Modeling Carbon Dioxide Vibrational Frequencies in Ionic Liquids: IV

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Modeling Carbon Dioxide Vibrational Frequencies in Ionic Liquids: IV. Temperature Dependence Clyde A. Daly, Cecelia Allison, and Steven A. Corcelli J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.9b01863 • Publication Date (Web): 03 Apr 2019 Downloaded from http://pubs.acs.org on April 8, 2019

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The Journal of Physical Chemistry

Modeling Carbon Dioxide Vibrational Frequencies in Ionic Liquids: IV. Temperature Dependence Clyde A. Daly Jr., Cecelia Allison, and Steven A. Corcelli* Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46656 Abstract In previous papers in the series, the vibrational spectroscopy of CO2 in ionic liquids (ILs) was investigated at ambient conditions. Here, we extend these studies to understand the temperature dependence of the structure, dynamics, and thermodynamics of CO2 in the 1-butyl-3methylimidazolium hexafluorophosphate, [bmim][PF6], IL. Using spectroscopic mapping techniques, the infrared absorption spectrum of the CO2 asymmetric stretch mode is simulated at a number of temperatures, and the results are found to be consistent with similar experimental studies. Structural correlation functions are used to reveal the thermodynamics of complete CO2 solvent cage breakdown. The enthalpy and entropy of activation for solvent cage reorganization are found to be 6.9 kcal/mol and 7.6 kcal/mol/K, respectively, and these values are similar to the same for spectral, orientational, and translational diffusion. Caging times for CO2 are calculated, and it is shown that the short time dynamics of CO2 are unaffected by temperature, even though the long-time dynamics are highly sensitive to temperature.

* Corresponding author email: [email protected]

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I. Introduction

Carbon capture and sequestration (CCS) is a promising approach for improving the longterm viability of fossil fuel based energy generation.1 In CCS, a material is used to absorb CO2 from flue gas through physical solvation or chemical reaction. Temperature variations can then be used to remove the CO2 for storage and recycle the CCS material. Many materials have been explored for this purpose, including amine-based compounds in aqueous solution and metalorganic frameworks.2–4 Ionic liquids (ILs) have also been studied extensively for CCS and many other applications.5–13 The unique combination of electrochemical, physical, and chemical properties of ILs motivates their continued study. In CCS, temperatures ranging from 250 to 500 K are often used to achieve the required conditions for CO2 absorption or release.1,4,10,14–16 Because of this, it is essential to understand how temperature affects the solvation of CO2 in ILs, which will aid in the design of ILs and other materials for CCS. There are numerous studies of the effect of temperature on the physical properties of ILs and on the solvation of small gas molecules, like CO2, in ILs.13,17–20 Despite these efforts, how temperature modulates the molecular-level interactions of solutes with the IL is not well understood. In recent years, infrared spectroscopy has revealed new physical insights about the structure and dynamics of ILs in the immediate vicinity of vibrational reporters.21–25 Because the vibrational frequencies of small molecules like CO2 are sensitive to their local solvent environment, infrared absorption and two-dimensional (2D-IR) spectra contain a wealth of structural and dynamical information.22,26,27 Molecular dynamics (MD) simulations and spectroscopic calculations are indispensable for the interpretation of these measurements, and the

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simulations benefit from the fact that the experiments probe time scales readily accessible in the simulations.28–31 Recently, Garrett-Roe and coworkers used two-dimensional infrared (2D IR) spectroscopy with a SCN- vibrational reporter and solute to understand the effect of temperature in a series of ILs with imidazolium cations having different chain lengths.32 The dynamics revealed in the 2D IR measurements was consistent with a random walk of the SCN- solute on a rough potential energy surface. Here, we complement this study by considering a different solute, CO2, in the 1-butyl-3-methylimidazolium hexafluorophosphate, [bmim][PF6], IL. Although CO2 is a neutrally charged solute, in contrast to the anionic SCN-, we have found that the CO2 electric quadrupole moment is sufficiently strong to interact directly with the charged regions of the ionic liquid.30,31 Thus, we expect the dynamical behavior of CO2 to be qualitatively similar to SCN-. In the three previous papers in this series, we (1) developed a robust method for calculating CO2 asymmetric stretch vibrational frequencies in [bmim][PF6] using DFT,33 (2) developed and validated a spectroscopic map that relates the CO2 asymmetric stretch frequency to electrostatic and Lennard-Jones interactions of the solute with the surrounding IL,34 and (3) used a combination of MD simulations and experiments to examine the solvation dynamics of CO2 in [bmim][PF6] at 300 K.30,31 Here, we seek to leverage these previous studies to understand how temperature affects the structure, dynamics, and vibrational spectroscopy of CO2 in [bmim][PF6]. In particular, we use the temperature dependence of the dynamics to determine, in more detail, the mechanism and thermodynamics of solvent cage reorganization and replacement around CO2. We then connect this to the previously studied systems in order to show that, for the case of CO2 in ILs, the solvation dynamics occupy a middle regime between the behavior of a

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charged solute, like SCN-, and a neutral solute with minimal electrostatic interactions with the IL, like N2.

II. Methods

The methods utilized in the present study are similar to those used in prior work.30,31,33,34 All of the MD simulations were performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS)35 package. The force field used to describe the [bmim][PF6] IL is adopted mainly from the generalized Amber force field and the work of Liu et al. It employs a 0.84 scaling factor on the atomic-centered partial charges of the cations and anions, and the CO2 solute is treated as a rigid body using a quaternion representation.24,35–40 Simulations were performed at 278, 293, 313, 333, and 353 K. The initial positions and velocities for 256 ions pairs of [bmim][PF6] and one CO2 solute were taken from a simulation at 300 K from previous studies.33,34 At each temperature, the systems were first equilibrated by a 5 ns run in the NPT ensemble, bringing the cubic simulation box to the density corresponding to the target temperature. Next, the systems were heated to 600 K over 1 ns then cooled for an additional 1 ns to the target temperature. 10 ns NVE simulations were performed at each temperature, and initial coordinates and velocities were generated for production simulations by sampling this trajectory every 1 ns. Each set of initial coordinates was again subjected to the equilibration procedure above, besides the NPT step. This ensures that each production run trajectory is distinct from the trajectory that generated the initial conditions. Each of these ten simulations was extended in the NVE ensemble for 100 ns, for a total of 1 𝜇s of production simulation data with a 4 fs collection frequency for further analysis. We check that our simulations have reached the correct density by

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simulating the densities of boxes of 200 ion pairs of the same IL (i.e., without the CO2 solute) at the studied temperatures and comparing to experiment (Table 1).41 There is less than 4% error in all cases, however it is important to note that even modest errors in density for a liquid can affect the dynamics. We are able to additionally verify our simulations by obtaining diffusion constants for the CO2 solute at the considered temperatures. Calculated diffusion constants range from 0.5 x 10-10 m2 s-1 at 278 K to 7.3 x 10-10 m2 s-1 at 353 K (Table 1). The experimental values are 0.3 x 10-10 m2 s-1 at 278 K and 2.8 x 10-10 m2 s-1 at 353 K, and are generally about half the magnitude of our values.42 However, the activation energy for diffusion obtained from our simulations (6.99 kcal/mol) is consistent with that obtained from the experiments (6.21 kcal/mol).42

Table 1. Diffusion constants of CO2 and density of neat IL from simulation and experiment. T (K)

DSim (10-10 m2 s-1)

DExp42 (10-10 m2 s-1)

rSim (kg/m3)

rExp41 (kg/m3)

278

0.5

0.3

1335

1388

293

1.0

0.5

1325

1371

313

2.0

0.9

1313

1358

333

4.1

1.6

1289

1341

353

7.3

2.8

1272

1323

The CO2 asymmetric stretching mode vibrational frequency was calculated every 4 fs using the previously developed empirical spectroscopic map,34

𝜔 = 𝜔$ + 𝑏' 𝐸)*+,-./ + 𝑏0 𝐸)+/-./ + 𝑐' 𝑈) + 𝑐0 𝑈3 (1)

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where 𝜔$ is the experimental gas-phase frequency (2349.1 cm-1), E is the electric field projected along each CO bond axis, and the Ui are the Lennard-Jones interactions between CO2 and the solvent. The subscripts C and O refer to the site on the CO2 solute where the interactions are calculated. For O, the interaction is taken as the average from the two oxygen atoms in the CO2 molecule. The contributions of the electric field of the surroundings are computed separately for the anions, 𝐸)+/-./ , and cations, 𝐸)*+,-./ . The weights are as follows: 𝑏' = 64.4 cm-1 au-1, 𝑏0 = 93.2 cm-1 au-1, 𝑐' = 4.70 cm-1 kcal-1 mol, 𝑐0 = −3.55 cm-1 kcal-1 mol.34 Linear IR absorption spectra, 𝐼(𝜔), are computed from the fluctuating frequency trajectory, 𝛿𝜔(𝑡) = 𝜔 (𝑡) − 〈𝜔〉 , where 〈𝜔〉 is the average value of the frequency, using the fluctuating frequency approximation (FFA),43

K

I

𝐼(𝜔) = < 𝑑𝑡 〈𝜇⃑'? (0)𝜇⃑'? (𝑡)𝑒 BC ∫J EFGH(F) 〉 𝑒 BL/0NO ?

(2) Lifetime broadening is captured empirically through the factor, 𝑒 BL/0NO , where 𝑇' is the vibrational population lifetime from experiment (58 ps).31 The transition dipole moment of CO2 is insensitive to its environment (i.e., the Condon approximation is valid for CO2 in ILs).34 Thus, the dipole moment integral factors, 𝜇⃑'? = ⟨1|𝜇⃑|0⟩, are assumed to be unit vectors along the axis of the CO2 solute.

III. Results and Discussion 6


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A. IR Absorption Spectrum

The simulated IR absorption spectrum computed using Eq. (2) changes very little with temperature (Figure 1) with both the peak frequency and the full width at half maximum (FWHM) slightly increasing (Table 2) at elevated temperatures. This is mostly consistent with a recently published experimental study of the vibrational spectroscopy of SCN- in a series of ILs with bistriflimide ([NTf2]) anions and imidazolium cations of varied chain lengths.32 There, the IR absorption spectra were mostly unchanged by temperature, exhibiting a slight red-shift and a modest broadening. The magnitude of both effects was less than 2 cm-1 across the whole temperature range, as it is here, although the peak shift is in the opposite direction. The line broadening in Figure 1 occurs mainly on the high frequency side of the peak. The growth of higher frequencies at increased temperatures indicates weaker solvent caging around the CO2 solute,31 which is likely related to the decrease in IL density with increasing temperature.

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1

278 K 293 K 313 K 333 K 353 K

0.75

I (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.5

0.25

0 2330

2340

-1

2350

2360

ω (cm ) Figure 1. Calculated IR absorption spectra for CO2 in the [bmim][PF6] at the indicated temperatures. Table 2. Peak frequency and full-width at half-maximum (FWHM) in cm-1 for calculated IR absorption spectra at indicated temperatures. T (K)

Peak

FWHM

278

2345.4

2.61

293

2345.4

2.77

313

2345.6

2.70

333

2345.6

2.95

353

2345.6

3.05

8


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B. Correlation Functions

Several different types of correlation functions are examined in this work, including solute-cage correlation functions (SCCFs), orientation correlation functions (OCFs), and frequency fluctuation correlation functions (FFCFs). Each correlation function provides a different measure of the dynamics of the IL from the perspective of the CO2 solute. SCCFs report upon the reorganization of the solvent directly around the CO2 molecule, OCFs describe the ability of the CO2 bond-axis to change direction in the IL, and FFCFs report the spectral diffusion of the asymmetric stretch of CO2. The FFCFs are directly comparable to dynamics extracted from 2D IR measurements. In order to facilitate comparison, the time correlation functions are fit to equation: \

𝐶 (𝑡) = 𝑎' exp(−𝑡 0 𝜏'B0 ) + [ 𝑎C exp(−𝑡𝜏CB' ) C]0

(3) which corresponds to a Gaussian function for the short-time dynamics and three exponential decays for the longer-time response. For the SCCFs, the Gaussian term is excluded because the functions were computed with a time resolution of 1 ps, which precludes full resolution of the short time dynamics. The Supporting Information (SI) contains tables of all 𝑎C and 𝜏C constants, and mean correlation lifetimes. Henceforth, we focus upon the long time dynamics of the correlation functions, and the values of 𝜏\ are shown in Table 3.

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Table 3. Long time constants for solvent-cage correlation functions (SCCFs), orientation correlation functions (OCFs), and frequency fluctuation correlation functions (FFCFs). T (K)

SCCF (ps)

OCF (ps)

FFCF (ps)

278

927

1400

1100

293

510

756

635

313

233

305

234

333

89.6

124

98.3

353

56.8

66.3

44.9

Our definition of the SCCFs is inspired by the work of Zhang et al.,44 and is the same that we utilized in prior work.30 Briefly, the full set of ions within the first solvation cylinder30,31 is continuously tracked. If any ion enters or exits the solvation cylinder, the solvent cage is considered broken, but cages are allowed to reform. The SCCF is a representation of the likelihood of a cage persisting for a specified period of time, and can be written as 𝑆𝐶𝐶𝐹 (𝑡) = 〈ℎ(0)ℎ(𝑡)〉/〈ℎ〉, where ℎ(𝑡) is one when the ion cage identified at t = 0 remains, and ℎ(𝑡) is zero if not.44 There is a distinct similarity between the timescales reported in the FFCFs and SCCFs (Table 3), which supports the interpretation that the 2D IR measurements are monitoring cage dynamics at long time.30 The decay of solvent cages increases greatly as temperature increases (Table 3 and Figure 2A).

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0.3 A 0.03 0.003

Cθ(t)

0.3 B 0.03 0.003

Cω(t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


CSC(t)

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0.3 C 0.03 0.003 0.01

278 K 293 K 313 K 333 K 353 K

0.1

1

10

t (ps)

100

1000

Figure 2. (A) solvent cage correlation functions (SCCFs), (B) orientational correlation functions (OCFs), and (C) frequency fluctuation correlation functions (FFCFs) at the indicated temperatures. The OCF tracks the randomization of the direction of a CO bond vector over time and is defined as

𝐶a (𝑡) = ⟨𝑃0 [𝑢e(𝑡) ⋅ 𝑢e(0)]⟩ (4) where 𝑢e(𝑡) is the unit vector along the CO2 bond axis at time 𝑡, and 𝑃0 is the second Legendre polynomial. For short time delays, the bond axis is likely to remain in the original direction (𝐶 (𝑡) = 1) and for longer time delays the bond axis is randomly oriented (𝐶 (𝑡) = 0). Figure 2B shows the rotational diffusion of the CO2 in solution at the sampled temperatures. At shorter times (< 0.1 ps), the correlation functions change very little with temperature. However, at longer

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times (> 1 ps), the OCF changes rather dramatically with temperature, slowing significantly as temperature is decreased. Even so, the qualitative profile of the orientational dynamics remains the same, regardless of temperature. The same general trend is observed in the FFCFs, which tracks the randomization of the CO2 asymmetric stretch vibrational frequency over time

𝐶H (𝑡) =

〈𝛿𝜔(𝑡)𝛿𝜔(0)〉 〈𝛿𝜔(0)𝛿𝜔(0)〉 (5)

where 𝛿𝜔(𝑡) = 𝜔 (𝑡) − 〈𝜔〉 and 〈𝜔〉 is the average frequency. The short-time dynamics are again almost identical at all temperatures, while the long-time dynamics vary greatly (Figure 2C). Features of the correlation functions are the same regardless of temperature, including the small oscillations between 0.1 and 1 ps. As discussed in prior work, these oscillations are due to interactions between the CO2 and the cation ring.31 Decomposition of these correlation functions (Figures S1 and S2) show that, regardless of temperature, the anions and cations make similar relative contributions to the full FFCFs.

C. Structural Dynamics

In order to understand the trends in the correlation functions, we use cylindrical distribution function (CDFs) to reveal the solvation environment of CO2. CDFs are analogous to radial distribution functions, but offer two advantages in the present context. CDFs respect the cylindrical symmetry of the CO2 solute, and they provide a more intuitive description of the three-dimensional structure of the solvent around the CO2 molecule and its evolution in time. In

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the CDFs, atomic charges are binned only if they satisfy both 𝑟 ≤ 5.0 Å and |𝑧| ≤ 5.0 Å at 𝑡 = 0 and at some later time, 𝑡 = 𝜏. Figure 3 shows that the CO2 solute has the same average environment at a waiting time of 0 ps, regardless of temperature. In Figure 3, the CO2 is centered along the z axis. The positive regions near the oxygens are populated by cations, and the negative regions near the carbon are populated by anions.31 As temperature increases, the cage of solvent surrounding the CO2 evolves more rapidly. However, the qualitative structure of the solvent around CO2 does not vary with temperature or the time delay. As expected, at each of the investigated temperatures, the CO2 maintains the same local solvent cage for about as long as the SCCFs persist. For instance, the solvent cage around CO2 at 278 K has a reasonable chance of persisting at 1 ns and the SCCF timescale is 927 ps. At 353 K, the solvent cage is mostly absent at 100 ps, and the SCCF timescale is 56.8 ps.

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Figure 3. Cylindrical distribution functions (CDFs) for CO2 in the [bmim][PF6] IL. At short times, there is a clear solvent cage around the CO2 that weakens as time increases (rows). As temperature increases, the rate of the solvent cage decay increases as well (columns). D. Arrhenius and Eyring Analysis

Recent 2D IR measurements of the solvation dynamics of SCN- in ILs showed that the long timescales exhibit Arrhenius behavior32

ln(𝜏 B' ) = −𝐸q r

1 u + ln (𝐴) 𝑘t 𝑇 (6)

where 𝐸q is the activation energy and 𝐴 is the pre-exponential factor. In the context of reaction rates A is usually interpreted as an attempt frequency, a measure of how often collisions occur that, if they had enough energy, could produce the reaction.45,46 We applied this analysis to each 14


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of our calculated correlation functions, and the results are shown in Figure 4 and Table 4. The activation energies for the SCCF, FFCF, and OCF are similar falling in the range 7.5 – 8.3 kcal/mol. The activation energy derived from the dynamics is similar, but slightly larger, than that determined from the diffusion constants, 7.0 kcal/mol. Furthermore, because the SCCF directly reports on the reorganization of the solvent cage around the CO2 solute, we can conclude that this activation energy is the energy required to break up the cages that surround the CO2. The FFCF and OCF have the same activation energy because the solvent cage breakup directly randomizes the vibrational frequency (meaning that the 𝜏C ’s for FFCF and SCCF are similar30) and creates an opportunity for the orientation to randomize as a new solvent cage is found.

0

0

FFCF A FFCF E OCF A OCF E SCCF A SCCF E

-5

-10

-10

-1

-1

ln(τ )

-5

ln(τ T )

-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-15

-15 1

1.2

1.4

1.6

-1

-1

1.8

2

(RT) (kcal mol) Figure 4. Arrhenius (A, left scale) and Eyring (E, right scale) plots for the long time scales of the SCCFs, OCFs, and FFCFs. The slopes are all similar, indicating similar activation energies and enthalpies. Table 4. Parameters of the Arrhenius and Eyring, Eqs. (6) and (7), fits to the long time constants of the indicated correlation functions.

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𝛥𝐻 ‡ (kcal/mol) 𝛥𝑆 ‡ (cal/mol/K)

𝐸q (kcal/mol)

𝐴 (fs-1)

OCF

8.1

1.6

7.5

8.9

FFCF

8.3

2.8

7.7

10.1

SCCF

7.5

0.8

6.9

7.6

If interpreted directly as the rate of molecular restructuring, the pre-exponential factors are anomalously high. These values tend to be on the order of 0.01 fs-1 for unimolecular chemical reactions involving the evolution of a single major complex.47 Experimental studies using FFCFs obtained from 2D IR spectra to study systems with especially slow dynamics also see large preexponential factors.32,48 In order to understand this, we use the Eyring equation,

ln(𝜏 B' 𝑇 B' ) = −𝛥𝐻 ‡ r

1 𝑘t 1 u + ln r u + 𝛥𝑆 ‡ r u 𝑅𝑇 ℎ 𝑅 (7)

where 𝛥𝐻 ‡ and 𝛥𝑆 ‡ are the enthalpy and entropy of activation, respectively.45,49,50 𝛥𝐻‡ and 𝛥𝑆 ‡ are reported in Table 4, and range from 6.9 – 7.7 kcal/mol and 7.7 – 10.1 cal/mol/K, respectively. These values represent the change in the given thermodynamic variable upon the formation of the transition state complex.45,49,50 In this context, these values are related to the solvent cage breakup when the CO2 is transitioning from one ion cage to another. The enthalpies of activation are all 0.6 kcal/mol smaller than the energies of activation. This is expected since 𝛥𝐻 ‡ = 𝐸q − 𝑅𝑇, and 𝑅𝑇 = 0.6 kcal/mol at 312 K, near the average of the temperatures examined here.45 The entropies are positive, implying that the transition state involves the described complex falling apart and increasing the solution disorder, which is consistent with a cage breaking event.49 Comparing Eqs. 3 and 4, we can determine that45

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𝑘t 𝑇 {| ‡ ~' 𝐴= 𝑒 } ℎ (8) At 312 K, our pre-exponential factors are consistent with our entropies of activation. Pre-exponential factors from the Arrhenius equation are normally used to describe chemical reactions, especially where collisions between particles are involved in supplying the needed energy for reaction.45,46 However, for a process that is not a chemical reaction, or where the transition entropy is positive, the pre-exponential factor loses its meaning as a collision frequency. The entropy of activation does, however, have a straightforward physical interpretation; it is a measure of the change in organization when moving from the reactant to the transition state. Unimolecular decomposition reactions involving ordered initial structures and disordered final structures, such as the breakdown of strained organic cycles, often have high pre-exponential factors and large positive transition entropies.47 For instance, the decomposition of cyclobutane into two ethane molecules, resulting in a significant increase in rotational and translational freedom, has a pre-exponential factor of 4.0 fs-1 and an energy of activation of 62.5 kcal mol-1.47,51 The cage breaking events in this case are similar in so much as that they involve a stable structure dissociating. The major difference is that the complex in question is held together by weak (compared to covalent chemical bonds) electrostatic interactions. The event begins with an increase in the ability of the CO2 and ions to translate and rotate. This results in comparable pre-exponential factors and transition entropies but much smaller energies/enthalpies of activation than the ring decomposition example. Considering SCN- in [bmim][NTf2], the unphysical pre-exponential factor for the FFCF corresponds to a reasonable entropy of activation of 8.5 cal/mol/K at 312 K.32 This is very

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similar to the value we have obtained for CO2, suggesting that the caging dynamics in the two liquids are similar. Since both solute molecules interact strongly with the IL, and the cations in both these cases are the same, the similarity of the solvent cage dynamics is reasonable. The energies of activation, however, seem to suggest that CO2 is holding onto its local cages more tightly than SCN-, by a small amount. However, the ILs are different, which could plausibly account for the differences.

E. Localization

Now we shall examine our mean squared displacement (MSD) data in the style of Larini et al. in order to explore the solvent cage dynamics of CO2 in [bmim][PF6].52 They examined the function

𝛥 (𝑡 ) ≡

𝜕 log〈𝑟 0 (𝑡)〉 𝜕 log 𝑡 (9)

where 〈𝑟 0 (𝑡)〉 is the MSD. When 𝑡 = 0, the diffusing particle will be in the ballistic regime and the value of 𝛥(𝑡) will be 2; when 𝑡 = ∞, the particle will be undergoing free diffusion and 𝛥(𝑡) will tend towards 1. Between these two extremes, 𝛥(𝑡) will find a minimum at 𝑡 = 𝑡 ∗ , where 𝑡 ∗ is interpreted as approximately the time it takes for the CO2 to change direction due to collisions with its solvent cage, which should be much smaller than the total amount of time it spends inside that cage. In Figure 5 we can see a clear minimum near 3 ps for every temperature. Because of the large sampling resolution, exact values for 𝑡 ∗ are found by fitting the lowest 4 18


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values of 𝛥(𝑡) to 𝛥(𝑡) = 𝑚0 𝑡 0 + 𝑚' 𝑡 + 𝑏 and finding the minimum of the resulting function (Table 5). The shape of 𝛥(𝑡) is not sensitive to temperature, and as such, neither is 𝑡 ∗ . This observation mirrors the insensitivity of the SCCF shortest timescale to temperature, the value of which is similar to 𝑡 ∗ . The value of the MSD at this time (〈𝑟 0 (𝑡 ∗ )〉) can be thought of as a localization area, and it is sensitive to temperature. MSD values from 1 ps to 5 ps are fit to ∗

〈𝑟 0 (𝑡)〉 = 𝑚𝑡 {(L ) in order to find exact values (Table 5). As the temperature increases, the CO2 explores more space as it rattles in its local cage. Less populated and softer solvent cages should promote higher CO2 frequencies, which we have seen in prior work and in the asymmetric broadening of the IR spectrum in Figure 1.31

1.5

278 K 293 K 313 K 333 K 353 K

1

∆(t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5

0 1

10

t (ps)

100

Figure 5. Plots of Δ(t), Eq. (9), at the indicated temperatures.

19


1000


Page 20 of 29

Table 5. Caging times, localization areas, and Δ(τ‰33Š ), Eq. (9), at the indicated temperatures. MSD (t*) (Å2) 𝛥(𝜏|‹‹Œ )

T (K)

t* (ps)

278

2.83

2.0

0.80

293

2.76

2.4

0.82

313

2.84

3.0

0.80

333

2.83

3.7

0.83

353

2.78

4.4

0.80

Looking at the long time behavior of 𝛥(𝑡), we can see that it takes a great deal of time (> 100 ps) for even the fastest system to reach a 𝛥(𝑡) = 1, designating free diffusion. To investigate the connection between this free diffusion and solvent cage decay, we find 𝛥(𝜏|‹‹Œ ), the value of 𝛥(𝑡) at the long SCCF timescale. These values are in Table 5 and are all near 0.8. Total relaxation of the SCCF is correlated with nearly free diffusion of the solute. It takes substantially different amounts of time for 𝛥(𝑡) to reach this value for each temperature. Overall, the short time translational dynamics are insensitive to temperature, although the structure is not, and the long time dynamics are very sensitive to temperature.

IV. Conclusions

In this work, we have examined the temperature dependence of the spectroscopy and molecular dynamics of CO2 in [bmim][PF6]. The linear absorption IR spectra change very little, but the FFCFs change a great deal as temperature is increased. This points to the importance of non-linear spectroscopy. We find that the FFCF is thermodynamically similar to the OCF and

20


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SCCF, and the positive enthalpy and entropy associated with these processes is understood as describing a cage breaking process. The CO2 dynamics in a local cage are found to be insensitive to temperature, unlike the structure. The solvent cage dynamics are linked to the beginning of free diffusion.

Supporting Information Description

•

Decompositions of the calculated FFCFs into contributions from electrostatic and Lennard-Jones interactions

•

Decompositions of the calculated FFCFs into contributions from anions and cations of the IL

•

Fits of the SCCFs, OCFs, and FFCFs to Eq. (3)

Acknowledgements

SAC is grateful for financial support from the National Science Foundation (CHE1565471). The authors are also thankful for high performance computing resources and support from the Center for Research Computing at the University of Notre Dame, and for helpful discussions with Sean Garrett-Roe.

21



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TOC Graphic 1

278 K

293 K Increasing 313 K Temperature 333 K 353 K

0.1

Cωω(t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.01

0.001 0.01

0.1

1

10

t (ps)

29

100

1000


Modeling Carbon Dioxide Vibrational Frequencies in Ionic Liquids: IV

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