Solvent Thermodynamic Driving Force Controls Stacking Interactions

Article pubs.acs.org/JPCC

Solvent Thermodynamic Driving Force Controls Stacking Interactions between Polyaromatics Aaron S. Rury,*,† Christine Ferry,† Jonathan Ryan Hunt,† Myungjin Lee,† Dibyendu Mondal,† Sean M. O. O’Connell,† Ethan N. H. Phan,† Zaili Peng,† Pavel Pokhilko,† Daniel Sylvinson,† Yingsheng Zhou,† and Chi H. Mak*,†,‡ †

Department of Chemistry and ‡Center of Applied Mathematical Sciences, University of Southern California, Los Angeles, California 90089, United States ABSTRACT: Polyaromatic dye molecules employed in photovoltaic and electronic applications are often processed in organic solvents. The aggregation of these dyes is key to their applications, but a fundamental molecular understanding of how the solvent environment controls the stacking of polyaromatics is unclear. This study reports initial results from Monte Carlo simulations of how various acene molecule dimers stack when they are dissolved in different solvents. Free energies computed using full dispersion interactions versus those with sterics only suggest that solvent entropy alone accounts for the majority of the stacking free energy in solvents with compact molecular geometries such as carbon tetrachloride. However, in contrast with carbon tetrachloride, we also observe significant variations in the stacking free energies of naphthalene, anthracene, and tetracene across other solvents such as toluene and cyclohexane. The weak attractive dispersion interactions between the acene solutes and planar and near-planar solvent molecules enable them to intercalate between the acene monomers, inducing extra stability beyond what solvent entropic driving force alone could predict. In all three solvents studied (carbon tetrachloride, cyclohexane, toluene) the solvent environment helps facilitate stacking of all three acenes studied (naphthalene, anthracene, tetracene), inducing a significant stabilization free energy between −4 and −8 kcal/mol. Extensive free energy umbrella sampling along the other orthogonal directions allows us to accurately calculate the dimerization equilibrium constants of all three acenes, which vary over several orders of magnitude in a way that depends intricately on the solvent they are in. Given the prevalence of solution-based processing techniques for organic electronic and photonic devices, these results provide useful insights into the critical role that solvent structure and characteristics play in the solution-based aggregation of organic dyes.

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INTRODUCTION

particular materials to aggregate and crystallize will greatly enhance solvent-based approaches to device fabrication. Current theoretical understanding of the interactions of aromatic molecules relies heavily on electronic structure calculations.12−17 In these approaches researchers used ab initio methods to estimate the energy scales of the interactions between gas-phase molecules. While it had been thought for many years that the delocalized electrons notable in aromatic molecules dominated the interactions between these systems, a more solvent-centric picture is emerging as an important, if not the dominant, driving force behind the stacking interactions between these molecules.12−14 While these studies have laid the groundwork for understanding some of the fundamental physical interactions of aromatic molecular systems, it remains unclear if these are the driving forces that dictate structure in the solvent environments necessary for efficient processing of organic semiconductors. Most notably, what is the role of the

Organic semiconductors remain materials of interest in a wide array of fields in both fundamental and applied science.1−5 This interest stems from the natural abundance of the elements used to synthesize these materials, as well as their novel physical properties found thus far only in organics, such as singlet fission.6,7 Several research groups have demonstrated the ability to fabricate functional devices such as field-effect transistors,8 optoelectronic components,9−11 and photovoltaic cells using solutions of organic semiconductors.5 While these solutionbased approaches may offer lower costs in both dollars and energy than some of the more traditional approaches for inorganic materials, using a solvent carrier for the active molecular unit introduces a level of complexity to processing not encountered using other methods. This complexity in one way can extend our ability to exercise fine control over the physical properties of the functional solute molecules but in another could also limit our capability to reproducibly fabricate these devices. Therefore, establishing a systematic understanding of how solvent characteristics impact the ability of © XXXX American Chemical Society

Received: August 16, 2016 Revised: September 21, 2016

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DOI: 10.1021/acs.jpcc.6b08292 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

From a physical perspective, it is somewhat unexpected that entropy could be responsible for stacking. After all, one expects entropy would drive two stacked molecules apart rather than bind them to each other. This expectation is correct in the gas phase, but within a solvent environment, the entropic driving forces coming from the solvent molecules can dominate the thermodynamics. For nucleobases in small-molecule solvents like water, the MC studies in ref 27 show that solvent entropy provides a weak but sufficiently negative free energy driving force for the nucleobases to stack. This entropic driving force comes from the natural tendency of solvent molecules to fill space, and any solvent will sustain an entropic penalty whenever any solute of any size and shape is to be solvated within it. However, depending on the molecular structure of the solvent and solute molecules, it is possible that the entropic cost of solvating a pair of solute molecules with certain geometric arrangements between them could be less than solvating two of the same molecules when they are not paired. This entropy difference is typically of the order of one to several kcal/mol for the nucleobases, which is sufficiently strong to drive base stacking in DNA and RNA. This role of the solvent entropy as the dominant driving force behind stacking interactions in nucleobases begs an important question: what is the role of direct interaction between the two solutes, such as London dispersion or π−π interactions? Clearly, the precise role of these direct interactions is different for different polyaromatics (primarily a function of size and secondarily shape). For DNA and RNA bases in water, it has been shown that the effects of the London dispersion force between the solutes are largely neutralized by their interactions with the solvent molecules, and solvent entropy alone accounts for the majority of the thermodynamic driving force (>80%) behind the stacking free energy between nucleobases. For the stacking of PAMs in solution, two major differences stand out in comparison to nucleobases in water. First, PAMs are processed predominantly in organic solvents. Typical organic solvent molecules are larger than water in size, according to both their atom counts and their electronic densities, and the molecular shapes of typical organic solvent molecules are also more complex than water. Therefore, the molecular-scale environment of the solvent within which the PAMs are embedded are structurally more diverse than water. Second, PAMs often consist of many fused aromatic rings, making them much larger than the DNA and RNA bases. This could influence the relative contributions of the direct versus solvent-mediated interactions between PAMs to their stacking propensity. As we will see below, these two major distinctions between the two classes of systems will be key to understanding the stacking of PAMs in organics, which turns out to be more sensitive to the size and shape of the solute/solvent pair than the stacking of nucleobases in water. To elucidate some of the general factors that control stacking between PAMs, we present results from umbrella-sampled Monte Carlo (MC) simulations of the arrangement of simple acene molecules using both the Weeks−Chandler−Anderson (WCA) and the full van der Waals (VDW) intermolecular potentials. We examined the structure and stability of three acene molecular dimers, naphthalene, anthracene, and tetracene, in three organic solvents, carbon tetrachloride (CCl4), cyclohexane (CYX), and toluene (TOL). We selected these acene solutes since they are structurally similar but vary in size systematically from having two fused rings in naphthalene to three in anthracene and four in tetracene. The three solvents

solvent in facilitating, in a constructive way, the stacking interactions between these molecules or conversely impeding them in a destructive way? A number of recent investigations have focused on this question.18−25 While the electronic coupling between polyaromatics via π−π interactions is no doubt responsible for the unique chromatic properties of their aggregates, questions have recently been raised as to whether π−π interactions are indeed the dominant force that stabilize their stacking in the first place.20−22,26 Recent studies, both experimental and theoretical,20−22,24,25 have begun to unravel the role of the solvent environment in facilitating the stacking interactions between polyaromatics. Thus far, it remains unclear what the dominate solvent forces that most substantially impact stacking of polyaromatic solute molecules are in organic solvents and whether there is even any universality in these forces across all solvents and solutes that are relevant to the processing of functional dye molecules for device appliations. Since polyaromatic molecules (PAMs) used in photovoltaic and electronics applications are routinely processed in solutions with organic solvents, a detailed molecular understanding of the solvent-mediated driving forces behind the stacking of these molecules is key to their technological applications. With the hope of seeding this effort, in this paper we investigate the stacking free energies of several representative acene molecules in three different organic solvents. Figure 1 shows structures of

Figure 1. Stacking interactions of three polyaromatic molecules (PAM) in the acene family (top row) have been examined in three different organic solvents (bottom row) using Monte Carlo simulations, yielding first-principle numerical estimates for the stacking free energies and dimerization equilibrium constants of nine different PAM solute/solvent combinations.

the solute and solvent molecules studied in this paper. This study is motivated by previous work on the thermodynamics of the geometric arrangements of biomolecules in water. In a recent paper,27 one of us described the dominant physical effects of the aqueous environment on the stacking of purine and pyrimidine bases in nucleic acids. While stacking interactions are known to be a key driving force behind the formation of secondary structures in DNA and RNA, the precise molecular origin and the magnitude of these stacking interactions between the nucleobases have been somewhat controversial. Using a systematic perturbation expansion of the interaction potential between the solvent molecules and the nucleobases in a series of large-scale Monte Carlo (MC) simulations,27 this study demonstrates unequivocally that the dominant driving force behind nucleobase stacking in water is solvent entropy. This entropic thermodynamic driving force appears to be universal regardless of whether the solvent is hydrophobic or not, providing a net stabilization free energy of the order of 1 kcal/mol for each stacked pair. B


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between them, which we call “stacked”, relative to those two unstacked individually solvated solute molecules. In the WCA solvent with no attractive van der Waals forces, the only operative solvent thermodynamic force is entropy, and if the stacked solute pair is found to be stable, solvent entropy alone must then be responsible for their stacking interactions. By comparing the dependence and magnitude of this stacking free energy to that in the solvent with full van der Waals interactions, we can ascertain what fraction of the full stacking free energy is actually controlled by solvent entropy. Furthermore, with additional computer simulations employing solvents with different molecular shapes and sizes, we can also find out what some of the other factors may be, apart from solvent entropy, that could potentially control the stacking interactions between PAMs in more complex solvents. For each Monte Carlo (MC) simulation a solute molecule dimer pair was immersed inside a large solvent box with periodic boundary conditions. Extensive umbrella sampling was carried out to compute the free energy of association of the dimer relative to two infinitely separated monomers. In order to definitively disentangle the effects coming from the direct interaction between the solute molecules from those produced by the solvent, two different models for each solute/solvent pair were simulated. First, a set of VDW (van der Waals) simulations was carried out for each solute/solvent combination with full Lennard−Jones (LJ) interactio ns 12 6⎤ ⎡ uLJ(rij) = ϵij⎢ aij /rij − 2 aij /rij ⎥ between every pair of ⎣ ⎦ heavy (non-hydrogen) atoms in the solute and solvent molecules using Amber ff99 parameters for the energies ϵij and diameters aij28 within a NVT ensemble at the target physical density of the solvent and 298 K. Then a second set of MC simulations was repeated for the same nine solute/solvent combinations using the Weeks−Chandler−Andersen (WCA) potential29 uWCA(rij) = uVDW (rij) + ϵij for rij ≤ aij and = 0 for rij > aij, which retains only the repulsive branch of each of the LJ potentials. Without the attractive branch of the LJ energy functions, there are no direct interactions due to London dispersion forces among the solutes and solvent molecules in these WCA simulations except for sterics. If a stacked solute dimer pair is observed to be stable within a WCA simulation, the effective attractive free energy between them must have (a) arisen entirely from the solvent and (b) been predominately entropic in nature, i.e., its origin is dictated by the intrinsic liquid-phase structure of the solvent but not by direct molecular interaction energy. Since a WCA solvent has no cohesive energy, a NPT ensemble at 298 K instead of NVT was used for each of the WCA simulation, where a constant external pressure was applied to the solvent box to reproduce the same physical solvent density as in the VDW simulations. Perturbation theory29,30 confirms that the structures of simple dense liquids are largely dictated by their repulsive interactions, and a WCA liquid has a very similar structure as its corresponding VDW liquid at the same density. II. Stacking in Carbon Tetrachloride. Umbrella sampling MC results for the stacking free energies of the three acenes in CCl4 are summarized in Figure 2. Figure 2A shows WCA free energies between two parallel stacked solute molecules as a function of z, the vertical intermolecular separation, with the minimum free energy ΔG(zmin) ≡ 0 as the reference. For all three solutes, naphthalene, anthracene, and tetracene, the minimum free energy separation is approximately 3.5 Å. At distances smaller than this, the direct repulsive interaction

selected for this study are all nonpolar, but they differ in size and geometry. Of the three solvents, CCl4 has the most compact molecular shape and highest symmetry. TOL, on the other hand, is the largest in size (about one-half of naphthalene) but has a planar molecular geometry. CYX, which exists predominantly in its chair conformation in the liquid phase, has a size similar to TOL but is nonplanar. The simulations find that the entropy difference between solvating individual and pairs of solute molecules acts as a thermodynamic driver for the nominal stacking of two solute molecules. However, the magnitude of this thermodynamic driving force depends sensitively on the solvent of choice. In the case of CCl4 a free energy approaching 5 kcal/mol due to solvent entropy stabilizes the formation of the stacked configuration of two tetracene molecules, while in CYX and TOL this stabilization free energy reduces to less than 2 kcal/ mol. Comparison of the simulations results with full dispersion interaction versus those with only repulsive steric interactions indicates that this difference results from the molecular structure of each solvent. These comparisons suggest that the relative flatness of CYX and TOL molecules in addition to their dispersive interactions with solute molecules creates a substantially lower free energy barrier for the solvation of individual solute molecules relative to pairs. Using the free energies derived from the MC simulations, we calculate their dimerization equilibrium constants, showing that the simulated geometric configurations qualitatively match the limited experimental data on the dimerization of naphthalene in CCl4. We briefly discuss further experiments that could be used to validate numerical values of the remaining 8 solute/solvent systems investigated. Examining the solvent effects on the stacking of the solute PAMs in these nine solute/solvent combinations provides a basis for understanding the key solvent driving forces controlling the aggregation of functional dye molecules during their processing in various solvent environments for application in next-generation electronics and photonics technologies. The research described in this paper was part of a team class project in a graduate statistical mechanics course taught in the Chemistry Department at the University of Southern California during the Spring semester of 2016. The course was cotaught by the two corresponding authors of this paper.

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RESULTS AND DISCUSSION A. Dimerization Equilibria Arising from the Stacking of Acenes. I. Simulation Study Design. To test the hypothesis of whether intrinsic solvent entropy alone could explain the stacking of polyaromatics in organic solvents, we carried out a series of large-scale computer simulations to determine the stacking free energy between two solute acene molecules within a number of different solvents. Computer simulations offer us the freedom to turn on or turn off any molecular interactions between the solutes and the solvent molecules. Exploiting this unique advantage, we simulated stacking for each solute/ solvent combination in two models: one with full dispersion interactions turned on (the “VDW solvent”) and another one with no attractive dispersion interactions at all (the “WCA solvent”). For each model, we calculate the probability of finding two solute molecules with a specific geometry between them as a function of six different intermolecular coordinates. From these probabilities we can then calculate the free energy of these two solute molecules necessary to exclude solvent molecules from the volume occupied by them and the space C

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taking two perfectly parallel solutes at infinite separation and reorienting them to any other relative orientation inside the solvent should incur a ΔGreorient(z∞) = 0, so umbrella sampling by unstacking along the z direction alone correctly establishes ΔGstack. Within the WCA model there is no direct attractive potential energy between the solutes, among the solvent molecules, or between the solute and the solvent at all. The stable ΔGstack values observed in Figure 2A for all three solutes in CCl4 must therefore owe their origins entirely to the structure of the solvent, i.e., the equilibrium configurations of the solvent molecules within its liquid state. This is corroborated by the nonmonotonic z dependence of the stacking free energy in Figure 2A. A ΔG(z) that is due to the direct interaction, such as dispersion or π−π interactions, between two solute molecules is expected to decay monotonically with the distance z. Figure 2A clearly shows that the computed stacking free energy has a nontrivial z dependence, with ΔG(z) increasing initially as z increases starting from zmin ≈ 3.5 Å but reaching a maximum at intermediate values of z before returning to a local minimum at still larger z. With further increases in z, this behavior repeats for a second time as z continues to increase for all three solutes until the free energies approach their asymptotic values. These nonmonotonic variations could not have come from the direct interactions between the two solute molecules. In essence, the observed free energy barrier at z1* ≈ 6 Å separating the stacked dimer at zmin ≈ 3.5 Å from the unstacked monomers at larger distances is a result of the entropic forces in the solvent. Entropy forces solvent molecules to spread out to fill space in such a way that accommodating solute molecules necessarily costs entropic penalty for the solvent. In terms of this entropic consideration, the free energy of stacking between two acenes is then controlled by whether the stacked dimer excludes the same number of solvent molecules as two separate monomers combined (in which case ΔGstack = 0 with no stabilization of stacked dimer configuration) or the stacked dimer excludes fewer solvent molecules than the two monomers (in which case ΔGstack < 0 and the dimer is stable). As we see in Figure 2A, ΔGstack is negative for all three acenes inside the WCA CCl4 solvent for intermolecular separations smaller than ∼4.5 Å, indicating that the stacked dimer must exclude fewer CCl4 molecules than two separate and unstacked monomers under those conditions. However, as the separation between the two stacked acene molecules increases beyond 5 Å the repulsive interaction between solute and solvent molecules does not permit solvent molecules to reinfiltrate the space between the two solutes. Within this range of separations, the combined volume of the dimer plus the void between them excludes more solvent molecules than the monomers. This results in the peak in the free energy profiles at z1* ≈ 6 Å in Figure 2A. The width of this peak should approximately correspond to the thickness of the void space, which according to Figure 2A develops roughly between z ≈ 5 and 9 Å. Curiously, there is a second peak in the free energy profiles in Figure 2A at larger distance at z2* ≈ 11.3 Å. This feature is a result of the microscopic liquid structure of the solvent.27 Free energy curves for unstacking in the z direction from MC simulations in CCl4 with full VDW interactions are shown in Figure 2B. With the inclusion of attractive VDW interactions, the free energy curves are surprisingly similar to the results in Figure 2A for the WCA potential. The two sets of free energies exhibit analogous nonmonotonic z dependence, and they have peaks and valleys in almost the same values of z as each other.

Figure 2. Stacking free energies as a function of intermolecular separation in the direction perpendicular to the plane of the molecules for naphthalene (red), anthracene (organe), and tetracene (green) dimers in carbon tetrachloride derived from umbrella-sampled Monte Carlo simulations of using the Weeks−Chandler−Anderson potential (A) compared to those using the full van der Waals potential (B).

between the two solute molecules prevents excessive steric overlap between them. As the separation between the two solute molecules increases beyond 3.5 Å, the free energy ΔG(z) first increases, reaching a maximum ΔG(z1*) at approximately z1* ≈ 6 Å for all three solutes, and then oscillates after that, passing through a secondary local maximum ΔG(z*2 ) at z2* ≈ 11.1 to 11.4 Å, before reaching an asymptotic free energy value ΔG(z∞). This free energy at infinite separation differs for each of the three solutes, beyond z ≈ 14 Å. For all three, the unstacked solutes are higher in free energy (i.e., less stable) than when they are stacked. The thermodynamic stability of a stacked solute dimer pair is defined by the free energy difference ΔGstack = ΔG(zmin) − ΔG(z∞). From Figure 2A we see that ΔGstack in CCl4 is approximately −3.3 kcal/mol for naphthalene, and it becomes increasingly stable for anthracene and tetracene at −4.6 and −5.1 kcal/mol, respectively, as the molecular size of the solute grows larger. Statistical error (∼0.4 kcal/mol) for the calculated energies is indicated by the error bar in Figure 2A. Throughout these simulations the two solutes were allowed to move along a reversible path where the separation z between them fluctuated at equilibrium, but the two solutes were confined to maintain maximal overlap in the x−y plane throughout. While the asymptotic value ΔG(z∞) in Figure 2 corresponds to two wellseparated solutes which remain perfectly parallel to each other, the final ΔGstack must contain another term ΔGreorient(z∞) describing the free energy difference between two parallel solutes at z∞ and two unstacked solutes with random relative orientation with respect to each other. It is easy to see that D


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The Journal of Physical Chemistry C Since these oscillatory behaviors of ΔG(z) are indicative of solvent control over stacking stability between the solutes, the similarity between Figure 2A and 2B suggests that even in the presence of direct dispersive interactions between the two solute molecules solvent entropy remains to be the dominant factor controlling how they stack. While the z dependence of the free energy curves in Figure 2A for the WCA-only and Figure 2B for the full VDW simulations are similar, the magnitudes of the stacking free energies derived from each potential do show differences. From the full VDW simulations, ΔGstack is predicted to be −4.1, −4.7, and −6.9 kcal/mol for naphthalene, anthracene, and tetracene, respectively, compared to −3.3, −4.6, and −5.1 kcal/mol in the WCA-only simulations. Incorporating direct attractive interactions between the solutes due to dispersion back into the WCA simulations renders them full VDW, and the magnitude of the free energies emerging from the full VDW simulations are stronger than WCA, as expected. However, given that the direct LJ interaction between two parallel naphthalenes at a distance of 3.5 Å provides close to −5.0 kcal/mol of stabilization, the slight differences in the free energies in Figure 2A compared to Figure 2B are far smaller than what can be attributed to direct dispersion interactions. This comparison suggests that the solvent plays a significant role in renormalizing the direct interactions between two solute molecules when they are immersed in a solvent, and this renormalization, as we will see later from the stacking free energy data of the same three solutes in CYX and TOL, is a complicated function of the size and shape of both the solute and the solvent molecules. To summarize the results in CCl4 in Figure 2, the stacking of all three solutes, naphthalene, anthracene, and tetracene, in the full VDW model of the CCl4 solvent is remarkably similar, in both functional form as well as magnitude, to their stacking free energy calculated in the WCA-only CCl4 solvent. This suggests that the physical origins of the driving force behind stacking in CCl4 are (1) primarily solvent induced, (2) predominantly driven by solvent entropic effects, and (3) sufficient to stabilize the stacked pair by several kcal/mol relative to the unstacked monomers. III. Stacking in Toluene and Cyclohexane. Umbrella sampling MC results for the stacking free energy for naphthalene, anthracene, and tetracene in toluene (TOL) are shown in Figure 3A as a function of intermolecular separation z. Solid lines show full VDW simulation results, with the corresponding WCA-only calculations displayed as dotted lines (naphthalene to tetracene, from bottom to top). Comparing the VDW results of the stacking free energies (solid lines) in Figure 3A to the corresponding WCA results (dotted lines), it is clear that they do not exhibit the same similarities that were observed between the VDW and the WCA models of CCl4 in Figure 2. While both the VDW and the WCA free energy profiles show a peak at around 6 Å, the precise position of this free energy maximum is different between the two solvent models. At separations beyond this peak, the WCA free energy profiles for the three different acenes (dotted lines) are rather featureless as they approach their respective large-z asymptotic values, whereas the VDW results exhibit nonmonotonic variations, which are similar among all three solutes and suggest that their origin is due to the solvent. Also, the stacking stability ΔGstack = ΔG(zmin) − ΔG(z∞) from the WCA-only results severely underestimates the VDW values for all three solutes, by ∼2 kcal/mol for

Figure 3. (A) Comparison of the stacking free energies as a function of intermolecular separation, z, for naphthalene (red), anthracene (orange), and tetracene (green) dimers in toluene derived from umbrella-sampled Monte Carlo simulations using the Weeks− Chandler−Anderson potential (dotted lines) to those derived from simulations using the full van der Waals potential (solid lines). (B) Comparison of the stacking free energies as a function of intermolecular separation, z, for naphthalene (red), anthracene (orange), and tetracene (green) dimers in cyclohexane derived from umbrella-sampled Monte Carlo simulations using the Weeks− Chandler−Anderson potential (dotted lines) to those derived from simulations using the full van der Waals potential (solid lines).

naphthalene up to ∼4 kcal/mol for tetracene. Taken together, these observations suggest that while intrinsic solvent entropic effects seem to control stacking within compact solvents like CCl4 (see Figure 2), the driving forces behind stacking in solvents like toluene are more complex. Toluene is a planar molecule (see Figure 1) with all of its carbon atoms lying on the same plane. Structurally, toluene is very similar in both shape and size to benzene, the smallest of the aromatic molecules in the acene family. On the basis of this consideration of molecular structure, it is possible that one or more toluene molecules can better intercalate between two acene solutes than CCl4 can. The MC simulations corroborate this. Figure 4 shows a representative configuration of the tetracene−toluene system calculation from the MC simulation using the VDW potential for an intermolecular dimer separation of ∼7 Å. This clearly shows that a toluene molecule can fit inside the void between the two tetracene molecules by aligning itself parallel to their stacking plane. The molecular structure of the toluene molecules therefore enables them to refill the space between the two solute molecules more efficiently than CCl4. This produces three effects manifested by the free energy profiles in Figure 3A. First, the width of the peak in the free energy profiles, which corresponds roughly to E


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Figure 4. Wall-eye stereogram of a representative configuration of toluene solvent molecules around two tetracene solute molecules separated by a distance of ∼7 Å from a Monte Carlo simulation with full van der Waals interactions, showing that the molecular structure of the solvent allows individual solvent molecules to intercalate between the solutes and to stabilize the two tetracenes via weak attractive dispersion forces.

Figure 5. Wall-eye stereogram of a representative configuration of cyclohexane solvent molecules around two tetracenes separated by a distance of ∼7 Å from the Monte Carlo simulations with full van der Waals interactions, showing that similar to toluene the molecular structure of cyclohexane allows them to intercalate between two tetracenes at this intermolecular separation.

the width of the gap between two acene solute molecules devoid of solvent, is narrower by ∼2 Å (z between ∼5 and 7 Å in toluene compared to between ∼5 and 9 Å in CCl4). Second, the peak in the VDW model of toluene (z1* ≈ 5.8 Å) is shifted to a smaller value compared to WCA toluene. One likely interpretation of this effect is that the attractive dispersion interactions between the solute and the solvent toluene molecules in the full VDW model help draw the solvents into the void space between the two solutes. Third, there are features in the VDW free energy profiles beyond the first peak that are not predicted by the WCA model. These features must be due to the details of the liquid structure of toluene, because their signatures are distinct in different solvents (compare the free energy profiles in Figure 3A in toluene to those in Figure 3B in cyclohexane). The precise molecular origin of these features will require more extensive investigations in the future to completely unravel. Umbrella sampling MC results for the stacking free energy for naphthalene, anthracene, and tetracene in cyclohexane are shown in Figure 3B as a function of intermolecular separation z. Solid lines show full VDW simulation results, with the corresponding WCA-only calculations displayed as dotted lines (naphthalene to tetracene, from bottom to top). Cyclohexane exists predominantly in the liquid in its chair conformation. The thermodynamic stability of the boat form of cyclohexane is approximately 6 kcal/mol higher than the chair conformation,31 which favors the chair by more than 15 000 to 1. In all our MC simulations, which consisted of approximately 4090 cyclohexanes in each, all cyclohexanes were fixed in their chair conformation. Although cyclohexane is not a completely planar molecule, its stacking free energy profiles for the three acenes (Figure 3B) show features similar to those in toluene. Figure 5 shows a snapshot from the MC simulation for a tetracene dimer at ∼7 Å separation inside cyclohexane from a VDW simulation. The MC snapshot in Figure 5 reveals that cyclohexane molecules, like toluene, can intercalate in the gap space between two tetracene molecules by aligning themselves predominantly parallel to the solute molecules. Similar to toluene, the free energies predicted by entropy-only WCA and

the full-VDW cyclohexane solvents differ substantially. The functional forms of the VDW and the WCA free energy profiles are also quite distinct. Furthermore, the VDW profiles contain extra molecular liquid structure signatures not observed in WCA model simulations. As in toluene, all these features suggest that the stacking of PAMs in solvents like cyclohexane cannot be fully attributed to solvent entropic effects, and the precise molecular origin of the stacking forces is somewhat more complex. B. Dimerization Equilibria Arising from the Stacking of Acenes. While the stacking free energy calculations presented above reveal tantalizing evidence that stacked acene dimers can be stabilized by solvent-mediated thermodynamic driving forces with a negative ΔGstack of up to several kcal/mol for naphthalene, anthracene, and tetracene, the question that has the most experimental relevance is how do these calculated free energies affect the dimerization equilibrium of these acenes in the different solvents. Equilibrium constants are directly measurable experimentally. Of the nine solute/solvent combinations we simulated, to the best of our knowledge, the experimental dimerization equilibrium constant of only one of them has been published. Park and Herndon32 studied the dimerization equilibrium of naphthalene in CCl4 by concentration-dependent NMR and reported a dimerization constant K for naphthalene + naphthalene ⇌ (naphthalene)2 of 0.071 M−1 at 37 to 38 °C. Estimating the equilibrium constant K requires more than just the stacking free energies in the z direction like those in Figures 2 and 3. The gradients of the free energies must also be known in the other directions too. Treating both solute molecules as rigid planes, the position and orientation for each solute is described by six degrees of freedom (three center-ofmass coordinates and three Euler angles). The two monomers together are therefore described by 12 degrees of freedom. In the dimer, six of these describe the center-of-mass coordinates and the global orientation of the stacked pair and the remaining six describe the internal motions of one acene molecule relative to the other. F


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solvent-induced interactions between the two naphthalene molecules in CCl4. From the free energy profiles in Figure 6 for θ rotation (top right panel) and for y and x sliding (middle left and right panels), it is clear that the naphthalene dimer within CCl4 is structurally rather fragile, and if two naphthalene molecules do form a dimer in CCl4, they must not be very tightly stacked against each other. This makes the direct computation of the dimerization equilibrium constant K difficult, since in addition to the stacking free energy ΔGstack, K also contains terms describing the volume of configuration space accessible to the dimer relative to the volume accessible to the two monomers. Since the naphthalene dimer is not tightly bound, its dimer configuration space volume cannot be accurately evaluated by considering just small deviations away from the free energy minimum. However, this small-deviation approximation nonetheless does provide a first-order estimate for K. Using this approximation and the data in Figure 6, we estimate that the configuration space volume available to a naphthalene dimer in CCl4 is reduced by a factor of 1.3 × 10−5 relative to two naphthalene monomers at 1 M concentration, which, in combination with ΔGstack = −4.1 kcal/mol from Figure 2B, produces an equilibrium constant K ≈ 1.3 × 10−5 M−1 × exp

Many additional umbrella sampling MC simulations were carried out along the other internal coordinates for all nine solute/solvent combinations using the full VDW solvent model. As an example, Figure 6 shows the free energy profiles obtained

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ΔGstack RT

−1

) = 0.014 M

. Notice that since the small-deviation

approximation necessarily underestimates K (i.e., the dimer will appear tighter than it should be), the computed value of K ≈ 0.014 M−1 at 25 °C is a lower bound to the true K and compares quite well with the experimental value of 0.071 M−1 obtained at 37 °C. A more direct alternative to evaluate the dimerization equilibrium constant K is to compute the average concentration of dimers in an equilibrium MC simulation containing a solution of monomers at 1 M concentration in a periodic box. This method does not require any special sampling scheme (e.g., umbrella) and can potentially produce a direct unbiased estimate of K. However, this direct method can also suffer from ergodicity problems and is generally not the method of choice for simulating dimerization equilibria that have either very large or very small K. The equilibrium constant for naphthalene dimerization in CCl4 is of the order of 0.1 M−1, and this direct method works reasonably well. Figure 7 shows a representative snapshot from an equilibrium MC simulation of a 1 M solution of naphthalene in a VDW CCl4 solvent (395 naphthalenes in a 6.55 × 105 Å3 periodic simulation box with 3206 CCl4 molecules) at 25 °C. From the simulation we then constructed a naphthalene−naphthalene radial distribution function based on the distance between the interior C atom on each naphthalene marked by the asterisk (*) in Figure 1 (atom C4a according to conventional numbering convention). This g**(r), which is displayed in Figure 8, shows the probability of finding another naphthalene molecule at a distance r, if one is already present at the origin, normalized by the differential volume of the shell and the concentration of naphthalene in the solution. The small shoulder in g**(r) at r ≈ 3.5−4.5 Å corresponds to the formation of dimers. Integration g**(r) from 0 to r = 4.5 Å yields an estimate that approximately 2.3% of the naphthalene molecules exist in dimers at equilibrium. Since the monomer concentration is 1 M in the simulation, this analysis yields a value for K ≈ 0.023 M−1, which lies quite close to the value of K ≈ 0.014 M−1 obtained above from the free energy profiles using the small-deviation assumption. Interestingly, there is a more prominent peak in the radial distribution

Figure 6. Comparison of the changes in the stacking free energy of naphthalene dimers in carbon tetrachloride upon changes in intermolecular separation (top left), rotation of one molecule in the dimer about their intermolecular axis (top right), sliding of one molecule along their short molecular axis relative to the other (middle left), sliding of one molecule along their long molecular axis relative to the other (middle right), rotation of one molecule about its short molecular axis (bottom left), and rotation of one molecule about its long molecular axis (bottom right), with all other coordinates held fixed.

for naphthalene in CCl4 for each of the six intermolecular degrees of freedom accompanied by a cartoon of the internal motion to which each profile corresponds. The top left panel shows the free energy profile for unstacking two naphthalenes in CCl4 in the z direction, as seen in Figure 2B. Next, the top right panel shows free energy as a function of rotating one naphthalene about the z axis, with the separation between them constrained to zmin = 3.5 Å. The two middle panels show free energy profiles for sliding one naphthalene molecule against the other in the y or x direction. The two bottom panels show free energy profiles for rotating one naphthalene about its short (y) or long its long (x) axis. These curves represent some of the cross sections of the free energy function describing the G


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Figure 7. Wall-eye stereogram showing a representative configuration of a number of naphthalene molecules (orange) within a CCl4 (gray) solvent taken from a Monte Carlo simulation with full van der Waals interactions, illustrating an equilibrium mixture of naphthalene monomers with a few dimers and trimers.

Similar results were observed for anthracene, which we will not show explicitly here. In addition to naphthalene in CCl4, dimerization equilibrium constants have been computed using the free energy curves for each of the nine solute/solvent combinations at 25 °C within the small deviation assumption. These results are summarized in Table 1. As the dimers are more rigid for larger solute molecules, the small-deviation assumption becomes more accurate for anthracene and tetracene in comparison to naphthalene. Table 1 shows that the equilibrium constant is significantly more favorable toward the stacked dimer for tetracene in comparison with naphthalene. Within all three solvents, the stability of the dimer increases from naphthalene to anthracene to tetracene, scaling roughly proportionately with the molecular contact area between the two monomers in the stack. However, across the three different solvents, there are significant variations in the magnitude of K for even the same solute. These variations are the results of complex molecular effects in the solvents as well as in how they solvate the solutes. In CCl4, the arguments presented in the previous section suggest that the dimerization of acenes is driven largely by solvent entropic effects. These effects are caused by the exclusion of solvent molecules from the space occupied by the solutes, and the stacked dimer pair appears to exclude fewer solvent molecules than the two monomers combined when they are separated. However, for toluene and cyclohexane, while solvent entropy remains an important factor, the shape and size of the solvent molecules, as well as the dispersion interactions with each other and with the solutes, affect the stacking of polyaromatic molecules in a highly nontrivial manner.

Figure 8. Radial distribution function g**(r) between the interior C atoms (marked by an asterisk) on different naphthalenes at 1 M concentration in CCl4.

function at distance r ≈ 7 Å. This C*−C* distance is too far for the two naphthalene molecules to be considered stacked, and the nature of this structure will require further investigations in the future to completely unravel. Figure 9 compares the free energy profiles of naphthalene in CCl4 to those of anthracene and tetracene, along several orthogonal directions away from the minimum. In the z direction, the stacking free energy for tetracene is most stable, as Figure 2B has already shown. In addition, the stacked dimer is increasingly more rigid from naphthalene to anthracene to tetracene. This is most clearly revealed by the free energy profiles along the z rotation angle θ (panels in the second column from the left in Figure 9), which show that two tetracene molecules are highly rotationally constrained near their maximally stacked conformation. This result contrasts to the case of naphthalene, which is almost rotationally free. In addition, the free energy profiles for sliding along both the y and the x directions are also narrower for the tetracene dimer as compared to naphthalene. To examine the difference among the different solvents, Figure 10 compares the free energy profiles of tetracene in CCl4 to cyclohexane and toluene. As described previously, the three solvents yield very distinctly different free energy profiles for changes in the intermolecular separation, z. Along the other cross sections, the free energy profile for a tetracene dimer shows relatively minor variations among the three solvents.

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CONCLUSIONS In this study, we examined the origin of the solvent thermodynamic driving forces that mediate stacking interactions between three polyaromatic solute molecules (naphthalene, anthracene, and tetracene) in three simple organic solvents (carbon tetrachloride, toluene, cyclohexane). Using large-scale Monte Carlo simulations and via a systematic comparison of the umbrella-sampled stacking free energy functions computed within various solvent models, we report compelling evidence that solvent entropy alone could play a predominant role in facilitating the aggregation between polyaromatic molecules inside organic solutions. In particular, H


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Figure 9. Comparison of the changes in the stacking free energy of dimers of naphthalene (top), anthracene (middle), and tetracene (bottom) upon changes in intermolecular separation (first column from the left) rotation of one molecule about the intermolecular axis (second column), sliding of one molecule along the short molecular axis (third column), and sliding of one molecule along the long molecular axis (rightmost column) derived from full van der Waals simulations in CCl4.

Figure 10. Same as Figure 9 but comparing the stacking free energies of tetracene among the three different solvents.

these simulations show that solvent entropy can largely account for the formation of dimers of naphthalene, anthracene, and tetracene, with stabilization free energies all exceeding 2 kcal/ mol upon solvation in CCl4. These results are also qualitatively

similar in toluene and cyclohexane, but replacing CCl4 with toluene or cyclohexane as the solvent produces more complex phenomena that cannot be attributed to solvent entropy alone, highlighting the role of the molecular structure of the solvent in I


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The Journal of Physical Chemistry C Table 1. Dimerization Equilibrium Constants K (in M−1) at 25 °C Computed from MC Free Energy Data naphthalene anthracene tetracene

in CCl4

in cyclohexane

in coluene

0.014 0.008 0.47

0.016 0.26 4.5

0.018 0.14 1.3

for CCl4, 0.779 g/cm3 for cyclohexane, and 0.867 g/cm3 for toluene). Approximately 4096 solvent molecules were used in each periodic solvent box, generating a unit cell volume of close to 660 000 Å3 for all three solvents. The neat solvents were equilibrated extensively before solute atoms were introduced and colliding solvent molecules removed. The resulting solute/ solvent mixtures were used as starting configurations for umbrella sampling. Each of the VDW simulations was performed in a constant NVT ensemble by using the equilibrated configurations from the WCA simulations as starting points but restoring the attractive branch to the WCA potential to reconstitute the full VDW potential. Liquid theory29,30 shows that the structures of simple liquids are largely controlled by the repulsive branch, and the radial distribution function of a WCA liquid and the corresponding VDW liquid are essentially indistinguishable at normal liquid densities. Figure 11 demonstrates this using data from the MC simulations to construct each of the g(r), showing that the WCA model produces almost exactly the same liquid structure as the full VDW model solvent. All solute molecules were assumed planar and both solute and solvent molecules were assumed rigid in both the WCA and the VDW models. Because all solute and solvent molecules considered in this study were nonpolar, no charge−charge interactions were active in the simulations. Simulations. Standard Metropolis sampling34 was used, with random translations and rotations applied to one molecule at a time at random. In the NPT simulations, an auxiliary constant pressure move35 was executed every 10 MC passes generating volume fluctuations, where a MC pass is defined as every particle having undergone one trial move on average. Translation and rotation displacements were adjusted to yield approximately 40% acceptance. The stacking free energies for each dimer were computed by umbrella sampling36 in the limit of low solute concentration by embedding a pair of monomers inside a solvent box consisting of typically 4096 solvent molecules under periodic boundary conditions. Each free energy profile was sampled along a single direction relative to the minimum free energy configuration of the dimer, with all other orthogonal degrees of freedom fixed. For example, for the free energy profiles in Figure 2, the two solute molecules were constrained to stay parallel to each other during the entire umbrella run, where only their vertical

the formation and stabilization of dimerized polyaromatic molecules as a complex function of both solvent molecular size and shape. In addition, calculation of the free energy function between two monomers along each of the orthogonal intermolecular coordinates allows us to also accurately estimate the dimerization equilibrium constant of the nine solute/ solvent systems we studied, which compared qualitatively well against available experimental data on the naphthalene/CCl4 system. Given the importance of solvent effects in the solution processing of organic devices, the results found in this study provide a fundamental contribution to the understanding of the drivers of molecular morphology paramount for the successful application of polyaromatic molecules to electronics and photonics applications.

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METHODS Solute and Solvent Models. Ideal geometries33 have been assumed for all solute and solvent molecules. In the full VDW simulations, solute−solute, solvent−solvent, and solute− solvent interactions were described by Amber ff99 Lennard− Jones interactions28 ⎡⎛ ⎞12 ⎛ aij ⎞6 ⎤ aij ⎢ ⎜ ⎟ uVDW (rij) = ϵij⎢⎜ ⎟ − 2⎜⎜ ⎟⎟ ⎥⎥ r ⎝ rij ⎠ ⎦ ⎣⎝ ij ⎠

where standard combination rules were used for the minimumenergy distances aij and for the energies ϵij. In the WCA simulations, these interactions were replaced by a Weeks− Chandler−Andersen (WCA) potential30 uWCA(rij) = uVDW(rij) + ϵij for rij ≤ aij and = 0 for rij > aij. All simulations were carried out with periodic boundary conditions. A constant NPT simulation was used for each of the WCA simulations, with the pressure P adjusted to produce an average solvent density matching its physical value at room temperature (1.594 g/cm3

Figure 11. Radial distribution functions from VDW and WCA solvent models for (A) CCl4 and (B) cyclohexane. In A g(r) between carbons on different CCl4 molecules in the VDW and WCA models are colored black and orange, respectively, and g(r) between chlorines in the VDW and WCA models are colored red and green, respectively. In B g(r) between in-plane carbons on different cyclohexane molecules in the VDW and WCA models are colored black and orange, respectively, and g(r) between out-of-plane carbons in the VDW and WCA models are colored red and green, respectively. The VDW liquid and the corresponding WCA liquid at the same density have essentially the same liquid structure. J


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The Journal of Physical Chemistry C separation z was allowed to fluctuate. To allow for accurate alignment, we ensured that each umbrella window had at least 50% overlap with the window before it and the window after it. Window sizes from 0.2 to 1.0 Å wide were used for the simulations, depending on the variation in the free energy across each window. For each solvent/solute combination, typically 0.5−2 million MC passes were used to converge the free energy in each window. Some of the umbrella runs consisted of more than 60 windows, amounting to a total of close to 30−120 million passes for the entire run. With these statistics, the accuracy of the free energy is better than 0.04 kcal/mol within each window. After alignment, we estimate that the statistical error at any point along the entire free energy profile is no larger than 0.5 kcal/mol, which is indicated by the error bars shown in Figures 2 and 3. The stacking free energies for each of the nine solute/solute combinations were computed individually for the WCA model and then repeated for the VDW model. For a direct estimate of the dimerization equilibrium constant for naphthalene in CCl4 at 1 M concentration, we also performed a series of equilibrium simulations in the NVT ensemble with a fixed number of naphthalenes inside a periodic solvent box of sizes similar to the stacking free energy simulations. Typically, 48 million MC passes were necessary to generate the kind of radial distribution function shown in Figure 8.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: 323-610-3261. *E-mail: [email protected]. Phone: 213-740-4101. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant No. CHE-0713981. A.S.R. acknowledges support from the Anton Burg Foundation through a postdoctoral teaching fellowship.

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Solvent Thermodynamic Driving Force Controls Stacking Interactions

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