Helical Transition and Growth of Supramolecular Assemblies of Chiral

of helical bonds, ϑ, becomes independent of the mean aggregation number30 ...... Nucleation−Elongation Mechanism, and Sergeants-and-Soldiers Pr...
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Helical Transition and Growth of Supramolecular Assemblies of Chiral Discotic Molecules 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

Luc Brunsveld and Rint P. Sijbesma Laboratory of Macromolecular and Organic Chemistry, Department of Chemical Engineering, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Aı¨ssa Ramzi Dutch Polymer Institute, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Received June 5, 2000. In Final Form: September 20, 2000 It has recently been established experimentally that the supramolecular assemblies formed by a particular class of chiral discotic molecules in solution exhibit a helical transition. To rationalize the experimental data, we apply the standard theory of linear self-assembly modified by a simple two-state model for the molecules in the aggregates, based on the Zimm-Bragg theory for the helical transition in conventional polymers. The theory provides a description of the amount of material in aggregates as well as their mean size and state of helicity as a function of the concentration of dissolved material, two binding strengths, and a parameter measuring the cooperativity of the helical transition. By fitting to the experimental data, all the model parameters are determined. It emerges that the helical transition is highly cooperative and that the growth spurt the aggregates exhibit at low temperatures is largely due to coupling of the selfassembly to the helical transition.

I. Introduction Linear self-assembly, that is, the reversible aggregation of material into quasi one-dimensional polymer-like objects, is observed in systems as widely varying as liquid sulfur, selenium, living polymers,1 electro- and magnetorheological fluids,2 proteins,3 dyes, antiasthmatic drugs,4 organometallic complexes,5 surfactant micelles,6 microemulsions,7 asphaltenes,8 and many more. Considering their widespread appearance, it is not surprising that a vast body of literature is devoted to the study of selfassembled (equilibrium) polymers. See, for example, the reviews in refs 9-14 and the work cited therein. A mere glance at the literature is sufficient to convince oneself that the types of interaction responsible for the emergence * Corresponding author. E-mail: [email protected]. (1) Greer, S. C. J. Phys. Chem. B 1998, 102, 5413. (2) Gast, A.; Zuksoky, C. F. Adv. Colloid Interface Sci. 1989, 30, 153. (3) Oosawa, F.; Asakura, A. Thermodynamics of the Polymerization of Protein; Academic: New York, 1975. (4) Boden, N.; Bushby, R. J.; Hardy, F.; Sixl, F. Chem. Phys. Lett. 1986, 123, 359. (5) Terech, P.; Schaffhauser, V.; Maldivi, P.; Guenet, J. M. Langmuir 1992, 8, 2104. (6) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (7) Schurtenberger, P.; Cavaco, C. J. Phys. France II 1993, 3, 1279. (8) Brandt, H. C. A.; Hendriks, E. M.; Michels, M. A. J.; Visser, F. J. Phys. Chem. 1995, 99, 10430. (9) Moore, J. S. Curr. Opin. Colloid Interface Sci. 1999, 4, 108. (10) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (11) Odijk, T. Curr. Opin. Colloid Interface Sci. 1996, 1, 337. (12) Taylor, M. P.; Herzfeld, J. J. Phys.: Condens. Matter 1993, 5, 2651. (13) Brunsveld, L.; Folmer, B. J. B.; Meijer, E. W. MRS Bull. 2000, 25, 49. (14) Lydon, J. Curr. Opin. Colloid Interface Sci. 1998, 3, 458.

of equilibrium polymers are as diverse as the systems in which they occur. The bonds linking the building blocks inside the linear assemblies can be covalent in nature1 but are more often of physical rather than of chemical origin.10 Sometimes these physical interactions are specific, involving, for instance, hydrogen bonds.15 Generic physical interactions, such as those of the dispersion type or those arising from the hydrophobic effect, are normally not directional but can lead to directional growth if permitted by the geometry of the particles involved.12 This is particularly true for discotic molecules, which exhibit quite a strong tendency to form cylindrical or rodlike aggregates in solution. Often, discotic molecules consist of a planar polyaromatic core with pendant side chains to enhance their solubility in certain types of solvents.4 In selective solvents, self-assembly into cylindrical stacks provides a mechanism for amphiphilic discotic molecules to shield their cores from the solvent and yet take advantage of the solubilization of the side chains. Phenomenologically, the reversible polymerization of dissolved molecules is reasonably well understood, in particular when ring closure and branching are suppressed.10,16-18 The simplest possible model description available balances an entropy of mixing against a binding (free) energy.10 The former promotes the generation of (15) 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. (16) Wheeler, J. C.; Kennedy, S. J.; Pfeuty, P. Phys. Rev. Lett. 1980, 45, 1748. (17) Wang, Z.-G.; Costas, M. E.; Gelbart, W. M. J. Phys. Chem. 1993, 97, 1237. (18) Wittmer, J. P.; Milchev, A.; Cates, M. E. J. Chem. Phys. 1998, 109, 834.

10.1021/la000794v CCC: $19.00 © 2000 American Chemical Society Published on Web 11/23/2000

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many small aggregates, whereas the latter favors few, large aggregates. The model contains a single, dimensionless phenomenological parameter, the so-called scission energy, E. The scission energy is defined as the free energy cost of breaking an aggregate into two parts, measured in units of thermal energy kBT, where kB denotes the Boltzmann constant and T is the absolute temperature. The mean-field theory based on this model predicts an almost exponential size distribution for the aggregates, with a number-averaged aggregation number, 〈N〉, that obeys12

〈N〉 )

1 1 + x1 + 4φ exp E 2 2

(1)

with φ representing the volume fraction of dissolved material. The fraction of material in the aggregated state, η, is simply

η ) 1 - 〈N〉-2

(2)

Although self- and mutual interactions do slightly modify eqs 1 and 2, mean-field theory proves sufficiently accurate for many practical purposes, in particular when the polymer-like assemblies are not too flexible.19 Because the mean size of the assemblies is an experimentally accessible quantity, either directly or indirectly, the scission energy is observable. E ) E(T) can be obtained by means of T-jump experiments,10 scattering methods,6 rheological measurements,20 and so forth. It is not a pure molecular constant, for it depends on the temperature as well as on the type of solvent, the ionic strength, and so on.10 Typically, E takes a value in the range between 5 and 25, but values up to 40 have been reported in the literature.21 A tacit assumption of the standard theory is that the scission energy does not depend on the position along the chain where the break is introduced. Clearly, this is reasonable whenever the assemblies are in the mean sufficiently large compared to the range of interaction between the molecules. The linear self-assembly of any actual experimental system can then be represented by a simplified picture in which the monomeric building blocks find themselves in either of two states, bound or not bound. In very many cases, this has proven to provide an adequate description of the state of affairs, particularly when electrostatic interactions along the chains are either absent or strongly screened.22,23 However, recently conducted experiments by Meijer and collaborators on solutions of a particular class of discotic molecule show that there are notable exceptions.24-26 The molecules that belong to this class self-assemble in selective solvents to form what presumably are rodlike stacks, with the remarkable property of exhibiting a helical transition not dissimilar to that found in some conventional polymers.27 (19) van der Schoot, P.; Cates, M. E. Langmuir 1994, 10, 670. (20) Duyndam, A.; Odijk, T. Langmuir 1996, 12, 4718. (21) Narayanan, J.; Urbach, W.; Langevin, D.; Manohar, C.; Zana, R. Phys. Rev. Lett. 1998, 81, 228. (22) Odijk, T. J. Phys. Chem. 1989, 93, 3888. (23) MacKintosh, F. C.; Safran, S. A.; Pincus, P. A. Europhys. Lett. 1990, 12, 697. (24) Brunsveld, L.; Zhang, H.; Glasbeek, M.; Vekemans, J. A. J. M.; Meijer, E. W. J. Am. Chem. Soc. 2000, 122, 6175. (25) Palmans, A. R. A.; Vekemans, J. A. J. M.; Havinga, E. E.; Meijer, E. W. Angew. Chem., Int. Ed. Engl. 1997, 36, 2648. (26) Palmans, A. R. A. Supramolecular Structures Based on the Intramolecular H-bonding in the 3,3′-Di(acylamino)-2,2′-Bipyridine Unit. Ph.D. Thesis, Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 1997. (27) Poland, D.; Scheraga, H. A. Theory of Helix-Coil Transitions in Biopolymers; Academic: New York, 1970.

Figure 1. Chemical structure of the C3 discotic discussed in the text. The asterisk marks the position of the chiral center in the side chains.

Figure 2. Schematic diagram showing the transition from the molecularly dispersed to the self-assembled, supramolecular state. The polymerization transition takes place at a temperature T*. At another crossover temperature, T**, the aggregates become helical.

Two features appear to be crucial for the understanding of the aggregation behavior of the discotic molecule (pictured in Figure 1) studied by Meijer and co-workers.24 The first is the presence of (nine) homochiral side chains, and the second is the ability of neighboring molecules in an assembly to become locked. A host of experimental evidence points at the existence of two bound states the molecules can have. In the weakly bound state, two neighboring molecules in an assembly can (more or less) freely rotate with respect to each other, but in the more strongly bound state this ceases to be the case. We speculate that although dispersion-type interactions dominate in both the weakly and the strongly bound states, steric interactions and possibly also intermolecular hydrogen bonds are responsible for the interlocking of molecules in the strongly bound state. The chiral nature of the side chains, combined with the possibility of a lockedin state, explains why a transition from a more or less randomly stacked structure to an ordered, helical structure of definite handedness is possible. In Figure 2, we present the physical picture that the experimental data seem to support.24 Both the transition of the molecularly dispersed to the “disordered” self-assembled state and that from the disordered to the helical state are not phase transitions in the true sense of the word, because of the essentially one-dimensional nature of the assemblies. The helical transition does resemble a phase transition, because of its strong cooperativity.

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The purpose of this paper is to present a minimal model that is able to offer a complete description of a wide range of experimental data, obtained for the system at hand. We combine the standard model for linear self-assembly, outlined above, with a Zimm-Bragg-type order-disorder theory allowing for bond cooperativity.28 All the model parameters are determined by comparison with experimental data, confirming that the helical transition of the self-assembled stacks, studied by Meijer and co-workers, is highly cooperative. We also find that the growth spurt that the aggregates exhibit at low temperatures, as evidenced by newly conducted small-angle neutron scattering (SANS) studies,29 is largely due to the coupling of the self-assembly to the helical transition. The remainder of this paper is organized as follows. Section II focuses on the presentation of a theory of selfassembly of quasi-unidimensional objects, which exhibit a configurational ordering transition that is potentially cooperative. In section III, we analyze the results of the circular dichroism measurements of Meijer et al.24 in the context of the theory describing the helical transition of the aggregates. With the help of additional data from microcalorimetry, we fix the two binding free energies that characterize the strongly bound state. Section IV deals with an analysis of the self-assembly process. By fitting the theory to results from experiments on fluorescence decay, we establish the binding free energy of the weakly bound state. Using the fitted model parameters, the temperature dependence of the aggregate size for various concentrations of dissolved material is predicted. We end the paper in section V with a brief discussion and with the conclusions. Confrontation of the theoretically predicted aggregation numbers with small-angle neutron scattering data29 supports our view that the minimal model presented in this paper captures the underlying physics of the problem. A plausible explanation for the cooperative character of the helical transition is given. II. Theory We consider a model system of amphiphilic discotic molecules dispersed in a solvent. The molecules, assumed to be homochiral, self-assemble in a linear fashion to form cylindrical supramolecular structures. Depending on the underlying driving force, the aggregates appear at temperatures either above or below a certain transition temperature, which actually is not sharp. For definiteness, we presume that they appear below that temperature, in line with the experiments discussed in the previous section.24 We define the transition temperature from the monomeric to the self-assembled polymeric state, T*, as that temperature at which half of the material resides in aggregates and the other half remains in the monomeric, molecularly dispersed form. According to this definition, η(T*) t 1/2 exactly at the polymerization temperature. Let us tentatively assume that for temperatures just below T* the molecules in the assemblies are predominantly in the weakly bound, freely rotating state. The (dimensionless) free energy change per formed bond is then given by an as yet unknown function fw(T). Postponing a discussion of fw(T) to section IV, we note only that fw(T*) < 0, for otherwise aggregates would not form. At a temperature T** < T*, another transition takes place, as a result of which the aggregated molecules go from the disordered to the helical state. Half of all the bonds are (by definition) in the helical state for T ) T**. (28) Zimm, B. H.; Bragg, J. K. J. Chem. Phys. 1959, 31, 476. (29) Ramzi, A.; Brunsveld, L.; Meijer, E. W. Manuscript to be published.

van der Schoot et al.

Details of the helical state, such as the pitch and the handedness, are irrelevant to the phenomenological description of the problem we advance. What is relevant is the possibility of the helical transition being cooperative. To allow for this, we distinguish between four types of consecutive bonds. A molecule that is weakly bound to both of its nearest neighbors has (as before) associated with it a free energy per bond of fw(T). A molecule that is strongly bound to both of its neighbors has attributed to it an additional free energy equal to fs(T) per bond, that is, in excess of fw(T). In this prescription, the weakly bound state acts as a reference state to the strongly bound one. The free energy fs can be either positive or negative. If fs > 0, the strongly bound state tends to be less stable than the weakly bound one, and if fs < 0, vice versa. When a strong bond is followed by a weak bond or the reverse, a free energy penalty fi(T) g 0 is introduced. This free energy penalty acts like an interfacial tension between regions of strongly and weakly bound molecules within a single aggregate. A large value of fi promotes the merging of many small helical regions into fewer large ones.30 As it happens, our model for the helical transition is equivalent to that of Zimm and Bragg describing the helixto-coil transition of polymers in dilute solution.28 The theory of Zimm and Bragg has been discussed at length in the literature, so we simply quote the results of the theory in the following and refer for an overview to the work of Poland and Scheraga.27 To simplify matters, we furthermore assume T** and T* to be well-separated. The assemblies are then in the mean already quite large before the helical transition occurs. In the limit of very large aggregates 〈N〉 . 1, the fraction of helical bonds, ϑ, becomes independent of the mean aggregation number30

ϑ)

1 + 2

s-1 2x(s - 1)2 + 4sσ

(3)

where s t exp[-fs] and σ t exp[-2fi] denote the wellknown Zimm- Bragg parameters.28 The former parameter dictates at what temperature the helical transition takes place, and the latter regulates the degree of cooperativity of the transition. By definition, half the bonds are helical exactly at the helical transition, ϑ(T**) t 1/2. According to eq 3, this occurs when s(T**) ) 1 or, equivalently, when fs(T**) ) 0. Finite-size corrections do modify this result, but because we expect these to be subdominant we do not explicitly deal with them here. For a helical transition that is not cooperative or is only weakly so, the parameter σ attains a value close to unity, σ f 1. On the other hand, for very cooperative processes σ becomes very small, that is, σ f 0. In principle, both parameters s ) s(T) and σ ) σ(T) are experimentally accessible, and therefore so also are the free energies fs and fi, provided that the temperature dependence of ϑ is known. In the limit where eq 3 is valid, the so-called groundstate approximation holds.30 The partition function of a single aggregate is then dominated by the largest eigenvalue of the matrix describing the transition probabilities between various kinds of bonds. Within the ground-state approximation, the partition function of an aggregate consisting of many discotic molecules can be cast in the following form:31

Z(N) ) (1 - ϑ)

(1 +2 s + 4ϑs --12)

N-2

× exp[-fw(N - 1)] (4)

Although strictly only applicable in the limit N . 1, eq 4 behaves sensibly for all N > 1. As a matter of fact, in

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the limit s f 0 the partition function of eq 4 reduces exactly to that of a nonhelical stack of an arbitrary number of weakly bound molecules, Z(N) f exp[-fw(N - 1)]. We therefore extrapolate eq 4 down to N ) 2 for all s, which is justified provided 〈N〉 . 1 at T ) T**. The free energy density of the solutions of aggregates can now be written as ∞

f)

∑ F(N)[ln F(N) - 1 - ln Z(N)] N)1

(5)

where Z(1) t 1, Z(N) for N > 1 is given by eq 4, and F(N) denotes the dimensionless number density of aggregates of size N. The first two terms on the right-hand side of eq 5 represent an ideal entropy of mixing, and the third gives the free energy change upon aggregation. The equilibrium size distribution Feq(N) functionally minimizes the free energy eq 5, subject to the conservation of mass: ∞

∑ N F(N) ) φ

(6)

N)1

The outcome of the minimization can be written as

Feq(N) ) (1 - 〈N〉-1)N exp(-E)

(7)

where the scission (free) energy takes the form

(

E ) -fw - ln(1 - ϑ) - 2 ln 2 + 2 ln 1 + s +

s-1 2ϑ - 1 (8)

)

and 〈N〉 is given by eq 1. Note that eq 7 asymptotes to an exponential distribution F(N) ∼ exp(-E - N 〈N〉-1) for 〈N〉 . 1, that is, in the strong-growth limit.10 In the next sections, we determine fw, fs, and fi by fitting to the experimental data. III. The Helical Transition The reader is referred to the publication of Meijer et al.24 for details on the experiments performed on the molecule in question, which we for brevity refer to as the “C3 discotic”.32 The chemical structure of the molecule is given in Figure 1. Experiments were performed in the polar solvents acetonitrile, methanol, ethanol, and nbutanol. However, because the most extensive measurements were done on solutions in n-butanol, we focus our analysis to the data taken in that solvent. As we shall see below, in n-butanol the helical transition occurs at a temperature at least 15 K below the polymerization transition, provided the concentration of C3 discotic exceeds ∼10-6 M. Because T* and T** will prove to be well separated, it is reasonable to assume that the groundstate theory holds and that the fraction of helical bonds (or molecules) obeys eq 3. The helicity of the aggregates was determined by means of circular dichroism (CD) spectroscopy for concentrations of material ranging from about 10-6 to 10-2 M. In Figure 3, we have reproduced the normalized helicity of the (30) Grosberg, A. Yu.; Khokhlov, A. R. Statistical Physics of Macromolecules; AIP: New York, 1994. (31) The power N - 2 in the second bracketed term of eq 4 stems from the fact that we need a minimum of three molecules in our model for the cooperative helical transition. (32) 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-tricarboamide.

Figure 3. Normalized helicity obtained from circular dichroism measurements as a function of temperature T for the C3 discotic dissolved in n-butanol at a concentration of 10-5 M. The drawn line is the fit to the theory, assuming the normalized helicity equals the fraction ϑ of helical bonds. The temperature scale is deg C.

solution at a concentration of 1.00 × 10-5 M material, as obtained by CD spectroscopy. (The data are normalized so as to approach the value of unity in the low-temperature regime.) If we identify the normalized helicity with the fraction of helical bonds in the solution, ϑ, we find that for this concentration the helical transition occurs at T** ) 294.5 ( 0.5 K. Experiments at other concentrations show that the transition temperature is weakly concentration dependent: T** = 292 K for 9.89 × 10-7 M, 295 K for 1.00 × 10-5 M, 297 K for 9.64 × 10-5 M, 297 K for 9.41 × 10-4 M, and 300 K for 9.21 × 10-3 M. We attribute the slight shift in T** with concentration to the effects of finite aggregate size discussed in section II. (Note that although accurate for 〈N〉 . 1, eq 3 is only formally exact in the limit 〈N〉 f ∞.) As the mean aggregate size and the transition temperature both increase with concentration, the experiments confirm the expectation that large aggregates lose their helicity (“melt”) at a higher temperature than smaller ones. To be able to describe the experimental data, the temperature dependence of the Zimm-Bragg parameters s ) exp[-fs] and σ ) exp[-2fi] needs to be made explicit. For this purpose, we make a Taylor expansion of the free energy fs about its value at the helical transition temperature, fs(T**) ) 0, giving to linear order

fs(T) =

|

∂fs ∂T

T**

(T - T**)

(9)

where the derivative ∂fs/∂Τ is evaluated at T ) T**, as indicated by the symbol |T**. The dimensionless quantity T∂fs/∂T |T** ) -∆hs/kBT** is the enthalpy change per molecule upon becoming helical, in units of thermal energy. As for the cooperativity parameter σ(T), we make the common approximation σ(T) = σ(T**) t σ.33 Although seemingly crude, this approximation is sensible for we then retain not only in fs but also in fi only the first nonzero term in the expansion about T**. We are left with two model parameters, ∆hs/kBT** and σ, which can be determined, for instance, by a simple two-parameter fitting procedure to the experimental curves. However, for practical reasons we follow a different, more discriminating procedure. The procedure we follow is inspired by the identity

|

|

∂fs 1 ∂ϑ ) - σ-1/2 ∂T T** 4 ∂T T**

(10)

which is easily obtained from eq 3. The significance of eq (33) Applequist, J. J. Chem. Phys. 1963, 38, 781.

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Figure 4. The fraction of helical bonds, ϑ, as a function of a reduced temperature ∆τ t - ∂ϑ/∂T|T** (T - T**) for concentrations ranging from 10-2 to 10-6 M. All the data collapse onto a single master curve.

10 is that it allows us to gain access to information about the cooperativity of the helical transition by simply taking the derivative of the helicity curve with respect to temperature at a single temperature T ) T**, with the proviso that we can determine ∆hs separately by microcalorimetry. As it turns out, a strong signal is indeed produced in differential scanning calorimetric measurement because of the cooperative nature of this transition. An advantage of our procedure is that once we have fixed both T** and the dimensionless group σ-1/2∆hs/kBT**, eq 3 becomes virtually independent of σ. In other words, by fitting the theory to the tangent in a single point, we should be able to accurately describe the entire helicity curve. From Figure 3, we see that ∂ϑ/∂T|T** ) -0.11(8) for the 1.00 × 10-5 M sample. For this sample, we thus find that ∆hs/kBT** = -139xσ, with an error of approximately 10% given an estimated uncertainty in the temperature of about 0.5 K. To fix σ, we combine this result with what is known from the microcalorimetry experiments.24 These have produced a value for ∆hs equivalent to -50 kJ/mol, independent of the concentration material, at least in the range from 10-4 to 10-2 M where the calorimetric experiments were done. (Note that ∆hs is defined as the enthalpy associated with transition from the weakly to the strongly bound state.) Assuming this value to extend to the concentration of 10-5 M, we get ∆hs/kBT** = -20 and σ = 0.022. The latter corresponds to an interfacial free energy of fi(T**) = 1.9(kBT**). Apparently, the helical transition of the equilibrium polymers formed by the C3 discotic is highly cooperative. Although not as cooperative as the helix-to-coil transition of, for example, the conventional polymer poly(γ -benzyl-L-glutamate), for which σ ) 2.4 × 10-4,28 this is quite a remarkable conclusion considering the labile nature of the aggregates. For comparison, we have in Figure 3 together with the experimental data the plotted theoretical curve for σ ) 0.022, using eqs 3 and 10, and the value for ∆hs/kBT** taken from microcalorimetry. The theoretical curve very accurately describes the data. Fitted curves of similar quality can be obtained for all concentrations in the experimental range from 10-6 to 10-2 M. The cooperativity parameter σ is found to decrease slightly (0.026, 0.022, 0.016, 0.016, 0.011) upon each 10-fold increase of the concentration starting from ∼10-6 M. We conclude that the cooperativity of the helical transition increases with increasing concentration, although only weakly so. A test of our conclusion that the helical transition is characterized by only two quantities, T** and σ-1/2∆hs/kB T2**, is given in Figure 4, where we plot the experimental

values for the normalized helicity in the concentration range from 10-2 to 10-6 M versus ∆τ t - ∂ϑ/∂T|T** (T T**). Both T** and ∂ϑ/∂T|T** we read off from the original data. All the data points collapse onto a single master curve, strongly supporting the idea that the Zimm-Bragg model does indeed apply to the problem at hand. We furthermore conclude that finite-size effects, not taken into account in our adaptation of the model but definitely relevant experimentally, influence the transition solely via a shift of T**. IV. Self-Assembly and Growth The relatively weak concentration dependence of T** suggests that the aggregates are already quite large at temperatures close to where the helical transition takes place. This implies that the polymerization transition must be decoupled from the helical transition. At temperatures well above T**, the self-assembly is regulated by the free energy associated with the weakly bound state, that is, E = -fw. See also eq 8. The temperature dependence of the free energy fw can be established from that of the fraction of aggregated material. Meijer and co-workers studied the aggregation behavior of the C3 discotic by means of time-resolved fluorescence spectroscopy.24 In solvents such as chloroform, where the molecules show no tendency to aggregate, the fluorescence decays with a single decay time. If the C3 discotic is dissolved in the solvent n-butanol, where the discotic molecules do selfassemble, two decay times are observed. One of these is of comparable magnitude to the one in chloroform, but the other is roughly an order of magnitude larger, at least in the low temperature regime, and is much more strongly temperature dependent. It was concluded that the fast decay time can be attributed to the free molecules and the slow decay time to those in the aggregated state. One may surmise that the relative amplitudes associated with the two relaxation modes of the fluorescence decay are directly proportional to the relative abundance of the molecules in the free and in the aggregated state. This seems quite reasonable in the high temperature regime, where the electronic states of the weakly bound and the free molecules are probably not all that different. However, in the low-temperature regime, where the aggregated molecules strongly interact, this assumption becomes rather more tenuous. We tentatively ignore complications of this sort and treat the relative amplitudes as relative abundances. The normalized amplitudes of the fluorescence decay are plotted in Figure 5. Unfortunately, data for only a single concentration of 2.35 × 10-6

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Figure 5. Relative amplitude of the fluorescence decay associated with aggregated molecules (η) and that associated with the free molecules (1 - η) as a function of the temperature T. The concentration is 2.35 × 10-6 M. The symbols represent the experimental data points. The drawn curves are based on the theory, assuming dominance of the weakly bound state. The arrows mark the temperature T** at which the onset of the helical transition takes place.

M are available. The curve clearly shows a fairly broad transition from the molecularly dispersed to the aggregated state. The crossover occurs at a temperature of T* = 316 K, which indeed is well separated from the estimated helical transition temperature of T** = 292 K. (See section III.) To attempt a fit of the theory to the data, we write

|

∂fw fw(T) = fw(T*) + ∂T

T*

(T - T*)

(11)

where

fw(T*) ) ln φ - ln(2 - x2)

(12)

showing that the polymerization temperature is concentration dependent. The reason that we now truncate the binding free energy after not the first but the second nonzero term is that the polymerization transition is much more gradual than the helical transition. If the polymerization transition is indeed dominated by the weakly bound state, information about the enthalpy associated with that transition can be extracted from the experimental data using the identity

|

|

x2 - 1 ∂fw ∂η )∂T T* 2x2 - 1 ∂T T*

(13)

where T∂fw/∂T|T* ) - ∆hw/kBT* is the (dimensionless) enthalpy of the polymerization transition. Equation 13 follows from eq 2, if we put E = -fw. The tangent to the experimental curve of Figure 5, taken at T ) T*, is consistent with an enthalpy of aggregation of ∆hw/kBT* = -27 per molecule. If we insert an estimated φ = 6.2 × 10-6 in eq 12,34 we obtain for the free energy of aggregation fw(T*) = -12. This in turn leads us to conclude that each molecule loses an entropy of about 15kB upon aggregation. An explanation for such a large entropy loss is the increased steric hindrance the side chains of each molecule must experience in the aggregated state. Because the C3 discotic has nine side chains (see Figure 1), this implies (34) To calculate the volume fraction from the molar concentration, we make use of the density of the molecule of 1.3 g cm-3, established by means of X-ray measurements in the solid state (ref 26). The molecular weight of the molecule is 3406 g mol-1. The solvent n-butanol has a density of about 0.81 g ml-1 at room temperature.

an entropy loss equivalent to about 1.5kBT per side chain, which is hardly excessive. The theoretical curve, based on the approximation E = -fw and our estimate for the enthalpy of association, is plotted in Figure 5 together with the experimental data. Agreement in the high-temperature regime is very good indeed. Less good is the agreement for temperatures T j T**, although this should not really surprise us for a number of reasons. The first, most obvious reason is that the curve ignores the transition to the strongly bound state, which starts to affect the self-assembly around T**. In clear disagreement with the observed trend, correcting for the omission would lead to an increase of the predicted fraction of material in the bound state for temperatures below T**. A more plausible explanation therefore is the breakdown of our assumption that the electronic properties of the molecules are unaffected by the transition to the locked-in, helical state. At present, we know of no way to correct for this. For the concentration of about 10-6 M, all the relevant model parameters are now determined, allowing us to predict the mean aggregate size, 〈N〉, as a function of the temperature. For the other concentrations, we have no access to direct experimental information concerning the location of T*. Fortunately, we can make use of a Taylor expansion of eq 12 to estimate T* at densities different from the one measured at 2.35 × 10-6 M. Recalling that we assume the onset of the self-assembly to be dominated by the weak binding process, we have

T*(φ2) = T*(φ1) +

kBT2* (φ1) φ1 ln ∆hw φ2

(14)

where we presume that ∆hw is only weakly concentration dependent, which is plausible because the system is dilute at the polymerization transition.18,35,36 Obviously, eq 14 can be expected to provide an accurate estimate only when the transition temperatures T*(φ2) and T*(φ1) at the concentrations φ2 and φ1 differ not too much, say, by at most 20%. Using T* = 316 K for the concentration of 2.35 × 10-6 M, we obtain T* = 306 K for 9.89 × 10-7 M, T* = 333 K for 1.00 × 10-5 M, T* = 359 K for 9.64 × 10-5 M, T* = 385 K for 9.41 × 10-4 M, and T* ) 412 K for 9.21 × 10-3 M. Clearly, extrapolation to higher densities than 10-4 M does not seem warranted. We now have all the information we need to predict the mean aggregation number of the C3 discotic in n-butanol for concentrations up to ∼10-4 M. In Figure 6, we have plotted our theoretical results for the temperature dependence of the mean size of the aggregates for the concentrations 9.89 × 10-7 M, 1.00 × 10-5 M, and 9.64 × 10-5 M, based on the input from the circular dichroism, fluorescence, and calorimetric measurements. The curves unequivocally predict a strong increase in the growth below T**, in other words, a growth spurt triggered by the helical transition. The reason is the vastly increased gain in binding energy the discotic molecules have in the strongly bound state. V. Discussion and Conclusions One way of gauging the mean size of the aggregates is by scattering methods. To confront our theoretical predictions for the growth of the aggregates with experiment, we present some of the results of recently conducted smallangle neutron scattering experiments. A full account of (35) van der Schoot, P. Europhys. Lett. 1997, 39, 25. (36) Gelbart, W. M.; Ben-Shaul, A.; McMullen, W. E.; Masters, A. J. Phys. Chem. 1984, 88, 861.

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Figure 6. Prediction for the number-averaged aggregation number, 〈N〉, as a function of the temperature for the concentrations 9.89 × 10-7 M (bottom curve), 1.00 × 10-5 M (middle curve), and 9.64 × 10-5 M (top curve). The arrows mark T** for the three concentrations.

Figure 7. Results of SANS experiments on a sample of concentration 2.39 × 10-3 M.29 Indicated is the zero-angle scattered intensity divided by the volume fraction of dissolved material, I/φ, as a function of the temperature. The quantity I/φ is given in arbitrary units.

the neutron scattering experiments will be presented elsewhere soon.29 As is well-known, the scattered intensity at vanishing scattering angles divided by the concentration of dissolved material, I/φ, is directly proportional to the mean aggregation number of the linear aggregates, I/φ ∝ 〈N〉, but only in the limit of dilute solutions. The relation between the scattered intensity and the aggregation number actually involves a weight rather than a number average, but for our purposes the distinction is immaterial. For reasons of contrast, useful data were obtained only at the relatively elevated concentration of 2.39 × 10-3 M. This causes several practical difficulties, an important one being that the aggregates then grow so large that interactions between aggregates start to influence the scattered intensity. This is particularly relevant below T** = 300 K, where the strongest growth occurs. The other difficulty is that for such a high concentration we cannot provide a reliable theoretical prediction for 〈N〉 (as was clarified in section IV). Satisfied with only a qualitative comparison, we give the experimental results in Figure 7. The experimental data of Figure 6 confirm the trend of the theoretical curves given in Figure 5, namely, that when the temperature drops below T** the aggregates undergo a growth spurt. This growth spurt is caused by the helical transition. The agreement between theory and experiments strongly supports our view that the self-assembly of the C3 discotic can be characterized by two very different aggregated states; see Figure 2. In the high-temperature regime, the molecule forms relatively small rodlike aggregates, dominated by a weakly bound state which

van der Schoot et al.

allows the molecules free rotation about the main axis. At low temperatures, the aggregates undergo a configurational transition to become helical. Associated with the helical configuration is a strongly bound state in which neighboring molecules are locked in and free rotation is no longer possible. The free energy gain upon becoming helical is so large that a growth spurt is induced in the low-temperature regime. The crossover from random to helical supramolecular stacks occurs over a very small temperature range because of the strong cooperativity of the helical transition. Some insight into the precise nature of the weakly and strongly bound states may be gotten from the computer modeling study of Meijer and co-workers.24 The study revealed that the C3 discotic is not necessarily a rigid, flat molecule but in fact has a shape reminiscent of a propeller. The three wedge-shaped moieties that make up outer part of the disk (the “blades”) are connected to the flat inner core of the disk by a single covalent bond (cf. Figures 1 and 8). We speculate that in free solution these moieties can freely rotate about this bond. If two molecules become weakly bound, the free rotation of the wedges or blades is in all likelihood lost. The blades then become coplanar with the core to maximize the intermolecular bonding through dispersion-type interactions. In other words, it is only in the weakly bound state that the molecules resemble flat disks. To further reduce the spacing between two bound molecules and increase the interaction strength to that found in the strongly bound state, another configurational change is necessary.24 The wedges have to be slightly twisted out of the plane of the central cores of the molecules, which have to rotate relative to each other to allow for a tight fit. The resulting locking in of the now propeller-screw-shaped molecules explains the loss of the rotational degrees of freedom in the strongly bound state. See Figure 8. The well-defined handedness of the helical bonds is imposed by the stereochemistry of the chiral side groups. It has been suggested that (bifurcated) hydrogen bonding may aid in holding the locked-in molecules into place,24 in a similar fashion as intrachain hydrogen bonds assist the stabilization of the structure of proteins in solution.37 True or not, the free energy gained in any hydrogen bonding is not likely to be significant compared to the contribution from the dispersion interactions. We infer this from the temperature invariance of ∆hs, noting that although T** depends on the concentration of discotic, ∆hs does not. The proposed transition from a flat, disklike shape of the molecules in the weakly bound state to a propellerscrew shape in the strongly bound state quite naturally explains the cooperativity of the helical transition. The reason is that when a molecule is weakly bound to a neighbor that itself is strongly bound to its next neighbor, the very shape of that neighbor necessitates (for steric reasons) a larger spacing than when both are weakly bound. As a consequence, a weak bond following a strong one is weaker than a weak bond that follows a weak bond, for a larger interparticle distance implies a weaker dispersion interaction. The difference in free energy is exactly the interfacial free energy fi introduced in section II. Although this free energy penalty amounts to a modest 2kBT for the C3 discotic dissolved in n-butanol, it is quite sufficient to make the helical transition very cooperative. The good description of the helicity data in terms of the theory strongly suggests the ground-state approximation (37) Pain, R. H. Mechanisms of Protein Folding; Oxford University Press: Oxford, 1994.

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Figure 8. Schematic diagram showing the propeller-like shape of the C3 discotic molecules (left) and their packing in the helical aggregates (right).

to be justified for the system at hand. On the other hand, inspection of the theoretical curves of Figure 5 shows that at least for concentrations up to ∼10-4 M, the mean aggregation number at the temperature T ) T** is actually not all that large, implying that finite-size effects should, if only in principle, be part of an accurate description. A partial explanation for this paradox was already given in section III, where we found the experiments to strongly suggest that finite-size effects influence only the precise location of T**. Because we used T** as input, we implicitly dealt with finite-size effects in our description, although admittedly in an uncontrolled manner. We expect that a more accurate theory would only slightly modify the theoretical curves of Figure 5 and then only around T**. The reason is that the theory is internally consistent below T**, where the aggregates are indeed large, as well as above T**, because then the weakly bound state dominates, and the standard theory applies. Work is currently underway to deal with the influence of finitesize effects on the growth and helical transition of

aggregates of chiral discotics in a more controlled fashion.38 An interesting problem we intend to investigate (also experimentally) is what happens when T** f T*. Our guess is that the helical transition then occurs only in the very few large random aggregates present, which then act as a bootstrap mechanism for the entire population to selfassemble into helical supramolecular stacks. Acknowledgment. We are grateful to M. Glasbeek and H. Zhang of the University of Amsterdam for providing us with the fluorescence data on the C3 discotic. We also gratefully acknowledge E. W. Meijer and J. Vekemans, both at the Eindhoven University of Technology, for discussions and a critical reading of the manuscript. The artwork of Figure 8 was kindly provided by K. Pieterse, Eindhoven University of Technology. LA000794V (38) van Gestel, J.; van der Schoot, P. Work in progress.