Impact of Steric Interactions on the Helical Transition in Assemblies of

446

Langmuir 2006, 22, 446-452

Impact of Steric Interactions on the Helical Transition in Assemblies of Discotic Molecules Jeroen van Gestel* Physical Chemistry and Molecular Thermodynamics Group, Technische UniVersiteit Delft, P.O. Box 5045, 2600 GA Delft, The Netherlands

Paul van der Schoot EindhoVen Polymer Laboratories, Technische UniVersiteit EindhoVen, P.O. Box 513, 5600 MB EindhoVen, The Netherlands ReceiVed August 11, 2005. In Final Form: October 19, 2005 We investigate theoretically the effect of excluded-volume interactions on the helical configurational transition of supramolecular assemblies in solutions of chiral disklike molecules. To this end, we set up a second-virial theory within the context of the helical self-assembly of rodlike objects. We find that interaggregate interactions shift both the helical-transition point and the sharpness of the transition. For realistic values of the model parameters, the helicaltransition temperature shifts by several degrees, and the more so the higher the concentration of assembling material. The mean aggregation number is also affected by the interactions, albeit only by a modest amount, unless the solution becomes very concentrated.

1. Introduction There is a range of synthetic molecules of quite diverse molecular weight and chemical structure that in solution selfassemble into helical superstructures.1-10 The current interest in assemblies of this type is not only due to their resemblance to those found in biology11-16 but also because helical aggregates tend to exhibit a strong growth that makes them potentially useful as gelators.17-19 The all-or-nothing theory of Oosawa and Kasai for helical aggregation12 as well as our recent generalization of it18,20 provide a good description for sufficiently dilute solutions * Corresponding author. E-mail: [email protected]. (1) Van der Schoot, P. In Supramolecular Polymers, 2nd ed.; Ciferri, A., Ed.; Taylor and Francis: Baton Rouge, LA, 2005. (2) Brunsveld, L.; Zhang, H.; Glasbeek, M.; Vekemans, J. A. J. M.; Meijer, E. W. J. Am. Chem. Soc. 2000, 122, 6175. (3) Hirschberg, J. H. K. K.; Brunsveld, L.; Ramzi, A.; Vekemans, J. A. J. M.; Sijbesma, R. P.; Meijer, E. W. Nature 2000, 407, 167. (4) Van Nostrum, C. F.; Bosman, A. W.; Gelinck, G. H.; Schouten, P. G.; Warman, J. M.; Kentgens, A. P. M.; Devillers, M. A. C.; Meijerink, A.; Picken, S. J.; Sohling, U.; Schouten, A.-J.; Nolte, R. J. M. Chem. Eur. J. 1995, 1, 171. (5) Yoshida, N.; Harata, K.; Inoue, T.; Ito, N.; Ichikawa, K. Supramol. Chem. 1998, 10, 63. (6) Brunsveld, L.; Meijer, E. W.; Prince, R. B.; Moore, J. S. J. Am. Chem. Soc. 2001, 123, 7978. (7) Gallivan, J. P.; Schuster, G. B. J. Org. Chem. 1995, 60, 2423. (8) Lovinger, A. J.; Nuckolls, C.; Katz, T. J. J. Am. Chem. Soc. 1998, 120, 264. (9) Fenniri, H.; Mathivanan, P.; Vidale, K. L.; Sherman, D. M.; Hallenga, K.; Wood, K. V.; Stowell, J. G. J. Am. Chem. Soc. 2001, 123, 3854. (10) Engelkamp, H.; Middelbeek, S.; Nolte, R. J. M. Science 1999, 284, 785. (11) Klug, A. Angew. Chem., Int. Ed. Engl. 1983, 22, 565. (12) Oosawa, F.; Asakura, S. Thermodynamics of the Polymerization of Protein, 1st ed.; Academic Press: London, 1975. (13) Korn, E. D. Physiol. ReV. 1982, 62, 672. (14) Ciferri, A. Prog. Polym. Sci. 1995, 20, 1081. (15) Friedhoff, P.; von Bergen, M.; Mandelkow, E.-M.; Davies, P.; Mandelkow, E. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 15712. (16) Watts, N. R.; Misra, M.; Wingfield, P. T.; Stahl, S. J.; Cheng, N.; Trus, B. L.; Steven, A. C.; Williams, R. W. J. Struct. Biol. 1998, 121, 41. (17) Van der Schoot, P.; Michels, M. A. J.; Brunsveld, L.; Sijbesma, R. P.; Ramzi, A. Langmuir 2000, 16, 10076. (18) Van Gestel, J.; Van der Schoot, P.; Michels, M. A. J. J. Phys. Chem. B 2001, 105, 10691. (19) Terech, P.; Weiss, R. G. Chem. ReV. 1997, 97, 3133. (20) Van Gestel, J.; Van der Schoot, P.; Michels, M. A. J. Langmuir 2003, 19, 1375.

of these aggregates.17,21 However, at higher concentrations of aggregating material, in the regime potentially most relevant to gelation, interactions between the aggregates should become important, e.g., through a modified mass action or through effects that stem from changes in the configurational statistics of the assemblies.22,23 In the present paper, we examine the effect of excluded-volume interactions on the conformational state and the size distribution of helical self-assembled polymers, focusing on concentrations that are low enough to prevent any liquidcrystalline states from becoming stable. For rigid rods, this occurs at packing fractions roughly equal to the ratio between their diameter and length and for semiflexible ones at packing fractions proportional to the diameter divided by the persistence length, irrespective of whether the rods are self-assembled or not.24 Note that for (self-assembled) polymers that may undergo a helixcoil type transition the situation is more complex, because the coil-to-helix transition is usually accompanied by a decrease in the flexibility of the aggregate and in the effective interaction length. This may lead to a coupling between the helical conformational transition and liquid-crystalline phase transitions. In the following, we focus solely on the isotropic case and postpone an analysis of liquid-crystalline states for future work. We shall presume the aggregates to be reasonably described as rigid and rodlike and to interact through steric repulsion alone, allowing us to apply Onsager’s second-virial theory for interactions between rigid cylinders.25 This is not always realistic, e.g., if the aggregates consist of alternating helical and coillike conformations or if they are not slender. However, molecules have been synthesized whose aggregates can indeed be described as rodlike, irrespective of their helical content. An example is the discotic compound described by Brunsveld et al.,2 which self-assembles into stacks that can undergo a cooperative helical conformational transition, as evidenced by circular dichroism (21) Brandt, H. C. A.; Hendriks, E. M.; Michels, M. A. J.; Visser, F. J. Phys. Chem. 1995, 99, 10430. (22) Kim, Y. H.; Pincus, P. Biopolymers 1979, 17, 2315. (23) Matsuyama, A.; Kato, T. J. Chem. Phys. 2001, 114, 3817. (24) Van der Schoot, P.; Cates, M. E. Langmuir 1994, 10, 670. (25) Onsager, L. Ann. N.Y. Acad. Sci. 1949, 51, 627.

10.1021/la0521903 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/23/2005

Helical Transition in Discotic Molecules

Figure 1. Cartoon of a rodlike aggregate of discotic monomers, containing helical and nonhelical bonds. Indicated are the lengths of the helical and nonhelical bond, l1 and l2, the thickness of the monomers D, their transverse dimension d, and the overall length of the aggregate, L. Helical and nonhelical regions are present in the aggregate, corresponding respectively to monomers that are “locked” in place, and monomers which are more or less free to rotate.

(CD) spectroscopy. In the nonhelical bonded state, there is plausibly a relatively large distance between two consecutive monomers allowing for freedom of rotation, whereas in the helical state, the monomers are locked into place; molecular chirality is then translated into supramolecular chirality. See Figure 1 for a schematic representation of this type of assembly. Interactions between “dead” polymers susceptible to helixcoil type transitions22,23 have been described earlier in the context of the isotropic-nematic transition, as have interactions between self-assembled, “living”, flexible chains.21,26,27 Here, we consider the effect of interactions on polymers that exhibit both properties, i.e., a helical configurational transition and an annealed (equilibrium) size distribution. Since excluded-volume interactions are predicted to influence the mean size of supramolecular polymers, both for rod-shaped ones and flexible coils,21,24,26,28-31 it stands to reason that they could have a similar effect in helical self-assembled aggregates. In fact, because there is a coupling between the size and the conformational state of (partially) helical aggregates,17,18 both of these properties are potentially affected in a nontrivial way by volume exclusion. As we shall see, the helical state may be enhanced or repressed depending on the difference in the binding length of the helical and nonhelical bonded states, causing a shift in the onset of the helical transition, as well as a broadening of this transition. This also influences the mean aggregate size, because the length of the interacting aggregates strongly depends on their conformational state. This latter effect is modest, unless the concentration of assembling molecules is high. The remainder of the paper is set up as follows. In section 2, we first describe a mean-field theory linking the conformational state and the size of the aggregates to the interaggregate interactions for the case of long, rigid aggregates dispersed in isotropic solution. Next, in section 3, we present the results of our calculations and find that the fraction of helical bonds can increase or decrease drastically under the influence of the interactions, depending on the difference in length between the helical and nonhelical bonds, the concentration of aggregating monomers, and the degree of cooperativity of the helical transition. Volume exclusion also shifts the onset of this transition to a (26) Gelbart, W. M.; Ben-Shaul, A.; McMullen, W. E.; Masters, A. J. Phys. Chem. 1984, 88, 861. (27) Van der Schoot, P.; Cates, M. E. Europhys. Lett. 1994, 25, 515. (28) Ben-Shaul, A.; Gelbart, W. M. J. Phys. Chem. 1982, 86, 316. (29) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matt. 1990, 2, 6869. (30) Odijk, T. J. Phys. France 1987, 48, 125. (31) Scha¨fer, L. Phys. ReV. B 1992, 46, 6061.

Langmuir, Vol. 22, No. 1, 2006 447

different value of the free energy for the formation of a helical bond, and in addition, it changes the sharpness of the helical transition. Finally, a change in the mean length of the aggregates and their mean degree of polymerization, while somewhat less pronounced, is also found. Unfortunately, at present, we cannot confront these predictions with experiments. This is because, to our knowledge, no experimental results have been reported that describe the conformational state and size of helical aggregates in the long-chain regime, where excluded-volume-type interactions become important. Our hope is that the present work may be validated through future experimental studies. We conclude the paper, in section 4, by summarizing our findings. Here, we also briefly discuss why we believe volume exclusion has no effect on a phenomenon that potentially plays a large role in helical supramolecular polymers, known as chirality amplification.

2. Theory We consider an isotropic solution of discotic molecules that self-assemble into long, rodlike aggregates, i.e., aggregates with a persistence length very large compared to their mean linear extension. The discotic monomers can bind to each other in two ways: one that leads to a helically ordered configuration and one that does not. See Figure 1. We presume the aggregates to be infinitely rigid and smooth and to interact with each other solely through a harshly repulsive, excluded-volume interaction. To calculate its effect on the self-assembly, we invoke the secondvirial approximation, which should be accurate if the mean length of the aggregates is much larger than their transverse dimension.25,32,33 This requires quite large aggregation numbers, because aggregates consisting of discotic molecules tend to have a large transverse dimension. Let us first deal with the ideal case of noninteracting aggregates: interactions turn out to only renormalize the various parameters of this model. The Hamiltonian of the one-dimensional model, which describes experiments in dilute solution remarkably well, may be written as17,18,34

1 N-2 1 N-1 Hid ) - R (sisi+1 - 1) + P (si + 1) - (N - 1)E 2 i)1 2 i)1 (1)

∑

∑

for the number of monomers that comprise an aggregate N > 2, where the subscript id reminds us that we are dealing with noninteracting, ideal chains. (We need not specify H for N ) 1 and 2, because we shall be assuming the mean aggregation number to be large.) In eq 1, P is the difference in free energy between a helical and a nonhelical bond, R is the free energy penalty of a molecule involved in one helical and one nonhelical bond along the chain, and -E is the free energy of a nonhelical bond. Note that these free energies, like all energies in this paper, are given in units of thermal energy kBT, with kB Boltzmann’s constant and T the absolute temperature. Finally, the quantity si ) (1 describes the state of a bond: +1 for a helical and -1 for a nonhelical one. The fraction helical bonds in a single aggregate of arbitrary configuration {si} is then given by θ ) θ({si}) ) N-1 (2N - 2)-1∑i)1 (si + 1). In the ground-state approximation, the canonical partition function Q(N) of a single chain can by standard methodology be shown to take the following form: (32) Vroege, G. J.; Lekkerkerker, H. N. W. Rep. Prog. Phys. 1992, 55, 1241. (33) Khokhlov, A. R.; Semenov, A. N. Physica 1981, 108A, 546. (34) Notice that in the Ising model R has the role of coupling constant and P that of magnetic field.

Van Gestel and Van der Schoot

448 Langmuir, Vol. 22, No. 1, 2006

Q(N) )

∑ exp - H ∼ (λ+ exp E)N exp - Ecap s )(1

(2)

i

where we insert for the Hamiltonian the ideal-solution one, eq 1, so H ) Hid. Formally, this equation is only valid if N . 1, but we may extrapolate it down to N g 1, provided the mean aggregate size remains large. Here, λ+ ) 1/2 + s/2 + 1/2[(s - 1)2 + 4sσ]1/2 with s ≡ exp(-P) the Boltzmann weight of a helical bond, and σ ≡ exp(-2R) the square of the Boltzmann weight of two unequal consecutive bonds (or a spin flip in Ising language); they have identical meanings in the Zimm-Bragg theory of the helix-coil transition of biopolymers.35 The latter parameter is often seen as a measure of the cooperativity of the helical transition.17,35 We furthermore have for the so-called endcap energy Ecap ) E - ln y + 2 ln λ+, where y is a function that depends on the boundary conditions imposed on the first and last bond of the assembly.20 In the current paper, we study two boundary conditions: one where one end is kept nonhelical and one is left free for which y ) (λ+ - s + sxσ)/[(1 - s)2 + 4σs]1/2, and one where one end is kept helical and the other unrestrained for which y ) (sλ+ - s + sxσ)/[(1 - s)2 + 4σs]1/2. The former is known to describe the conformational state of Brunsveld’s discotic molecules well,20 whereas we include the latter to gauge the impact of the choice of boundary conditions on the size and conformation of the interacting assemblies. See ref 20 for a discussion of the importance of the boundary conditions for ideal assemblies. From the partition function, eq 2, we calculate the mean fraction of helical bonds 〈θ〉 in the long-chain limit, by taking the derivative of ln Q(N) with respect to ln s and dividing it by the number of bonds. This gives17,36

1 1 〈θ〉 ) + (s - 1)[(s - 1)2 + 4σs]-1/2 2 2

(3)

independent of the aggregation number in the limit N . 1 where the ground-state approximation holds.17 We can also determine the mean aggregation number, given by ∞

〈N〉 ≡

∑ N)1

∞

NF(N)/

∑ F(N) N)1

(4)

where F(N) denotes the number density of aggregates of degree of polymerization N, given by F(N) ) ν-1Q(N) exp(µN) with µ the chemical potential of the discotic molecules and ν the volume of a monomer.1 Equation 4 reduces to

〈N〉 ∼ xφ exp Ecap . 1

(5)

in the long-chain regime, where the conservation of the concentration of dissolved material, φ, defined as ∞

φ≡ν

∑ NF(N)

(6)

N)1

is used to eliminate the chemical potential, µ. Now, we turn on steric interactions between the rodlike assemblies. The excluded volume of two rods with lengths L and L′ is to leading order equal to 2LL′d| sin γ|, where d is the width of the rods (in this case, the diameter of the discotics) and γ is the angle between them.27,32 The correction terms of order Ld2 (35) Zimm, B. H.; Bragg, J. K. J. Chem. Phys. 1959, 31, 526. (36) Grosberg, A. Yu.; Khokhlov, A. R. Statistical Physics of Macromolecules; AIP Press: New York, 1994.

and d3, corresponding to the interaction of an aggregate end with the cylindrical part and that with the end of another aggregate, are small at sufficiently low concentrations. The correction of order Ld2, for instance, renormalizes the end-cap energy by an amount of κφ, with κ a constant of order unity and φ again the volume fraction of discotic material in solution.27 The scission energy is typically 10-20 times the thermal energy, whereas this correction is very much smaller (of the order of one-tenth of the thermal energy), so we ignore this renormalization here (see also below.) Since we limit ourselves to isotropic solutions, we average the excluded volume over all angles, which gives πLL′d/2 for this quantity.24,32 To specify the excluded volume of the aggregates, we now need to define L and L′ in the context of our self-assembled, helical assemblies. The total length of a self-assembled (test) rod can be expressed as

L(N) ) ND + (N - 1)l

(7)

with N again the number of monomers comprising the aggregate, l the mean bond length, and D the monomer thickness along the aggregate axis, see Figure 1. As indicated, in our aggregates there are two types of bond, helical and nonhelical. Their lengths, measured along the main aggregate axis, may be unequal. If we take this into account, we can write

l ) l1θ + l2(1 - θ) ) l1(θ + q - qθ)

(8)

where l1 is the effective length of a helical bond, l2 is the length of a nonhelical bond, q ≡ l2/l1 is the ratio between these two bond lengths, and θ denotes the fraction of helical bonds in the test aggregate. We give estimates for the various lengths below. We now invoke a mean-field approximation and replace the length and fraction of helical bonds of the assemblies surrounding a test aggregate by their mean values. Within this approximation, the free energy of interaction, Hint, of the test aggregate with the other rods is given at the level of a second-virial approximation by half the excluded volume multiplied by the number density of rods, and becomes

π Hint ) FL(N)〈L〉d 4

(9)

∞ Here, F ≡ ∑N)1 F(N) is the number density of the aggregates.37 The quantity 〈L〉 ) 〈N〉 D + (〈N〉 - 1)〈l〉 gives the mean length of the aggregates, averaged over all conformations of the aggregates, as well as over their size distribution; 〈N〉 denotes again the mean degree of polymerization and 〈l〉 ) l1(〈θ〉 + q - 〈θ〉q) the mean length of a bond in these aggregates, where 〈θ〉 is the mean fraction of helical bonds in the solution. This gives

[

〈l〉 π Hint ) φdν-1[ND + (N - 1)l] D + 〈l〉 4 〈N〉

]

(10)

with φ ) νF〈N〉 the aforementioned volume fraction of aggregating material (see also eq 6). In eq 10, the first term between square brackets describes the length of the test aggregate and the second the mean length of the others (scaled to 〈N〉). The (mean-field) Hamiltonian of a test rod interacting with all of the others is given by H ) Hid + Hint. Combining eq 1 with N-1 eqs 8 and 10 and inserting θ ) (2N - 2)-1∑i)1 (si + 1), defined (37) Formally, we should take into account that the number density F is influenced by volume exclusion. However, we expect this effect to be modest. This is because F is given by φ/〈N〉ν, and the effect of interactions on the mean aggregate size is modest, whereas φ and ν do not change due to volume exclusion.

Helical Transition in Discotic Molecules

Langmuir, Vol. 22, No. 1, 2006 449

above, we find that the interactions only renormalize the helical and nonhelical bond energies. Hence, eqs 2 and 3 hold for interacting supramolecular chains too, provided we replace the excess helical-bond energy P, and the energies E and Ecap, by the renormalized ones. The renormalization of P is given by

P′ ) P + wφ(1 - q)

(

D + 〈θ〉 + q - 〈θ〉q l1

)

(11)

where 〈θ〉 now equals

E′ ) E - wφ

( )(

D D +q + 〈θ〉 + q - 〈θ〉q l1 l1

(12)

In eq 11, we have dropped the last term from eq 10, as it is small in the long-chain limit 〈N〉 . 1. The dimensionless interaction volume w is defined as πdl12/4ν. Note that P′ depends on the mean fraction of helical bonds. As we see in eq 12, this fraction in turn depends on P′ through s′ ≡ exp(-P′), which implies that the problem must be solved self-consistently. Equation 11 establishes that the helical transition is shifted to a different value of P by the interactions. According to eq 12, the fraction of helical bonds equals one-half at s′ equal to unity. From eq 11, it then follows that for 〈θ〉 equal to one-half, P ) wφ(q - 1)(Dl1-1 + 1/2 + q/2). The shift of P (from its value of zero in the noninteracting case, φ f 0) can be positive or negative, depending on whether q ) l2l1-1 is larger or smaller than unity. The amount is determined by the values of D/l1, q, and φ. Similarly, it follows from eq 12 that for each value of 〈θ〉 there is a unique value of s′, independent of concentration, bond lengths, or dimensionless volume; the impact of these is absorbed in the renormalized parameter s′. Substitution of 〈θ〉 and s′ into eq 11 then shows that the shift of the transition at a low fraction of helical bonds differs from that at a high fraction for fixed q and concentration. Hence, we can expect to see a shift not only in the onset of the helical transition but also in its sharpness. That this is indeed so is shown in the next Section. Note that similar shifts in onset and sharpness of the helicity curve are also known to be caused by finite-size effects.18,35 Let us now investigate to what extent the shift in the helical transition manifests itself in a change of the helical-transition temperature, Th, defined such that 〈θ〉(Th) ) 1/2. This is of course an experimentally relevant quantity. Equation 11 gives us P(Th′) ) w(q - 1)φ(Dl1-1 + 1/2 + q/2), as argued above, with Th′ the helical-transition temperature for the interacting case. We can approximate the temperature dependence of P by Taylor expansion. This gives at the helical-transition temperature for the interacting case

∆H (Th - Th′) kBTh2

(13)

Combining eqs 11 and 13 yields

)

(15)

The end-cap energy29 (the part of the free energy intensive in the assembly size) becomes

E′cap ) E - ln y′ + 2 ln λ′+ - wφq

1 1 〈θ〉 ) + (s′ - 1)[(s′ - 1)2 + 4σs′]-1/2 2 2

P(Th′) ≈

size N (see eq 2) and is renormalized due to the interactions according to

(

)

D + 〈θ〉 + q - 〈θ〉q l1 (16)

Here, the quantities y′ and λ+′ are identical to y and λ+ as defined above, but with s′ ≡ exp(-P′) substituted for s ≡ exp(-P). We find that the renormalization of the bond energy E given in eq 15 depends on the fraction of helical bonds. Away from the helical transition, this free energy would be rescaled by an amount that is of the same order of magnitude as that which arises from the terms proportional to Ld2 that we ignored (as mentioned earlier). Hence, corrections to E due to interaggregate interactions are small (typically of the order of 0.1 kBT on a free energy of order 10-20 kBT), unless the product φdl12/ν is very large. This being unlikely for discotic molecules (see below), we can reasonably ignore the corrections entirely, and write E′ ≈ E. The end-cap energy Ecap, on the other hand, is renormalized by the interactions in two ways: through the shift in the helicalbond free energy in the terms y and λ+, and more directly through the last term of eq 16. This last term predicts a shift in the endcap energy (and hence also in the mean aggregate size) even if the helical and nonhelical bond lengths are identical, q ) 1. It turns out, however, that (for realistic values of φdl12/ν and q) the former contribution is the only one that has a significant effect on the end-cap energy. Hence, we may write E′cap ≈ E ln y′ + 2 ln λ′+. Apparently, the coupling between interactions and self-assembly predominantly occurs through the renormalization of the excess helical-bond energy. So, it appears that we can describe the interacting case with our model for noninteracting rods, provided we replace s by s′ and Ecap by E - ln y′ + 2 ln λ′+. Even without resorting to a numerical analysis of the selfconsistent equations (eqs 3 and 11 and 15 and 16), we infer that there must be two limiting cases, one where the mean bond length is much larger than the thickness of a monomer, 〈l〉 . D, and one where the opposite is the case, D . 〈l〉. We shall focus most of our attention on the case where the bond length dominates and set D/l1 ) 0 in eqs 11, 15, and 16, simplifying these. (Note that this simplification is only valid for large enough q ≡ l2/l1, as for small q, the D/l1 term dominates in eqs 11, 15, and 16. Hence, we focus in the following on relatively high q values.) At the end of the next section, we give a brief summary of the results for the limit where D . 〈l〉.

3. Results

(

)

kBTh2 D 1 q Th′ ) Th + wφ(1 - q) + + ∆H l1 2 2

(14)

Based on this, we expect a shift in the helical transition temperature as a result of the interactions, dependent on the values of w, φ ∆H, and Th and the dimensions of the monomers. That this indeed occurs, we show in the next section. The free-energy gain of a nonhelical bond, E, is also shifted by excluded volume interactions. It is given as the part of the (nonhelical) free energy that is extensive in the aggregate

To gauge the effect of the interaggregate interaction on the helical transition, we first need to estimate the dimensionless volume w, as this quantity, together with the dimensionless concentration φ, sets the strength of the coupling between the fraction of helical bonds and the volume exclusion (see eq 11). Its numerical value depends on the geometry of the monomers. For the case of the discotic molecules of Brunsveld and coworkers,2 ν is of the order πDd2/4 with a thickness D ≈ 0.1 nm and a diameter d ≈ 5 nm, so w is likely of the order of magnitude of 0.3, if we take for l1 a value of 0.4 nm.38,39 For the discotics, the ratio q presumably has a value larger than unity since the

450 Langmuir, Vol. 22, No. 1, 2006

Figure 2. Mean fraction of helical bonds 〈θ〉 as a function of the volume fraction φ of assembling monomers, for the case where the cooperativity is high with Zimm-Bragg parameter σ ) 10-3, a similar value to that found in the helix-coil transition in biopolymers.35 We set the Boltzmann weight of a helical bond s ≡ exp(-P) equal to unity, a value which would give a helicity of one-half in the noninteracting case. The curves are shown for four values of the ratio of the lengths of a nonhelical and a helical bond, q, as indicated. For the dimensionless monomer volume w we choose a value of unity. In the inset: The mean fraction of helical bonds as a function of φ, for three different values of the cooperativity parameter σ. Here we choose a large value of the ratio of a nonhelical and a helical bond length q ) 5 to amplify the effect of this parameter, and set s ) 1, w ) 1, and E ) 20kBT. The latter value is typical for this type of self-assembling system.17

bond strength of a helical bond should be larger than that of a nonhelical one. On the other hand, it seems likely that q is not larger than, say, two, due to the details of the molecular architecture.40 In the following, we shall show that the effect of volume exclusion on the helical transition can be quite substantial for the values given above, and that the helical-transition temperature is shifted by several Kelvin for high enough concentrations. The interactions, however, have a modest effect on the mean aggregate size at realistic values of q, unless the volume fraction of assembling material is very high, φ f 1. We now calculate the mean fraction of helical bonds by solving eqs 11 and 12 in a self-consistent way. In Figure 2, we plot the mean fraction of helical bonds 〈θ〉 against the volume fraction of self-assembling material φ, for σ ) 10-3 and several values of the ratio between the lengths of a nonhelical and a helical bond, q. The value for σ that we choose is one close to that found in discotics17,18 and indicates a relatively high degree of cooperativity. Note that, if the aggregates were not to interact, the conformational state would depend only on the values of s and σ and 〈θ〉 would equal one-half at s ) 1.17 The figure shows that 〈θ〉 < 1/2 for q < 1, 〈θ〉 ) 1/2 for q ) 1, and 〈θ〉 > 1/2 for q > 1. This means that, perhaps not entirely surprisingly, fewer helical bonds form if a helical bond is longer than a nonhelical bond. This is because the excess free energy due to the excluded-volume interaction scales with the square of the length of the aggregates (see eq 9). The longer the aggregates are, the larger the effective (repulsive) interaction between the chains.25 Therefore, the shorter (in this case, the (38) Palmans, A. R. A. Ph.D. Thesis; Technische Universiteit Eindhoven, 1997. (39) Note that this gives a value of D/l1 of 1/4, rather than zero. Taking D/l1 ) 1/4 into account only introduces a constant factor into the equations, which does not change the observed trends. The error its omission introduces is largest if the helical bond is longer than the nonhelical one, which is not the case for Brunsveld’s discotics. In the opposite case, q > 1, the error in the mean fraction of helical bonds (Figure 2), as well as that in the value of s at which the helical transition occurs (Figure 3), induced by neglecting D/l1, amounts to no more than a few percent, for realistic values of the other parameters. (40) Sijbesma, R. P. private communication.

Van Gestel and Van der Schoot

Figure 3. Fraction of helical bonds 〈θ〉 as a function of the Boltzmann weight for the helical transition in the absence of interactions, s, for three different values of the volume fraction φ, as indicated, and for q ) 2, w ) 1, and σ ) 10-3. Inset: 〈θ〉 as a function of the temperature for three different values of the volume fraction φ, as indicated. Parameter values are identical to those in the main figure with excess helical bond enthalpy ∆H/kB ) -6011 K and helical transition temperature Th ) 300 K taken from refs 2 and 17.

nonhelical) type of bond is preferentially formed if the assemblies interact appreciably. The condition q ) 1 corresponds to the case where both types of bonds have the same length, and gives the same results as the model in which there are no interactions, since s′ ) s in this case (see eq 11). For q > 1, we find an increase in the fraction of helical bonds, for the same reason. We conclude that interaggregate interactions can indeed have a large effect on the conformational state of the aggregates if the concentration of dissolved material and the difference in the helical and nonhelical bond lengths are both large enough. It appears that the value of the dimensionless volume and that of the cooperativity parameter σ both play important roles. The role of the latter is shown in the inset to Figure 2: the effect of interactions becomes stronger the smaller σ. If the cooperativity is low, at σ close to unity, or if dl12 is small compared to the reference volume ν, a high concentration is required for the interactions to have a significant impact on the fraction of helical bonds. Now, let us investigate to what extent volume exclusion impacts upon the onset and the sharpness of the helical transition. To this end, we plot in Figure 3 the fraction of helical bonds as a function of the bare Boltzmann weight for a helical bond, s ≡ exp(-P). As argued in the previous section, combining eqs 11 and 12 gives us the shift in the bare Boltzmann weight s for given w, q, and φ. It turns out that s is shifted to lower values for the ratio between bond lengths q larger than unity. This makes sense, because the interactions cause the (shorter) helical bond to become more favorable for q > 1, making it possible for the helical transition to occur at lower values of s. For the same reason, a value of q between zero and unity causes the curve to shift toward higher s (not shown). Note that the shift of the helical transition caused by volume exclusion is greater, the lower the fraction of helical bonds (see eq 11). This causes the broadening of the helical transition seen in Figure 3. Specifically, for a high fraction of helical bonds 〈θ〉 ≈ 1 and values of q ) 2 and w ) 1, we obtain P′ - P ) -φ, whereas for 〈θ〉 close to zero, this amounts to -2φ. Given that P′ - P equals zero in the absence of volume exclusion, the difference we calculate here corresponds to that between the noninteracting and the interacting case. As discussed above, we can also determine if the helical transition displays a similar shift and broadening if expressed in

Helical Transition in Discotic Molecules

Figure 4. Mean size of the aggregates 〈N〉 as a function of the volume fraction of assembling material φ for two different values of q, as indicated, and for w ) 1, s ) 1, E ) 20kBT and σ ) 10-3. Inset: 〈N〉 as a function of the Boltzmann weight for the helical transition in the absence of interactions, s, for two different concentrations φ, as indicated, for w ) 1, q ) 5, E ) 20kBT, and σ ) 10-3. The arrows indicate the onset of the growth spurt.

temperature units, applying eqs 11 and 13 (see the inset to Figure 3). The shift of the transition temperature Th is given by eq 14. For a value of the enthalpy of the helical transition ∆H/kB ≈ -6011 K and the transition temperature in the absence of interactions Th ≈ 300 K, corresponding to results found for Brunsveld’s discotics in dilute solution,17 we find that the shift in the transition temperature can be quite substantial. For instance, for φ ) 0.1, q ) 2, and w ) 1, it equals 2.3 K. As shown in the inset to Figure 3, the broadening of the transition (calculated with eq 13) as a function of temperature is not as pronounced as that in terms of the Boltzmann weight s. The reason is that the Boltzmann factor depends on the temperature in an exponential way, so that any change in the temperature scales with the logarithm of the corresponding change in s and hence is much less pronounced. We now turn to the coupling of the helical transition and growth. This coupling leads to a growth spurt in the regime where the helical conformation dominates.17 Given that interaggregate interactions can have a significant effect on the fraction of helical bonds, we must assume that we can also observe a growth spurt for interacting aggregates and that its strength depends on the lengths of the helical and nonhelical bonds. That this is indeed so can be seen in Figure 4. For the case where the ratio between the helical and nonhelical bond lengths, q, is unity, we see a steady increase of the mean aggregate size with concentration, corresponding to the expected square-root law of eq 5. However, for high values of q (we have chosen here the unrealistically high value of q ) 5 so that the change in the aggregate growth is more apparent), we see that the size of the aggregates increases faster with increasing concentration. For values of q lower than unity (not shown), the growth of the aggregates with increasing concentration is less strong than for the case where q ) 1, that is, the size increases less strongly than expected from the square-root law. That the strength of the interactions also shifts the onset of the growth spurt associated with the helical transition to a lower value of the (bare) Boltzmann weight of a helical bond is shown in the inset to Figure 4. At low concentrations, where volume exclusion has little effect, we regain the case without interactions, where the helical transition (and hence the growth spurt) sets in at s ) 1 (P ) 0, see eq 11). We can conclude that an increase in concentration not only results in an increase of the mean size

Langmuir, Vol. 22, No. 1, 2006 451

Figure 5. Ratio 〈N〉/〈N〉0 between the mean aggregate sizes with and without interactions versus the ratio between the lengths of a nonhelical and a helical bond, q, for two different boundary conditions: one with one end fixed to be nonhelical and the other free (NF) and one with one end fixed to be helical and the other free (HF), as indicated. Here, E ) 20kBT, s ) 1, σ ) 10-3, φ ) 0.1, and w ) 1. In the inset: Same for the ratio 〈L〉/〈L〉0 between the mean aggregate lengths with and without interactions.

of the aggregates by mass action, but also causes the growth spurt to occur more readily, i.e., at a lower value of s. In addition, we can see here that the sharpness of the transition decreases with increasing concentration. Both observations agree with those of Figure 3. Another factor with a potentially large effect on the mean degree of polymerization is the description of the aggregate ends imposed through their influence on the end-cap energy, see section 2.20 We do not expect the boundary conditions to affect the fraction of helical bonds, since the prefactor y only appears in the equation for the end-cap energy and not the energy for a helical bond. The impact of two different boundary conditions on the ratio 〈N〉/〈N〉0, with 〈N〉0 the mean aggregate size in the absence of the interaggregate interaction, is shown in Figure 5. Clearly, the theory predicts a qualitatively different behavior as a function of q for the boundary condition in which one end is helical and one is free (denoted HF) than for that with one end nonhelical and one free (denoted NF). (For boundary conditions with both ends free, the effect of a change in q is very small, results not shown.) The effects of volume exclusion in the descriptions with the two sets of boundary conditions in fact show opposite trends. An explanation for this finding may be provided as follows. As we concluded from Figure 2, for q > 1, a helical bond is favored over a nonhelical bond, in terms of excluded-volume interaction, and the larger q, the more so. However, for the boundary condition with one end free and the other fixed nonhelical, at least one bond of an aggregate is always nonhelical. This type of bond being unfavorable, the system tends to form as few of these as possible and hence increases the mean aggregate size for this boundary condition, thereby reducing the number of chains at equal concentration of monomers. For the boundary condition with one end free and one helical, there is hardly any effect in the regime q > 1, for exactly the same reason. For q ) 1 and s ) 1, the values of y for the two sets of boundary conditions become identical, so in that case the aggregate size does not depend on the boundary conditions at this point. For q < 1, we see the opposite trends. Here, the nonhelical bond is more favorable, and hence the boundary conditions with one end helical show a strong aggregate growth.

Van Gestel and Van der Schoot

452 Langmuir, Vol. 22, No. 1, 2006

Although Figures 4 and 5 show us that, at relatively high concentrations, the mean size of the aggregates can be affected somewhat by the interaggregate interactions, at lower volume fractions, this is no longer the case. For φ equal to 0.01, for instance, the mean aggregate size changes at most by a factor of 1.1 over the range of q from zero to two (results not shown). Apparently, the effect of interactions on the mean degree of polymerization is modest at fixed s ) 1, at most, say, a factor of 2. For s larger or smaller than unity, the effect on the mean size becomes even smaller. For instance, for s ) 2, q ) 2, φ ) 0.1, w ) 1, σ ) 0.001, E ) 20, and the boundary condition with one end nonhelical, 〈N〉/〈N〉0 takes a value of approximately 1.2, whereas for s ) 10, it equals 1.1. For s < 1, the assemblies decrease slightly in size, e.g., 〈N〉/〈N〉0 lies just below unity for s ) 0.1. It is of interest to investigate if the same conclusions hold for the mean length of the assemblies. The renormalized mean length of the aggregates as a function of q is shown in the inset to Figure 5. Here, we have approximated the mean length as 〈L〉 ≈ 〈N〉l1(〈θ〉 + q - 〈θ〉q), again assuming that the aggregates are large and that the bond length is much larger than the thickness of a monomer. The change in the mean length of the interacting aggregates we find is fairly small too, but the effect of the interactions is somewhat different from what we saw for the mean degree of polymerization. This indicates that the conformational state of the molecules, and not the mean degree of polymerization, provides the dominant contribution to the mean length of the interacting aggregates. This implies that experimental methods probing, e.g., (hydrodynamic) radii and molecular weights will respond very differently to the effects of excludedvolume interactions on the aggregate size. Finally, we now briefly address the case that the bond length is much shorter than the thickness of a monomer (D . 〈l〉). The effect of interactions on the mean aggregate size and the fraction of helical bonds is then much smaller, although the trends turn out to be similar to those for the opposite case of 〈l〉 . D. This is because in this case the renormalized free energy of a helical bond does not depend on the fraction of helical bonds. This means that the problem is no longer a self-consistent one, and that the coupling between the helicity and the interactions becomes passive. As a consequence, the dependence on q is weaker than in the opposite case where 〈l〉 . D, and therefore, at the same value of q, a higher concentration is needed to obtain the same shift in the mean fraction of helical bonds. The transition temperature also shows a less pronounced dependence on q but its shift can also amount to several degrees for the same parameter values as in the case where 〈l〉 . D.

4 Conclusions and Outlook We have shown that hard-core interactions potentially shift the helical transition of self-assembled helical aggregates in isotropic solution. This manifests itself in a shift in the onset and the sharpness of the helical transition. These effects should be strong if the lengths of the helical and nonhelical bonds are sufficiently different and the concentration high enough. The former shift also expresses itself in a change of the helical transition temperature that can amount to several degrees. The latter shift is noteworthy, because it does not occur through a change of the free energy penalty on a frustrated monomer (that is, a monomer that is bound in a helical way to one neighbor and in a nonhelical way to the other), the parameter usually connected to the sharpness of the helical transition. Rather, the shift is due to the self-consistent relationship between the mean fraction of helical bonds in the interacting aggregates and the excess free energy of such a bond. In keeping with these results, we find that excluded-volume interactions cause the growth spurt associated with the helical transition to become more pronounced and to occur at less favorable values of the excess free energy of a helical bond. The effect of interactions on the mean aggregate size is modest, and depends very strongly on the state of the aggregate ends. It would be interesting to speculate on whether interactions are able to couple to the related phenomenon of amplification of chirality in self-assembled polymers. Chirality amplification occurs in helical copolymers formed from chiral and achiral monomers, or from the two enantiomeric forms of the monomers. A small amount of chiral material (or a small excess of one of the enantiomers) can then induce a disproportionately large number of bonds of a specific helical screw sense to be formed.41-45 We speculate that, since the right-handed and lefthanded helical bond are in a way symmetrical, differing only in the screw sense, they have the same length, and the coupling of the interaggregate interactions should be the same for both types of bond. This would mean that interactions of the type described here most likely have no effect whatsoever on the chirality amplification. LA0521903 (41) Van Gestel, J.; Van der Schoot, P.; Michels, M. A. J. Macromolecules 2003, 36, 6668. (42) Teramoto, A. Prog. Polym. Sci. 2001, 26, 667. (43) Brunsveld, L.; Lohmeijer, B. G. G.; Vekemans, J. A. J. M.; Meijer, E. W. J. Incl. Phenom. Macrocycl. Chem. 2001, 41 , 61. (44) Green, M. M.; Park, J.-W.; Sato, T.; Teramoto, A.; Lifson, S.; Selinger, R. L. B.; Selinger, J. V. Angew. Chem., Int. Ed. Engl. 1999, 38, 3138. (45) Green, M. M.; Garetz, B. A.; Munoz, B.; Chang, H.; Hoke, S.; Cooks, R. G. J. Am. Chem. Soc. 1995, 117, 4181.