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Reversible Association of Telechelic Molecules: An Application of Graph Theory Yardena Bohbot-Raviv,* Thomas M. Snyder,† and Zhen-Gang Wang* Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125 Received January 12, 2004. In Final Form: May 23, 2004 We develop a method for calculating the exact free energy of tree clusters formed from associating telechelic molecules. The method uses the concept of rooted trees from the graph theory to enumerate all topologically distinct trees having a maximum degree of branching; it recursively separates the trees into different classes based on their connectivity, thus enabling the exact summation of the trees weighted by their respective Boltzmann factors. We apply our method to studying the pregel properties in pure telechelic solutions and in mixed telechelic and single-associating-end polymer solutions. We highlight the effect of energetic tendency for branching in the former and the effect of competitive association in the latter.
1. Introduction Many different soft materials exhibit interesting processes involving the reversible association of small molecules into large linear and/or cross-linked clusters.1-4 These include, for example, reversible polymerization, associating block copolymers, and, on a higher molecular hierarchy, self-assembly of elongated micelles often referred to as “living polymers”.5,6 In all these materials, the molecules, that is, the constituents of the cluster, consist of “functional groups” that are capable of reversibly associating into physical clusters. Functional groups refer, for example, to different chemical compositions along polymer molecules and particular local shapes at the edges of linear micelles. Beyond a certain concentration of molecules in solution, the formation and growth of crosslinked clusters can lead to reversible gelation where space spanning networks with long relaxation times are manifested by the large increase in the viscosity of the solution.7,8 In this work, we consider the reversible association of telechelic molecules with functional end-groups that have a fixed maximum degree of functionality. Such systems are typical for association processes that are due to directional interactions such as hydrogen bonding. Our aim is to develop a statistical thermodynamic theory for * Corresponding authors. E-mail:
[email protected] (Y.B.-R.);
[email protected] (Z.-G.W.). † E-mail:
[email protected]. (1) Russo, P. S. Reversible Polymeric Gels and Related Systems; ACS Symposium Series 350; American Chemical Society: Washington, DC, 1987. (2) Schulz, D. N.; Glass, J. E. Polymers as Rheology Modifiers; ACS Symposium Series 462; American Chemical Society: Washington, DC, 1991. (3) Gelbart, W. M.; Ben-Shaul A. J. Phys. Chem. 1996, 100, 1316913189 and references therein. (4) Jain, S.; Bates, F. S. Science 2003, 300, 460-464. (5) Porte, G.; Gomati, R.; El Haitamy, O.; Appell, J.; Marignan, J. J. Phys. Chem. 1986, 90, 5746-5751. Appell, J.; Porte, G.; Khatory, A.; Kern, F.; Candau, S. J. J. Phys. II 1992, 2, 1045-1052. Khatory, A.; Kern, F.; Lequeux, F.; Appell, J.; Porte, G.; Morie, N.; Ott, A.; Urbach, W. Langmuir 1993, 9, 933-939. (6) Filali, M.; Ouazzani, M. J.; Michel, E.; Aznar, R.; Porte, G.; Appell, J. J. Phys. Chem. B 2001, 105, 10528-10535. Appell, J.; Porte, G.; Rawiso, M. Langmuir 1998, 14, 4409-4414. (7) Rubinstein, M.; Dobrynin, A. V. Curr. Opin. Colloid Interface Sci. 1999, 4, 83-87. (8) Kumar, S. K.; Panagiotopoulos, A. Z. Phys. Rev. Lett. 1999, 82, 5060-5063 and references therein.
the cluster size distribution in the pregel state from which the gelation point can be determined. Insofar as meanfield theory remains the most useful approach for relating thermodynamic properties to molecular parameters, the most essential input in the present theory is the partition function, or the free energy, of tree clusters formed from associating molecules. Much to our surprise, however, a correct expression for the cluster free energy does not exist in the literature. The widely used partition function that was derived by Stockmayer9 nearly 60 years ago contains counting errors that cannot be easily corrected and is in any case inapplicable to the present systems studied. In addition, all known methods for calculating the cluster free energy separate the energetic from the entropic contributions and, therefore, limit the validity of the theory to a few special cases, which cannot be easily adapted to the present study.10-14 The latter limitation essentially amounts to counting all clusters of the same size with equal weights, without regarding the different degrees of branching, and hence different energetics, among the clusters. As a result, such methods are unsuitable for predicting the gelation behavior of associating telechelic molecules with different molecular properties related to their propensity for branching. In this paper, we present a new method based on the graph theory for calculating the exact free energy of tree clusters of associating telechelic molecules. The method uses the concept of rooted trees taken from the graph theory15 to enumerate all non-isomorphic (i.e., topologically distinct) trees, given a maximum degree of association at the chain ends.16 It recursively separates the trees into different classes based on their connectivity, that is, degree of branching, that correspond to the same energy. As a result, the summation over different classes, weighted by their respective Boltzmann factor, yields the exact partition function of tree clusters. The method that we develop (9) Stockmayer, W. J. Chem. Phys. 1943, 11, 45-55. (10) Lubensky, T. C.; Isaacso, J. Phys. Rev. A 1979, 20, 2130-2146. (11) Tanaka, F. Macromolecules 1990, 23, 3790-3795. (12) Ishida, M.; Tanaka, F. Macromolecules 1997, 30, 3900-3909. (13) Semenov, A. N.; Rubinstein, M. Macromolecules 1998, 31, 13731385. (14) Semenov, A. N.; Nyrkova, I. A.; Cates, M. E. Macromolecules 1995, 28, 7879-7885. (15) Davis, R. E.; Freyd, P. J. Chem. Educ. 1989, 66, 278-281. (16) Balaban, A. T.; Kennedy, J. W.; Quintas, L. V. J. Chem. Educ. 1988, 65, 304-313.
10.1021/la049906z CCC: $27.50 © 2004 American Chemical Society Published on Web 07/30/2004
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Figure 2. Schematic illustration of tree graphs and their molecular analogues. A tree graph consisting of v vertices (or v - 1 edges) corresponds to a cluster with m ) v - 1 molecules.
Figure 1. Schematic illustration of end-associating telechelic polymers forming linear, cross-linked, large network clusters, as found in solutions.
in this work is similar to the alkane counting method described by Davis and Freyd15 which involves counting all non-isomorphic trees. However, the subdivision of nonisomorphic trees into classes based on their connectivity is an important new feature of the present work and is essential for correctly counting the inseparable contributions of the entropy and energy to the free energy of associating molecules. Furthermore, we extend the graph theory method to account for mixtures of two associating species. The latter extension allows us to address new physical features arising from competitive association. In the next section, we present the thermodynamic models used for studying the pregel properties of pure and mixed solutions. In section 3, we describe the basic graph theory tools for generating the recursive algorithms that calculate the exact free energy of pure and mixed clusters. We then describe their application and present results on solutions of one telechelic species in section 4 and on mixed solutions in section 5. We make some concluding remarks in section 6. 2. The Model Consider an ideal solution of reversibly associating telechelic polymers. Each polymer molecule consists of a linear chain carrying at each end a functional group capable of associating with others. Each functional group may interact with one up to a maximum of D other groups residing on different molecules to form linear and/or crosslinked clusters of different sizes and shapes (see Figure 1). Mathematically, a cluster is represented by a graph consisting of edges and vertices (Figure 2). A vertex of degree (i.e., functionality) d represents the association formed by d end-functional groups, and an edge represents a polymer chain. The energy of a cluster consisting of m molecules with a given number distribution of vertices, {vd}, can be written as
Em({vd})/kBT )
∑
vdd
(1)
d)1,2,3,...,D
Here, vd denotes the number of vertices with degree d and d is the energy (in units of kBT) of a vertex of degree d, measured relative to the energy of free molecules, i.e., 1 ≡ 0. For simplicity, we assume that the energy of an edge depends only on its local functionality and not on the global
topology of the cluster. The formation of gel in a telechelic solution is possible only if the maximum functionality is larger than 2 (D > 2). The equilibrium cluster size distribution in a pregel solution at temperature T follows from the law of mass action17
φm ) mφm 1 qm
(2)
where φm denotes the volume fraction of clusters of size m and qm stands for the partition function of an m-cluster consisting of m indistinguishable molecules. The volume fraction of unassociated molecules is φ1 ) eµ/kBT, where the de Broglie thermal wavelength and other contributions to the free energy due to the internal degrees of freedom of the molecule have been absorbed into the definition of the chemical potential, µ. The latter contributions should be of the order of the volume associated with the size of the polymer which does not depend on m, and therefore, chain length and flexibility are not expected to influence the gelation in ideal solutions. The total volume fraction of polymer in solution is given by φ ) ∑∞0 φm. Though recent computer simulation studies highlight the nonequilibrium nature of gelation,18 in this work, we follow the more traditional view where gelation corresponds to the formation of a space-spanning network. That is, gelation occurs at a volume fraction, φ*, and a temperature, T, for which the weight-averaged cluster size, 〈m〉w, defined as ∞
〈m〉w )
∑ mφm m)1
(3)
φ
diverges. Nevertheless, for sufficiently large association energies that make the lifetime of the clusters sufficiently long, these two views of gelation should be equivalent. The key quantity that appears in eq 2 is the partition function, qm, for a cluster of m indistinguishable molecules. This partition function accounts for all the degrees of freedom of the cluster that are not included in the overall translation and rotation. Its evaluation consists of first summing over all the internal degrees of freedom of a cluster with a given topological structure and then summing over topologically distinct structures with the proper statistical weights.19 It is the second aspect that (17) Reichl, L. E. A Modern Course in Statistical Physics; University of Texas Press: Austin, TX, 1991. (18) Kumar, S. K.; Douglas, J. F. Phys. Rev. Lett. 2001, 87, 188301188304.
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is the focus of this paper, and in order not to be encumbered by secondary effects, we assume that the dominant contribution to the free energy for a cluster of a given topology is the association energy given in eq 1. Thus, the partition function, qm, can be written as
qm(T) )
∑
{vd}∈m
Ωm({vd}) exp[-Em({vd})/kBT]
(4)
Here, Ωm({vd}) denotes the number of topologically distinct clusters for a given distribution of vertices, {vd}, and the summation is over all possible distributions consistent with a fixed number of molecules, m, in the cluster. The analytical evaluation of Ωm({vd}) allowing for arbitrary topological structures is intractable. On the other hand, exact enumeration is possible only for small cluster sizes, since Ωm({vd}) grows exponentially with m. Following a mean-field approach, we make the tree approximation which has been extensively used by other researchers in the past.9,20,21 The fundamental aspect of this approximation assumes that clusters grow into branched trees that contain no closed loops and that their growth is not limited by neighboring branches; for example, intracluster excluded volume effects are neglected. Studies in percolation theory suggest that, below the gelation point, the fraction of loops is negligible.14,22 In the following sections, we develop a graph theoretical method that recursively generates the exact Ωm({vd}) value for any m with a finite maximal degree of association, D, at the vertices. Telechelic polymer solutions for which D ) 2 can exhibit only linear association. For any linear m-sized cluster, v1 ) 2, v2 ) m - 1, and Ωm ) 1. If 2 < 0, the size distribution is simply φm ) mφm 1 exp[(1 - m)2].23 In such systems where association energies are finite, there does not exist a finite volume fraction for which an infinite weight-averaged cluster would exist. This is because the energy gain involved in one-dimension association can never compensate for the entropic loss due to association. Gelation can therefore occur only for D g 3.24 In the main body of this paper, we will explicitly work out the recursive algorithms for computing Ωm({vd}) for which D ) 3; an extension to D ) 4 is provided in Appendix B. In addition to solutions containing a single telechelic polymer species, we also consider a mixed system which contains both telechelic end-associating polymers (hereafter referred to as A) and polymers with only one associating end (hereafter referred to as B). Since the B molecules consist of only one functional group, they are unable to form a gel phase in solution by themselves. However, by participating in the association with the A molecules, they alter the gelation behavior of the telechelic polymers. It is this effect we wish to address. We will assume for simplicity that the functional groups and polymer chains of both A and B have the same chemical composition. Consequently, the association energy of an (mA, mB) cluster with mA + mB ) m is still given by eq 1. However, the calculation of the partition function of an (mA, mB) cluster is different from the single telechelic case (19) Mayer, J. E.; Mayer, M. G. Statistical Mechanics; John Wiley and Sons: New York, 1940. (20) Flory, P. J. J. Am. Chem. Soc. 1941, 63, 3083-3096. (21) Tanaka, F. Macromolecules 1989, 22, 1988-1994. (22) Cohen, R.; Erez, K.; Ben-Avraham, D.; Havlin, S. Phys. Rev. Lett. 2000, 85, 4626-4628. (23) Gelbart, W. M.; Ben-Shaul, A.; Roux, D. Micelles, Membranes Microemulsions and Monolayers; Springer-Verlag: New York, 1994. (24) Bohbot, Y.; Ben-Shaul, A.; Granek, R.; Gelbart, W. M. J. Chem. Phys. 1995, 103, 8764-8782.
but follows on similar grounds and requires no additional concepts from the graph theory. In the next section, we present the graph theoretical method and develop the recursive algorithms for computing Ωm({vd}), for both the pure and mixed telechelic systems. We advise readers that are not interested in the counting procedure to read the introductory part of section 3 and skip from there to the following sections (sections 4 and 5), where we demonstrate the application of such algorithms to the study of the gelation properties in pure and mixed solutions. 3. Graph Theory Method 3.1. Introdution. We first describe several useful concepts of the graph theory that are necessary for understanding the recursive algorithms for counting trees. The “tree” nature of the graphs permits a simple relation between the number of edges and vertices in a tree graph. In any tree cluster formed by m end-associating polymers, there are a total of m edges (polymers) and v ) m + 1 ) ∑d vd vertices with at least two vertices of degree 1. Since each molecule contributes two vertices of degree 1, the sum of all the degrees of vertices is ∑d dvd ) 2(v - 1) ) 2m. Whenever a tree admits an additional vertex of degree 3 or 4, the number of vertices of degree 1 must also increase by 1 or 2, respectively. For the case of D ) 3, the number distribution of vertices, {vd}, is uniquely determined by only one type of vertex, either of degree 1, 2, or 3. For example, knowing v3, the number of vertices of degree 3 in a given m-tree defines v1 ) v3 + 2 vertices of degree 1 and v2 ) m - 1 - 2v3 vertices of degree 2. For the general case where D > 3, D - 2 different vertices are required for uniquely determining {vd}, and hence the energy of the cluster. Note that trees bounded by a maximum vertex degree of 4 (i.e., D ) 4) can be mapped to alkanes in organic chemistry. In the “alkane counting problem”, vertices and edges represent carbon atoms and carbon-carbon bonds, respectively.16 However, the alkane counting problem is concerned with only the total number of alkanes containing a given number of carbon atoms.15 In the present problem, the different distributions correspond, in general, to different cluster energies. Therefore, a subdivision of the non-isomorphic trees into classes based on their connectivity (i.e., {vd}) is necessary for correctly calculating the inseparable contributions of the energy and entropy to the free energy of associating clusters. The correct counting of tree graphs is made possible by considering a unique element of every tree, the centroid. A centroid is defined for a tree with v vertices (or m molecules; v ) m + 1) as either (i) an edge that can be removed to separate the tree into two rooted trees of exactly v/2 vertices each or (ii) a vertex that can be removed (together with the edges connected to it) to leave a collection of rooted trees, each containing less than v/2 vertices. The root in a rooted tree is the vertex that “used to be” connected to the centroid. Note that only case i or case ii but not both can occur for a given tree. Several examples of trees and rooted trees resulting from the removal of their centroid are illustrated in Figure 3. The scheme for counting all non-isomorphic trees relies on the fact that the centroid is a unique element of any tree and that there are a limited number of ways that rooted trees can result from removing a centroid. The recursive counting scheme thus consists of first enumerating the different rooted trees in a recursive manner, from which the final tree is then built. The subdivision of trees of a given size into classes based on their distribution of degrees of vertices requires a
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Figure 3. Illustration of the basic concepts in the graph theory used for counting trees. The centroid, as defined in the text, may either be an edge or a vertex (dashed line) of any given tree graph. Removing the centroid results in two or more rooted trees, where the vertices that used to be connected to the centroid represent the roots (hollowed circles).
classification system. For the case of D ) 3, we represent the number of rooted trees, Rv, and trees, Tv, of a given size, v, by a one-dimension array of the form Av ) [a0, a1, a2, ...]. a0 is the number of unique trees with no vertices of degree 3, a1 is the number of trees with 1 vertex of degree 3, etc. The total number of v-trees is given by the sum ∑i)0 ai. For D > 3, arrays of higher dimension are necessary due to the increase in the degrees of freedom specifying the distribution of vertices. For example, D ) 4 requires the classification of trees in the form of twodimension matrices where each element corresponds to the number of trees with, say, v3 and v4 vertices. Next, we present the schemes for counting rooted trees and trees that correspond to single and mixed telechelic solutions for which D ) 3. At this point, readers that are not interested in the counting procedure can skip to section 4 and/or 5. 3.2. Counting Trees (D ) 3). The method for counting rooted trees relies on the fact that, by removing the root of a rooted tree, a smaller set of rooted trees results. As this process may be continued to the smallest possible rooted trees, a set of rooted trees can be built up recursively from smaller rooted trees. An example in Figure 4 illustrates how smaller rooted trees can be combined into larger rooted trees. Each “old” root is connected to the “new” root by introducing new edges. Notice that, since D ) 3, the maximum number of rooted trees resulting from removing the root is two, because the third edge is reserved for the centroid. Accordingly, the degree of the root in a rooted tree will always be one more than its apparent degree. In the example shown in Figure 4, rooted trees of a size (number of vertices) of k ) 3 are combined with the rooted trees of size j ) 4 to form rooted trees of size v ) k + j + 1 ) 8. The counting operations that result from combining rooted trees are defined as follows. Consider the elements of the array Av to be the coefficients of the following generating series: a0y0 + a1y1 + a2y2 + ... , where i, the exponent that appears in yi and as an index in ai, represents the number of vertices of degree 3 and ai is the number of trees having i vertices of degree 3. For two or more arrays, each corresponding to clusters of different sizes (k * j), a simple product of the polynomials yields the correct new array of coefficients which corresponds to the distribution of the larger (i.e., combined) clusters. As an example, consider the two rooted trees of size 3, R3 ) [1, 1] and the three rooted trees of size 4, R4 ) [1, 2], which are all illustrated in Figure 4. Combining the two sets yields six rooted trees of size 8, distributed according to the product of the polynomials of each array, times y, that
Figure 4. Graphs of rooted trees with v ) 3 and 4 vertices that correspond to R3 ) [1, 1] and R4 ) [1, 2], respectively. ac and bd are two examples, out of the six rooted trees, resulting from the combination of R3 and R4. According to the combination scheme described in the text, R8 ) R3R4 ) ac + ad + ae + bc + bd + be ) [1, 3, 2].
is, yR3R4 ) y(1 + y)(1 + 2y) ) [0, 1, 3, 2]. The additional multiplication by y accounts for the new root being of degree 3 which shifts the position of each element in the product by one. However, a simple multiplication of identical arrays produces redundancies. For example, multiplying R3 by itself according to the above rule does not give the desired distribution of [1, 1, 1]. In fact, this is the very reason for the redundancies involved in the expressions given by the lattice-based mean-field percolation theory.25 The correct distributions, [b0, b1, b2, ...], resulting from the square power, A2v, and the cube power, A3v, of identical arrays in terms of the coefficients [a0, a1, a2, ...] are given, respectively, by
bv )
∑ acad + 2c)v ∑( c+d)v
ac + 1 2
)
(5)
c
