J. Phys. Chem. 1982, 86, 3696-3714
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FEATURE ARTICLE Equillbrlum and Kinetic Theory of Polymerizatlon and the Soi-Gei Transition Rlchard J. Cohen and George B. Benedek' hparhnent Of physb, center fw h&tM@ls sckmce 8nd EnghWkJg, 8nd h W 8 d - M I T l?Msbn Of Haelth SClences and T&tlObgy, Massachusetts Institute of Technobgy, Cam-, Massachusetts 02139 (Receive& Febrwy 23, 1982; I n Fin81 ~ m JUIY: 8, 1982)
Analysis of recent experimental determinations of cluster size distributions has led us to reconsider the fundamental Flory-Stockmayer theory for the polymerization of multifunctional units. We first consider the reversible self-condensationof multifunctional units at thermodynamic equilibrium. A statistical mechanical analysis is presented which enables one to compute the equilibrium n-mer distributions. The fundamental independent thermodynamic quantities of the problem are shown to be temperature, standard free energies of bond formation, and mole fraction of monomer initially present. In the special case where the standard free energy of bond formation is the same for all similar bonds in the system, then the n-mer distribution is of the same form as that given by Flory and Stockmayer. However, the present treatment permits an explicit determination of the degree of reaction (a), in terms of the fundamental independent thermodynamic quantities. The statistical mechanical treatment presented clearly elucidates the physical basis of the sol-gel transition and enables one to compute the exponents which characterize the divergence of the moments of the n-mer distribution as the sol-gel transition is approached. Our analysis shows that these exponents are markedly dependent on the rules of chemical bonding between combining units, but appear to be independent of the magnitude of the functionality (f L 3) for the cases considered. We also consider the kinetically controlled self-condensation of multifunctional units. We show that, contrary to widely held assumptions, the form of the kinetically evolving n-mer distribution may differ markedly from the form achieved at equilibrium. In both our equilibrium and kinetic analyses, we consider the interesting case of systems with high functionality. We show that in the high functionalitylimit the combinatoric complexity of the polymer problem is dramatically simplified, and the physical content of the theory is clearly revealed. Moreover, the high functionality limit may provide a unified theoretical basis for considering both the problem of units combining at discrete reactive sites (as in the organic polymer case), and the problem of units interacting under the action of continuous potentials (as in the case of colloidal flocculation). The theory presented here clearly identifies the crucial experimental quantities that need to be measured in order to satisfactorily characterize the n-mer distributions and the sol-gel phase transition.
I. Introduction The F l ~ r y ~ ~ - S t o c k m a ytheories e P ~ for the condensation of multifunctional units constitute the conceptual framework for the understanding of polymerization reactions and sol-gel transitions. A renewed interest in these theories has emerged recentlf'O because of their applicability to many isomorphic problems in the areas of phase transitions and critical phenomena, percolation processes, aerosol physics, colloid physical chemistry, and immunology. Despite the fundamental importance of the FloryStockmayer theory, there have been lamentably few experimental investigationsof the principal prediction of this theory-namely, the detailed mathematical form of the n-mer distributions, particularly in the vicinity of the (1)G.K.von Schulthess, G. B. Benedek, and R. W. DeBlois, Macromolecules, 13,939 (1980). (2)P.J. Flory, J. Am. Chem. SOC.,63,3083 (1941). (3)P.J. Flory, J. Am. Chem. SOC.,63,3091 (1941). (4)P. J. Flory, J. Am. Chem. SOC.,63, 3096 (1941). (5)P.J. Flory, "Principlesof Polymer Chemistry",Comell University Press, Ithaca, 1953. (6) W. H. Stockmayer,J. Chem. Phys., 11, 45 (1943). (7)W.H.Stockmayer,J. Chem. Phys., 12, 125 (1944). (8)D. Stauffer, J. Chem. SOC.,Faraday Trans. 2, 72, 1354 (1976). (9)P.G.DeGennes, J.Phys. Lett., 40L,197 (1979). (10)T. Lubeneky and Iaaaceon, Phys. Reu. A , 20,2130 (1979). 0022-3654f 82/2086-3696$01.25/0
sol-gel transition. One of the reasons for this paucity of data is the experimental difficulty in resolving different size n-mers when the monomeric units have molecular m a w s on the order of a few hundred daltons. The work of Clarke et al." represents one of the few experimental attempts to measure the n-mer distributions for small size monomers. Using gel permeation chromatograpy, they succeeded in measuring the monomer and dimer weight fractions. However, the concentration of larger n-mers could not be directly determined. Recently von Schulthess et al.' demonstrated that it is experimentally possible to determine directly the n-mer distribution by studying a polymerizing system in which the monomeric units are themselves macromolecules. These investigators'J2 employed the model system previously studied by Cohen and Benedek13 and others'"16 using quasielastic light scattering and light scattering an(11) N. J. Clarke, C. J. Devoy, and M. Gordon, Brit. Polym. J., 3,194 (1971). (12)G.K.von Schultheaa, Ph.D. Thesis, M.I.T. (unpublished),1979. (A limited number of copiee of this thesis is available upon request.) (13)R. J. Cohen and G. B. Benedek, Immunochemistry, 12, 349 (1975). (14)G.K.von Schultha, R. J. Cohen, N. Sakato, and G. B. Benedek, Immunochemistry, 13,955(1976).
0 1982 American Chemlcal Society
Feature Article
isotropy methods. This model system involves the cross-linking of antigen-coated polystyrene microspheres by means of antibody molecules. In this system the antigen-coated microspheres constitute the monomeric uni* the molecular mass of these monomeric units is on the order of l O l s daltons. von Schulthess et al.l utilized the nanopar resistive pulse analyzer”J8 to study this model system, and thereby made possible the first detailed determination of the n-mer distribution in a polymerizing system. Moreover, the temporal evolution of the detailed n-mer distribution was measured without perturbing the polymerization process. The existence of this experimental data has prompted us to r e e ~ a m i n e ’ ~the * ~physical ~ basis of the FloryStockmayer polymerization theory. This theory essentially involves first a specification of each type of monomeric unit in terms of the number (fA,fB, ...) of reactive chemical groups of each type (A, B, ...) found in the monomeric unit. Then the bonding rules are stated which define which types of chemical group pairs may react to form bonds (e.g., AA, BC, ...). For a particular system, one specifies the total number of monomeric units of each type initially present (before polymerization) and the fraction of chemical groups of each type which have reacted (cYA, CY^, ...I as the reaction proceeds. The assumption of “a priori equal chemical reactivity of identical chemical groups” is then introduced. This principle states if a bond of type AB is allowed, then a given chemical group A has an equal probability of reacting with any B group in the system. By applying the principle of “a priori equal chemical reactivity”, Flory and Stockmayer developed elegant methods for determining the detailed n-mer distribution. Furthermore, Flory was able to show that in some systems a gel phase consisting of macroscopic size polymers would appear in coexistence with the sol phase consisting of finite size n-mers. He was able to calculate the gel fraction as well as the n-mer distribution in the sol phase. We have reformulated the problem of the polymerization of multifunctional units on a statistical mechanical basis.21 We have considered both equilibrium systems in which bonding is thermodynamically reversible, and also considered nonequilibrium systems in which the polymerization process is under kinetic control. In equilibrium systems, one can formulate the polymerization problem in terms of multiple chemical equilibria. In this formulation the natural determinants of the distribution are temperature, the standard free energies of bond formation, and the concentration of free monomers of each type. We show that in this formulation the assumption of “equal standard free energies of bond formation” for all bonds of the same type is thermodynamically equivalent to the assumption of “a priori equal chemical reactivity”. Under the as(15)G.K.von Schulthess, R. J. Cohen, and G. B. Benedek, Zmmunochemistry, 13,963 (1976). (16)G. K. von Schulthem, M. Giglio, D. S. Cannell, and G. B. Benedek, Mol. Immunol., 17,81-92 (1980). (17)R. W. DeBlois and C. P. Bean, Rev. Sci. Instrum., 41,909 (1970). (18)R. W. DeBlois, C. P. Bean,and R. K. Wesley, J. Colloid Interface Sci., 61 (2),323 (1977). (19)R. J. Cohen, G. K. von Schulthess, and G. B. Benedek, Ferroelectrics, 30, 185 (1980). (20)G. B. Benedek, Lectures on the Theory of the Sol-Gel Transition. Presented at E.T.H. Zurich, June 1980,unpublished. (A limited number of copies of these lectures are available upon request.) (21)It should be pointed out that Stockmayer (ref 6) recognized that his calculation for the most probable distribution resembles the maximization of entropy at constant energy. More recently,Gordon and Temple (M. Gordon and W. B. Temple in ‘Chemical Applications of Graph Theory”, A. T. Balaban, Ed., Academic Press, New York,.1976,.pp 300-32)combined graph theory with statistical mechanical considerations in the calculation of the distribution functions.
The Journal of Physical Chemistry, Vol. 86, No. 19, 1982 3697
sumption of equal standard free energies of bond formation, then indeed the predicted n-mer distribution is identical with that of the Flory-Stockmayer theory. Next, we examine the case of nonequilibrium systems where the polymerization process is under kinetic control. Heretofore, it had been assumed that, at each instant in time, the functional form of the n-mer distributions was given by the Flory-Stockmayer distribution appropriate for the bonding rules of the system. The kinetic aspect of the problem was entirely specified by the temporal evolution of the extents of reaction.6 Our analysis shows that in fact the form of the kinetically evolving n-mer distribution may be quite different from the FloryStockmayer distribution appropriate for the system’s bonding rules. The functional form of the actual n-mer distribution is in fact determined by the magnitudes of the bimolecular association rate constants and unimolecular dissociation rate constants. However, if the system should approach equilibrium, then the n-mer distribution will approach the form predicted by the equilibrium theory, which corresponds to the Flory-Stockmayer distribution under the assumption of equal standard free energies of bond formation. Of course, it is possible to choose a particular set of interaction rate constants under which the kinetically evolving distribution would have the same functional form as the equilibrium distribution! However, the assumption of “equal standard free energy of bond formation” by no means requires this particular set of rate constants. In particular, we show that, for diffusion-controlled processes, we would generally obtain kinetic distributions which are not of the same form as the equilibrium distribution, even when no steric hindrance is present so that all functional groups on a given n-mer are “equally reactive”. This occurs because, in diffusion-controlled reactions, the “reactivity”of an unreacted functional group located on a large n-mer is reduced relative to the “reactivity” of the same group found on a smaller n-mer. In this sense the principle of equal chemical reactivity is violated, and thus the original formulation of the FloryStockmayer theory cannot apply. Our motivation in undertaking the analysis presented here was the experimental datal mentioned above. In these experiments the functionality of the monomeric units is quite large. Therefore, we were particularly interested in examining the theory of equilibrium and nonequilibrium polymerization in the high functionality limit. We found that a dramatic simplification of the combinatoric complexity of the problem is obtained in this limit. This simplification helps reveal the underlying physical principles which govern the polymerization and gelation processes. For this reason we introduce the high functionality limit in our analysis and provide a discussion of equilibrium and nonequilibrium polymerization in this limit. Finally, we show that the high functionality limit provides a natural means of unifying the theory of polymerization of multifunctional molecules with the theory of aggregation in colloidal systems, aerosol physics, and the theory of crystal growth. 11. Self-Condensation of Multifunctional Units at Thermodynamic Equilibrium
To determine the equilibrium polymer distribution of reversibly self-condensing identical multifunctional units in solution, we first examine the partition function Z of such a system. Consider the system containing N solvent molecules and M multifunctional units which constitute the solute molecules. The multifunctional units can react with each other to form different size n-mers. The n-mers
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The Journal of Physai Chemistry, Voi. 86, NO. 19, 1982
Cohen and Benedek
are presumed to be of a branched treelike structure.22 We designate m, to be the number of n-mers present in solution. The form of the partition function 2 for a specified set of (m,) is
z(V,T,{mdJV) = [zn(v,T)lNln [ z n ( V,TJV)Im
Q(bnJ)
n
l
~
(11-1)
kT Em,(ln
where m
M = Cnm,
(11-2)
- 1) (Ib5b)
where
n=l
This form for 2 assumes that the solution is sufficiently dilute that nonspecific interactions between n-mers can be neglected. Here V,T represent the volume and temperature of the overd system. 2, is the partition function of a single solvent molecule in pure solution. 2, is the partition function of a single n-mer in solution. Q((m,)) is defined as the total number of distinct ways of combining M distinguishable units to form ml monomers, m2 dimers, .... This factor is needed because the factor (Z,)% corresponds to choosing a specific group of nm, units and combining them in a unique way to form the m, n-mers. The partition function is divided by M! to express the actual indistinguishability of the units. We now evaluate Q((m,])
(g)
p,"(P,T) = k 1 a( In4 V - 1) In Z,(V,T)
Here pso(P,T) is the standard chemical potential of the solvent molecules in pure solution. In order that 5'be a linear homogeneous function of the extensive variables N a n d (m,), the quantity of a,(P,T,.ZV) must be of the form23 a,(P,TJV) = $,(P,T) - kT In N (11-6)
*,(P,T) represents the local free energy change obtained upon adding a single n-mer of a given configuration to the system. Combining eq 11-5 and 11-6 we obtain
The first factor is the number of ways to designate of the M units, ml units for monomers, 2m2units for dimers, etc. The second factor represents the number of ways to distribute each set of nm, units among m, n-men The third factor indicates that in general there may be W, distinct ways to combine n distinguishable units with n - 1 indistinguishable bonds to form a single n-mer. The magnitude of the statistical factors W, depend on the functionality, f , of the monomeric units and the specific bonding rules for combining the units. Flory and Stockmayer have calculated the factors W , for a variety of systems with different bonding rules (see Table I). Simplifying eq 11-3we obtain
Equations 11-1and 11-4 now completely specify the form of the partition function Z(V,T,(m,}) in terms of the individual partition functions for the solvent molecules and the n-mers. The forms of these individual partition functions depend on the detailed Hamiltonian, HI,which characterizes the internal quantum structure of the molecule, and the Hamiltonian, HE, which characterizes the interaction of the molecule in question with the surrounding solvent molecules. We can now compute the Gibbs free energy of the system from the partition function as follows: S(P,T,{m,J,I?) = 3 + P V (II-5a) where (22) Falk and Thomas (ref 27) showed that formation of branched
treelike molecules, without cyclization, need not be made a separate assumption of the Flory-Stockmayer theory. It indeed follows as a result of the assumption of 'equal chemical reactivity". This issue is discussed in section IV of this article.
I t is now straightforward to compute the chemical potentials of the solvent and the n-mers (II-8a) p,(P,T,(X,j) = d g / d N = p:(P,T) - kTCX, n
p,(P,T,X,,) = a$/am, = \k,(P,T)
X,n!
+ k T In - (II-8b) Wn
where X, = m,/N X, is to an excellent approximation just the mole fraction of n-mer in the system since the number of solvent molecules, N, is much greater than the number of solute molecules. Equation II-Sa is the standard result obtained for the solvent chemical potential in the theory of dilute solutions. In this equation the second term represents the decrease in the solvent chemical potential due to the presence of solute molecules. Equation II-8b can be compared with the conventional formula for expressing the chemical potential of a solute molecule in dilute solution p,(P,T) = pcL,"(P,T) + k T In X, (II-9a) where pno(P,T) is the standard part of the chemical potential. Comparison of eq II-Sa and II-8b reveals that the standard part of the chemical potential may be expressed in terms of \k,(P,T) as follows:
Wn
p,"(P,T) = \k,(P,T) - k T In n!
(II-9b)
(23) L. D. h d a u v d E. M. W t z , 'Course of TheoreticalPhysics", Vol. 5, 'Statistical Phyma",Tradated by J. B. Sykea and M. J. Kearsley, Pergamon Press, London, and Addison-Wesley, Reading, MA, 1958,1969.
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TABLE I: Degeneracy Factors, D n ( f ) Cluster , Size Distributions X,, and Transformation Equations between {X,g} and { X , , a } for RAf, ARBf-_,and ARB Type Monomer Units system type
ARBf-
R Af wncn
Ql(f)
=
7
P(fn- n)! ( f n - 2n t 2 ) ! n !
( f n - n)! ( f n- 2n + l ) ! n !
The first term in eq -1I-9bis the local free energy change obtained upon adding a single n-mer of a given configuration to the system. The second term in eq 11-9b represents the decrement in free energy associated with the degeneracy of the n-mer-the fact that there are many possible configurations in which n monomeric units can be combined to form an n-mer. In fact, it will be convenient to define D , 3 W,/n! as the degeneracy factor. D, depends on the number of reactive groups of each type present on the monomer (Le., the functionalities fA, fB, ...) and the bonding rules of the system. Henceforth we will express the degeneracy factor's dependence on functionality by writing it as D,(f, where f may in fact represent the set of functionalities (fA, fB, ...). In Table I we tabulate D,cf) for the case of units of the type RAj, ARB,,, and The free energy of the entire system can now be expressed in terms of the chemical potential of solvent, ps, and solute n-mer, p,, given in eq 11-8a and IMb, as follows: 9 (P,T,(m,JN = NPAP,T,IXnl) +
ARB
Level
Standard Free Energy Difference
1
0
2
3 4
5
+5
- 5J;
n
Cm+,(P, T,Xn) n
(11-10) Up to this point we have assumed that (m,) are fixed. To determine the equilibrium situation, we have to choose the set of m, that minimize 9,subject to the constraint M = E, nm,. This equilibrium condition simply implies p , = Wl (11-11) Substituting eq 8b into eq 11 we obtain X, = Xlne-(4.-n4i)/kTWn/n!
(11-12)
where we have used W, = 1. Equation 12 yields the equilibrium mole fraction of n-mer in terms of the equilibrium mole fraction X,. We have now derived from first principles an expression for the equilibrium distribution of n-mers in solution. It is useful to introduce the energy ladder diagram (Figure 1) in order to obtain a physical understanding of the structure of eq 11-12. In the diagram, the nth level corresponds to the n-mer. The occupancy of the nth level corresponds to the mole fraction (X,) of n-mer. The spacing between the ground state (free monomer) and the nth level is the free energy difference 9,- nq1. The occupancy X,of each level is a product of the three factors given in eq 11-12. The first factor (X,),, which is always much less than unity in magnitude, represents the entropic (24) Each monomer unit of the form R.Af containa f functional groups of type A. Bonding can take place between pairs of A groups on different monomers. ARB+l monomers each contain one A type and f - 1 B type functional groups. Bonding can take place between an A group on one monomer unit and a B group on a different monomer. A-R-B monomer unita each contain one A group and one B group. Bonding can take place only between A and B groups on different monomers.
Figure 1. Energy ladder diagram for cluster size dlstributlon. For a cluster containing n units, the associated free energy advantage is 9, - n 9 , . The occupancy of the nth level is X,.
cost of localizing n monomeric units in one region of space. This factor results from the entropy of mixing. The second factor in eq 11-12 is a Boltzmann factor, where the relevant "energy" is 9,- nql, which is the standard free-energy change associated with the removal of n monomers from the solution and the addition of one n-mer of a particular configuration to the solution. The final factor D, W,/n! is in essence a degeneracy like factor measuring the number of distinct ways of assembling the monomers into an n-mer. The occupancy of successive chemical potential levels is determined by the relative sizes of the (X,)" factor which reduces the occupancy of successive levels, the Boltzmann factor which increases the occupancy of successive levels, and finally the degeneracy factor whose precise dependence upon n, as we shall see, plays a crucial role in determining the total number of monomeric units which can be accommodated in the n-mer distribution. Indeed, systems which have a finite capacity for accommodating monomeric units in the n-mer distribution are precisely those which exhibit the sol-gel transition. A particularly simple situation results when the energy level spacings in the diagram are equal. In this case (11-13) 9,- n9, = (n - 1)g and since \kl can be identified with the standard chemical potential of monomer plo (see eq 11-9b)
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The Journal of Physical Chemktty, Vol. 86, No. 19, 1982
9, = nplo + (n - 1)g
(11-14)
where g is here determined as the free energy spacing of the levels. In fact, the assumption that the energy levels are equally spaced is equivalent to assuming that the standard free energy of bond formation is equal to g for every bond in the system. Under these conditions, eq 11-12 reduces to x, = X,ne-(n-l)#/kTW,/n! = Xlne-(n-l)#/kTD, (11-15) Quation 11-15allows us to determine the equilibrium mole fraction of n-mer X,, in terms of the equilibrium mole fraction of free monomer XI, the standard free energy of bond formation g, and the degeneracy factor D,. It is useful to compare our basic eq 11-15 with the n-mer distribution predicted by the original Flory-Stockmayer theory. In their approach they use as independent variables not X1 and g, but rather M, the total number of monomeric units present prior to polymerization and a the extent of reaction. In the case that the monomeric units are of the RAf form, then as Stockmayer6has shown, the mole fraction of n-mer can be written as Xoan-’(l - ~ t ) f ” - ~ ~ + ~ D ” ( f ) x, = (11-16)
P-’
where we have used X, = m,/N and Xo = M/N. To compare eq 11-16 with our result eq 11-15 we must make a transformation of variables from (XI&)to (Xo,a). In section IV we derive using physical arguments the necessary transformation equations. Here we shall confine ourselves to obtaining these relations by requiring in the cases n = 1and n = 2 that eq 11-16 and 11-15 be identical. Using this we obtain XI = Xo(1 - a)’ (11-17a) g = -kT In
a
X d ( l - a)2
(11-17b)
Using these in eq 11-15we find for all n that eq 11-16 and 11-15 are identical. This agreement between the equilibrium theory (our eq 11-15),assuming a constant standard free energy of bond formation) and the Flory-Stockmayer formulation is not limited to the RAf.system. The agreement occurs regardless of the specific bonding rules of the system. We tabulate in Table I the formulae for X, as a function of the variables {X,,g]and alternatively as a function of the variables (Xo,a]for the RAj, ARB 1, and ARB systems. The transformation equations reiiting the two sets of dependent variables are also indicated for each system. In the case of the RAj system a denotes the fraction of reacted A sites; in the ARB,, system, ‘YB denotes the fraction of reacted B sites; and in the ARB system, a denotes the fraction of reacted A or B sites. In summary, the Flory-Stockmayer theory is based on the intuitively reasonable assumption of equal a priori probabilities of reaction. Here we have presented an equilibrium statistical mechanical formulation of the polymerization problem, assuming that the bonding is thermodynamically reversible. In our formulation, the crucial variables are XI, the mole fraction of free monomer, and the bond energies-rather than the extents of reaction and the total number of monomeric units. Also, the role of temperature in determining the degree of polymerization is clearly identified. The theory presented here does not require that the standard free energy of bond formation be constant; however, in the case where this is true, the theory reduces to the Flory-Stockmayer result. The im-
plication is that, in a thermodynamically reversible polymerization process a t equilibrium, the condition of a constant free energy of bond formation for all identical bonds in the system is the thermodynamic equivalent of the assumption of equal a priori chemical reactivities for all identical reactive groups. Our basic equation (eq 11-15) shows that, at each temperature, the cluster size distribution is determined by D,, g, and XI.The first of these is fixed by the bonding rules and functionality of the monomeric units; the second, the bonding free energy, is regarded as a parameter. The third quantity, X,, the mole fraction of monomer at equilibrium, is physically determined by the mole fraction of monomers Xo = M / N originally placed in solution. In subsequent sections we shall investigate how to obtain the relationship between X1 and X,,. Once this functional relationship is discovered, the cluster size distribution and all its physically important moments can be fully specified as a function of X,. We shall regard Xo as the fundamental independent variable which controls the distribution of n-mers. The simple structural form of eq 11-15 permits us to express conveniently the various moments of the cluster size distribution in the form of derivatives of a generating function 2. First, consider the total number of clusters A regardless of size. According to eq 11-15 m
m
(11-18)
A = Em, = NCX, n=l
n=l
The final sum, which corresponds to the zeroth moment of the distribution, we shall denote as 2, the generating function. m
-
Z(X1) I C X , n=l
-
(11-19)
Notice that as Xo 0 the ratio (2/Xo) 1from below. At low concentration of clusters, 2 determines the osmotic pressure II of the solution according to the Van’t Hoff law II = ( A / V ) k T = (N/V)kTZ Consider next the first moment of the cluster size distribution. If we denote as Xs the mole fraction corresponding to the total number of units contained in all the clusters, then clearly m
Xs = C n X ,
(11-20a)
n=l
Since X, is known (eq 11-15)as a function of X1, if Xs were known as a function of Xo,eq 11-20a would in principle provide a means of obtaining the fundamental relationship between XI and X,,. Indeed for small Xo, Xs and Xo are identical with one another. However, as we shall see in sections I11 and IV,for certain bonding rules, if X, exceeds a critical value (X,) the cluster size distribution cannot accommodate all the material (Xo)added and instead the total mole fraction Xo splits into a part (X,)associated with the clusters of eq 11-15 called the sol phase and a part XGcalled the gel phase. Even when Xoexceeds X, eq 11-20a still holds, but in this case Xs # Xoand a separate line of reasoning must be employed to obtain Xs and XG as functions of X,,. Once Xs (Xo)is known, eq 11-20a can be used to obtain XI by summing the right-hand side. Indeed the sum in eq 11-20a can be related to the generating function Z(XJby noting from eq 11-15 that Xl(dX,/dX,) = nX, Thus we see that
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The Journal of phvsicai Chemistry, Vol. 86, No. 19, 1982 3701
a ax1
m
XS = C n X , = X1-Z(X1) n=l
(11-20b)
An important measure of the cluster size distribution is ii the mean number of monomeric units per cluster. ii =
5 (nXn)/ exn= X ~ / Z
(11-21a)
n=l
n=l
Using eq 11-20b we have ii = a In Z/a In X1
(11-21b)
Hence this derivative of the generating function determines ii.
A second, experimentally important characterization of the distribution, which is related to the first moment, is the mean number of bonds per unit (bs). bS is defined as the total number of bonds in all the clusters divided by the total number of units in the clusters. For the branched clusters of eq 11-15 m
b~ = C (n - 1)X,/Xs
(11-22)
n=l
since a branched cluster of n units has n - 1bonds. Using eq 11-2Ob and 11-19 we find bs = (XS - Z)/XS (11-23a) = 1-
z/x, = 1- (l/ii)
(11-23b)
Consider finally the second moment of the cluster size distribution. This is given by
The mean square cluster size 3 is
n2
m
m
C n2Xn/ C X, n=l
(11-24b)
n=l
Thus (11-24~) The second moment of the cluster size distribution is closely connected with the intensity of light scattered from the clusters in solution. If the monomer units and clusters are small compared to the wavelength of the incident light, then the intensity (I)of light scattered independently by the various clusters will be proportional to the square of the number of particles in each cluster, i.e. m
I = (constant). C n2Xn n=1
If Io is the intensity of light scattered from the units placed in solution before the polymerization occurs then Io = (constant).Xo Thus the relative intensity (Illo)of light scattered by the clusters is
I =-Cn2Xn Io
xo
Thus (11-25) The quantity I l l o is identical with the degree of polym-
erization DPw as defined by Stauffer.* The moments of the cluster size distribution are particularly interesting from the point of view of critical phenomena because they diverge as X X, with certain characteristic critical exponents. These critical exponents are in fact not universal but depend markedly upon the chemical rules which govern the bonding between the polyfunctional units. -+
111. Finite Capacity of the Sol Phase. The Physical Basis of the Sol-Gel Transition under Conditions of Thermodynamic Equilibrium In section I1 we showed that the mole fraction of n-mer is given by
X, = (Xl)ne-(n-l)g/kTDn
(111-1)
Since g is a physical parameter of the system, and D, is determined entirely by the bonding rules, the entire n-mer distribution is fully determined once the mole fraction of free monomer, XI, is known. We now addreas the question of how X1 is determined as a function of Xo xo = M / N (111-2) the mole fraction of monomer units initially put into solution. Naively, we would expect that the relationship between X1 and Xois fully determined by the conservation of mass condition m
m
Xo = C nX, = C n(Xl)ne-g(n-l)/kTDn (111-3) n=l
n=l
which assumes that all the monomeric units originally placed into solution are found in n-mers. By performing the sum in eq 111-3, the desired explicit relationship between Xl and Xo can indeed be obtained in the case of system ARB and ARB,,. However, in the case of the system RAf, this procedure fails. The failure occurs when Xoexceeds a critical value X,. In this case no value of X1 can be found which satisfies eq 111-3. That is to say, at most a finite amount of material X, can be accommodated in the n-mer distribution. To understand the physical and mathematical origin of this phenomenon, we must examine the behavior of the sum S m
S(Xl) = C n(Xl)ne-g(n-l)lkTDn
(111-4)
n=l
as a function of X1. Clearly, S is zero when XI = 0. S will converge when X1 lies within the radius of convergence Xlc 0 I x1 Ix1c
Let us further define X,as the asymptotic value of S in the limit as X1 approaches XIcfrom below X, = lim S(Xl) (111-5) xpx1c
As we shall show below, in the cases of ARB and ARB,, monomers, X, is infinity. Therefore, for any value of Xo, regardless of how large, a corresponding value X1 less than Xlc can be found which satisfies eq 111-3. However, in the case of the RAf system, X, is finite. Thus when Xoexceeds X,,there is no value of X1 that satisfies eq 111-3. The question of whether X, is finite or infinite is determined solely by the behavior of the terms in S for large n. To examine this large n behavior, we need to have an expression for D, for large n. In the three cases considered above, the asymptotic form of D, for large n (see Table 11) is of the form
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The Journal of Physlcal Chemlstry, Vol. 86, No. 19, 1982
TABLE 11:
Degeneracy Factors D n ( f )in the Case n >> 1 for RAf, ARBt-,, and ARB Type Monomer Units system type ARBf-,
D, N Aqnn-'
(111-6)
where A, q, and T are constants independent of n. A and q depend only on the functionality f. Inserting this asymptotic form for D, into the expression for nXn, which is the nth term in S, we find 1 nXn N A(Xl)nqne-(n-l)g/kT- nT-l =
We now may identify the quantity XlCdefined above as xlc = egIkTq-1 (111-7) Clearly, the s u m S(Xl) will converge only for values of X1 less than Xlc. The magnitude of q (which depends on the functionality f , serves only to fix the magnitude of X1. The question of the finiteness of X, is determined entirely by the exponent 7. In the limit as X1 approaches Xlc, by comparison with the harmonic series we see that X, will be finite only if T is greater than two. Conversely, X, will be infinite if T is less than two. Indeed, for the systems ARB and ARBf-! T equals zero and 1.5, respectively. Therefore, X, is infinite so that an infinite amount of material can be accommodated in the n-mer distribution. Conversely, for the RAf system 7 equals 2.5, X,is finite, and only a finite amount of material can be accommodated in the n-mer distribution. More generally, our analysis can be applied to a broader class of polyfunctional units having bonding rules different from those considered above. Provided that the asymptotic form of the degeneracy factor has the general structure of eq 111-6, it is possible from the magnitude of the corresponding exponent T to immediately determine whether the sol phase has finite or infinite capacity.% The finite capacity of the sol phase is a sufficient condition for the existence of a sol-gel transition, which occurs when Xo exceeds X,.
IV. Postgelation Relationships and Moments of the RAf Distribution under Conditions of Thermodynamic Equilibrium We have seen in the previous section that the capacity of the acyclic, branched tree, structures can be finite. It is natural to inquires as to the disposition of the system when the total amount of material exceeds this finite capacity. Stockmayer6 and more recently Donoghue and Gibbs26have analyzed an RAf system containing a finite number M of units for which the assumption of acyclic structures is strictly maintained. They conclude that a distinct peak corresponding to the gel phase may occur in the n-mer distribution for large but finite n < M. Flory,s however, concludes that the material not accommodated in the sol phase appears in the gel phase (25) The magnitude of
T
not only determines whether the sol phase
has a f i i t e or infiiite capacity: the magnitude of T also determines how the momenta of the n-mer distribution diverge as X1approaches XlC(see Stauffer, ref 8). In this connection it is important to notice that the critical exponents describing the divergence of these momenta are different for systems with different bonding rules. (26) E. Donoghue and J. Gibbs, J. Chem. Phys., 70, 2346 (1979).
ARB
which is composed of macroscopic cyclic structures. Falk and Thomas2' showed that the assumption of acyclic structures in the sol phase is not required in the Flory theory. Their work demonstrates that the assumption of equal a priori probabilities of reaction by itself is sufficient to produce a sol phase composed of finite-sized acyclic structures and a gel phase containing macroscopic cyclic structures. Essentially, finite size clusters are acyclic because the probability that a unit already present on a cluster will react with another unit on that same cluster is negligible, compared to the probability of that unit reacting with any of the other units in the entire system. However, a macroscopic structure (i.e., a structure containing a significant fraction of all the units in the system) can form cycles because the probability that a given unit on the structure reacts with another unit on that same structure is significant. The Falk and Thomasn computer calculations show that both the n-mer distribution in the sol phases and the gel fraction rapidly converge to the Flory result as the total number of elements grows large. Our analysis implicitly follows the approach of Flory, and Falk and Thomas. Up to this point, we have only dealt with acyclic structures, which necessarily correspond to the sol phase. In the present section we will show how the total number of units is partitioned between sol and gel phases for large M systems. The analysis presented in section I1 yields the equilibrium distribution of n-mers; the mole fraction of n-mer Xn is given as a function of the mole fraction of free monomer X1. This relationship between X n and X1 remains correct whether or not a macroscopic gel phase coexists with the sol phase. However, under conditions where a gel phase does exist, the quantity Xs defined in section I1 as m
Xs
= CnX,
(IV-1)
n=l
is less than Xo,the mole fraction of all monomeric units put into solutions. The remaining material XG Xo- Xs is found in the gel phase. In section I11 we demonstrated that a sufficient condition for the existence of a gel phase is xo xc (IV-2)
'
where X, is the maximum value that Xs can achieve. To actually determine the phase equilibrium for a given X, (Le., the relative magnitudes of Xs and XG) it would appear necessary to analyze the structure of the gel itself and to determine the chemical potential of a monomeric unit within the gel phase. The condition for equilibrium between sol and gel phases would then be where bls is the chemical potential of a monomeric unit in the sol phase and plG is the chemical potential of a monomeric unit in the gel phase. pls is a function of Xs and plGis a function of X G . Of course, Xs and X G must satisfy the condition
(27) M. Falk and R. E. Thomas, Can. J. Cheh., 52, 3286 (1974).
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In principle these two equations (eq IV-3 and IV-4) should permit the determination of X s and X G individually. Once X s is known then the normalization condition eq IV-1 and eq III-1can be used to determine X1,and thus the entire n-mer distribution. In fact, this direct thermodynamic procedure for determining the phase equilibrium will not be employed, because the statistical mechanics of the gel phase is not sufficiently understood to provide plGas a function of XG In view of this obstacle, we shall proceed to obtain the requisite quantities X s and X , using an alternate procedure equivalent to that introduced originally by Flory, which was based upon the assumption of equal a priori probabilities of reaction. We have already shown that this assumption is equivalent to the assumption of equal standard free energies of bond formation. We illustrate this alternate procedure by considering the case of the RAf system which shows a sol-gel transition. In this approach the quantity X 1 is first determined directly in terms of the quantity X,,. Once X 1 is known, X s is determined by summing the n-mer distribution (eq IVl),and XG follows immediately as the difference between Xo and Xs. We calculate X 1 as follows. Let us denote by p1 the probability that a given polyfunctional monomeric unit is present as a free monomer, unbonded to other units. Alternatively, p1 is equal to the probability that none of the f functional groups on a monomeric unit have reacted. According to the principle of equal a priori probability we can assign the symbol a to the probability that any given functional site has reacted.% Therefore p1is given simply by (IV-5) p1 = (1- a)f The expected mole fraction of free monomers, XI, is given by x 1 = p1Xo x1 = (1- a)fX0 (IV-6) Equation IV-6 gives a relationship between X 1 and XW However, the problem is not yet solved since a itself depends on X,,. To obtain this functional dependence we once again use the principle of equal reactivity of groups regardless of their locations. Under this assumption we may conceptually strip all the functional sites off the monomeric unita and allow them to react individually. These “stripped” functional sites will either be present as monomers of mole fraction t1 or dimers of mole fraction t2. a is the fraction of functional sites which have formed bonds, i.e. a = 252/(51 + 252) (IV-7a) Of course the total mole fraction of functional sites is (IV-8) 51 + 252 = X d
f2
(IV- 11) Carrying this result one step further we can obtain an expression for a explicitly in t e r m Xo and the bond energy g. To simplify the expression of this result, we shall use the definition
KX0) = 4&fe-glkT On solving eq IV-11 for a we find 2 1- = -[(1 + f(Xo))1/2- 11
reaction, and used in eq 11-16.
(IV-13)
Using this result for a,the extent of reaction, in terms of X o and g we can obtain an explicit formula for X1as a function of Xovalid for all X o by using eq IV-6. This gives
This relationship between Xland X o is the key to the determination of the cluster size distribution and all its moments as described in section 11. In addition, eq IV-6 and eq IV-11 represent the physical basis for the transformation equations between (Xo,a)and (Xl,g)which we obtained previously by a purely formal device. Indeed, the transformation equations, eq II-17a and II-l7b, obtained previously, are identical with eq IV-6 and IV-11. It is important to note that eq 11-14 is valid both for X o < X,, in which case only the sol phase is present, and for Xo> X,,in which case both sol and gel coexist. We can utilize the X 1 vs. X o relationship eq IV-14 to determine the cluster size distribution and its moments and to determine X s (when Xo> X,).We observe that X,, X 1 decreases with increasing X,,. Maximization of eq IV-14 shows that X l c and X , are given by
-(
-)f-
X l c = f(f-1) f - 2 (f-2I2
f-&IkT l
f2
1
(IV-15a)
It is useful to define normalized variables x1 and xodirectly proportional to X 1 and Xo, respectively, by the equations x1
= X1/XIC
xo = xo/xc
(IV-16a) (IV-16b)
We can reexpress eq IV-12 and IV-14 in terms of the normalized variables (IV-17a)
On the other hand, f 2 and f 1 are related to one another by the basic relationship (eq 11-15) which in this case gives (28) a = (totalnumber of reacted A sitea)/(totalnumber of A sites on all the polyfunctionalunita initially placed in solution). This is the same aa the quantity a introduced by Flow and Stockmayer aa the extent of
(IV- 12)
U O )
(IV-7b)
Similarly, the fraction of sites which have not formed bonds is 1 - ff = h/Xd (IV-9)
(IV-10)
Using eq IV-8, IV-9, and IV-7b, we see that a can be related to X o and g by
Thus a = 2[2/Xd
= [?e-glkT(1/2)
x1
= xo
[
(f - 2)((1 + KXo))1’2
2x0
- 1)
I
(IV-17b)
Using these normalized coordinates the X 1 vs. X o relationship can be plotted in such a way that, regardless of
The Journal of Physlcal Chemism, Vol. 86, No. 19, 1982
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Cohen and Benedek
for 0 4 xo- 1 1 the dotted lines show how one determines the sol fraction xs for a given amount of RA, monomer (xo) initially present.
the functionality f , each curve of x1 vs. xo has the same maximum value of x1 = 1 at x0 = 1. In Figure 2 we plot x1vs. xo for a range of values off. We observe from Figure 2 and analytically from eq IV-17b that for each value of xo = Xo/X, > 1there is a single value of xp For this single value of xl,however, there are two values of xo,namely, xs and x,. The smaller of these, xs = Xs/X,, shown in Figure 2 representa the total amount of material contained in the cluster size distribution. Thus,we see how eq IV-17a and IV-17b can be used to determine both x1 and xs as a function of x,. As xo varies in the range 0 < xo < 1,xs = xo in the entire range and x1 increases monotonically until its maximum value of unity. As xo increases for values greater than unity, both xs and x1decrease monotonically. This figure points out the fact that for large values off the x1vs. xocurve assumes a remarkably simple asymptotic form which can be shown from eq IV-17a and IV-17b to be in the limit f >> 1 (IV-18) This great simplification in the x1vs. xo relationship reflects the fact that the degeneracy factor D,(f, loses its complex factorial character in the limit of large f and separates into two distinct product factors each of which has a simple physical meaning. We shall, in fact, exploit this simplification in the following section in which we show that the central features of both the RAf and the ARBS1 cluster size distributions emerge quite clearly in the high functionality limit. A second feature which emerges from an inspection of Figure 2 is that around xo = 1the x1 vs. xo relationship is quadratic. Indeed it follows from eq IV-17a and IV-17b that for all f
x1 = xoe170
l - x 1 = [ (f - 2)(f 2fz - 1)
]
(1- xo)2
(IV-19)
for 11 - xol X,, the Xscan be determined from Xoby using eq IV-12 and IV-14 or equivalently Figure 2. In fact, eq IV-12 and IV-14 are necessarily valid when XSis substituted for Xoand may be rewritten 2(1 +
ls)1’2
-2
f
(IV-22a)
Cs = 4XSfe-gfkT (IV-22b) We now can use eq 11-2Ob to obtain an expression for the generating function 2. Since we actually know the X1vs.
XSrelationship, we can use the chain rule in eq 11-20b to express Z as a function of Xs,rather than X1.Indeed using the fact that as Xs 0,Z N Xs 0,we see that as a simple eq 11-20b enables us to expression Z(Xs) quadrature, viz. d In X, dXs (IV-23)
-
-
Using now eq IV-22a and IV-22b in eq IV-23 and the change of variable 1 + ls = y2,we obtain after an elementary integration the result:
The quantity 2/Xs, the mean number of clusters per monomer placed in solution, decreases monotonically from unity as xs ranges between 0 I xs I 1. In the limit of large f, Z/Xs 1/2 as xs 1. Since A, the mean number of monomeric units per cluster is equal to Xs/Z (according to eq 11-2la) we see that eq IV-24 also determines the A, viz.
-
-
Also, eq IV-24 enables us to determine bs the mean number of bonds formed per monomer in the sol phase, since according to eq II-23b, bs is equal to 1- l/ii which is the left-hand side of eq IV-24. Notice that in the limit of large f , 0 i bs I 1 / 2 when 0 i xs i 1. The quantities, 2, ii,and bs all vary smoothly as xs approaches the sol-gel transition at xs = 1. The second moment 2,however, diverges as xs approaches 1. Indeed,
The Journal of physical Chemistry, Vol. 86, NO. 19, 1982 3705
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from eq II-24a, II-24b, and 11-21a we can show that
- -
-
For small X s , X I N X s and hence Xs(d In X l / d X s ) 1. Thus for xs > 1. Using Stirling's approximation once again, we see that
single phase is present. As a result, we may set C n X , = X o on the left-hand side of eq V-5 to obtain
This equation can be compared with the identity of eq V-7 which can be written for X S 1 as nn-l
m
X = C(Xe-Vz
Since we see that the factor can also be absorbed into the Boltzmann factor of eq 11-15, the "unclothed degeneracy factor" which we define as the term in eq V-3 in fact decreases with increasing n with an exponent 7 = 512. This value of the exponent is a direct result of the rules of chemical combination between RAf units. The fact that D,' varies as n-5/2is responsible for the fact that only a finite amount of material can be accommodated in the sol phase constituted of RAf monomers (see section 111). Hence, a sol-gel transition occurs for this system. The precise features of the cluster size distribution can be obtained in the high functionality limit as follows. According to eq V-2a, in the high functionality limit the RAf cluster size distribution takes the form (V-4a) where
l / X , = fLe-glkT
(V-4b)
Let us now examine the amount of material which can be contained in this sol phase as the total mole fraction X o of units is increased from X o = 0. For this purpose we consider the sum which describes the total mole fraction of monomers contained in the distribution of clusters:
As the left-hand side increases from zero, X 1 increases monotonically until X 1 = X A / e , which is the radius of convergence of this series. Thus we see at once that the critical value X l Cis equal to X A / e ,consistent with the general result of eq IV-15a in the limit of large f . When X 1 takes its maximum value, the corresponding total mole fraction X of initially added monomer is given by
In Appendix I we prove the following result: For 0 I X I 1
5
n=l
n!
(V-9)
n=l
This equation shows that eq V-8 can be satisfied exactly provided that we choose = XO/XA (V-loa) and
_ x1 - -e-(xO/xA) xo
(V-lob) XA Indeed this result is in exact agreement with the form of the X 1 vs. X o relationship obtained previously (eq IV-18) for f >> 1. (Note that x1/xA = Xl/e and xO/xA xo.) The present analysis produces this X 1 vs. X o relationship in a very simple and direct manner. In Figure 3, we present a plot of X 1 / X Avs. A. The physical arguments presented in section IV demonstrate that eq V-loa and V-lob are still valid even when Xo/& > 1. In this regime, however, Cn,.lmnXn= X s and X s is less than Xo. We now show how one can compute XS. First we note that by substituting eq V-loa and V-lob into eq V-4a, we can obtain an expression for X , directly in terms of X o and XA X0e+"(An)n-l xn= (V-11) nn! where X = X o / X k Since eq V-loa and V-lob are valid for all positive values of X = X o / X A ,eq V-11 is valid for all positive values of A. To compute X s , we use eq V-11 XA
m
-n-1
m
xS= n=l C nXn = XA C [he-xlnn=l n!
I&- -
(V-12)
We observe that the function Xe-Xis double valued. For each value of X > 1 there is a corresponding value of As < 1. We rigorously define As in terms of X:
XSe-Xs = Xe-A
(V-13a)
Xs I 1
(V-13b)
In the upper half of Figure 3, we show graphically how As can be obtained from A. By comparing eq V-12 with eq V-9 (which is valid only for A S l),we see that
=1
Using this result in eq V-6 we see at once that the maximum mole fraction ( X , ) of monomer units which can be contained in the finite size cluster is X , = X k This result is also consistent with our previous finding (eq IV-15b) for the finite possible occupancy of the sole phase in the case of RAf monomers. We may now go on to obtain the X 1 vs. X o relationship in the following way. In the domain 0 IX o IX , only a (30) Specificauy the factor nw2 in the expreasion for D,,'repreaenta the number of distinct ways of linking n distinguishable particles with n 1 indistinguishable bonds (see Appendix 11).
m
nn-l
= C[Xse-Vn! n=l
= As
(V-14)
Thus we have identified the quantity X = &/XA and the quantity As = xS/xA. Since AS can be determined from A, similarly X s can be determined for each value of X o when X o exceeds X k In addition, we can easily compute the sol fraction S s = x s / x , = XS/X (V-15) S equals unity for X I1; S < 1 when X > 1. The result
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-
4r
Figwe 3. Upper graph shows the relationship between X , / X A and X = X o / X Afor the RA, distribution In the high functionality IlmR. The dotted lines show how X, can be determined graphicaWy from A. Lower graph shows S defined as X,/X,, the sol fraction, as a function of X over the entire range of X for the RA,dls~utkinIn the high functknauty limit.
of this computation is shown graphically in the lower graph of Figure 3. Equation V-11 is valid for all positive values of X = Xo/XA. By taking eq IV-13 to the high functionality limit, we can show that X represents the mean number of reacted A sites per monomeric unit, counting all monomeric units in both the sol and gel phases X = af (V-16) As represents the mean number of reacted sites per monomeric unit in the sol phase alone. We can define a quantity b which is the mean number of bonds per monomeric unit initially put into solution b = h/2 (V-17a)
We observe that as the sol-gel transition is approached bs 1/2. Also we note that $ / i i diverges simply as (1X s / X J 1 as X S X k This is again in agreement with the result of eq IV-29 evaluated in the high functionality limit. In analyzing the RAf system we initially evaluated a system with a finite number of reactive sites, and then extended our analysis to the continuum limit. The key parameters characterizingthis high functionality limit refer to interactions between monomeric units rather than to interactions between individual sites. That is g' = g - kT In fL refers to an effective energy of interaction between units, D,' refers to the number of distinct ways of linking n monomeric units with n - 1bonds, X refers to the number of reacted sites per unit, b is the number of bonds per unit, etc. Thus, the high functionality limit is an appropriate means of characterizing equilibrium distributions of colloidal particles, which interact on the basis of continuum forces (such as van der Waals forces and electrostatic forces) rather than binding at distinct sites. In our discussions of the high functionality limit we have indicated no upper limit on f, despite the fact that steric factors may of course limit the number of sites on a unit which can be simultaneously reacted (e.g., this limit is 12 for close-packed spheres). The number of available sites is critical in calculating the statistical weighting factor D,. The steric limitation is relevant in regard to precluding certain configurations which require many simultaneously reacted sites on a single monomeric unit. Since gel formation begins when X = 1corresponding to a mean of one reacted site per unit, the steric limitations should not become practically significant except under conditions where the system is nearly fully gelled. We close our discussion of the high functionality limit by considering the case of polyfunctional monomers of the form ARBf_,. In this case when f >> 1the use of Stirling's approximation shows that the degeneracy factor (Table I and Florf) becomes
In terms of b, the cluster size distribution is written
Xoe-Pbn( 2 bn)n-l x, = (V-17b) nn! This equation is valid for all positive values of b. Similarly, the quantity bs is defined bs = Xs/2 (V-17~) and represents the mean number of bonds per monomeric unit in the sol phase. Finally, we can use the X I vs. X o relation eq V-lob to obtain all the moments of the distribution using the arguments of section IV. These are as follows:
2 _ - I - - =XS XS
2XA mean number of clusters per initial monomer unit (V-18a)
bs = X s / 2 X , = mean number of bonds per monomeric unit in the sol phase (V-18b) = 1/(1- (xS/2xA)) = mean number of units per cluster (V-18c)
. relative scattered intensity I / I o when X o < X A (V-18d)
-
nn-l
D,@ = - (f- I),-'
n! The distribution X , now can be written as
(V-19)
where 1 / x A ' = (f - l ) e - g / k T
(V-20b) We observe that in this case the renormalized rung spacing as described above can be defined as g' = g - kT In (f l),and the renormalized degeneracy factor is 0,'= n"'/n!. In the present case the renormalized degeneracy factor is just n times larger than that for the RAf distribution. As a result for large n, D,' [e/(2?r)1/2]en-1n-3/2. The fact that D,' asymptotically varies as n-3/2rather than as n6l2 permits an arbitrarily large amount of material to be accommodated in the solution phase. As a consequence, there is no gel phase in the polycondensation of this ARBk1 system. Nevertheless, the theory does show characteristic divergencies in the momenta of the distribution as the sol phase approaches full occupancy. We now obtain XIvs. X o for this system. If all the material X o originally placed in solution can be accommodated in the sol phase then X I and X o should satisfy the equation
-
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The Journal of Fhyslcal Chemistry, Vol. 86, No. 19, 1982
By differentiating both sides of eq V-9 relative to X it follows (for X I 1)that
e-nb(n b)”-l for 0 I b I 1 (V-27) XO n! The distribution observed by von Schulthess et al.’ had a form quite close to that given in this equation.
Upon comparing this equation with eq V-21 we see that if we chose
VI. Kinetic Analysis of the Cluster Size Distribution
(V-23a) then
X l / x A ’ = he+
(V-23b)
Thus we see that the relationship between X 1 and X o is
X 1 monotonically increases with X,,. When X 1 increases to ita largest value, namely, XA’/e,the corresponding value of X o is infinite. The normalization condition (eq V-21) is therefore satisfied by eq V-24 for any value of X, There is no transition to the gel phase. Basically this is a result of the fact that the basic degeneracy factor falls off sufficiently slowly (as n-3/2)with increasing n. The moments of the ARBf-l distribution in the high functionality limit can now be obtained readily by using eq V-9 and V-22 and the further identity:
One then finds the following results:
z -- 1 - xO/(xO+ XA’) _
(V-26a)
b = Xo/(Xo + XA’)
(V-26b)
n2 = (1- xO/(xO+ xA’))-3
(V-26e)
XO
-
These resulta are of considerable interest particularly when compared with those presented above for the RAf system. Notice here that the parameter b which is equal to the mean number of bonds per initial monomer units has as ita entire range 0 I b I 1. In the limit b 1we approach the full occupancy of the sol phase. In the RAf case, b = 1/2 corresponds to the maximum occupancy of the sol phase. In the ARBf-, distribution, ii and the relative scattered intensity (Illo)diverge respectively as (1- b)-’ and (1b)-2. In contrast in the RAf case, ii does not diverge as the sol-gel transition is approached, and the relative scattered intensity (I/&,)diverges only as strongly as (1- 2bs)-’. The critical exponents describing the divergence of the momenb are thus quite different for these two distinctly different types of combining units. We conclude that the rules of chemical binding between monomer units play a central role in determining the critical exponents which characterize divergence of the momenta of the cluster size distribution. We also present the form of the cluster size distribution for the ARB,, distribution in the high functionality limit:
-
Xn _ - (1- b)
von Schulthess et al.’ have measured experimentally the cluster size distribution of antigen-coated microspheres cross-linked by bifunctional antibodies. In the language of polymer condensation, the former unit is a polymer of the form RAf, where f is the number of active antigens on each sphere (f >> l), and the latter unit is of the form B2 Links can be formed only between A and B sites due to the specificity of the antibody-antigen interaction. Thus, they were in effect examining the condensation of a twocomponent system of the form RA,-B2 with f >> 1. Their experiments showed that the cluster size distributions changed with the time. Nevertheless, at each instant of time the parameter b, the mean number of cross-linking double bonds per RAf unit, completely characterized the distribution in rather close accordance to eq V-27. If the equilibrium analyses presented above are applied to their system, the distribution of microspheres would correspond precisely to that of the simple RAf self-condensingsysteml9 according to eq V-11 and V-l7b, and a sol-gel transition should have occurred at X = 2b = 1. However, no such transition took place over a considerable range of antibody concentration, microsphere concentration, and time. In searching for an explanation for the great deviation between the experimental results and the prediction of the equilibrium theory it is natural to consider whether kinetically controlled deviations from the equilibrium state could yield the observed distributions. The approach to equilibrium can, in the lowest order of approximation, be regarded as the result of two primary processes. The first is a collision between two clusters resulting in a new cluster containing the sum of the particles in each. This is designated as “bimolecular combination”. The second process is the decomposition of a cluster into two smaller clusters containing the same total number of particles as was contained in the original cluster. This is designated as “unimolecular decomposition”. In harmony with this designation, the unimolecular decomposition fate coefficient bkl can be defiied as follows. If xk and X Iare the rates of formation of k-mers and Z-mers by the breakup of a (k + I)-mer, then x k
= x, = bklXk+l = - x k + l
(VI-1)
with bkl = blk. Furthermore, since the bimolecular combination of a k-mer with an I-mer depletes the I-mers and k-mers equally and at a rate proportional to xkxl one defines the bimolecular reaction rate constants akl as follows: -xk = = aklxkxl = x k + , (VI-2) with Ukl = U l k . using th- definitions Of the matrim akl and bkl we may write the kinetic equation for the time dependence of the mole fraction X n by considering all the reactions in which an n-mer can be generated or consumed. This gives the full kinetic equations:
8, = “-1
. .
(VI-3)
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In this equation the first two terms represent respectively the contribution to X, from the coalescence of smaller clusters and from the loss of n-mers due to breakup into smaller size clusters. The final two terms represent respectively the loss of n-mers due to coalescence of the n-mer with any other size cluster and the gain due to breakup of larger size clusters into fragments one of which is an n-mer. The set of eq VI-3 describe the temporal evolution of a coupled series of reversible chemical reactions. From eq VI-3 it follows that equilibrium is achieved (i.e., X, = 0) for all n when the cluster size distribution satisfies the condition lim
(")
=
X,X,
t--
amn b,,
(VI-4)
The equilibrium analyses given in section I1 explicitly shows the left-hand side of this equation is determined, a t equilibrium, by the free energy of bond formation (g) and the combinatorics which determine the degeneracy factors. Thus, using eq 11-15 in eq VI-4 we see that if the system is ultimately to reach the thermodynamic equilibrium state, then the ratio amn/bmn must be given by Dm+n - = -e-g/kT amn
bmn
(VI-5)
DmDn
This condition on the ratio, however, does not specify the dependence of say a,, on m and n. It is this dependence which determines the form of the cluster size distribution during its temporal evolution. The precise functional dependence of a, upon m and n is determined by the physics which governs the approach, the accessibility, and the likelihood of bonding of the functional groups on the clusters involved. Thus, for example, clusters made up of monomers of the RAf type can in principle have bimolecular reaction rate coefficients which depend on m and n in accordance with any one of or a linear combination of the following forms: (1)a, = A , ( 2 ) a, = B ( m + n), ( 3 ) a,,, = Cmn. Here A , B, and C are all constants independent of m and n. We shall presently describe the conditions under which each of these forms for a, can arise. At this point, however, we wish to stress that each choice for the form of a, will produce a different cluster size distribution during the early evolution of the distribution. The presence of the b,, terms, as given in eq VI-5, will of course ensure that ultimately the distribution will go over into that appropriate for thermodynamic equilibrium. It is possible to see quite clearly the dependence of X, upon the am in the limit that the bonding energy is large. In this case we see from eq VI-5 that (for fixed a,) b,, 0 exponentially when g is negative and large compared to kT. This corresponds physically to the formation of strong chemical bonds between functional groups, which strongly inhibits the breakup of clusters into smaller units. Indeed under these conditions the kinetic equations become unidirectional due to the irreversibility of the chemical reactions. A second reason for the investigation of this case is the experimental finding by von Schulthess et al. that the antigen-antibody bond was not reversible under conditions in which the free antibody concentration was drastically changed. In the case that b,, 0, the kinetic equations become the well-known unidirectional equations
-
-
n-1
8, = 72 an-k,kXn-kXk k=l
m
c ankXnXk
k=l
(VI-64
along with the initial conditions X,(t=O) = Xo, X,(t=O) = 0,
n
>1
(VI-6b)
This equation was first used to study linear polymer kinetics by Dostal and Raffa31 It is a generalization of the equation used by von S m o l u ~ h o w s k isee , ~ ~also ~ ~ Chan~ d r a ~ e k h a rto , ~study ~ the coagulation of colloids. Stockmayer used it in his analysis of the condensation of RAf polymers. A particularly clear analysis of this equation and its solutions has recently been published by ZifP and the kinetics examined beyond the gel point by Ziff and Stell.% (See also Lushnikov3' and Leyvraz and Tschudi.=) Equations VI-6 can, in fact, be solved rigorously to obtain X,(t) when the bimolecular reaction rate coefficients have the following form: a, = A + B ( m + n) + Cmn (VI-7) where A , B, and C are constants independent of m and n. We present below the solutions of eq VI-7. The methods used for these solutions are to be found in Drake3-l and Ziff.35 See also von Schulthess,12Benedek,20and Cohen et al.19 Using the methods above X, can be calculated for arbitrary A , B, and C in eq VI-7. Here we present, in the interest of clarity, the results for three interesting cases: viz. (1)am = A, (2) am = B(m + n), (3) a, = Cmn. These solutions may be verified by direct substitution into eq VI-6, with the aid of the identity proven in Appendix 111. m-1 (m - s)m-s-1s8-1 2"-3 e(VI-8) ( m - s)!s! ( m - 2)!
c
Case 1. The bimolecular reaction rate is independent of the size of the interacting clusters: a, = A. As in the equilibrium case, it is useful to characterize the distribution by the mean number of bonds formed at time t per unit initially added, viz. m
b
E
C ( n - l)X,(t)/Xo
(VI-9)
n=1
In terms of this characteristic parameter, the solution to eq VI-6a and VI-6b is X,(t) = Xo(1 - b)'b"-' (VI-loa) where the time dependence of X, is contained entirely in the time dependence of b, viz. (VI-lob) b(t) = (t/TA)/(1 + ~ / T A ) where, 1/7A
= 72AXo
(VI-l0c)
As the time varies 0 C t / r A C m, b varies from 0 to 1. Equation VI-loa is of the same form as that of the ARB system at equilibrium, where b corresponds to the quantity a (see Table I). Case 2. Bimolecular reaction rate constants are proportional to the sum of the numbers in the interacting clusters: a, = B(m 4- n). (31)H.Dostal and R. Raff,Z . Phys. Chem. B , 32, 117 (1936). (32)M.von Smoluchowski, Phys. Z., 17, 585 (1916). (33)M.von Smoluchowski, 2.Phys. Chem., 92, 129 (1917). (34)S.Chandraaekhar, Rev. Mod. Phys., 15, 1 (1953). (35)R. M.Ziff, J. Stat. Phys., 23, 241 (1980). (36)R. M.Ziff and G . Stell, J. Chem. Phys., 73, 3492 (1980). (37)A. A. Lushnikov, J. Colloid Interface Sci., 65, 276 (1978). (38)F. Leyvraz and H. R. Tschudi, to be published. (39)R. L. Drake in "Topicsin CurrentAerosol Research",Vol. 3,Hidy and Brock, Ed., Pergamon Press, New York, part 2, 1972. (40)R. L. Drake and T. S. Wright, J. Atmos. Sci., 29, 537 (1972). (41)R. L. Drake and T. S. Wright, J. Atmos. Sci., 29, 548 (1972).
3710
Cohen and Benedek
The Journal of Physical Chemistry, Vol. 80, No. 19, 1982
Again we characterize the distribution by the mean number of bonds (b(t))formed per unit initially placed in solution in accordance with eq VI-9. Then the solution to eq VI-6a and its initial condition VI-6b in the present case is X,(t) = Xo(l - b)e-nb(nb)n-l/n! (VI-lla) The time dependence of X, is contained entirely in that of b, which in this case is b = 1 - e-t/m (VI-11b) with I/TB= BXo
(VI-llc)
In the present case,b approaches its ultimate value of unity much more rapidly than in case 1, as a result of the enhanced likelihood for the formation of larger size clusters produced by the linear increase of am, with m and n. The cluster size distribution is normalized for all values of 0 Ib I1. The cluster size distribution eq VI-lla is in fact the form found experimentally by von Schulthess et al.‘ It should be pointed out, however, that in those experiments the time dependence had the form b ( t ) = b,(l where b, changed with the concentration of crosslinking antibody. The presentation and analysis of that datal2 will be published elsewhere.43 Finally we note that eq VI-lla for the cluster size distribution has the same form as that predicted for the equilibrium distribution in the case of units of the form ARB,,, in the high functionality limit. Case 3. Bimolecular reaction rate constants are proportional to the product of the numbers in the interacting clusters: am, = Cmn. In this case a sol-gel transition occurs when the mean number of bonds (b)reaches 1/2. For values of b greater than 1 / 2 the normalization of the finite size clusters fails. This case has been recently discussed extensively by Ziffj5 and Ziff and Stell.% As they point out, past the gel point it may be necessary to add additional terms to the kinetic equations (eq VI-3) to properly account for the interactions of the n-mers with the gel phase. In the present work we shall confine our attention to the case b < 1/2. In this regime, eq VI-6a and VI-6b have the solution (VI-12a) The time dependence of b is given by b = t/rc 0 Ib < Y2
(VI-12b)
where 1/Tc = cxo/2
(VI-12c)
In this case the sol-gel transition occurs as a result of the great preference for the formation of large size clusters produced by the strong product dependence of am, on m and n. The distribution given in eq VI-lla is of the same form as that of the system RAj at equilibrium. It is interesting to note that am, equal to A , B(m + n), and Cmn, respectively, leads to distributions of the form achieved at equilibrium by ARB, AFIBFl (large f limit), and RA, (large f limit), respectively. In each case the form of the distribution is specified in terms of the parameter b. In the kinetic distributions b is a function of time; in the equilibrium cases b is determined by the thermodynamic parameters of the system. Furthermore, if one makes the following two assumptions, one can show that the coefficients A, B(m + n),and Cmn would be expeded to apply to the ARB, ARBkl (large
f ) ,and RAj (large f ) systems, respectively. Assume, first of all, that each unbonded functional group on an n-mer is equally likely to bond to each complementary group on an m-mer regardless of the location of the unbonded groups. Further, one must assume that the probability of an n-mer and m-mer forming a bond is governed solely by the statistical weight corresponding to the number of ways a bond could be formed between complementary sites on the two clusters. Under this assumption the bimolecular reaction rate coefficients will be proportional to the product of unreacted functional sites on each of the encountering clusters. Under the two suppositions above one can readily calculate the form of the coefficients am,. In the case of linear polymers, each cluster regardless of size has only two unbonded groups, one at each end. Thus for linear polymers we expect am = A. In the case of units of the form ARBf-l, a branched n-mer contains m - 1A-B bonds. Thus it has 1A unit and mu- 1)- (m - 1)= m u - 2) + 1B sites. Thus the bimolecular reaction rate constant describing the linkage of the one free A unit on either cluster with the numerous B sites on the other cluster is given by am,, = b[m(f - 2) + 1 + n(f - 2) + 11. In the limit of high funcb ( f - 2)(m + n) which has the form tionalityf >> 1, am B(m n). In the case of units of the form RAf, a branched cluster containing m units possesses mf - 2(m - 1) unbonded functional groups. In this case we expect am, = c(m(f - 2) + 2)(n(f - 2) + 2) which is the form used by Stockmayer.6 Notice that in the high functionality limit where f >> 1, am c(f - 2)2mn,which is of the form Cmn. However, the two assumptions listed above are not necessarily valid, particularly in diffusion-controlled reactions. In particular the assumption that the association rate constant is just proportional to the product of unreacted sites on the two clusters is clearly violated in diffusion-controlled reactions. For example, as shown by von S m o l u c h ~ w s k ifor ~ ~ the * ~ ~unidirectional diffusioncontrolled association of spheres, to first order the interaction coefficients am, are independent of m and n. Yet this system is fully equivalent to the RAj system in the high f limit. This occurs because, in diffusion-controlled reactions, the “reactivity”of sites on large clusters is reduced due to the slow translational diffusion of the large cluster. The “reactivity” of an entire cluster is to first order independent of its size because the decrease in “reactivity” of individual sites on a large cluster is just balanced by the increased number of sites on the large cluster. Thus RAj units interacting via a diffusion-controlled unidirectional mechanism yield a distribution which is identical in form with that expected for linear polymers at equilibrium. Similarly, von Schulthess et al.’ in their system which is similar to an RAf system observed a distribution which one would expect for an ARB,, system at equilibrium. Therefore, the form of the kinetic coefficients is not determined solely by the nature of the monomeric units and the bonding rules. The dependence of the kinetic coefficients on the sizes of the interacting clusters is determined by the detailed physical mechanisms governing the interaction kinetics. In connection with the kinetic evolution of sol-gel transitions, Ziffj5 has provided a very provocative conjecture as to the structure of the am, matrix needed to produce a sol-gel transition. In cases for which the am,, matrix is a well-behaved function, then the diagonal elements amm can be used to characterize the formation rate of large size clusters in the following way. For large m let a m m have the form
-
+
-
amm
-
md
for m >> 1
(VI-13)
Feature Article
Ziff then makes the conjecture that when d I1the kinetic equations yield regular solutions, i.e., there is no sol-gel transition. On the other hand, for d > 1, after a certain critical time, or critical b value, the transition to the gel state will begin. Thus from the kinetic point of view the transition from the sol to the gel state is the direct consequence of the physical mechanism governing the dependence of the rate coefficients on the size of the interacting clusters. In this context the absence of the sol-gel transition in the case of the experiments of von Schulthess et al.' is a consequence of a relatively weak increase of the bimolecular reaction rate constants as the size of the interacting cluster increases. Indeed the finding that the experimental distribution takes the form of eq VI-lla implies at least that d = 1in that case, i.e., that ummincreases only linearly with the size of the interacting clusters. VII. Summary and Future Prospects The linking of polyfunctional units into clusters of various sizes and the formation of a macroscopic phase of connected units serves as the conceptual basis for the understanding of a wide variety of important phenomena in physics, chemistry, and biology. Perhaps the most familiar example is to be found in the polymerization and gelation of polyfunctional organic molecules which is the basis of polymer chemistry. Perfectly analogous phenomena also occur in colloid chemistry, aerosol and atmospheric physics, critical phenomena and phase transitions, percolation theory, immunology, and hematology. The pioneering advances in the theory of the distribution of cluster sizes and the gel phase were made by Flory and Stockmayer in the early 1940's. In their theory, once the bonding rules between units is specified, the theoretical distribution is calculated as the most probable one consistent with an empirically determined a priori probability (a)of bond formation. In their approach neither statistical mechanics nor thermodynamics is employed. The theory contains no reference to the temperature or to the free energy of bond formation. Such dependences are in effect buried within the a priori probability a which is assumed to be experimentally determined. Stockmayer' buttressed that approach by demonstrating that the distribution obtained probabilistically for the case of the bonding of RAj units could also be obtained by the solution of a set of unidirectional kinetic equations (generalized Smoluchowski equations) provided that the bimolecular reaction rate coefficient was determined not by diffusion, but by the number of available binding sites associated with each of the combining clusters. As a result of his finding in this single special case, it became widely accepted that, given the basic bonding rules, the distribution of clusters was fully determined by the experimental parameter a in accordance with the probabilistic calculation regardless of the reaction pathway which produced that a. Thus the impetus to develop an equilibrium statistical mechanical theory for this problem was diminished and the intuitively sound notion of a distinction between the equilibrium and kinetically achieved distributions was blurred and neglected. von Schulthess et al. determined experimentally the form of the cluster size distributions for an RArB, system evolving under unidirectional kinetic control. Application of the Flory-Stockmayer probabilistictheory to this system predicts that the distribution of clusters containing n RAj units is of the same form as obtained for a simple selfcondensing system of RAf units alone. However, the experimental data demonstrate quite clearly that the actual
The Journal of Physical Chemistry, Vol. 86, No. 19, 1982 3711
kinetically evolving distribution was quite different. Stimulated by those experimental findings we have developed and present here an equilibrium statistical mechanical theory of polymerization and gelation. We also present the kinetic theories for the temporal evolution of the cluster size distributions with particular emphasis on the case of unidirectional kinetics. The essential and controlling role of the structure of the bimolecular reaction rate coefficients as the determinant of the form of the cluster size distribution emerges quite clearly from this analysis. The equilibrium statistical mechanical theories can be shown to be equivalent to the Flory-Stockmayer results in certain important special cases, viz., that the standard free energy for bond formation is a constant independent of the formation of other bonds on the same cluster, and that the clusters are Cayley trees (acyclic or branched trees). Under these conditions the cluster size distributions are determined in our theory entirely by the concentration of monomers (X,)originally added, by the temperature (0,and by the free energy of bond formation g. In effect the a priori probability a can be predicted theoretically from X,,T, and g. We have examined the various moments of the cluster size distributions corresponding to different bonding rules, in the immediately vicinity of the sol-gel transition. Following StaufferSwe have shown that, analogous to critical phenomena in systems such as the gas-liquid or magnetic order-disorder transition, the moments of the cluster size distributions diverge with certain critical exponents. Surprisingly, however, we have found that the critical exponents themselves are very different for different rules of chemical bonding between units. The exponents are clearly not invariant (universal) under changes in the bonding rules. Clearly this finding of the equilibrium theory merits further examination from the point of view of the renormalization group theories. More importantly, however, it is clear that these predictions of the theory should stimulate experimental measurements of the critical exponents. These critical exponents can be found by using measurements of the scattered light intensity, the diffusivity, and the pulse height distributions for the passage of clusters through single pores. In view of the fact that the theoretical predictions for these exponents follow in a most direct way from the fundamental assumptions (branched clusters and equal bond energies) of the theory, deviations from these predictions found experimentally should provide a very powerful means of testing and perhaps revising those assumptions. Indeed the sol-gel transition remains as the outstanding order-disorder transition which has not been examined experimentally in the important region around the critical point. A particularly interesting simplification of the equilibrium analysis emerges upon considering the limit of high functionality. This limit may at first sight appear unreasonable, particularly in the case of the polymerization of organic molecules. However, our analysis shows that while the functionality does affect the critical concentration at which the gel phase appears, the functionality does not affect the critical exponents which characterize the divergence of the moments and the growth of the gel fraction near the gel point. These exponents are fixed entirely by the bonding rules. A great virtue of the high functionality limit is a dramatic simplification of the intricate combinatorial factors which describe the statistical weight (or degeneracy factor) associated with all possible distinguishable clusters having the same number (n)of units. In the high functionality limit it becomes clear that in effect these weighting factors
3712
The Journal of Physical Chemistry, Vol. 86, No. 19, 1982
decrease with n as n-l, where T is a constant fixed by the bonding rules. As a result of this decrease it follows that the sol configuration can only accommodate a finite number of units when T > 2. The gel phase results when more units than this finite amount is placed initially in the solution. When T < 2, the sol phase can accommodate an arbitrary number of units. As a result no sol-gel transition need occur. The exponent T also determines the exponents characterizingthe divergence of the moments of the cluster size distributions. Finally, the high functionality limit provides a clear theoretical pathway for the passage from the organic polymer problem to problems such as colloidal flocculation and the condensation of aerosol particles. In the polymer problems the interacting units have a discrete number of reactive sites. In the colloid and aerosol problem the particles interact under the influence of continuous potentials such as the van der Waals or the electrostatic potentials. The continuum problems and the discrete problems have previously been treated quite independently. The high functionality limit presented here provides a unified conceptual framework for the analysis of both classes of problems. The kinetic analysis of the unidirectional reactions presented here shows quite clearly that the essential determinant for the form of the cluster size distributions is the structure of the bimolecular reaction rate coefficients aii. By structure we mean the dependence of aij upon the number of units (ij3 in each of the coalescing clusters. The analysis shows that, depending upon this structure, the kinetically evolving distributions can have a form very different from that predicted by the equilibrium theory in accordance with the chemical bonding rules. On the other hand, there are special cases in which the kinetically evolving and the equilibrium distributions can have the same form. From the physicochemical point of view one can state that if the reaction rate coefficients aij are controlled by factors (such as diffusion) other than simply the number of available bonding sides on each of the interacting clusters, then the kinetic and equilibrium distributions will be different. We do not here comment on still more subtle issues that apply to the kinetic evolution of the sol and gel in the two-phase region. Nevertheless, this two-phase system could prove to be a most interesting one to investigate important additional phenomena such as metastability and spinodal decomposition. Acknowledgment. The authors acknowledge with thanks stimulating discussions with Dr. Gustav K. von Schulthess. His emphasis on the usefulness of extant kinetic theories for the analysis of the temporal evolution of cluster size distributions markedly influenced and informed our discussions in section VI. G . B. Benedek is most grateful to Professor W. Kanzig and to the Department of Physics at the Eidgenossische Technische Hochschule, Zurich, for providing the opportunity to work out a number of elements of this paper in a series of lectures presented at the E.T.H. R. J. Cohen is grateful for support from a John A. and George L. Hartford Foundation Fellowship. This work was supported in part by Contract No. N00014-80C0500 from the Office of Naval Research, by the National Science Foundation under Grant No. PCM8013659, by the Center for Materials Science and Engineering at M.I.T. under National Science Foundation Grant No. 78-24185DMR, and by a grant from the Whitaker Health Sciences Fund. Appendix I We will prove42the identity
Cohen and Benedek
-
s =,=Ic
=1
m!
(AI- 1)
forOIX 0) 121 = 1 + 6 (AI-10) Then utilizing eq AI-9 we see that, since Re ( z ) - 1 I6
+ 6)
(AI-11)
Provided we choose 6 to satisfy h < In (1 + 6)/6
(AI-12)
rI
then r will be less than unity. For any value of 0 5 h < 1, we can find a suitable positive value of 6 to satisfy eq AI-12 because the function In (1 + 6)/6 is monotonic in 6 with the following limits In (1 + 6) lim =1 (AI-13a) 6-0 6 lim 6-m
In (1 + 6)
.
=o
(AI-13b)
(42) R. M. Ziff (private communication) has developed alternative proofs of the identities in Appendices I and 111, based on the Lagrange expansion of the function y ( x ) where x = ye?. (43) G . K. von Schultheea, G. B. Benedek, and R. W. DeBlois, Mucromolecules, in press.
The Journal of Physical Chemistry, Vol. 86, No. 19, 7982 3713
Feature Article
Now that we have shown that for any value of 0 I X < 1 we can find a contour for which the geometric series in eq AI-7 converges, we will perform the sum and obtain
This integral has simple poles at z = 0 and z = 1, which both lie within the integration contour. To compute the integral we evaluate the residues R at z = 0 and z = 1 (AI-15a)
Thus
x d(XS) - 27ri(R(z=O) + R(z=l)) = --1-X
dX
1-X
(AI-16)
or
d(XS)/dX = 1
(AI-17)
Integrating eq AI-17 and noting that S(X=O) = 1 from inspection of eq AI-1, we obtain S=l forOIX 0 we must close the contour in the lower half-plane so that the expression eiY(l*) tends to zero as 1x1 a. In this case the poles are excluded from within the contour of integration. When y < 0 we must
-
= -- 2"-3 lim (m - 2)! y ~ =-
1:
e-Ykl
e-iymm-3
1
sin y dy
+
2"-3
( m - 2)!
(AIII-7)
Q.E.D.
ARTICLES Absorption Spectra and Photochemical Rearrangements of 1-, 2-, and 3-Chlorocycloheptatriene, 7-Chloronorbornadlene, and Benzyl Chloride Cations in Solid Argon Lester Andrews,' Benuel J. Kekall, Chrslopher K. Payne, m a r R. Rodlg, and Helmut Schwarz Chemlsby Dspartment, Unlverslty of Vkgln&, Charbttesvik, Vkgln& 2290 1 end lnstitut tiK Organlsche Chemie, Technischen Unlversitat M l n , D l O O O Berlln 12, West Germany (Rec8lved: August 25, 1981; I n Flnal Fonn: June 14, 1982)
Matrix photoionization of chlorocycloheptatriene (CCHT)during condensation with excess argon at 20 K produced new broad 484-, 517-, and 544-nm bands, a sharp 469.3-nm absorption, and weak 453- and 468-nm bands. The broad bands are assigned to the 1-, 2-, and 3-chlorocycloheptatrienecations. Photolysis with 500-1000-nm radiation decreased the broad bands and increased the sharp 469.3-nm absorption. The 469.3-nm band and a sharp 707.8-nm counterpart in benzyl chloride experiments suggest a conjugated triene cation identification with 6-chloro-5-methylene-1,3-cyclohexadiene cation (3)as the most reasonable possibility. The 453- and 468-nm bands, which markedly increased on near-UV photolysis in benzyl chloride experiments, were strong products in similar studies with 7-chloronorbornadiene. Photolysis with 470-1000-nm radiation decreased the 468-nm band and increased the 453-nm absorption; this interconversion was reversed with 370-460-nm radiation. The latter bands are probably due to E and 2 isomers of 5-(chloromethylene)-1,3-cyclohexadienecation (6), which isomerize upon photoexcitation.
Introduction The ring-expanded chlorocycloheptatriene cation has been postulated as an intermediate in gas-phase rearrangment processes involving chlorotoluene (1) Tait, J. M.S.; Shannon, W. T.; Harrison, A. G. J.Am. Chem. SOC. 1962,84,4.
(2) Yeo, A.
N.H.; Williams, D. H. Chem. Commun. 1970, 886. 0022-3654/82/2086-37 14$01.25/0
This ring-expansion process is thought to be the rate-determining Step for Chlorine atom 10SS from chlorotoluene cations.3 Very recent matrix photoionization experiments have produced cycloheptatriene cation; photolysis into the cy(3) Stapleton, B. J.; Bowen, R. D.; Williams, D. H. J. Chem. SOC., Perkin Trans. 2 1979, 1219.
@ 1982 American Chemical Society
