Accurate Characterization of Ion Transport Properties in Binary

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The Journal of Physical Chemistry

Accurate Characterization of Ion Transport Properties in Binary Symmetric Electrolytes Using In-Situ NMR Imaging and Inverse Modeling Athinthra Krishnaswamy Sethurajan,† Sergey A. Krachkovskiy ,‡ Ion C. Halalay,P Gillian R. Goward,∗,‡ and Bartosz Protas∗,S School of Computational Science & Engineering, McMaster University, Hamilton, Ontario, Canada L8S-3K1, Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S-3K1, General Motors Global R&D, Warren, Michigan, United States of America 48090-9055, and Department of Mathematics, McMaster University, Hamilton, Ontario, Canada L8S-3K1 E-mail: [email protected]; [email protected] Phone: (905)-525-9140 x 24176; (905)-525-9140 x 24116. Fax: (905)-522-2509; (905)-522-0935

∗

To whom correspondence should be addressed School of Computational Science & Engineering, McMaster University ‡ Department of Chemistry, McMaster University P General Motors S Department of Mathematics, McMaster University †

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Abstract We used NMR imaging (MRI) combined with data analysis based on inverse modeling of the mass transport problem to determine ionic diffusion coefficients and transference numbers in electrolyte solutions of interest for Li-ion batteries. Sensitivity analyses have shown that accurate estimates of these parameters (as a function of concentration) are critical to the reliability of the predictions provided by models of porous electrodes. The inverse modeling (IM) solution was generated with an extension of the Planck-Nernst model for the transport of ionic species in electrolyte solutions. Concentration-dependent diffusion coefficients and transference numbers were derived using concentration profiles obtained from in-situ

19 F

MRI measurements. Material

properties were reconstructed under minimal assumptions using methods of variational optimization to minimize the least-square deviation between experimental and simulated concentration values with uncertainty of the reconstructions quantified using a Monte Carlo analysis. The diffusion coefficients obtained by pulsed field gradient NMR (PFG-NMR) fall within the 95% confidence bounds for the diffusion coefficient values obtained by the MRI+IM method. The MRI+IM method also yields the concentration dependence of the Li+ transference number in agreement with trends obtained by electrochemical methods for similar systems and with predictions of theoretical models for concentrated electrolyte solutions, in marked contrast to the salt concentration dependence of transport numbers determined from PFG-NMR data.

Introduction Lithium-ion batteries (LIBs) are increasingly used for a large variety of applications ranging from portable electronics and electric vehicles to stationary energy storage systems. While several types of electric cars have been already commercialized, significant improvements in LIB performance are needed to achieve higher charging rates, extended lifetime, wider operating temperature range, lower cost, and sustained abuse tolerance. Since many of these requirements are closely associated with electrolyte properties, the design, develop2


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ment, and optimization of electrolytes are key factors for improving battery performance. In this context, improved measurements of mass transport properties, such as salt chemical diffusion coefficients (Ds ) and ionic transference numbers (t+ ), including their concentration dependence, are needed for the accurate modeling of Li-ion batteries performance and cell design. The transference number is defined as the fraction of the total ionic current I carried by ions of a particular charge sign. Thus, defining the currents carried by the respective species I + and I − , t+ = I + /I is the current fraction carried by cations (positive ions) and t− = I − /I that carried by anions (negative ions). In contrast, we will designate the ratios τ + = D+ /(D+ + D− ) and τ − = D− /(D+ + D− ), where D+ and D− are the diffusion coefficients of the cations and anions, as the transport numbers of the respective ions. We stress here that transference and transport numbers are not identical quantities because the electric field affects only the translational motion of charged solution species, while both ions and neutral ion pairs can migrate in the presence of a concentration gradient. Furthermore, we note that the two quantities represent ratios of physical quantities with different physical dimensions, [L2 ]/[T] and [Q]/[T] for the transport number and transference number, respectively. We make this point here since the interchangeable (and evidently less than judicious) use of the two appellations is still a source for confusion in the peer-reviewed literature. There exist several approaches to measuring transference numbers, such as Hittorf’s, 1 moving boundary, 2 electromotive force, 3 galvanostatic polarization, 4 electrophoretic nuclear magnetic resonance (ENMR), 5 and electrical ac impedance methods. 6 While each approach has advantages and disadvantages, the various methods often yield rather different results at salt concentrations of practical interest. 7,8 The most common cause for such discrepancies is the extensive ion association existing at high salt concentrations. In particular, it is difficult for most NMR-based techniques to distinguish free ions from ions in clusters, which is necessary to accurately fit a proposed theory to experimental results. Furthermore, such

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experiments can only provide transport number values because they are performed in the absence of an applied electric field, i.e., in the absence of electro-migration. Note that, while transport numbers coincide with transference numbers in the case of negligible ion pairing (i.e., for dilute solutions) as pointed out by Stolwijk et al., 9 such agreement is merely fortuitous. Another often encountered issue arises because transport parameters are derived under the assumption that uniform (bulk) electrolyte solution properties exist throughout the solution volume probed by the experimental technique. However, according to Hafezi and Newman, 10 the quantities t+ and D may be considered to be constant (i.e., at their bulk or as-prepared values) only when the concentration profiles which develop in the electrolyte solution under application of an electric field do not extend far from the electrode surfaces, and when the initial charge carrier concentration is maintained at distances far from the 1

electrodes. This condition is satisfied only as long as (4Dti ) 2 ≪ L2 , where ti is the duration of the electric current application and L is the cell length. Since ionic diffusion coefficients in most common organic electrolytes are in the 10−11 –10−10

m2 s

range and the inter-electrode

distance in Li-ion batteries is typically tens of micrometers, it is clear that this condition is violated in real-life batteries within seconds from the application of an electrical field. Therefore, to correctly model the electrolyte behavior in lithium-ion batteries, one needs to know the dependencies of D and t+ on the salt concentration. From the point of view of mathematical modeling, material properties such as the ionic diffusion coefficient and transference number appear as coefficients in the partial differential equations (PDEs) describing the evolution of the ionic species’ concentration. Given some measurements of concentration profiles as functions of time, these material properties can be recovered by means of the model equation through the solution of the so-called inverse problems. Inverse modeling is a well-established area of applied and computational mathematics with numerous applications to fields such as medical imaging, geoscience and biology. 11 Reconstruction of material properties when they are constant throughout a system is fairly well

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understood and this simplifying assumption has already been used for solving such problems for electrochemical systems analogous to the one considered here. 12 Motivated by the issues described above, we introduce and validate an approach which enables the determination of the concentration dependencies for the diffusion coefficient D(c) and the transference number t+ (c) through the analysis of NMR imaging (MRI) data 13 by inverse modeling (IM). The proposed approach extends the applicability of the Planck-Nernst formalism for describing ion transport (valid in the idealized limit of dilute solutions) to moderately concentrated electrolyte solutions, by allowing diffusion coefficients and transference numbers to depend on spatially resolved species concentrations. We demonstrate that such an extended model describes the experimental measurements more accurately than the standard data analysis with constant material properties. Concentration profiles of T F SI − anions were determined from in-situ, one-dimensional 19

F MRI profiles, which take advantage of the high sensitivity of NMR for the fluorine

nucleus. The main advantage of the in-situ MRI technique, when compared with electrochemical methods, is its ability to directly provide ion-specific information. There exist several reports on the in-situ use of both standard NMR and MRI for monitoring the performance of materials in energy storage devices. 14–16 In particular, Klett et al. 12 have recently demonstrated the applicability of in-situ 7 Li MRI to the characterization of mass transport in lithium ion battery electrolytes in the presence of an applied electric field. MRI allows the mapping of concentration gradients which arise in an electrochemical cell in the presence of an applied electric current. Mass transport parameters are then determined from fitting these data to a theoretical model for the experiment. The main advantage of the MRI approach over the more direct electrophoretic NMR (E-NMR) technique is that it does not require the application of a large electrical potential for signal detection. It thus avoids electrolyte decomposition with gas generation at electrode surfaces as well as convection currents due to heating of the electrolyte solution, both of which produce serious artifacts in the E-NMR signal. 17 MRI has therefore a wider applicability than E-NMR and may well

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become the standard method for determining the mass transport properties of electrolyte solutions in the presence of low electrical potentials, as encountered in Li-ion and other batteries. All measurements from our study were carried out with standard NMR equipment and an in-situ electrochemical cell design based on a conventional 5 mm NMR tube. Thus the proposed technique can be fairly easily implemented in any modern R&D facility. From the mathematical point of view, the inverse problem considered in the present study is non-standard, since the unknown material properties explicitly depend on a state variable (i.e., concentration of ions). A robust computational technique for the solution of such inverse problems was developed by Bukshtynov et al. 18 Although it has already been applied to a number of problems in physics, 19,20 this study represents a first such application in chemistry. The main advantage of the inverse modeling approach developed here is that it does not rely on a priori assumptions about the type and nature of the material properties to be reconstructed, except for some constraints on their regularity (understood as smooth dependence on the concentration variable). The concentration dependence of transport properties was thus determined in the present work by combining the ionic concentration profiles in an operating electrochemical cell obtained by MRI with a data analysis through inverse modeling in conjunction with a robust computational method. Furthermore, the reconstructed diffusion coefficients were compared against and shown to agree with the results obtained for homogeneous samples with the conventional method of pulsed-field gradient nuclear magnetic resonance (PFG-NMR) spectroscopy. 21 In contrast, transference numbers are extracted using the MRI-IM method and follow the expected trends with concentration. This indicates that MRI+IM provides a more robust methodology relative to the standard PFG-NMR alone, which determines transport numbers and is prone to inaccuracy due to the invisibility of ion-pairing effects in the latter measurements.

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Experimental Measurements of the anion and cation diffusion coefficient were performed by PFG-NMR on ten calibration solutions of lithium bis(trifluoromethanesulfonyl)imide lithium salt (LiT F SI) in propylene carbonate (PC, Sigma Aldrich) with salt concentrations ranging from 0.25 M to 2.0 M. The 1.0 M LiTFSI/PC solution was chosen as the model electrolyte for the in-situ one-dimensional NMR imaging experiments conducted in a DC-NMR cell (also referred to as electrophoretic insert) constructed according to a design by Hallberg et al. 22 (Figure 1a). We used a standard 5 mm NMR tube (inner diameter 4.2 mm) and two copper wires insulated with Teflon heat-shrink tubing. Pieces of metallic lithium were placed on the uncovered ends of the wires to form non-blocking electrodes. Polyimide (KaptonT M ) tape covered the current leads inside the cell to ensure that the current in the electrolyte solution flowed only between the Li electrodes. The distance between the electrodes was 17 mm. The maximum available sample volume with undistorted NMR signal was determined by calibrating our probe with non-conductive components (Teflon disks) inserted into the cell volume. It is well-known that electrically conductive measurement cell components cause distortions in the applied static magnetic field B0 , the radio frequency magnetic field B1 , and the magnetic field gradients. 23,24 In order to avoid the influence of such distortions on our results, only a 14 mm axial segment of the MR images with both ends a distance of 1.5 mm away from the electrodes was used in the data analysis by inverse modeling. Preparation of the electrolyte solutions and the cell assembly were performed inside an argon-filled glove box (1.2 ppm oxygen, ≤ 0.1 ppm water contents). Anaerobic conditions were maintained throughout the duration of the NMR measurements by using epoxy for sealing the electrophoretic insert and by flowing dry nitrogen through the bore of the magnet and over the cell. A constant current of 50 µA was applied to the sample cell using an Autolab PGStat 30 instrument operating in galvanostatic mode (the electric potential was varied between 0.28 and 0.32 V over the duration of the in-situ MRI experiment). The direction of the current was chosen such that lithium was stripped from the bottom electrode and deposited on the top 7



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diffusion coefficients measured by PFG-NMR is estimated as ±1% based on the fit quality. The concentrations of Li+ cations and T F SI − anions are the same in any infinitesimal volume of the sample due to the local electroneutrality condition. Therefore, the concentration profile of 19 F nuclei will be the same as that of the 7 Li nuclei, since LiT F SI is a binary symmetric electrolyte. We chose to monitor the the relative sensitivity for the

19

19

F nuclei in the in-situ experiments, since

F nucleus is about 3 times higher than for the 7 Li nucleus,

which shortens significantly the duration of MRI profile measurements. One-dimensional 19

F NMR images were obtained using a gradient spin-echo pulse sequence with the magnetic

field gradient applied along the x-direction (i.e., along the axis of the cell), 27 with a 3 ms echo time and a 20 G/cm reading gradient. 256 frequency-domain points were collected over the 200 kHz spectral width. The combination of magnetic field gradient and spectral resolution yielded a spatial resolution of 40 µm. 64 scans with a relaxation delay of 3.5 s were collected for each image, resulting in an acquisition time of 4 minutes per image. The imaging measurement sequence was repeated at 2-hour intervals over the 16 hours duration of the DC current application. The NMR intensity measurements thus obtained are shown in Figure 1b. The concentration of ions at any particular point of the sample can be calculated from the measured profiles by normalizing the images with respect to the data obtained in the absence of the electric current and under the assumption of a uniform salt distribution. The resulting concentration profiles obtained this way are shown in Figure 2.

Modeling and Computational Methods Assumptions and Transport Problem The following modeling assumptions were made in order to obtain the mathematical description of the mass transport during the in-situ NMR experiment 28 A1: isothermal conditions;

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A8: the cation flux at the two boundaries (x = 0 and x = L) corresponds to the applied electric current and results in lithium deposition and stripping, respectively. 29,30 We will therefore consider a 1D problem with the spatial variable x ∈ [0, L], where L is the length of the electrolyte region in the cell, and time t ∈ [0, T ], where T denotes the duration of the experiment. The domain was chosen in such a way that its boundaries are at 1.5 mm distance away from both electrodes and therefore the double layers formed in the vicinity of the electrodes are not included in the model. The above assumptions lead to the following partial differential equation describing mass transport in the electrolyte (1a), subject to the boundary conditions (1b) and the initial condition (1c): ∂c ∂c (1 − t+ ) I ∂ D = + ∂t ∂x ∂x FA + ∂c (1 − t ) I =− ∂x x=0,L DF A c|t=0 = ci

in (0, L) × (0, T ],

(1a)

in (0, T ],

(1b)

in (0, L),

(1c)

where ci the initial concentration, A the cross-sectional area of the cell, and F Faraday’s constant. Note that the effective Fickian diffusion coefficient D and the transference number t+ are considered unknown 10 and will be reconstructed from the experimental data using the inverse modeling approach described below. The particularly simple form of system (1) is made possible by the assumptions specified above, especially A6. We note that the problem could be equivalently described in terms of the Maxwell-Stefan diffusion model, 12 however, the advantage of the present formulation is that we do not need to explicitly account for the chemical activity which is absorbed into the dependence of the effective diffusivity D on the concentration c. Despite the simplifying assumptions, system (1) is relevant to the vast area of Li-ion batteries research and development.

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Inverse Model The unknown material properties will be reconstructed based on the concentration profiles c˜(x, t) obtained in the in-situ NMR experiment. A widely used formulation of an inverse problem is to frame it as an optimization problem which aims to minimize a cost functional representing the least-squares deviation between the concentration values predicted by the model (denoted c in eq. (1)) and the experimentally determined concentration values (denoted c˜). The cost functional can be represented as 1 J (D, t ) = 2 +

Z

T 0

Z

L 0

c(x, t; D, t+ ) − c˜(x, t)

2

dx dt.

(2)

We will consider two distinct formulations, corresponding, respectively, to constant and to concentration-dependent diffusivity and transference number. The first one pertains to the standard Planck-Nernst theory, whereas the second one represents the extension discussed in the present study. We will henceforth distinguish the two cases by using the following notations: D and t+ for the former case, D(c) and t+ (c) for the latter case. When both D and t+ are constant, then we obtain the simple optimization problem (which is exact in the limiting case of an ideal solution, i.e., at very dilute salt concentrations)

P1 :

b b [D, t+ ] = argmin J (D, t+ ) [t+ ,D]∈R2

which is rather well understood and can be solved in a straightforward manner using commercially available software tools (carets “b·” denote optimal reconstructions henceforth). Problem P1 was in fact solved in the recent study by Klett et al. 12 and is also solved here as a preliminary step in a more complete analysis. A more complicated optimization problem arises when both D and t+ are concentration-

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dependent, which reflects the physics of the problem in more realistic fashion

P2 :

b b [D(c), t+ (c)] =

argmin [t+ (c), D(c)]∈X

J D(c), t+ (c) ,

where X denotes a suitable function space to which D(c) and t+ (c) belong. We emphasize that, apart from smoothness, no other a priori assumptions are made about the functional forms of D(c) and t+ (c). In contrast to the simplified case (problem P1), the computational approach required to solve the more realistic case (problem P2) with concentration dependent material properties is more involved and necessitates specialized tools. This approach has the general form of iterative gradient-based minimization (n)

D(n+1) (c) = D(n) (c) − ξD ∇D J D(n) (c), t+(n) (c) (n)

+

t+(n+1) (c) = t+(n) (c) − ξt+ ∇t J D(n+1) (c), t+(n) (c) b b [D(1) (c), t+(1) (c)] = [D, t+ ],

n = 1, 2, . . . ,

(3a)

n = 1, 2, . . . ,

(3b) (3c)

where ∇D J and ∇t+ J are the gradients (sensitivities) of cost functional (2) with respect to (n)

(n)

perturbations of, respectively, D(c) and t+ (c), whereas ξD and ξt+ are the corresponding lengths of the descent steps in the two directions. The initial guess for problem P2 in (3c) b and tb+ which are the optimal reconstructions obtained is given by the constant values D

from problem P1. (Problem P1 also requires an initial guess, however, this problem tends to have a unique minimum and therefore this initial guess may be arbitrary. 31 Thus, the advantage of solving P1 first is that it provides a very robust initial guess for problem P2.) The optimal concentration-dependent properties can then be computed using (3) as b D(c) = limn→∞ D(n) (c) and tb+ (c) = limn→∞ t+(n) (c). A key element of the iterative process

(3) is the evaluation of cost functional gradients ∇D J and ∇t+ J , and further details are provided in the Supporting Information.

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Computational Algorithm The approach introduced above has been implemented as a computational framework in the MATLAB environment. Solution of the inverse modeling problem involves the following main steps: i. Processing of the in-situ MRI concentration profiles to eliminate any systematic errors (in the form of band-limited fluctuations in the MRI signal which appear due to a phenomenon referred to as “Gibbs ringing,” an artifact consisting of spurious signals near sharp transitions between media with different electrical conductivity, as is the case at the surfaces of electrodes 32 ). ii. Analysis of noise levels in experimental data necessary to establish the stopping criterion for the iterative reconstruction algorithm. First, noise in the experimental data is isolated using the Savitzky-Golay noise filter available in MATLAB. Then, based on the variance of this noise, an absolute lower bound and bounds corresponding to 95% confidence intervals are established for the cost functional (2). In the optimization procedure (3) the cost functional can only be reduced down to this level, as otherwise the reconstruction will fit the noise in the data. iii. Solution of the reconstruction problem with constant material properties (problem P1) using MATLAB’s inbuilt optimization function fminsearch. During this step, system (1) is repeatedly solved using a standard finite-difference numerical method. iv. Solution of the more realistic problem P2, which is the central task of our data analysis, using iterative algorithm (3) with the initial guess provided by the solution of problem P1 (in fact, in practice, we use a variant of the conjugate gradient algorithm 33 which is a more advanced version of (3)). In order to evaluate gradients ∇D J and ∇t+ J , in addition to the governing system (1), we also need to solve the adjoint system (see Appendix A for its definition). The adjoint system is solved applying essentially the

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same numerical technique as used for solving the governing system (1). The step sizes (n)

(n)

ξD and ξt+ are computed using Brent’s method for line minimization. 34 v. Estimation of error bounds for the reconstructed material properties using a MonteCarlo analysis. The inverse problem is solved repeatedly for a large number (500 in the present study) of independent normally-distributed noise samples added to the experimental concentration data c˜. The noise amplitude in each sample is set to match the noise level in the original data. Error bars are then obtained from the standard deviation of the reconstructed material properties corresponding to different noise samples. Algorithm 1 given below is used to reconstruct the material properties D and t+ from the experimental in-situ MRI data . Hereafter this approach will be referred to as MRI+IM. We refer the reader to Ref. 31 for additional details concerning the derivation, implementation and validation of the proposed reconstruction approach.

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Algorithm 1 : Iterative minimization algorithm for optimal identification of concentrationdependent diffusion coefficient and transference number from experimental data. Input: Jnoise — cost functional value corresponding to filtered noise, εJ — adjustable tolerance, ℓ — degree of smoothing, c˜(x, t) — concentration values obtained from experiments, ˜ t˜+ — constant initial guesses for problem P1 D, b Output: D(c) and b t+ (c) ˆ ← argmin J (D, t+ ) using initial guesses D ˜ and t˜+ Solve problem P1: [tˆ+ D] t+ ,D

ˆ (constant initial guess) D ←D +(0) ˆ t ← t+ (constant initial guess) n←1 repeat solve governing system (1) solve adjoint system (S12) (see Supporting Information) evaluate ∇D J compute the conjugate direction G [∇D J ] perform line minimization: (n−1) ← argmin J D(n−1) (c) − ξ G [∇D J ] , t+(n−1) (c) ξD (0)

(n)

ξ (n−1)

(n−1)

G [∇D J ] update: D (c) ← D (c) − ξD solve governing system (1) solve adjoint system (S12) (see Supporting Information) evaluate ∇t+ J perform line minimization: (n−1) ξ t+ ← argmin J Dn (c), t+(n−1) (c) − ξ G [∇t+ J ] +(n)

ξ +(n−1)

(n−1)

update: t (c) ← t (c) − ξt+ G [∇t+ J ] n←n+1 until |J (D(n) (c), t+(n) (c)) − J (D(n−1) (c), t+(n−1) (c))| < εJ |J (D(n) (c), t+(n) (c))| or J (D(n) (c), t+(n) (c)) < Jnoise

Results and Discussion Analysis of Ionic Diffusion Coefficients Obtained by PFG-NMR Figure 3a displays the room temperature diffusivity of Li+ cations and of T F SI − anions for the set of ten calibration samples, plotted versus the salt concentration, which is also listed in Table 1. Note that in neutral solutions PFG-NMR determines self-diffusion coefficients. 35 The concept of self-diffusion assumes that all solution species move independently of each 16


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other, an approximation valid only in the (infinitely) dilute limit in the case of electrolyte solutions. On the other hand, in concentrated electrolyte solutions, the local electroneutrality condition ensures that positive and negative ions move from the region of higher to that of lower concentration at the same speeds. 36 Therefore, the concept of self-diffusion in concentrated electrolyte solutions represents a considerable simplification of the interactions among ions and should be used with caution. We will show in a future publication that the role of ion pairing can be included in a rigorous way into the IM analysis of MRI data to correctly evaluate both diffusion coefficients and lithium transference numbers for such solutions. Several trends are readily visible from the data. First, the diffusivities of both cations and anions decrease monotonically with increasing salt concentration. This is due to the increased viscosity of the electrolyte solution upon salt addition and to the inverse proportionality between the diffusion coefficient and the viscosity (η) described by the Stokes-Einstein equation D=

kB T , 6πηr

(4)

where kB is Boltzmann’s constant, T is the absolute temperature and r is the hydrodynamic radius of the diffusing particle. Second, despite Li+ being a much smaller ion than T F SI − (their van der Waals radii are 0.076 nm and 0.326 nm, respectively), the diffusivity of Li+ cations is smaller than that of the anions over the examined salt concentration range, since the Li+ solvation shell significantly increases its hydrodynamic radius. (For example, Ganesh et al. reported that Li+ is tetrahedrally coordinated by four EC molecules in LiP F6 /EC solutions. 37 ) A third conclusion can be drawn by examining the ratio of anion to cation diffusion coefficients (Figure 3b, Table 1). The main disadvantage of PFG-NMR is its impossibility to distinguish between free ions and ions in aggregates. Therefore, the PFG-NMR only yields values of diffusion coefficients averaged over all ionic species. Under the assumption that the

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Table 1: Diffusivity Data obtained using ex-situ PFG-NMR experiment. C [mol m−3 ]

D+ [ × 10−10 m2 /s]

D− [ × 10−10 m2 /s]

D− /D+

τ+

250 500 750 900 1000 1100 1250 1500 1750 2000

1.79±0.02 1.41±0.01 1.04±0.01 0.89±0.01 0.80±0.01 0.70±0.01 0.61±0.01 0.480±0.004 0.370±0.003 0.265±0.002

2.95±0.03 2.33±0.02 1.72±0.02 1.44±0.01 1.27±0.01 1.08±0.01 0.91±0.01 0.68±0.01 0.51±0.01 0.350±0.003

1.65±0.03 1.65±0.03 1.65±0.03 1.62±0.03 1.59±0.03 1.54±0.03 1.49±0.02 1.42±0.02 1.38±0.02 1.32±0.02

0.38±0.01 0.38±0.01 0.38±0.01 0.38±0.01 0.39±0.02 0.39±0.02 0.40±0.02 0.41±0.02 0.42±0.02 0.43±0.02

ionic aggregates higher than ion pairs can be neglected, this can expressed as

DN M R = (1 − α) Dion + αDpair ,

(5)

with α the association constant or degree of ion pairing. In the case of a lithium cation, ion paring leads to the replacement of a solvent molecule from its solvation shell by an anion. This substitution does not cause any drastic change in the overall size of the diffusing entity and therefore cannot lead to significant changes in the Li+ cation diffusivity. On the other hand, the hydrodynamic radius of a T F SI − anion changes significantly between its free state and its complex with a solvated Li+ cation. As the consequence, the anion diffusivity will be more sensitive to changes in the degree of ion pairing with salt concentration. Figure 3b suggests that ion paring is negligible at low salt concentrations and becomes detectable by PFG-NMR at concentration larger than 0.8 M, since the ratio between the anion and cation diffusion coefficients is constant at the lowest three concentrations. These observations correlate very well with the data obtained by Wang et al. in their investigation of ion association in a LiT F SI/P C electrolyte using Raman and IR spectroscopy. 38 Our interpretation also agrees with the results of Takeuchi et al. obtained for LiP F6 in propylene solutions, 39 according to which the coordination numbers of Li+ by O and Li+ by F change from 3.97 and 0.04 to

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3.72 and 0.32, respectively, when the LiP F6 concentration in PC increases from 0.5 to 1.0 M. Lithium transport numbers can be calculated from the ions diffusivities according to the following equation τ+ =

D+ . D+ + D−

(6)

and the values are shown in Table 1. By combining Eqs. (5) and (6) it becomes apparent that transport numbers will tend toward 0.5 as the degree of ion association increases (α → 1). At the same time, the increase in ion-ion interactions and especially the formation of higher order ion aggregates, in particular close to the positive electrode, manifest themselves by a decrease in the lithium transport number. This is due to a significant drop in the electrophoretic mobility of ion clusters, as pointed out by Dai and Zawodzinski. 40

Analysis of Salt Concentration Profiles Measured by In-Situ NMR Imaging The concentration data collected during the application of a 50 µA constant current are shown in Figure 2. As can be readily seen, the concentration of Li+ ions increases near the positive electrode during the experiment while their concentration decreases near the negative electrode, which leads to a concentration gradient inside the cell. It is also clear that the shape of the concentration profile changes systematically over time. The most significant changes in anion and cation concentrations occur closest to the electrodes, while their concentrations are fairly constant over time in the central part of the cell (i.e., they exhibit more shallow gradients). Concentration profiles obtained from model (1) under the simplified assumption of constant diffusivity and transference number (problem P1) and in the more realistic case with concentration-dependent material properties (problem P2) are presented in Figure 4. Their comparison shows that the latter approach provides systematically improved fits between

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the experimental data and the predictions of the model, with the difference between the results obtained by solving problems P1 and P2 increasing with time. As is evident from Figure 5, this difference is larger near the positive electrode during the late stages of the experiment and in regions of the cell where the electrolyte has higher salt concentrations. These observations are quantified in Figure 6 which demonstrates that allowing the diffusivity and transference number to depend on the concentration (problem P2) results in a reduction of the cost functional by a factor of approximately 2 with respect to the simplified case with constant material properties (note that the value of the cost functional at the first iteration corresponds to the optimal solution of problem P1). The fact that the terminal value of the cost functional is close to the noise level confirms a posteriori that the simplifying assumptions A1-A8 are indeed satisfied and that the corresponding model can accurately represent the data, provided it is equipped with the correct concentrationdependent diffusivity and transference number. We add that solution of problem P2 places the model prediction within the 95% confidence band already after three iterations. Note that the values of the cost functional obtained after the third iteration change very little with respect to the 95% confidence bound and significantly exceed the noise level in the measurement data, indicating that the fits reflect the behavior of the data, rather than the noise. Given the slow decrease of the error functional (2) during subsequent iterations, we may conclude that the reconstructed material properties have reached the forms warranted by the experimental data after as few as three iterations. The solution of problem P1 does not depend on the choice of the initial guess for D and t+ . In Figure 5 we note that the simplified model with constant material properties fits the experimental data fairly well for small concentration changes, as is the case throughout the cell at relatively early times (≤ 10 h) and near the center of the cell at all times. However, near the electrodes, where concentration gradients are significantly steeper beginning with 10 h from the inception of the experiment, the fits with constant parameters deviate systematically from the data. A significantly refined reconstruction in the regions of high

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salt diffusivity calculated from the diffusivities of cations and anions listed in Table 1 with the results obtained by MRI+IM. Figure 7a shows that PFG-NMR and MRI+IM provide very similar effective diffusivities over the concentration range measured during the in-situ experiment and that any differences between them are well within experimental errors (less than 5%). The agreement between salt diffusivity values determined by two independent methods is due to the fact that in LiT F SI/P C electrolyte solutions ionic association is negligible up to concentrations close to 0.8 M. Moreover, as discussed before, the cation diffusivity is less sensitive to ion paring than the diffusivity of anions. Therefore, according to eq. (7), the effective salt diffusivity Ds will not be highly sensitive to ion pairing and aggregation even at concentrations higher than 0.8 M. This conclusion is in good agreement with our previous results obtained by in-situ slice-selective NMR diffusion measurements, for which we also observed a good correlation with measurements performed by PFG-NMR on calibration samples with several different salt concentrations. 13 At the same time, the increased ion association at the high-concentration end of the cell leads to a decrease in the average electrophoretic mobility, because neutral ion pairs do not drift in an applied electric field, but move only by diffusion along the concentration gradient. A significant increase of Li–F coordination number in an electrolyte solution near the positive electrode was previously reported by Vatamanu et al. using a molecular dynamics simulation. 42 Therefore, one can expect that the value of the lithium transference number will be smaller in regions of the cell with high salt concentrations. This effect was previously observed using electrochemical methods, while PFG-NMR results yielded the opposite trend for the transport numbers 8 as also observed in our present study. We show in Figure 7b that an analysis of 1D NMR imaging data based on inverse modeling in which the value of t+ depends on concentration (problem P2) allows one to quantify this effect and to observe the expected decreasing trend in the value of the transference number from the MRI+IM method. The combination of in-situ NMR imaging and mathematical modeling provides not just an average value for t+ , but also a dependence of t+ on the salt concentration in agreement with

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predictions of electrostatic theories for electrolyte solutions. 7,40 While lithium transference and transport numbers obtained by the various methods used in this work are close to each other (differences are less than 10%), we stress again that the numerical similarity between t+ and τ + values is merely fortuitous, since the physical quantities involved in the two ratios have different physical dimensions, as already pointed out in the Introduction.

Conclusions We have demonstrated in this study that NMR imaging can be combined with a data analysis based on inverse modeling to produce new insights regarding the transport of lithium ions in concentrated electrolyte solutions under an applied potential. Specifically, we showed how models based on the Planck-Nernst equation, which are applicable in the limit of dilute electrolyte concentrations, can be extended to accurately describe mass transport in electrolyte solutions at concentrations used in commercial batteries. This was achieved by allowing both the diffusion coefficient and the transference number to depend on the salt concentration, while optimal forms of these dependencies are found under minimal assumptions by solving the relevant inverse modeling problem. Such more realistic models are capable of representing experimentally obtained concentration profiles with a significantly higher accuracy than the standard model with constant material properties, especially near electrodes where gradients in ion concentrations are high. Movement of ions through this area is significantly restricted because of increasing interactions between them, therefore this is an important limiting factor for ion transfer between the electrodes. In this regard, our method has crucial advantage over common techniques used for measuring t+ and D which neglect information about the local concentration of ionic species through averaging over the whole electrolyte volume involved in the experiment. While it is well known that models with a larger numbers of free parameters may in principle represent data more accurately, the assumption of concentration-dependent material properties is justified from the physical

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point of view, as diffusion coefficients do depend on the solute concentration. Furthermore, this assumption leads to a robust inverse model with solutions that exhibit limited sensitivity to uncertainty and noise in the data. (In fact, the assumption of a constant diffusion coefficient is an approximation used for the sole purpose of obtaining a closed-form solution to the transport problem and is known to be approximately valid only in situations in which concentration gradients are small.) The results obtained for the diffusion coefficients using PFG-NMR agree with the results of MRI+IM analysis within the 95% confidence bounds for latter. However, the MRI+IM method is preferable to PFG-NMR measurements, because the former yields a dependence of the transference numbers on concentration which agrees both with the results obtained by electrochemical methods and with the predictions of concentrated electrolyte theory, in contrast with the concentration dependence of the transport numbers derived from PFGNMR. An interesting question which arose in the present study concerns the thermodynamic consistency of the material properties reconstructed via the inverse modeling approach. In the context of the results presented here, this is related to enforcing positive-defined diffusion coefficients and transference numbers over the considered concentration range. While this can be achieved in a straightforward manner by the inclusion of suitable constraints in optimization problems P1 and P2, 19 we chose not to do so in the present study. The reason for our choice is that information about violations of thermodynamic consistency by the derived material properties is very useful, as it signals the loss of the model’s validity for describing the data at hand. This may occur, for example, in electrolytes with high degree of ionic association, where the concentration of ion pairs must be included as another state variable in the model. 3 Such questions are already under investigation and results will be reported in a follow-up publication.

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Acknowledgment The authors are grateful to Dr. Jamie Foster for many helpful discussions and feedback on the manuscript. The team acknowledges financial support through the NSERC Automotive Partnerships Canada program, as well as General Motors of Canada.

Supporting Information Available The Supporting Informtion demonstrates the uniqueness of the minimizers obtained in Problem P1. In addition, it offers some technical details concerning the evaluation of cost functional gradients used in the solution of Problem P2, cf. Algorithm 1. A discussion of the accuracy of PFG-NMR data is also provided. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Accurate Characterization of Ion Transport Properties in Binary

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