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Local probing of ionic diffusion by electrochemical strain microscopy: spatial resolution and signal formation mechanisms

Electrochemical insertion-deintercalation reactions are typically associated with significant change of molar volume of the host compound. This strong coupling between ionic currents and strains underpins image formation mechanisms in electrochemical strain microscopy (ESM), and allows exploring the tip-induced electrochemical processes locally. Here we analyze the signal formation mechanism in ESM, and develop the analytical description of operation in frequency and time domains. The ESM spectroscopic modes are compared to classical electrochemical methods including potentiostatic and galvanostatic intermittent titration (PITT and GITT), and electrochemical impedance spectroscopy (EIS). This analysis illustrates the feasibility of spatially resolved studies of Li-ion dynamics on the sub-10 nanometer level using electromechanical detection.


Introduction
Energy storage systems are the key enabling component of hybrid and electric automotive systems, portable electronics, and renewable-energy based energy technologies. 1,2,3 Crucial for progress to understand, develop, and optimize battery materials is the capability to decipher individual mechanisms responsible for battery functionality, including Li-ion and electron transport and electrochemical kinetics locally, at the level of grain assemblies, sub-micron grains, and ultimately at the nanometer scale of individual structural and morphological defects.  temperatures. 9 Overall, the capability of AFM to resolve the minute details of surface structure is invaluable in observation electrochemical processes; however, the chemical identification is generally unavailable and information is limited to surfaces only (as opposed to the details of ion insertion and dynamics within the material).
(b) Static strain mapping: The AFM can be used to map the strains developed in material during the ion intercalation that manifest as the shape changes of intrinsic or fabricated topographic features. The Dahn group used the AFM to measure volume changes during voltage cycles by using patterned electrode structures. 10,11,12 Beaulieu et al. and Matsui et. al used the AFM to measure roughness evolution during charge-discharge cycle of non-patterned electrode films of Si-Sn 13 and LiCoO 2 [14] where direct height changes cannot be determined. The elegant work of Shao Horn group has demonstrated direct measurement of step height on LiCoO 2 surface as a function of Li concentration. 15 This approach, however, is slow and is applicable only to a limited number of materials.
(c) DC conductance mapping: The SPM can be used as a moving current electrode, potentially extending well-known time domain electrochemical methods as PITT, GITT, and charge-discharge measurements, to the nanoscale. Recently, Semenov et al. [16] used a biased tip to image spatial distribution of conductance on V 2 O 5 on top of Li 3 PO 4 electrolyte. Kuriyama et al. [17] measured currents on a bare LiMn 2 O 4 surface in air and measured locally current changes with a slowly increasing electrical field (0.05 V/min). The Li-ions in LiMn 2 O 4 are extracted tip field from the tetrahedral sites, resulting in current increase and topographical changes ascribed to a relaxation of Jahn-Teller instability. The primary limitation of the SPM-based current detection is that the sum of ionic and electronic currents is measured. Given that standard (e.g. Pt or Au coated) SPM tip is blocking electrode and the fact that ionic impedance are typically very large compared to electronic, the information on ionic flows is essentially lost.
(d) AFM based impedance measurements. A number of authors 18,19,20 have demonstrated the use of an AFM as a probe for local electrochemical impedance spectroscopy.
However, the simple comparison of the tip-surface junction and cantilever surface impedances illustrates that direct measurements are possible only for well-defined mesoscopic objects (i.e. single-crystalline conductive grain with insulating grain boundaries), but not local volume of material below the tip.
Overall, the existing strategies for SPM-based probing of electrochemical processes in solids do not allow high-veracity studies due to the limits in spatial resolution, lack of quantitativeness, and multiple mechanisms contributing to measured signal and hence spatiallyresolved contrast. The use of ion-sensitive local electrodes (similar to liquid electrochemical SPM) to directly probe ionic currents in solids is limited by slow diffusion rates and large contact impedances that effectively limit spatial resolution to 10s of microns. Hence, the capability for probing electrochemical functionality in solids has been elusive.
Recently, we have proposed that local electrochemical dynamics in solids can be studied using bias-strain coupling mediated by ionic diffusion. 21,22,23 In this method, the periodically biased conductive SPM tip concentrates electric field in a small volume of material, resulting in redistribution of mobile ions through diffusion and electromigration mechanisms. The associated changes in molar volume and strains results in periodic surface displacement detected by an SPM tip. This approach is further referred to as electrochemical strain microscopy (ESM), and is similar to the well-known Piezoresponse Force Microscopy [24,25,26,27] of ferroelectrics and multiferroics based on converse piezoelectric effect. Here, we analyze the image formation mechanism in ESM for the case of a single-step diffusion process, derive the local strain responses in frequency and time domains, and analyze the sensitivity and resolution limits.

II. Principles and Image Formation mechanism of E-PFM
The fundamental principles of electrochemical PFM are illustrated in Fig. 1 Here, we analyze image formation mechanism for the case of fully reversible Li-ion electrode, corresponding to the case of Li-ion conductive material or process mediated by the liquid droplet in the tip-surface junction (i.e. Fig. 1 b or c for localized electric field). We adopt the method equivalent to the decoupled approximation previously used for ferroelectric materials. 31,32,33,34

II. 1. Mechanical displacement caused by ionic diffusion
The problem of mechanical stresses developing in the electrochemical systems have been recently addressed by a number of authors, including both the cases of macroscopic material and case of spherical particle [35,36,37,38,39]. For the latter, both decoupled [35] and coupled [37][38][39] numerical solutions are available. Importantly, the error induced by decoupling approximation is shown to be proportional to the squire of the molar expansion tensor and generally does not exceed 30% [38], well below the uncertainty of tip-surface contact radius in a typical SPM experiment.
Here, the mechanical stresses that develop during AFM-induced local electrochemical process were simulated for plane-stress conditions by solving the mechanical equilibrium equation, and mathematically describing the elastic contributions to the deformation of each volume element of material, as detailed by several authors [40,41]. For the particular case when the chemical contribution is the dominant active mechanisms for strain, the equations of state (Hooke's law for the chemically active solid) for isotropic elastic media, subjected to the ionic flux relates concentration excess δC, mechanical stress tensor σ ij and elastic strain u ij are the following [42,43]: Here s ijkl is the tensor of elastic compliances, β ij is the Vegard tensor of chemical expansion, describing the lattice deformations under the small changes of composition δC.
In subsequent analysis we note that the typical contact area in SPM experiment is well below micron-scale. The corresponding intrinsic resonance frequencies of material are thus in the GHz range, well above the practically important limits (both in terms of ion dynamic, and SPMbased detection of localized mechanical vibrations). Hence, we consider the general equation of in the quasi-static case that leads to the equation for mechanical displacement vector u i in the bulk of the system (see Appendix A) as: Boundary conditions on the free surface S are where c ijkl is the tensor of elastic stiffness, and n j is the components of the surface normal.
General solution of the problem (2)-(3) is 0   3  2  1  3  2  1   3  3  2  2  1  1  3  2  1   3   ,  ,  ,  ,  ,  ,  ) , The essential condition for the existence of well-defined solution existence is the absence of δC at the infinity, well satisfied in an SPM experiment with local excitation (i.e. Fig. 1 a-c). G is appropriate tensorial Green function (see [44] and Appendix A). Here we approximate the symmetry of elastic properties as isotropic (well justified to 3D compounds such as spinels and olivines), albeit numerical schemes for Eq. (4) can be developed for lower symmetries in straightforward fashion. We further restrict the analysis to the transversally isotropic Vegard Here ν is the Poisson coefficient.

II.2. Constitutive equations with boundary conditions
Electrostatic potential at the tip electrode x 3 = 0 and potential vanishing at infinity or remote bottom electrode. When the current density at the tip electrode x 3 = 0 appeared only due to the Li cations, a local deviation from equilibrium of the surface electrostatic potential, i.e. the overpotential, constitutes the driving force for the reaction to take place. For lithium such reaction is given by equation The rate of the ions transfer from the electrolyte to the active material phase is controlled by the Butler-Volmer relation [45,46]. The transport kinetics model has been analyzed using a phase-field formulation that resulted in a set of coupled Cahn-Hilliard equations [45].
Theoretical description proposed by García et al. [47,48] is based on the free-energy density formulation and includes variational principles.
Generally, Li-ion dynamics can be reduced to the ambipolar diffusion equation [49]: Diffusion coefficient is D. The diffusion constant may depend on the ionic concentration. Here, we ignore this dependence in order to develop analytical description of the process. Furthermore, this approximation is rigorous in the high-frequency regime of ESM, when the changes of ionic concentrations are minute.
Note that electrostatic potential does not contribute to the equation in the case of ambipolar diffusion (see [49] and Appendix B). In decoupling approximation the concentration is calculated ignoring the strain effects, i.e. the term proportional Here is the electrostatic potential distribution at the tip electrode x t , 3 = 0. This boundary conditions reduces to the case of either fixed concentration or fixed ionic flux at phenomenological exchange coefficient λ = 0 or η = 0, correspondingly.
is determined by the reaction rate constant χ and the solubility limit of lithium in the tip electrode C S . Here, F is Faraday's constant, R is the universal gas constant, T is the absolute temperature, α a is anodic empirical constant, α c is cathodic empirical constant, R c is the total contact ohmic resistance, is the current, χ is the reaction rate, C S is the solubility limit of lithium in the electrode, S is the cross-section area.
Using Laplace transformation on time t, and Fourier transformation on transverse coordinates, the solution of problem (12) was found as Here vector , its absolute value

III.1. Mechanical response frequency spectrum
In particular case of periodic bias variation Here the condition should be valid to ensure the stability.
To define the boundary conditions in a generalized ESM experiment, we assume that the potential (and thus ionic flux) spatial distribution V is kept constant inside the circle of radius R ) , , ( 2 1 0 t x x 0 and zero outside. This condition provides an approximate description of the probe tip having a well-defined characteristic size. We further utilize the fact approximate solutions developed here are insensitive to the details of the probe shape. Hence, Fourier image can be taken as ( 22 11 . ( Hereinafter (unless specifically indicated) we consider the simplest case of the isotropic Vegard tensor β . Then allowing for Eq.(11), Eq. (7) for the surface mechanical displacement acquires the form where the polar radius . Below, we explore the specific limiting cases of Eq. (12) that define frequency response and losses in the ESM signal ) , 0 (  Table 1, which specifically indicates the limiting cases for low frequency and high frequency for fluxcontrolled and concentration-driven processes. Hereinafter the characteristic diffusion time The comparison of the exact and approximate expressions from the Table 1 is shown in Substituting the tip potential Fourier image ( , the flux distribution at the surface x 3 = 0 and in the point r = 0 acquires the form The latter approximation is exact in the limit of flux-controlled process, η→0.
The comparison of exact and approximate expressions for flux in the point r = 0 is shown in Fig. 3.
, for the concentration-and flux-controlled processes is shown in Fig. 4. and fixed ionic flux When generating all subsequent plots we used exact integral in Eq. (12). In the case of concentration-driven process, the response is constant for small frequencies ( ), and is on the boundary. This independence of response on frequency (despite the fact that the diffusion length diverges for low frequencies) is related to the finite signal generation volume in ESM (i.e. strain contribution from the parts of material far from the tip-surface junctions is small). In comparison, the total flux is For high frequencies ( ), the response decays as . 1

>> ωτ
In comparison, flux-controlled process ( 0 = η ), the response logarithmically diverges for low frequencies, since the amount of transferred material increases linearly with cycle time.
However, as a consequence of strain transfer effect this divergence is only logarithmic, as opposed to a power-law expected for current detection. For high frequencies, the response decays as , and the corresponding phase angle becomes π/2.
It is instructive to analyze the dynamic response in terms of the hysteretic loop behavior, providing direct link to the observables in the SPM experiment. The complex quantities   As expected, the concentration driven process the hysteresis loop is closed and linear for small frequencies for which the ion-dynamics in the signal generation volume of material (of the order of R 0 ) follows probe bias exactly. For intermediate frequencies, 1 ≈ ωτ , the response is still large, but the ionic response becomes delayed compared to probe bias, resulting in hysteresis loop opening. Finally, for high frequencies strain response is small, and hence the hysteresis loop becomes effectively closed as well. In the flux-limited case, the hysteresis loop area diverges in the low-frequency limit, while in the high-frequency limit it is qualitatively similar to λ=0 case.
Analytically, the loop area S is given by expression ( ) Loops area frequency dependence is shown in Fig. 7. As expected, the hysteresis loop area is maximum for concentration-driven case is maximal for 1 ≈ ωτ . Note that this analysis is strictly valid only for small concentration changes, i.e. within the applicability limit of the linear diffusion equations. Experimentally, the response is expected to saturate once the concentration change becomes comparable to the total capacity of the material.

10
Some of these effects are considered qualitatively in Section V.

III.3. Spatial resolution in ESM
The spatial resolution in ESM can be estimated from the characteristic size of the area affected by the concentration change, i.e. lateral displacement profile on the surface. Simplified analytical expressions for the response radial dependence ) , ( 3 ω r u are listed in the Table 2, which specifically indicates the limiting cases for low frequency and high frequency for fluxcontrolled and concentration-driven processes.

Table 2. Spatial resolution in ESM measurements
The radial dependence of the response is shown in Fig. 8 for different frequencies. For low frequencies the lateral strain is generated over large areas, and far from the tip-surface junction the response decays as ~r R 2 2 0 for concentration driven process and ~r R 2 3 0 for flux driven process. As expected, in both cases the decay length is given by the characteristic tip size, Radius r/R 0 Here, we define the lateral resolution as the half width of the excited spatial region at half maximum (HWHM), normalized on tip size R 0 . Fig. 9 shows

IV. Mechanical response in time domain
The ESM measurements can be performed in the time-domain, in which the response is  7) and (10) as The analytical results were derived for the case of concentration-determined (λ=0) and fluxdetermined (η=0) process (see Appendix C). For the case λ=0 the relaxation of the surface displacement is: where ( ) For a rectangular voltage pulse, , with pulse duration t 0 , approximate expressions were obtained from Eq.(20) as . The response amplitude increases with t 0 as 0 0 3) , 0 ( t t u and decay when the pulse is off according to the power law For the particular case η=0 the displacement signal obeys the law: where I 1 is the modified Bessel function.
For a rectangular voltage pulse, , with pulse duration t 0 , the response increase with the bias pulse duration increase is logarithmic        Based on these considerations, we distinguish three characteristic frequency regimes of the ESM imaging and spectroscopy: I. Static regime, in which the driving frequency is below the inverse diffusion time and well below cantilever resonant frequency.

II. Low frequency regime, in which the driving frequency is larger then the inverse diffusion
time, but is well-below the cantilever resonance frequency.

III. Dynamic (or high-frequency) regime, in which the driving frequency is well above the
inverse diffusion time and is comparable to the resonant frequency of the cantilever.
Below, we briefly discuss the expected aspects of response in these limits.

V.1.1. Response in static and low-frequency regimes
The relationship between the amplitude of surface vibrations and measured cantilever deflection is given by the transfer function. The latter is dominated by the cantilever dynamics in the Hz-Mhz ranges, while at higher frequencies the photodetector and electronic behavior becomes significant. Here, we discuss cantilever dynamics effects only, since the other effects are a part of the measurement system and can be established and calibrated independently.
In the static and low-frequency regimes, the transfer function of the SPM cantilever is (ideally) linear. Practically, the response function exhibits significant dispersion below 10 kHz due to multiple (typically narrow) tip-holder resonances. However, these are generally unrelated to the variation of mechanical properties of the tip-surface junction, and hence do not result in topographic cross-talk (since transfer function at each frequency can be calibrated). Thus, the frequency dependence of the ESM response is effectively controlled by the Li-ion dynamics.
In the static regime, the Li-diffusion length significantly exceeds the signal detection volume of the SPM probe, i.e. the bias-induced changes occur on the length scale larger then SPM probe size. In the limit of dc bias ramp, electrochemical transformation can occur in the full volume of the material, with the possible onset of irreversible dynamics and limits on the number of charge-discharge cycles. The advantage of this regime is that the surface displacement can be measured quantitatively (albeit note that due to uncertainty in the tip size and relationship between probe potential and Li-concentration change, this information may be insufficient for characterization of electrochemical functionality).
In the low frequency regime, the Li-diffusion length is comparable to the probe size, allowing for the optimal compromise between the potential to study process in detail (i.e. control the degree of lithiation by changing the voltage range) and bandwidth of measurements (i.e. the spatially resolved measurements are possible). Furthermore, the small area affected by electrochemical process increases the degree of reversibility of the process. The transfer function of the SPM in this frequency interval is ideally constant, i.e. measured response will be defined only by Li-ion dynamics. Practically, the transfer function has significant dispersion (esp. below 10 kHz) due to the tip-holder resonances. However, these are position independent, and result only in the change of the absolute value of response signal and can be calibrated. Fig. 13 shows the frequency dependence of the electromechanical response real, imaginary part, absolute value and phase for several Li-containing materials Li x CoO 2 , LiMn 2 O 4 and LiC 6 . In all cases, we assume that that driving voltage is small, so that the change in Li concentration is small, the diffusion coefficients are constant and linear theory is applicable.

V.1.2 Imaging in the high frequency regime.
In the high frequency regime, the excitation frequency is comparable to the resonance frequency of the cantilever (0.03 -1 MHz) and well above the inverse diffusion time of lithium even on the length-scales of 1 -30 nm (i.e. characteristic probe size of SPM tip).
Correspondingly, even high driving forces will result only in minute changes in Li concentration, as analyzed in Section III.2. Note that the applicability of continual models on the length scales below Li diffusion length presents an interesting problem not addressed in this manuscript. MHz range. Note that while the intrinsic materials response rapidly decays above ~ 1 Hz, this decay is relative slow power of frequency (~1/f). Correspondingly, the presence of sharp (Q ~100) cantilever resonances results in the strong amplification of the ESM signal. Directly at the resonance, the high-frequency ESM signal is comparable to that in the static regime.
The advantageous features of high-frequency ESM can be illustrated once the sources of noise in the SPM detection system are taken into account. 67 For sufficiently high frequencies, the ultimate noise level of SPM is given by the thermomechanical white noise (practically, most commercial AFMs with appropriate vibrational isolation operate within 1-1.5 order of magnitude from this limit). However, for low frequencies the noise is 1/f character, in which case averaging the signal over large times does not allow more precise measurements. The 1/f noise corner is typically in the 1-10 kHz range. To estimate the noise effect on ESM signal, plotted in Fig. 16d is the signal/noise ratio, with the noise being approximated as N = (f 0 /f+1). Notice that while 1/f noise strongly affects the static and low-frequency ESM measurements, the high-frequency ESM allows imaging with high signal-noise ratios despite that only small changes of Li concentration can be induced.  Table 3.
Note that the use of high-frequency regimes necessitates the use of the resonant-frequency tracking methods, since cantilever resonance frequency is strongly dependent on surface topography through the tip-surface spring constant. 68,69 Similarly to PFM, the use of standard phase-locked loop circuitry for ESM is unlikely to succeed, since the phase of the local electromechanically response is dependent on materials functionality, and is hence not constant (unlike SPMs with acoustic excitation, for which this condition is semi-quantitatively satisfied).
Correspondingly, the use of amplitude-based resonance tracking 70 or band excitation method is required. 71

V.2. The role of electrochemical reactions and limitations
The linear theory developed in Section II-IV describes the signal formation mechanism in ESM in the linear limit, in which the change in Li concentration is relatively small. Here, we qualitatively consider the effects of finite Li concentration and electrochemical reaction on responses.
The saturation effect is related to the limit of Li concentration in real materials. In this case, the concentration cannot exceed the intercalation limit, and change in Li concentration induced by progressively high bias amplitude will saturate, as shown in Fig. 16 a. The expected shape of the hysteresis loop in this case is shown in Fig. 16 b.

Bias
Bias δC(Li) The second factor potentially affecting the ESM signal formation mechanism is the electrochemical reaction in the tip-surface junction. In this case, the Li insertion does not start until the critical overpotential is achieved. We note that for most processes the process is reaction limited for small overpotentials, and diffusion limited by large overpotentials due to exponential Tafel-like dependence of reaction rate on the overpotential. The width of transition is typically of the order of ~0.1-0.3 V. 72 Correspondingly, the response as a function of bias and hysteresis loops can be expected to behave as shown in Fig. 17. Note that the fine shape of hysteresis loop at sufficient resolution may provide the information on the phase transition sequence in the material, providing electromechanical analog of cyclic voltammetry.

Bias
Bias δC(Li) Expected shape of the ESM hysteresis loops.

V.3. Sensitivity limits for ESM
One of the interesting issues in ESM is the ultimate resolution limit of the ESM. The simple estimate illustrates that vertical displacement of the material surface due to intercalation can be detected down to the nanometer scales, as a consequence of extremely high vertical sensitivity of AFM (note that even in the static regime the work by Shao-Horn group has demonstrated direct measurements of step height in LiCoO 2 as a function of intercalation degree).
Here, we estimate the resolution limit for the ESM, following the detailed analysis for PFM reported in Ref. 73 .
Briefly, in the absence of electrostatic forces the detection limit in PFM is set by the signal transfer from material to the tip, determined by the cantilever spring constant, , and tipsurface spring constant, . The contact stiffness , and in the simple Hertzian model can be found as , or where h is the indentation depth, is the tip radius of curvature, and P is load. Note that the key aspect of the electromechanical detection principle in ESM, as opposed to the current detection in conventional electrochemical characterization techniques is that the electronic transport does not directly contribute to strain. Hence, electrochemically induced strains provide the information on ion dynamics even in the presence of large electronic currents, and hence are provide relevant information even at high frequencies and small contact areas.

VI. Summary
The image formation mechanism in the electrochemical strain microscopy is analyzed. The

Appendix A. Elastic problem solution
For the quasi-static case, the equations of state δ for isotropic media can be rewritten as: Here δC is the concentration redistribution. Here comma separated subscript means the derivative on corresponding coordinate, e.g.
Using the definition of strain components Using the general equation of equilibrium, written in terms of displacement vector u is where vector denotes the point where we look for the solution, x ξ is the point were point force ), one can see that the considered case of equations (A.6) and Eq.(A.7) can be considered as the usual problem of the elasticity theory with bulk force density and surface force density, acting in the elastic media (or/and on its surface) .
Mechanical displacement (7) is: (A.14) Integration on ξ 3 leads to Here, R is the Debye screening radius, ε d ( ) 0 is the dielectric constant, ε is relative dielectric permittivity, e is the absolute value of the electron charge, h is the cell thickness. Electric charge and electric current densities of the anions and cations are i.e. the overpotential U , constitutes the driving force for the reaction to take place. For lithium such reaction is given by equation Li + + e − ↔ Li . The rate of the ions transfer from the electrolyte to the active material phase is controlled by the Butler-Volmer relation [45]. Thus the total charge flux density [46]: Here, F is Faraday's constant, R is the universal gas constant, T is the absolute temperature, α a is anodic empirical constant, α c is cathodic empirical constant, R c is the total contact ohmic resistance, is the macroscopic electron current (S is the battery cross-section area), is the prefactor (χ is the reaction rate constant, C The boundary conditions for ionic fluxes can be analyzed as following. When the current density at the tip electrode  The analytical results were derived for the case λ=0 and η=0 for a rectangle-like temporal dependence of the voltage pulse, , with pulse duration t 0 .