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6753b4db7be152b1d031afe9
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The quantity ϕ sol (r) is computed on a real-space Cartesian grid and eq. 2 is solved iteratively (because ϕ sol polarizes the solute's charge density), using a multigrid algorithm. Following previous work by others, implementation of Poisson boundary conditions for a smoothly varying (but otherwise arbitrary) permittivity function ε(r) is accomplished by reformulating eq. 2 as a vacuum-like Poisson equation, 21
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| 14 |
The quantity ρ iter (r) is updated iteratively in solving eq. 16 for a fixed solute charge density (ρ sol ), the latter of which is obtained via self-consistent field (SCF) iterations. For the permittivity function in eq. 1 that forms the basis of PCM theory, the factor [1 -ε(r)]/ε(r) in eq. 18 vanishes inside of the solute cavity. Following iterative solution of eq. 16, the polarization density is used to augment the one-electron contributions to the solute's Fock operator, to include the interaction between ρ sol (r) and ρ pol (r). The total energy is
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| 15 |
represents the interaction of the SCF density (ρ sol ) with the polarized continuum. For a PCM, this polarized electrostatic interaction is the entirety of the solvation energy. The permittivity function in eq. 1 is the one that defines the PCM but the equations outlined in this section can be solved for any smoothly varying function ε(r), subject only to numerical limitations. For example, it is not possible to use an absolutely sharp dielectric boundary (as in eq. 1), and for direct comparison to PCM results this boundary must be smoothed somewhat. In practice this must be done carefully. Too much smoothing causes the model to deviate from the sharp dielectric boundary that defines the PCM and other reaction-field solvation models that are predicated on the use of a sharp spherical boundary for the continuum. In the opposite limit, an absolutely sharp boundary causes wild oscillations in ρ iter (r), leading to convergence failure, unless the discretization grid is made extremely dense (at greatly increased cost). These features make the PEqS approach cumbersome in practice, even if its generality is appealing. HetPCM, on the other hand, is just as computationally simple and robust as any other PCM, requiring relatively inexpensive two-dimensional discretization of the cavity surface rather than three-dimensional discretization of all space. This makes it an attractive alternative to PEqS, provided that it realistically models the same phenomenology.
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The HetPCM method was implemented in a locallymodified version of Q-Chem, where it can be used alongside the PEqS algorithm from Ref. 21 and the isotropic SwiG-PCM algorithm from Ref. 76. PEqS solvation energies for various molecules and basis sets can be found in Tables , which serve to document the implementation for reproducibility, and analogous data for HetPCM can be found in Tables S7-S10. A code to generate the Voronoi-PEqS grids (introduced in Section 3 .1 .2 below) is available. 3.1. Permittivity Models. PEqS calculations require specification of the permittivity function ε(r) in threedimensional space. This procedure was originally introduced for use with a van der Waals (vdW) molecular cavity, i.e., a sharp dielectric interface (as in eq. 1), defined by atomic spheres, somewhat smoothed for the numerical reasons discussed in Section 2 .3. We will call this the "vdW-PEqS" approach, for which the construction of ε(r) is described in Section 3 .1 .1. Alternatively, the "Voronoi-PEqS" method introduced in Section 3 .1 .2 uses atom-centered dielectric regions to define ε(r), as a model that can be compared to HetPCM.
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| 17 |
Illustration of the atomic spheres (A, B, . . .) that comprise the vdW molecular cavity surface, along with distances used to define switching functions in eqs. 24 and 29. For the indicated point r, min X r -RX is obtained for the sphere centered at RA. Therefore, Ā = A in eq. 32.
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| 18 |
where the values R Bondi,A are taken from Ref. 100 and differ from Bondi's original vdW radii 101 in that R Bondi,H is reduced from 1.2 Å to 1.1 Å. The use of eq. 23 is a standard cavity construction for PCM calculations. For a cavity consisting of a single sphere, however, we find that better agreement with the Born ion model is obtained using a switching function based on tanh(x) rather than erf(x). For a single sphere of radius R vdW , centered at R 0 , we take the permittivity function to be
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| 19 |
For fair comparison, this function must correspond reasonably well with a model in which a dielectric constant (either ε solv or ε nonp ) is assigned to each atomic sphere at the user's discretion. To construct a corresponding function ε(r), we use Becke's definition of "fuzzy" Voronoi cells, 102 as described in Appendix A, to define ε(r) in pointwise fashion over the real-space grid that is used in PEqS calculations. The weight function w A (r) that is defined in eq. A14 is used to smooth the boundaries of the Voronoi cells defined by atom (nucleus) A, and then we impose that ε(r) = 1 for all points inside of the molecular cavity. The result is a "primitive" permittivity function
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| 20 |
The effect of F cav (r) in eq. 28 is to make sure that ε ≈ 1 for points near the vdW cavity surface, which is constructed using the atomic radii defined in eq. 23. We take L = 0.5 Å and α = 4/L, as for the single-sphere cavity, so that the switching function is centered L/2 = 0.25 Å outside of the vdW surface as shown in Fig. .
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| 21 |
As an example, Fig. presents two different versions of the function ε(r) for several different solutes. These two versions are labeled "scheme 1" and "scheme 2" and correspond to different portions of each solute cavity being exposed to solvent (ε = ε solv ) or classified as hydrophobic (ε = ε nonp ). These two schemes are discussed in Section 4 .3. For now, it suffices to note, that the Voronoi-based permittivity functions that are plotted in Fig. can easily be mapped onto atomic choices for ε A in a HetPCM calculation.
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| 22 |
3.2. Computational Details. Unless otherwise noted, all calculations were performed using HF theory for the solute, because solvation energies computed with continuum models are no more accurate when DFT is used instead. An exception is that B3LYP+D3(BJ) is used for applications to protein pK a prediction in Section 4 .5, as a more representative use case. Here, D3(BJ) indicates Grimme's dispersion correction based on Becke-Johnson damping. These calculations were performed using Q-Chem with integral screening thresholds and shell-pair drop tolerance set to τ ints = 10 -12 a.u., as appropriate for medium-size systems with diffuse basis functions. The SCF convergence threshold was set to τ SCF = 10 -8 E h . For molecular PCM calculations, the cavity surface is constructed using radii in eq. 23 and discretized using 110 Lebedev points per atomic sphere, in the case of hydrogen, and 194 points per sphere for other atoms.
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| 23 |
Solution of Poisson's equation requires discretization of three-dimensional space, to a distance sufficiently far from the molecule such that ϕ tot (r) ≈ 0. In PEqS, this discretization is performed on a dense Cartesian grid in order to facilitate finite-difference evaluation of the requisite Laplacian, ∇2 ϕ tot (r), and to make use of a Cartesian multigrid algorithm. In Sections 4 .1 and 4 .2, where the solute cavities are spherical, we use a relatively coarse grid spacing ∆x = ∆y = ∆z = 0.2877 Å, which is comparable to the spacing used in previous work on excitation and ionization energies. The latter quantities are relatively insensitive to the grid spacing as compared to ∆G elst , which is sensitive to how quickly ε(r) changes at the boundaries of the molecular cavity. As such, in the present work we use a very fine grid spacing of ∆x = ∆y = ∆z = 0.05 Å when the cavity is constructed from atomic radii, as in Section 4 .3. (See Table for tests of how ∆G elst converges as a function of the grid spacing.) The coordinate origin is placed at the centroid of the atomic coordinates and the grid extends to a total length of 8 Å in each direction. In most cases, this means that all nuclei are at least 2 Å away from the edges of the grid. The multigrid PEqS algorithm described in Ref. 21 is used to solve Poisson's equation at each SCF iteration.
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3.3. Protein Systems. Previously, Su and Li 107 used a heterogeneous PCM to investigate the active sites of five type-1 copper proteins (1EY5, 1ID2, 1KDI, 1PZA, and 2CAK), and we will explore the same systems in this work. Relaxed structures were not provided in Ref. 107 so we started from crystal structures. Following previous work, we first protonated these structures using the H++ web server (pH = 7.0, salinity of 0.15 M, ε in = 10, and ε out = 80), 108,109 then relaxed the geometries using the GFN2-xTB 110,111 in conjunction with a generalized Born/surface area implicit solvent model for water. The relaxed structures were trimmed prior to further relaxation at the DFT level, following the protocol in Ref. 107. A significant number of atoms were fixed (a) In 1E5Y, 1KDI, 1PZA, and 2CAK the copper is in its reduced form (Cu + ), whereas in 1ID2 it is in its oxidized form (Cu 2+ ). These charge states were specified as part of the SCF initial guess. For HetPCM calculations, a dielectric constant ε solv = 78.4 was assigned to the solvent-exposed histidine side chains, whereas the remaining atoms were assigned a value of either ε nonp = 4 or 10. The solute cavity was constructed and discretized as described in Section 3 .2.
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4.1. Evaluating Discretization Errors. As a test of the PEqS setup, including the use of smooth Voronoi cells to construct ε(r), we first consider solvation energies for monatomic ions in spherical cavities, where the Born ion model affords an analytic result for ∆G elst , 3 assuming a sharp dielectric interface and an isotropic medium, as in eq. 1. As such, comparison to PEqS results is an assessment of discretization errors in the latter. This comparison is presented in Table . Isotropic PCM results are included for completeness, and this method exactly reproduces the analytic Born model, whereas PEqS errors average about 1% of ∆G elst .
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| 26 |
These include a standard vdW cavity construction, as in the original PEqS approach, which simply means smoothing the sharp dielectric boundary according to eq. 21. (For the present calculations, that boundary surface consists of a single sphere.) In addition, we consider pointwise construction of ε(r) according to the Voronoi scheme in eq. 28. These calculations use large atomic radii (R = 3.12R vdW ) to ensure that there is no escaped charge, since the analytic Born model is only exact under that assump- tion. A 21 Å × 21 Å × 21 Å grid is employed, with ∆x = ∆y = ∆z = 0.29 Å, which is typical of earlier PEqS calculations. For the PEqS calculations using the standard vdW cavity construction, the discretized Poisson model solvates the ions slightly more strongly than the exact analytic solution to Poisson's equation, although the discretization errors are smaller than 0.3 kcal/mol or about 1% of ∆G elst . For the Voronoi construction of ε(r), the errors are in the other direction but also about 1% of ∆G elst .
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| 27 |
Another case where an analytic solution to Poisson's equation is available is Kirkwood's solution for an arbitrary point multipole in a spherical cavity, embedded in an isotropic medium. To use this model, ρ sol (r) is represented by a multipole expansion (up to = 20 for these examples), then the analytic result is applied for each spherical harmonic function. In Table , we present results for two polar diatomic molecules and one polyatomic ion, using spherical cavities with R = 5.5-6.5 Å so that there is no escaped charge. Because the solvation energies for the neutral species are quite small, the PEqS grid spacing is set to ∆x = ∆y = ∆z = 0.05 Å in order to provide well-converged results.
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| 28 |
For the standard vdW cavity construction, PEqS calculations reproduce analytic results to < 0.01 kcal/mol accuracy for the molecules FH and ClH, although the solvation energies themselves are |∆G elst | = 0.1-0.2 kcal/mol. For the Voronoi construction, the error for FH is large in relative terms but is only 0.1 kcal/mol in absolute terms, while the error for ClH is negligibly small. The NO - 3 ion has a much larger solvation energy and larger absolute errors for PEqS as compared to the analytic Kirkwood result, on the order of 0.3-0.4 kcal/mol and thus similar to discretization errors for the monatomic ions. As in that case, this error is only about 1% of |∆G elst |. This level of agreement suggests that we can use the PEqS code, in conjunction with the Voronoi construction for ε(r), to generate near-exact benchmarks that can be used as reference values for other methods. As such, only the Voronoi construction of ε(r) will be considered in subsequent PEqS calculations.
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| 29 |
The Voronoi-PEqS solvation energies are systematically smaller than PCM values, with absolute differences ranging from 0.8-3.0 kcal/mol and averaging 1.8 kcal/mol. However, these differences amount to no a Calculations at the HF/def2-TZVPD level. b Permittivity construction; see Fig. . c Voronoi construction of ε(r) using L = 0.5 Å, k = 10, and a 0.05 Å grid spacing.
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| 30 |
In order to justify the use of HetPCM, we next compute solvation energies using different dielectric setups and compare HetPCM results to Voronoi-PEqS values, as the Voronoi construction of ε(r) best maps onto the concept of "different dielectric constants for different atoms". We use ε solv = 78.4 for all of these calculations but vary ε nonp . Results are provided in Tables 4 for ε nonp = 1 and 2, and in Table for ε nonp = 4 and 10.
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For each combination of dielectric constants, we examine two different Voronoi-based constructions of ε(r) that are called "scheme 1" and "scheme 2" in Tables and. This is a reference to the numbering scheme introduced in Fig. , which shows how three-dimensional space is partitioned into polar and nonpolar regions characterized by ε solv and ε nonp , respectively. Taking NO - 3 as an example, the difference between schemes 1 and 2 lies in whether a single N-O moiety is exposed to the high-dielectric region (scheme 1, Fig. ) or else two N-O moieties are immersed in ε solv (scheme 2, Fig ). One may object that this is not a physically realistic setup for a strongly polar species such as NO - 3 , although it is a well-defined model problem. Considering glycine as another example, the difference between schemes 1 and 2 lies in whether the carboxylate moiety is solvent-exposed (scheme 1, Fig. ) or whether the amino group is instead solvent-exposed (scheme 2, Fig. ). For other solutes, the meaning of scheme 1 versus scheme 2 can be gleaned from Fig. , and in each case amounts to exposing different parts of the vdW cavity surface to ε solv = 78.4 versus a smaller dielectric value, ε nonp . In the Voronoi-PEqS calculations this is done in a smooth way-and Fig. is actually a color map of the permittivity function ε(r) that is constructed-whereas for HetPCM, ε solv and ε nonp are assigned at the atomic level, in a manner that should be obvious from Fig. .
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| 32 |
The most extreme example of heterogeneity is to set ε nonp = 1, for which results are provided in Table . Here, the mean absolute difference between the HetPCM and Voronoi-PEqS values of ∆G elst is 11.2 kcal/mol (or 64% of ∆G elst ) with a maximum deviation of 35.2 kcal/mol. This is clearly an unacceptably large deviation, reflecting the ad hoc nature of the HetPCM construction. It is notable, however, that the largest errors are incurred for the least realistic solvation environments. For example, in the case of NO - 3 there is a 56% difference between HetPCM and Voronoi-PEqS values of ∆G elst when one N-O moiety is in contact with the solvent (represented by ε solv = 78.4), which is reduced to 19% error when the high-dielectric environment surrounds two N-O moieties. Similar remarks can be made for the other ions, for which scheme 1 is a less realistic solvation model because it puts less ion's vdW surface in contact with ε solv . In contrast, scheme 2 surrounds more of the vdW surface with ε solv , which is closer to a homogeneous solvation model.
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| 33 |
For the neutral species (NaCl, glycine, benzene, and phenol), absolute differences between HetPCM and Voronoi-PEqS solvation energies are 1 kcal/mol even for the case where ε nonp = 1 and ε solv = 78.4. These solvation energies are much smaller as compared to the ions, a Calculations at the HF/def2-TZVPD level. b Permittivity construction; see Fig. . c Voronoi construction of ε(r) using L = 0.5 Å, k = 10, and a 0.05 Å grid spacing.
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and in several of the charge-neutral cases the percentage errors in ∆G elst are rather large. For example, phenol exhibits the largest differences absolute differences in ∆G elst amongst the neutral solutes (at 2 kcal/mol), which translates into relative errors as large as 118% since the Voronoi-PEqS benchmarks lie in the range |∆G elst | = 4-6 kcal/mol. However, these deviations are substantially reduced simply by increasing ε nonp from 1 to 2 (Table ). For the combination of ε nonp = 2 and ε solv = 78.4, the maximum deviation between the HetPCM and Voronoi-PEqS methods is no greater than 1.7 kcal/mol (41% of ∆G elst ) for the neutral solutes. (Including the ions, the average deviation is 4.8 kcal/mol.) Although the change from ε nonp = 1 to 2 seems quite modest, there is precedent for such a change inducing relatively large changes in reaction energies in QM/PCM calculations on cluster models of enzymes. Those changes are often large upon passing from vacuum dielectric boundaries to ε = 2, with the effect saturating by ε = 4. The agreement between HetPCM and Voronoi-PEqS improves even further for ε nonp = 4 or 10, as demonstrated in Table . For ε nonp = 4, which is often used to represent the hydrophobic interiors of proteins, the absolute differences for charge-neutral solutes are all < 2 kcal/mol, which is no more than a 32% error, while the errors for ions are all < 10%. For ε nonp = 10, this difference drops to 2% for both neutral and ionic solutes.
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For context, we note that absolute errors in C-PCM solvation energies average 1.6 kcal/mol for small, chargeneutral solutes in water, if one compares calculated value of ∆G elst to the experimental free energies of solvation (∆G • ), with no corrections for nonelectrostatic interactions. (The experimental data have estimated uncertainties of only ±0.2 kcal/mol. 121,122 ) As such, discrepancies of 1-2 kcal/mol between HetPCM and Voronoi-PEqS are the same order of magnitude as the intrinsic accuracy of continuum solvation models themselves. For ions, the accuracy of C-PCM drops to 7-8 kcal/mol for aqueous solvation, although the uncertainties associated with the experimental data are ±3 kcal/mol for ions. In our judgement, HetPCM behaves reasonably well as assessed by comparison to the Voronoi-PEqS method and in comparison to the inherent accuracy of continuum solvation models, provided that ε nonp ≥ 4. The explanation for this is likely the rapidity with which the continuum solvation energy converges to the ε → ∞ limit, which is at least part of the reason why values ε > 4 have so little effect in enzyme modeling. In contrast, the combination of ε nonp = 1 with ε solv = 78.4 affords much larger discrepancies between HetPCM and Poisson electrostatic solvation energies.
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Calculation of ∆G elst is not actually our motivation to develop anisotropic continuum solvation models. A more compelling rationale for development of HetPCM is simply to have heterogeneous dielectric boundary conditions that can be applied to sizable biomolecular models, where it does not make sense to use the same dielectric constant for all residues because some of them may be solventexposed while others are buried within the hydropho- bic interior of the protein. What we desire is a qualitative means to provide compensating charge when an ionic residue is solvent-exposed, while not over-polarizing the hydrophobic regions with a large dielectric constant. In this way, HetPCM boundary conditions (combined with an atomistic quantum chemistry description of the solute) may provide a reasonable, qualitative description of the electrostatic potential in a QM model of a biomolecule. We explore that possibility directly in the present section.
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Figure compares surface electrostatic potentials for a protonated His(+)-Ile-His-Ile tetrapeptide, computed using either a conventional PCM with ε s = 78.4, or else HetPCM with ε solv = 78.4 and ε nonp = 4. The first histidine in the sequence is protonated and positively charged, so we select ε solv for the atoms of this cationic histidine and ε nonp for the other three residues.
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For the protonated histidine residue, there is essentially no difference in the surface charge predicted by the isotropic PCM and HetPCM methods (Fig. ). However, subtle differences emerge on the other three residues, which are described with nonpolar boundaries in HetPCM but are solvent-exposed in the conventional PCM calculations. In the latter case, there is buildup of charge associated with the nitrogen and oxygen atoms that is not observed with HetPCM, where the residues in question are exposed to a low-dielectric environment that apparently cannot support such a charge buildup. We expect this to be important in biomolecular modeling, in order to prevent anomalous buildup of charge on what are supposed to be hydrophobic parts of the surface. In contrast, and not unexpectedly, the surface charge at the alkyl groups is more similar between these two models. Whereas tests against exact Poisson boundaries in Section 4 .3 suggest that HetPCM should probably not be used to predict absolute solvation energies, results below demonstrate that the use of an appropriate reference state does allow calculation of a relative solvation energy, and thus a pK a . This methodology builds upon work by others to compute relative ∆G • and pK a values using a heterogeneous PCM. We also provide an assessment of how the HetPCM results change with respect to the value of ε nonp that is used for the surface boundary of the hydrophobic portions of the protein model.
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The type-1 Cu centers considered here are critical to electron transfer in biological systems. The Cu center is coordinated by two histidines and one cysteine residue in a trigonal planar structure, and mutations or modifications of this site are known to cause disorders related to copper homeostasis. Two protonated forms of these centers exist in solution, which are depicted for the enzyme fern plastocyanin (PDB code 1KDI) in Fig. . In the HetPCM models, a singular histidine residue is exposed to a higher dielectric, as indicated. Following Ref. 107, 1KDI is used as a reference for relative pK a calculations in other enzymes. The free energy of each protein (Pro), relative to that of plastocyanin (Plc), is estimated using the reaction
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| 40 |
where PlcH is the protonated form of fern plastocyanin. The standard-state free energy change ∆G • will be approximated using the electronic energies for each system, neglecting entropic effects. From this, the pK a of each protein is calculated (at T = 298 K) is computed using 1KDI as a reference (pK a = 4.4 ± 0.1), 124 so the pK a for the Pro in eq. 33 is pK a = 4.4 + ∆G • RT ln (10) .
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Relative energies of two different acid forms, analogous to the 1KDI structures in Fig. and Fig. , are listed in Table with a sign convention such that the imidazolium-flipped form is more stable when ∆E < 0. These calculations, performed at the B3LYP+D3(BJ)/ 6-31+G* level, suggest that the flipped form is preferred for the enzymes 1E5Y, 1ID2, and 2CAK, as it emerges lower in energy regardless of the boundary conditions. For the other two examples (1KDI and 1PZA), it is unclear which form is preferred because different boundary conditions afford different relative energies.
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Experiments on 1ID2 suggest that the imidazoliumflipped acid form is more stable than the unflipped form by about 1 kcal/mol, in solution at T = 297 K. Focusing on 1ID2, the conventional PCM with ε s = 78.4 is closest to that experimental value, predicting that the flipped from is 1.75 kcal/mol more stable, although the two HetPCM methods predict that the flipped form is 2.0-2.5 kcal/mol more stable, which is much closer to experiment than the gas-phase value (4.9 kcal/mol more stable).
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Relative energies reported in Table are not consistent in sign with B3LYP-based heterogeneous PCM calculations reported previously by We have not attempted to mimic their computational paradigm exactly; for example, calculations in Ref. 107 omit diffuse functions and any dispersion correction. Experiments suggest that π-π interactions near the active sites of Cucontaining metalloenzymes have a significant impact on pK a values, 126 suggesting dispersion effects may be important. Another difference, relative to the calculations reported in Ref. 107, is that the latter employ a unitedatom cavity construction with no explicit atomic spheres for the hydrogen atoms. Although this seems like an odd choice for pK a calculations, we have performed calculations at the B3LYP+D3(BJ)/6-31+G* level using the same cavity construction, for comparison. Results in Table S11 demonstrate that the relative energies are rather erratic in sign and magnitude.
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The polarized electrostatic interaction ∆G elst is a key factor in determining amino acide residues interact with their surroundings, which is essential for understanding protein stability in biological environments. Table compares ∆G elst for the three forms of each protein (acid, base, and imidazolium-flipped acid), computed using different PCM-based boundary conditions. We now understand that absolute HetPCM solvation energies do not always agree with Poisson benchmarks, and there is significant variation in ∆G elst across different solvation models in Table . The isotropic PCM affords the largest values of |∆G elst |, because it allows the most polarization charge to accumulate at the solute cavity surface.
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One might anticipate that relative solvation energies (comparing different isomers of the same protein) are likely to be more accurate than absolute solvation energies, and indeed we find that difference in ∆G elst are more consistent, comparing HetPCM with ε nonp = 4 versus ε nonp = 10 (Table ). These results are consistent in trend (but not necessarily magnitude) with those presented by Su and Li. In general, our calculations afford larger values of |∆G elst | as compared to those reported in Ref. 107, consistent with the use of smaller atomic radii in the present work. Results obtained with unitedatom radii (Table ) are more consistent with those in Ref. 107.
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Finally, we compute pK a values for these Cucontaining proteins. Prediction of pK a values is a notoriously difficult computational problem, but one where continuum solvation has historically played a role. Figure reports pK a values for each system in its flipped and unflipped form using various PCM-type boundary conditions. (Numerical data are provided in Table .) Experimental values, used here and in Ref. 107 to assess errors, are: pK a < 2 for 1E5Y and 2CAK, 131-135 pK a = 7.2 for 1ID2, 136 pK a = 4.4 for 1KDI, 124 and pK a = 4.8 for 1PZA. For each system except 2CAK, we see a considerable decrease in the predicted pK a upon application of any solvent model. Reduction in pK a upon solvation is expected based on stabilization of the charged species, and the one exception is also the only system to exhibit a negative pK a . This indicates that 2CAK is a much stronger acid than fern plastocyanin. Looking at the different solvation models, the best-performing one (as compared to experiment) is HetPCM with ε nonp = 10. However, all of the solvation models reduce the error as compared to the gas-phase result. Excluding fern plastocyanin (1KDI), which is the reference value, the average absolute difference from experiment, expressed in pK a units, is 4.6 (gas phase), 1.3 (isotropic PCM), 1.4 (HetPCM with ε nonp = 4) and 1.1 (HetPCM with ε nonp = 10). Expressed in energy units (∆G • ), both versions of HetPCM predict these values within 1.3 kcal/mol of experiment.
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Results obtained with gas-phase (vacuum) boundary conditions are effectively useless, with average pK a errors that exceed 2 in most cases and exceed 6 in several cases. The largest differences with respect to experiment are for the enzyme 1PZA, which was also true in Ref. 107. (In the present work, the relaxed structure for 1PZA exhibits coordination distances to the Cu center that differ by only about 0.1 Å from those reported in Ref. 107.) Moreover, gas-phase calculations cannot rationalize differences in pK a upon imidazolium flipping. The differ-ence between the two acid forms (∆pK a ), averaged across the data set in Fig. , is ∆pK a > 1 for gas-phase calculations but ∆pK a < 1 for solvated calculations.
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On average, HetPCM with ε nonp = 4 shows a smaller difference from the experimentally measured pK a values for the flipped acid form (1.16 pK a units) and the HetPCM model using ε nonp = 10 has a smaller difference for the acid form (1.28 pK a units). For the unflipped acid, the homogeneous PCM has a very slightly smaller average difference (1.26 pK a units) as compared to the HetPCM methods. To obtain better agreement with experiment, one might construct a larger QM model of the active site, or else include a larger number of low-energy configurations in the averaging.
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Overall, HetPCM with ε nonp = 10 is in better agreement with the homogeneous PCM representation with ε s = 78.4 across all three metrics investigated for these protein systems as compared to the HetPCM with ε nonp = 4. This behavior highlights the diminishing sensitivity of the model to differences in dielectric constant as the values approach those of bulk water, effectively mirroring a homogeneous representation. At higher dielectric constants, the effects of using dielectric boundary conditions become saturated, and the differences between internal and external dielectric environments are less pronounced. This convergence suggests that the choice of dielectric constant within a reasonable range may not significantly impact the predictions for solvation free energies or pK a values. These findings emphasize the importance of using a PCM to improve the accuracy of cluster-QM enzyme models, while demonstrating a range of ε nonp values within which the results are not too sensitive to the specific choice.
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The Voronoi-PEqS method has been shown to replicate exact (Kirkwood multipolar) solvation results for spherical cavities, and near-exact PCM results for molecular cavities, which justifies our smooth Voronoi construction of the permittivity function ε(r). The latter can be used as a test of the ad hoc HetPCM approach. The HetPCM and Voronoi-PEqS methods afford solvation energies (∆G elst ) within 2 kcal/mol of one another when the nonpolar dielectric constant is ε nonp = 4 or 10, but agreement deteriorates considerably for ε nonp ≤ 2.
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An an exemplary application, we tested HetPCM with ε nonp = 4 and 10 for prediction of pK a values in a set of Cu proteins, using ε nonp to represent hydrophobic parts of a protein model and ε solv = 78.4 for solvent-exposed parts, for the purpose of DFT calculations. Reasonable agreement with experiment is obtained, which is vastly superior to the use of vacuum boundary conditions. HetPCM may thus represent a simple means to implement heterogeneous dielectric boundary conditions for cluster-QM enzymatic modeling with electronic structure theory.
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The use of PEqS requires a dense three-dimensional grid with ε(r) defined at every point. For the model environments considered here, where the dielectric constant is defined atomwise, this requires smooth interpolation between different atomic values of the dielectric constant (ε A in eq. 28). values of ε. For this we use Becke's version of Voronoi cells with smooth boundaries. This scheme was developed by Becke for numerical evaluation of three-dimensional integrals,
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where {r i } and {w i } are a set of quadrature points and weights, respectively. For molecular integrals, it makes sense to partition the sum in eq. A1 into atomic regions and to use nuclear-centered grids for each region. The partition is accomplished using Voronoi polyhedra, 138 defined by the nuclei. The idea is to define a weight function w A (r) for each atom A, such that the normalization condition A w A (r) = 1 (A2) holds at every point r in three-dimensional space. This condition implies that the integrand in eq. (A1) can be decomposed into atomic contributions,
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Following Becke, 102 we define the Voronoi polyhedra using a two-center coordinate system, via confocal elliptical coordinates r = (λ AB , µ AB , φ AB ) for the pair of nuclei situated at R A and R B . In this coordinate system, φ AB is the angle between r and the internuclear axis (A10)
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Anisotropic molecules, whose shape deviates significantly from spherical, are hugely important in many technologies. In many cases, such molecules can form orientationally ordered liquidcrystal phases that can be controlled by temperature or concentration, and are the basis of a multi-billion dollar display industry, with an extensive history dating back to the late nineteenth century. Additionally, organic molecules with extended π-conjugation, many of which form liquid-crystal phases, have been shown to display interesting opto-electronic properties, and devices based on these organic semiconductors (OSCs) are growing in importance as their performance improves. As important electronic processes often happen in the vicinity of interfaces, controlling the alignment of molecules at these interfaces is important for optimising the performances of OSC-based devices and liquid-crystal displays (LCDs). Due to the anisotropic shape of these molecules, symmetry breaking at both solid and vapour interfaces often leads to a preferred orientation, even when the bulk phase is isotropic. This interface alignment has broad implications for device performance.
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For a uniaxial molecule (in which two of the principal axes are equivalent), alignment can vary between the extremes of the non-equivalent molecular axis aligned parallel (planar anchoring; side-on orientation for an oblate ellipsoid) or perpendicular Department of Chemistry, School of Physical Sciences, The University of Adelaide, Adelaide, Australia, Tel: +61 8 8313 5580; Email: david.huang@adelaide.edu.au † Electronic Supplementary Information (ESI) available. See DOI: 00.0000/00000000. end/side-on face-on Fig. Face-on versus end/side-on alignment at an interface for diskshaped particles. Interface orientation of rod-like particles can be similarly dened by the angle of their long axis with respect to the interface. For uniaxial particles, end-on and side-on orientations are equivalent.
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In OSCs in particular, orientation at the solid and vapour interface is important for improving the performance of a variety of devices, with different functionalities requiring different alignments. In organic field-effect transistors (OFETs), for example, charge mobility is greater when molecules are aligned side-on at the solid dielectric interface, as charge transport can occur in the π-stacking direction rather than having to proceed through the often insulating side-chains. Conversely, in organic photovoltaics (OPVs), the face-on orientation at the electron donoracceptor bulk heterojunction (BHJ) interface is favoured as it reduces charge recombination by allowing efficient charge transport away from the interface, while also ensuring better charge generation by increasing donor-acceptor orbital overlap. More specifically, charge separation at the donor-acceptor interface has been shown to be favoured by a displaced planar (slipstacked) molecular arrangement rather than a perfectly stacked (cofacial) arrangement because it leads to weakly bound charge-transfer states that can readily separate but are less susceptible to recombination. Organic light-emitting diode (OLED) performance also depends on the orientation of the emitter's transition dipole moment with respect to the substrate, with planar alignment preferred. The design of molecules for these devices should therefore focus not only on the electronic properties of the individual molecule, but also on how the molecules align at important interfaces in the device in order to maximise performance.
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Although the importance of controlling interface orientation in OSC devices is generally well understood, and there is no shortage of experimental or computational examples of preferential alignment of anisotropic molecules at interfaces, there remain few general rules for predicting alignment at either the solid or vapour interface from the chemical structure of OSC molecules. We have recently extensively reviewed the subject in the context of OSCs, and despite a number of factors being implicated in controlling molecular orientation, in many cases an understanding of the general physical principles that dictate this behaviour was lacking, especially at the solid interface.
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There is an extensive history of simulations of liquid crystals using simple computational models (see ref 33 for a review), which have been used to provide a basic understanding of the behaviour of these anisotropic molecules. A smaller subset of these studies have examined the interplay of repulsive and attractive interactions in controlling the interfacial orientation. At a free interface (such as that with a vapour phase), the orientation of anisotropic particles has been studied at various levels, ranging from mean field theories, to particle-based molecular dynamics (MD), and Monte-Carlo (MC) simulations. It has been found that switching between the face-on and side-on orientation at this interface can be achieved by tuning the anisotropy of the attractive interactions. In the absence of attractive interactions, the long axis of an anisotropic molecule has been predicted to align perpendicular to the interface due to excluded-volume entropic effects. MD simulations of the Gay-Berne (GB) fluid have indicated a dependence of the orientation at the vapour interface of nematic fluids on the ratio of the shape anisotropy to the interaction anisotropy of the particles. Fluids that are isotropic in the bulk have also been shown to display preferential alignment at a vapour interface. In contrast, at the solid interface, free volume is maximised in the face-on orientation, for which the long molecular axis is parallel to the interface. Density functional theory calculations of hard platelets, and theoretical calculations of hard spherocylinders and spheroplatelets at hard impenetrable walls show a preference for alignment with the short axis of the molecule perpendicular to the wall due to entropic effects and minimisation of the surface tension. The inclusion of attractive interactions between fluid particles has been predicted to introduce a temperature dependence to interface orientation, with systems at high temperatures showing the same alignment as the purely repulsive systems due to the dominance of entropy, and a transition to the opposite alignment at lower temperatures as energetic effects become more significant. Many computational studies of specific systems have shown similar dependences on temperature and interaction anisotropy. In general, the orientation at the solid interface depends on both the strength and anisotropy of the fluid-substrate interactions, and the strength of the fluid-fluid interactions. A number of experimental studies report control of orientation at this interface for specific small molecules, and the strength of the interactions with the substrate has been shown to directly influence the orientation, again for a specific set of interaction parameters. An interesting illustration of the contrasting behaviour of hard particles at solid and vapour interfaces is the case of a penetrable wall, from which a particle's centre-of-mass is repelled but which the rest of the molecule can penetrate to varying degrees. As the wall interacts isotropically with the centre-of-mass of the anisotropic particle, a highly penetrable wall is qualitatively similar to a free interface, whereas a impenetrable wall represents a solid interface. MC and density functional theory calculations of these interfaces show a transition between face-on alignment at an impenetrable surface and side-on alignment at a fully penetrable wall, consistent with previous theoretical predictions at solid and vapour interfaces, respectively, due to excluded-volume entropic effects. Although a number of general rules that appear to be relatively predictive for determining the orientation of anisotropic particles at both solid and vapour interfaces exist, and many specific examples of orientation control can be found in the literature, a more detailed understanding of how molecular shape and interactions control the orientation in general, and the ability to predict orientation based on combined energetic and entropic contributions using a simple analytical expression, is lacking. In this work, we investigate a series of disk-shaped OSC-like systems using equilibrium coarse-grained MD simulations of GB particles between solid and vapour-like interfaces. These simulations allow for systematic tuning of the strength of the interactions, shape of the anisotropic molecule, and its interactions with the substrate, to gain an understanding of how the interplay of these features influences the orientation at both the solid and vapour interfaces. The use of GB particles, which have a simple ellipsoidal shape, means that these results should be generally applicable to a wide range of molecules, whether OSCs or not, that share similar shapes and interactions, rather than specific to a single type or class of chemical structure.
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We focus our work here on systems that are isotropic in the bulk in order to isolate the effects of the interfaces, to remove the additional complications associated with alignment of additional layers, and to broadly understand what happens under experimental conditions in which a thin film of anisotropic molecules is annealed from the melt, deposited from solution, or subjected to high operating temperatures. It is important to note, however, that device fabrication from an isotropic liquid phase is uncommon, and not expected to be possible at room temperature. The behaviour of isotropic liquids is therefore more applicable to post-processing of thin-film devices, which may involve treatment with high temperatures, than to the deposition process. Though studying only simple single-component systems, this work also presents an initial step towards understanding the behaviour of components of mixtures (such as solution-phase OSCs) at various interfaces, which has significant importance for the fabrication of OSC devices. Generalisation to interfaces with bulk fluids that are anisotropic is possible. We also focus on simulations of equilibrium fluid structure.
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The paper is organised as follows. We first describe the simulation and analysis methods in section 2. We then show that molecular orientation of anisotropic fluid particles with both repulsive and attractive interactions at the solid and vapour interface of an isotropic fluid cannot generally be predicted using a previously proposed metric, and propose a semi-empirical meanfield estimate of the free-energy difference between the perfectly face-on and side-on orientations as an alternative. We use simulations of purely repulsive particles confined between a solid and vapour interface to obtain quasi-universal relationships for the scaling of the entropic component of the free energy with the shape anisotropy and bulk density of the fluid particles (section 3.1). Combining this estimate of the entropy with an estimate of the energetic component of the free energy from nearestneighbour pair interactions (section 3.2), we show that this free energy parameter accurately predicts molecular orientation at the solid (section 3.3.1) and vapour (section 3.3.2) interfaces. In section 3.4 we generalise these results into a number of practical design principles for OSCs, with the goal of providing guidelines towards rational design of OSC interfaces.
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The orientation of a fluid of anisotropic particles at both solid (impenetrable) and vapour-like (penetrable) interfaces was studied using classical molecular dynamics (MD) simulations. A range of uniaxial oblate Gay-Berne (GB) ellipsoids, representative of typical small OSC molecules, were studied to elucidate the general effects of shape and interaction anisotropy. These simple models greatly increase computational efficiency compared with all-atom simulations, allowing a wide variety of parameters and conditions to be examined. Simulations were carried out using LAMMPS (version 3 March 2020), and visualisation and analysis of simulation trajectories were conducted using OVITO. The GB potential is an anisotropic form of the widely used Lennard-Jones (LJ) pair potential, and captures the short-ranged excluded-volume repulsion and longer ranged van der Waals attraction between uncharged anisotropic molecules. The potential is characterised by parameters σ and ε that define the length and energy scales of the potential, respectively, parameters ν and µ that tune the shape of the potential, and, for uniaxial particles, the short and long principal diameters, σ F and σ S , and dimensionless relative well depths, ε F and ε S , along the corresponding axes (the subscripts "F" and "S" denote "face" and "side", respectively). The anisotropy of the uniaxial particles is described by the shape anisotropy parameter κ = σ F /σ S and the interaction anisotropy parameter κ = ε S /ε F . A full description of the GB potential and parameters is given in the ESI, section S1.
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In all simulations, we have used ν = 1 and µ = 2, as is common for the GB potential, in line with its original parameterisation and previous parameterisations of OSC systems, and and have set ε S = 1 and σ F = 1.03σ . A non-bonded interaction cutoff of 3σ S was used for all systems. In order to match exper-Table Shape anisotropy parameter κ and interaction anisotropy parameter κ for uniaxial models of some typical organic semiconductors (OSCs) and benzene based on published parameters for biaxial models. The sideside parameters used to calculate the anisotropies in each uniaxial model were obtained as the average of the sideside and endend parameters in the biaxial model, as described in the main text. were used as a starting point to define the interactions in the systems studied here. The relationship between σ F and σ was taken from the published perylene model, with a similar relationship found for a range of other simple OSC systems. To obtain a simpler uniaxial representation of each molecule, for which the end-end and side-side interactions and dimensions are equivalent, the published biaxial principaldiameter and well-depth parameters for end-end and side-side interactions were averaged. This typically resulted in changes to these parameters of less than 10 and 20%, respectively, from their original published values. The resulting shape and interaction anisotropies of several small organic molecules are given in Table and inform the values of κ and κ explored.
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We have used LJ reduced units throughout this work, with energies, lengths, temperatures, pressures, and times given in units of ε, σ , ε/k B , ε/σ 3 , and τ ≡ m 0 σ 2 /ε, where m 0 is the unit of mass and k B is the Boltzmann constant. We note that setting ε = 1.8 kcal/mol and σ = 3.2 Å gives behaviour consistent with the experimental behaviour of perylene, i.e. a density of 1.1 g/cm 3 at 560 K and 1 atm for a system with κ and κ approximately those of perylene in Table , and so the parameter range studied should be representative of the behaviour of a variety of OSCs given this choice of ε and σ .
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To mimic the scaling of molecular mass with molecular size typical of OSCs, the particle mass m was scaled in proportion to the particle volume v f = πσ F σ 2 S /6 (calculated as the volume for an ellipsoid with principal diameters σ F , σ S , and σ S ) to give a constant mass density within the particle. Although approximate, this behaviour is reasonably representative of real systems, for which the mass density within the particle generally varies between 1.4 and 1.7 g/mol/Å 3 when the particle volume is calculated from published biaxial GB parameters (ESI Table ). To facilitate comparison between systems with different sized particles, we have analysed results in terms of the volume fraction, φ , rather than the number density, ρ b , in the bulk fluid. The two are related as
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To encompass the region of parameter space in Table for typical OSCs, values of κ of 0.30, 0.35, 0.40, 0.45, and 0.50, and κ of 0.15, 0.20, 0.25, and 0.30 were used. Additional κ values of 0.50 and 0.70 were examined for certain systems to access points with κ > κ, as interfacial behaviour has previously been predicted to exhibit a transition around κ/κ = 1 in some circumstances. Several systems with κ = 1.2 were also studied to examine the behaviour when side-side interactions are stronger than faceface ones. While side-side interactions that are stronger than the face-face ones are not typical of OSCs-like molecules, they can potentially be accessed through, for example, functionalisation of the aromatic core and may present an interesting means of tuning interfacial orientation. The systems with κ > 1 in this work had weaker face-face interactions than the systems with κ < 1 in order to maintain an isotropic bulk fluid as the side-side interactions were increased, meaning face-face interactions were weaker than would be the case for a typical OSC. As the alignment at both solid and vapour interfaces is expected to depend on temperature, due to competition between excluded-volume entropic effects and attractive intermolecular interactions, several temperatures between 0.56 and 0.84ε/k B were studied, corresponding to a temperature range of 507-761 K for the representative OSC systems described above, encompassing a range of temperatures where the OSCs-type molecules that are represented by these GB parameters are liquid. A full list of the specific systems studied is given in the ESI, Table .
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Simulations were conducted with the fluid confined in the z direction between two fixed interfaces parallel to the (x, y)-plane, as illustrated in Fig. : a particle-based solid surface at the bottom and a perfectly flat wall at the top that interacts with the center of each fluid particle with a repulsive harmonic potential,
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where z w is the wall position, ε w describes the strength of the wall-particle interaction, and z is the vertical position of the centre of the fluid particle. This repulsive wall was implemented at the top surface of the fluid to maintain an approximately constant average fluid density between systems with particles of the same shape and to prevent evaporation. Similar to a vapour interface, this wall constrained only the vertical position of the centre of a fluid particle and placed no constraints on the position of the particle surface defined by the repulsive part of the GB potential.
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The solid substrate was modelled as a single layer of either atomistic graphene or silicon, positioned with atoms centred at z = 0. These substrates have been shown experimentally to interact strongly (graphene) or weakly (silicon) with OSCs and to give rise to face-on and side-on orientations, respectively, of the common OSC polymer poly(3-hexylthiophene) (P3HT). Parameters (non-bonded interactions and bond length) for graphene were taken from the OPLS-AA force field for aromatic carbons (ε LJ = 0.039ε, σ LJ = 1.11σ , bond length = 0.44σ , for ε = 1.8 kcal/mol and σ = 3.2 Å) and atoms positioned in the hexagonal lattice of graphene with separation taken from the OPLS-AA force field as the bond length for aromatic carbons. OPLS parameters were also used for silicon (ε LJ = 0.056ε, σ LJ = 1.70σ ) and atoms positioned to give the fcc(001) plane of silicon's diamond cubic lat-
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Example of the system setup, with GayBerne (GB) uid particles conned between two surfaces: the impenetrable solid surface consisted of a single layer of Lennard-Jones (LJ) particles, centered at z = 0 (black dash-dot line), while a repulsive harmonic potential acted on the centre of each uid particle at the penetrable vapour-like interface (dashed black line) to prevent evaporation. The plot at the top right is representative of the form of this potential. The bulk region was dened as being within ±3σ S of the z coordinate of the centre of the box (z mid , black solid line). z max is the maximum z coordinate of any particle during the simulation, σ sub the diameter of a substrate particle, and dotted black lines indicate the bounds of the bulk region. This region was suciently far from both interfaces that, except for a couple of the systems that formed anisotropic phases, the uid had uniform density here. The position of the vapour interface, z v (green line), was dened as the z-coordinate where the average density was equal to half the average bulk value. The average molecular orientation at the vapour interface, s vap , was calculated as the average of s(z) from eqn (2) over particles whose centres were at z coordinates within z v ± σ S /2 (region indicated in gure). The average molecular orientation at the solid interface, s sub , was calculated by averaging s(z) over uid particles whose centres were within σ S of the surface of the substrate (i.e. at z < σ sub /2 + σ S , cyan line). These denitions selected, to a good approximation, the particles within the rst uid layer at either interface. Particles are coloured by the orientational order parameter s(z) (which is -0.5 and +1, respectively, for fully side-on and face-on orientations), averaged over the entire simulation of this system (κ = 0.4, κ = 0.2, T * ≡ k B T /ε = 0.73, substrate = graphene). The particles in the two interfacial regions are coloured by the average order parameter calculated over all particles within this region, while all other particles are coloured according to the value of s(z) corresponding to their z coordinate shown in (c). (b) Fluid density ρ(z) and (c) orientational order parameter s(z) proles calculated over the entire simulation of this system. The shaded areas correspond to the interfacial uid regions at the solid or vapour interface. Dashed and dotted horizontal lines in (c) correspond to the values of s sub and s vap respectively. Additional ρ(z) and φ (z) proles can be found in the ESI, section S5.
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tice (lattice spacing of 1.69σ ). Although ε LJ is larger for silicon, the larger lattice spacing means that the overall attraction of a single GB fluid particle to the substrate is weaker for silicon than graphene. An additional substrate with the same structure as graphene but with interactions that were twice as strong (ε LJ = 0.078ε, σ LJ = 1.11σ ) was also examined, which we will call the "strong" substrate. The structures of these substrates are shown in the ESI, Fig. . The mixing rules for combining GB parameters of two dissimilar particles, defined by eqns (S3)-(S8) of the ESI, were used to model the fluid-substrate interactions so as to maintain the correct shape dependence of the interactions. The interaction range σ in eqn (S2) of the ESI was replaced by the arithmetic mean, (σ + σ LJ )/2, of the fluid and substrate interaction ranges for the fluid-substrate interactions. The substrate parameters, and the influence of the type of substrate on the orientation at both interfaces are given in full in the ESI (Table , Fig. ).
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To give comparable steepness for the repulsive potential at the vapour interface to the interaction with the solid substrate, the value of ε w was determined by fitting eqn (1) to the repulsive part of the potential energy versus z of a single fluid particle representative of perylene (κ = 0.36, κ = 0.19) interacting with the solid substrate in the face-on orientation for the region where the potential energy was < 2k B T at T = 0.62ε/k B . This gave a value of ε w = 17.8ε for the graphene substrate and ε w = 8.9ε for the silicon substrate. The same values for each substrate were used for both the attractive and repulsive surfaces, and the graphene value also used for the "strong" substrate, as the strength of the wall potential did not significantly affect the behaviour at the vapour interface (ESI Fig. ).
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All simulations were conducted at constant volume and temperature with the positions of the substrate particles fixed. Each system consisted of 6000 fluid particles and 4680 (graphene) or 800 (silicon) substrate particles. The simulation timestep was 0.012 τ, which corresponds to 5 fs if the unit of mass m 0 is taken to be the molecular mass of perylene. Temperature was controlled with a Nosé-Hoover thermostat and set to values of 0.56, 0.62, 0.73, or 0.84ε/k B . Most systems were constrained to a constant overall volume fraction φ av (that is, the volume fraction calculated from the number density of all particles between z = σ sub /2, the surface of the substrate, and z = z w , the position of the harmonic wall) of 0.39, corresponding to a real density on the order of 1.1 g/cm 3 if ε = 1.8 kcal/mol and σ = 3.2 Å. The z coordinate of the harmonic wall, chosen to achieve the desired overall volume fraction for the different values of κ, are given in the ESI Table . Additional systems were examined at lower (0.28) and higher (0.49) φ av values to determine the dependence (if any) of interfacial orientation on the system density. The pressure in each system was measured as the normal force per unit area acting on the solid substrate. The average pressure at equilibrium was generally < 200ε/σ 3 , but was up to 1200ε/σ 3 for φ av = 0.49.
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To initialise the simulations, fluid particles were packed with random positions and orientations into a large box (periodic in x, y, and z, with x and y dimensions 10× the target values and z dimension equal to the target value). For systems to be simulated with the graphene substrate, the target x and y dimensions were 34.48σ and 34.125σ , respectively, to enable periodic packing of the substrate particles; for the silicon substrate, the final dimensions were x = y = 33.94σ for the same reason. The size in the z dimension was used to control the volume fraction of fluid particles and so depended on fluid particle size; the values for various systems can be found in the ESI Table . A soft potential U soft (r) = A 1 + cos πr r c for r < r c , where r c is is a cutoff distance and A an energy pre-factor that varies the "hardness" of the potential, was applied to remove particle overlaps (r c = 5σ , A increased linearly from 0 to 16.7ε over 20,000 timesteps). Interactions were changed from the soft potential to the GB potential, and the box dimensions shrunk linearly over 50,000 timesteps at a higher temperature of T = 0.88ε/k B to give the final dimensions outlined above. To initialise the system in an isotropic phase, the fluid particles were then allowed to equilibrate in the bulk system at T = 0.88ε/k B for another 50,000 time steps. The periodic boundary conditions in the z direction were then removed, and the box extended in the z direction to give a region of vacuum above the bulk fluid. The solid substrate was introduced below the fluid, and the harmonic wall at the vapour interface was positioned at the z coordinate required to give the desired overall volume fraction (ESI Table ). The energy of the system was then minimised to remove any overlaps between fluid and interfaces. The system was simulated for a further 510,000 timesteps at the specified temperature, with the equilibration and correlation time for the orientation of the top and bottom fluid layers determined using pymbar's timeseries module, which estimates the equilibration time as the time that maximises the number of uncorrelated samples (see ref. 71 for details). Only the data after the equilibration time was used for further analysis. Although a number of systems were simulated that were anisotropic in the bulk (typically for the most anisotropic shapes and interactions; see ESI Table for a list of systems that formed anisotropic bulk phases), the slow dynamics in these systems did not allow equilibrium properties to be measured reliably over the time scales that were simulated. Results are therefore only reported in the main paper for systems with an isotropic bulk fluid phase.
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As their interactions have, to a good approximation, no energetic component, purely repulsive particles were used to determine the contribution of entropy to molecular alignment at the solid and vapour interfaces. For these purely repulsive particles, the shape parameters described above (with κ = 0.30, 0.35, 0.40, 0.45, 0.50) were used, and κ was fixed at 0.2 (ε S = 1, ε F = 5). The GB potential was cut-off and shifted to 0 at the minimum for each orientation to obtain a purely repulsive potential, analogous to the Weeks-Chandler-Andersen (WCA) truncation of the LJ potential. Likewise, a cut-off and shifted form of the fluid-substrate interactions, using the mixing rules for combining GB parameters of two dissimilar particles discussed previously, was used for the interactions with the solid substrate. These simulations were conducted at T = 0.62ε/k B at overall volume fractions between 0.1 and 0.5 using either the silicon or graphene substrate. Although these systems are expected to be athermal, simulations were also carried out for the silicon substrate at T = 0.8ε/k B to verify this fact. A full list of the systems studied is given in ESI Table . Data for the different substrates and temperatures was combined to determine the scaling of interface orientation with system parameters (see ESI, section S4).
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where θ is the angle between the short (unique) particle axis and the z axis, and the average • • • z is over all fluid particles whose centres were in the histogram bin centred at position z. A value of 1 indicates fully face-on alignment, and -1 2 indicates fully sideon. For isotropic orientations, s(z) = 0. Throughout this work, the terms face-on and side-on will be used to describe any orientation that is either predominantly face-on (s(z) > 0) or side-on (s(z) < 0), not just the fully aligned extremes. Representative density and orientation profiles are shown in Fig. , and selected additional profiles are given in the ESI, section S5.
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For a bulk nematic GB fluid, the orientation at the vapour interface has previously been shown to scale with the ratio of the shape and interaction anisotropy parameters, κ/κ , with a faceon orientation observed for κ/κ < 1, and an side-on orientation otherwise. For the systems studied in this work, in which the bulk phase is isotropic rather than nematic and in which solid interfaces are also considered, Fig. shows that κ/κ is not a good predictor of orientation at either the solid or vapour interface. At the vapour interface, the orientation does not appear to depend on κ/κ for κ/κ > 1, although there seems to be a transition to very marginally face-on orientations at κ/κ < 1, as predicted previously for bulk nematic fluids. The systems that display a slightly face-on orientation are those with the most isotropic interactions studied here (κ = 0.7), resulting in very slight orientation preference, while those with the more face-on orientation are where side-side interactions are stronger than face-face (κ = 1.2). Stronger alignment could be achieved by decreasing κ (increasing interaction anisotropy), but the corresponding increase in the shape anisotropy κ required to maintain κ/κ < 1, coupled with the more anisotropic interactions, would likely result in systems that are anisotropic in the bulk. At the solid interface, there is a weak trend towards a more face-on orientation as κ/κ decreases within each set of substrate/temperature conditions, but there is a strong dependence on temperature and κ/κ = 1 does not correspond to the transition from side-on to face-on orientation. This dependence on temperature points to a significant entropic contribution, which is only accounted for implicitly in the κ/κ parameter through the shape anisotropy κ.
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face-on (θ = 0) and side-on (θ = π/2) orientations, which explicitly accounts for the roles of temperature and fluid-substrate interactions. The overall free energy includes an entropic term, ∆ S, described in section 3.1 and calculated based on the shape anisotropy, κ, and bulk volume fraction, φ , of the system, and an energetic term, ∆ Ū, described in section 3.2 and based on the interactions between nearest-neighbour particles. The free energy difference per interface particle between perfectly face-on and side-on orientations is calculated as
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Although a number of theories exist to quantify the entropy difference between face-on and side-on interface configurations, a simple, accurate, analytical expression for the entropy as a function of system parameters is lacking, and a nematic bulk phase is often assumed. Instead, we have taken a semi-empirical approach, in which the entropy difference is measured in a comparatively small number of MD simulations of purely repulsive particles, to obtain a universal scaling relationship at the solid and vapour interfaces. Due to the absence of attractive interactions, any alignment of the purely repulsive particles at the solid or vapour interfaces can, to a good approximation, be attributed entirely to entropy. The Helmholtz free energy difference between two states of a system can be calculated as ∆F = ∆U -T ∆S, which in the absence of an energetic contribution (∆U = 0) gives ∆F = -T ∆S. The entropy difference per interface particle between states of a system in the NVT ensemble with perfectly face-on and side-on orientations at an interface can then be estimated as
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| 25 |
where P F and P S are the probabilities of finding a fluid particle in the solid or vapour interfacial region (defined in Fig. ) in the perfectly face-on (cos(θ ) = 1) and side-on (cos(θ ) = 0) orientations, respectively. The orientational probability distribution P(cos(θ )) versus cos(θ ) was approximated as a normalized histogram of observations of the value of cos(θ ) for fluid particles found in the interfacial region during the simulation, using evenly spaced histogram bins of width ∆ cos(θ ) = 0.04 between cos(θ ) = 0 and cos(θ ) = 1. P S and P F were taken as the histogram values of the first (probability of 0 ≤ cos(θ ) ≤ ∆ cos(θ )) and last (probability of 1 -∆ cos(θ ) < cos(θ ) ≤ 1) histogram bins, respectively. It was verified that the histogram bin size was sufficiently small and the number of histogram counts sufficiently large that further decreasing it had little effect on the calculated entropy difference.
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| 26 |
The orientation at the solid interface was found always to be predominantly face-on, while a predominantly side-on orientation was exclusively found at the vapour interface (Fig. ). In both cases, the degree of alignment was to depend systematically on the volume fraction of the bulk fluid and the shape anisotropy of the fluid particles, with higher densities and shape anisotropies giving a greater degree of alignment. It has previously been predicted that a variety of hard anisotropic particles align with their long axis parallel to a solid substrate, as this alignment maximises the free volume and hence is the entropically favoured orientation. The opposite argument can be applied at the vapour interface, where the particles are able to extend over the interface, increasing the free volume when their long axis is perpendicular to the interface. Studies of hard ellipsoids at penetrable walls also show this behaviour. Thus, for oblate particles, a face-on orientation at the solid interface and a side-on orientation at the vapour interfaces are predicted to be entropically favoured, which is qualitatively consistent with the results presented here.
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| 27 |
Based on expectations from free-volume arguments that the entropy difference between face-on and side-on interface configurations of purely repulsive particles is controlled by the fluid density (or volume fraction) and molecular shape anisotropy, with the entropy difference increasing with increasing density and shape anisotropy (decreasing κ for oblate particles), we have fit the entropy difference at each interface for all systems to a power law with a positive exponent for φ and a negative one for κ. The entropy difference was found to have a quasi-universal power-law dependence on φ /κ at the solid interface and on φ /κ 1/2 at the vapour interface, where φ is the bulk fluid volume fraction and κ is the shape anisotropy, as shown in Fig. , although there is some scatter in the data, particularly for the solid interface. The scaling relationships that best fit the data in Fig. were
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| 28 |
While the scaling at the vapour interface with φ /κ 1/2 is the same for both substrates, the relationship between the entropy difference and φ /κ for the silicon substrate is not as straightforward as that for the graphene substrate (Fig. ). Due to the spacing of the substrate particles (ESI Fig. ), the fluid-substrate potential has more significant lateral corrugations at the silicon interface compared with graphene. This induces a greater dependence of the potential on the in-plane (x, y) coordinates compared to graphene, which is much closer to an "ideal" planar solid surface for which the potential depends only on the z coordinate. Despite the additional variability, the overall scaling behaviour . (The missing points at low κ and high φ formed anisotropic bulk phases and could not be equilibrated on the simulation time scale, and so are not included.) follows the same trend at both substrates, and the fit to all the data was used.
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| 29 |
Overall, if entropy is the only contributing factor and the system is isotropic in the bulk region, the orientation preference at the solid interface is expected to be face-on, and side-on at the vapour interface, consistent with previously published theories and calculations. Increasing the effect of entropy, such as by using higher temperatures, more anisotropic particles, or higher pressures (to increase density), could be used to enhance this alignment if desired at either interface.
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| 30 |
The alignment of particles with attractive interactions at the solid interface is complicated by attractive interactions of the fluid with the substrate and within the fluid itself. Qualitatively, the faceon orientation should be energetically favoured when the facesubstrate interactions are strong (although this will also depend on the relative strength of the fluid face-face, side-side, and sidesubstrate interactions), and the side-on orientation should be energetically favoured when the fluid particle's face-face interactions are stronger than both their side-side and face-substrate in-teractions. Assuming that the fluid is isotropic everywhere except for the layer adjacent to the interface, and that a fully face-on particle interacts with six nearest neighbours through side-side interactions (hcp packing) while a side-on particle has two faceface interactions and two side-side interactions (cubic packing with inequivalent lattice spacing) with its nearest neighbours (as found in simulations in which the interface layer was close to fully aligned, as shown in ESI Fig. ), the difference in energy per interfacial fluid particle between a completely face-on orientated interfacial fluid layer and a completely side-on oriented one can be estimated to be
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| 31 |
where ŪF ( ŪS ) is the interaction energy of a particle in the face-on (side-on) orientation with its nearest neighbours in the interfacial layer and with the substrate. (Note that the nearest-neighbour fluid interactions are halved in these equations because each such interaction contributes to the energy of two interfacial fluid particles.) Ūi j is the interaction energy in the face-on/face-face orientation (for i = F) or side-on/side-side orientation (for i = S) with the substrate (for j = s) or another fluid particle (for j = f).
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| 32 |
For fluid-fluid interactions, the interaction energy in either the face-face or side-side orientation was taken as the minimum of the GB pair potential in the specified ŪFs and ŪSs were calculated as the minimum in the interaction between a single GB ellipsoid and the substrate in the face-face and side-side orientations, respectively. As a particle-based substrate was used in this work rather than being perfectly flat, the energy minimum was averaged over 100 random positions on the substrate. Equation (7) also applies for the energy difference per fluid particle at the vapour interface, but with the fluid-substrate interactions set to zero, i.e. ŪFs = ŪSs = 0.
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| 33 |
Although the consideration of only nearest-neighbour interactions in the calculation of the interfacial energy difference is a major simplification, it accounts for the most significant inter-particle interactions due to the rapid decay of interaction strength with inter-particle separation; for example, next-nearest-neighbour interactions are only a few percent of the nearest-neighbour interactions in the face-face orientation for the systems studied. Thus, even though a more complex description of interfacial energetics that includes more inter-particle interactions may be more quantitatively accurate, it is not expected to qualitatively affect our findings.
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61f8a5d4191637746dfb103e
| 34 |
To predict the behaviour at both the solid and vapour interfaces, we have used the free energy described in eqn (3), with the entropic component determined from the bulk volume fraction and shape anisotropy of each system as described by eqns ( ) and ( ) for the solid and vapour interfaces, respectively, and with the energetic component determined from the nearest-neighbour in-teractions in the interfacial layer as described by eqn (7).
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61f8a5d4191637746dfb103e
| 35 |
The average molecular orientation at the solid substrate, measured by the orientation order parameter s sub , shows excellent universal scaling with the free energy difference defined by eqn (3) for the entire range of systems studied, covering temperatures from 0.56 to 0.84ε/k B , bulk volume fractions from 0.29 to 0.50, strongly and weakly interacting substrates, and shape and interaction anisotropies corresponding to molecules as varied as benzene, perylene, and porphine (Fig. ; see ESI Table for a full list of systems). Interestingly, although systems with a predominantly side-on orientation at the solid interface are observed, they are relatively rare, being obtained in less than 15% of cases. These side-on systems generally correspond to situations for which the fluid particles have low shape anisotropy (high κ), the system is at low temperature, and to a lesser extent the fluid particles have high interaction anisotropy (low κ ). The rarity of the side-on orientation points to a dominant entropic contribution in most systems studied.
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| 36 |
On separating the free energy into its energetic (Fig. ) and entropic (Fig. ) components, the reason for the preference for the face-on orientation at this interface becomes clear. Over the range of parameters studied, the energetic component varies from favouring the side-on orientation by values typically in the range of approximately 2ε, to the face-on orientation by a similar amount. Over the same range of parameters, the entropic component of the free energy always favours the face-on orientation by between 1 and 5ε, increasing in magnitude as the shape becomes more anisotropic. The combined result of these effects is an orientation that is only side-on under conditions where the entropic contribution is lowest (low temperature, low shape anisotropy) or the energetic contribution highest (highly anisotropic interactions). Although highly anisotropic interactions shift the energetic term significantly towards favouring the side-on orientation, high temperatures would be required in many cases to maintain an isotropic bulk fluid, which would increase the mangitude of the entropic term in the opposing direction, giving little change to the overall free energy. This separation of the free energy into its components also highlights that neither the energy nor the entropy is sufficient to completely explain the observed interface orientation, again emphasising the importance of both contributions for accurately predicting the orientation.
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61f8a5d4191637746dfb103e
| 37 |
When examining the effect of the substrate on the interface orientation, a slight dependence of the scaling of orientation with free energy difference is observed, with the observed orientation at the silicon substrate generally being shifted towards side-on relative to systems at the graphene-structured substrate (with either regular or strong interactions) for the same free energy difference (ESI Fig. ). As described previously in the context of the purely repulsive systems, this slight substrate dependence is likely due to the looser packing of substrate particles in the silicon substrate compared with the graphene substrate. This increases the dependence of the fluid-substrate potential on the in-plane coordinates of the fluid particle at the silicon substrate, which behaves as a slighly penetrable surface, relative to the graphene substrate, 6 Orientational order parameter s sub at the solid interface as a function of (a) overall, (b) energetic, and (c) entropic components of the interfacial free energy dierence. A least-squares best t to a logistic function, constrained to go to -0.5 and 1 at high and low ∆ Ū -T ∆ S respectively, is shown in (a). The energetic component is coloured by the interaction anisotropy parameter, the entropy component by the shape anisotropy parameter, and the overall free energy by the ratio of the two parameters. Filled symbols correspond to the graphene-structure substrate (graphene or "strong") and unlled symbols to the silicon substrate. Shaded grey regions highlight where the points should fall if obeying the predicted scaling with the free energy or free energy component.
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| 38 |
which is much closer to an "ideal" smooth impenetrable solid substrate. No dependence of the scaling relationship on the strength of the fluid-substrate interactions was observed, however, with the results at the graphene and "strong" (same particle arrangement but double the interaction strength) substrates showing the same behaviour. This result indicates that patterning of the substrate may be an effective method to favour a more side-on alignment, as has been previously observed, although further study would be required to fully understand this behaviour.
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61f8a5d4191637746dfb103e
| 39 |
Although it appears difficult to significantly influence the interfacial orientation by tuning fluid-fluid interaction strength within the realm of reasonable OSC parameters while the bulk fluid remains in an isotropic liquid phase, the energetic component of the free energy difference does have a strong dependence on the strength of the fluid-substrate interactions. Switching from a strongly (e.g. graphite) to weakly (e.g. silica) interacting substrate has been shown experimentally and computationally to influence the orientation at the solid interface for a range of OSCs. Fig. highlights the relative importance of the fluid-substrate interactions: the difference in energy on switching from a strongly to weakly interacting substrate (e.g. graphene to silicon) is comparable to changing the interaction anisotropy from values close to those representative of benzene to perylene (a significant structural change), and is sufficient to switch the free energy from favouring face-on to side-on orientation in some cases.
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61f8a5d4191637746dfb103e
| 40 |
Although increasing the strength of the substrate-fluid interactions will increase both the side-substrate and face-substrate interaction strength, which make equal and opposite contributions to the energetic component of the interface free energy different in eqn (7), the face-substrate interactions are expected to dominate for most cases studied in this work due to two factors. First, the stronger face-face interaction compared with sideside interaction (κ < 1) typical of OSCs leads to the strength of face-substrate interactions increasing more rapidly than that of side-substrate interactions when the substrate-substrate interaction strength is increased isotropically, given the mixing rules for fluid-substrate interactions. Second, for oblate (κ < 1) particle shapes typical of OSCs (all the systems studied in this work), a face-on particle interacts with more substrate particles than a side-on particle, and so the strength of the interaction of a fluid particle with the whole substrate increases more rapidly for face-on particles than for side-on particles when the substratesubstrate interaction strength is increased isotropically, even if the fluid particles have no interaction anisotropy (κ = 1). Overall, while changing the fluid-fluid or fluid-substrate interactions are both plausible ways of influencing the energy, large structural changes would likely be required chemically to significantly change the fluid-fluid interaction anisotropy, which will likely also change the shape anisotropy and bulk phase behaviour of the fluid. It should generally be much simpler to change the nature of the solid substrate.
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| 41 |
side-chain (one being a conjugated π-system, the other an alkyl chain) means that changing the substrate to favour interactions with either one or the other should enhance either the face-on (strong backbone-substrate interactions) or side-on (strong sidechain-substrate interactions) orientations at the substrate interface. Although we do not consider any molecules with sidechains here, the impact of side-chains can be approximated in the coarse-grained modelling framework used in this work by considering their effect on the overall fluid-substrate interactions: a substrate that is more strongly attracted to the side-chains (particle side) will energetically favour the side-on orientation. This behaviour has been observed experimentally, for example though treatment of a substrate with a self-assembled monolayer of octadecyltrichlorosilane (OTS).
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| 42 |
In contrast to alignment at the solid interface, which depends on the fluid-fluid and fluid-substrate interactions, the energetic driver for orientation at the vapour interface is simply expected to be related to the relative strength of the interactions for a face-on and side-on fluid particle with other fluid particles in the interfacial layer. Face-face interactions are maximised in the side-on orientation, so increasing the strength of these interactions is expected to promote the side-on orientation. Compared with the solid interface, the vapour interface does not show as clear universal scaling of the molecular orientation with the interface free energy parameter in eqn (3), but a strong trend with the free energy is still observed (Fig. ). Where κ < 1, the orientation at this interface converges towards isotropic at the vapour interface as the free energy approaches zero, and becomes more aligned as the free energy magnitude increases. At high values of κ (low interaction anisotropy), the energetic component (Fig. ) shifts towards very slightly favouring a face-on orientation, qualitatively consistent with previous predictions of ordering of bulk nematic fluids at the free interface. Several systems were also examined with κ > 1, corresponding to stronger side-side than face-face interactions, as it is expected to promote the face-on orientation at the vapour interface. respectively is shown in (a). The energetic component is coloured by the interaction anisotropy parameter, the entropy component by the shape anisotropy parameter, and the overall free energy by the ratio of the two parameters. Filled symbols correspond to simulations with the graphenestructure substrate (graphene or "strong") and unlled symbols to the silicon substrate. Shaded grey regions highlight where the points should fall if obeying the predicted scaling with the free energy or free energy component.
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| 43 |
Even with reduced face-face interaction strength, bringing them outside the range of typical OSCs, most of these systems did not remain isotropic in the bulk. The two that did were the least anisotropic in shape and are the face-on outliers at ∆ Ū -T ∆ S ≈ 2ε in Fig. . While this result shows that it is possible to achieve a slight face-on orientation at the vapour interface for anisotropic particles while still maintaining an isotropic bulk phase, these points do not fall in the region where the simple free-energy predictor presented here predicts a face-on orientation. The deviation from universal scaling at the vapour interface is likely due to several factors. Firstly, for a number of the systems, particularly those at high density, the pressure, measured as the normal force on the substrate, was relatively high -up to 1200ε/σ 3 in some cases. Examining the dependence of the relationship between s vap and the free energy on system pressure (ESI, Fig. ) shows more extreme side-on orientation (lower values of s vap ) at higher pressures for the same value of the free energy. While the orientation at the solid interface also shows a slight pressure dependence (ESI, Fig. ), the effect at the vapour interface is more pronounced. As the vapour interface is constrained by a repulsive harmonic wall, interacting with the centers of each fluid particle, high pressures correspond to a strong interaction between the fluid particles and this wall, artificially suppressing fluctuations and in some cases (e.g. ESI, Fig. ) resulting in a density enhancement relative to the bulk and oscillations in the density profile that are not representative of a true vapour interface.
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| 44 |
Secondly, as the energetic component of the interfacial free energy in eqn ( ) is calculated assuming an idealised packing structure for a fully face-on and side-on interfacial layer, variations in the packing structure at the interface reduce the accuracy of the estimated free energy. The positions of fluid particles are less strongly constrained by the vapour interface than the solid substrate, and the side-on packing that predominates at the vapour interface appears to be less robust to disorder than the face-on packing that predominates at the solid interface, due to the tendency of nearest neighbours to be displaced parallel or angled slightly with respect to another. Weaker ordering at the vapour interface results in molecular packings that deviate from the idealised structures, as shown in the ESI in Fig. . Similarly, the calculation of the energetic component of the free energy based solely on the fully face-on and fully side-on configurations means that the predictability of the model is likely to be poorer for systems with interfacial orientation that is closer to isotropic.
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| 45 |
Despite these complications, the results indicate that it is difficult to obtain a face-on orientation under equilibrium conditions of isotropic bulk GB fluids for parameters representative of OSCs. For the systems studied here, the interaction energy favours the face-on orientation when the interaction strength is the most isotropic (κ = 0.7) of the values studied (Fig. ), though only marginally, and more significantly when κ is increased to > 1. In all cases, however, the entropic term (Fig. ) is sufficiently large that the overall free energy still predicts the side-on orientation. While significantly lowering the temperature would help to transition to the face-on orientation by reducing the entropic contri-bution, this is likely to result in a transition to a liquid-crystal or crystalline bulk phase. On the other hand, reducing the entropic component by reducing the shape anisotropy is likely to lead to a more isotropic interface orientation.
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| 46 |
It has previously been shown that the orientation of similar GB particles at a nematic-vapour interface can be easily switched between face-on for κ/κ < 1 to side-on κ/κ > 1, 37 whereas only a very slight preference for the face-on orientation was observed if κ/κ < 1 for the isotropic-bulk systems studied here (Fig. ). The greater propensity for a nematic bulk fluid to give the face-on orientation at the vapour interface compared with an isotropic bulk fluid can be understood qualitatively in terms of differences between the interfacial energy of the face-on relative to the side-on orientation in the two cases. For the isotropic fluid, this energy is described by eqn (7), whereas for the nematic fluid, interactions with the adjacent anisotropic fluid layer must also be taken into account. Assuming that, compared with the isotropic fluid, each fluid particle in the interfacial layer of the nematic fluid interacts with an additional particle of the same orientation in the next fluid layer, the face-on orientation is predicted to be more energetically stabilised relative to the side-on orientation in the nematic fluid than in the isotropic fluid for the systems studied here with κ < 1 and in ref 37, for which the face-face interactions were always stronger than the side-side ones (κ < 1).
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| 47 |
In all manner of OSC-based devices, orientation at interfaces has been correlated with device performance. Reliably controlling the orientation at these interfaces is therefore an important step towards designing high-performing molecules for commercial applications. Based on the results presented in the previous sections, we propose several general guidelines, which are supported by empirical evidence, for how the molecular structure, interface, and processing conditions may be tuned to shift towards the desired alignment at these interfaces.
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| 48 |
Increased shaped anisotropy promotes a face-on orientation at the solid interface, and side-on at the vapour. A number of experimental studies have shown that backbone planarisation (induced by fluorination of the backbone, or extending the conjugation of the backbone, such as by adding aromatic components between the backbone and side-chains, both of which can be associated with an increase in shape anisotropy) results in a more face-on orientation at the donor-acceptor interface in OPVs. It should be noted that both of these chemical modifications will also influence the energetics, so an interplay between changes in interaction strength and shape will control the orientation at the interfaces.
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| 49 |
Stronger fluid face-face (or weaker side-side) interactions promote side-on orientation at both interfaces. Enhancing the hydrogen-bonding capabilities of a small molecule in the sideside direction has recently been shown to promote a more face-on orientation at a solid substrate, consistent with the predictions of our model. However, other methods to tune the interaction anisotropy, such as introducing longer/bulkier side-chains that may block the face of the molecule or deplanarising the backbone, also change the shape anisotropy, which may have the opposite effect on interface orientation.
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| 50 |
Increasing the interaction strength of a solid substrate promotes the face-on orientation at the solid interface. There are many published examples of substrates that interact more strongly with the semiconductor shifting the orientation towards face-on, consistent with the behaviour observed here for different substrates. Choosing a substrate that interacts favourably with either OSC backbone or side-chains may therefore be an effective means of tuning the orientation at this interface.
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| 51 |
Processing conditions that favour entropy promote face-on orientation at the solid interface, and side-on at the vapour. In cases where the equilibrium orientation is face-on at the solid interface (which appear to be many), processes such as melt annealing may allow the equilibrium orientation to be obtained, even if deposited through a non-equilibrium method in the opposite orientation. However, when considering high temperatures, care should be taken to ensure that the orientation remains in the entropically favoured orientation as the system is cooled down, which can be achieved by rapid quenching, as the degree of alignment will be a function of temperature.
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| 52 |
Conditions for which the bulk fluid is anisotropic, or sideside interactions are stronger than face-face ones, can give the face-on orientation at the vapour interface. From our predictions based on the behaviour of a bulk isotropic fluid, the face-on orientation at the vapour interface is difficult to achieve, although it could be obtained by systems with stronger side-side than face-face interactions (e.g. by chemical functionalisation of the aromatic core of the OSC). The interface orientation of liquid crystal fluids has previously been shown to be much more variable. Under appropriate conditions, many OSCs form liquidcrystal phases, meaning the predictions of ref 37 based on the shape and interaction anisotropy can be applied to tune the orientation at the vapour interface.
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| 53 |
Non-equilibrium processing can result in different behaviour than that predicted here. The above guidelines were proposed based on equilibrium behaviour, but could be circumvented using non-equilibrium methods to potentially reach the more difficult to obtain orientations (face-on at the vapour interface, side-on at the solid interface). Many common device processing techniques, such as vapour deposition (see refs 80 and 81 for extensive reviews of this method and how processing temperature can influence the orientation), spin coating, or blade coating, are non-equilibrium in nature and so will not necessarily result in the equilibrium structures predicted in this work.
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| 54 |
Through the use of classical simulations of oblate ellipsoidal fluid particles, this work provides a general analysis of how particle shape and interaction anisotropy influence the equilibrium orientation of a wide variety of isotropic liquids at interfaces with both a solid and vapour-like phase. For a range of parameters spanning a subset of organic semiconductor (OSC) space, different substrate types, and different thermodynamic conditions, we find remarkable universal scaling of interface orientation with a simple interfacial free energy parameter based on nearest-neighbour pair interactions and the interfacial entropy of purely repulsive particles of the same shape. Based on these results, we propose several practical methods for achieving the desired orientation at both interfaces, which we hope provide a pathway towards the rational design of high performing OSC-based devices.
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| 55 |
At the interface with a solid, the entropically favoured orientation is face-on, consistent with previous theories based on excluded volume entropic effects. Even when attractions are introduced, the orientation is largely influenced by the entropy and can be tuned by modifying the shape anisotropy. The side-on orientation is more difficult to obtain, but can be achieved most simply by using a substrate that interacts weakly with the face of the fluid particle (or strongly with its side).
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| 56 |
The entropically favoured orientation at the vapour interface is side-on, in contrast to the alignment at the solid substrate, but again can be rationalised through excluded-volume entropic effects. Importantly, in all cases studied here, the orientation at the vapour interface of a fluid that is isotropic in the bulk is predicted, and generally observed, to be side-on. Conditions for which the face-on orientation is favoured appear to be difficult to achieve for an isotropic bulk liquid phase for interactions typical of OSCs, but can be obtained if the system has a bulk liquid-crystal phase. Tuning of molecular shape and interaction parameters can result in a shift towards a more isotropic interfacial layer.
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| 57 |
Many open questions remain in this area, such as how sidechains, biaxiality, degree of polymerisation, multiple fluid components, or non-equilibrium processes such as pre-aggregation of molecules in solution during solvent evaporation, influence the orientation, but we hope that this work provides a useful starting point for understanding some simple methods for how interfacial orientation can be systematically tuned. Additionally, although this work focuses on oblate ellipsoidal particles, representative of molecules such as perylene, similar arguments should be applicable to prolate molecules such as pentacene. Further studies that account for interactions with the surrounding layers may allow this analysis to be extended to the non-isotropic systems that may be more typical of certain OSC molecules.
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