good. It coflirms that eq 13 or eq 16 is adequate to describe lZ9XeNMR of xenon in polymers at the glass transition. Most importantly, they give the correct sign for A in relation to A. That is, cq 13 and eq 16 predict that the slope of the chemical shift-T' curve must decrease as one goes from the high temperature side of the glass transition to the low temperature side if the thermal expansion of the polymer follows that shown in Figure 3a near the glass transition. It is interesting to note that eq 16 overestimates the value of cp for I-PP considerably while only slightly underestimates that of i-PMP. One of the reasons is that xenon samples only the free
-2~T,2[~1(d~/dn11T+/~lo~
It is clear that Q(TJ will have a negative sign only if the following two conditions are satisfied: (1) V>> Nv[exp(Ulo/&Tt) 11, and (2) Ulo> 2kT,2[(dV/dT)/VllT+. A typical thermal expansion coefficient, (dV/dlT)(l/V), of polymers is about 2 X 10-'/K. Therefore, the seamd amdition restricts the transition temperature Tt to Tt < 5 o ( U 1 0 / k ) ~ / ~ . A discontinuity in the volume as a function of temperature implies a fmtsrder phase transition. In polymers, it would have been observed at the liquid-solid (crystalline phase) transition,l2 if there had been no glassy phase present. It has been reported1* on some occasions that such a transition is observed at a temperature above the melting temperature of the crystalline phase. For solid polymers, a discontinuity in the free volume may occur at the transition from a less stable crystalline phase to a more stable one. The discontinuity in Vat Tt also leads to a Dirac delta function in Kl(T,). Although the delta function does not contribute to A, it leads to an abrupt jump in the chemical shift by the amount of (~/V)€(T-)(U(F))~+. A rough estimate with c/Vin the range 1, and ( U ( F ) ) ~ + 200 ppm shows that of 10-3-10-2, (T-) the jump is about 0.2-2 ppm. If the change of the free volume during the transition is not as abrupt as that depicted in Figure 3b but spreadsover a narrow range of temperature, the abrupt jump in the chemical shift will not occur. Instead, one will fvst observe a rapid increase in the chemical shift at the beginning of the transition (at the high temperature side). Near the end of the transition (at the low temperature side), the increase of the chemical shift will level off sharply. Therefore, the transition leads to two discontinuities in the slope of the chemical shift-T1 curve: one at the beginning of the transition and the other at the end. These two discontinuities have the same origin as that due to an abrupt change in dV/dT as described by eq 13; the discontinuity at the beginning is due to a sudden increase in dV/dT as the temperature is lowered while the discontinuity at the end results from a sudden decrease in dV/dT. To a large extent, the transition with Vchanging rapidly over a narrow range of temperature can be considered as two transitions of the type of Figure 3a compressed together. One of the transitions follows what is sketched in Figure 3a, while the other is the inverse with a steeper slope at the low temperature side. The discontinuities in the slope of the chemical shift-T1 curve at Tb = 233 K and T, = 200 K in Figure 2 seem to fit this description. The difference in temperatures between these two discontinuities is about 33 K,and they occur just above and below the transition at 220 K observed by dynamic mechanical measurements. Since (u(F))T. ( € / V ) ( U ( ~ ) results )~~, 0.035. The fraction of the volume in Figure 2 yields (t/V)
+
-
-
-
-
9554 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992
contraction over the same temperature range based on the thermal expansion coefficient of amorphous poly(4"thyl- 1-pentene) of 3.20 X l P / K is 0.01 1. Therefore, the volume contraction of 3.5% between Tb and T, is consistent with the interpretation of densification of the amorphous phase at low temperatures. In a related but equally important phenomenon of physical aging of polymers below their glass transition temperatures, the free volume of the polymer diminishes19with increasing aging time at a fued temperature. The relations between the free volume shrinkage and 129XeNMR can be derived using the two basic assumptions that we outlined before eq 4. The results are summarized in the Appendix. So far in our investigation of the discontinuities in the chemical shift-r' curve, we have only considered the situation where N, the total number of interaction sites on the surface of the enclosure, is a constant, independent of the changes in the free volume V. However, our treatments above can be extended to other situations where N is not a constant but varies with surface area A of the enclosure. Let us consider the case that N is proportional to the surface area A, and the number of interaction sites per unit area, N/A, is a constant, independent of the changes in V. Then by defining X = V/A, and using the two basic assumptions outlined before eq 4, one again obtains eq 8 with KI(T ) and K2(T ) given by eqs 9 and 10, respectively, except that [(dV/@)/Y] in eq 9 is replaced by [(dh/d@)/X]. Derivations of the rest of the equations for the two typea of discontinuities in V are straightforward. It suffices to summarize the main results here: (I) Discontinuity in dV/d@resulting in a discontinuity in dA/d@ (Vand X remain continuous). Let A' be the discontinuity of dX/@ at TI, Le. A' = (dX/d@l~+ - dX/d@b-I/X ( 14') Then one obtains A = -A'(
4r3 ) Tt€( TI)
( 13')
(2) Changing of X from X to A - e' as V changesfrom V to V
- e and T goesfrom
T+ to T. (dV/d@and dA/d@remain constant aboue and below T,). Equation 17 is translated to
Cheung and Chu the free volumes of each type of domain and compare them with those of the pure polymers. In summary, we have presented 129XeNMR results of xenon dissolved in i-PMP and i-PP and shown that transitions in polymers can lead to discontinuities in the slope of the chemical shift-T' curve. The discontinuities are amenable to theoretical investigations based on the free volume concept. In particular, we find that, at the glass transition, the discontinuity in the slope of the chemical shift is directly related to the discontinuity in the thermal expansion coefficient of the polymer at the transition. Acknowledgment. The authors would like to thank Dr. S.K. Doun and Dr. Eric T. Hsieh, both of Phillips Petroleum Co., for discussions on properties of polymers. Appendix
Noting that Vis a function of the aging time 7 and using the two basic assumptions outlined before eq 4, we obtain = [r + v ( - w [ r + V(T)I (AI where
(m)T,r/(m)T,,.+-
r = N~h3i(exp[@Ul(FF,)] - 11 Equation A1 can be rewritten into a more useful form [(a(r3)~,roo/(.(r3)~,,1 [(d?))~,~ - (a(7))~,,.+-I/[(a(r3)~,~=0 - ( 4 r 3 ) ~ , =J [V(T) - V(m)l/[V(O) - V(m)1 (A2 Thus, within the free volume model, one can monitor the physical aging of a polymer using '29XeNMR. Equations A1 and A2 are valid for the situation where N is a constant, independent of the free volume V. On the other hand, if N/A is a constant, independent of the changes in the free volume V, we have instead (m)T,r/(m)T,l-= [ r / A + V-")l/[r/A + X(T)1 (A 1'1 which can be rewritten as
- ( di?) T,,.+-.] / [ ( ) T,r=O [VT)- X(m)l/[X(O) - Um)l (A23
( a(i? ) T,r=O/ ( d9 ) T , r l [ ( a(fl)T,r
- (4r3)~,,---.] =
R e g k y No. PMP (isotactic), 24979-98-4; PP (isotactic),25085-53-4; XC, 7440-63-3.
While the question of whether a constant Nor a constant N/A is more suitable to describe *29XeNMR in polymers remains to be resolved by experimmts, it should be emphasized that the results of these two descriptions are not that different because (e/V) and (d/X) are related, and so are (dV/@)/Vand (dX/@)/X. In fact, for an enclosure with a simple geometry, one can obtain the simple relationships E/V = y(d/X) and (dV/d@)/V= y(dX/d@)/X. y is a numerical constant. For example, y = 1 when the enclosure is defined by two parallel layers with the dimension of the layers much larger than the layer separation and changes in the layer separation responsible for the changes in V. y = 2 when the enclosure has a long cylindrical shape with the length of the cylinder much larger than the radius and changes in the radius responsible for the changes in V. Finally,y = 3 when the enclosure takes on the spherical or cubic shape. Therefore, equations such as eq IS and eq 16, which relate the discontinuity of the slope of the chemical shift to the discontinuityof the thermal expansion coefficient, can still be applied to the situation where N/A is a constant provided that one includes a factor of l / y in those equations.
IV. Conclusioe It is clear that NMR may provide information about the free volume in solid polymers. This information may be complementary to those obtained by other traditional techniques such as dilatometry, which actually probe the volume-temperature curve. One of the advantages of using '%e NMR is that in highly heterogeneous systems such as block copolymers or polymer blends, xenon atoms located in different spatial domains may produce well-resolved 129XeNMR peak4 This allows one to investigate
References and Notes (1) For a general review,see: Dybowski, C.; Bansal, N.; Duncan, T. M. Annu. Rev. Phys. Chem. 1991,42, 433. (2) Sefeik, M. D.; Schaefer, J.; Desa,J. A. E.; Yelon, W. B. Polym. Prepr. (Am. Chem. Soc., Diu. Polym. Chem.) 1983,24, 85. (3) Stengle, T. R.; Williamson, K. L. Mocromolecules 1987, 20, 1428. (4) Walton, J. H.; Miller, J. B.; Roland, C. M. J. Polym. Sci.: Polym. Phys. 1992, 30, 527. ( 5 ) Cain, E. J.; Wen, W. Y.; Jost, R. D.; Lui, X.; Dong, Z. P. J . Phys. Chem. 1990, 94,2128. (6) See, for instance: Struik, L. C. E. Physical Aging in Amorphous Polymers and Other Moteriols; Elsevier: New York, 1978. (7) Chen, Q. J.; Fraissard, J. J . Phys. Chem. 1992,96, 1814. (8) Puleo, P. K.; Paul, D. R.; Wong, P. K. Polymer 1989, 30, 1357. (9) Stricklen, P. M.;White, S. A.; Wilka, G. L.; Lopez,L. C. Submitted for publication in J. Mocromol. Sci.: Rev.Mocromol. Chem. Phys. (IO) Pratt, C. F.; Geil, P. H. J . Mocromol. Sci. Phys. 1982, B21, 617. (11) Bair, H. E.; Schilling, F. C. Proc. 15th Notos Con/. 1986, 15, 32. (12) Petrie, S. E. B. Polymeric Moteriols: Relotionrhips Between Structure ond Mechonicol Behovior; American Society for Metals: Metal Park, OH, 1975. (13) Kovacs, A. J.; Hutchinson, J. M.; Aklonis, J. J. In The Structure of Non-Crystolline Moteriols, Gaskell, P. H., Ed.; Taylor and Francis: London, 1977. (14) Tant, M. R.; Wilka, 0. L. Polym. Eng. Sci. 1981, 21, 874. (15) See,For instance: McKinney, J. E.; Goldstein, M. J. Res. Notl. Bur. Stond. 1974, A78, 331. Also: Kovacs, A. J. J . Polym. Sci. 1958,30, 131. (16) See also: refs 6 and 12-14. (17) Simha, R.; Boyer, R. F. J . Chem. Phys. 1%2,37, 1003. S h a m , S. C.; Mandelkem, L.; Stehling, F. C. J . Polym. Sci. B 1972, 10, 345. Boyer, R. F.; Simha, R. J . Polym. Sci.: Polym. Lett. Ed. 1973, I ! , 33. Beck, D. L.;Hiltz, A. A.; Knox, J. R. SPE Trons. 1963,3.279. (18) Boyer, R. F. Polymeric Moteriols: Relationships Between Structure ond Mechonlcol Behmior, American Society for Metals: Metal Park,OH, 1975. (19) For example, see: Goldbach, G.; Rehage, G. Rheol. Acto 1967, 6, 30.
