J . Phys. Chem. 1989, 93, 6919-6926 other than three, the two Percus-Yevick results continue to bracket the true results. In five dimensions, for example, the exact fourth virial coefficient is very close to being the same two-thirds compressibility, one-third virial, combination that generates the Carnahan-Starling equation in three dimensions. Our five-dimensional formula, however, does not predict a fourth virial coefficient between the two Percus-Yevick coefficients. The two-thirds/one-third “rule” may thus be more of a statement about
6919
the Percus-Yevick approximation than about the CarnahanStarling equation. Acknowledgment. We thank Jan Tobochnik for helpful comments and for providing us with his computer simulation data. This work was supported by the National Science Foundation under Grants CHE-8420214, CHE-8815163, CHE-8509416, and CHE-88 19370.
Kinetics of Crystallization and Morphology of Poly(piva1oiactone): Regime I I Transition and Nucleation Constantst
-
III
Daniel B. Roitman,*Hew6 Marand,# Robert L. Miller, and John D. Hoffman* Michigan Molecular Institute, Midland, Michigan 48640 (Received: February 7, 1989; I n Final Form: April 24, 1989) Growth rates of the crystalline objects formed from the subcooled melt in poly(pivalolactone),PPVL, were measured optically at various temperatures T, between 180 and 218 OC. Spherulites were formed over most of this range, but axialite-like objects were formed at the highest T,; the growth data refer to the dominant a-crystal form. The results provided clear evidence of a regime I1 I11 transition near 203 OC. The regime behavior was in good agreement with expectation; K,(IIi$Kg(II) was 1.99 & 0.05. From the Kg values, and a determination of the layer thickness bo based on microbeam X-ray studies and related data, it was found that uue = 1748 erg2 cm-“. (The growth front corresponds to 120.) With a u, value of 58 erg cm-2 obtained independently from a T,’ vs 1/1 plot, it was determined that u = 30 erg cm-*, leading to an a of 0.25 in the modified Thomas-Staveley expression. These values of u and a are discussed briefly. The work of chain folding derived from uc is 7.5 kcal mol-’ which is within the anticipated range. WAXD studies confirmed the antiparallel arrangement of the chains in both regimes I1 and 111. Evidence based in part on the presence of cracks in the crystals corresponding to 120 cleavage planes indicates that (i) the chain folding which occurs in melt-crystallized PPVL is rather regular and (ii) in accord with nucleation theory the folding is more regular in regime I1 than regime 111. Finding (i) is supported by a recent “gambler’s ruin” type calculation by Mansfield which shows that an enhanced degree of “tight” folding is demanded when antiparallel packing is present.
-
I. Introduction Interest in poly-/3-lactones has increased in recent years. Substitutions in the 2- and 3-positions relative to the ester group lead to series of materials spanning a wide range of physicochemical pr0perties.I Research in the areas of blending and copolymerization of polylactones in general, and poly(piva1olactone), PPVL, in particular, has been active for some time.24 The chemical structure of PPVL is H CHsO
[ IiH:
-c-c-c-0-
1”
One major aim of this work is to check certain aspects of the kinetic theory of crystallization with chain folding by using spherulite growth rate, melting point, and X-ray data on PPVL. This theory of chain folding, initiated in 19605 and often termed “LH”, has been extended and refined over the years and accounts for a number of aspects of the crystallization behavior and morphology of highly flexible m a c r ~ m o l e c u l e s . ~According ~~ to this approach, the initial lamellar thickness, fg*, is kinetically determined. An important feature of this model is that the growth rate G varies as exp[-K,/T(AT)], where the nucleation constant K,,which can be determined experimentally with considerable precision, contains the factor bouu,. Here bo is the layer thickness, Dedicated with deep respect to Professor R. W. Zwanzig in recognition of his outstanding contributions to our understanding of the nature of condensed systems. * To whom correspondence should be addressed. *Present address: Dow Chemical Company USA, 2800 Mitchell Drive, Walnut Creek, CA 94598. *Present address: Virginia Polytechnic Institute and State University, Chemistry Department, Blacksburg, VA 2406 1.
0022-3654189 12093-6919$01.50/0 , I
I
-
u the lateral surface free energy, u, the fold surface free energy, AT = Tm0- T, the undercooling, T, = T the isothermal crystallization temperature, and Tm0the melting temperature of a very large extended-chain crystal of the molecular weight under consideration. The quantities u and a, are of fundamental interest. A specific aim of the present study is to determine these quantities for PPVL, beginning with the known value of bouue. In the present work we determine c and u, for PPVL by the following approach. The quantity K , is determined for meltcrystallized PPVL from the crystal growth rates by appropriate plots, and from this one finds bouue. The relationship between bouu, and Kg is certain in the case of PPVL because the crystallization regimes are known (see below). By microbeam X-ray and other studies we determine bo. This permits an accurate value of uue to be obtained. The value of a, was determined in a parallel study using a T,’ vs 1/ I plot,I5 which also gives a reliable estimate
(1) Koleski, J. V.; Lundberg, R. D. J . Polym. Sei., Part A-2 1972, 10, 323. (2) Allegrezza, A. E.; Lenz, R. W.; Cornibert, J.; Marchessault, R.H. J . Polym. Sci., Polym. Chem. Ed. 1978, 16, 2617. (3) Bluhm, T. L.; Hamer, G. K.; Marchessault, R. H.; Fyfe, C. A.; Veregin, R. P. Macromolecules 1986, 19, 2871. (4) Borri, C.; Briickner, S.; Crescenzi, C.; Della Fortuna, G.; Mariano, A.; Scarazzato, P. Eur. Polym. J . 1971, 7, 1515. ( 5 ) Lauritzen, J. I., Jr.; Hoffman, J. D. J . Res. Natl. Bur. Stand., Sect. A 1960, 64, 73. (6) Hoffman, J. D. SPE Trans. 1974, 4 , 315. (7) Lauritzen, J. I., Jr.; Hoffman, J. D. J. Appl. Phys. 1973, 44, 4340. (8) Hoffman, J. D.; Davis, G. T.; Lauritzen, J. I., Jr. In Treatise on Solid Stare Chemistry; Hannay, N. B., Ed.; Plenum Press: New York, 1976; Vol. 3, Chapter 7. (91 Hoffman, J. D. Polymer 1982, 23, 656. (10) Hoffman, J. D. Polymer 1983, 24, 3. (11) Clark, E. J.; Hoffman, J. D. Macromolecules 1984, 17, 878. (12) Hoffman, J. D.; Miller, R. L. Macromolecules 1988, 21, 3038. (13) Hoffman, J. D.; Miller, R. L. Macromolecules, in press. (14) Hoffman, J. D. Macromolecules 1986, 19, 1124. (15) Marand, H.; Hoffman, J. D.; Briber, R. M.; Barnes, J. D., to be
submitted to Polymer for publication.
0 1989 American Chemical Society
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The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
of Tmo.Then u is calculated as uu,./u,. The route just described to find both u and ue is a direct way to obtain dependable values for these quantities, as well as Tmo.It has been employed in the case of polyethylenes and avoids the use of empirical expressions that are sometimes used to estimate u. In fact, one of our objectives is, in light of the known value of u for PPVL as determined above, to determine the constant a in the modified ThomasStaveley equation, which relates u to the heat of fusion and the cross-sectional area of the chain. Both u and a for PPVL are considerably larger than were found for polyethylene. The consequence of this finding are discussed briefly here, and in detail elsewhere.16 The value of the work of chain folding for PPVL calculated from uc is well within the expected range. Appropriate extensions of the L H model of crystallization with chain folding predict well-defined changes of slope in plots of In G vs l/T(A7‘), or appropriate variations thereof.sJO These changes of slope are denoted “regime transitions”. We shall show that PPVL exhibits an unmistakable regime I1 111 rate transition. which theory gives as 2.1° Here we examine the ratio Kg(III)/Kg(I!), In determining the Kg’s,some attention is given to the effect of changes in W and T, in the transport term in the growth rate, T,)]. It will be shown that which v a r i e ~ ~as * ~exp[-U*/R(T~” the ratio Kg(III)/Kg(rI) is affected but little even by rather drastic changes in u* and T,. The effect on K g of variations of Tmois also examined. The overall result is that regime I1 I11 theory is amply confirmed in the case of PPVL. Moreover, the presence of the regime transition renders the numerical relationship between b o w c and Kg free of significant uncertainty. One of the main points of interest concerning PPVL centers on the fact that it has a chain backbone directionality. Thus previous X-ray studies have indicated that the a-form, which is the dominant one in the present experiments (see below), has stems in each lamella that exhibit antiparallel packing; i.e., they alternately point up-down-updown, etc.18 We were interested in seeing if specimens crystallized in regime I1 and regime I11 showed any differences in packing. We shall show here that antiparallel stem packing does indeed occur in the a-crystal structures in both regimes I1 and 111. This information, when combined with theoretical considerations based on a version of the “gambler’s ruin” calculation appropriate to antiparallel packing, provides insights concerning the nature of the chain-folded surface in each regime, particularly with respect to the degree of “tight” folding. Additional information concerning the fold surface is provided by the presence in certain specimens of active cleavage planes that lead to cracks in the crystals. Three crystal structures of PPVL have been identified. Two variations, a and y , are obtained by melt crystallization under quiescent condition^,'^ while a third one, @, is obtained20 by cold stretching. We note that the a-phase is the only one found at high T,, and the main one found at the lowest T, employed in this research. Thus the a-phase is dominant between 180 and 218 “C,which range covers both regimes I1 and 111. The crystal structure of the a-phase was determined by Carazzoloisa and Perrego et a1.lSbwith later refinements by Briickner et a1.21 Both the a-and y-phases exhibit the antiparallel chain packing mentioned above.19 The @-phase,which is formed by cold stretching, exhibits essentially random chain packing. It is emphasized that in this work we have concentrated our efforts on the dominant a-phase spherulites formed by isothermal crystallization in regimes I1 and 111.
-
-
Roitman et al.
where i is I, 11, or 111, and AT = (Tmo- T,). The preexponential factor G, has only a weak dependence on temperature. The factor exp[-W/R( T - T,)] accounts for the temperature dependence on the transport or “reeling in” of the polymer chain through the subcooled melt to the liquid-solid interface.’@12 In general, the Kgi are given
YI = 4; Y*I= 2; YIII = 4 (2b) The quantities u, uc, and bo have been defined previously. The symbol k is the Boltzmann constant, Ahf the enthalpy of fusion, and 3(T ) = 2T/( Tmo 7‘) is a correction* for the temperature dependence of Ahf. The value of Y depends on the relationship between the surface nucleation rate i and the growth rate G: justification of the values in eq 2b is discussed for lamellar systems in ref 8-10. It is convenient to write eq 1 in logarithmic form:
+
Experimental values for the Kg(,)’scan be obtained from the slopes of a plot of the left-hand side expression vs 1/T(AT)3. In principle, such a plot could show two slope changes, corresponding to Kg(III)/Kg(II)
-
=2
-
Kg(IdKg(11) =
2
(4a) (4b)
In practice, both the I I1 and I1 111 transitions have been reported in the same polymer only for melt-crystalized PE,l0 cis-polyisoprene,22and poly(3,3-dimethylthietane).23 The I1 111transition has been found in other melt-crystallized polymers and including it-poly(propylene),l poly(3-hydroxyb~tyrate),~~ I1 transition has been poly(p-phenylene s ~ l f i d e ) . ~ ’The I reported in melt-crystallized poly( 1,3-dio~olane),~ and in melt-8-27 and solution-crystallized28polyethylene, and in melt-crystallized poly(L-lactic acid).29 In order to plot eq 3 for PPVL, Tmo,u*,and T , must be either estimated or measured. Though there are some exceptions, it is commonly found thats T , z Tg - 30 K, where Tg is the glass transition temperature of the polymer. The activation energy W is ordinarily abouts 1500 cal mol-’. We shall demonstrate that, in the range of crystallization temperatures considered in this work, uncertainties in W and T, in the transport term have negligible effects on Kg(i).Uncertainties in Tmo,on the other hand, can have a larger effect on KgCi), but this can be minimized by appropriate measurements (see below). The value of Tmois usually obtained from either one of two well-known methods. The first method is based on the fact that the melting temperature T’, of lamellar crystals of thickness I is given by the thermodynamic relation8
-
-
11. Growth Rate and Melting Point Expressions The rate of isothermal growth Gi in regime I, 11, and 111 of
in which it is assumed that the other lamellar dimensions are much
spherulites or axialites is given by the expression8-I0
(22) (23) 2076. (24) (25) (26) (27)
(16) Marand, H.; Roitman, D. B.; Hoffman, J. D., to be submitted to
Polymer for publication. (17) Suzuki, T.; Kovacs, A. J. Polym. J. (Tokyo) 1970, I , 82. (18) (a) Carazzolo, G. Chim. 2nd. [Milan] 1964,46,525. (b) Perego, G.; Melis, A.; Cesari, M. Makromol. Chem. 1972, 157, 269. (19) Meille, S . V.; Konishi, T.; Geil, P. H. Polymer 1984, 25, 773. (20) Clark, E. S. Bull. Am. Phys. SOC.1979, 24, 479. (21) Bruckner, S.; Meille, S. V.; Porzio, W. Polymer 1988, 9, 1586.
Phillips, P. J.; Vantansever, N. Macromolecules 1987, 20, 2138. Lazcano, S.; Fatou, J. G.; Marco, C.; Bello, A. Polymer 1988, 29,
Barham, P. J.; Keller, A.; Otun, E. J. Mater. Sci. 1984, 19, 2781. Lovinger, A. J.; Davis, D. D.; Padden, F.J. Polymer 1985, 26, 1595. Alamo, R.; Fatou, J. G.; Guzman, J. Polymer 1985, 26, 1595. Hoffman, J. D.; Frolen, L. J.; Ross,G.S.; Lauritzen, J. I., Jr. J. Res. Natl. Bur. Stand., Sect. A 1975, 79, 61. (28) Organ, S. J.; Keller, A. J. Polym. Sci., Part B: Polym. Phys. 1986, 24, 2319. (29) Vasanthakumari, R.; Pennings, A. J. Polymer 1983, 24, 175.
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6921
Crystallization of Poly(pivalo1actone) larger than the lamellar thickness I. A plot of T,' vs 1/1 has intercept Tmoand slope -2uCTmo/Ahf,from which a, can be determined if Ahf is known. In this way Marand et al.15 determined that Tmo= 269 "C (542.2 K) f 2 "C and uc = 58 erg cm-2. The second procedure is the familiar T,' vs T,. plot,30 based on the assumption that the lamellar thickness in the grown spherulites I is approximately proportional to the kinetically controlled "initial" lamellar thickness Ig* according to I ylg*, where y > 1. (The fact that y > 1 is the result of isothermal thickening.) When the small contribution 6 in 1,. = 2ucT,/ (Ahf)(AT) 6 is neglected, it has been shown thatasso
=
+
T,' = T,O
[
1-
:I
-
'I'
+-
Under the additional implicit assumption that y is constant or nearly so in the T,' vs T, plot, the intersection of the linear least-squares fit of T,' vs T, with the 45" line (T,' = T,) gives an estimate of Tm0.(Because of the approximation that y is constant, the T,' vs T, plot is inherently less accurate in producing a reliable value of Tm0than is the T,' vs 1/ I plot.) Mostly because of varying values of y, particularly in regime 11, the T,' vs T, method for the case of PPVLI5 leads to curved lines making difficult an accurate extrapolation to Tm0. Once the regimes have been established, the growth rate data can be used to obtain uue by using eq 2 provided that Ahf and bo are known. Borri and co-workers4 have determined Ahf and values of bo and a. (the stem width) are determined for PPVL in section IV. From this, u can be determined as uue/ue and the work of chain folding q can be calculated froma q = 2aobouc (7) In all published X-ray diffraction studies of the a-crystal structure, there is agreement with a T2G2 chain conformation resulting in antiparallel 2l helix structure, as proposed by Carazzo1o.l" The unit cell is monoclinic with dimensions1aba = 0.905 nm, b = 1.158 nm, c = 0.603 nm, and y = 121.5" (y here is the angle between a and b). There are two chains per unit cell. Details of the positions and directions of the chains will be subsequently described after the X-ray microcamera experiments are discussed. As noted earlier, we shall find here that the a-crystal form is dominant in isothermal growth between T, = 180 "C and T, = 218 "C. The y crystals form less birefringent spherulites at T, < 180 "C; these grow slower than those belonging to the aphase. Pratt and Geil" showed that the a-crystal form is also dominant in samples that are crystallized from the glassy state, Le., at very large undercoolings. They indicated that the electron diffraction patterns of the a and y phases are consistent with antiparallel 21 helical chains, with little evidence of random directions between adjacent chains. On the other hand, the chains in the &crystal modification, which is formed by cold drawing, have nonhelical zigzag conformations and random chain directiomm This latter molecular morphology has suggested to previous investigators that a high fraction of adjacent or very near adjacent folds present in the original undrawn samples were removed by the cold-drawing p r o ~ e s s . ~ ~Certain J ~ - ~ ~details of the positions and directions of the chains will subsequently be described after the X-ray microcamera experiments are discussed.
-
111. Experimental Section
The PPVL samples were prepared from a batch synthesized by the Tennessee Eastman Co. and were kindly supplied to us by Professor P. H. Geil, University of Illinois. Spherulites as large as 2 mm in diameter were easily obtained on the hot stage with the material as received. Additional sample cleaning was considered to be unnecessary. The samples for optical microscopy (30) Hoffman, J. D.; Weeks, J. J. J . Res. Natl. Bur. Stand., Secr. A 1962, 66,13. (31) Pratt, C. F.;Geil, P. H. J . Macromol. Sci. Phys. 1982, B21, 617. (32) Geil, P. H. Faraday Discuss Chem. SOC.1979, No. 68, 440. (33) Prud'homme, R. E.; Marchessault, R. H. Macromolecules 1974, 7, 541. (34) Cornibert, J.; Marchessault, R. H. Macromolecules 1975, 8, 296.
TABLE I: Growth Rates of a-Phase PPVL Spherulites Tx,a"C G,bcm/s Tx! "C G,b cm/s Tx! "C 180.0 7.93 X lo4 197.5 1.90 X lo-' 210.0 185.0 3.60 X lo4 200.0 9.50 X 10" 213.0 190.0 1.37 X lod 202.Y 4.02 X 10" 215.0d 192.5 7.25 X 10" 205.0' 2.35 X 10" 218.0d 195.0 3.65 X 207.5 1.22 X 10"
G,bcm/s
6.38 3.88 2.18 9.90
X X lo-' X X lo-*
-
a Isothermal crystallization temperature. 'Radial growth rate. CThe 111 transition occurs at 203 OC. dThe morphology for regime I1
these samples is axialite- or lathlike.
studies were prepared as follows. They were degassed by melting a few milligrams of PPVL between a clean slide and a cover slip at 255 "C in a vacuum oven for short times (1-2 min). After reaching room temperature under vacuum, the samples were melted again on top of an auxiliary heating plate in an N z atmosphere and rapidly brought into a Mettler FP2 hot stage which had been preset at the desired crystallization temperature T,. Optical micrographs of the growing crystals were obtained with a 35 mm camera mounted on a Leitz Dialux polarizing microscope. A Nikon photo flash and an Olympus intervalometer were also added to the system for precision timing and unmanned photography. The radii of the spherulites were measured directly on the 35 mm films using a Nikon profile projector. The calibration of the hot stage was checked by melting indium ( T , = 156.6 "C) and tin ( T , = 231.96 "C) at the heating rate of 2 "C/min. The observed melting changes appeared well within 1 "C of the reported melting points. The reproducibility and stability of the hot stage was checked by comparing the kinetics of repeated PPVL growth cycles, and from the linearity of plots of spherulites radii vs time. Good temperature stability of the hot stage was observed in the entire range of times and temperatures of the experiments. Cyclic growth rate reproducibility also indicated that thermal degradation, if present, did not affect the results. A t lower temperatures, T, < 170 "C, on the other hand, the spherulites grew rapidly and impinged one another in short times (e.g., 40 s at T, = 160 "C) so that reliable growth rate measurements were not possible. At the same time it was difficult to achieve thermal equilibration, and self-heating effects could not be ruled out. For these reasons, neither the crystallization behavior of the a-form for T, < 180 "C nor the behavior of the y-form, which appears when T, < 180 "C, were included in the present communication. A wide-angle X-ray diffraction technique (WAXD) was used in the transmission mode with nickel-filtered Cu K a radiation for detecting changes in the crystal structure of the a-phase spherulites isothermally crystallized at 190.6 "C (regime 111) and 207.6 OC (regime 11). A Philips microcamera was used for the determination of the crystallographic planes of growth in the radial direction of the spherulites. Since the spherulite radii (- 1 mm) were many times larger than the bore of the microcamera (- 50 Km), fiberlike diffraction patterns of radially oriented lamellar bundles were readily obtained. The mean orientation of the crystal axes with respect to the radial direction could be determined to within about *loo. IV. Results A. Growth Rates. The growth rate vs crystallization temperature data are given in Table I and plotted in Figure 1. In Figure 1, the solid circles are in regime I1 and the open circles are in regime 111. Figure 2 shows a plot of log G U*/2.303R(T - T,) vs 1/(T(AT)3) assuming T , = ( T g- 30 K) = -33 "C = 240.2 K and W = 1500 cal/mol. The value Tm0= 269 "C = 542.2 K was employed. The K g values and the ratio Kg(III)/Kg(II) derived from the plot in Figure 2 are shown in Table IIA for with various assumed values of Tm0.The variation of Kg(III)/Kg(II) W and T , is shown in Table IIB; the ratio is seen to be very insensitive to even rather large changes in W and T,. The slopes, are, however, somewhat more sensitive and their ratio, Kg(III)/Kg(II), to the value of Tmoas indicated in Table 11. The data show a transition between regime I11 crystallization at large undercoolings and regime I1 at lower undercoolings. This
+
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The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Roitman et al.
n
Tx(-I -1
0 0 -.?
t
I
190
200
210
218
I
I
I
I
I
-4
25
I
I
35
1
45
lOW(A71 a Figure 2. Plot of log G + Lr/2.303R(T - T,) vs 1/T(AT)3 for PPVL growth rate data given in Table I (G in cm s-l). The values Tm0= 269 O C = 542.2 K, Lrc = 1500 cal mol-', and T, = T, - 30 OC = -33 " C = 240.2 K were employed. Solid circles, regime 11; open circles, regime 111.
rates were measured. Spherulitic structures were found at all but the highest temperatures, where axialite-like (and occasionally T, ( " C ) lathlike) objects were seen (Figure 4). The lathlike morphology and the more representative axialite-like crystals (Figures 3d and Figure 1. Growth rate in cm s-l of crystalline objects in PPVL as a 4) were observed in regime I1 crystallization; this is interesting, function of isothermal crystallization temperature (a-crystal form). Solid circles, regime 11; open circles regime 111. Solid lines calculated for since it is sometimes assumed for crystallization from the melt regimes I1 and 111 with eq 1 using "best" values from Table 11. that axialitic and single-crystal-like structures are representatives only of regime I. All these structures were in the a-crystal form. TABLE 11: Values of Nucleation Constants for PPVL in Regimes I1 The PPVL samples contained very low concentrations of efand 111: (A) Ks(II),K,,), and K m)/K,(n) for Various Assumed fective nucleating particles. As a consequence, large a-phase ) Variations in U sand T, Values of Tma;"(B) Ratio K g ( m ) h s ( nfor spherulites were formed in the range of temperatures investigated. for Tmo= 269 "C It was obvious that the spherulites were heterogeneously nucleated, though short delay times were noted. Characteristic delay times A could not be statistically determined since usually very few spherulites could be seen in the field of the microscope. At T, < 180 "C it was evident that heterogeneous nucleation of the 2696 4.32 x 1056 8.59 x 1056 1.996 y-crystal form spherulites began to occur frequently, while the 27 1 4.67 x 105 9.11 x 105 1.95 a-crystal spherulites that were also present grew at a faster rate. B The a-form spherulites also nucleated on the growing edges of the y-spherulites, obstructing the growth of the latter. At higher temperatures, only the a-spherulites were present. On quenching to room temperature, the axialite-like crystals 1.98 1.99 1500 formed in regime I1 at T, = 218 "C exhibited cracks that were 2.00 2.04 3000 often more or less parallel to the growth front (Figure 4), sug2.00 4120 gesting that chain folding takes place parallel to that front. These Lr = 1500 cal mol-!; T , = T, - 30 OC = 240.2 K. The value T, = cracks formed spontaneously on cooling; they were not present -3 "C from ref 31 was employed in estimating T,. bBest values. The at the isothermal crystallization temperature. Lathlike crystals error in each K , is about 5% if Tmois within *1.5 " C of 269 OC. The were occasionally observed at these high growth temperatures. value of Go(,,,is 1.5 X lo2 cm s-l and of GO(III) is 4.3 X 10' cm s-'. These exhibited a similar crack structure on quenching (not 'For T , = TB- 30 OC = 240.2 K. dFor T, = T, - 51.6 OC = 218.6 K. shown). Spherulites formed at high temperatures in regime I1 that were then quenched to room temperature also exhibited is indicated in Figure 1 but is more clearly evident in Figure 2. similar cracks in the form of concentric rings (not shown). 111 transition is a t 203 f 1 "C and is clearly present The I1 Spherulites formed at a low temperature in regime 111 ( T , = 185 irrespective of the particular values chosen for W , Tmo,and T , "C) and quenched to room temperature did not exhibit cracks. within a reasonable range. A spherulite grown partly at 210 "C (regime 11) and partly at With the preferred Tmoof 269 "C, the least-squares lines in 190 "C (regime 111) exhibited cracks on quenching to room Figure 2 give Ke(III)= 8.59 X lo5 K2, K g ( l ~=) 4.32 X lo5 K2, and temperature only in the portion grown in regime 11. The presence Ke(III)/Ke(I~) = 1.99 f 0.05. A value of Tmo= 269 f 2 OC could of spherulites with circular crack patterns in PPVL is not an have been inferred directly from the kinetic data by assuming that isolated case. For example, Barham and Keller found such crack Ke(III)/Ke(II) = 2 and treating Tmoas a variable. This approach patterns in spherulites of poly(3-hydroxyb~tyrate).~~ can be used as an alternative method to determine Tmo,with the C. Determination of Fold Plane and Layer Thickness bo: proviso that the slopes be insensitive to W and T,. Cleavage Planes. In samples crystallized in regime 111, the It is concluded that PPVL exhibits a classical I1 I11 transition that closely obeys the theoretical predictionlo that K e ( ~ ~ ~ ) / K e ( l l )microcamera WAXD patterns (Figure 5a) showed t h e (120) planes to be essentially perpendicular (within 10") to the radial = 2. This lends added confidence to the values of and Ke(lll) direction of the spherulites (at T, = 185 "C). The (020) reflection that will be employed subsequently to obtain values of boas, and was split into two arcs symmetrically placed on either side of the thence u, a,, q, and a. B. Morphological Aspects. Figure 3a through 3d shows typical optical micrographs of the crystalline objects found in the tem(35) Barham, P. J.; Keller, A. J . Polym. Sci., Polym. Phys. E d . 1986, 24, perature range ( T , = 180 "C to T, = 218 "C)where the growth 69 175
185
195
205
215
-
-
*
Crystallization of Poly(pivalo1actone)
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6923
-
Figure 3. Spherulite and axialite-like structures in PPVL (a-form). Regime I11 cases: (a) T, = 195 "C, (b) T, = 200 "C. Regime I1 cases: (c) T, = 210 "C, (d) T, = 218 "C. The I1 I11 regime transition occurs at 203 "C (AT = 66 "C).
(120) arc which is consistent with a radial orientation of the (120) normal coupled with lamellar twisting around the radius commonly encountered in spherulitic structures (cf., the reciprocal lattice diagram in Figure 5b). On the other hand, since the angle between the (120) and (100) planes is less than 8', detectable splitting of the (100) reflection into two arcs was not expected or seenbasically the (100) reflection occurred as an arc roughly perpendicular to the 120 reciprocal lattics direction. In the original X-ray photograph, traces may be seen also of reflections from (101) and (121) planes (based on the assignment of Perego et a1.18b).These reflections are located appropriately according to the reciprocal lattice diagram of Figure 5b. This orientation information indicates that growth in the acrystal form takes place predominantly by the successive addition of stems on (120) planes (Figure 6). This finding is clearly compatible with the alternating "up-down" directions of the chain axes on these planes as proposed by Carazzolo18aand Cornibert and M a r c h e ~ s a u las t ~ indicated ~ in Figure 6. Alternatively, one might try to assign the chain-fold plane to (020), the only other viable possibility based upon the WAXD data. In that case, the growing lamellae would be chisel-shaped with an unacceptable large angle of roughly 50' between the growing face and the spherulite radius. Additionally, such a plane would not satisfy the up-down requirements of the space group. Accordingly, one may confidently conclude that the 120 planes are the principal "fold planes". In PPVL specimens crystallized at high T, in regime 11, it was more difficult to establish the direction of the bundle axes by WAXD because of effects attributable to twinning and to a more "single-crystal" character of the crystalline bodies. Nevertheless,
the results were generally consistent with the assignments of (120) as the fold plane. On this basis, we interpret the cracks exhibited by axialite-like crystals (Figure 4) and by spherulites grown@ regime I1 as being associated to a substantial degree with 120 cleavage planes. As will be noted later, the presence of such cracks implies the presence of rather regular chain folding in regime 11. Recall that such cracks do not occur in material crystallized in the main body of regime 111. Accepting that chain folding in the a-crystal form takes place along the 120 planes as shown in Figure 6, the thickness of the molecular layer bo and the molecular width a. are found to be bo = ( b / 2 ) sin 8 = dp0 = 0.574 nm and u0 = area/bo = 0.779 nm where 8 = 82.17' is the angle between the 120 and 100 planes, and the area is aobo = 0.447 nm2. D. Determination of aa,, a, q, and a. With the above value of bo,the preferred value of Tmo,the corresponding values of Kg from Table IIA, the value Ahf = 1.83 X lo9 erg cm-3 as determined by Borri et a1.: and the definition of the Ke)s of eq 2a, one obtains aa, = 1753 erg2 cm4 for regime I1 and aa, = 1743 erg2 cm4 for regime 111. Accordingly, we employ the mean value aa, = 1748 erg2 cm4 in what follows. The value a, = 58 erg cm-2 was determined independentl~'~ as mentioned. Thus, one obtains a = aa,/a, = 30 erg cm-2. Judged in comparison with the case of polyethylene, this is an unusually high value. From the value of a,, the work of chain folding is q = 2aoboa, = 5.2 X erg fold-' = 7.5 kcal mol-'. This is a normal value for a work of chain folding, being only slightly larger than that of polyethylene. This work of chain folding represents an average of adjacent and very near adjacent tight folds.
6924
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Roitman et al.
Figure 4. Axialite-like crystal grown at T, = 218 "C and quenched to room temperature (a-form, regime 11). The longest dimension of the axialite is 0.2 mm. Note cracks approximately parallel to growth front in many parts of specimen. The growth front corresponds to the 120 crystallographic plane, i.e., the plane along which folds form during substrate completion. The 120 plane forms a natural cleavage plane in the crystal interior provided that nonadjacent events are minimal.
To complete the calculations, we employ the deduced value of a to obtain the dimensionless empirical constant a in the modified
Thomas-Staveley equation appropriate to polymers:*
The value of a for PPVL is much larger than that for polyethylene, which is nearer to 0.1. Table I11 provides a comparison of aa,, a,, a, q, and a for PPVL and polyethylene. More detailed remarks concerning the significance of the large a and a found for PPVL are reserved for the Discussion. It suffices at this juncture to note that there appear to be two classes of polymers, one epitomized by PPVL, which has an a centering 0.25 f 0.03, and another class typified by polyethylene that exhibits an a in the vicinity of 0.1. E. WAXD Patterns of a-Phase Spherulites in Regimes 11 and III. Wide-angle X-ray diffraction (WAXD) patterns for PPVL for a-phase spherulitic specimens crystallized in regime I1 and regime I11 are given in Figure 7. These specimens were crystallized at the T,values noted to a high degree of crystallinity and then cooled to room temperature where the diffraction patterns were obtained. The complete absence of (hkO) peaks with k odd for the specimens is consistent with a very high degree of antiparallel packing in both regimes I1 and 111. The two most prominent peaks, (100) and (lzO), are quite similar for T, = 190.6 "C and T,= 207.6 "C,but significant line broadening occurs in the regime I11 ( T , = 190.6 "C) specimen for the other peaks. This broadening appears to be largely associated with lines involving the c axis of the unit cell., i.e., 01 1, 111, 101, 121, 021, 111, 211, and 221. Probably much of this line broadening results from the fact that the lamellar thickness is smaller at T, = 190.6 O C than for T, = 207.6 "C. It is evident from the diffraction patterns that the specimens are highly crystalline.
Figure 5. Determination of growth plane in a-phase PPVL spherulites (T, = 185 "C) with WAXD microcamera. Upper: Diffraction pattern with direction of spherulite radius shown by arrow. Lower: Reciprocal lattice corresponding to WAXD pattern. The mean positions of the strongest reflections present in the X-ray pattern are indicated by the solid circles. The growth plane is 120.
Figure 6. Molecular projection in ab plane of a-phase PPVL (after Cornibert and M a r c h e ~ s a u l t ~ ~The ) . symbols +, -, +, -, ... represent alternating "up" and "down" directions of the polymer chains in the 120 growth plane. In the ideal folded structure, folds connect the + -, + -, .... sites on one side of the crystal, and - +, - +, .... sites on the obver_se side. Growth at a rate G takes place by addition of stems on the 120 plane, these stems creating a new layer of thickness bo = d12,-,.
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6925
Crystallization of Poly(pivalo1actone)
TABLE 111: ComDarison of Nucleation Theory Parameters for PE and PPVL polymer H
CH,O
I I II
Tma,OC
Ahf, erg/"
uc, erg/cmz
me,erg2/cm4
269 f 1.5
1.83 X
145.5 f 1
erg/"
a
q, kcal/mol
lo9
58
1748
30
0.25
7.5
2.80 & lo9
90
1063
11.8
0.097
4.9
u,
PPVL. +c-c-c-o*
I I
H
CH,
H H
PE.'-(-
I I
C-C%
t i
H H
"The values of me,u, ue,and q for polyethylene are summarized from ref 12. References to the original sources are given in the same publication.
T, = 190.6 ' C
-
(2 e)
Figure 7. WAXD patterns for a-phase PPVL spherulites at T, = 190.6 O C (regime 111) and T, = 207.6 O C (regime 11).
-
V. Discussion and Conclusions The dominant a-form of PPVL exhibits a typical regime I1 I11 transition that closely obeys theoretical expectation,I0 Kg(II1)/Kg(I1) being 1.99 f 0.05. This result covers all the values in Table 11, including those resulting from variations in Tmo,u*, and T,. The fit of the growth rate data in both regimes I1 and I11 corresponds closely to eq 1 (see Figure 1); that is, the data accord well with the growth rate expressions derived on the basis I1 regime of the kinetic theory of chain folding. A regime I transition was not detected, most likely because the highest growth temperature employed (218 "C) still corresponded to the quite I1 transitions large undercooling of AT = 51 O C ; regime I generally occur only at considerably lower undercoolings. The presence of the I1 I11 transition permitted an unambiguous determination of the value of Y for each regime in eq 2a. Consequently the K g ( I ~ and l K g ( I Ivalues I) led to accurate values of beau,. The absolute Kgvalues were only weakly dependent on changes in u* and T,, and only moderately on Tmo(Table 11). Microbeam X-ray data, together with crystallographic information from the literature, permitted a positive identification of the lattice plane associated with chain folding, which proved to be 120. The layer thickness, bo,was 5.74 A. The 120 plane (which is the most densely packed plane) appears to correspond to a cleavage plane, as suggested by the presence of macroscopic cracks in approximately this orientation in the crystals in quenched specimens (Figure 4). From the known bo, a value of uu, of 1748 ergZcm4 was found. It is believed that this figure is correct within about
-
-
5%.
-
From this value of uu, known from the kinetic studies, and the value of u, determined by a thermodynamic method," it was determined that u for PPVL was close to 30 erg cm-*. This is seemingly a quite high value. The corresponding value of a in the modified Thomas-Staveley equation proved to be 0.25. Table 111 shows a comparison of u and a for PPVL and polyethylene. We regard the difference in the a values to be definite. Elsewhere it has been shown that there are in fact two basic classes of crystalline polymeric systems, one of which has an a centering about 0.25 f 0.03, and another which has an a centering about 0.10 f 0.02.16 The dipolar polyesters with a distinct chain direction, namely PPVL, poly(3-hydroxybutyrate), and poly(L-lactic acid), all exhibit the higher a. (In one sense this higher a is not unusual since it corresponds closely to that found for many small organic molecules.I6) Nonpolar or weakly polar polymers based on the substituted 44-backbone structure such as polyethylene, poly(chlorotrifluoroethylene), and it-poly(styrene) all exhibit an a of close to 0.1.l6 Thus the details of the physics associated with the lateral surface free energy u are evidently somewhat different in the two classes. The magnitude of u as well as a is clearly dependent on chain structure. The fact that u tends to be considerably larger in one class than in the other in no way obviates the applicability of nucleation theory to either class. It is clear that both polyethylene and PPVL conform closely to the basic predictions of the kinetic theory of chain folding. However, it is evident a t the same time that caution is required in making estimates of u using the empirical modified ThomasStaveley expression. If a 0.1 had (erroneously) been employed to estimate u for the analysis of the PPVL rate data, the absurd value q 18.5 kcal mol-' would have resulted. The work of chain folding of 7.5 kcal mol-I for PPVL is close to what one would expect. It is somewhat higher than that of polyethylene (Table 111), PPVL having a rather stiffer chain as suggested by its higher melting point. We observe however that the relatively slow growth rate of PPVL as compared to polyethylene is not to be attributed mostly to the higher work of chain folding, but rather more so to its considerably larger u contribution to Kg. The changes in morphology in PPVL correspond to the normal trend where spherulites form at low growth temperatures and axialite-like or lathlike structures appear at high temperatures (Figures 3 and 4). Observe, however, that axialite- and lathlike structures appear in regime I1 in PPVL, whereas axialites appear in polyethylene only at quite low undercoolings in regime I. The changes in morphology in PPVL as observed optically are gradual as the system passes through the I1 I11 transition near 203 OC. Regime theory does not demand an abrupt change in macroscopic morphology at either the I1 I11 or the I I1 regime transitions. We consider now information pertaining to the nature of the chain-folded surface. DSC measurements have shown that specimens of PPVL crystallized to essential completion in regime 111 exhibited a somewhat lower degree of crystallinity than those crystallized to completion in regime 11; the crystallinities ranged from -0.7 to -0.85.15 The implication is that there is somewhat more disorder in the fold plane in regime I11 than in regime 11. While the existence of an amorphous component in a lamellar system provides indisputable evidence for the occurrence of some nonadjacent reentrant events (e.g., a traverse of an emergent chain to a distant point in the same lamella, or interlamellar links), it
=
=
-
-
-
6926
J . Phys. Chem. 1989, 93, 6926-6928
is equally certain that a relatively high crystallinity ensures that very considerable tight folding must also be present. Regions of the lamellar surface involving tight folds, many of which are strictly adjacent, do not contribute to the amorphous component (see below). Thus the crystallinity measurements imply an even higher degree of tight folding in regime I1 than in regime 111. This is in accord with expectations based on nucleation theory, regime I11 clearly presenting the maximum opportunity for nonadjacent reentry with its concomitant amorphous component.1° It is relevant to emphasize that there exist definite and irreducible lower bounds of topological origin on the degree of tight folding. The special point here is that more tight folding is required to avoid a density paradox at the fold surface in the case of systems with antiparallel chain packing than for the case where there exists no chain direction. In the latter case, the gambler’s ruin and related calculations3638 demonstrate that a lower bound of about 2/3 of the emergent stems must reenter a lamella as tight folds, whereas in the case of antiparallel packing, M a n ~ f i e l dhas ~ ~recently shown in a corresponding calculation that a lower bound of 5/6 of the emergent stems must reenter as tight folds in order to prevent a density paradox. (A tight fold involves a short traverse to an adjacent or very near adjacent site that has no normal amorphous character. For normal fold energies, approximately half or more of the tight folds involve strictly adjacent reentry.38 The mixture of strictly adjacent tight folds and very near adjacent tight folds at the lamellar surface occurs within a quite thin (ca. 10-15 A) boundary layer.38) The WAXD results leave little doubt that a high degree of antiparallel chain packing exists in the a-form of PPVL in both regimes I1 and 111. The foregoing supports Geil’s contention32 that antiparallel chain packing in melt-crystallized PPVL tends to enhance the probability of regular chain folding with considerable adjacent reentry. It is necessary to point out, however, that the gambler’s ruin restrictions are somewhat relaxed when the chain is helical in the crystal as it is in the a-form of PPVL. Thus somewhat less tight folding might be expected because of this factor. The results to be discussed below suggest that in PPVL this effect is at least balanced, and probably more than balanced, by the antiparallel chain packing effect that promotes tight folding. Another way of visualizing the origin of (36) Guttman, C. M.; DiMarzio, E. A., Hoffman, J. D. Polymer 1981.22, 1466. (37) Guttman, C. M.; DiMarzio,
E. A. Macromolecules 1982, 15, 5 2 5 . (38) Mansfield, M. L. Macromolecules 1983, 16, 914. (39) Mansfield, M. L. J . Phys. Chem., following paper in this issue.
an extra tendency for adjacent reentry in PPVL is to note that the nearest site accessible to an emergent chain is at the adjacent “niche”, many segments of the emergent chain being present at this energetically favorable position. In a system with antiparallel chain packing, however, the next-nearest site is forbidden to the emergent chain, and moreover, is not energetically favored because of the absence of a niche. Another indication that rather regular chain folding can occur in melt-crystallized PPVL is afforded by a consideration of the cracks that appear in the crystals on quenching after crystallization at high temperatures in regime I1 (Figure 4). The implication is that many of these cracks represent active cleavage planes corresponding to the 120 fold plane (Figure 6). (Recall that the growth front is itself 130.) From the failure of cracks to appear in spherulites formed at a low temperature in regime I11 ( T , = 185 “C) and then quenched, we infer that crystallization in this regime leads to less regular folding than that obtained in regime 11, where such cracks do appear. This trend in fold surface perfection with the regime of crystallization (and thus crystallization temperature) is in accord with theoretical expectationL0 and is also consistent with the conclusion drawn above on the basis of crystallinity measurements. Finally, we observe that the cracks that appear in PPVL crystallized in regime I1 and then cooled to room temperature may well have an important bearing on the mechanical properties of the polymer. It is clear that crystallization in regime 111, where such cracks do not appear on cooling, leads to a somewhat larger number of nonadjacent events. These events result specifically in an increased number of molecular connections between the layers of thickness bo and most importantly a larger number of interlamellar links. Mainly because of the latter, the implication is that tougher polymer would be obtained by processing PPVL at a temperature corresponding to regime I11 rather than one corresponding to regime 11. Acknowledgment. Thanks are due to Dr. M. L. Mansfield for helpful discussions and for making his calculations on chain reentry in lamellar systems with antiparallel packing available to us prior to publication. Also we are grateful to K. P. Battjes for obtaining the microbeam WAXD data and to M. F. Rozniak for the photographic work. This research was supported in part by grant DMR 86-07707, Polymers Program, Division of Materials Research, National Science Foundation. Registry No. PPVL (homopolymer), 24969-1 3-9; PPVL (SRU), 24931-5 1-7.
Gambler’s Ruin Model of Semicrystalline Polymer Systems with Antiparallel Chain Packingt Marc L. Mansfield Michigan Molecular Institute. 1910 W. St. Andrews Road, Midland, Michigan 48640 (Received: February 7, 1989)
The simple cubic lattice version of the gambler’s ruin model of semicrystalline polymer systems has been modified to treat the case in which polymers having a specific head-to-tail sense crystallize in such a way that neighboring stems in the crystal lie in alternating directions. The amount of adjacent reentry predicted by the model increases from 2/3 to 5 / 6 . The long-range properties (Le., net fraction of tie chains, average length of loops and ties) of the model remain unchanged to leading order in the domain thickness.
Introduction A number of lattice models of the amorphous domains of semicrystalline polymers have appeared in recent years.I4 These models vary in the sophistication with which chain packing is ‘Dedicated to Prof. Robert Zwanzig on the occasion of his 60th birthday.
0022-3654/89/2093-6926$01 S O / O
treated. The simplest, and therefore most mathematically tractable of these models, are sometimes called “gambler’s ruin” (1) Guttman, C. M.; DiMarzio, E. A.; Hoffman, J. D. Polymer 1981, 22, 1466. (2) Guttman, C. M.; DiMarzio, E. A. Macromolecules 1982, 15, 5 2 5 .
0 1989 American Chemical Society
