Ion clustering in aqueous solutions probed with vibrational energy transfer

Abstract

Despite prolonged scientific efforts to unravel the hydration structures of ions in water, many open questions remain, in particular concerning the existences and structures of ion clusters in 1∶1 strong electrolyte aqueous solutions. A combined ultrafast 2D IR and pump/probe study through vibrational energy transfers directly observes ion clustering in aqueous solutions of LiSCN, NaSCN, KSCN and CsSCN. In a near saturated KSCN aqueous solution (water/KSCN molar ratio = 2.4/1), 95% of the anions form ion clusters. Diluting the solution results in fewer, smaller, and tighter clusters. Cations have significant effects on cluster formation. A small cation results in smaller and fewer clusters. The vibrational energy transfer method holds promise for studying a wide variety of other fast short-range molecular interactions.

The solution properties of ions in water are relevant to a wide range of systems, including electrochemistry, biological environments, and atmospheric aerosols (1, 2). For more than 100 yr, tremendous scientific efforts have been devoted to unravel the hydration structures of ions in water (1–11). However, many fundamental questions remain open, in particular concerning the existence, concentration, and structure of ion clusters in 1∶1 strong electrolyte aqueous solutions. Whether strong 1∶1 electrolytes (especially salts of Na⁺ and K⁺) form ion pairs or clusters in water has been considered a key issue for understanding many important problems, e.g., the excess ionic activity in 1∶1 electrolytes (12), ion dependent conformational and binding equilibria of nucleic acids (13), the concentration difference between Na⁺ and K⁺ in living cells, protein denaturation by salts (14, 15), and ion concentration dependent properties of ion channels (16).

The properties of aqueous solutions of 1∶1 strong electrolytes deviate from the ideal dilute solution at extremely low concentrations (< 10^-5 M). The deviations were generally believed to be caused by the attraction between ions of opposite charge and the repulsion of ions of the same charge, leading to the development of the Debye-Hückel theory (17, 18). However, this theory begins to fail at a very low concentration (∼10^-3 M), as the assumptions upon which the theory was based become invalid. The formation of ion pairs containing two ions of opposite charge has been proposed to be primarily responsible for this failure (1, 2). Recently, calculations from molecular dynamics (MD) simulations, suggested that, clusters with more than one ion of the same charge which are traditionally viewed as unlikely, could be a major factor contributing to the nonideality of solutions at medium or high concentrations (12, 19). However, these predicted ion clusters cannot be investigated by the usual tools for probing molecular structures and particle sizes in liquids, e.g., X-ray or neutron diffraction (20), or the dynamic light scattering (19, 21), because the contribution of ion-ion correlations to the total scattering pattern is too small compared to the contributions from water-water and water-ion interactions (19). In addition, the clusters are expected to be small, containing only a few ions which exchange rapidly with ions in the water phase (12).

In this work, by monitoring intermolecular mode-specific resonant and nonresonant vibrational energy transfers (22–25) using ultrafast 2D IR and pump/probe techniques, we were able to directly probe ion clustering in a series of 1∶1 strong electrolyte (LiSCN, NaSCN, KSCN and CsSCN) aqueous solutions. We obtained through these experiments clear evidence that a significant portion of the ions form clusters in unsaturated solutions. In highly concentrated solutions, almost all anions (> 90%) form ion clusters, suggesting water/ion microphase separations. In other words, in an apparent “homogeneous” SCN^- aqueous solution, both clustered and water-solvated anions simultaneously exist, as illustrated in Fig.. 1A. Even at a medium concentration (1M) with a salt/water molar ratio only 1∶50, ∼27% of anions are clustered under the ambient condition. As expected, diluting the solution shifts the dissolution equilibrium to fewer and smaller clusters, but, surprisingly, dilution makes the clusters tighter. The ion clustering is cation-size-dependent with smaller cations tending to form smaller and fewer clusters.

Fig. 1.

(A) A snapshot of a 1.8M KSCN aqueous solution from a molecular dynamics simulation (Details are in SI Appendix). O (red), H (white), C (light blue), N (deep blue), K (green), and S (yellow). An ion cluster is visible at the center of the picture. Some water molecules are removed from the original file to better display the cluster structure. (B) FT-IR absorption spectra of the CN and ¹³C¹⁵N stretches of SCN^- and S¹³C¹⁵N^- of a 10M 1∶1 KSCN/KS¹³C¹⁵N aqueous solution (solution C). (C) The time dependence of the 2D IR spectrum of solution C. As T_w increases, the off-diagonal peaks grow in because of energy exchange between SCN^- and S¹³C¹⁵N^-. The vibrational coupling and the heat effect (22, 23) (pump/probe data are in SI Appendix: Fig. S1) are too weak to show up in the spectra of the time range.

Results and Discussion

Anions in a Cluster Can Exchange Vibrational Energy.

We first describe the vibrational energy transfer method for studying ion clustering in a concentrated KSCN solution, and then present the concentration and cation dependent results. Fig. 1B is the FTIR spectrum of the CN and stretches of SCN^- and in a 1∶1 KSCN/KS¹³C¹⁵N mixed aqueous (D₂O) solution with a salt/water molar ratio 1/2.4 (10M, which we name as solution C). The isotope labeling shifts the CN stretch frequency from 2,064 cm^-1 (SCN^-) down to 1,991 cm^-1 (). Theoretical predictions for other concentrated or melt electrolyte solutions (26, 27), suggest that the probability of forming contact clusters of the general formula K_n(SCN)_m(S¹³C¹⁵N)_p is statistically very high in solution C. In these clusters, the anions SCN^- and can be considered as “ligands” to the cation K⁺. The anions thus held in close proximity can exchange vibrational energy through their overlapped orbitals or via dipole-dipole interactions, in a manner similar to that observed for metal carbonyl compounds (28). Vibrational energy exchange between SCN^- and in the clusters can be monitored with fast 2D IR methods (Fig.. 1C, here the solution is C, i.e., 10M). From these 2D IR measurements, not only the vibrational energy exchange rates, but also the cluster concentration and the exchange dynamics between clustered and separated anions can be obtained.

The intermolecular mode-specific vibrational energy transfer 2D IR technique has been previously described in detail (22, 23). Very briefly, energy exchange 2D IR measurements allow the energy exchange between the CN and ¹³C¹⁵N stretches of SCN^- and to be followed in real time through the growth of the two cross peak pairs (Peaks 5 ∼ 8) as marked in the 50-ps row of Fig.. 1C (for the 10M solution named C). The anions which have not exchanged their energy (including anions which have exchanged but received energy from reverse transfers and the resonance acceptors of the same isotope) produce the two diagonal peak pairs (Peaks 1 ∼ 4). Now we consider all six waiting-time (T_w) dependent 2D IR spectra of solution C at room temperature. The 200-fs row corresponds to a very short T_w, at which negligible vibrational energy exchange has occurred. During the T_w period, the initial and final energy carriers in the sample are unchanged. Therefore, the ω_pump (the excitational frequency) and ω_probe (the detected frequency) values of each red peak (the 0–1 CN or ¹³C¹⁵N stretch transition) are identical, and the peaks appear only on the diagonal. The two blue peaks 2 and 4 are the 1-2 transitions, which shift to lower frequencies along the ω_probe axis because of vibrational anharmonicity. Peaks 1 and 2 represent SCN^- and Peaks 3 and 4 are for . With the increase of probe delay time (T_w = 200 fs ∼ 20 ps), vibrational energy begins to flow between the two anions. Cross peak pairs begin to grow. After a long reaction period (T_w = 50 ps), vibrational energy has exchanged to a substantial degree as shown by the additional peaks (Peaks 5 ∼ 8) that have appeared on the off-diagonal. These new peaks arise from the vibrational energy exchange. The vibrational energy transfer from SCN^- to produces peaks 5 and 6 at positions with ω_pump = 2,064 cm^-1 and ω_probe = 1,991 cm^-1 and 1,966 cm^-1. ω_pump = 2,064 cm^-1 is the 0–1 transition frequency of the CN stretch, representing the vibrational energy is originally from SCN^-. ω_probe = 1,991 cm^-1 and 1,966 cm^-1 are the 0–1 and 1–2 transition frequencies of the stretch, respectively, representing that at the end of T_w the vibrational energy has transferred to . Likewise, peaks 7 and 8 are produced by energy transfer from to SCN^-.

In contrast to the chemical exchange 2D IR methods (29–33), the cross peak intensities in the energy exchange method are not equal. The ratio of the cross peaks’ growth rates is determined by the Boltzmann distribution. The energy mismatch between the CN and ¹³C¹⁵N stretches is 2,064-1,991 = 73 cm^-1, making the energy up-pumping rate constant from to SCN^- ∼70% of the down-flowing rate constant from SCN^- to . As in Fig.. 1C, Peaks 5 and 6 are always bigger than Peaks 7 and 8 at the same T_ws. In solution C, some anions are clustered and can transfer energy efficiently. Some anions are well separated from each other and less able to exchange energy with other anions. These two types of anions are not frequency resolvable. Both produce diagonal peak pairs in 2D IR spectra in Fig.. 1C. Because the clustered anions can exchange energy much more efficiently, the cross peak pairs are mostly from the clustered anions. Therefore, simultaneous analysis of diagonal and cross peaks provides not only the rate constants for energy exchange, but also the ratio of clustered to separated anions and the rate constants for the exchange of anions from separated to clustered.

In solution C, based on the liquid density, the nominal average anion distance is calculated to be 5.5 Angstrom (Å). Can the energy exchange observed in Fig. 1C be simply because of the short average distance rather than ion clustering? The speculation can be directly tested by diluting the solution. Previous experiments suggest that the intermolecular vibrational energy transfer can be described by the dipole-dipole interaction (25, 34–36):

[1]

where k is the energy transfer rate constant and r is the donor/acceptor distance. According to [1], diluting solution C with a water/salt ratio from 2.4/1 to 25/1 will increase the average anion distance for ∼1.9 times and therefore slow down the energy transfer rate for 51 times. However, from the concentration dependent 2D IR measurements (Fig.. 2), in such a dilute solution (1.8M), the energy exchange cross peaks at 50 ps are still clearly visible, with a normalized intensity about of those of solution C at the same waiting time (Fig.. 1C). This result indicates that the apparent energy transfer rate in the 1.8M solution is only about four times slower than solution C, one order of magnitude smaller than the predicted 51 times. The contradiction between experiments and the prediction based on the speculation suggests that the observed energy transfer is probably from some anions with average distance much smaller than the nominal 5.5 Å and this distance is not affected by dilution significantly. The only plausible explanation of this dilution experiment is that the anions responsible for energy transfer are in clusters. In a control experiment, no energy exchange was observed between the SCN and S¹³C¹⁵N groups in a 1∶1 C₂H₅SCN and C₂H₅S¹³C¹⁵N mixed liquid (Fig.. 3). In the control sample, the average molecular distance is 5.2 Å, and the energy mismatch (78 cm^-1) between the two isotope-labeled CN stretches is very similar to that (73 cm^-1) between the two anions in C. If anions in the electrolyte solutions are not clustered, based on [1] without considering other factors, the energy transfer rates in the control sample would be 67 times faster than those in the 1.8M solution, which has assuming no clustering an average anion distance of 10.5 Å. However, at T_w = 30 ps, the energy transfer cross peaks in the 2D IR spectrum of the 1.8M solution (Fig.. 2) are already visible, while no cross peaks are observed for the control sample, indicating that energy transfers faster in the 1.8M solution than in the control sample. Ion clustering which holds anions within a very close distance in the aqueous solutions is the most probable reason for this discrepancy. (More details and supporting experiments are in the control experiment part of the SI Appendix.)

Fig. 2.

The concentration dependences of the 2D IR spectrum of 1∶1 KSCN/KS¹³C¹⁵N aqueous solutions at different T_ws.

Fig. 3.

The time dependence of the 2D IR spectrum of a 1∶1 C₂H₅SCN/C₂H₅S¹³C¹⁵N mixed liquid. No cross peaks have grown in up to 30 ps, indicating negligible energy has exchanged between the CN and ¹³C¹⁵N stretches of the two molecules.

The main reason for the smaller cross peaks in the spectra of a more dilute solution (Fig.. 2) is a lower concentration of clustered ions instead of a slower energy transfer rate. Quantitative analysis based on the nonresonant energy transfer data and the following kinetic model shows that only 35% of anions (Data and calculations are in SI Appendix) are clustered in the 1.8M solution while 95% of the anions form clusters in solution C. Surprisingly, diluting the solution in fact decreases rather than increases the average distance between two clustered anions, as observed from our energy transfer experiments presented in following paragraphs.

95% of Anions Form Clusters with Average Anion Distance 3.7 Å in a 10M KSCN Aqueous Solution.

In solution C, some and SCN^- form clustered anions (denoted as and ), and the rest of the anions are separated (denoted as and ). These two types of anions are in dynamic equilibrium: they can exchange locations with rate constants k_clu→iso and k_iso→clu with a ratio equal to an equilibrium constant . The and anions can exchange vibrational energy with rate constants and with a ratio determined by detailed balance: at experimental temperature T = 295 K. We assume that the energy transfer rate for the separated anions is negligibly small, i.e., a separated anion cannot transfer vibrational energy to another anion. The vibrational excitations of and SCN^- decay with their own vibrational relaxation rate constants and k_SCN^- (here we assume the clustered and separated ions have the same k. The assumption with different vibrational relaxation rate constants for clustered and separated ions is also tested. See SI Appendix for more details). From these considerations, we construct a kinetic model for the observations as illustrated in the following scheme 1.

Scheme 1.

In these experiments, we obtained molecular-rotation-free data (22). Therefore, the model doesn’t contain any orientational component. From the model, four differential equations are derived. By solving the equations (Details are in SI Appendix), the energy transfer rate constants, the clustered/separated ion equilibrium constant, and the location exchange rate constants are obtained. Because the ratio of the rate constants for location exchange is the equilibrium constant and the ratio of the energy transfer rate constants are determined by the difference in the CN stretching energy of the isotopic species, there are only three unknown parameters to be determined. The vibrational relaxation rate constants, and the time dependent concentrations of excited species are experimentally determined. Calculations simultaneously fit the four experimental curves very well (Fig. 4 A and B) with the three parameters: , the equilibrium constant K = 19 ± 3 (95 ± 1% of anions are clustered), and the clustered and separated ion exchange time constant 1/k_clu→iso = 12 ± 7 ps. Based on the energy transfer rate constant, and the energy transfer equation from a previous publication (23), the experimentally determined vibrational coupling constant, and the assumption of dipole-dipole interaction for vibrational energy transfer, the average distance between the C ≡ N groups of two SCN^- anions in a cluster is determined to be 3.7 ± 0.3 Å. The value is very close to the shortest C ≡ N distance 3.8 ∼ 4.0 Å in KSCN crystals (37, 38). (More details are in SI Appendix.)

Fig. 4.

Peak intensity data (dots) and fits to the data (lines) of the 2D IR spectra of solution C. Calculations details are in SI Appendix.

One Energy Donor has Seventeen Acceptors in a 10M KSCN Solution.

In addition to the concentration of ion clusters revealed by the nonresonant energy transfer experiments, the cluster sizes can also be evaluated by resonant energy transfer measurements.

In these experiments, the resonant energy transfer from one donor to any acceptor, as well as molecular rotations, can cause the anisotropy of vibrational excitation to decay (25, 36). During resonant energy transfers, the energy can be transferred back from acceptors to the original donor. The probability of reverse transfer is inversely proportional to the number of acceptors: more acceptors resulting in statistically less likely reverse transfers. When a reverse transfer occurs, the anisotropy is recovered. Therefore, fewer acceptors for one donor (corresponding to a smaller cluster) will result in slower energy-transfer-induced anisotropy decay. Based on the physical picture, we derive an equation which can be used to extract the number of anions involved in the effective energy transfer (Details are in SI Appendix):

[2]

where τ_or is the molecular rotational time constant in a cluster, and c is the fraction of (the energy carrier) among the isotope-labeled anions in a cluster. Changing the ratio of in a solution can change the number of resonant energy acceptors for one donor and therefore the resonance-energy-transfer-induced anisotropy decay rate, while the chemical properties of the cluster are unchanged. n_tot is the number of anions (both and SCN^-) within an effective energy transfer unit. τ is the resonant one-donor-to-one-acceptor energy transfer time constant. In Eq. 2, only two parameters (n_tot and τ) are experimentally unknown. Calculations with the two adjustable parameters simultaneously fit the experimental results of six different isotope ratios very well (Fig.. 5). The calculations show that τ = 54 ± 8 ps and n_tot = 18 ± 3. The number 18 is the same as the number of first shell SCN^- anions surrounding one anion in the KSCN crystals (37, 38). (More details are in SI Appendix). At this point, we don’t have any solid evidence to show that the energy acceptor number 18 - 1 = 17 ± 3 obtained from our experiments represents the number of anions surrounding one anion in the crystal. However, from the similarity of the anion distance and this acceptor number between the ion clusters and the crystal, we believe that some structural aspects of a big cluster in solution C is probably similar to those in the crystal, e.g., the shortest anion distance and the number of anions in the first solvation shell of an energy donor. In experiments, K⁺ doesn’t produce any signal. K⁺’s number in any cluster was therefore not determined. In highly concentrated solutions, the clusters could be large and contain many energy transfer units. Thus n_tot is not the same as the number of anions in a cluster. At lower concentrations, the numbers and sizes of clusters become smaller. n_tot is expected to become closer to the number of anions in a cluster.

Fig. 5.

The anisotropy decay data (dots) and calculations of Eq. 2 (lines) of the ¹³C¹⁵N stretch of in 10 M salt aqueous solutions with different KS¹³C¹⁵N/KSCN ratios. Adjusting the KS¹³C¹⁵N/KSCN ratio changes the number of resonance energy acceptors for the excited donor. The calculations yield n_tot = 18 ± 3, τ = 54 ± 8 ps.

Diluted Solutions Have Fewer, Smaller, and Tighter Clusters.

According to the thermodynamic principle (39), diluting solution C with water shifts the dissolution equilibrium to fewer clusters. Diluting solution C with water shifts the dissolution equilibrium to fewer clusters, this can be revealed by simple inspection of the growth of cross peaks in 2D IR spectra of solutions with different salt concentrations (Fig.. 2). At higher concentrations, the intensities of cross peaks are higher at the same T_ws. As described above, the growth of cross peaks is from the energy exchange of clustered anions, while the diagonal peak pairs are from both clustered and separated anions. The cross/diagonal peak ratio represents not only how fast the energy exchange is, but also how many of the anions form clusters. A higher cross/diagonal peak ratio indicates a faster energy transfer and/or more clusters. Quantitative analyses based on the above methods show that fewer and smaller clusters form in a lower concentration (Fig. 6A. Numerical values are listed in SI Appendix: Table S1). The fraction of anions in clusters is unexpectedly large in all studied concentrations. In highly concentrated solutions (10 and 8.8M) almost all anions are in clusters (> 90%). Even for a relatively dilute solution (1M) whose salt/water ratio is only ∼1/50, there still ∼27% of anions in clusters which contain three anions on average. The results suggest microphase separation in these solutions.

Fig. 6.

(A) The concentration dependences of clustered ion percentage and the number (n_tot) of anions in an energy transfer unit of KSCN aqueous solutions. (B) The concentration dependence of the one-donor-to-one-acceptor resonant energy transfer time constant in ion clusters of KSCN aqueous solutions. (C) The cation dependences of clustered ion percentage and the number (n_tot) of anions in an energy transfer unit of 4M aqueous solutions of LiSCN, NaSCN, KSCN, and CsSCN. (D) The cation dependence of the one-donor-to-one-acceptor resonant energy transfer time constant in ion clusters of 4 M aqueous solutions of LiSCN, NaSCN, KSCN, and CsSCN.

SCN^- is one of the strongest and most frequently used protein denaturants. SCN^- has been the subject of intense investigations and debates for its “salt-in” effect for many years (40). The high clustering tendency of KSCN observed in these experiments suggests another avenue for understanding SCN^-’s high effectiveness in denaturing proteins: it is conceivable that the interaction between SCN^- and water is indeed not very strong so that SCN^- prefers to associate to amino acid residues of a protein over water in a protein aqueous solution.

Another interesting result of the concentration dependent experiments is that the one-donor-to-one-acceptor energy transfer rate is faster at a lower concentration (Fig.. 6B). According to [1], this result suggests that a smaller cluster is tighter. The exact molecular mechanism giving rise to this phenomenon is not clear at this point. Instead, we propose a qualitative explanation. In a bigger cluster, more anions are close to each cation so that the average radius of the anion shells could be larger because of the geometry constraint and electrostatic repulsion.

Salts with Bigger Cations Form More Clusters.

It has long been recognized that the size and charge density of a cation have profound effects on the properties of electrolyte solutions and their biological activities (2). Theoretical calculations suggest that cations may affect the formation of ion pairs and clusters in aqueous solutions (12, 41). To explore cation specific effects, we performed energy transfer measurements on 4M (salt/water ratio = 1/10) aqueous solutions of LiSCN, NaSCN, KSCN, and CsSCN. These experiments show that in solution, smaller cations form smaller and fewer clusters (Fig.. 6 C and D. Numerical values are listed in SI Appendix: Table S1). In solution with the smallest cation Li⁺ (LiSCN), ∼50% of the anions form clusters which contain ∼4 anions on average. In solution with the biggest cation Cs⁺ (CsSCN), ∼70% of the anions form clusters containing ∼9 anions. This trend may be qualitatively understood with the theoretical description of “matching cation and anion sizes” for some electrolyte solutions (41, 42): small-small and large-large easily associate, while small-large readily dissociate. SCN^- is large and polarizable, and therefore, more readily associates with the large and polarizable Cs⁺ than the small Li⁺.

The Energy Exchange Method Can Be General for Short-Range Molecular Interactions.

The mode-specific vibrational energy exchange method can be used to study many molecular interactions if the interactions are strong enough and the probe vibrational lifetimes are long enough. The energy exchange method may or may not need isotope labeling, which in general does not perturb the molecular interactions. The only requirement is that the ions (or molecules) have IR active modes and the vibrational lifetimes of the modes are comparable to the energy transfer time scales (which mostly range from a few ps to a few hundred ps). Many important anions in biology or electrochemistry, e.g., CN^-, , , , , , SCN^-, and , have strong IR active vibrational modes. These modes typically have lifetimes of a few to tens of ps, overlapping with the energy transfer time scales. In addition, high salt concentrations are not necessarily required. The method can be applied to solutions of any concentration if the percentage of the clustered ions is high enough to provide a sufficient signal/noise ratio. In principle, the requisite clustering percentage can be as low as ∼0.1%. The method is not limited to ions. The energy exchange method can be also applied to the investigation of other short-range molecular interactions, e.g., those of peptide/sugar, DNA/protein, and drug/protein complexes, as long as the complexes have vibrational active modes fulfilling the requirements.

Concluding Remarks

The results presented here demonstrate that in the 1∶1 electrolyte aqueous solutions with medium to high concentrations, a significant portion of the ions form clusters. Diluting the solution results in fewer, smaller, and tighter clusters. Cations have significant effects on cluster formation. A small cation results in smaller and fewer clusters. The vibrational energy transfer method holds promise for studying a wide variety of other fast short-range molecular interactions.

Materials and Methods

Materials.

Unless specified, chemicals were purchased from Sigma-Aldrich and used without further purification. KS¹³C¹⁵N and NaS¹³C¹⁵N were purchased from Cambridge Isotope Laboratory. D₂O was from C/D/N ISOTOPES INC. LiS¹³C¹⁵N was synthesized by precipitating KClO₄ out of the KS¹³C¹⁵N and LiClO₄ mixed aqueous solution. CsS¹³C¹⁵N was synthesized by precipitating LiF out of the LiS¹³C¹⁵N and CsF mixed aqueous solution.

Methods.

A ps amplifier and a fs amplifier are synchronized with the same seed pulse. The ps amplifier pumps an optical parametric amplifier (OPA) to produce ∼1 ps mid-IR pulses with a bandwidth ∼21 cm^-1 in a tunable frequency range from 900 cm^-1 to 4,000 cm^-1 with energy 10 ∼ 40 μJ/pulse at 1 KHz. The fs amplifier pumps another OPA to produce ∼140 fs mid-IR pulses with a bandwidth ∼200 cm^-1 in a tunable frequency range from 900 cm^-1 to 4,000 cm^-1 with energy 10 ∼ 40 μJ/pulse at 1 KHz. In 2D IR and pump/probe experiments, the ps IR pulse is the pump beam (pump power is adjusted based on need). The fs IR pulse is the probe beam which is frequency resolved by a spectrograph yielding the probe axis of a 2D IR spectrum. Scanning the pump frequency yields the other axis of the spectrum. Two polarizers are added into the probe beam path to selectively measure the parallel or perpendicular polarized signal relative to the pump beam. Vibrational lifetimes are obtained from the rotation-free 1–2 transition signal P_life = P_∥ + 2 × P_⊥, where P_∥,P_⊥ are parallel and perpendicular data respectively. Rotational relaxation times are acquired from .