### 1. Introduction

*Q*

*, and rate capability,*

_{0}*Q*(

*t*), along with the number of discharge-charge cycles, are important target parameters of the battery use. Various analytical methods are used to describe

*Q*(

*t*): Peukert’s rules [1], Lebenov’s [2], hyperbolic functions [3], taking into account the structural and electrochemical parameters [4–10]. These are important specialized tools for improving the electrode, the component and the battery technology in general [11–13].

_{4}cathodes, technologies (shape engineering) are being developed, aimed to reduce the size of crystallites along the direction of Li diffusion, the [010] axis, up to giving them plate- and bar-like shapes [14–16]. In these extreme cases, scanning (SEM) or transmission (TEM) electron microscopy is sufficient to determine the anisotropy of particles. However, studies of industrial samples of cathode powders show that the most demanded is the smaller anisotropy of their particle sizes, no more than 5 times. Here, complementary X-ray diffraction (XRD) and TEM measurements are possible [17].

Statistical parameters of various particle shapes were determined: cuboid, ellipsoid, and possible others. Their relationships were established for XRD and TEM measurements.

The parameters of 3D function including the use of a correlation matrix were determined.

An analytical model of rate capability

*Q*(*t*) was developed using the 3D function of particle sizes and correlations between them.3D functions were used to describe the experimental dependence

*Q*(*t*), with the determination of the electrical relaxation time (*τ*_{el}) and Li diffusion coefficient (*D*).The obtained values of

*D*were compared with other methods, in particular, cyclic voltammetry according to the Randles-Sevcik equation.Examples of crystallite shape engineering were given to optimize cathode rate capability and increase their capacity.

### 2. Experimental

### 2.1 XRD crystallite and TEM particle sizes: how to connect them

*L̄*

_{i}_{XRD}are a result of the averaging of the column length over the crystallite volume [18,19]. One of these, with column

*M*along the [010] direction, is shown in Fig. 1.

*L̄*

_{i}_{TEM}values can be obtained from the decomposition of two distributions of the longitudinal and transverse dimensions of TEM particle images into their three Lognormal distributions

*L*

*along three crystallographic axes [17]. As a result, the average linear particle sizes*

_{i}*L̄*

*and their variance*

_{i}

*σ**, used to describe electrochemical measurements, as well as their volume-averaged sizes*

_{i}*L̄*

_{i}_{TEM}(i.e., with their volume as a weighting factor), can be obtained, which are required to match the results of TEM and XRD measurements. For a LiFePO

_{4}lattice with an orthorhombic olivine-like structure and orthogonal axes between these dimensions, the following relationship can be used

*i*= 1, according to Fig. 1, corresponds to the direction [010], etc., the last equality is applicable for the Lognormal size distribution with mean values

*L̄*

*and variances*

_{i}

*σ**, with K = 1 for a rectangular parallelepiped (cuboid) and K = 3/4 for spheres. The latter is also applicable to ellipsoids. Since this is not an obvious fact in the available sources, the corresponding conclusions are included in Supporting Information. The case of non-orthogonal lattices is described in [20]. It is also important that the above procedure of averaging along the columns will also be used for a 1D diffusion of Li, in an analysis of capacity rate of the crystallite.*

_{i}*L̄*

_{i}_{XRD}and averaged over the volume

*L̄*

_{i}_{TEM}, which for log-normal distributions can be related to the average sizes of real measurements

*L̄*

*. The first sizes can be obtained from XRD measurements, the second and third are obtained from TEM measurements using the first, which will be discussed in the next section. This requires preliminary information about the particle shape, which is established only with the use of microscopic measurements. Ellipsoid and cuboid shapes, in fact, include all possible cases of particles (with a closed 0-th kind surface) because the available precision of XRD studies makes it impossible to reliably detail more complex particle shapes, for example in the form of superellipsoids (see Supporting Information).*

_{i}### 2.2 XRD and TEM measurements of LiFePO_{4} powder to get 3D Lognormal crystallite distributions

_{4}Phostech Lithium P2 as an example [21]. It turned out, and as can be seen from Fig. 1, there are noticeable differences in the shape of large and small particles on SEM images. The study of such mixed powders is a more difficult task, but relevant in connection with works in which a denser mass of electrodes is achieved by mixing particles of different sizes [16,22–24]. In this case, smaller particles are located in the voids between large ones.

_{6}powder. The values of

*L̄*

_{V}_{[}

_{hkl}_{]}and of the errors, reported in Table 1 (in parentheses), characterize the reproducibility.

*L*

*,*

_{s}*L*

*into components and determine the required values of*

_{b}*L̄*

*,*

_{i}

*σ**, we use the values of*

_{i}*L̄*

_{V}_{[}

_{hkl}_{]}and the relationship between these:

*L̄*

_{V}_{[010]}<

*L̄*

_{V}_{[100]}<

*L̄*

_{V}_{[001]}. We assume that the probability of a crystal face to be aligned with the object plane of the microscope is proportional to its area. For example, the normalized probability for the (100) face is given by

*L̄*

*consists of two parts: one with the size*

_{b}*L̄*

_{V}_{[001]}and the probability

*P*

_{(100)}+

*P*

_{(010)}, and second with the size

*L̄*

_{V}_{[100]}and probability

*P*

_{(001)}. Similarly,

*L̄*

*consists of two parts: with the size*

_{s}*L̄*

_{V}_{[001]}and the probability

*P*

_{(001)}+

*P*

_{(010)}and with the size

*L̄*

_{V}_{[010]}and probability

*P*

_{(100)}. The rationale for making these assumptions is contained in [17]. Further, assuming that large particles are blocky, we describe the histograms of

*L̄*

*and*

_{b}*L̄*

*by Lognormal distributions with mean*

_{s}*L̄*

_{V}_{[001]},

*L̄*

_{V}_{[010]}and

*L̄*

_{V}_{[100]},

*L̄*

_{V}_{[010]}, respectively, we obtain the parameters of the marginal Lognormal distributions

*L̄*

*,*

_{i}

*σ**(Table 1). To these parameters it is necessary to add the correlation coefficients*

_{i}*r*

*, which have the following form, for example, for two arbitrary data arrays*

_{ik}*r*

_{12}

*f̄*, obtained by discretizing the function

*f*(

*L̄*) (4). The results of these inverse calculations are shown in Fig. 4. Comparing the obtained parameters with the data in Table 1, we can see that the fitting and discretization errors are less than 0.3%.

*f*(

*L̄*), described by equation (4) and including the correlation matrix with the parameters specified in Table 1 and in Fig. 4, were determined. Details of the program used are described in Supporting Information.

### 2.3 Electrochemical measurements of LiFePO_{4} cathodes

^{2}aluminum plate, 0.4 mm thick, after which it was dried at 120°C in air for 12 hours. To reduce the errors of galvanostatic measurements of rate capability

*Q*(

*t*) at small

*t*, the cathode thickness was minimal, about 8 μm (see Fig. 5).

_{4}electrodes. Ti oxidation is also not observed, which may be due to the presence of a thin passivation layer. Therefore, to improve contact of Li metal counter electrode with the current collector, a perforated Ti substrate with a deposited Li layer was used.

_{4}composite on aluminum plate) and the counter electrode (in our case the lithium metal on a perforated Ti substrate) [31]. The measuring circuit includes a separate third electrode. The material of this reference electrode in our case is also lithium.

^{2}. Fig. 6 shows the results of galvanostatic measurements.

_{4}powders was determined using an ASAP-2020 instrument from Micromeritics by adsorption-structural analysis. Degassing of the sample is carried out according to a given program, evacuation at 300°C for 10 hours, then measurement of the nitrogen adsorption and desorption isotherm at 77 K and calculation of the specific surface area of the sample using BET methods. A value of 23.1 m

^{2}/g was obtained.

### 3. Results and Discussion

### 3.1 Analytical model of rate capability *Q* (*t*)

*Q*(

*t*) on the particle shape of the initial cathode powders, on the function

*f*(

*L̄*) and its statistical parameters, we develop an analytical model using the following simplifications:

*M·dx*

_{2}*·dx*

*column (see Fig. 1). By analogy with [8], to describe the rate capability*

_{3}*q*

*(*

_{se}*t*) of a separate particle column shown in Fig. 1, we will use the following equation

*t*is the time constant. In [8] the exponent values

*n*are defined for batteries and supercapacitors, 0.5 and 1.0, respectively.

*q*

_{su}(

*t*) is not realized with probability

*P*in time

*t*. That is, the event is described by the equation

*P*approaches 0, and

*q*(t) asymptotically approaches the limit value

*q*

*. As t → 0, it approaches the dependence*

_{M}*n*is equal to the slope tangent of the dependence

*q*(

*t*) in double logarithmic coordinates at small

*t*.

*τ**and capacitor described by the electrical time*

_{d}

*τ**, both normalized to the discharge time*

_{el}*t*. An analysis of equations (5–8) makes it possible to draw the following 3 conclusions:

*t*decreases, a large slope in double logarithmic coordinates, about

*n*= −1.5 (see Supporting Information), should be observed, which was not observed by us and rarely earlier in experiments [8].

*n*= −1.5 is possibly due to experimental errors. To reduce the probability of this strong dependence in our experiments, the cathode thicknesses were minimally possible, and the large volumes of the electrochemical cell reduced cathode overheating under study at low

*t*(high currents).

*t*decreases, and ends with −1.5 at extremely small

*t*.

*D*on the coordinate along the column

*M*. The real mechanism of FePO

_{4}/LiFe-PO

_{4}phase boundary motion [34–36] can be taken into account at least in a phenomenological way. In this study, the dependence

*D*= Const. and which describes the process of diffusion (desorption) from a finite size

*M*with associated boundary conditions [37], as well as the process of homogenization of the chemical segregation [38].

*Q*(

*t*) using the 3 D function

*f*(

*L̄*), we assume that it consists of the sum of the crystallite rate capabilities in the form

*q*

*(*

_{cr}*t*,

*L*

_{1},

*L*

_{2},

*L*

_{3}), which can be obtained by integrating (9) over the volume of a crystallite with dimensions

*L*

_{1},

*L*

_{2},

*L*

_{3}

*x*

*coordinate.*

_{3}*Q*(

*t*) can be obtained by multiplying the elements of the N-bit 3D matrix

*f̄*and the matrix of their rate capability

*Q*(

*t*) is developed, described by equations (9–11) and taking into account the anisotropic 3D distribution function of powder particles.

### 3.2 Determination of the experimental parameters of rate capability *Q*(*t*): *D* and *t*_{el}

_{el}

*Q*(

*t*) are, in fact, phenomenological [39]. Especially the part described by equations (6–8) and associated with the use of probability theory. However, this developed analytical model does not yet correspond to the above-described task of studying mixed powders [16,22–24], including the LiFePO

_{4}powder chosen by us. Therefore, to describe the mixture of cuboid and ellipsoidal particles shown in Fig. 2, we make the following phenomenological assumptions:

*Q*(

*t*

*,*

_{nr}*D*,

**), obtained using equation (11), with its experimental value**

*τ**Q*

*(*

_{nr}*t*

*) at the discharge*

_{nr}*t*

*. That is, we consider*

_{nr}*Q*(

*t*,

*D*,

**) is piecewise-continuous at the point**

*τ**t*

*. At*

_{nr}*t*>

*t*

*, it is determined by large particles and the parameters of the cuboid distribution function given in Table 1 can be used, and at*

_{nr}*t*<

*t*

*, it is determined by small ellipsoidal particles.*

_{nr}*q*

*(*

_{cr}*t*,

*L*

_{1},

*L*

_{2},

*L*

_{3}) dependent on crystallite sizes and on the experimental parameter

*t*and two adjustable parameters,

*D*and

*τ**. Fig. 9 shows the results of fitting the dependence of the calculated sum (12) on the discharge time with reference to the experimental value of the rate capability at*

_{el}*t*

*= 1800 s (2 C), which is intermediate between the dominant contributions. It can be seen that if*

_{nr}

*τ*

_{el}*=*90 s is used, then a good result of the combined cuboid-ellipsoid model is achieved. In this case, the value of the second adjustable parameter

*D*is greater, by about 7 times, for large cuboid particles,

*D*= 2.1 nm

^{2}/s, in comparison with small ellipsoid particles,

*D*= 0.3 nm

^{2}/s. This trend does not correspond to that described in [42,43] when comparing micro- and nanosized particles and is related to the number of particle defects. However, the presence of local regions of a more stoichiometric composition in the melt during the growth of particles and, as a consequence, a reduced concentration of defects in the places of nucleation and growth of large particles is quite real.

^{2}/s) we have chosen is more preferable, since it allows us to more clearly correlate

*D*value with the particle sizes. For example, at

*D*= 1 nm

^{2}/s, using the expression for the relaxation time for one-dimensional diffusion

*D*is greater, by about 7 times, for large cuboid particles,

*D*= 2.1 nm

^{2}/s, in comparison with small ellipsoid particles,

*D*= 0.3 nm

^{2}/s. The value of

*D*strongly depends on the particle discharge degree and can vary by 4 orders of magnitude [44], from 10

^{−11}to 10

^{−15}sm

^{2}/s (10

^{3}–10

^{−1}nm

^{2}/s). The value of obtained

*D*is some averaging and is within this range. In this regard, it is important to compare them with other ways to get value of

*D*, obviously also averaged, which, in particular, can be obtained as a result of galvanostatic measurements and using the Randles-Sevcik equation.

### 3.3 Comparison with the results of using the Randles-Sevcik equation

*i*

*- the peak current in amperes, constant 2.69 × 10*

_{p}^{5}with units of C/mol

*n*

^{1/2},

*C*

_{Li}- the initial concentration of Li in LiFePO

_{4}in mol cm

^{−3}, taken as the total amount of Li in a particle before delithiation and is 0.0228 mol cm

^{−3},

*n*- potential sweep rate in V/ s,

*S*

*- the electrode area in cm*

_{el}^{2}, defined by us as the sum of the areas of the (010) planes of all cathode particles,

*D*- diffusion coefficient in cm

^{2}/s [45,46]. Since the access of electrons to the particle must be ensured for the electrochemical reaction to take place, Li can escape from it only in one direction. That is why we assume that the particle interaction area is equal to the particle projection area on the (010) plane.

*S*

*. Using the weight of the powder in the cathode, 0.0015(3) g, and its density, 3.6 g/cm*

_{el}^{3}, we can find the total volume of all cathode particles,

*V*

*= 4.2 × 10*

_{exp}^{−4}cm

^{3}. On the other hand, similarly to (12), we find the average particle volume

*f̄*is normalized to 1 in this case,

*v*- matrix of particle volumes, each element of which has volume

*V*

*by*

_{exp}*N*

*in the cathode and the value*

_{ct}*S*

*by calculating an equation similar to (14). Instead of*

_{el}*v̄*, it uses the matrix

*s̄*, particle area projections onto the (010) plane, and it has the form

*S*

*for cuboids and ellipsoids equal to 90 cm*

_{el}^{2}and 50 cm

^{2}(specific surface 0.6 m

^{2}/g and 0.33 m

^{2}/g), respectively. These specific surface values are more than 1 order less than the BET value and almost 2 orders of magnitude greater than the area of the cathode electrode. Using Fig. 7 data, equation (13) and the values of

*S*

*, we obtain diffusion coefficients of 0.9 nm*

_{el}^{2}/s and 0.5 nm

^{2}/s in cuboid and ellipsoidal particles, respectively.

*D*have obvious sources of error. But the developed method has a clearly lower value of systematic errors due to the ability to really take into account the shape and statistics of crystallites, and it is also useful for improving the accuracy of the Randles-Sevcik equation.

### 3.4 Shape engineering of crystallites to optimize rate capability and increase the cathode capacity

_{4}cathodes, efforts are mainly directed to the development of technology (shape engineering) to reduce the size of crystallites along the direction of Li diffusion, along the [010] axis. Some examples of LiFePO

_{4}developments and studies of plate-like particles are shown in Table 2 [47–51], in which for [51] the results of 5 types of samples with different aspect ratio c/a (2.4–6.9) are briefly noted. The approach developed in our work makes it possible to quantitatively describe these and other technological experiments. First of all, let’s illustrate in Fig. 10 how the Lognormal distribution functions look like in double logarithmic coordinates. The statistical parameters hereinafter are close to the values obtained above in magnitude, and the dimensional parameters - are close to those occurring in Table 2.

*τ**to increase the rate capability at small times, which is not directly solved by the technology of particles, but is necessary by improving the quality of their coating. Examples of this dependence will be given below, but in calculations it is sometimes assumed that*

_{el}

*τ**»*

_{d}

*τ**. We mean that*

_{el}

*τ**is of the order of magnitude about minutes.*

_{d}*Q*

_{1C}= 135 mAh/g and

*Q*

_{0.01C}= 170 mAh/g, respectively. (Sometimes other values will be used to better illustrate the

*Q*(

*t*) dependencies.) A similar principle of rate capability rationing, in other words, linking theoretical curves to experimental results, has already been used in equation (12) above.

*t*= 3600 s). It can be seen that an increase in rate capability at big times leads to its decrease at small times and vice versa. At

*t*→ ∞, rationing integration normalizations (11) and crystallite size distribution function (4), the quantity

*Q*(

*t*,

*Dif*,

**), defined by equation (5), will be equal to the theoretical limit capacity LiFePO**

*τ*_{4}cathode

*Q*

_{0}= 170 mAh/g. This means that in some cases, in particular, in Fig. 11 for the parameters of the curve with average values of cuboid crystallites

*L̄*

_{1}= 300 nm, a region of times t >

*t*

_{crt}arises that is greater than some critical value

*t*

_{crt}, for which

*Q > Q*

*. This means that for these values of the dependence parameters, its inflection will occur at the point*

_{0}*t*

_{crt}. If, for other parameters, the limiting values

*Q < Q*

*, it can be concluded that there are non-optimal stages in the powder technology with the presence of inactive impurities in them, or a slow approach of*

_{0}*Q*to its asymptotic value

*Q*

*.*

_{0}#### 3.4.2 Task specification

*τ*

_{el}

*τ**= 0.09 s and three orders of magnitude larger*

_{el}

*τ**, respectively, for guaranteed fulfillment of the inequality*

_{el}

*τ**»*

_{d}

*τ**in the first variant. The ranges of coordinates are chosen so that the differences in the dependences of the curves on the values of the calculation parameters would be visually observable. In particular, it is seen that the slopes tangents of the falling dependence parts in Fig. 12a and 13a are close to −0.5 and −1.0, respectively. The inset of Fig. 12a highlights the results shown in Table 2, obtained in [47–51] and which were normalized to the value of rate capability at 0.1 C. The latter was done to equalize their possibly different quality of particle coverage.*

_{el}*Q*

_{2C}= 139 mAh/g, then opposite parameter changes are needed to increase rate capability at big times. In particular, cuboid-shaped crystallite powders with large dispersion of their size distributions are preferable and the magnitude of the electrical relaxation time is insignificant.

#### 3.4.3 Influence of covariances

*t*= 360 s.

#### 3.4.4 Influence of size along the [010] axis

_{4}cathodes was achieved, and in [50], its 6-fold increase was achieved with a decrease in the particle size along the [010] direction of diffusion of lithium ions. Fig. 15 shows calculations using the distributions shown in Fig. 10. It can be seen that such a multiplicity can be achieved and is strongly dependent on the value of the diffusion coefficient. To a much lesser extent, the dependence on the shape of particles is manifested in comparison with the experimental dependence Fig. 9. This is due to the following calculation feature. In Fig. 15, the same values of statistical parameters are used, in particular, the average particle sizes along 3 crystallographic axes. The dimensions of Fig. 9 are different, shown in Table 1 and obtained as a result of the described features of processing real XRD measurements.

*L̄*

_{1}= 15 nm and = 450 nm with the parameters of the correlation matrix close to the experimental are shown in Fig. 15. This dependence turns out to be significant for small values of

*D*and cuboid crystallite shapes. The Supporting Information contains additional calculations on the influence of the

*D*,

*τ**and particle face area (010) on the rate capability.*

_{el}### 4. Conclusions

_{4}powder: between the averaged over the crystallite columns

*L̄*

*, obtained from XRD measurements, and averaged over the volume*

_{i XRD}*L̄*

*, obtained from TEM measurements, which for Lognarmel distributions can be related to the average sizes of real measurements*

_{i TEM}*L̄*

*. In this way, a full 3D Lognormal distribut ion function can be obtained, including its correlation matrix of the distribution between the crystallite sizes in their three crystallographic directions.*

_{i}_{4}powder was chosen for testing the developed model, consisting of big cuboid and small ellipsoid crystallites. Bearing in mind that the study of such mixed powders, although a difficult task, is very important in connection with a promising technological direction in which a denser electrode mass is achieved by mixing particles of various sizes. In this case, smaller particles are located in the voids between large ones. To reduce the errors of galvanostatic measurements of rate capability

*Q*(

*t*) at small

*t*, the cathode thickness was minimal, about 8 μm.

*Q*(

*t*) is developed, described by equations (9–11) and taking into account the anisotropic 3D distribution function of powder particles. The main model simplification is that electrochemical processes are described as probabilistic events, in which the rate capability of the crystallite column

*q*

*(*

_{su}*t*) is an event and is realized with probability (1–

*P*) dependent on time

*t*. That is, the event is described by the equation

*q*(

*t*) =

*q*

*(1 –*

_{M}*P*), and at

*t*→ ∞ the probability

*P*→ 0, in particular.

*Q*(

*t*) (

*D*and

*τ**), the following important simplifications have been made. We assume that*

_{el}*Q*(

*t*,

*D*,

**) is piecewise-continuous with some intermediate point**

*τ**t*

*. At*

_{nr}*t*>

*t*

*, it is determined by large particles and the parameters of the cuboid distribution function given in Table 1 can be used, and at*

_{nr}*t*>

*t*

*, it is determined by small ellipsoidal particles.*

_{nr}*D*adjusted to the experimental dependence is greater, by about 7 times, for large cuboid particles,

*D*= 2.1 nm

^{2}/s, in comparison with small ellipsoid particles,

*D*= 0.3 nm

^{2}/s. These values are within the limits usually obtained earlier, but then they were compared with the values obtained as a result of galvanostatic measurements of the test sample and using the Randles-Sevcik equation. On this path, the area of the electrodes

*S*

*was determined, defined as the sum of the areas of the (010) planes of all cathode particles. So, we obtain diffusion coefficients of 0.9 nm*

_{el}^{2}/s and 0.5 nm

^{2}/s in cuboid and ellipsoidal particles, respectively. Inconsistencies between those results with the results of the developed method can be considered quite acceptable.

Achieving a large rate capability (and capacity) at big times, or increasing the rate capability at small times.

Decreasing electrical relaxation time

*τ*to increase the rate capability at small times, which is not directly solved by the technology of particles, but is necessary by improving the quality of their coating._{el}