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J. Phys. Chem. B 2008, 112, 4662-4665
Ring-Expanding Polymerization by Reversible Ring Fusion. A Fascinating Process Driven by Entropy Gianfranco Ercolani*,† and Stefano Di Stefano‡ Dipartimento di Scienze e Tecnologie Chimiche, UniVersita` di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy, and Dipartimento di Chimica, UniVersita` di Roma La Sapienza, P.le Aldo Moro, 00185 Roma, Italy ReceiVed: September 6, 2007; In Final Form: January 23, 2008
Grubbs et al. reported a successful method for synthesizing high-molecular weight cyclic polymers without the intermediacy of linear chains (Science 2002, 297, 2041). In spite of its practical significance, there are, however, important questions about this process that remain open. Here it is presented a theory of ringexpanding polymerization mediated by ring-ring equilibria, and driven by entropy only, that addresses the observed results. In particular the theory predicts the existence of a critical concentration (CC) below which only low-molecular weight cyclic oligomers are formed and above which the entire system is predicted to collapse into a single giant ring molecule, the process resembling an irreversible one. Dilution of the system below the CC is predicted to restore the mixture of low-molecular weight cyclic oligomers.
Introduction
SCHEME 1
Cyclic polymers have a variety of physical properties that distinguish them from their acyclic counterparts, making them attractive targets for the development of new materials (lower viscosities, higher glass transition temperatures, and smaller hydrodynamic volumes and radii).1 The synthesis of highmolecular weight (MW) cyclic polymers starting from low-MW monomers, however, is not trivial.1 Typically, cyclic polymers are prepared by ring-chain competitions occurring under either thermodynamic control or kinetic control. In both cases, the final concentrations of large cyclic i-mers decrease as i-5/2,2,3 implying that the number average degree of polymerization of the cyclic fraction is very low.4 Recently, Grubbs and co-workers reported an alternative procedure for synthesizing cyclic polymers employing a cyclic ruthenium complex (1) capable of catalyzing the metathesis of cyclic olefins without passing through the formation of linear intermediates.5 Polymerization is believed to proceed through the formation of a transient macrocyclic complex (2) in which both ends of the growing polymer chain remain attached to the Ru center. Subsequent intramolecular chain transfer later releases cyclic polymer (3) from the intermediate complex. The synthesis of cyclic polyoctenamers using cis-cyclooctene as the monomer is illustrated in Scheme 1. This revolutionary approach, that, to the best of our knowledge, is the only one that excludes the presence of linear chains by design, allowed the preparation of large quantities of pure cyclic polyoctenamers with a number average degree of polymerization up to ca. 10 000 (average MWs 1200 kD),5a in marked contrast with the results obtainable with ring-chain competitions.6 Apart from the extraordinary experimental result of high practical importance, there are some important questions about * To whom correspondence should be addressed. E-mail: ercolani@ uniroma2.it. † Universita ` di Roma Tor Vergara. ‡ Universita ` di Roma La Sapienza.
this process that remain open and await to be enlightened by a deeper theoretical understanding of the reaction. It is well-known that the olefin metathesis reaction is reversible;7 thus, the cyclic ruthenium complex 1 is a genuine catalyst8 which only promotes ring fusion and ring fission of the various cyclic oligomers giving rises to a ring-ring equilibrium mixture as illustrated in Scheme 2. Considering that in the course of ring fusion no additional bond is either formed or broken, and that large rings, with, say, more than 25-30 skeletal bonds, are strainless,1,9,10 ring fusion of two large rings involves no enthalpy change and is therefore driven by entropy only. Then, why does entropy favor the formation of relatively few, very large, polymeric rings instead of many cyclic oligomers, large enough to be strain free, but not exceedingly larger? Moreover, why does the reaction show a critical initial monomer concentration below which only low molecular weight cyclic oligomers are obtained (see note 12 of ref 5a)? From the above considerations, it is evident that entropy plays a key role in the equilibrium state of the system. To focus on this aspect and to make things easier, we will consider a system
10.1021/jp711389t CCC: $40.75 © 2008 American Chemical Society Published on Web 03/26/2008
Ring-Expanding Polymerization
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4663
SCHEME 2
in which all of the equilibrating cyclic oligomers are large enough to be strainless. Before evaluating the entropy change accompanying ring fusion, let us consider the entropy change accompanying the equilibrium shown in eq 1, where Pj and Pi+j are linear polymer chains of degree of polymerization j and i + j, respectively, and Ci is a large cyclic oligomer of degree of polymerization i.
Pi+j h Ci + Pj
(1)
The equilibrium constant, made up of the equilibrium concentrations relative to the standard state of 1 M, has been dubbed as either the molar cyclization constant2a or the effective molarity (EMi).2b,10 It represents a measure of the ease of formation of a given cyclic oligomer under thermodynamic control. The problem of finding a theoretical expression for such a constant was solved by Jacobson-Stockmayer (J-S) as early as 1950.2a According to the J-S theory, the effective molarity decreases as shown in eq 2, where B coincides with the EM of the monomeric ring.
EMi ) Bi
-5/2
(2)
The factor B depends on the conformational character of the chain forming the monomeric ring upon ring closure. Although B can be calculated on the basis of the geometric parameters of the chain,2a,11 its evaluation is irrelevant for the present treatment. The factor i-5/2 may be regarded as the product of i-3/2 and i-1. The former relates to the probability that a Gaussian chain of i repeating units has its ends co-incident and the latter to the number of equivalent bonds available for the ring-opening of a cyclic i-mer. The J-S theory is based on several assumptions: (1) the thermodynamic reactivity of end groups is independent of the length of the chain; (2) all of the rings are strainless and thus there is no heat of cyclization; (3) the probability of end-to-end closure of a chain depends on the fraction of configurations for which the ends coincide without considering angle correlations; (4) the end-to-end distribution function of a chain in solution is Gaussian; and (5) the mean square end-to-end distance is proportional to the number of skeletal bonds. All of these approximations must fail for not sufficiently long chains.11 The last approximation also fails when the effect of the excluded volume is important. This effect causing chain expansion is expected to be maximal for very long chains at high dilutions in good solvents.12 Although the deviation from eq 2 due to the effect of the excluded volume can be eliminated by judicious selection of solvent and temperature (the so-called theta conditions),12 the effect of a good solvent in fact is merely to make the exponent appearing in eq 2 slightly less than -2.5 for the higher members of the series (see, for example, ref 6a). Thus the effect of solvent is expected to be uninfluential with respect to the conclusions drawn here assuming a strict adherence of the EMi values to eq 2.
Application of the equilibrium law to eq 2, yields
µC0 i + µP0 j - µP0 i+j ) - RT ln Bi-5/2
(3)
where µC0 i , µP0 j , and µP0 i+j are the standard chemical potentials of Ci, Pj, and Pi+j.13 By analogy with the Benson group additivity scheme for estimation of thermochemical properties,14 it can be safely assumed that the difference µP0 i+j - µP0 j is equal to the product of iµ0M where µ0M is the contribution of one internal monomeric unit to the standard chemical potential of the polymeric chain. Accordingly, eq 3 can be rewritten as eq 4.
µC0 i ) iµ0M - RT ln Bi-5/2
(4)
Let us consider now the equilibrium in eq 5 in which i monomeric rings C1 undergo ring fusion to form a cyclic i-mer Ci. Ki
i C1 y\z Ci
(5)
Taking into account eq 4, the equilibrium constant Ki is given by eq 6.
[Ci] [C1]i
) B1-i i-5/2
(6)
Rearranging eq 6 and defining the variable x ) [C1]/B, eq 7 is obtained
[Ci] ) Bi-5/2 xi
(7)
Let us consider now the ring-expanding polymerization of a large monomeric ring initially present at the concentration [C1]0. When equilibrium is attained, the system contains, in principle, an infinite number of cyclic oligomers in equilibrium with each other. For such a system, the mass balance equation is ∞
[C1]0 )
i[Ci] ∑ i)1
(8)
Substitution of eq 7 into eq 8 yields ∞
[C1]0 ) B
i-3/2xi ∑ i)1
(9)
The series in eq 9 is convergent for x values in the range [0,1]. When x ) 1, the series is given in closed form by the Riemann zeta function ζ(3/2) ) 2.612.4 Since the series in eq 9 diverges for x values larger than 1, eq 9 is only valid in the range of initial monomer concentrations from 0 to 2.612B mol L-1. Let us call the latter as the critical concentration (CC). Before investigating what happens above the CC, it is useful to analyze the ring-size distribution below the CC, where eqs 7 and 9 hold. The easiest way to build plots of the cyclic oligomer concentra-
4664 J. Phys. Chem. B, Vol. 112, No. 15, 2008
Ercolani and Di Stefano ∞
G1 )
nC µC ∑ i)1 i
(10)
i
Assuming the solution is ideal, µCi ) µC0 i + RT ln[Ci] where, if the rings follow the J-S theory, µC0 i is given by eq 4. Accordingly, eq 10 becomes ∞
G1 )
nC (iµ0M - RT ln Bi-5/2 + RT ln [Ci]) ∑ i)1 i
(11)
Substituting eq 7 for [Ci], and the initial moles of monomer, n0, for ΣinCi, eq 12 is obtained.
G1 ) n0(µ0M + RT ln x) Figure 1. Plots of the equilibrium concentrations of the first three cyclic oligomers against the initial monomer concentration for a reversible ring-expanding polymerization. Reported data are for B ) 1 mol L-1 and for [C1]0 up to the CC.
tions as a function of [C1]0 is to (i) fix the value of B, (ii) compile a list of x values in the range0,1, (iii) calculate for each x, the concentration of the given cyclic i-mer by eq 7, and the initial monomer concentration by eq 9,15 and (iv) plot the two results against each other. For the sake of illustration, plots of the first three cyclic oligomers vs the initial monomer concentration are shown in Figure 1 for the case B ) 1 mol L-1. It is apparent from Figure 1 that the concentration of each cyclic i-mer increases on increasing the initial momomer concentration up to a maximum value coinciding with the corresponding EMi value given by eq 2. The number average degree of polymerization of this distribution, that from now on we call the J-S distribution, is 1.95.4 The distribution is exactly the same as that which would be obtained in a reversible ringchain competition in the limit of an infinite value for the equilibrium constant of linear propagation.2b Indeed, according to the phenomenology of the J-S theory, in a reversible ringchain competition in which the extent of reaction is practically equal to 1, there is a critical concentration of the initial monomer (cutoff point), identical to the CC defined above, below which the system is virtually composed of cyclic species only and above which the concentration of each cyclic species remains constant and the excess monomer produces linear polymer only.2 Now the question is: what happens above the CC in a ringexpanding polymerization where the formation of linear polymer is not allowed? The equilibrium condition expressed by eq 9 is not compatible with a finite amount of the initial monomer concentration larger than the CC. Thus, any real mixture of cyclic oligomers, being in the impossibility of following the equilibrium constraint imposed by eq 9, cannot be in a state of thermodynamic equilibrium. The logical consequence of this statement is that the series of consecutive equilibria of ring fusion, exemplified in Scheme 2, must be completely shifted to the right, i.e., toward the formation of a single giant ring molecule. As we are going to show, this view is supported by the comparison of the total Gibbs energy of the J-S distribution (G1) with that of a system composed of a single ring molecule obeying eq 4 and having an infinite degree of polymerization (G2). Indeed, the total Gibbs energy for a mixture of cyclic oligomers is given by eq 10, where nCi is the number of moles of a given cyclic i-mer in the mixture and µCi is its chemical potential.
(12)
In correspondence of the CC, x ) 1 and thus G1 ) n0µ0M. If all of the monomeric units underwent ring fusion to form a specific cyclic i-mer only, its number of moles would be n0/i and its concentration in solution [C1]0/i, thus, the Gibbs energy for such a system would be
G2 )
(
)
n0 0 [C1]0 iµM - RT ln Bi-5/2+ RT ln i i
(13)
After some manipulation, eq 13 becomes
G2 )
(
n0 µ0M
RT [C1]0i + ln i B
)
3/2
(14)
Of course, since in the range 0 e [C1]0 e CC the equilibrium ring-size distribution is univocally determined by eqs 7 and 9, the corresponding total Gibbs energy G1 will be lower than G2, no matter the value of i in eq 14. But what about [C1]0 values larger than the CC? For values of the initial monomer concentration larger than the CC ([C1]0 > 2.612B mol L-1), the minimum value of G2 is given by the limit of eq 14 for i tending to infinity, i.e., G2 ) n0 µ0M. The physical meaning of this limit is that the most stable system composed of a cyclic i-mer only is that in which i assumes the largest possible value; this value corresponds to the formation of a single giant ring molecule incorporating all the available monomeric units. Note that the Gibbs energy G2 coincides with the value of G1 in correspondence of the CC showing that at this concentration the collapse of the system into a single giant ring molecule is consistent with the continuity of the Gibbs energy function. The abrupt change of the system composition above the CC must be understood in terms of entropy. Consider the equilibrium in eq 5; upon ring fusion of i molecules of C1 to form Ci, 3(i - 1) translations and 3(i - 1) overall rotations are converted into 6(i - 1) vibrations having lower entropy content. This loss of entropy is compensated by a gain of entropy due both to the greater conformational mobility of Ci relative to C1, and to concentration effects (entropy of mixing). All of these entropic factors are taken into account by the equilibrium constant Ki establishing the equilibrium concentrations of the two species. The i-5/2 dependency of Ki, however, implies that there is a critical concentration above which the entropy associated to a single infinite ring, which benefits from both the greatest conformational mobility and the greatest possible dilution, is larger than that associated to any possible mixture of cyclic oligomers. Although above the CC the thermodynamically more stable system is constituted by a single infinite ring, this is not a sufficient condition for its actual formation. Indeed its formation
Ring-Expanding Polymerization would require an ideal system, completely free of linear contaminants, and an infinite time of reaction. In fact the expected behavior of the system above the CC would be analogous to that of an irreversible step-growth linear polymerization whose hypothetical outcome, given an infinite amount of time, is the formation of a single chain molecule incorporating all of the available monomeric units.17 This analogy is substantiated by the polydispersity indices (PDIs) of the resulting cyclic polymers recorded by Grubbs and co-workers that were around 2.0,5 identical to the theoretical value expected for step-growth polymerization.17 However, in contrast to the irreversible stepgrowth polymerization, the behavior of the system here reported would not be strictly irreversible but only apparently irreversible, because dilution below the CC would restore the thermodynamically more stable J-S distribution. The possible presence of linear contaminants is expected to have a detrimental effect on the obtainment of a high molecular weight cyclic polymer. This effect would be especially important in the latest stages of the polymerization process. In principle the presence of a single molecule of linear monomer would be sufficient to convert the single infinite ring into a J-S distribution of low molecular weight cyclic oligomers in equilibrium with a single linear chain. Since in practice it is impossible to have a cyclic monomer completely free of linear contaminants, this conclusion points to the importance of avoiding prolonged reaction times that inevitably would lead to a J-S distribution as the thermodynamically more stable final state. In conclusion a theory of reversible ring-expanding polymerization has been presented that addresses the results observed by Grubbs and co-workers.5 In particular the theory predicts the existence of a critical monomer concentration below which only low-molecular weight cyclic oligomers are formed whereas, above which, the system undergoes ring-expanding polymerization up to fuse into a single giant ring molecule. The theory, besides providing a deeper understanding of the process, points out to the importance of removing traces of linear contaminants as well as of avoiding prolonged reaction times as possible sources of failure in the obtainment of high-molecular weight cyclic polymers. References and Notes (1) (a) Semlyen, J. A. Cyclic Polymers, 2nd ed.; Kluwer Academic: Dordrecht, The Netherlands, 2000. (b) Semlyen, J. A. Large Ring Molecules; Wiley: New York, 1996. (2) For equilibrium ring-chain competitions, the -5/2 exponent of the ring-size distribution is theoretically predicted, and observed, above the critical monomer concentration: (a) Jacobson, H.; Stockmayer, W. H. J. Chem. Phys. 1950, 18, 1600. (b) Ercolani, G.; Mandolini, L.; Mencarelli, P.; Roelens, S. J. Am. Chem. Soc. 1993, 115, 3901. (3) In kinetically controlled polycondensations, the -5/2 exponent of the ring-size distribution is theoretically predicted, and observed, when propagation is much faster than cyclization (i.e., in concentrated monomer
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4665 solutions): (a) Craven, J. R.; Nicholas, C. V.; Webster, R.; Wilson, D. J.; Mobbs, R. H.; Morris, G. A.; Heatley, F.; Booth, C.; Giles, J. R. M. Br. Polym. J. 1987, 19, 509. (b) Yuan, X.-F.; Masters, A. J.; Nicholas, C. V.; Booth, C. Makromol. Chem. 1988, 189, 823. (c) Fawcett, A. H.; Mee, R. A. W.; McBride, F. V. Macromolecules 1995, 28, 1481. (d) Ercolani, G.; Mencarelli, P. Macromol. Theory Simul. 1997, 6, 1139. (4) The number average degree of polymerization (DP) of the cyclic fraction is by definition equal to the ratio Σi[Ci]/Σ[Ci]. If the concentrations of all the rings are proportional to i-5/2,2,3 DP ) Σi-3/2/Σi-5/2. Series of the type ∑ ∞ i- s where s > 1 are given in closed form by the Riemann zeta i)1 function, ζ(s), implemented in most software programs for symbolic computation. Accordingly, since ζ(3/2) ) 2.612 and ζ(5/2) ) 1.341, DP ) 1.95. Arfken, G. B.; Weber, H. J. Mathematical Methods for Physicists, 6th ed.; Elsevier: Amsterdam, 2005. (5) (a) Bielawski, C. W.; Benitez, D.; Grubbs, R. H. Science 2002, 297, 2041. (b) Bielawski, C. W.; Benitez, D.; Grubbs, R. H. J. Am. Chem. Soc. 2003, 125, 8424. (6) (a) Reif, L.; Ho¨cker, H. Macromolecules 1984, 17, 952. (b) Chen, Z.-R.; Claverie, J. P.; Grubbs, R. H.; Kornfield, J. A. Macromolecules 1995, 28, 2147. (7) For recent reviews on olefin metathesis, see: (a) Grubbs, R. H. Tetrahedron 2004, 60, 7117. (b) Handbook of Metathesis; Grubbs, R. H., Ed.; Wiley-VCH Verlag: Weinheim, Germany, 2003; 3 Volumes. (c) Schrock, R. E.; Hoveyda, A. H. Angew. Chem., Int. Ed. 2003, 42, 4592. (d) Connon, S. J.; Blechert, S. Angew. Chem., Int. Ed. 2003, 42, 1900. (g) Trnka, T. M.; Grubbs, R. H. Acc. Chem. Res. 2001, 34, 18. (h) Fu¨rstner, A. Angew. Chem., Int. Ed. 2000, 39, 3012. (i) Ivin, K. J.; Mol, J. C. Olefin Metathesis and Metathesis Polymerization; Academic: New York, 1997. (8) Indeed most of the catalyst (over 80 %) can be recovered from the polymerization by column chromatography.5a (9) Winnik, M. A. Chem. ReV. 1981, 81, 491. (10) Mandolini, L. AdV. Phys. Org. Chem. 1986, 22, 1. (11) For a discussion of the assumptions underlying the J-S theory, see: Flory, P. J.; Suter, U. W.; Mutter, M. J. Am. Chem. Soc. 1976, 98, 5733. (12) Flory, P. J. Statistical Mechanics of Chain Molecules; WileyInterscience: New York, 1969; p 32. (13) Since in the present case ∆H h 0) 0, ∆Sh0 ) -∆µ0/T. Thus the same arguments could be carried out in terms of the standard partial molar entropy, Sh0. However, we have preferred to use the standard chemical potential because it is the partial molar quantity most familiar to chemists. (14) Benson, S. W.; Cruickshank, F. R.; Golden, D. M.; Haugen, G. R.; O’Neal, H. E.; Rodgers, A. S.; Shaw, R.; Walsh, R. Chem. ReV. 1969, 69, 279. (15) Series of the type φ(x,s) ≡ ∑∞i)1 i-sx i, where s is a rational number and x is a real number between 0 and 1 have been studied by Truesdell (Truesdell, C. Ann. Math. 1945, 46, 144). The series φ(x,s) is rapidly convergent for x values near 0 but slowly convergent for x values near 1. Truesdell showed that the series φ(x,s) as defined above can be also calculated by the series φ(x,s) ) Γ(1 - s)(-ln x)s-1 + ∑∞i)0 ζ(s i)(ln x) i/i!, where Γ(z) and ζ(z) are the Gamma function and the Riemann zeta function, respectively (both the functions are implemented in most software programs for symbolic computation). The second series, in contrast to the first one, is rapidly convergent for x values near 1 but slowly convergent for x values near 0; thus, the two series complement each other for the evaluation of φ(x, 3/2) in the x range [0, 1]. Values accurate to the fourth decimal figure or better, have been obtained by summing the first five terms of the first series for x in the range [0, 0.3], and the first five terms of the second series for x in the range [0.3, 1]. (16) It is worth pointing out that this conclusion is not affected by the possible presence of strain in the smaller cyclic oligomers. This possibility, as discussed in our previous work on reversible ring-chain competitions,2b would only decrease the value of the CC. (17) Flory, P. J. Chem. ReV. 1946, 39, 137.
