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
Gambler’s Ruin Model of Polymer Systems The gambler’s ruin models treat chain packing only by requiring the amorphous domains to have the correct, uniform density. Other mean-field models take more detailed account of the local structure,+8 while the most sophisticated treatment of the packing is afforded by Monte Carlo c a l c ~ l a t i o n . ~ These models consider random walks between the two planes z = 0 and z = M + 1 for M some integer. The region 0 < z < M 1 represents the amorphous domain layer, and the regions z < 0 and z > M + 1 represent the two crystalline domains lying on either side of the amorphous domain. The two planes z = 0 and z = M 1 represent, therefore, the crystal-amorphous interface. Consider a crystalline stem lying in the crystalline region below the z = 0 surface. The sequence of points ..., (0, 0, -2), (0, 0, -l), (0, 0, 0) represents this crystalline stem. When the stem reaches the crystal-amorphous interface at z = 0, we assume that it is followed by a noncrystalline sequence of one of three types: 1. Strictly-adjacent loops. The chain steps from (0, 0, 0) to ( f l , 0, 0) or (0, f l , 0). It then begins a new crystalline stem going back down into the crystal. These loops are assumed to have unit length, and make no contribution to the amorphous layer. 2. Amorphous loops. The chain enters the amorphous domain by stepping to (0, 0, 1). From there it performs a random walk, terminating when it reaches the z = 0 plane at some point (i, j , 0). The total length of the chain is assumed to be the number of steps performed in going from (0, 0, 0) to (i, j , 0). 3 . Ties. The chain also steps to (0, 0, 1) and begins a random walk, but terminates at the opposite face, at some point ( i , j ,M+1). The total length is defined as the number of steps in going from (0, 0, 0) to (i, j , M+1). Amorphous loops and ties are collectively referred to as amorphous walks. Strictly-adjacent loops and amorphous loops are collectively referred to as loops. Obviously, similar definitions hold for walks entering the amorphous domain from other points on the z = 0 plane, and also from the opposite side of the layer. The statistical properties of these different types of walks are of interest in all these models. Many polymers, unlike polyethylene, have a specific head-to-tail sense. Many of these crystallize with adjacent stems lying in antiparallel directions. This fact places restrictions upon the types of chain folds that are possible, since any particular molecule must, of course, maintain its head-to-tail sequence. Within the context of the simple cubic lattice models, we can treat this effect by assuming that each of the two planes representing the crystalamorphous interface has a chessboard coloring, and that chains leaving a crystallite through a black square must return through a white square and vice versa. In this paper, we consider the gambler’s ruin model in this case.
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6927 CHART I
116 213 116 116 213 116
+
116 213 116 116 213 116
+
Calculation of the Amount of Strict Adjacency The requirement that the two ends of any loop lie on opposing colors is completely equivalent to requiring that all loops contain an odd number of steps. The selection of a phase between the two chessboards representing the two interfaces is arbitrary; we remove that arbitrariness by requiring that ties also contain an odd number of steps. Therefore, the color patterns of the two planes are either in phase or out of phase depending on the parity of the distance separating the two planes. Consider a random walker tracing out the trajectory of some amorphous walk. We let P, be a column vector giving the probability of finding the random walker at a given level between the two absorbing barriers representing the two interfacial boundaries. In other words, (P,),, f o r i = 1, 2, 3 , ..., M , is the probability of finding the walker at level j after n steps. Let f represent the fraction of all walks that are amorphous walks. ( 3 ) Mansfield, M . L. Macromolecules 1983, 16, 914. (4) Flory, P. J.; Yoon, D. Y . ;Dill, K. A. Macromolecules 1984, 17, 862. (5) Yoon, D. Y . ;Flory, P.J. Macromolecules 1984, 17, 868. (6) Leermakers, F. A. M.; Scheutjens, J. M. H. M.; Gaylord, R. J. Polymer 1984, 25, 1577. (7) Marquee, J . A.; Dill, K. A. Macromolecules 1986, 19, 2420. (8) Kumar, S. K.; Yoon, D. Y . Bull. Am. Phys. Soc. 1988, 33, 249.
CHART I1
213 116 116 213 116 116 213 116 116 213 116 R =
(5)
116 213 116 116 213 116 116 213 116 116 213
Therefore, 1 -fis the fraction of all walks that are strictly-adjacent loops. We write
Pl = ( f / 2 ) D (1) where we let D represent the mth-order column vector whose 1st and Mth elements are unity and all other elements are zero. This is equivalent to assuming that in the first step a fraction f / 2 of the walks step from level M 1 to level M , another fraction f / 2 steps from level 0 to level 1. The remaining 1 -ffraction of walks does not enter the amorphous domain; these walks are the strictly-adjacent loops. The fractionfof walks continues walking randomly until reaching either level 0 or M 1, at which point they are removed, although as explained above, steps to levels 0 or M 1 are only permitted on odd jumps. One of our goals is to determine the correct value off. We use transfer matrices to generate all the additional P, vectors. For example, we may write
+
+
+
where Q is the M X M matrix (eq 3 , Chart I). Note that Q sends a walker at levelj, whenj # 1 a n d j # M , to level j f 1 with probability 1/6 and leaves it at level j with probability 2 / 3 , as expected, since those walks remaining at level j actually represent steps in one of the four orthogonal directions. It sends a walker at level 1 to level 2 with probability 1 - x and leaves it in place with probability x. Similar probabilities hold for walkers at level M . Note that since each column sums to unity, Q does not send any walkers out of the system, consistent with our requirement that no walks are permitted to terminate after an even number of steps. As of now, the parameter x is arbitrary; its value will be fixed below. Now for P3 we may write
P3 = RPI (4) for R the M X M matrix (eq 5, Chart 11). R differs from Q only in the first and last columns. For example, R sends a walker from level 1 to level 2 with probability 1/6, and leaves it in place with probability 2 / 3 , but since 2 / 3 + 1/6 is still only 5/6, it removes walkers originally at level 1 from the system with probability 1/6. These represent the walks steping from level 1 to 0, or the walks which are permitted to exit after an odd number of steps. Of course, a similar process is occurring at level M . We can, of course, generalize eq 2 and 4: Pj = QPj-,
if j is even
(6)
Pj = RPj-,
i f j is odd
(7)
6928
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Other Statistical Properties of the Random Walks In this section we consider several statistical properties of the walks in this model. To provide a framework for comparison, we also quote the results’*2for the standard gambler’s ruin model, in which no restrictions are placed on the reentry point of any walk. The following properties are listed in Table I for both models:
TABLE I: Results of the Two Models “standard” gambler’s ruin “chessboard” gambler’s ruin
- .. fT
(j,v + 31-1
fAL
M(3M
+ 3)4
(1/6) 1
BA
BT BAL
BL Bw
( M + 1)’ 2M+ 2 2(M + 1)2(3M+ M+1
- ( 1 / 3 ) W 1 + O(M-’)
f
fraction of fraction of fA = 1 - f fraction of fT = ffAT fraction of fAL = 1 - fraction of
Mz + y M + O(1)
+
4M 2 + O(M-1) ( 2 / 3 ) M + O(1) M+ 1
2)-l
Mansfield
fAT
fTX
BA BT BAL BL BW
The sum
2pj
j=1
represents the density of the amorphous domain. Requiring the amorphous domain to have a uniform density equal to the density of the crystal implies m
z = j=CPj 1
+ QP1 + P3 + QP, + Ps + QPS + ... 01
= (1 + Q).CPzj+i j=O
average length average length average length average length average length
(9)
where 1 is the order-M identity matrix. The above becomes j-0
(10)
or Z = (1
+ Q)*M-’.P,
a strictly-adjacent loop a tie an amorphous loop all loops any walk
Any walk of length L visits L - 1 lattice sites in the amorphous domain. Therefore, the average length of any walk, as a direct consequence of the requirement that the amorphous domains have the correct density, must obey
(11)
where
M = 1 - R-Q For the time being, let us assume that calculations indicate that Q Z = Z and
M*Z = D / 6 Equation 14 implies
M-’*D= 6 2 Combining eq 1, 11, 13, and 15 yields Z = 6fL which is satisfied when f = 1/6 Any other value of x would not have yielded eq 13 and 14, so we conclude that x = 5 / 6 is the only appropriate value of x. This model predicts that only 1 /6 of the crystalline stems emit random walkers that enter the amorphous domain. Compare this with the value 1/3 which is obtained if no restriction is placed on the reentry point of any walk.’S2 This calculation therefore predicts that the restrictions placed on the chain folds when stems of a definite head-to-tail sense must alternate in the crystal decrease the amount of random reentry.
(18)
in either model. Note also that Bw must be given rigorously as Analytically rigorous results for the current model, beyond those given in the previous section, would require the analytically exact inversion of the matrix M of eq 12, and analytically exact eigenvalues and eigenvectors of the matrix R.Q. We have been unable to do either calculation exactly and so have resorted to numerical calculations to obtain many of the expressions in Table I. Because of numerical uncertainty, we could not calculate precisely the three subdominant coefficients, x , y , and z, that appear in Table I. We note that eq 18 and 19 require x
Z = (1 + Q).[2(RQYI.Pl
of of of of of
Bw=M+l
where Z is an order-M column vector each of whose elements is 1. The two parameters f and x must be chosen to satisfy eq 8. The model would be fundamentally flawed (Le., unable to predict a uniform domain density) if no values off or x existed that satisfied eq 8. Note that we may write eq 8 as
Z = PI
- fT
all walks that are amorphous walks amorphous walks that are ties all walks that are strictly-adjacent loops all walks that are ties all walks that are amorphous loops
+ 2y + z = 3/2
(20)
=
and that the calculations suggest x Y -1 1/2, y 11/2, z Y 7/2. An interesting fact becomes apparent from Table I. Note that even though the fraction of tight loops has changed, all long-range properties of the two models agree to highest order in M . In other words, the fraction of tie chains is 1/(3M) for both models, and the average lengths of loops and ties are respectively 2M/3 and Mz for both models. The constraint that allows chains only between opposing colors tightens up a number of the loops, but aside from that, makes minimal changes to the long-range structure. It is interesting to note that this behavior has come to be expected from all such models.9J0
Summary In this paper we have modified the simple cubic lattice version of the gambler’s ruin model’s2to permit only chains with an odd number of steps. For this particular lattice, that is equivalent to assuming that the polymer has a specific head-to-tail sense and that the polymer crystallizes with neighboring stems lying with an alternating sense. This change increases the adjacent reentry prediction of the gambler’s ruin model from 2/3 to 5/6. This increase results from the tightening up of a number of the loops, since no change is predicted (to leading order in the domain thickness) in such long-range properties as the fraction of tie chains or the average length of loops and ties. Acknowledgment. The author thanks Prof. John D. Hoffman for suggesting this problem and for several helpful discussions. Partial support of this research was provided by the National Science Foundation, Grant No. DMR-8607708. (9) Mansfield, M. L.; Guttman, C. M.; DiMarzio, E. A. J. Polymn. Sci., Polym. Lett. 1986, 24, 565. (10) Mansfield, M. L. Macromolecules 1988, 21, 126.
