Helical Transition of Polymer-like Assemblies in Solution - The Journal

Oct 9, 2001 - Seda Cantekin , Yoko Nakano , Jeffrey C. Everts , Paul van der Schoot , E. W. Meijer , Anja R. A. Palmans. Chemical Communications 2012 ...
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J. Phys. Chem. B 2001, 105, 10691-10699

10691

Helical Transition of Polymer-like Assemblies in Solution Jeroen van Gestel,* Paul van der Schoot,† and M. A. J. Michels‡ Polymer Physics Group, Department of Applied Physics, Technische UniVersiteit EindhoVen, P.O. Box 513, 5600 MB EindhoVen, The Netherlands ReceiVed: May 8, 2001; In Final Form: July 25, 2001

A minimal model is presented for helix-coil type transitions in solutions of self-assembled supramolecular polymers. The diagram of states calculated using this model shows that there are essentially two ordering regimes, one where the polymerization transition and the helical transition coincide and one where they are separate. In both regimes a strong coupling exists between the growth and the helical transition of the supramolecular polymers. Comparison of the theory with experimental data confirms that the minimal model provides an adequate description of helix-coil type transitions in self-assembled polymers and that these transitions can be highly cooperative.

I. Introduction There are many types of macromolecules that, under appropriate conditions, attain a helical shape in solution. These include biologically and technologically important molecules such as RNA, DNA, polypeptides, and polysaccharides.1 Not surprising, therefore, is the large body of work, both experimental and theoretical, devoted to elucidating the physics underlying the helix-coil transition, which is now reasonably well understood.1-7 It appears that the helical state can be stabilized not only through hydrogen bonds but also by solvent effects or by electrostatic interactions.1 Evidence exists8-13 that helical transitions are not restricted to conventional polymers but that these may also occur in what are known as self-assembled polymers, supramolecular polymers, or equilibrium polymers.14-20 For instance, Hirschberg et al.12 found a transition between the monomeric form of a certain bifunctional material and helical aggregates of this material, while Brunsveld et al.13 reported separate polymerization and helical transitions with varying temperature in solutions containing chiral discotic molecules. In Figure 1 we schematically depict two examples of helical transitions that may occur in self-assembling systems. The first example shows self-assembly and a subsequent helical transition in a system of discotic molecules, whereas the second shows aggregation and a helical transition in a system of bifunctional molecules. These two examples can be seen as schematic representations of, respectively, the systems of Brunsveld et al.13 and Hirschberg et al.12 In the first example the aggregation and helical transition are expected to be caused mainly by nonspecific solvent-solute interactions, while in the second example specific ones (such as hydrogen bonding) are required to generate the transitions. One might also argue that the self-assembly of the protein g-actin into f-actin filaments and that of tobacco mosaic virus in the absence of RNA are regulated by a helical transition.21-24 The presumably most exhaustively studied supramolecular system exhibiting a helical transition, with the exception of actin, is that of Brunsveld et al.,13 which we describe in more detail in section V. These authors were able to determine not only * To whom correspondence should be addressed. E-mail: j.a.m.v.gestel@ phys.tue.nl. Fax: +31402445253. † Also at Dutch Polymer Institute, P.O. Box 902, 5600 AX Eindhoven, The Netherlands. E-mail: [email protected]. ‡ Also at Dutch Polymer Institute. E-mail: [email protected].

Figure 1. Schematic diagram of the transition of the monomeric state to the helical self-assembled state by way of a disordered self-assembled state (a) for a discotic system and (b) for a system of bifunctional molecules consisting of two reactive groups separated by a spacer. The polymerization transition takes place at temperature T/, and the helical transition takes place at a temperature T//.

the fraction of aggregated material and the fraction of helical bonds as a function of temperature and concentration of dissolved material but also the enthalpy associated with the helical transition. Van der Schoot et al.25 used these data as input for a highly approximate theoretical model that was set up to describe helix-coil type transitions in solutions of selfassembled polymers and concluded the transition to be remarkably cooperative. Here, we extend that theory to allow for a more accurate description of this transition, explicitly including finite-size effects that were dealt with implicitly in ref 25. As a result, we are now able to present a diagram of states, indicating the existence of two ordering regimes, one where the polymerization transition and the helical transition coincide and one where they do not. In both cases there is a strong coupling between the growth of the aggregates and the helical transition. The theory implies that in the experiments of refs 12 and 13 only a part of the possible aggregation behavior was explored and that by changing (for instance) the solvent conditions, a more complete picture of the physics of the systems may be obtained. In section II we briefly discuss the theory of equilibrium polymerization in terms of the partition function of the individual

10.1021/jp011733o CCC: $20.00 © 2001 American Chemical Society Published on Web 10/09/2001

10692 J. Phys. Chem. B, Vol. 105, No. 43, 2001 aggregates. We calculate this partition function exactly in section III, applying the well-known treatment of Zimm and Bragg for the helical transition in polymers.3 Because the validity of our theory extends to very small aggregates, we are able to study the coupling of the helical and the polymerization transitions. This was not possible in the earlier work of ref 25 because it relied on the so-called ground-state approximation, valid for large aggregates only. The predictions of the combined selfassembly and helical-transition models are presented in section IV, where we focus on the presentation of a diagram of states. The theoretical diagram of states is fully described by the concentration of dissolved material and three binding energies, one of which regulates the cooperativity of the helical transition. By fitting the theory to the experimental data of Brunsveld et al.,13 we fix all parameters for that system in section V, achieving remarkable agreement. Our theory not only is capable of accurately describing the concentration dependence of the helical-transition temperature but also reproduces the growth spurt of the aggregates observed when the temperature drops below the helical-transition temperature. We end this paper with a discussion and with conclusions in section VI, where we also compare in some detail our approach with the earlier mentioned work.25 Also discussed will be the comparison of our work with the theory of Oosawa et al. which was constructed to model the helical polymerization of actin.21,22 II. Theory of Equilibrium Polymerization We consider a dilute solution of molecules or particles that, under the appropriate conditions, form quasi-one-dimensional, polymer-like aggregates. The driving force of the self-assembly need not be specified, but one may think of specific interactions, such as hydrogen bonding26,27 and ionic bonding,15 or nonspecific ones, such as those arising from the hydrophobic effect.13,20 Important is that the polymerization is reversible so that the material in the monomeric and polymeric forms remains in thermodynamic equilibrium. The monomers are of an as yet undetermined shape and size (although large compared to the solvent molecules, allowing us to regard the solvent as a structureless continuum) but should display an order-disorder type transition in the aggregated state. Depending on whether the transition is enthalpy- or entropy-driven, the polymeric state appears either above or below a certain transition temperature that, because of the essentially one-dimensional nature of the aggregation, is not sharp. The same applies, at least in principle, to the appearance of the ordered state, or more specifically, the helical state.28 Without loss of generality and in keeping with the experimental findings, we assume the transitions from monomers to aggregates and from disordered aggregates to helical aggregates to be enthalpy-driven, implying that these take place upon a lowering of the temperature. The self-assembly of linear equilibrium polymers can be described at different levels of approximation and within very different theoretical approaches,29-38 as can be the helical transition.1-3,5,7 Here, we apply the often used density functional theory17,18,38,39 and combine it with the well-known ZimmBragg theory for the helical transition of conventional polymers.3,4 The potential usefulness of the Zimm-Bragg model in equilibrium polymerization was first suggested by Oosawa and co-workers22 in their description of the polymerization of actin. We postpone a description of the Zimm-Bragg theory to the next section. The free-energy density ∆F for any type of noninteracting (i.e., dilute) system of aggregating molecules can be written as the sum of an ideal entropy of mixing and the free energy

van Gestel et al. associated with the aggregated state: ∞

∆F )

∑ F(N) [ln F(N) - 1 - ln Q(N)]

(1)

N)1

where the sum is taken over all aggregate sizes N, F(N) is the dimensionless number density of aggregates of degree of polymerization N, and Q(N) is the partition function of an individual aggregate. In this and all following equations, the thermal energy kBT has been set to equal unity. For reasons of convenience, we choose the free energy of a disordered aggregate as the reference state, i.e., we put

Q(N) ≡ Qh(N) exp[-fn(N - 1)]

(2)

where fn represents the dimensionless free energy of the formation of a nonhelical bond between two molecules, assumed to be negative, and where the factor N - 1 arises because of the fact that an aggregate of N molecules contains N - 1 bonds. Ignoring a possible folding back of the aggregates on themselves to form rings,40,41 we treat the aggregates as one-dimensional objects that may undergo a potentially cooperative helical transition. The partition function Qh (where the subscript h stands for helical) then describes this transition; it will be evaluated in the next section. In thermodynamic equilibrium, the free energy is at a minimum. This minimum may be found by taking the functional derivative of ∆F with respect to the distribution F(N) and equating it to the chemical potential µN of an aggregate, thus enforcing the condition of conservation of mass, ∞

∑ N F(N) ) φ

(3)

N)1

with φ being the volume fraction of aggregating material, present in both monomeric and polymeric forms. This gives for the number density of aggregates of size N

F(N) ) λN Qh(N) exp fn

(4)

with λ ≡ exp(µ - fn) the fugacity of the monomers. Note that the number density F(N) is directly proportional to the partition function Qh(N), making it extremely sensitive to any configurational changes. Given Qh(N), various properties of the system, such as the mean aggregation number and the fraction of material in aggregates, can be calculated. For reasons to become clear below, we introduce number- and weight-averaged quantities, defined as ∞

〈‚‚‚〉n )

∑ (‚‚‚) F(N) N)1 ∞

(5)

∑ F(N)

N)1

and ∞

∑ N(‚‚‚) F(N)

〈‚‚‚〉w )

N)1

∞

∑ N F(N)

N)1

(6)

Helical Transition of Polymer-like Assemblies

J. Phys. Chem. B, Vol. 105, No. 43, 2001 10693

where ‚‚‚ represents the quantity to be averaged. As will be shown later, the mean aggregation numbers 〈N〉n and 〈N〉w are very sensitive to the occurrence of an ordering transition in the chains. Specifically, the size of the aggregates increases dramatically below the ordering transition. (Of course, it is not necessary to have an ordering transition to attain a large degree of polymerization. If the free energy associated with aggregation is large enough, very long aggregates can be formed without the presence of a cooperative ordering transition.18) Like the degree of polymerization, the fraction of molecules present in aggregated form, η, is also sensitive to an ordering transition and obeys (by definition)

η)1-

F(1) φ

(7)

III. Theory of the Helical Transition To calculate the partition function of the quasi-onedimensional aggregates, we first have to specify the model. The theory we apply is that of Zimm and Bragg,3 originally set up to describe the helix-coil transition in conventional polymers, i.e., polymers of fixed length. The model presumes the existence of two relevant free-energy scales fh and fi. The former gives the free-energy change of a disordered bond becoming helical; if fh > 0, the nonhelical bond is stronger than the helical bond, and if fh < 0, the helical bond is stronger. Thus, in this model, the nonhelically bound state functions as a reference state to the helically bound state. When a helical bond is followed by a nonhelical bond or vice versa, a free-energy penalty fi g 0 is introduced. This free-energy penalty can be seen as an interfacial tension between regions possessing different levels of order (hence the subscript i). The system will preferentially form long helical and nonhelical stretches for large fi, implying that fi can be seen to induce cooperativity in the bonding of helical and nonhelical stretches. In practice, not free energies but rather their Boltzmann factors are the relevant quantities, giving rise to the so-called Zimm-Bragg parameters.3 These are defined as s ≡ exp(-fh) and σ ≡ exp(-2fi), where s regulates the helical transition and σ describes the degree of cooperativity of this transition. In the limit N f ∞, the helical transition takes place at s ) 1,25 shifting to higher values for finite N. If σ f 1, the transition is not cooperative. If σ , 1, it is. It is only in the limit N f ∞, σ f 0 that the helical transition is believed to become a true phase transition in the thermodynamic sense.1,42 We are now able to construct the so-called transfer matrix, which describes the probabilities qij that a particular bond of type j follows a bond type i along the chain, with i, j ) n, h indicating the type of bond, n standing for “nonhelical” and h for “helical”. By construction qnn ) 1. For the other transition probabilities we use qhh ) s, qhn ) sxσ, and qnh ) xσ, which results in the same matrix as used by Grosberg and Khokhlov for the helix-coil transition of conventional polymers.7 The transfer matrix then becomes

(

) (

q q 1 xσ M ) qnn qnh ) sxσ s hn hh

)

bonds. A logical choice is to let the first bond always be of the disordered type; i.e., assign the Boltzmann factors 1 and 0 for the nonhelical and helical bond, respectively. We choose the second vector such that the last bond of the aggregate is not influenced by the absence of a following bond, i.e., we set both a priori probabilities equal to unity.43 From transfer matrix theory it follows that the partition function of a single chain of N g 3 monomers is given by

Qh(N) ) u‚MN-2‚u+

Here u+ ) (1, 0) and u ) (1, 1) are the bond probability vectors describing the beginning and end of the aggregate. The factor N - 2 arises from the matrix formalism, which requires a minimum of two bonds. By construction, Qh(1) ≡ Qh(2) ≡ 1. The evaluation of eq 9 is simplified considerably by diagonalization of the transfer matrix,

M ) T‚Λ‚T-1

Because the helix in our case has a definite handedness, the problem has an intrinsic directionality. The asymmetrical nature of the matrix mirrors this directionality. To finalize our description, we need to define the possible states of the first and last bond of an aggregate, which we will do by introducing bond probability vectors describing these

(10)

where Λ is a diagonal matrix containing the eigenvalues of the matrix M and where T is the matrix of column eigenvectors. T-1 is the inverse of T, so T-1‚T ) I, with I being the unit matrix. Note that because M is not Hermitian, the inverse of T is not equal to its transpose. For the eigenvalues of M, we find

1 1 1 λ1,2 ) + s ( x(1 - s)2 + 4σs 2 2 2

(11)

where the + sign of ( defines λ1 and the - sign of ( defines λ2. These eigenvalues are identical to the ones found by Zimm and Bragg.3 The eigenvectors are determined only to within an arbitrary prefactor, which we fix at xσ. This prefactor turns out to be irrelevant to the final form of the partition function (as it should). With this choice of normalization, we obtain

T)

(

xσ xσ λ 1 - 1 λ2 - 1

)

(12)

The partition function Qh(N) can now be determined by combining the above equations, giving the exact result

Qh(N) )

λ1 - s + s xσ N-2 s - λ2 - s xσ N-2 λ1 + λ2 λ1 - λ2 λ1 - λ2 (13)

This expression resembles the one found by Zimm and Bragg but is not identical to it because of a slightly different choice of transfer matrix.7 With Qh(N) determined, the size distribution F(N) is determined too (cf. eq 4). Our size distribution is much more complex than that of classical equilibrium polymers, which obey an exponential distribution.18 With Qh(N) and therefore also Q(N) known, the fraction of helical bonds in a single aggregate of size N can be determined using the equality3

θ(N) ) (8)

(9)

∂ ln Q(N) (N - 2)-1 ∂ ln s

(14)

valid for N g 3, where we note that N - 2 is the maximum number of possible helical bonds, assuming, as we have, that the first bond of an aggregate is always nonhelical. Equation 14 is easily understood if one realizes that, in a mean-field picture, the free energy will be proportional to the number of helical bonds times the bond energy fh for each helical bond. Experimentally, θ(N) cannot be determined, only some average

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van Gestel et al.

over all the aggregate sizes. As shall be discussed in section V, the experimentally relevant average is the weight average defined in eq 6. From eqs 2, 6, 13, and 14, we find for the weight-averaged fraction of helical bonds

〈θ〉w ) 2

[(ar - 2cr)(dr + er) + sλ′r dr λ-1 ∑ r (2ar + br) + dr fr(2cr - ar)] r)1 2

λexp(fn) + 2λ2exp(fn) +

dr(2ar + br) ∑ r)1 (15)

where

λ3λr exp(fn) 1 - λλr

br ≡

cr ≡ λ2 exp(fn) ln(1 - λλr)

dr ≡

ar ≡

er ≡

λ′rs2xσ

fr ≡

(λr - 1) + σs 2

ar 1 - λλr s xσ(λr - 1) + σs (λr - 1)2 + σs s[2(λr - 1)λ′r + σ] (λr - 1)2 + sσ

(16)

with λ′r ≡ ∂λr/∂s. For the fraction of material in aggregates and their mean size, we obtain 2

2λ2exp(fn) + η)

(2ardr + brdr) ∑ r)1 (17)

2

λ exp(fn) + 2λ2exp(fn) +

(2ardr + brdr) ∑ r)1 2

λ exp(fn) + 2λ2exp(fn) + 〈N〉n )

(2ardr + brdr) ∑ r)1 2

λexp(fn) + λ exp(fn) + 2

(18)

ardr ∑ r)1

Equations 15-18 contain the as yet unknown fugacity, which ∞ we fix by demanding that F(N) obeys ∑N)1 N F(N) ) φ. Unfortunately, we have not been able to eliminate λ analytically and have had to take recourse to (standard) numerical methods to fix λ. IV. Results With the combined self-assembly and helical-transition theories a diagram of states can be constructed. This we do by defining the polymerization transition to take place when half of the available material is present in aggregates, η ) 1/2, and by defining the helical transition to take place when half of the bonds are helical, 〈θ〉w ) 1/2. We stress again that the transitions are not sharp; in the monomeric region there will be a small amount of polymer present, gradually increasing as the η ) 1/2 line is crossed from below. The same applies to the helical transition. A natural parametrization of the diagram is in terms of the parameters Φ ≡ φ exp(-fn) and s ≡ exp(-fh) because for s f 0 the polymerization transition can be shown to take place roughly at Φ = 1.17 Figure 2 shows the diagram of states

Figure 2. Theoretical diagram of states for σ ) 1 (solid line) and σ ) 1.5 × 10-3 (dashed line) with the quantity Φ ) φ exp(-fn) on the vertical axis and s ) exp(-fh) on the horizontal axis. φ is the volume fraction of solute molecules, fn is the free energy for the formation of a nonhelical bond, and fh is the free energy for the formation of a helical bond from a nonhelical bond. The lines indicate conditions where the helical transition takes place, 〈θ〉w ) 1/2 (top), and where the polymerization transition occurs, η ) 1/2 (bottom). Note that these are not true phase transitions in the thermodynamic sense but gradual crossovers.

within this parametrization for the case where the helical transition is not cooperative, with σ ) 1, and the case where it is, with σ ) 1.5 × 10-3. The results for both cases are very similar, from which we conclude that our parametrization with only Φ and s is indeed a sensible one. As it turns out, the parameter σ in effect only influences the sharpness of the transitions. We learn from Figure 2 that there are two regimes. In the low-s regime, the polymerization transition and the helical transition are clearly separate transitions, while in the high-s regime they, for all practical purposes, coincide. In the latter regime, growth and helical transition are evidently strongly coupled. We shall see below that they are strongly coupled even in the low-s regime, for crossing the helical transition induces a fairly dramatic increase in the size of the assemblies. Although difficult to map directly onto an experimentally accessible diagram of states, there is experimental evidence to support the main features of Figure 2.12,13 Hirschberg et al.12 found for their bifunctional molecules a polymerization transition in one solvent and a combined polymerization and helical transition in another. In the work of Brunsveld et al.13,44,45 separate transitions were observed for their discotics in the solvents n-butanol and dodecane. In dodecane, however, the two transitions were closer together for comparable concentrations.45 This suggests that a change in the solvent quality can lead to the exploration of a different area of the diagram of states. To illustrate some of the salient features of the behavior of the system at hand, we take cuts through the diagram of states and show how the fraction of helical bonds, the aggregation number, and the fraction of aggregated material respond to changing conditions. In Figures 3 and 4 we take vertical cuts through the diagram of states at fixed s ) 3 and at fixed s ) 10. Shown are η and 〈θ〉w as a function of Φ, which may be seen as an effective density. The insets in these figures give the dependence of 〈N〉n on Φ, where we have indicated with arrows the locations of the polymerization transition, Φ/, for which η ) 1/2, and the helical transition, Φ//, for which 〈θ〉w ) 1/ . Figures 3 and 4 illustrate the merging of the two transitions 2 with increasing value of the parameter s, i.e., with increasing

Helical Transition of Polymer-like Assemblies

Figure 3. Predicted fraction of molecules in the aggregated state and the mean fraction of molecules in the helical state, respectively η and 〈θ〉w, as a function of effective density Φ for exp(-fh) ) s ) 3. The inset displays the mean aggregate size as a function of Φ. The arrows approximately indicate the locations of the aggregation transition Φ/ and the helical transition Φ//.

J. Phys. Chem. B, Vol. 105, No. 43, 2001 10695

Figure 5. Predicted fraction of molecules in the aggregated state and the mean fraction of molecules in the helical state, respectively η and 〈θ〉w, as a function of s for Φ ) 3. The inset displays the mean aggregate size as a function of s. The arrow approximately indicates the location of the helical transition.

in this case the polymerization line has long been passed, this demonstrates that it is indeed the helical transition that causes this growth spurt. It is easy to show that for large s, 〈N〉n ≈ s,25 which is indeed observed in the figure. Large assemblies are feasible below the helical transition but only if Φ . 1. So far, we have discussed the predictions of the theory in terms of the “theoretical” control variables σ, s, and Φ. To test the theory against experiment, we need to translate these control variables into experimentally accessible ones, such as the temperature, the concentration of dissolved material, and the transition enthalpies. This we do in the next section, where we compare the theory with the observations of Brunsveld et al.13 V. Comparison to Experiment

Figure 4. Predicted fraction of molecules in the aggregated state and the mean fraction of molecules in the helical state, respectively η and 〈θ〉w, as a function of effective density Φ for exp(-fh) ) s ) 10. The inset displays the mean aggregate size as a function of Φ. The arrows approximately indicate the locations of the aggregation transition Φ/ and the helical transition Φ//.

strength of the helical bond. In the limit of infinite helical-bond strength, the polymerization transition is completely dominated by the helical transition. Note the strongly enhanced growth for Φ J Φ//; below Φ//, the growth of the assemblies is a relatively weak function of Φ. In the absence of a helical transition, 〈N〉n ≈ 1 + Φ if Φ , 1 and 〈N〉n ≈ xΦ if Φ . 1.17,25 The helical transition shifts the crossover from the weakto the strong-growth regime to smaller values of Φ as it becomes more important, i.e., for larger values of s. In Figure 5 we take a horizontal cut through the diagram of states in Figure 2 for fixed Φ ) 3. Illustrated in the main figure are η and 〈θ〉w as a function of the helical-bond strength s and in the inset 〈N〉n vs s. For this choice of Φ, the polymerization and the helical transition are widely separated; the polymer fraction is already at a high 0.8 at the point where the fraction of helical bonds is still zero. The trend that was already visible in Figures 3 and 4 presents itself even more prominently in this figure: for low values of s, where the system is present largely in a nonhelical state, the aggregates are in the mean relatively small and their size is fairly constant, while near the helical transition a sudden, accelerated growth sets in. Because

To determine the quality of the theoretical model, a comparison was made with experimental work on solutions of the so-called C3 discotic13,46,47 in n-butanol. This molecule, like most discotics,9,10,32,48-51 consists of an aromatic core, attached to which are (nine) side chains. In the work of Brunsveld et al. these side chains were chosen to be homochiral and of a polar nature. This combination of an apolar core with a polar rim causes the molecule to self-assemble in solution in order to shield the apolar core from the more polar solvent. The discotic molecules can also undergo a helical transition, which is caused by the increased binding free energy of the helical aggregates when compared to the nonhelical aggregates. A model relating the shape of the molecule to its ability to aggregate and form helices has been described in some detail earlier.13,25 The homochiral side chains bias the twist sense of the intrinsically helical columns, causing the formation of helices of a single handedness instead of a mixture of right-handed and left-handed helices, an example of molecular chirality inducing macromolecular chirality.11 A host of experimental techniques were used to study dilute solutions of these molecules.13 We will concentrate on results from circular-dichroism spectroscopy (CD), timeresolved fluorescence spectroscopy, and differential scanning calorimetry (DSC) experiments. Because of the large amount of experimental data that was gathered on this system, it turns out to be possible to fix all the control parameters of our theory. We now make the temperature dependence of our control variables explicit. Following van der Schoot and co-workers,25 we Taylor expand fn around T 0/, the (hypothetical) polymerization temperature associated with the nonhelical bond in the

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van Gestel et al.

absence of a helical transition, and truncate this expansion after the second term, giving

fn(T) ≈ ln

φ 2 - x2

- (T - T 0/ )

∆hn kB(T 0/ )2

(19)

with ∆hn being the dimension-bearing enthalpy for the formation of a nonhelical bond. If the helical transition and polymerization are widely separated, T/ f T 0/. This happens to be the case for the single measurement of η(T) done by Brunsveld and co-workers, performed at a volume fraction of φ ) 6.2 × 10-6 (equivalent to 2.35 × 10-6 M 52). Earlier,25 it was concluded that for that particular concentration, but presumably also for other concentrations, ∆hn/(kBT 0/) ) -27 per bond. Note that the value of fn does not go to zero for T ) T 0/ but remains a function of the volume fraction φ, i.e., T 0/ is a function of concentration. The function T 0/(φ) for arbitrary concentration can also be calculated by Taylor expansion from the value of 316 K found at φ ) 6.2 × 10-6, giving T 0/(φ) ≈ 456 + 11.7 ln φ. Because no other experimental data are available, we shall use this expression in our data-fitting procedure (following the procedure of van der Schoot et al.25). For the temperature dependence of the free energy for the helical bond fh, we use the Taylor expansion

∆hh fh(T) ≈ (T ∞// - T) kB(T ∞//)2

Figure 6. Fraction of molecules in the aggregated state as a function of the temperature T in °C. The line gives the theoretical result, and the symbols indicate the experimental data for the C3 discotic in n-butanol at a concentration of 2.35 × 10-6 M.13 The arrows mark the transition temperature to the helical state T// and the transition temperature for the self-assembly T/.

(20)

where T ∞// is the helical-transition temperature of an infinitely long aggregate and ∆hh is the enthalpy for the formation of a helical bond from a nonhelical bond. Here, we use the fact that fh(T ∞//) ) 0.3 The enthalpy of the helical transition was determined experimentally by means of differential scanning calorimetry (DSC)13 and found to amount to -50 kJ/mol (-20kBT at T ) 292 K), independent of the concentration for the range 10-2 to 10-4 M. We assume that ∆hh is independent of concentration for all concentrations down to 10-6 M. The parameters T ∞// and σ were fixed by fitting the theory to the experimental data on the temperature dependence of the fraction of helical bonds, as determined by CD spectroscopy performed at five different concentrations (9.89 × 10-7, 1 × 10-5, 9.64 × 10-5, 9.41 × 10-4, and 9.21 × 10-3 M). For each curve we obtained the optimal values of T ∞// and σ, which we then averaged. As usual,2,3 we assumed σ to be temperature- as well as concentration-independent. We found T ∞// ) 300.3 K and σ ) 1.5 × 10-3. These averaged values were inserted back into the equations to produce the theoretical curves for the various concentrations. In Figure 6 we compare the measured temperature dependence of the fraction of aggregated material with what we find from the theory for the concentration of 2.35 × 10-6 M. The fraction of aggregated material was deduced from time-resolved fluorescence spectroscopy.13 Indicated in the figure is the earlier quoted polymerization temperature of T/ ≈ T 0/ ) 316 K for the concentration 2.35 × 10-6 M. For high temperatures the agreement between the theoretical and the experimental curves is excellent. Below T// , also indicated in the figure, theory and experiment appear to diverge. As was discussed already by van der Schoot and collaborators,25 the divergence may be due to the interpretation of the experimental data, in which the possible difference in electronic states of the molecules between a helical aggregate and a nonhelical aggregate was not taken into

Figure 7. Mean helicity as determined by circular-dichroism measurements as a function of temperature T for a concentration of 9.64 × 10-5 M of the C3 discotic in n-butanol.13 The drawn line is the theoretical prediction, and the symbols are the experimental results.

account.13 We therefore believe the apparent discrepancy at low temperatures to be unimportant. Figure 7 gives the temperature dependence of the average fraction 〈θ〉w of intermolecular bonds in the helical state for the concentration of 9.64 × 10-5 M, as obtained by circulardichroism (CD) spectroscopy. The reason the CD signal corresponds to a weight average, as opposed to a number average, is that both the weight average and the intensity of a CD signal are proportional to the number of helical bonds present in the system.53 Also indicated is the theoretical fit of 〈θ〉w, which was calculated in a slightly different manner than described earlier in that monomers and dimers were ignored whereas earlier they were taken into account. The reason is that this definition of the mean helicity corresponds more closely to what is experimentally determined. The agreement between theory and experiment is remarkably good for all concentrations in the range from 10-2 to 10-6 M (results not shown for all concentrations). Indeed, the theory provides a quantitative description of the experimental data. Now that all parameters are fixed, we can see how well the theory describes the concentration dependence of the helicaltransition temperature T// . This is done in Figure 8, showing

Helical Transition of Polymer-like Assemblies

Figure 8. Transition temperature T// for helix formation as a function of the concentration of the C3 discotic in n-butanol. The drawn line is the theoretical result, and the data points are experimentally found values.13

Figure 9. Calculated mean fraction of helical bonds of the C3 discotic in n-butanol, 〈θ〉w, as a function of a reduced temperature ∆τ ≡ -∂〈θ〉w/ ∂T|T//(T - T//) for concentrations ranging from 10-2 to 10-6 M. For clarity’s sake, lines connecting the data points have been omitted.

once again a more than adequate agreement between the experimental data and the theoretical results, especially when the scatter in the experimental values of T// is taken into consideration. The increase of the transition temperature with the concentration essentially shows that larger helical aggregates are more difficult to “melt” than smaller ones. A similar effect is seen in polymer crystals.54 As is shown in Figure 9, the theoretical helicity curves (lines connecting the points were omitted in the figure for clarity) obey the same scaling function as the experimental data were found to obey.25 Indeed, all predicted curves collapse onto a single master curve if we follow the same prescription as was used in the paper of van der Schoot et al.25 and rescale the temperature such that ∆τ ≡ -∂〈θ〉w/∂T|T//(T - T//), where we read off the slope of the helicity curve at T// from the theoretical curves. The circumstance that the helicity curves conform to a simple scaling function implies that, for the material discussed, the difference in concentration and hence in self-assembly merely shifts T// , albeit in a nontrivial manner. In the ground-state approximation the shape of the curve is only a function of ∆hh and T// ;25 apparently this remains true in our theory. Because all parameters are fixed, we are able to predict the number-averaged stack size 〈N〉n of the C3 discotic aggregates in n-butanol. Results as a function of temperature can be seen for different concentrations in the inset of Figure 10. All of the curves predict a strong growth of the aggregates that sets in

J. Phys. Chem. B, Vol. 105, No. 43, 2001 10697

Figure 10. Results of SANS measurements on the C3 discotic in n-butanol at a concentration of 2.39 × 10-3 M.55 Indicated is the dimensionless zero-angle SANS signal divided by the volume fraction of solute molecules, I0/φ, as a function of the temperature. The drawn line gives the predicted weight-averaged aggregate size, scaled to best fit the data. In the inset is the prediction for the mean aggregation number 〈N〉n as a function of the temperature for the concentrations 9.89 × 10-7, 1.00 × 10-5, 9.64 × 10-5, 9.41 × 10-4, and 9.21 × 10-3 M, from bottom to top.

below T// . The cause of this is the increase in the strength of the bond between the molecules. Also shown in Figure 10 are data from earlier published, preliminary small-angle neutron scattering (SANS) measurements performed on the C3 discotic in n-butanol at the relatively high concentration of 2.39 ×10-3 M.25,55 Plotted is the intensity I0 of the SANS signal, extrapolated to zero momentum transfer, divided by the volume fraction φ of the solute. From scattering theory of polymers we know that this quantity I0/φ is directly proportional to the weight-averaged molecular weight, albeit only in dilute solution.54 Lacking the proportionality factor between I0/φ and 〈N〉w, our prediction for I0/φ is only known to within a numerical constant, which we fix by fitting to the experiments. The result of this one-parameter fit, which only influences the magnitude of our theoretical I0/φ, is indicated in Figure 10 by the drawn line. The mean aggregate size increases from 〈N〉w ≈ 4 at 90 °C to 〈N〉w ≈ 123 at 20 °C. While the agreement between the experimental and theoretical data seems only fair, the trend of a sudden increase in aggregate size is reproduced at roughly the right temperature, which we identify as T// . A full report on the SANS measurements will be presented elsewhere.55 Despite the fact that we extrapolate to rather high concentration, where interactions between the aggregates may start to play a role, we still see a semiquantitative agreement between theory and experiment. VI. Discussion and Conclusions The agreement between theory and experiment provides a strong indication that our view of the linear self-assembly in dilute solution of a class of chiral materials exhibiting a potentially cooperative conformational transition is accurate. At high temperatures, the molecules that belong to this class aggregate into short polymeric assemblies in which they retain a relatively large amount of configurational freedom. When the temperature is lowered, the aggregates undergo the transition to a helical state of definite handedness; they, loosely speaking, “crystallize internally”. The free-energy gain associated with the transition to the helical state causes a growth spurt of the aggregates at low temperatures. As it turns out, because of its strong cooperativity, the transition from the random to the helical state can occur over a very small temperature range. As a result

10698 J. Phys. Chem. B, Vol. 105, No. 43, 2001 of the coupling between growth and the helical transition, there exist two regimes. In one, the helical transition and the polymerization are separated, and in another, they coincide. The theory presented is exact within the model assumptions and extends earlier work based on the same model.25 Although the previous work, which employed the so-called ground-state approximation, was also able to accurately describe the experimental data discussed in the previous section, it could only do so by dealing with finite-size effects in an ad hoc way. Indeed, in the earlier model experimental values of T// had to be put in by hand. Here, we only need to fix T ∞// to obtain T// at arbitrary concentrations. We find that the helical transition temperature is a function of concentration, in effect because the mean length of the aggregates depends on the concentration and because the helical-transition temperature in turn depends on the aggregate size. Our more exact theory gives for the cooperativity parameter σ of the C3 discotic in n-butanol a value of 1.5 × 10-3, more than 10 times smaller than was found within the ground-state approximation. This amounts to a doubling of the interfacial free energy between helical and nonhelical stretches along a chain, from 1.6kBT to 3.2kBT per molecule. Apparently, the helical transition of the material is much more cooperative than was previously thought, although it is still not as cooperative as the helix-coil transition of, for instance, the conventional polymer poly(γ-benzyl-L-glutamate) in a 4:1 mixture of dichloroacetic acid and ethylene dichloride, for which σ = 2 × 10-4.2 The theory developed here also bears some resemblance to the theory of Oosawa and Kasai in the fact that both theories describe the existence of a region where the helical form of the aggregates is predominant and a region where the nonhelical form dominates.21,22 However, there is a significant difference between the theories. Our model is a two-state model at the bond level, while Oosawa and Kasai’s theory is a two-state model at the polymeric level. In other words, in our theory it is possible to have helical and nonhelical regions within one polymer. This is not so in their theory, although the authors do briefly touch upon an extension of their model wherein this would become possible. Even with this extension, the theory agrees more closely with the theory of van der Schoot et al. than the current one, because both rely (in essence) on a groundstate approximation. The theory in its present form is believed to be suitable for any system that undergoes self-assembly coupled with an ordering transition, provided the system is in the dilute regime. The theory is also potentially unsuitable for processes requiring an activation step. Despite its apparent accuracy in describing experimental data, there is room for improvement. For instance, the possibility of ring closure has been neglected, as have excluded-volume-type interactions. For the C3 discotic at the considered concentrations, these are presumably reasonable approximations. The reason ring formation can be neglected is that we found the aggregates to be relatively small for T > T// . In the helical state, the aggregates do become long, but at the same time they are also more rigid than in the nonhelical state because the packing of the molecules is expected to be tighter.13 Excluded-volume interactions can be neglected if the volume fraction φ j D/L, with D being the diameter of an aggregate and L its length.56,57 Because the diameter and height of the C3 disks are approximately known, it is possible to determine values of φ and 〈N〉n for which there will be discrepancies between our theory and the experimental results. We find that interaggregate interactions will only play a role for concentrations exceeding 10-2 M, and then only at temperatures below

van Gestel et al. approximately 0 °C. Obviously, there may be systems for which ring closure and interaction are important. In those cases excluded-volume effects between randomly oriented stacks of disks can be included in a simple mean-field way for concentrations up to the close-packing regime.58 We deal with effects of these kinds in future work. Acknowledgment. We gratefully acknowledge R. P. Sijbesma, L. Brunsveld, E. W. Meijer, and A. Ramzi for useful discussions and for the sharing of experimental results. References and Notes (1) Poland, D.; Scheraga, H. A. Theory of Helix-Coil Transitions in Biopolymers; Academic Press: New York, 1970. (2) Applequist, J. J. Chem. Phys. 1963, 38, 934. (3) Zimm, B. H.; Bragg, J. K. J. Chem. Phys. 1959, 31, 526. (4) Qian, H.; Schellman, J. A. J. Phys. Chem. 1992, 96, 3987. (5) Hairyan, Sh. A.; Mamasakhlisov, E. Sh.; Morozov, V. F. Biopolymers 1995, 35, 75. (6) Dupre´, D. B. Biopolymers 1990, 30, 1051. (7) Grosberg, A. Yu.; Khokhlov, A. R. Statistical Physics of Macromolecules; AIP Press: New York, 1994. (8) Yan, G.; Lubensky, T. C. J. Phys. II (France) 1997, 7, 1023. (9) 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.s Eur. J. 1995, 1, 171. (10) Gallivan, J. P.; Schuster, G. B. J. Org. Chem. 1995, 60, 2423. (11) Maltheˆte, J.; Jacques, J.; Tinh, N. H.; Destrade, C. Nature 1982, 298, 46. (12) Hirschberg, J. H. K. K.; Brunsveld, L.; Ramzi, A.; Vekemans, J. A. J. M.; Sijbesma, R. P.; Meijer, E. W. Nature 2000, 407, 167. (13) Brunsveld, L.; Zhang, H.; Glasbeek, M.; Vekemans, J. A. J. M.; Meijer, E. W. J. Am. Chem. Soc. 2000, 122, 6175. (14) Moore, J. S. Curr. Opin. Colloid Interface Sci. 1999, 4, 108. (15) Lehn, J.-M. Supramolecular Chemistry: Concepts and PerspectiVes, 1st ed.; VCH: Weinheim, Germany, 1995. (16) Ciferri, A. Mechanisms of Supramolecular Polymerizations. In Supramolecular Polymers, 1st ed.; Ciferri, A., Ed.; Marcel Dekker: New York, 2000. (17) Taylor, M. P.; Herzfeld, J. J. Phys.: Condens. Matter 1993, 5, 2651. (18) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (19) Brunsveld, L.; Folmer, B. J. B.; Meijer, E. W. MRS Bull. 2000, 25, 49. (20) Whitesides, G. M.; Mathias, J. P.; Seto, C. T. Science 1991, 254, 1312. (21) Oosawa, F.; Asakura, S. Thermodynamics of the Polymerization of Protein, 1st ed.; Academic Press: London, 1975. (22) Oosawa, F.; Kasai, M. J. Mol. Biol. 1962, 4, 10. (23) Greer, S. C. J. Phys. Chem. B 1998, 102, 5413. (24) Klug, A. Angew. Chem., Int. Ed. Engl. 1983, 22, 565. (25) van der Schoot, P.; Michels, M. A. J.; Brunsveld, L.; Sijbesma, R. P.; Ramzi, A. Langmuir 2000, 16, 10076. (26) Sijbesma, R. P.; Beijer, F. H.; Brunsveld, L.; Folmer, B. J. B.; Hirschberg, J. H. K. K.; Lange, R. F. M.; Lowe, J. K. L.; Meijer, E. W. Science 1997, 278, 1601. (27) Corbin, P. S.; Zimmerman, S. C. Hydrogen-bonded Supramolecular Polymers. In Supramolecular Polymers, 1st ed.; Ciferri, A., Ed.; Marcel Dekker: New York, 2000. (28) The helical transition does become sharp if it is infinitely cooperative and the polymers are infinitely long. (29) Scott, R. L. J. Phys. Chem. 1965, 69, 261. (30) Tobolsky, A. V.; Eisenberg, A. J. Am. Chem. Soc. 1960, 82, 289. (31) Scha¨fer, L. Phys. ReV. B 1992, 46, 6061. (32) Henderson, J. R. J. Chem. Phys. 2000, 113, 5965. (33) Wittmer, J. P.; Milchev, A.; Cates, M. E. J. Chem. Phys. 1998, 109, 834. (34) Wheeler, J. C.; Kennedy, S. J.; Pfeuty, P. Phys. ReV. Lett. 1980, 45, 1748. (35) Wang, Z.-G.; Costas, M. E.; Gelbart, W. M. J. Phys. Chem. 1993, 97, 1237. (36) van der Schoot, P. Europhys. Lett. 1997, 39, 25. (37) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (38) Ben-Shaul, A.; Gelbart, W. M. Statistical Thermodynamics of Amphiphile Self-Assembly: Structure and Phase Transitions in Micellar Solutions. In Micelles, Membranes, Microemulsions, and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer-Verlag: New York, 1994. (39) van Roij, R. Phys. ReV. Lett. 1996, 76, 3348. (40) Porte, G. J. Phys. Chem. 1983, 87, 3541.

Helical Transition of Polymer-like Assemblies (41) Cates, M. E. J. Phys. (France) 1988, 49, 1593. (42) Poland, D.; Scheraga, H. A. J. Chem. Phys. 1966, 45, 1456. (43) Different theoretical approaches, such as models where both ends are necessarily nonhelical or both ends are free to adopt a nonhelical or a helical conformation, will be compared to the current theory by us in future work. (44) Palmans, A. R. A.; Vekemans, J. A. J. M.; Havinga, E. E.; Meijer, E. W. Angew. Chem., Int. Ed. Engl. 1997, 36, 2648. (45) Brunsveld, L. Private communication. (46) The chemical name of the C3 discotic is N,N′,N′′-tris{3[3′-(3,4,5tris{(2S)-2-(2-{2-[2-(2-methoxyethoxy)ethoxy]ethoxy}ethoxy)propyloxy})benzoylamino]-2,2′-bipyridyl}benzene-1,3,5-tricarbonamide. (47) Palmans, A. R. A.; Vekemans, J. A. J. M.; Fischer, H.; Hikmet, R. A.; Meijer, E. W. Chem.sEur. J. 1997, 3, 300. (48) Zhang, J.; Moore, J. S. J. Am. Chem. Soc. 1992, 114, 9701. (49) Ecoffet, C.; Markovitsi, D.; Jallabert, C.; Strzelecka, H.; Veber, M. J. Chem. Soc., Faraday Trans. 1992, 88, 3007. (50) Sheu, E. Y.; Liang, K. S.; Chiang, L. Y. J. Phys. (France) 1989, 50, 1279.

J. Phys. Chem. B, Vol. 105, No. 43, 2001 10699 (51) Boden, N.; Bushby, R. J.; Hardy, C.; Sixl, F. Chem. Phys. Lett. 1986, 123, 359. (52) To calculate the volume fraction from the molar concentration, a density for the molecule of 1.3 g mL-1 was used, determined from X-ray measurements in the solid state.47 Other characteristics of the system are that the molecular weight of the discotic is 3406 g mol-1 and that the solvent n-butanol has a density of 0.81 g mL-1 at room temperature. (53) Strictly speaking, the average that corresponds to the CD data should have the a form similar to the weight average as defined in eq 6 but with N - 2 substituted for N. However, in practice, the effect of this improvement is negligible. (54) Gedde, U. W. Polymer Physics, 1st ed.; Chapman and Hall: London, 1995. (55) Ramzi, A.; Brunsveld, L.; Meijer, E. W. To be published. (56) Gelbart, W. M.; Ben-Shaul, A.; McMullen, W. E.; Masters, A. J. Phys. Chem. 1984, 88, 861. (57) van der Schoot, P.; Cates, M. E. Langmuir 1994, 10, 670. (58) Brandt, H. C. A.; Hendriks, E. M.; Michels, M. A. J.; Visser, F. J. Phys. Chem. 1995, 99, 10430.