J . Phys. Chem. 1992, 96,60834086
References and Notes (1) (a) Kalyanasundaram, K. Photochemistry in Microheterogeneous Systems; Academic Press: Orlando, 1987; Chapter 2 for micelles, Chapter 8 for polymers. (b) Thomas, J. K. The ChemistryofExcitation at Interfaces; ACS Monograph Serica; American Chemical Society: Washington, DC,1984, and references therein. (2) (a) Meisel, D.; Matheson, M. S . J . Am. Chem. Soc. 1977,99.6577. (b) Meyastein, D.; Rabani, J.; Matheson, M. S.; Meisel, D. J. Phys. Chem. 1978,82, 1879. (c) Meisel, D.; Rabani, J.; Meyerstein, D.; Matheson, M. S. J . Phys. Chem. 1 W , 82,985. (d) Sassoon, R. E.; Rabani, J. J. Phys. Chem. 1980,81, 1319. (e) Kelder, S.; Rabani, J. J . Phys. Chem. 1981, 85, 1637. ( f ) Duveneck, G. L.; Kumar. C. V.; Turro, N. J.; Barton, J. K. J. Phys. Chem. 1988,92, 2028. (3) (a) Kumar, C. V.; Barton, J. K.; Turro, N. J. J. Am. Chem. Soc. 1985, 107, 5518. (b) Barton, J. K.; Goldberg, J. M.; Kumar, C. V.; Turro, N. J. J. Am. Chem. Soc. 1986, 108, 2081. (c) Pyle, A. M.; Rehmann, J. P.; Meshoyrer, R.; Kumar, C. V.; Turro, N. J.; Barton, J. K. J. Am. Chem. Sm. 1989. 111, 3051. (d) Friedman, A. E.; Chambron, J.-C.; Sauvage, J.-P.; Turro, N. J.; Barton, J. K. J . Am. Chem. Soc. 1990, 112, 4960. (e) Kelly, J. M.; Tossi, A. B.; McConnell, D. J.; Ohuigin, C. Nucleic Acids Res. 1985, 13,6017. (f)GBmer. H.; Tossi, A. B.; Stradowski, C.; Schulte-Frohlinde,D. J . Photochem.Photobiol. E 1988,67. (g) Kirsch.De Mesmaeker, A,; Orellana, G.; Barton, J. K.; Turro, N . J. Photochem. Photobiol. 1990, 52, 461. (4) (a) Moreno-Bondi, M.; Orellana, G.; Turro, N. J.; Tomalia, D. A. Macromolecules 1990,23,910. (b) Caminati, G.; Turro, N. J.; Tomalia, D. A. J . Am. Chem. SOC.1990,112,8515. (c) Gopidas, K. R.; Leheny, A. R.; Caminati, G.; Turro, N. J.; Tomalia, D. A. J. Am. Chem. Soc. 1991, 113, 7335. ( 5 ) Meyer, T. J. Pure Appl. Chem. 1990,62, 1003, and references therein. Rodgers, M. A. J. J. Phys. Chem. 1982,86,4906. (6) (a) Baxendale, J. H.; (b) Baxendale, J. H.;Rodgers, M. A. J. Chem. Phys. Lot?. 1980,72,424. (c) See also: Tominaga, T.; Matsumot, S.; Koshiba, T.; Yamamoto, Y. Chem.
6083
Lett. 1988, 1427, for evidence of premicellar aggregates using Fe(phen), as a probe and tracer diffusion as an experimental technique. (7) (a) Drtssick, W. J.; Cline, J., III; Demas, J. N.; DcGraff, B. A. J . Am. Chem. Sm. 1986,108,7567. (b) Hauensteh, B. L., Jr.; Dressick, W. J.; Buell, S.L.; Demas, J. N.; DeGraff, B. A. J. Am. Chem. Soc. 1983, 105, 4251. (8) Sano, H.; Tachiya, M. J . Chem. Phy. 1981, 75, 2870. (9) Turro, N. J.; Barton, J. K.; Tomalia, D. A. Acc. Chem. Res. 1991,24, 332. (10) Details of the systhesis and characterization will be published elsewhere. (1 1) (a) Ottaviani, M. F.; Baglioni, P.; Martini, G. J. Phys. Chem. 1983, 87, 3146. (b) Baglioni, P.; Ferroni, E.; Martini, G.; Ottaviani, M. F. J . Phys. Chem. 1984,88, 5107. (c) Baglioni, P.; Ottaviani, M. F.; Martini, G. J . Phys. Chem. 1986, 90, 5878. (12) (a) Kivelson, D. J . Chem. Phys. 1960,33, 1094. (b) Jolicoeur, C.; Friedman, H. L. Eer. Bunsen-Ges Phys. Chem. 1971,75,248. (c) Schreirer, C. F.; Polnaszek, C. F.; Smith, I. C. Eiochim. Eiophys. Acta 1978,515,395. (d) Goldman, S. A.; Bruno, G. V.; Polnaszek, C. F.; Freed, J. H.J . Chem. Phys. 1972,56,716. (13) Schneider, D. J.; Freed, J. H. In Eiological Magnetic Resonance; Berliner, L. J., Reuben, J., Eds.;Plenum Press: New York, 1989; Vol. 8, Spin Labelling Theory and Applications, pp 1-76. (14) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffman, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J . Phys. Chem. 1976, 80, 905. (1 5 ) Schreier-Muccilo,S.;Marsh, D.; Dugas, H.; Schneider, H.; Smith, I. C. P. Chem. Phys. Lipids 1973, 10, 11. Freed, J. H. In Spin Labelling Theory and Applications; Berliner, L. J., Ed.; Academic Press: New York, 1976; Vol. 1, p 85. ( 16) Martini. G.: Ottaviani. M. F.: Romanelli. M. J. Colloid InrerfhceSci. 1983, 105, 94. Martini, G.;'Ottaviani, M. F.: Romanelli, M. J.*Colloid Interface Sei. 1987, 115, 81.
Remarks on the Assoclatlon of Rodlike Macromolecules In Dilute Solution Paul v a n der Scboot Department of Polymer Technology, Faculty of Chemical Engineering and Materials Science, Delft University of Technology, P.O. Box 5045, 2600 GA Deut, The Netherlands (Received: February 3, 1992; In Final Form: March 31, 1992)
It is argued that van der Waals attraction can induce the parallel association of rodlike macromolecules in solution. Theories describing micellization and condensation are applied to analyze the stability of these aggregates. We conclude that aggregates consisting of only a small number of rods can be stable, at least in principle.
I. Introduction Association of polymers in solution is often discussed in terms related to the chemical details of the molecules, focusing on mechanisms that involve intermolecular hydrogen bonding, interactions between charged moieties, structural transitions, and so forth. A perhaps less specific mechanism may on the other hand be given by bare van der Waals interactions. Initially one expects these to be too weak to provide the physical bonds that keep the macromolecules in the aggregate together-for, if they were not weak the solution would most likely phase separate. But when the polymers are rigid, rodlike,' and of large aspect ratio the situation is slightly more delicate. As was pointed out in ref 2, dispersion forces between such long rodlike macromolecules are essentially short-ranged and, as a result of that, highly anisotropic. So, even when the attraction between two (nearly) touching c r d rods is weak, its effect can easily become sizeable in case the rods adapt a parallel c~nfiguration.~The strong anisotropy displayed by the dispersion interaction expresses itself for instance in the second virial coefficient, where nearly parallel configurationsof the macromolecules are weighted heavily.* Note that this is in spite of the tiny angular phase volume that can be attributed to two almost aligned rods. The van der Waals attraction may very well induce the association of rodlike polymers in solution for similar reasons. In this paper we investigate the aggregation or association of rodlike macromolecules into "bundles" as a result of van der Waals'
type of attractive interactions. The relevant thermodynamic quantity determining the size distribution is the chemical potential of the rods, which we calculate approximately by applying notions from the theory of micellizationc6 and the (physical cluster) theory of the condensation of gases?+ First the basic expression for the chemical potential is determined without explicitly specifying the contributions from the internal degrees of freedom of the aggregates (section 11). In section I11 this internal free energy is estimated in the continuum limit, i.e. for aggregates consisting of a large number of rods. There we discuss the known result that two-dimensional growth of aggregates is in principle ~ n b o u n d . ~ So, if stable or metastable aggregates of finite size are feasible, these can probably only be found in the opposite l i t , at relatively small aggregation numbers. The analysis of section IV confirms that aggregation of rods in parallel configurations is possible for values of the model parameters that appear not unreasonable. Some concluding remarks are given in the last section. 11. The Chemical Potential of the Rods
Consider a solution of mutually impenetrable rods that interact via a short-ranged attractive potential. The length L of the rods is much greater than the width D. We follow the route prescribed by physical cluster theory and formulate a plausible criterion whereupon the distinction between free and bound particles is made."' Two rods are defined neighbors of the same cluster or bundle when (1) their center lines have a relative angle smaller
0022-3654/92/2096-6083$03.00/00 1992 American Chemical Society
6084 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992
than a unspecified but tiny value, ( 2 ) the surface to surface distance of the rods is also smaller than an arbitrary but very small
value, and (3) the overlap of the rods along the center lines is large enough. We are deliberately vague because the conccpt of a cluster is somewhat arbitrary? Also, and more importantly, a precise intermolecular potential is at present unknownn2 Intuitively one would expect that an association-dissociation equilibrium exists between clusters of different sue (and shapeL2) and immediately write down the equilibrium condition
= sp ( s 1 1) (2.1) where p denotes the chemical potential of a monomer or single rod and p ( s ) that of a s-mer, Le., of a bundle consisting of s rods. An (approximate) statistical mechanical justification of eq 2.1 can be given, albeit that the concept of a cluster has to be introduced explicitly and does not follow from the analysis (see for instance refs 4 and 1 1 ) . For the simplified case of a gas of rods the derivation can be outlined as follows. (i) Starting point is the grand partition function of the gas of rods. (ii) The cluster concept is introduced. We define for each configuration of rods a subdivision into a distribution of clusters in accordance with the previously given criteria. The partition function of a given number of rods is then written as a summation over all possible subdivisions and configurations consistent with each subdivision. (iii) Next the phase space coordinates are transformed to center-of-mass and orientation coordinates relative to the cluster the individual rods belong to. (iv) The internal coordinates of the clusters and the coordinates describing the clusters as a whole are decoupled. A cluster is thus regarded as supramolecular particle with independent internal degrees of freedom. (v) Interactions between the aggregates are dealt with in the virial approach. This leads to slightly different definitions of the usual (mathematical) cluster integrals, since we have to exclude codigurations that interfere with a given subdivision into bundles. (vi) The grand partition function has now obtained a form in which the absolute activities and cluster integrals pertain to aggregates instead of individual rods. The chemical potential related to the absolute activity of a cluster of size s is given by eq 2.1 and results directly from the use of the grand partition function. We furthermore obtainL3 j3p(s) = j3po(s) In ~ ( s ) virial terms (2.2) with (2.3) B P O ( ~ =) -In (Qint(s) Qrot(s) QtranAS)/ v) and /3 = l/kBT. Here kB denotes the Boltzmann constant, T the absolute temperature, and Y the volume of the system. The Q s represent the partition functions for the internal, rotational, and translational degrees of freedom of an s-mer and u(s) its number density, i.e. the number of s-mers per unit volume. We drop the virial terms in eq 2.2 for the sake of simplicity. As association this appears not already occurs at extremely low concentrati~ns,'~ too grave an approximation. The basic equations governing the cluster size distribution ~ ( s ) of the gas of rods are given by eqs 2.1 to 2.3. It has bem arguedM that the presence of solvent does not alter these equations much and that solvent effects are in fact naturally absorbed into the internal partition function Q&). Here we also suppose that eqs 2.1 to 2.3 describe the aggregation of rods in solution, except for the modified internal partition function. The translational part of the standard chemical potential k ( s ) gives p(s)
+
Qt,,(s)
+
/ V = s3/2A-3
(2.4)
where A is the thermal wavelength of a single rod of mass m A = (Bh2/2~m)L/2
(2.5) and Planck's constant is denoted by h. In order to determine the rotational contribution, Qrot,we require, at least in principle,
van der Schoot information about the precise shape of the aggregate. But if the aggregate consists of very long rods, Qra can only have the form
irrespective of its detailed structure. The power q depends on the shape of the aggregate and&) is some complicated function of the number of rods and the way in which they are arranged in the aggregate. For a cylindrical bundle of a large number of perfectly aligned rods q = 2 andfls) s2 (seealso the Appendix). We will simply assume that q is a constant and thatfls) is proportional to an unspecified power of s for all s 1 1. This implies that possible changes in the growth type. are suppressed. It should be noted that a transition from two-dimensional to three-dimensional growth is not entirely inconceivable: rods may for instance adsorb onto the ends of bundles once these have become wide enough. Effects of this kind will however not alter the conclusions of our qualitative analysis. Deferring a discussion of Qintfor a moment, eqs 2.2 through 2.6 yield j3p(s) -n In s + In #(s) - K + F(s) (2.7) Q:
=
for s 1 1 , with #(s) s v ( s ) l r U P / 4 the volume fraction of s-mers present in the solution and F(s) = -In Qinttheir (internal) free energy in units of kBT. It is expedient to subtract a constant F(1) from the chemical potential of a rod p(s)/s or, equivalently, substitute F(s) = F(s) - sF( 1 ) . This merely constitutes the introduction of a reference state. The parameters n and K are to be taken as adjustable in view of the approximate character of the analysis, with n z 0(1) and K O(10). We refer to the Appendix for a justification of these estimates. Equation 2.7 has to be supplemented with an expression for the internal free energy that somehow reflects the complexity of an aggregate. But at the same time our inexact definition of an aggregate does not allow for too great a detail in its description. In any event, there is no simple recipe available by which an account of all the internal degrees of freedom can be given to any degree of accuracy. Following theories in the related fields of micellizati~n~*~ and nu~leation,'.~ we instead rely on semiintuitive arguments and formulate what appears a plausible form for F(s). HI. The Continam Limit The internal free energy can be written as F(s) = V(s)- T S ( s )
(3.1)
where U(s)and S(s) denote an internal energy and entropy (both in units of k e n . U is roughly equal to the (free) energyI5difference of an s-mer of average configuration and a collection of s noninteracting rods. The entropic term S accounts for the internal vibrations, rotations, and so forth of the s-mer. Let the aggregate contain a large number of rods (s >> 1). The internal energy per rod, U(s)/s,is proportional to the average number of nearest neighbors a rod has in a cluster times the average energy of interaction of two neighboring rods. Assume that the macromolecules are packed together in a twodimensional hexagonal array. Those in the center of the aggregate have six neighbors while the ones on the surface have three or four. Hence
U(s)z -3Es(l - constant X d2) (3.2) with E the energy gain of dimerization in solution. E is a statistical average over the overlap lengths consistent with the definition of a cluster and the constant associated with the surface term s-II2 has a value close to unity. To estimate the entropic contribution to the free energy, we use the fact that for very large aggregates S(s) should be extensive in s TS(s)2 s s (3.3) with S > 0 an adjustable parameter. We guess that S @lo) (see the Appendix). Clearly, since surface effects are neglected, eq 3.3 represents an ideal entropy. This is not as inconsistent with our previous handling of the internal energy as it seems, because rotation5 of rods about their main axis may for instance produce
=
Association of Rodlike Macromolecules in Dilute Solution
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 6085
a term linear in s. Equations 3.2 and 3.3 now give
= - F'S + E'S'/' (3.4) where E' = 3E and F' = 3E + S should be regarded as free F(s)
parameters. We infer from refs 2 and 16 that a reasonable value for the (average) attractive energy is E = 10,so F'and E'are probably of order 10. Equation 3.4 is the continuum limit to the internal free energy of a bundle. When polydispersity is suppressed, the optimal cluster size can be foundI7 by minimizing p(s)/s. It is not difficult to see that for eqs 2.7 and 3.4the derivative a(p/s)/as = 0 yields an optimal s with a2(p/s)/as2 0+, indicating that the clusters tend to grow to macroscopic structures. This happens when the chemical potential per rod is smaller in the aggregated state than in the nonaggregated state (at equal total volume fraction 4). If we set 4(s) = and p( 1) z In 4 - K the condition for aggregation is given by limFm s-'p(s) - p( 1) - F'- In 4 + K < 0. In other words, F' < -In 4 + K if the solution is to be stable against seggregation at volume fractions lower than a volume fraction 4. In addition to the minimum we find a maximum in the chemical potential per rod when 1 n In s - n - In I#J - -E's'/' K =0 (3.5) 2 It is equivalent to the free energy barrier found in the nucleation of droplets from the gas phase and usually discussed in terms of supersaturation phenomena and the emergence of (metastable) pretransitional aggregate^?.^ In our cast both stable and metastable aggregates are found outside of the continuum limit, i.e., at low aggregation numbers.
--
-
=
+
IV. Clusters Consisting of a Small Number of Rods The continuum description of the internal free energy breaks down for clusters consisting of only a few rods, say s = O(1). We modify eq 3.4 by replacing the continuum limit of the internal energy by
=
U(S) -Ep(s) (4.1) with p ( s ) the number-of interacting pairs in a cluster of size s. We setp = 1, 3, 5 , 7 , 9 , 12,..., for s = 2, 3,4,5,6, 7, ..., giving the maximum number of neighboring pairs in a two-dimensional hexagonal lattice (so again effects of isomerism are absorbed in the values of the parameters). Although we realize that the expression for the cluster entropy given by eq 3.3 is highly approximate in the limit of small s, we nevertheless apply this expression for the reasons stated at the end of section 11. Equations 3.3 and 4.1 thus give F(s) 1-Ep(s) - SS (4.2) for s > 1, Recall that due to our definition F( 1) 0. In the limit >> 1 eqs 4.2 and 3.4 should give (virtually) identical results. Now consider the reversible work that must be expended in order to create clusters of a certain size from the single rods. The relevant quantity is s
G ( 4 = B(S'IL(4- 4 1 ) ) (4.3) and gives the change in (Gibbs) free energy upon aggregation. G is given in units of kBTand on a per rod basis. Note that p(s) and p(1) pertain to an identical volume fraction 4. Typical features of G versus s are plotted in Figure 1. Each curve represents a class of curves we found by systematically varying the model parameters in the indicated ranges. A negative value of G indicates a preference for the aggregated state and a positive value indicates a preference for the nonaggregated state. Curves I and I1 are of regimes where the system is stable against macroscopic aggregation. At the conditions of curve I most rods are in the monomeric state, while those of w e I1 are primarily bound into (say) dimers and trimers. Curve I11 represents, in principle, an unstable system. However, if the maximum is high enough, condensation may be inhibited (the nucleation rate7.9is roughly proportional to exp[-s*G(s*)], where s* maximizesI8 G). This makes the small aggregates that form under these conditions
+ I
I
t -I
I
V
1
Figure 1. Schematic representation of the function G(s) as defined in eq 4.3. Results for a number of regimes are indicated. For the parameters we chose: n = @I), K = @lo), -log 6 = 0(1),S = 0(10), and E L 0. The uppermost curve represents a system which has high value of K and low values of the other parameters. The lowest curve is one where K is low and the other parameters have high values. For a discussion of the various regimes, see the text.
I
-6 0
6
10
16
20
S
Figure 2. G as a function of s for three volume fractions of rods: 4 = lod (drawn curve), 4 = 10-4 (dashed curve), and 6 = 1V2(dotted curve). K = 37, S = 18, E = 8, and n = 4.5. The arrows indicate the limiting values when s =*
m.
metastable. Curves IV and V are those of from the outset unstable systems. The maximum in G versus s is found somewhere in the range 5 S s S 25 for our choice of parameters, while negative values of G pertaining to stable or possibly metastable clusters are restricted to aggregation numbers between unity and roughly ten. It is interesting to note that a choice of S / K
