4346
J . Phys. Chem. 1989, 93, 4346-4351
Lattice Treatment of Thermodynamic Properties of Ring Polymers in Solution J. Leonard CERSIM and DZpartement de chimie, UnicersitZ Lacal, QuZbec, GI K 7 P 4 . Canada (Received: May 24, 1988; I n Final Form: November 23, 1988)
Expressions for the thermodynamic functions of ring polymers in solution are derived. The configurational entropy is obtained from the number of ways of arranging N , flexible ring molecules and N , solvent molecules in a honeycomb lattice, taking into account the number of segments in a ring, the number of ways of placing the successive ring segments in the lattice, and the expectancy of finding a ring segment in an adjacent site. The entropy of mixing disoriented ring polymer with solvent is readily deduced. From the lattice model, an expression for the enthalpy of mixing, much similar to the one for linear polymer, is obtained. The free energy of mixing and the chemical potentials are deduced from the above functions. The equations for various thermodynamic functions are evaluated in terms of the available experimental data.
Introduction In recent years, reactions leading to the formation of large cyclic or ring molecules have aroused much interest. These rings may contain from about ten to many hundred skeletal bonds. Ring polymers are formed in the course of a number of polymerizations where ring-chain equilibrium occurs. Among these are certain polyesters, polyamides, polyethers,l and polyalkenes.* Recently large cyclic polystyrene containing up to 4000 monomer units has been prepared by the reaction of two-ended living poly(styry1sodium) with d i m e t h y l d i c h l ~ r o s i l a n eand ~ 1,4-bis(bromomethyl) b e n ~ e n e . ~ Properties of ring polymers are not all well defined, but it is expected they will differ from the properties of small cyclic molecules in about the same way the properties of linear polymers differ from those of their monomers. Moreover it is not known how different are most of the physical properties of ring polymers compared with linear polymers. The question may be put forward for ring polymers in the solid state as well as in solution. Concerning solution properties, theories dealing with the conformation and size of cyclic macromolecules in dilute solutions are rather well developed5-I3 and have been verified by diffusion coefficient, viscosity, and radius of gyration measurements on a few system^.^,'^ On the other hand, theoretical works on statistical thermodynamics of ring polymer in solution are scarce, and the Guggenheim theory15 remains the most important work on the subject. However, expressions are given only for three- and four-membered rings. The approach to this problem is similar to his treatment of open linear chains, and the resolution of the final equations requires the knowledge of a certain number of parameters related to the type of lattices and the type of molecules. Generalization of the Guggenheim equations has been attempted by Brzostowski.I6 Because the same types of parameters ought to be known. application of the generalized Guggenheim equation has been restricted to computation of thermodynamic functions (I)Semlyen. J. A. Pure Appl. Chem. 1981, 53, 1797.
(2) Ivin, K. J . Olefin Metathesis; Academic Press: London, 1983; Chapter I I. (3) Roovers, J.; Toporowski, P. M . Macromolecules 1983, 16, 843. (4) Lutz, P.: Strazielle, C.; Rempp, P. Polym. Prepr. 1986, 190. ( 5 ) Kramers, H . A. J . Chem. Phys. 1946, 14, 415. (6) Zimm, B. H.; Stockmayer, W. H. J. Chem. Phys. 1949, 17, 1301. ( 7 ) Casassa, E. F. J . Polym. Sci. A 1965, 3, 605. (8) Bloomfield. V . ; Zimm, B. H. J . Chem. Phys. 1966, 4 4 , 315. (9) Hess. W.: Jilge, W.; Klein, R. J . Polym. Sci.. Polym. Phys. E d . 1981. 19. 849. (10) Des Cloizeaux. J . J. Phys. Lert. 1981. 4 2 . 433. ( I I ) Edwards. C . J. C . ; Rigby, D.; Stepto, R. F. T.: Semlyen, J . A . Polymer 1983. 2 4 , 395. ( 1 2 ) Cherayil. B. J.; Bwendi. M . G.; Miyake, A,; Freed, K. F. Macromolecules 1986. 19. 2770. ( 13) Tanaka. F. J. Chem. Phys. 1987, 87, 4201 ( 1 4 ) Edwards. C. J. C.; Stepto, R . F. T.: Semlyen, J . .A. Po/vmer 1980, 21. 781 ( 1 5 ) Guggenheim, E. A . Mixrures: Oxford Lniversity Press: London, 1952; ChaDter I O . ( I 6 ) Brioatowski, u’.Bull. Pol. Acad. Sci 1963. 1 l , 407
0022-3654/89/2093-4346$01 50/0
for mixtures containing cyclic molecules such as cyclohexane and benzene.” In the present work, thermodynamic functions for mixtures containing ring polymers are derived by using a lattice model. Once the model has been defined, the general treatment of the problem follows the Flory-Hugginsla theory for mixtures of linear polymers. The Flory-Huggins approach leads to simple expressions and, despite its shortcomings, remains most valuable and is still widely used. It is found that with the same type of argumentation, simple expressions can also be obtained for ring polymers in solution.
The Model Except for the cyclic nature of the solute, the mixture under consideration is essentially the same as the one described by Flory15 for relatively concentrated solutions of linear polymer molecules. It consists of cyclic molecules, all of the same size, uniformly distributed in the solution. Each cyclic molecule is made of x segments, the size of each segment being equal to the size of a solvent molecule. Each segment is freely jointed to another segment so that the rings are as flexible as they can be. It is obvious that a ring of x segments is not as flexible as the equivalent linear chain. However, in the present case the restricted flexibility of the solute comes solely from its cyclic nature and not from bond-angle hindrance. The solution is represented by a lattice that consists of an array of regular hexagons as shown in Figure 1. Among many advantages of the honeycomb lattice over the Miller’s lattice used in the Flory-Huggins theory, it allows for a lattice coordination number of six for first adjacent neighbors to a given site, a number in accordance with the suggestion made by Flory,’*and it can be used for ring as well as linear molecules. A segment is represented by a “stick” that spans from the center of a site to another one for two adjacent sites. Because of the ring structure, the number of segments is strictly equal to the number of sites occupied in this fashion. It should be pointed out that with the honeycomb lattice, a cyclic trimer, the smallest ring that may be considered, can be placed in the lattice without stretching one of the segments, which is not the case for square-site lattices. The rings are distributed in the lattice by placing a first segment in any of the available sites and the remainder, by placing a segment i n a site next to a center occupied by the end of a segment, going always in the same direction. I t should be reminded that during the placement of the segments, they are always linked together and they should be placed in such a way that the length of the segments remains unchanged. Compliance with these requirements has an effect on the number of ways of placing certain segments, and i t is quite obvious that a random-walk approach cannot be applied here, although the segments are loosely linked together. Once all the ring molecules have been set in the ( I7 ) Brzostowski. W.: Magiera. B. Bull. Pol. Acad. Sci. 1964, l 2 , X99. ( 1 8 ) Florq. P. J . Principles of Polyn7er Chemistry; Cornell University Precs. Ithaca. ZY. 1953: Chapter I ? .
F, I989
Qniericdn Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. IO, 1989 4347
Ring Polymers in Solution
Figure I . Flexible-ring polymer molecules in the lattice. The right-hand side illustrates the setting of the last segment.
lattice, all the remaining sites are filled with solvent molecules. The configurational entropy of the mixture, Sc, is estimated from the number of ways of distributing N 2 cyclic molecules and N, solvent molecules in a lattice made of No sites so that
No = N1
+xN~
(1)
Figure 1 represents a lattice containing cyclic molecules. Assuming i rings are in the lattice already, the number of ways is of arranging the x segments of the ( i 1)th ring,
+
The function fG,( 1%)) represents the effect of ( 1 - A ) on the number of ways of arranging a segment, taking into consideration the relative position of j with respect to x. In the case of a flexible linear chain,'* once the first segment is set in place, the number of ways of arranging each other segment is ( z - I ) ( 1 - A ) . For a ring, if x is large, A, for the very first segments is practically identical with A, for a linear chain. However, as j approaches x in the ordering position, the number of ways of arranging a segment becomes smaller and smaller because the segments must be set in place in such a way that the ring is closed with the placing of the last segment. Because of this, the presence of segments of neighboring rings in the vicinity of the ring being placed, expressed by ( 1 -f),has a very small effect on A, and Ax is totally independent of ( 1 -f)as stated in ec, 5 . This means that fG,( 1%)) varies from a value of ( 1 - A ) at j = 2 to a value of 1 for j = x. Accordingly the requirements of the model are satisfied with
where the exponent B ( x j ) ranges from 1 for j = 2 to 0 for j = x. In the case of small rings, the molecule is less flexible than a large ring, and fU,(l-f)) tends very rapidly toward unity. Since X2 = z'( 1 -f)and A, = I , taking into consideration the requirements imposed by the model and eq 7, it follows that at j = 2, f(z',xj) = z'and a t j = x, f(z',xJ) = z Q = 1. Since the requirements are the same as for BU), then eq 6 can be rewritten in the following form: AJ = [z'(I
where X is the number of ways of arranging a segment. For the first segment set in the lattice, X is identical with X obtained in the case of linear chains, that is A, = N o - ix
-I;)]B'"J'
(8)
Several types of equations can meet the requirements imposed on B ( x j ) . Examples of three types of equations are given here:
(3)
X for the second segment is of the same form as for the case of linear chains and is given by A2
= z'(1 -f)
(4)
wheref; is the expectancy that an adjacent site is occupied by a segment of a ring. At this point the model departs from the model proposed by Flory and Huggins for linear chains as z , the lattice coordination number, is replaced by z'. z'is the maximum number of sites adjacent to one end of a segment already in place that are available to the next segment of the ring, taking into account the reduced flexibility of the ring under consideration. The introduction of z' is justified since the degree of freedom of the segments is smaller for a ring than for a linear chain. It is expected that z'will change with ring sizes. In the case of very large rings, z'will be practically equal to z but will be much smaller for small rings. Because of the lower degree of freedom in a ring, A, decreases as the number of segments of the ring being set in the lattice approaches x. Another consequence of this lack of freedom with increasingj is that X becomes more independent of the term ( I -5) as x is approached. In other words, the number of ways of arranging the segments as j x is determined by the fact that a t the end, the xth segment must necessarily link the ( x - 1)th segment with the first one, and the presence of other segments in the neighborhood has no effect on this number. As a matter of fact, once (x - I ) segments are set in place, there is only one way to place the xth segment as it is illustrated in Figure I , and it follows that
-
A, = I
(5)
Taking into account these considerations, a general expression can be written for A:, 'J
=f(z'*xL/)f6.(1
-f))
(6)
The function f(z',xj) represents the combined effect of the ring size, through x, a n d j , the relative position of the segment being placed with respect to the first segment of the ring already placed in the lattice and the remaining ( x - 1) segments to be placed.
Variations of Ab for j varying from 2 to x, computed through eq 8 withf. = 0 and z ' = 12 are shown in Figure 2 for the three types of B ( x j ) functions. Figure 2a shows this variation for a SOsegment ring, which may be considered as 2 large ring. For a type I l l function, deviation from the case of linear chain OCCUR only for the last I O segments. It is expected that the cyclic nature of the molecule will begin to show up about halfway in the process of placing the segments. In the case of a 100-segment ring, this deviation occurs only for the last 5 segments. Because of the unrealistic behavior of the type 111 functions, this type of function will not be discussed any further. For type I 1 functions, a sharp decrease of A, is observed for the very first values of j . This is most unlikely in the case of a totally flexible ring. This kind of behavior is to be encountered with serniflexible rings where the number of ways of arranging segments in the lattice is smaller than for the cases of linear chains and flexible rings. Because of the analogy with this particular type of ring, for the sake of comparison type I I functions will be discussed along with a type I function. In the latter case A, decreases very slowly up to about two-thirds of .Y and then decreases very rapidly to the limiting value of I , as is expected for a flexible ring. Figure 2b shows the variation of XJ for a IO-segment ring. Whatever the function may be, AJ decreases quite rapidly sincc after thc second segment has been set, the ring has to be closed in eight steps. Entropy of Mixing
With use of Flory's approximation and mathematical approach for linear chains," the configurational entrop) of mixing, Sc. ia readily obtained. With the expression for A, given by eq 3 and 8. it is nov. possible to evaluate from cq 2: IJ,+,
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989
4348
Leonard
a
I
I
20
I
I
40
,
I
I
60
I
J
80
100
X
I
1
I
1
I
10
20
30
40
50
Figure 3. Values of (S, + l ) / x for flexible rings with numbers of segments, x, varying from 3 to 100.
I
for x 2 3, where $ is the $ function. The numerical values of the functions are found in appropriate tables.2’ In default of tables, eq 15b is readily computed with the help of a microcomputer. If x is large, eq 15 can be written as
1
s, = (x - 1111 - ( I / x - l ) ) [ + ( X ) With $(x) large x
N
-$
011
(16)
In x and with $(2) = 0.4228, eq 16 becomes for
S , = x - In x - 0.5772 (17) Figure 3 shows the value of the function (S, I ) / x for values of x from 3 to 100. With (1 -fi) = ( N o- ix)/No,the number of ways of arranging N 2 identical rings in the lattice is given by
+
N2
R = (1 /Nz)!rIv, i= 1
(18)
and with a very good approximation,’8 eq 14 may be rewritten as
I
I
I
I
I
2
4
6
8
10
so that eq 18 becomes
1
Figure 2. Variation of A, with increasing values o f j using type I (eq 9), I I (eq I O ) , and I l l (eq 12) functions for rings with (a) 50 segments and (b) I O segments.
From eq 9 and IO, specific expressions for type I and I 1 functions may then be derived. Type I Functions. For flexible rings (type I), eq 13 becomes u i f I = ( N o - ix)[z’(I - f ; ) ] s r
Since S , represents the configurational entropy of mixing the perfectly ordered pure ring polymer with pure solvent, it follows that S, = k In R (21)
(14)
where X
s, = JC= 2B ( X j )
(15)
with B ( x j ) being obtained from eq 9. An explicit expression for S, can be obtained in terms of the 4 function, as the limit of a formula given by Bailey” for the sum of n terms of the hypergeometric series. The explicit sum for eq 15 iszo
-[
2)
.Y(X S A
=
.Y -
-
I
[
x*
$(
- .Y
x’ -
- 1 1
) ..i,r]]) 2x - 3
-
1 x- I
,=o(x - I ) j
where k is the Boltzmann constant. The entropy of disorientation of the ring polymer, ASD, is obtained by setting N , = 0 in eq 2 2 , and the entropy of mixing disoriented rings and solvent is As, = sc - AsD (23)
+ 2x - 3
( 15b)
( I 9) Bailey, W. N. Generalized Hypergeomerric Series; Cambridge University Pres, that is
4350
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 AHv = Aw12P12
(39)
where p12is the number of 1-2 contacts in a solution. The number of possible contacts with a segment is (z - 2) and with a ring, x ( z - 2). The probability that a cell adjacent to a polymer segment so that for the whole is occupied by a solvent molecule is simply solution PI2 = N2x(z -
2141 = N , ( z - 2142
(40)
From eq 39 and 40 it follows that AHM = (Z - Z ) A W , ~ N , ~ ~
(41)
LEonard the logarithmic term of eq 47 in series, with values of 42not too high, eq 47 and 50 yield for type I
With ij being the partial specific volume of the polymer and c, the concentration in grams per volume unit, then d2/xVI= c~/xF‘[ = c/M2,and the expression for the osmotic pressure can be written in the familiar form, retaining only the terms in lower powers of C:
which can be recast in the form AHM = ~ T x ‘ N= R ~ T~x ’~~ , @ ~
(42)
where X’ =
(Z
- 2)AwI,/kT
(43)
I f in the course of the derivation (z - 2)x is assumed to be approximately equal to zx, than eq 42 is identical with the expression obtained for linear chains.18 Combining eq 42 with eq 25 for flexible rings (type I ) and with eq 31 for “semiflexible” rings (type II), the free energy of mixing
where M2is the molar mass of the solute. In the case of linear - x). For type 11 rings polymers,’* the term in parentheses is the equations so obtained are
-[
n = R T y42 +
(e
+(
- x’)622
e l 4 2+
...I
(53)
Vl
AGM = A H , - TASM
is for type I rings
The chemical potential of the polymer in solution y2 relative to the pure amorphous polymer y t is obtained by differentiating eq 44, for type I rings, with respect to n2, bearing in mind that and 42 are related to n2 through eq 26: ~2
and for type 11 rings AGM = R T [ n 2 In 42
+ ((x + 1)/2x)nl
In 61 + ~ ’ n l 4 2 l (45)
In the case of linear chains, the free energy of mixing isla AGM = R T [ n 2In 42
+ n , In 41 + xnI$2]
(46)
which is similar to eq 44 and 45 except for the second term.
Chemical Potentials The physical quantities available for the evaluation of thermodynamic parameters of solutions are more closely related to chemical potentials than is AGM. The chemical potential of the solvent in the solution I.(, relative to the chemical potential of the pure solvent fila is obtained by differentiation of eq 44 or 45 with respect to n , (or Nl), bearing in mind that @I and 42are functions of n , according to eq 26. Making use of the definition of x for linear chains ( x = P2/ PI, where and PI are the partial molar volumes of polymer and solvent, respectively), the differentiation is for eq 44
v2
and for eq 45
The corresponding expression for linear chains isi8
The chemical potential of the solvent is directly connected with available physical quantities through the relationships pl - pIo=
RT In a , = RT In ( P I / P I o )= -xPI
(50)
where a , is the activity of the solvent, R is the osmotic pressure of the solution, and PI and P I oare the saturated vapor pressures of the solution and the pure solvent, respectively, assuming ideal behavior of the vapor. Vapor pressure measurements allow for J direct evaluation of x’ through eq 47, 48, and 50. Expanding
- w2O = R T b
42 -
-
42) + x’x(l -
(55)
For type I 1 rings, the result is w 2 - w20
= RTVn
@2 -
i/z(x - 1)(1 - 42)
+ x’4l
- 42)21 (56)
Equations 5 5 and 56 may be compared with the expression for linear polymers w2
-
w20
= R T b 42 - (x
-
1)(1
- 42) + xx(1 - G2l2I
(57)
Discussion The main object of the present work is to derive expressions that would describe the thermodynamic properties of ring polymers in solution. An approach similar to Flory-Huggins in their treatment for linear polymers in solution is used. The chief difficulty with the proposed model lies in the evaluation of how the number of ways of placing segments in a lattice varies from the second to the last segment of a ring. Considering the limitations imposed by the model, several functions may be used to estimate the variation of A, with j . Since such functions ought to be related to the physical reality of a ring, from the three types of functions shown in Figure 2, only two are considered. The type I function corresponds to a flexible ring. Because of a strong :yclic effect observed with a type 11 function, this type of functions is associated with a “semiflexible” ring. Even if this type of ring does not comply with the model of a flexible ring, it is interesting :o consider it in order to evaluate the effect of loss of flexibility on the properties of ring polymer mixtures. All the curves shown in Figure 2 are plotted for z ’ = 12. However, depending on the type of ring, a small ring or a “semiflexible” ring, z’can be smaller than 12 and so are the values of A. From eq 25 and 31 (or eq 35 and 36) and 37 (or eq 38) and from Figures 4 and 5 it appears that the entropy of mixing is lower for ring molecules than for linear ones. This is in accordance with the remark by DiMarzio and Gibbs,22 who state that the effect of ring closure is to reduce the total number of configurations of a molecule to below the maximum number associated with a flexible linear chain. The discrepancy between linear and ring polymers increases with increasing values of d2, ASM being more sensitive to 4~~in the case of linear polymer. Although the model proposed i n this work is intended for large flexible rings, since ~~
( 2 2 ) DiMdrno, E A , Gibbs. J H J Chem P h j s 1958. 28, 807
King Polymers in Solution
The Journal
previous works deal only with rings of x = 3 or 4, comparison is made between the Guggenheim equations (eq 32 and 33), the equation for linear chains (eq 38), and the equations derived in this work (eq 35a, 35b, 36, and 36b) and is illustrated in Figure 4. There is hardly any difference between curves obtained from the Guggenheim equations and the equation for linear chains. This is rather surprising since in the Guggenheim treatment the molecules are stiff. In the case of molecules with x = 3, the molecules are associated with triangles. Assuming a lattice coordination number of 6, once the first segment is set in place, the number of ways of arranging the second segment a t one end of the first segment is about two, and once the second one is in place there is only one way of arranging the third segment. In the case of flexible chains, the number of way of arranging the second segment is about five, and it is the same number for the third one. According to the lattice model, the number of ways of arranging the first segment is the same for either a chain or a ring so that, neglecting the ( I - J ) term vChaln = (first segment) X 5 X 5
u,,,,~= (first segment) X 2 X 1
In the case of a flexible ring, the number of ways of arranging the second segment should be in fact higher than for a stiff ring and the entropy of mixing should be higher for the former than the latter. It is interesting to note that the expression for the numbers of ways of arranging x segments of a cyclic molecule is of the same general form, that is A B ( x )(eq 19), as for the numbers of ways of setting an ( x I)-sided polygon,23 namely,
+
p(x),
As for larger rings, it should be pointed out that deviations from the entropy of mixing for linear polymers depend on the model chosen for the ring polymers. Equation 9 yields values of ASM that are much closer to ASM for linear polymers than does eq 11. As expected, the entropy of mixing of flexible rings is larger than for less flexible rings. Equations 37 (linear chain) and 25 (flexible ring) differ only by the ( S , l ) / x term. Figure 3 shows that (S, l ) / x tends slowly toward 1 with increasing x , the number of segments. As can be seen from Figure 5, the result is that the entropy of mixing of a very large flexible ring would be about the same as for a very long flexible chain, which is not the case for a less flexible ring or a small ring. The evaluation of the enthalpy of mixing is based on the contact number of a polymer molecule. For large molecules, this number is about the same irrespective of the cyclic or linear nature of the solute. Because of this, the expression for AHM is practically the same for chains as for rings. Consequently, if the interaction parameter x is solely of enthalpic nature, x would be the same for both types of molecules. However, from eq 44 to 46, x may be considered as a free energy paramcter. Because it is used to measure how a system departs from ideal behavior, it is evaluated from excess functions. I t is now recognized that x is made of an entropic contribution as well as an enthalpic contribution, and one may write x = xH
+
+
(23) Hammersley, J . M. Proc. Camb. Philos. Soc. 1961, 57, 516.
of Physical Chemistry, Vol. 93, No. 10, 1989 4351
+ xs. The enthalpic contribution xH may be the same for large flexible rings or for chains. However, the excess entropy of mixing is not necessarily the same for these two types of molecules and not necessarily the same for a flexible ring and a rigid or semiflexible ring. Consequently xs may differ and hence the use of x’ instead of x for the mixing of ring polymers is justified. The effect of ring configurations on the entropy of mixing is found in the expressions for the chemical potentials of solvent and solute and hence in the physical properties of the solutions such as the saturated vapor pressure and the osmotic pressure of solutions. Equations 52 and 54 predict that the second virial coefficient, A,, in the expression of the osmotic pressure of a ring polymer solution may differ to a various extent from an equivalent linear polymer solution. This has been verified for solutions of pdy(dimethylsi1oxane) in toluene at 298 K24 and for solutions of polystyrene in toluene a t 308 K.25 From light-scattering measurements (A2,/A2?),the ratio A, for linear polymer over A, for ring polymer is found to vary from 2.1 I for A, = 4000 to 3.19 for &fW = 400000 for the poly(dimethylsi1oxane) system whereas this ratio is equal to 1.1 for the polystyrene system. Equations 52 and 54 also predict that under 8 temperature conditions the value of x’ will be approximately 0.50 for a large flexible ring and 0.25 for a “semiflexible” one. These equations also show that irrespective of the variation of x’ with the molecular weight of the polymer, A , will vary with x through the terms (S, I)/x and (x 1)/4x. Here it should be pointed out that predictions about A2 and its related parameters are all of qualitative nature and have to be considered with great caution. Little can be said about the comparison of x’ with x, and it should be borne in mind that the present work deals with relatively concentrated solutions of polymer and not with dilute solutions. No prediction can be made regarding the absolute value of A,, its variation with temperature, and hence the 8 temperature (where A , = 0).
+
+
Conclusion
With use of a lattice model and the mathematical treatment for linear flexible chains, thermodynamic functions for ring polymer solutions have been derived. The model is simple and so are the expressions derived from it. Several functions, corresponding to different types of rings, can be used, and functions describing flexible and “semiflexible” rings are retained. Expressions for both types of rings are compared with similar equations obtained for linear polymers. The result is in agreement with the few available data showing that solutions of ring polymers behave differently from solutions of linear polymers. However, equations show that the extent of this deviation depends on the size and the flexibility of the rings. Because of its simplicity, the present approach to the problem of ring molecules in solutions may be considered as the first step to a more elaborate treatment. Acknowledgment. 1 acknowledge financial support from the Nationa! Sciences and Engineering Research Council of Canada. (24) Edwards. C. J. C.; Stepto, R. F. T.; Semlyen, J. A . Polymer 1982, 23, 869. ( 2 5 ) Roovers. J. J . Polym. Sci., Polym. Phys. Ed. 1985, 23, I 1 17.
