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J. Phys. Chem. C 2008, 112, 7818–7828
Aggregation of Borate Salts in Hydrocarbon Solvents† B. Endeward,‡,§ P. Brant,| R. D. Nielsen,‡ M. Bernardo,‡,⊥ K. Zick,# and H. Thomann*,‡ ExxonMobil Corporate Strategic Research, Annandale, New Jersey 08801, ExxonMobil Chemical Company, 5200 Bayway DriVe, Baytown, Texas 77522, and Bruker BioSpin GmbH, Silberstreffen, 76287 Rheinstetten, Germany ReceiVed: NoVember 19, 2007; ReVised Manuscript ReceiVed: February 9, 2008
The aggregation of ammonium borate salts in hydrocarbon was investigated by 1H, 19F, and 11B pulsed field gradient (PFG) NMR. The molecular self-diffusion coefficients of the borate salts, [C(C6H5)3][B(C6F5)4], [N(CH3)2(n-C18H37)2][B(C6F5)4], [N(n-C18H37)4][B(C6F5)4], a neutral borane compound, and a siloxane model compound were measured in toluene, cyclohexane, hexane, and a solvent of mixed alkanes. Diffusion coefficients were determined from the echo attenuation of the stimulated echo pulsed field gradient NMR signal as described by Stejskal and Tanner. In all of the samples studied, the echo decay was observed to be a pure exponential decay, corresponding to a single diffusion coefficient within the resolution of the NMR experiment. The hydrodynamic radius of an equivalent diffusing sphere was calculated from the experimental diffusion coefficients using the Stokes-Einstein relation. We found that the neutral borane and siloxane model compounds are monomeric (nonaggregated) in both the aliphatic and aromatic solvents. In contrast, the borate salts exist as simple ion pairs in aromatic solvents, and as larger aggregates in aliphatic solvents for concentrations above approximately 1 mM. In the aliphatic solvents ion pair aggregate numbers are found which range from 5 ( 1 to 11 ( 2 ion pairs. Energy minimized structures of ion-pair multiplets were obtained using molecular mechanics simulations, and were used to establish the dependence of molecular volume on aggregate size. The aggregation of ions in nonpolar solvents with low dielectric constant is consistent with the known chemistry of electrolyte solutions. Ethylene-propylene copolymerizations were carried out in hexanes diluent where 0-2 molar equivalents of [N(n-C18H37)4][B(C6F5)4] was added to a metallocene catalyst and the concentration of the metallocene catalyst was held constant at 0.2 µM. Catalyst productivity decreases and ethylene incorporation increases with increasing ratio of [N(n-C18H37)4][B(C6F5)4] to metallocene catalyst. From these observations, it is inferred that, even under typical catalytic conditions, the ammonium borate salt is in close contact with the metallocene catalyst during polymerization. I. Introduction It is generally accepted that group IVA single site olefin polymerization catalysts are ionic in the active form, and that they consist of a coordinatively unsaturated cation and a weakly coordinating anion. Borate salts are commonly used as discrete activators in single site metallocene chemistry.1–4 For example, the reaction mechanism of bis(dialkylcyclopentadienyl)zirconiumdimethyl with dimethylanilinium perfluorinated tetraphenylborate activator assumes the formation of an ionic single site catalyst (Scheme 1). The degree to which the activator anion (or other activator products) interacts with the 14e- metallocenium cation is a remarkably sensitive function of the activator and metallocenium structures.1–4 The nature of this interaction has a major impact on catalyst activity. Solvent, temperature, and catalyst concentration are several factors that impact the nature of the metallocenium cation-stabilizing anion interaction.5–8 † Part of the “Larry Dalton Festschrift.” * To whom correspondence should be addressed. ‡ ExxonMobil Corporate Strategic Research. § Current address: J.W. Goethe University, Max-von-Laue-Str. 7, 60438 Frankfurt, Germany. | ExxonMobil Chemical Company. ⊥ Current address: SAIC-Frederick, National Cancer Institute, National Institutes of Health, Bethesda, MD 20892. # Bruker BioSpin GmbH.
Virtually all solution polymerizations practiced commercially employ hexane(s), ISOPAR E (a mixture of aliphatic molecules), or similar aliphatic hydrocarbons. However, in practice, metallocene and activator are not typically soluble in aliphatic hydrocarbons at the higher concentrations used to inject them into a reactor. In cases where they are sufficiently soluble, the ionic catalyst product often “oils out” of solution. Consequently, aromatic hydrocarbons often are used initially to dissolve, activate, and inject catalyst into a reactor. Additionally, aromatic hydrocarbons such as toluene or benzene are often used for both activation and polymerization reaction medium in laboratory polymerization studies. In solution phase polymerizations, the properties of the cation-anion interaction could impact many important polymerization properties such as the polymerization rate, the relative probabilities of insertion of different monomers, and molecular weight. If the cation-anion pair aggregate to form ion-pair multiplets, these polymerization properties could be even more strongly affected. For commercial applications of supported catalysts, it is also common to prepare the catalyst using toluene as the solvent before depositing one or more components on a support such as silica. In this case, the potential aggregation of participating cations and anions in the solvent, or during the drying step, could impact the homogeneity of the dispersion of the activated metallocene deposited on the support. Ionic catalysts and activators have marginal solubility in hydrocarbon solvents such as hexane and ISOPAR E, and even toluene. This low solubility is a consequence of the low
10.1021/jp710981g CCC: $40.75 2008 American Chemical Society Published on Web 05/08/2008
Aggregation of Borate Salts in Hydrocarbon Solvents
J. Phys. Chem. C, Vol. 112, No. 21, 2008 7819
SCHEME 1: Reaction of Metallocene with Discrete Activator
dielectric constant of these hydrocarbon solvents. The low dielectric constant also results in a limited capability of the solvent to screen the electric field from each of the ions. The solubility limitation has stimulated research in synthetic chemistry to improve the solubility of activators and catalysts in hydrocarbons.9–13 However, even if the solubility increases, the use of low dielectric hydrocarbon solvents implies that electrostatic interactions between ions will be strong. These electrostatic interactions determine the strength of the ion-pair interaction and can result in the formation of larger ion-pair multiplets. In the present work, we report on the aggregation properties of borate salts. The borate salts serve as a model for the aggregation properties of metallocenes activated with discrete activators; the tetraalkyl amine cation assumes the role of the metallocene metal center. An advantage of studying the aggregation properties of such model borate salts is that they are chemically more stable than metallocenium cations. Furthermore, the borate salts studied here have been added to homogeneous and heterogeneous metallocene catalysts and have been shown to affect catalyst performance and alter the properties of the polyolefin polymerization products. For example, the addition of [N(n-C18H37)4][B(C6F5)4] has been shown to depress molecular weight and productivity, while increasing ethylene incorporation during ethylene-propylene copolymerization with rac-dimethylsilyl(bisindenyl)hafniumdimethyl activated with 1 equiv of dimethylanilinium perfluorotetraphenylborate (see Supporting Information). Similar effects have been reported with other metallocene systems, but no mechanistic interpretation has been given.14,15 Thus, insights into the aggregation properties of these borate salts may prove useful for understanding their potential role in altering the polymerization properties of the metallocene catalyst system. One method to probe the formation of molecular aggregates is pulsed field gradient NMR (PFG NMR).16 The PFG NMR approach to measurement of the diffusion coefficient was first introduced by Stejskal and Tanner.17 The Stokes-Einstein relationship provides a simple method to determine the radius of the diffusing species from the diffusion coefficient. Studies of molecular aggregation in organometallic systems using the PFG NMR approach have recently been reported.18–26 Beck and co-workers used PFG NMR to investigate a series of metallocenes in perdeuterated benzene solution with ammonium borate activators.18 They studied bridged and unbridged bis(cyclopentadienyl)- or bis(indenyl)-zirconium methyl metallocenium cations with either [CH3B(C6F5)3]- or [B(C6F5)4]borate anions, and with catalyst concentrations in the range of 1-20 mM (mmol dm-3). Based on the self-diffusion data recorded at 300 K, Beck and co-workers concluded that, in benzene, these catalysts form ion quadrupoles, i.e., a pair of ion pairs. Song and co-workers have observed aggregates of 3 ion pairs for a borate zirconocene complex at 10 mM in toluene.24 The studies of Marks and co-workers have emphasized the importance of metallocene concentration for aggregation.21,25 The latter studies found no evidence for ion-pair multiplets in the sub mM concentration range for solutions of metallocenes.21,25 All of the cited studies were conducted in aromatic solvents.
In the present work, we investigate the ion-pairing and ionpair aggregation properties of borate salts, as well as neutral borane and model compounds. The study encompasses a range of solute concentrations in both linear and cyclic aliphatic and aromatic solvents. The diffusivities of both solutes and solvent were determined independently from measurements of the 19F, 1H, and 11B PFG NMR signals. The molecular diffusion coefficient of ionic aggregates was determined from PFG measurements on both the anion and cation. The PFG measurements on the solvent provide an internal standard for solvent viscosity. Solvent PFG measurements are also useful for determining whether the solvent is strongly associated with the aggregate. The diffusivities for a neutral borane compound and a model siloxane compound were also determined as standards for the PFG NMR method of determining molecular size. We estimate the number of ion pairs in a molecular aggregate from the energy minimized structures of ion-pair multiplets using molecular mechanics simulations. To our knowledge, this is the first example in which molecular mechanics simulations have been used to determine the aggregation number for ions using molecular diffusivities determined from PFG NMR. The paper is organized as follows: section II describes the materials used and the PFG NMR data collection; section III presents results and analysis of the PFG data, the method for obtaining the hydrodynamic radius of ionic aggregates, and molecular modeling of the ionic aggregates; section IV discusses the applicability of hydrodynamic boundary conditions for the calculation of the hydrodynamic radius, concentration effects due to solute-solute interactions, and the assumptions used in modeling aggregation. Section V presents the conclusions. II. Experimental Details Materials. All solvents used were purified according one of the following methods: vacuum distillation over lithium aluminum hydride or nitrogen sparging followed by passing the solvents through a column of activated basic alumina inside the drybox. ISOPAR E is a commercial solvent available from ExxonMobil Chemical Company. Perfluorotriphenylborane, B(C6F5)3, was purchased from Boulder Chemical and recrystallized from pentane. Its purity was established by 11B and 19F NMR. Octavinyl-T8-silsesquioxane [(C2H3)8(Si8O12)] (T8, Gelest SIO6706.0 CAS 69655-76-1) was purchased from Gelest and used as received. A 1H NMR spectrum was recorded to establish the purity of this sample prior to PFG NMR experiments. Perfluorotetraphenylborate salts were synthesized as described previously.27 For simplicity, throughout this paper we use the following abbreviations: [N(n-C18H37)4][B(C6F5)4] is referred to as (AB-1), [N(CH3)2(n-C18H37)2][B(C6F5) 4] is referred to as (AB-2), and [(n-C6H11)4N][B(C6F5)4] is referred to as (AB-3). See Table 1. 1H, 11B, and 19F NMR and TOF-SIMS were used to establish the purity of each ammonium borate salt prior to the preparation of solutions. Additionally, the melting point of AB-1 was measured by differential scanning calorimetry and found to be 54.5 °C (∆Hmelt 90.7 J/g). NMR sample tubes (J. Young 5 mm NMR tubes, Wilmad Glass) were dried at 130 °C prior to transfer to the dry box for
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TABLE 1: Solutes Used in This Study and Abbreviations solute
MW of solute
[(C2H3)8(Si8O12)] (T8) [N(C18H37)4][B(C6F5)4] (AB-1) [N(CH3)2 (C18H37)2][B(C6F5)4] (AB-2) [(n-C6H11)4N][B(C6F5)4] (AB-3) [C(C6H5)3][B(C6F5)4] (trityl D4) B(C6F5)3
633.07 1706.12 1229.23 977.6 922.28 511.92
TABLE 2: Solution Properties and Nucleus Observed in PFG Experimentsa solute
concn (mM)
solvent
T8 AB-1 AB-1 AB-1 AB-1 AB-1 AB-1 AB-2 AB-2 AB-3 B(C6F5)3 B(C6F5)3 trityl-D4
5 0.1 0.5 5 5 5 10 9.3 10 5.25 5 5 5
toluene hexane hexane hexane ISOPAR E cyclohexane toluene hexane toluene toluene toluene cyclohexane toluene
a
nucleus observed 1
H F 19 F 19 F, 1H 19 F, 11B 19 F, 1H 19 F 19 F 19 F 19 F 19 F, 1H 19 F 19 F 19
All PFG measurements performed at 25 °C.
TABLE 3: Viscosities of Solvents solvent
viscosity in centipoise (mPa s) at 25 °C
toluene28 hexane29 ISOPAR E30 cyclohexane29
0.552 0.294 0.518 0.894
loading with sample solutions. All solutions were prepared and sealed under inert conditions in a glovebox which was equipped with both moisture and oxygen sensors. The molecular weights of the solutes are summarized in Table 1. The solutions used in the present study are summarized in Table 2. The viscosities of the solvents are listed in Table 3. NMR and PFG Data Collection. All NMR experiments were performed on a Bruker Instruments AVANCE series spectrometer operating at either 300 or 400 MHz for 1H NMR and equipped with the Diff-30 PFG accessory. PFG data were recorded using a Stejskal-Tanner stimulated echo sequence.17 Magnetic field gradient pulses are applied during periods of transverse magnetization free precession, but no magnetic field gradients were present during the echo detection period which allowed chemical shift resolved 1H, 19F and 11B spectra to be recorded. Typical experimental parameters were as follows: τπ/2 ) 6 µs; dephasing time ) 2 ms; diffusion time (the time between magnetic field gradient pulses) ) 20 ms; magnetic field gradient pulse widths ) 1 ms; magnetic field gradient magnitudes, g e 10 T/m. For PFG data acquisition the phase quadrature NMR signals (with the peak of the echo intensity as the time origin) were recorded for a series of gradient amplitudes with all other time delays held constant. The digital resolution was 128 points along the gradient dimension and 16k points along the time dimension. Experiment repetition times were typically selected to be at least 3 multiples of T1. Total experimental data acquisition times per sample varied from 30 min up to 6 h depending on the sample concentration and the nucleus observed.
Figure 1. 19F NMR spectrum (top) in toluene and composite PFG of AB-1 in cyclohexane, hexane, ISOPAR E, and toluene (bottom).
III. Results and Analysis Pulsed Field Gradient NMR. The PFG NMR spin-echo attenuation depends on molecular diffusivity as follows:17
Iecho ) I0 e-γ Dg δ (∆-δ⁄3) 2
2 2
(1)
where I0 is the initial magnetization, γ is the gyromagnetic ratio, D is the diffusion coefficient, g is the magnitude of the magnetic field gradient, δ is the magnetic field gradient pulse length, and ∆ is the time between gradient pulses (diffusion time). Equation 1 predicts a single exponential decay for a nucleus in a molecule whose diffusion can be described by isotropic Brownian motion in an isotropic medium. In the isotropic case, D can be determined directly from the slope of ln(I/I0) vs ∆ or g2, using the values of the gyromagnetic ratio and the experimental pulse width. PFG NMR data for the salt AB-1 (see Table 1) in the hydrocarbon solvents are shown in Figures 1 and 2. The 19F NMR spectrum of AB-1 in toluene is shown in Figure 1 (left). Three 19F NMR lines are observed in the spectrum. The three lines are assigned to the ortho (-45 ppm), meta (-11 ppm), and para (-41 ppm) fluorine nuclei in [B(C6F5)4]-, and are referenced to the 19F NMR chemical shifts of C6F5. A similar spectrum is observed for all borate and borane compounds shown in Table 1, and no other 19F resonances were observed. The PFG data in Figure 1 (right) are the maximum signal amplitude for the meta 19F lines of AB-1 in hexane, cyclohexane and toluene solvents (see figure legend) plotted as a function of the square of the applied magnetic field gradient (g2). The PFG intensities in Figure 1 (right) were normalized by the signal intensity of the 19F spectra recorded using the stimulated echo pulse sequence with smallest applied magnetic field gradient.31 The 19F signal intensity is observed over approximately 2 orders of magnitude. The lines through the 19F PFG data points are the least-squares fits to
Aggregation of Borate Salts in Hydrocarbon Solvents
J. Phys. Chem. C, Vol. 112, No. 21, 2008 7821 TABLE 4: Experimental and Calculated Radii of 5 mM T8 in Toluene radius/Å standard
solvent
experimenta
free volume
free surface
T8
hexane
5.5
4.9
6.3
a
Experimental error is estimated to be (4%.
TABLE 5: Radii from PFG NMR Data for B(C6F5)3 and Borate Salts 19
sample solute AB-1
solvent toluene cyclohexane ISOPAR E hexane
AB-2
toluene hexane AB-3 toluene trityl-D4 toluene B(C6F5)3 toluene cyclohexane
Figure 2. 1H NMR and PFG of AB-1 in hexane and cyclohexane, with 1H spectra taken at two different gradient values, illustrating the solvent line suppression (top). PFG for indicated line position (bottom).
F NMR
1
H/11B NMR
C radiusa diffusiona radiusa diffusiona (mM) (Å) (10-9 m2/s) (Å) (10-9 m2/s) 10 5 5 5 0.1 0.5 5 10 9.3 5 5 5 5
13.4 12.4 20.9 24.2 10.3 12.1 20.5 9.2 20.6 6.9 8.4 4.1 3.5
0.295 0.319 0.116 0.174 0.715 0.609 0.359 0.430 0.357 0.573 0.471 0.965 0.690
20.4 21b
0.121 0.20b
20.1
0.366
a Experimental error ) (5% unless noted otherwise. mental error ) (15%, 11B nucleus.
b
Experi-
The size of a diffusing sphere is related to the isotropic diffusion constant D by32,33 the Stejskal-Tanner equation (eq 1). A floating baseline fitting parameter was used in the least-squares fit to this and all other PFG data reported in this manuscript. The 1H NMR spectrum of AB-1 in hexane at two gradient strengths is shown in Figure 2 (top). In the absence of an applied magnetic field gradient, the solvent proton lines completely mask the methylene and methyl proton resonance lines for the cation of AB-1. PFG spectra were collected with an initial gradient value of g ) 0.65 T/m (Figure 2, top, dashed line). At this value of g, the methyl and methylene resonances are both still composed of a mixture of the solvent and AB-1 protons. At g ) 1.21 T/m, only the methyl and methylene protons from the ammonium cation of AB-1 are observed (Figure 2, top, solid line). The integrated area of the methyl resonance at 0.9 ppm is approximately 10% of the area of the methylene resonances between 1.25 and 1.3 ppm, as expected from the chemical structure of the cation. A single point at 1.272 ppm for the methylene resonance was fitted to determine the diffusion coefficient. This position in the spectrum is identified by the arrow in Figure 2, top. The 1H PFG NMR amplitude as a function of gradient strength for AB-1 in hexane and cyclohexane is shown in Figure 2 (bottom). The PFG signal attenuation is observed over 3 orders of magnitude. The least-squares fit of eq 1 to the PFG data is shown. The scatter in the data at the high field gradient is partially an artifact of the logarithmic scale used to display the data, which magnifies the noise where signal has decayed nearly to zero. The 1H resonance from the vinyl protons of the model T8 compound did not overlap with the 1H resonance of the hydrocarbon solvents and was used to collect diffusion data on T8. Hydrodynamic Radii. The diffusion coefficient, D, is related to the dimensions of the diffusing molecular species bytheStokes-Einstein(SE)relationofclassicalhydrodynamics.32,33
D)
kT 6πηr(1 - ξ)
(2)
where k is the Boltzmann constant, T is the temperature, η is the viscosity of solvent, and r is the radius of the sphere. The parameter ξ (0 e ξ e 1/3) defines the hydrodynamic boundary conditions. The limit ξ ) 0 gives the conventional SE relation, and corresponds to “stick” or “no slip” boundary conditions. The limit ξ ) 1/3 corresponds to pure “slip” boundary conditions. An intermediate value of ξ corresponds to a mixed slip-stick boundary condition. The experimental radii reported in Table 4, Table 5, and Table 6 were obtained from the PFG-determined diffusion coefficient by using eq 2 to calculate an effective spherical hydrodynamic radius of the diffusing species. The experimental solvent viscosities that were used are given in Table 3. All hydrodynamic radii reported use the ξ ) 0 “no slip” boundary condition. The applicability of the “no slip” boundary condition will be discussed in detail in section IV. The experimental hydrodynamic radius of the model T8 compound is shown in the third column of Table 4. The fourth and fifth columns of Table 4 are discussed below in the Molecular Mechanics Simulations section. The radii determined from analysis of the PFG data for the borate salts, neutral borane molecule, and trityl-D4 are shown in Table 5. Measurements of the radii for the neutral borane molecule, B(C6F5)3, in toluene and cyclohexane agree to within 15% (see the last two rows of Table 5). The [B(C6F5)4]- borate anion contains no protons while the [N(C18H37)4]+ ammonium cation contains no fluorine atoms. In the case of the AB-1 borate salt, PFG data was recorded for both 19F and 1H nuclei (see the fifth column of Table 5). The hydrodynamic radius values determined from the 19F and 1H PFG measurements for 5 mM AB-1 agree, to within the experimental fitting error. The same radius is
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TABLE 6: Radii Calculated from Free Volume and Free Surface Area of Structures from Molecular Mechanics free-volume calculation (radii (Å)) experiment in AB-1 AB-2b AB-3 trityl-D4 B(C6F5)3 a
toluenea
(Å)
12.4 9.2 6.9 8.4 4.1
free-surface calculation (radii (Å))
ion pair
cation
anion
ion pair
cation
anion
7.3 6.3 5.6 5.3 4.2
6.7 5.4 4.4 3.8
4.6 4.6 4.6 4.6
11.8 9.5 7.3 7.3 5.2
10.3 7.7 4.6 4.6
5.8 5.8 5.8 5.8
Experimental error ) (5%. b 10 mM.
obtained in both hexane and cyclohexane (20.5 ( 0.4 Å). The agreement between the AB-1 19F and 1H PFG measurements indicates that the borate anion and ammonium cation are constituents of solute species that are of the same molecular size. The hydrodynamic radii of 5 mM AB-1 aggregates in hexane and cyclohexane are greater than the size expected for the cation or anion alone, as will be shown below. The hydrodynamic radii data, in Table 5, exhibit dependence on both the nature of the hydrocarbon solvent and on the concentration of the borate salt in the solvent. Smaller radii are observed for the borate salts dissolved in toluene, even at the higher concentrations. Larger radii are observed for the borate salts dissolved in the saturated hydrocarbon solvents at higher concentrations. However, the radius observed at lower concentrations in hydrocarbons approaches the value observed in the aromatic solvent. See, for example, the AB-1 radii in hexane for the concentrations of 5, 0.5, and 0.1 mM (Table 5, rows 5, 6, and 7). Molecular Mechanics Simulations. Molecular mechanics (MM) simulations of salt aggregation were performed using the COMPASS34 force field in the Cerius2 program from Accelrys.35 Electrostatic and van der Waals forces are included in the interaction potential. The free volume and free surface of the ionic aggregates were determined from molecular mechanics simulations as follows. First, an energy minimized geometry was obtained for each ion, individually. The energy-optimized ions were then structurally fixed and ion pairs were formed by minimizing the electrostatic interactions between the two counterions, starting from several initial interionic geometries. The ion pairs were then further energy optimized by allowing both the inter- and intraionic structures to relax. Aggregates of ion pairs were formed by docking smaller ion pairs, as fixed structural units. For example, a quadrupole (tetramer) aggregate was formed by docking two ion pairs. Once docked, the structural constraints were removed and all intraionic geometries were optimized. We used two different measures to determine the effective spherical radius from the molecular mechanics minimized structures: the free-volume and free-surface methods. One approach is to calculate the molecular free volume by summing all spherical volumes occupied by each nucleus of the molecule of interest using the van der Waals radii. Intersecting volumes from distinct nuclei are counted only once. A second approach uses the free-surface calculation, which sums the exposed van der Waals surface areas of the nuclei. The radius calculated from both free volume or free surface is that of an equivalent sphere, with the same volume or area, respectively. The radius calculated from the free volume gives a lower limit to the radius, because internal voids are excluded. However, the free-surface calculation is prone to overestimation of the surface area because of the contribution of internal voids within the aggregate to the surface area. Free-volume and free-surface estimates of the spherical radius calculated from the energy minimized molecular mechanics structures are listed in Table 4, Table 6, and Table
TABLE 7: Radii for Ammonium Borate Ion-Pair Aggregates Calculated Using Molecular Mechanics Simulations molecules
ion pairs/cluster
free-volume radii (Å)
free-surface radii (Å)
AB-1
1 4 5 6 1 4 6
7.3 11.6 12.5 13.3 6.3 10.1 11.6
11.8 22.1 24.6 27.1 9.5 18.0 21.6
AB-2
7, for the T8 molecule, the individual ions, and aggregates, respectively. The free-surface and free-volume estimates of the radius of T8 are given in Table 4. The T8 molecule possesses high symmetry and has an almost spherical shape. The experimental value lies between the radius calculated from the free volume and the free surface and is within 10% of either value. Table 6 gives the free-volume and free-surface estimates for the energyminimized single ion pairs (the estimates for the individual cation and anion and neutral borane are also shown). The experimentally determined radii for the 5 mM borate salt data in toluene are reproduced in Table 6 for reference. The calculated radii for the neutral borane are in good agreement with the experimental results, suggesting that the neutral borane molecule does not aggregate in these hydrocarbon solvents. A comparison of the experimental radii for the borane and borate salts in dilute solution or in toluene where no aggregation is expected shows that the radius from the free-surface calculation agrees within ∼5% of the experimental radius derived from the PFG data and analysis (see Table 6 columns 1 and 5). The better agreement with the free-surface model compared to the free-volume model is perhaps not surprising since the surface of the ion pair is not smooth and does not form a compact sphere. The exception is the borane molecule where the radius from the free volume model is in much better agreement with the PFG data analysis (see Table 6, last row). The ligands of the neutral borane molecule are more compact and have less conformational freedom compared with the cation in the ion pairs. The free-volume and free-surface estimates for the energyminimized clusters of up to six ion pairs of AB-1 and AB-2 are given in Table 7. Figure 3 shows the free surface (squares) and free volumes (circles) of aggregates of AB-1 (open symbols) and AB-2 (closed symbols) as a function of the number of ion pairs in the aggregate. The free-surface and -volume values in Figure 3 are normalized to the free surface and free volume of a single ion pair. Examples of MM-simulated models for an AB-1 ion pair and ion-pair aggregate are shown in Figure 4 (monomer on the right, and [AB-1]6 on the left). Figure 3 establishes that the free volume of the aggregates is approximately an additive function of the free volume of a single
Aggregation of Borate Salts in Hydrocarbon Solvents
J. Phys. Chem. C, Vol. 112, No. 21, 2008 7823 IV. Discussion
Figure 3. Free volume and free surface as a function of ion pairs. Circles: free volume. Squares: free surface. Open: AB-1. Closed: AB2.
Figure 4. Model of the AB-1 aggregate calculated using molecular mechanics simulation: [AB-1]6 (right), [AB-1]1 (left).
TABLE 8: Aggregate Size of AB-1 and AB-2 Determined from Additivity of Free Volumea sample solute
solvent
C (mM)
aggregate size
AB-1
cyclohexane ISOPAR E hexane hexane
5 5 5 9.3
5(1 7(2 8(2 11 ( 2
AB-2 a
See Figure 3.
ion pair, with some small relaxation of ligand packing, and small internal voids in the larger aggregates. The free-surface values are not additive in the number of ion pairs. As the number of ion pairs increases, more of the surface area of the participating ion pairs is excluded due to overlap with neighbors. Using the additivity of the free volume, demonstrated by the MM results in Figure 3, the approximate number of ion pairs in the AB-1 and AB-2 aggregates can be estimated. For example, when the experimental hydrodynamic volume of AB-1 in hexane at 5 mM is divided by the experimental hydrodynamic volume of the single ion pair in hexane at 0.1 mM, an aggregate size of 8 ( 2 ion pairs is obtained. The aggregate sizes of 5 mM AB-1 and AB-2, determined by additivity of free volumes, are shown in Table 8 for the various solvents used. Because the experimental single ion pair volume was not available in all of the solvents, the experimental toluene radius was used to normalize the hydrodynamic volumes for all cases but AB-1 in hexane.
Ion pairing in transition metal organometallic chemistry has been recently reviewed by Macchioni.36 PFG NMR studies of metallocenes and related model compounds have been reported by several groups. 18–26 The contributions of the present work to the knowledge base of these compounds include studies of diffusion in alkane solvents, the measurement of the diffusion coefficient for both the anion and cation using the NMR signals from 1H and 19F nuclei, and the use of molecular mechanics simulations to determine the aggregation number from the diffusion coefficient. Studies using alkane solvents are important because nonaromatic solvents are widely used in industrial application of homogeneous metallocene polymerizations. Measurement of the diffusion coefficient for both the cation and anion using different nuclei provides an independent corroboration of the measured diffusion coefficient values. Molecular mechanics simulations provide a structural foundation for the interpretation of the hydrodynamic radius of multiple ion-pair aggregates, where molecular packing has a potentially large impact on interpretation of the number of participating ion pairs. PFG studies of the association of simple borate salts have been previously reported. For example, Mo and Pochapsky measured the concentration dependent diffusion coefficients for [N(C4H9)4]+ and [BH4]- dissolved in CDCl3 over the concentration range of 4 × 10-4 M to 2 × 10-1 M using 1H PFG NMR.37 They estimated aggregation numbers in the range of 2 to 5 by using the diffusion coefficient for a neutral tetrabutylsilane molecule dissolved in the same solution as the ammonium and borate ions as a reference for molecular size. Beck and co-workers recently reported 1H PFG measurements on zirconocene borate ion pairs in benzene in the concentration range of 1 to 20 mM.18 They calculated an effective hydrodynamic radius from the measured diffusion coefficients and deduced the formation of ion quadrupoles for the borate zirconocene complex by using diffusion data for the analogous borane zirconocene complex as a reference for molecular size. Marks and co-workers recently reported studies of zirconocene borate ion pairs in benzene, but found no evidence for aggregation of multiple ion pairs.21 However, in the latter study the maximum concentration used was 0.1 mM. The conclusion of Marks and co-workers is consistent with the borate salt aggregation data in the present study where we find only single ion pairs of borate salts at 0.1 mM concentration, even in nonaromatic solvents (see Tables 5 and 6). Another study of Marks and co-workers on a different borane zirconocene complex in benzene exhibited quadrupole aggregate at a concentration of 1.5 mM.25 Song and co-workers have recently reported aggregates of 3 ion pairs for a borate zirconocene complex at 10 mM in toluene.24 This is consistent with the data in Tables 5 and 6 for AB-1 at this concentration. At 10 mM the PFG-determined radius of AB-1 in toluene exceeds the free-surface upper bound for a single ion pair but is less than the free-surface estimate for a quadrupole aggregate (see Tables 5 and 7). Our PFG results for the borate salts in hexane solvent indicate aggregates exceeding six ion pairs (AB-1 at 5 mM in hexane), which is a larger aggregation number than is reported in the PFG metallocene literature for aggregation in aromatic solvents. Besides PFG NMR there are three traditional methods for assessing ionic aggregation: conductance, freezing point depression, and vapor pressure osmometry. Conductance studies have been carried out on the decyl analogue of AB-1 in cyclohexane.38,39 These studies identify the triple ion aggregate as the major charge carrier in the mM nominal salt concentration regime, and indicate the possibility of quadrupole aggregates.38,39
7824 J. Phys. Chem. C, Vol. 112, No. 21, 2008 Assessing the concentration of higher aggregates becomes difficult with conductance methods, because the equilibrium dissociation constants for each aggregate species must be modeled. Furthermore, the interpretation of conductance minima in terms of a constant dielectric medium has been criticized, due to the impact of the ion-pair dipole moment on the effective dielectric constant of the solution.40 Freezing point depression (cryoscopic) techniques determine the average molecular weight of salt aggregate, and are useful for estimating larger aggregates. Cryoscopic measurements were compared with PFG data in the work of Marks and co-workers, although only for systems exhibiting single ion pairs.21 The extensive literature of cryoscopic measurements shows that the tendency to form ionic aggregates is greatest if the dipole moment is large and one of the ions is small.41 For example, tetraisoamylammonium thiocyanate (Am4NSCN), which has a dipole moment of 15.4 × 10-18 esu, has a cryoscopically determined aggregation number, n, of 5.47 at 3.8 × 10-3 M, which is beyond the quadrupole stage.41,42 For comparison, tetraisoamylammonium picrate (Am4NHPi), has a large dipole moment of 15.4 × 10-18 esu, but at a concentration of 4 × 10-3 M only 40% of the ion pairs are associated into quadruples (aggregation number of n ) 1.40).41,42 Such dramatic effects of ion size on association are not observed for the borate salts reported here, but the larger aggregation number for AB-2 than AB-1 in hexane is consistent with this trend (see Table 8). Vapor pressure osmometry has been used to study quaternary ammonium salts in organic solvents.43 This technique, like cryoscopy, provides only the mean aggregation number. Alkyltrimethyl- and alkyltriethyl ammonium bromide salts with C10 to C15 alkyl hydrocarbon chains were studied in chloroform solution (dielectric constant 4.8) by vapor pressure osmometry. The data demonstrates that ion-pair multiplets form in the 100 mM regime.43 The salts that contain lower molecular weight ligands have a larger aggregation number.43 The latter observation is consistent with the aggregation numbers for AB-1 and AB-2 reported in Table 8 of this work. The versatility of the PFG NMR method for determining aggregate size through diffusion measurements makes the method attractive for general use, but there are several sources of uncertainty that are important to address. The accuracy of the PFG measurement of diffusivity and the applicability of the Stejskal-Tanner equation (eq 1) has been reviewed.44 To ensure the simple Gaussian dependence of eq 1 it is important that the diffusion time interval between gradient pulses, ∆, is longer than the fundamental microscopic time scale of the statistical process underlying the macroscopic diffusion. Because ∆ is on the order of milliseconds, the PFG measurement interval is well outside of the time scale associated with the molecular position autocorrelation function, at room temperature. Two other factors which affect the applicability of eq 1 are the dispersity of the distribution of aggregates and the anisotropy of diffusion. In the case of nonspherical molecules it is necessary to introduce three diffusion coefficients which correspond to the principal axis of a diffusion tensor. Experimental evidence for such anisotropic diffusion effects would be expected to manifest as multicomponent or nonlinear decay of ln(I/I0) as a function of g2. Likewise, polydispersity of the aggregate size distribution would be evident as nonlinear spin-echo intensity vs field gradient plot. No evidence for such multicomponent nonlinear decays is observed in the present data, providing support to use of an isotropic diffusion coefficient. Translation of the diffusivity into an aggregate size also involves assumptions that are attendant with the use of the SE
Endeward et al. relation and with the method of molecular modeling. We now turn to a discussion of the uncertainties associated with using the SE eq 2 to determine aggregate size and then address the molecular modeling. The influence of the following will be discussed: solvent-solute hydrodynamic interactions (“slip” vs “no slip” boundary conditions), electrostatics, and solute-solute hydrodynamic interactions (concentration effects). “Slip” vs “No Slip” Hydrodynamic Boundary Conditions. Several previous studies have found that the SE relation (ξ ) 0 limit of eq 2) is valid in the regime where the diffusing particle approaches a molecular size, despite the fact that the SE relation is derived for a continuum model and further assumes that (a) the diffusing particle is spherical, (b) the diffusing particle is much larger than the solvent molecules, and (c) that there is no “slip” between the moving sphere and the fluid in contact with it.45–47 The ξ ) 0 limit of eq 2 physically corresponds to “no slip” between the moving particles and the surrounding medium; that is, the diffusing particle is always in contact with a large number of solvent molecules. The “no slip” condition does not necessarily imply that individual solvent molecules are physically bound to the solute. The “no slip” condition is appropriate for a large spherical macromolecule in a solvent of low molecular mass. The “slip” condition (ξ ) 1/3) is approached when the solute and solvent are of comparable size. The hydrodynamic radius that is obtained in the “slip” limit is larger than the “no slip” limit by a factor of 3/2. Consequently, the “no slip” condition sets a lower bound on the hydrodynamic radius, and any misapplication (within the context of eq 2) can only lead to an underestimate of the radius. The validity of the “no slip” condition is questionable for the smaller nonaggregated molecules such as the T8 model compound and the neutral borane because these are closest in size to the solvent. However, as shown in Table 4, the radius of the T8 model compound determined from the PFG data assuming the “no slip” condition is equal to the mean of the free-volume and free-surface estimates of the radius. In contrast, applying the “slip” boundary condition produces a radius that significantly exceeds the free-surface upper bound. The T8 data, therefore, supports the use of the SE relation with the “no slip” condition. In addition to the relative size differences between the solute and solvent, another molecular property that impacts the interpretation of the SE radius is whether the solvent physically associates with the solute. The physical association of solvent with the solute molecules can be interpreted either as an increase in local solvent viscosity (modified η) or as an increase in the hydrodynamic radius (because associated solvent diffuses along with the solute species). The slip factor cannot account for this apparent increase in viscosity and eq 2 breaks down. There are two internal methods for assessing the validity of the SE radius. First, if the solvent is associated with solute on the time scale of the PFG NMR experiment, then PFG NMR of the solvent peaks will reflect a population of solvent species that possess a smaller diffusion coefficient than the bulk solvent (a multiexponential echo attenuation will be observed). For example, diffusion of methylaluminoxane (MAO) activated metallocenes has been studied by PFG NMR.22 Comparison of MAO 1H PFG NMR with toluene (solvent) 1H PFG NMR has shown that a component of the solvent diffuses with the MAO/metallocene complex (H.T., unpublished data). A second means of testing the hydrodynamic boundary conditions is to assess any variation of the hydrodynamic radius in different solvents, as this would
Aggregation of Borate Salts in Hydrocarbon Solvents
J. Phys. Chem. C, Vol. 112, No. 21, 2008 7825
be evidence for variation of the hydrodynamic boundary conditions due to solvent association. Toluene self-diffusion 1H PFG measurements were made in the presence of B(C6F5)3. If the solvent is tightly associated with B(C6F5)3, a second toluene diffusion component with a D value similar to the D for B(C6F5)3 solute (known from the 19F PFG data) would be observed. However, no evidence for a smaller D value for the 1H of toluene was observed. The selfdiffusion constant for toluene was determined to be (2.3 ( 0.1) 10-9 m2/s, which is in agreement with the value of D in the absence of B(C6F5)3. A similar study of cyclohexane diffusion produced no evidence of a second component of solvent diffusion. For the case of borate salts in toluene, the diffusivity of the solvent can be directly determined from the PFG NMR measurements of the aromatic 1H resonance. No evidence for a second solvent diffusion component was observed for the borate salts in toluene. In the aliphatic solvents, the methyl and methylene resonances overlap with those from the tetraalkyl amines, making the analysis of a potential solvent associate difficult. Further evidence supporting the SE radius is obtained from the PFG NMR data for the diffusion of B(C6F5)3 measured in different solvents, as shown in Table 5. In the case of B(C6F5)3, no difference (to within experimental error) in the molecule size can be detected when comparing the solute dissolved in the toluene and the cyclohexane. The AB-1 data possess a similar trend, but the trend is indirect because of the onset of ion-pair aggregation. For AB-1, no difference in the size of the ion-pair aggregate is observed when comparing AB-1 dissolved in the hexane and cyclohexane at 5 mM. Hexane and cyclohexane have similar dielectric constants, but they differ in their solubility parameters and in molecular shape. Therefore, differences in the local viscosity might be expected if solvent were affected through association with the solute. Concentration Effects: Electrostatics and Hydrodynamics of Solute-Solute Interactions. The presence of solute-solute interactions via multibody hydrodynamic effects or direct Coulomb electrostatic interactions can cause a deviation of the radius calculated from the diffusion constant using the SE relation (eq 2). In the context of the borate salts, where the mass of the solute exceeds the solvent molecules, the deviation of the diffusion from SE behavior is expected to be manifest as a slower diffusion compared to the limiting value at infinite dilution. Consequently, the hydrodynamic radius could potentially be overestimated, if multibody effects are important. It has been argued that charge and solvation effects may be responsible for differences in self-diffusion between neutral and ionic species of comparable molecular size and shape.48 PFG NMR has been used to measure the effect concentration and ionic strength on the diffusivity of charged polyelectrolytes.49 In the case of finite concentrations or in the presence of strong attractive or repulsive Coulombic fields the motion of the particles is no longer independent, and Brownian dynamics simulations must be modified by the addition of both hydrodynamic and Coulombic interactions.32,48 The measured diffusion coefficient, D, differs from the diffusion coefficient, D0, at infinite dilution. To first order, a linear expansion in the density of the solute leads to deviations given by the equation
D ) D0(1 + Rφ)
(3)
where R is a dimensionless quantity that depends on the total solute-solute interaction, and φ is the volume fraction of the solute.
PFG measurements do not distinguish between charged or neutral aggregates. The effect of Coulombic interactions on the measured PFG signal depends on the distribution of ionic species and can depend directly and indirectly on the ionic strength of the solution. Charged aggregates, for example a triple ion cluster, are influenced by direct electrostatic interactions. Neutral aggregate can still be influenced by the total ionic strength of the solution because the neutral aggregate is coupled to charged species through multibody hydrodynamic interactions (both solvent and solute mediated). Because the later link is indirect, an estimate of the effect of direct Coulombic interactions on diffusion between charged solute species sets an upper bound for the effect on diffusion of neutral aggregate. There are three basic length scales that determine the importance of the electrostatic interaction: the average separation of dissociated electrolyte, the distance at which the electrostatic interaction energy is comparable to the mean thermal energy (the Bjerrum length), and the screening length of the Coulomb interaction (the Debye length). It is instructive to assume initially that the borate salts are completely dissociated. With complete dissociation, the average ion-ion distance at 5 mM would be 30 Å, if the ions are randomly distributed in solution (rj ) 0.55/n1/3, where n is the number density50). The Bjerrum length is defined by
lB )
z1z2e2 εkT
(4)
where e is the unit of elementary charge, and z1, z2 are the charges of the interacting ionic species in units of fundamental charge. lB ) 569 Å, at 20 °C, and dielectric constant ε ) 1, for ions bearing unit charge. Hexane and cyclohexane have ε ≈ 2, which gives lB ≈ 300 Å. This is much larger than the mean distance between ions, assuming no aggregation. The concentration of free ions would have to be 1000 times less than 5 mM to have a mean ion separation equal to the Bjerrum length. The Debye length is l/lD ) 2.93(c/)1/2 at 20 °C, for a concentration c in mM units. For 5 mM salt, the Debye length is lD ) 7 Å. The condition jr > lD means that the electrostatic effects are well screened, but lB. jr means that the bare electrostatic interaction is extremely strong. Ohtsuki and Okano have calculated R, in eq 3, for a model system containing attractive and repulsive hard spheres.51 For 5 mM salt concentrations the results of Ohtsuki and Okano (eqs 3.9, 4.6, and Figure 2 of ref 51) predict |R| .10 in eq 3, assuming a hard sphere radius of 5 Å for the ions. The electrostatics effects are outside the predictive regime of the linearized theory for 5 mM concentrations (φ ∼ 10-2). The ions are, in fact, so strongly associating that there is condensation and aggregation. Bjerrum’s theory of ion association gives a simple criterion for ion pairing based upon the magnitude of lB.41,52,53 The Bjerrum expression for the dissociation constant, K, of an ion pair is41
1 ⁄ K ) 4πNlB3Q(lB ⁄ a)
(5)
The equilibrium constant, K, has the length units of lB. The parameter a is the distance of closest approach for the ions, and N is Avogadro’s number. Q(x) is the integral Q(x) ) ∫x2ex/x4 · dx. When Q(x) approaches zero, then K becomes infinite, and the ions are completely dissociated. The critical Bjerrum length l/B, for which ions completely dissociate, is defined by the condition l/B ) 2a because Q(2) ) 0. The critical Bjerrum length varies for solvents of different dielectrics (see eq 4), giving rise to a definition of a critical dielectric constant. Complete dissociation is predicted for the critical value of the
7826 J. Phys. Chem. C, Vol. 112, No. 21, 2008
Figure 5. Family of ion-pair dissociation curves from Bjerrum’s equation (eq 5).
dielectric constant given by, for example, ε* ) 285/a at 20 °C (for a in Å). The critical dielectric constant for ions with 10 Å closest approaches is 30, an order of magnitude larger than the solvents used here. Figure 5 shows the Bjerrum ion-pair dissociation constant plotted as a function of the dielectric constant of the solvent for a family of a. The dielectric constants of hexane and toluene are indicated, for reference. Using the value of lB for hexane and cyclohexane (see above), the Bjerrum theory predicts that the concentration of unpaired ions, assuming a closest approach of 10 Å, is 1 µM for a nominal concentration of 5 mM salt. According to the simple Bjerrum ion-pair theory, the true concentration of ionic species is greatly reduced from the nominal concentration because aggregation occurs. The Bjerrum expression (eq 5) only applies to ion-pair formation. The theory has been extended by Fuoss and Kraus to include triple ion association.53 However, experimental conductivity measurements give a more direct estimate of the total concentration of ionic species in solution, and provide another means to estimate of the importance of electrostatic interactions.39,54 The factor by which conductance is reduced from the free molar conductivity is of the same order of magnitude as the degree of ionic dissociation, assuming activity coefficients of unity.54 The conductivity of the decyl analogue of AB-1 has been studied over a range of concentrations which encompass the present study.38 At a 5 mM concentration, the conductance of the decyl solute is reduced by approximately a factor of 10-4 from the limiting conductance of free ions (see Figure 1 in ref 38). The lower conductance is consistent with a concentration of ionic species in the µM range for a nominal 5 mM borate concentration. This conductance estimate of total ion concentration agrees with the estimate from Bjerrum theory. The strong Coulombic interaction in low dielectric solvents requires a theory of strongly interacting Brownian particles.49,55,56 Dilution of ionic species, due to aggregation, reduces the mean distance between ions, but at the same time weakens the ionic strength, and increases the Debye length. At 1 µM concentration lD ) 500 Å for ε ≈ 2. The Debye length exceeds the Bjerrum length, so that electrostatic interactions are potentially important. Ohtsuki has treated the strong regime of Coulombic interaction and has calculated the reduction in diffusion due to electrostatics alone (neglecting hydrodynamic interaction) for a system of
Endeward et al. charged hard spheres that possess twice the Coulombic interaction energy of the borate salts in hexane and cyclohexane, and a mean separation that is four times the value of jr at 1 µM.55 For a Debye length of 500 Å the calculation of Ohtsuki gave a 10% reduction of diffusivity (see Figure 3 in ref 55). In charged hard sphere models the inclusion of hydrodynamic interactions can increase the diffusivity.49,55 Therefore, with hydrodynamic interactions added the reduction in the diffusivity due to Coulombic interactions is not expected to be as large as 10%. Furthermore, the effect of electrostatics on the diffusivity of charged species represents an upper bound on the on the PFG measured diffusivity, which includes primarily neutral aggregates. Both the multibody hydrodynamic interactions, which couple neutral and charged aggregate and electrostatic interactions increase with concentration. Diffusion measurements for concentrations in excess of the 10 mM used here may exhibit an electrostatic induced reduction of the diffusion coefficient, which could erroneously be interpreted as aggregation. Summary. The major sources of error in the interpretation of the Stokes-Einstein radius are anisotropy of diffusion, polydispersity of aggregate, slip in the hydrodynamic boundary conditions, solvent-solute association, and multibody electrostatic and hydrodynamic interactions. The linearity of the ln(I/ I0) vs g2 plots for the data reported here indicate a negligible influence of anisotropy, and aggregate polydispersity. The agreement of the molecular radius of the model compound T8 with the predictions of the SE relation using the “no slip” condition supports the choice of hydrodynamic boundary conditions that were used to obtain the SE radius in Table 5. Furthermore, the “no slip” condition provides a lower bound for the hydrodynamic radius calculated from the SE eq 2. The linearity of the solvent diffusion data supports the conclusion that solute-solvent interactions are transient on the NMR time scale. Theoretical estimates of electrostatic multibody effects, in conjunction with conductance data, indicate that these effects are not a major source of error in the calculation of the SE radius for the concentrations used herein. Modeling Aggregation. The hydrodynamic volume obtained from the PFG measurements either can be interpreted directly as an absolute measurement of aggregate size or can be interpreted relative to the hydrodynamic volume of another diffusing molecule. A direct conversion of the experimental hydrodynamic volume into an aggregation number requires complete simulation of the possible aggregation states, with detailed modeling of the solvent. In addition to the electrostatic and van der Waals forces of the ions, solvent dynamics or polarization should be included in order to fully account for the balance of enthalpy and entropy that controls the hydrodynamic cluster sizes. We have used the experimental single ion-pair volume as a reference for the size of the ion-pair aggregates which are observed at high concentration in aliphatics. To do so requires molecular modeling at an intermediate level, including electrostatic and van der Waals forces. This intermediate level of modeling precludes an accurate estimate of the relative stability of aggregates with different numbers of ion pairs. Consequently the relative population of the ion-pair multiplets in solution is not determined by the MM techniques used here. The monoexponential dependence of PFG NMR amplitude on g2 is consistent with the findings of previous PFG NMR studies on smaller quaternary ammonium salts in low dielectric solvents, and implies a narrow distribution of aggregation numbers that fall within the resolution of the exponential fitting (see Table 8).37,57 The apparent absence of a significant population of single
Aggregation of Borate Salts in Hydrocarbon Solvents ion pairs of AB-1 or AB-2 in the aliphatic hydrocarbons above 1 mM cannot be addressed by the level of molecular modeling used here. A possible physical basis for the observation of the narrow aggregate distributions is a pseudo phase transition that mimics critical micellization of aqueous ionic surfactants. For example, solubility and NMR chemical shift data for tetrabutyl ammonium halide salts in benzene solvent support the formation of a reverse micelle-type aggregate.58 The application of MM in the present work addresses how the hydrodynamic size of the aggregate depends on the number of ion pairs in the aggregate. To use the single ion-pair volume as a reference for the multiple ion-pair aggregates requires an understanding of the packing of ion pairs in the multiplets. The packing of the aggregate core is driven by electrostatic and van der Waals forces. The presence of internal voids in the ion-pair aggregate determines whether the total volume of an aggregate is simply related to the volume of a single ion pair. Refinement of the molecular modeling by including solvent forces will impact finer details, such as ligand structure at the solvent exposed surface of the aggregate, and would allow for the relative stability of different aggregate sizes to be assessed. There are two ways in which the aggregate can deviate from additivity of ion-pair molecular volumes. If the aggregate possesses significant internal voids or if the ion ligands in the aggregate have relaxed geometry relative to a single ion pair, then the aggregate will occupy a larger volume than the sum of the single ion-pair volumes. Assuming addition of ion-pair volumes as a model of ion pairing then leads to an overestimate of number of participating ion pairs in the aggregate. Conversely, if aggregate ligand geometry exhibits tighter packing in the aggregate, relative to the single ion pair, then the number of participating ion pairs will be underestimated by assuming additivity. The additivity of the ion-pair volumes was tested with molecular modeling at the level of electrostatic and van der Waals forces. The molecular modeling results (see Figure 3) show only small deviations from additivity of ion-pair free volumes. For example, the free volume of the hexamer [AB2]6 aggregate differs from the sum of six ion-pair volumes by less than one-quarter the free volume of a single ion pair. The experimental hydrodynamic volumes of AB-1 and AB-2 in aliphatic solvents possess errors of the order of two ion-pair molecular volumes. Consequently, the molecular modeling shows that the use of an additive scale for aggregate free volume introduces negligible error relative to the accuracy of the experimentally determined PFG hydrodynamic volumes. V. Conclusion The tendency of alkyl ammonium borate salts to form ion pairs and macromolecular aggregates in hydrocarbon solvents has been observed by PFG NMR. Single ion pairs of alkylamonium borate salts are present in toluene over the concentration range 0.1 to 10 mM. In aliphatic solvents, single ion pairs predominate at low concentration (
