Method for calculating diffusion coefficient of electrode material

Abstract

The invention discloses a method for calculating the diffusion coefficient of an electrode material, which comprises the following steps: establishing a Fick diffusion law-based constant current gap titration (GITT) theoretical model, and describing the concentration distribution of ions in the electrode material; solving a partial differential equation in the GITT theoretical model through Laplace transformation to obtain an analytical expression of ion concentration; obtaining a diffusion coefficient calculation formula suitable for a full time range; verifying the accuracy of the diffusion coefficient calculation formula through numerical simulation; and a concentration-based diffusion coefficient expression is converted into a voltage-based diffusion coefficient expression, so that processing and analysis of experimental data are facilitated. The method for calculating the diffusion coefficient of the electrode material, provided by the invention, is based on a constant current gap titration technology, is suitable for full-range data, extracts experimental data, calculates a voltage difference, selects an optimal formula and generates a visual image through an automatic program, and remarkably improves the calculation precision of the diffusion coefficient and the processing efficiency of the experimental data.

Description

Technical Field

[0001] The present invention relates to the field of electrochemical testing, and relates to a method for calculating the diffusion coefficient of an electrode material, and particularly relates to a method for calculating the diffusion coefficient of an electrode material based on the galvanostatic intermittent titration technique (GITT). Background Art

[0002] The galvanostatic intermittent titration technique (GITT) is an electrochemical testing method widely used in the study of the kinetic properties of electrode materials. By applying a constant current pulse and observing the change of the electrode potential with time, GITT can be used to measure the diffusion coefficient of active substances such as ions in the electrode material. However, the existing GITT diffusion coefficient calculation methods have significant limiting conditions and logical errors in use, which limit their accuracy and efficiency in practical applications.

[0003] The existing GITT diffusion coefficient calculation formula is usually based on the limiting condition (t << L 2 / D), and the following problems are brought by this limiting condition in practical applications:

[0004] 1. Logical error of the limiting condition: The existing method requires users to judge the condition (t << L 2 / D) before using the formula, and the judgment condition itself depends on the value of the diffusion coefficient D. However, the diffusion coefficient D is exactly the unknown quantity that needs to be calculated by the formula. This logical contradiction makes it difficult for users to ignore the limitation of the judgment condition in actual operation, thus affecting the reliability of the calculation results.

[0005] 2. The range of the limiting condition is fuzzy and the applicable range is limited: The limiting condition (t << L 2 / D) existing in the existing formula is very fuzzy, and the condition far less than has no clear definition; in addition, there is no clear solution for how to calculate the diffusion coefficient of GITT data outside this range. Summary of the Invention

[0006] In view of the above defects of the prior art, the technical problem to be solved by the present invention is the problems such as the fuzzy limiting conditions and limited application range of the existing galvanostatic intermittent titration technique (GITT). The present invention provides a method for calculating the diffusion coefficient of an electrode material, which is based on the galvanostatic intermittent titration technique, can be applicable to full-range data, and extracts experimental data, calculates the voltage difference, selects the optimal formula and generates a visualization image through an automated program, significantly improving the accuracy of the diffusion coefficient calculation and the efficiency of experimental data processing.

[0007] To achieve the above object, the present invention provides a method for calculating the diffusion coefficient of an electrode material, including the following steps:

[0008] S1. Establish a GITT theoretical model based on Fick's diffusion law to describe the concentration distribution of ions in the electrode material;

[0009] S2. Solve the theoretical model in S1 through Laplace transform to obtain the analytical expression between the diffusion coefficient D and the concentration C;

[0010] S3. Based on the analytical expression of the ion concentration, obtain the calculation formula D(C) of the diffusion coefficient applicable to the full time range;

[0011] S4. Verify the accuracy of the calculation formula of the diffusion coefficient of the concentration through numerical simulation;

[0012] S5. Convert the calculation formula of the diffusion coefficient based on the concentration into the calculation formula of the diffusion coefficient based on the voltage to facilitate the processing and analysis of experimental data.

[0013] Furthermore, use partial differential equations to describe the GITT theoretical model based on Fick's diffusion law,

[0014]

[0015] where C(x, t) represents the ion concentration at position x and time t, D is the diffusion coefficient; the initial condition C(x, 0) = C0 sets the uniform concentration at the initial moment of the diffusion region to C0; in terms of the boundary conditions, the no-flux condition at x = 0 indicates that this position is a symmetric boundary and no ion diffusion exchange occurs; the condition at the other end x = L reflects the constant current charge-discharge condition, where I0 is the applied current, n is the number of electron transfers, F is the Faraday constant, and A0 is the electrode surface area; this condition shows that at x = L, due to the current driving ion diffusion, there is a concentration gradient at this boundary; this set of equations is used to solve the concentration distribution of ions in the GITT test and further calculate the diffusion coefficient D and related kinetic parameters.

[0016] Furthermore, solve the theoretical model in S1 through Laplace transform to obtain the analytical expression between the diffusion coefficient D and the concentration C. The analytical expression between the diffusion coefficient D and the concentration C is

[0017]

[0018] where δ = I0 / nFA0D, I0 is the applied current, n is the number of electron transfers, F is the Faraday constant, A0 is the electrode surface area, D is the diffusion coefficient, L is the electrode thickness, C(L, t) represents the ion concentration at the electrode surface x = L and time t, and C0 is the ion concentration at the initial moment.

[0019] Further, the diffusion coefficient calculation formula includes the diffusion coefficient calculation formula obtained by using the traditional inverse Laplace transform method and the diffusion coefficient calculation formula obtained by using the improved inverse Laplace transform method. Further, the diffusion coefficient calculation formula for voltage obtained by using the traditional inverse Laplace transform method has the following expression

[0020] D = 4L 2 / (πt)·(ΔE s / ΔE t ) 2

[0021] where, ΔE t is the transient voltage change, ΔE s is the steady-state voltage change, L is the electrode thickness, and t is the single-step current application time.

[0022] Further, the diffusion coefficient calculation formula for voltage derived by using the improved inverse Laplace transform method has the following expression

[0023] D = -L 2 / (π 2 t)·ln(ΔE t / (4ΔE s ) - 1 / 2)

[0024] where, ΔE t is the transient voltage change, ΔE s is the steady-state voltage change, L is the electrode thickness, and t is the single-step current application time.

[0025] Further, according to the ratio of voltage change ΔE t / ΔE s to select whether to use the traditional inverse Laplace transform method or the improved inverse Laplace transform method to calculate the diffusion coefficient: when ΔE t / ΔE s ≥2.869, use the traditional inverse Laplace transform method to calculate the diffusion coefficient; when 2 < ΔE t / ΔE s < 2.869, use the improved inverse Laplace transform method to calculate the diffusion coefficient; when ΔE t / ΔE s ≤2, it is determined that the diffusion coefficient cannot be calculated by the GITT theoretical model.

[0026] Further, an automated program written using Python's Streamlit framework extracts voltage data from GITT experiments and calculates the diffusion coefficient. Specifically, the automated program reads GITT experiment data, extracts the key node voltages in each charge-discharge cycle, and calculates the transient voltage change ΔE t and the steady-state voltage change ΔE s , selects to use the traditional inverse Laplace transform method or the improved inverse Laplace transform method to calculate the diffusion coefficient, and generates a visualization image to display the calculation results of the diffusion coefficient.

[0027] Further, the automated program also includes an interactive interface that allows users to upload GITT experiment data and set parameters, identify charge-discharge cycles, and supports exporting calculation results and images for easy analysis and reporting.

[0028] Further, the accuracy of the diffusion coefficient calculation formula for concentration is verified through numerical simulation, including the following steps:

[0029] Preset a specific value of the diffusion coefficient D pre for subsequent verification;

[0030] Use the finite difference method to numerically solve the original partial differential equations (1)(2) to obtain the numerical solution for concentration C;

[0031] According to the numerical solution for concentration C obtained by the finite difference method, substitute it into the diffusion coefficient calculation formula D(C) to calculate the estimated value D of the diffusion coefficient est ;

[0032] Calculate the relative error between D est and D pre using the relative error calculation formula;

[0033] Compare the relative errors of the traditional solution and the improved solution, and evaluate whether the improved solution can improve the calculation accuracy within different ranges of tD / L 2 outside the control range of the traditional solution, that is, in the region where tD / L 2 is close to 1 or greater than 1.

[0034] Technical effects

[0035] The present invention provides a method for calculating the diffusion coefficient of an electrode material, having the following technical effects:

[0036] 1. Improve the calculation accuracy of the diffusion coefficient: Through the improved GITT diffusion coefficient calculation formula, the present invention can provide more accurate calculation formulas and results within the entire time range. The traditional formula performs well within the restricted conditions (t << L 2 / D), but when t and L2 Errors occur when the value of / D is close. Therefore, the present invention improves the GITT theoretical model and its solution, provides a calculation method applicable to the full range, and improves the accuracy of the diffusion coefficient calculation.

[0037] 2. Solve the limitations and logical errors of existing formulas: The existing GITT diffusion coefficient calculation formula is usually based on the limitation condition (t << L 2 / D), and requires the user to judge the time condition before using the formula, and the judgment condition itself depends on the value of the diffusion coefficient D. This logical contradiction causes the user to often ignore the determination of the limitation condition in actual operation, thus affecting the reliability of the calculation result. The present invention introduces a piecewise calculation formula, and automatically selects the applicable formula according to the ratio of voltage differences ΔE t / ΔE s to avoid the logical error of the limitation condition in the existing method.

[0038] 3. Achieve efficient processing of experimental data: Through an automated program written in the Streamlit framework of Python, the present invention can automatically extract the key node voltages in GITT experimental data, calculate the voltage differences, select the optimal formula, and generate visual images. The automated program also provides an interactive interface, allowing users to upload GITT experimental data and set parameters, automatically identify the charge and discharge cycles, and support the export of calculation results and images, significantly improving the efficiency of experimental data processing and the accuracy of calculation results.

[0039] The following will further illustrate the concept, specific structure and technical effects of the present invention with reference to the accompanying drawings to fully understand the purpose, features and effects of the present invention. Description of the Drawings

[0040] Figure 1 is a schematic diagram of the battery composition of the GITT test in a preferred embodiment of the present invention;

[0041] Figure 2 is the voltage change curve and the calculation method of the voltage difference in a preferred embodiment of the present invention;

[0042] Figure 3 is a schematic diagram of the establishment and solution method of the theoretical model in a preferred embodiment of the present invention, used to illustrate the similarities and differences between the traditional solution and the improved solution;

[0043] Figure 4 is the comparison between the numerical simulation results and the analytical expression in a preferred embodiment of the present invention, used to verify the accuracy of the formula;

[0044] Figure 5 is a schematic diagram of the processing result of the actual experimental data in a preferred embodiment of the present invention;

[0045] Figure 6 Schematic diagram of the function and interface design of the automation program of a preferred embodiment of the present invention;

[0046] Figure 7 Schematic diagram of the calculation and image generation results of the automation program of a preferred embodiment of the present invention. Detailed implementation manners

[0047] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention, and are not used to limit the present invention.

[0048] In the following description, specific details such as specific internal programs and technologies are presented for the purpose of illustration rather than limitation, so as to thoroughly understand the embodiments of the present invention. However, those skilled in the art should clearly understand that the present invention can also be implemented in other embodiments without these specific details. In other cases, the detailed descriptions of well-known systems, devices, circuits and methods are omitted to avoid unnecessary details from interfering with the description of the present invention.

[0049] As Figure 1 shown, in the embodiment of the present invention, a typical GITT test system is provided. The test device is composed of a constant current power supply, a counter electrode, an electrolyte and a working electrode. Among them, the constant current power supply provides a constant current for the battery or electrode material, and the direction of electron flow is indicated by the circuit. The counter electrode is mainly used to complete the current loop and usually uses a material with high stability. The electrolyte acts as a medium for ion transport to ensure that ions or other active ions can migrate between the working electrode and the counter electrode. The working electrode is the electrode to be tested, and its kinetic characteristics are measured through GITT tests.

[0050] The embodiment of the present invention provides a method for calculating the diffusion coefficient of an electrode material, including the following steps:

[0051] S1, establish a GITT theoretical model based on Fick's diffusion law to describe the concentration distribution of ions in the electrode material:

[0052]

[0053]

[0054] The above partial differential equation is a common theoretical model used to describe the diffusion behavior of ions in solid electrode materials during the constant current intermittent titration technique (GITT) test. The coordinate system is established as Figure 1As shown, C(x, t) in the equation represents the ion concentration at position x and time t, and D is the diffusion coefficient. The initial condition C(x, 0) = C0 sets the uniform concentration at the initial moment in the diffusion region to C0. In terms of the boundary conditions, the no-flux condition at x = 0 indicates that this position is a symmetric boundary where no ion diffusion exchange occurs; the condition at the other end x = L reflects the constant current charge-discharge condition, where I0 is the applied current, n is the number of electron transfers, F is the Faraday constant, and A0 is the electrode surface area. This condition shows that at x = L, due to the current-driven ion diffusion, there is a concentration gradient at this boundary. This system of equations can be used to solve the concentration distribution of ions in the GITT test and further calculate the diffusion coefficient D and related kinetic parameters.

[0055] S2. Solve the theoretical model in S1, i.e., the partial differential equation (1), by Laplace transform to obtain the analytical expression between the diffusion coefficient D and the concentration C.

[0056] By Laplace transform Obtain the corresponding control equation and initial and boundary conditions:

[0057]

[0058] The general solution expression of the control equation (3) is:

[0059]

[0060] Substitute the boundary conditions (4) to calculate the coefficients A and B, and then we can get:

[0061]

[0062] In the traditional GITT theoretical model, by expanding formula (6) using the binomial theorem of negative powers, we can get:

[0063]

[0064] Perform the inverse transform according to the following Laplace inverse transform formula:

[0065]

[0066] After the inverse transform, we can get:

[0067]

[0068] Then the concentration at the boundary x = L is:

[0069]

[0070] Formula (11) is the analytical expression of concentration C with respect to diffusion coefficient D obtained by the traditional inverse Laplace transform method. In fact, when solved by different inverse Laplace transform methods, formula (6) is also equivalent to:

[0071]

[0072] Write the hyperbolic sine / cosine function in formula (12) in the form of Taylor expansion:

[0073]

[0074] Based on the Heaviside expansion theorem, for the frequency-domain concentration expression in the form of p(s) / q(s), it is solved by classification according to the specific forms of p(s) and q(s) (polynomial degree, repeated roots). The denominator in formula (13) is a higher-order polynomial of s, and the corresponding roots are s = 0 (double root) and all solutions of sin(λ k ) = 0, where k = 1, 2, 3,.... According to the Heaviside expansion theorem, the inverse-transformed expression is:

[0075]

[0076] Therefore, the concentration at the boundary x = L is:

[0077]

[0078] Formula (15) is the improved analytical expression of concentration C with respect to diffusion coefficient D obtained by using the Heaviside expansion theorem.

[0079] S3. Based on the analytical expression C(D) of ion concentration, obtain the calculation formula of the diffusion coefficient of concentration applicable to the full time range;

[0080] By observing the traditional formula (11) and the improved formula (15), it is found that the traditional formula and the improved formula show a conjugate relationship. Specifically, when the traditional formula (11) satisfies the condition t << L 2 / D, approximate the series terms, only keep the first term, so as to simplify the expression and obtain the explicit analytical form of the diffusion coefficient with respect to concentration. At this time, the contribution of the high-order terms of the series to the overall result can be ignored. The explicit expression of the diffusion coefficient is:

[0081]

[0082] For the improved formula (15), through the corresponding condition t >> L 2 / D can make the terms of the infinite series approximately zero as a whole, and the corresponding expression for the diffusion coefficient is:

[0083]

[0084] S4. Verify the accuracy of the diffusion coefficient calculation formula through numerical simulation.

[0085] To verify the accuracy of the explicit expressions of the diffusion coefficient (Equations (16) and (17)) under the traditional solution method and the improved solution method, a comparative analysis was carried out through numerical simulation. The specific steps are as follows:

[0086] 1. Preset the diffusion coefficient: First, preset a specific value of the diffusion coefficient D pre for subsequent verification. The specific parameters used are the Faraday constant F = 96485 C / mol, the current density I0 / A0 = 1 A / mol, the electrode thickness L = 2×10 -3 cm, the current pulse time t = 600 s, the initial concentration C0 = 0.1 mol / cm 3 , and the preset value range of the diffusion coefficient is D pre = 10 -12 ~10 - 7 cm 2 / s;

[0087] 2. Finite difference solution: Use the finite difference method to numerically solve the original partial differential equations (1) and (2) to obtain the numerical solution of the concentration C, which ensures a reliable reference value in the verification process;

[0088] 3. Calculate the estimated value of the diffusion coefficient: Substitute the numerical solution of the concentration C obtained by the finite difference method into Equations (16) and (17) respectively to calculate the estimated value of the diffusion coefficient D est ;

[0089] 4. Calculate the relative error: Calculate the relative error between D est and D pre : Relative error = |D est - D pre | / D pre ;

[0090] 5. Comparative analysis: Compare the relative errors of the traditional solution method and the improved solution method, and evaluate their performance in different ranges (tD / L 2 ), especially outside the control range of the traditional solution method, that is, in the region where tD / L 2 is close to 1 or greater than 1, to analyze whether the improved solution method can improve the calculation accuracy.

[0091] By presetting the specific value \(D\) of the unknown diffusion coefficient \(D\) in formulas (1) and (2), and using the finite difference method to solve, the numerical solution of the concentration \(C\) is obtained. Here, the concentration \(C\) is a discrete value, and then it is substituted into the approximate calculation formula \(D(C)\) of the diffusion coefficient to calculate \(D\). pre , and the relative error between \(D\) and \(D\) is compared to determine whether the approximate calculation formula \(D(C)\) is accurate. est Compare \(D\) pre and \(D\) est to judge whether the approximate calculation formula \(D(C)\) is accurate by comparing their relative errors.

[0092] Through the above numerical simulation method, the performance of the traditional solution method and the improved solution method under different conditions is systematically verified, thus providing a more reliable basis for the calculation of the diffusion coefficient. This method can not only quantify the accuracy difference between the two solution methods, but also provide guidance for selecting the appropriate calculation method in practical applications. The comparison results are as Figure 4 shown.

[0093] S5. Convert the concentration-based diffusion coefficient calculation formula to a voltage-based diffusion coefficient calculation formula to facilitate the processing and analysis of experimental data. In practical applications, the GITT technique obtains data by measuring the voltage change of the electrode under constant current conditions and calculates the diffusion coefficient of the electrode using these data. Therefore, in order to combine theoretical derivation with actual measurement, we need to convert the concentration-based diffusion coefficient calculation formula \(D(C)\) to a voltage-based diffusion coefficient calculation formula \(D(E)\) for more direct application in the analysis and calculation of experimental data. This conversion process will help to better understand the diffusion behavior of electrode materials and provide more accurate theoretical support for experiments.

[0094] In the derivation process, it mainly involves the approximate derivative of the transient voltage change \(\Delta E\) t and the steady-state voltage \(\Delta E\) s . The transient voltage change within a short time is due to the diffusion-limited ion concentration change, while the long-term steady-state voltage change reflects the response after the concentration reaches a new quasi-equilibrium state.

[0095] The calculation formula of the diffusion coefficient obtained by using the traditional inverse Laplace transform method is expressed as:

[0096]

[0097] where \(\Delta E\) t is the transient voltage change, \(\Delta E\) s is the steady-state voltage change, \(L\) is the electrode thickness, and \(t\) is the single-step current application time. When \(t << L\) 2 / D, the infinite series term of formula (64) is approximately zero, and the calculation formula of the diffusion coefficient is:

[0098] \(D = 4L\) 2 / (πt)·(\(\Delta E\)s / ΔE t ) 2 #(19)

[0099] This formula is the most commonly used formula for calculating the diffusion coefficient of electrode materials in GITT tests.

[0100] The diffusion coefficient calculation formula derived using the improved inverse Laplace transform method has the following expression:

[0101]

[0102] When t >> L 2 / D, approximate the infinite series in formula (20) as its first term, that is, take n = 1. Then the corresponding diffusion coefficient calculation formula is:

[0103] D = -L 2 / (π 2 t)·ln(ΔE t / (4ΔE s ) - 1 / 2)#(21)

[0104] By comparing the diffusion coefficient calculation formulas (19) and (21) under the two methods, it is found that by equating the two formulas and solving, the respective more suitable value ranges can be determined according to the intersection point ΔE t / ΔE s = 2.869. The diffusion coefficient can be calculated using the traditional formula (19) or the improved formula (21) according to the ratio of voltage change ΔE t / ΔE s : When ΔE t / ΔE s ≥2.869, use the traditional inverse Laplace transform method to calculate the diffusion coefficient; when 2 < ΔE t / ΔE s <2.869, use the improved inverse Laplace transform method to calculate the diffusion coefficient; when ΔE t / ΔE s ≤2, since it exceeds the value range of formula (21), it is determined that the diffusion coefficient cannot be calculated using the GITT theoretical model. Based on this discovery, the respective more suitable ranges of the traditional solution and the improved solution in calculating the diffusion coefficient can be determined, and a piecewise formula can be used to calculate the diffusion coefficient:

[0105]

[0106] The usage conditions for all three cases can be expressed by the ratio of voltage differences ΔE t / ΔE s .

[0107] An automated program written using Python's Streamlit framework extracts voltage data from GITT experiments and calculates the diffusion coefficient. Specifically, the automated program reads GITT experiment data, extracts the key node voltages in each charge-discharge cycle, and calculates the transient voltage change ΔE t and the steady-state voltage change ΔE s . It selects to use the traditional inverse Laplace transform method or the improved inverse Laplace transform method to calculate the diffusion coefficient and generates a visualization image to display the calculation results of the diffusion coefficient.

[0108] The automated program also includes an interactive interface that allows users to upload GITT experiment data and set parameters, identify charge-discharge cycles, and supports exporting calculation results and images for easy analysis and reporting.

[0109] As Figure 2 shown, the voltage change curve in the GITT test includes the charging process (a) and the discharging process (b). During the charging process, the voltage gradually increases over time. After applying a constant current for a period of time, it reaches the instantaneous voltage E3, and then enters a rest stage where the voltage gradually decreases due to the relaxation effect of the concentration gradient and finally approaches the steady-state voltage E4. During the discharging process, the voltage rapidly drops to E3 during constant-current discharging and then gradually recovers to the new steady-state voltage E4 during the rest stage. Two key parameters in the voltage change are the transient voltage difference ΔE t and the steady-state voltage difference ΔE s . Among them, the transient voltage difference ΔE t is determined by the change in voltage after applying a constant current, and the steady-state voltage difference ΔE s reflects the voltage change within a single charge-discharge cycle. In the GITT test, the calculation of the diffusion coefficient usually depends on the relationship between the steady-state voltage and the transient voltage change. By measuring ΔE t and ΔE s , combined with Fick's law, the ion diffusion coefficient of the material can be deduced, providing important experimental data for studying the kinetic properties of electrode materials.

[0110] In the solution process of the embodiments of the present invention, first, the partial differential equation in the time domain is transformed into a partial differential equation in the frequency domain through Laplace transform to solve the concentration distribution, and then the analytical expression of the concentration varying with time is obtained through the inverse Laplace transform. Compared with the traditional method, the present invention has made improvements in the processing of the inverse Laplace transform. The traditional solution method uses the binomial theorem of negative powers, which can handle exponential term formulas in specific forms, while the improved solution method used in the present invention converts the concentration expression in the frequency domain into the form of hyperbolic functions and uses the Heaviside expansion theorem for inverse transform, making the final analytical expression present different series forms. Although the concentration expressions obtained by the traditional solution method and the improved solution method are different in form, when the series terms tend to infinity, the two show exactly equal numerical results, that is, the two are equivalent analytical solutions of the original equation.

[0111] As Figure 4 shown, the concentration expressions under the two solution methods obtained as in Figure 3 are approximated according to the limiting conditions (t << L 2 / D and t >> L 2 / D) to obtain an explicit expression of the diffusion coefficient. Subsequently, the discrete values of the concentration of the original partial differential equation under specific parameters (D pre ) are obtained through finite difference calculation, and they are substituted into the diffusion coefficient calculation formula to obtain the estimated value D est . By calculating the relative error between D pre and D est , the accuracy of the two solution methods is judged. As can be seen from Figure 4 , in the range of t << L 2 / D, the explicit formula obtained by the traditional solution method has almost no error, indicating that it is a good approximation formula in this region. However, the performance of the improved solution method is poor in this range; on the contrary, when t << L 2 / D does not hold, that is, tD / L 2 ≥1, at this time the improved solution method has better effects, the error is close to zero, and as tD / L 2 increases, the error of the traditional solution method will gradually increase, while the improved solution method always maintains a high degree of accuracy. This result verifies the effectiveness of the improved solution method, which can make up for the deficiencies of the traditional solution method when tD / L 2 is large. By combining the functions of the two solution methods in different ranges, a GITT diffusion coefficient calculation formula applicable to the full range can be jointly constructed. This combined method not only expands the applicable range of the formula but also improves the accuracy and reliability of the calculation results.

[0112] Subsequently, the original analytical expression based on ion concentration was converted into an expression based on voltage to facilitate the processing and analysis of experimental data. Through reasonable approximation, an explicit calculation formula for the diffusion coefficient was obtained. The formula includes the traditional formula and the improved formula: the traditional formula is applicable to the restriction condition (t<<L 2 / D), its expression is D=4L 2 / (πt)·(ΔE s / ΔE t ) 2 , and the improved formula is applicable to the constraint condition (t>>L 2 / D), the expression is D=-L 2 / (π 2 t)·ln(ΔE t / (4ΔE s )-1 / 2), where ΔE t is the transient voltage change, ΔE s is the steady-state voltage change, L is the electrode thickness, and t is the single-step current application time. By combining the two equations, we can obtain the boundary conditions of the two, namely ΔE t / ΔE s =2.869. In addition, the improved formula itself has a domain, namely ΔE t / ΔE s >2, so according to the voltage difference ratio ΔE t / ΔE s Choose to use the traditional formula or the improved formula to calculate the diffusion coefficient: When ΔE t / ΔE s ≥2.869, the traditional formula has higher calculation accuracy. When 2<ΔE t / ΔE s <2.869, the improved formula has higher calculation accuracy, and when ΔE t / ΔE s ≤2, the diffusion coefficient cannot be calculated using the GITT theoretical model. This restriction does not contain any logical errors and is better than the original condition (t< <L 2 / D) is clearer and easier to calculate.

[0113] like Figure 5 As shown, an example is used to illustrate the application of the above three-stage formula: the experiment uses a lithium cobalt oxide electrode as the working electrode, and the electrode thickness is L = 2×10 -3 cm, the single-step charge / discharge time is t=1440s. The corresponding voltage changes in a complete charge / discharge process are as follows: Figure 5 (a) and Figure 5 As shown in (b), the transient voltage change (ΔE t ) and steady-state voltage change (ΔE s) The calculation method is as Figure 2 shown. Based on this, all the parameters required to calculate the diffusion coefficient in formula (22) are obtained. According to different values of ΔE t / ΔE s , different formulas can be selected to calculate the diffusion coefficient. Figure 5 (c) and Figure 5 (d) respectively show the changes in the diffusion coefficient during the charge / discharge process. In the figure, the diffusion coefficient formula under the traditional solution method is used as a benchmark, and the results obtained by other calculations are compared. When ΔE t / ΔE s ≥2.869, the diffusion coefficient formula D = 4L 2 / (πt)·(ΔE s / ΔE t ) 2 can accurately calculate the GITT diffusion coefficient, as shown by the triangular scatter points in Figure 5 (c) and Figure 5 (d); when 2 < ΔE t / ΔE s < 2.869, there will be errors in the traditional solution method. At this time, the improved formula D = -L 2 / (π 2 t)·ln(ΔE t / (4ΔE s ) - 1 / 2) can calculate a more accurate GITT electrode material diffusion coefficient. The improved data is shown by the rectangular scatter points in Figure 5 (c) and Figure 5 (d); when ΔE t / ΔE s ≤2, the actual ion diffusion coefficient of the electrode material cannot be calculated by this GITT diffusion model. Figure 5 (c) and Figure 5 (d) The red dots are represented by the formula obtained by the traditional solution method, but they are not the true values of the diffusion coefficient.

[0114] As Figure 6 shown, the present invention writes an automated program based on the Streamlit framework of Python. Users can upload experimental data according to the data format shown in the interface and select whether to calculate the charging process or the discharging process. After the selection is completed, the program will automatically calculate the changing data of the diffusion coefficient according to the formula.

[0115] Figure 7 Taking Figure 5 the data in the example as the basis for the image generation process, users can save the schematic diagram of the diffusion coefficient change according to their own needs for use.

[0116] The preferred specific embodiments of the present invention have been described in detail above. It should be understood that those of ordinary skill in the art can make many modifications and variations based on the concept of the present invention without creative efforts. Therefore, all technical solutions that can be obtained by those skilled in the art in the technical field based on the concept of the present invention through logical analysis, reasoning, or limited experiments on the basis of the prior art should fall within the protection scope determined by the claims.

Claims

1. A method for calculating the diffusion coefficient of an electrode material, characterized in that, It includes the following steps: S1. Establish a GITT theoretical model based on Fick's diffusion law to describe the concentration distribution of ions in the electrode material; S2. Solve the theoretical model in S1 through Laplace transform to obtain the analytical expression between the diffusion coefficient D and the concentration C; S3. Based on the analytical expression of the ion concentration, obtain the diffusion coefficient calculation formula D(C) applicable to the entire time range for the concentration; S4. Verify the accuracy of the diffusion coefficient calculation formula of the concentration through numerical simulation; S5. Convert the diffusion coefficient calculation formula based on the concentration into the diffusion coefficient calculation formula based on the voltage to facilitate the processing and analysis of experimental data.

2. The calculation method of the diffusion coefficient of an electrode material according to claim 1, wherein Use partial differential equations to describe the GITT theoretical model based on Fick's diffusion law, where C(x, t) represents the ion concentration at position x and time t, and D is the diffusion coefficient; the initial condition C(x, 0) = C0 sets the uniform concentration at the initial moment of the diffusion region to C0; in terms of the boundary conditions, the no-flux condition at x = 0 indicates that this position is a symmetric boundary where no ion diffusion exchange occurs; the condition at the other end x = L reflects the constant current charge and discharge condition, where I0 is the applied current, n is the number of electron transfers, F is the Faraday constant, and A0 is the electrode surface area; this condition shows that at x = L, due to the current driving ion diffusion, there is a concentration gradient at this boundary; this set of equations is used to solve the concentration distribution of ions in the GITT test and further calculate the diffusion coefficient D and related kinetic parameters.

3. The method for calculating the diffusion coefficient of an electrode material according to claim 2, wherein Solve the theoretical model in S1 through Laplace transform to obtain the analytical expression between the diffusion coefficient D and the concentration C. The analytical expression between the diffusion coefficient D and the concentration C is where δ = I0 / nFA0D, I0 is the applied current, n is the number of electron transfers, F is the Faraday constant, A0 is the electrode surface area, D is the diffusion coefficient, L is the electrode thickness, C(L, t) represents the ion concentration at the electrode surface x = L and time t, and C0 is the ion concentration at the initial moment.

4. The method for calculating the diffusion coefficient of an electrode material according to claim 1, wherein The diffusion coefficient calculation formula includes the diffusion coefficient calculation formula obtained by using the traditional inverse Laplace transform method and the diffusion coefficient calculation formula obtained by using the improved inverse Laplace transform method.

5. The calculation method of the diffusion coefficient of an electrode material according to claim 4, characterized in that, The calculation formula for the diffusion coefficient obtained by using the traditional inverse Laplace transform method is expressed as D = 4L 2 / (πt)·(ΔE s / ΔE t ) 2 Among them, ΔE t is the transient voltage change, and ΔE s is the steady-state voltage change. L is the electrode thickness, and t is the single-step current application time.

6. The calculation method of the diffusion coefficient of an electrode material according to claim 4, wherein, The calculation formula for the diffusion coefficient derived by using the improved inverse Laplace transform method, and its expression is D = -L 2 / (π 2 t)·ln(ΔE t / (4ΔE s ) - 1 / 2) Among them, ΔE t is the transient voltage change, and ΔE s is the steady-state voltage change. L is the electrode thickness, and t is the single-step current application time.

7. The method for calculating the diffusion coefficient of an electrode material according to claim 5 or 6, characterized in that According to the ratio of voltage change ΔE t / ΔE s to select whether to use the traditional inverse Laplace transform method or the improved inverse Laplace transform method to calculate the diffusion coefficient: when ΔE t / ΔE s ≥2.869, use the traditional inverse Laplace transform method to calculate the diffusion coefficient; when 2 < ΔE t / ΔE s <2.869, use the improved inverse Laplace transform method to calculate the diffusion coefficient; when ΔE t / ΔE s ≤2, it is determined that the diffusion coefficient cannot be calculated by the GITT theoretical model.

8. The calculation method of the diffusion coefficient of an electrode material according to claim 7, characterized in that, An automated program written using the Streamlit framework in Python extracts voltage data from GITT experiments and calculates the diffusion coefficient. Specifically, the automated program reads GITT experiment data, extracts the key node voltages in each charge-discharge cycle, and calculates the transient voltage change ΔE t and the steady-state voltage change ΔE s , selects to use the traditional inverse Laplace transform method or the improved inverse Laplace transform method to calculate the diffusion coefficient, and generates a visualization image to display the calculation results of the diffusion coefficient.

9. A method for calculating the diffusion coefficient of an electrode material according to claim 8, characterized in that The automated program also includes an interactive interface that allows users to upload GITT experimental data and set parameters, identify charge and discharge cycles, and supports the export of calculation results and images for easy analysis and reporting.

10. The method for calculating the diffusion coefficient of an electrode material according to claim 1, wherein Verify the accuracy of the diffusion coefficient calculation formula of the concentration through numerical simulation, including the following steps: Preset a specific numerical value D of the diffusion coefficient pre for subsequent verification; Use the finite difference method to numerically solve the original partial differential equations (1)(2) to obtain the numerical solution of the concentration C; The numerical solution of the concentration C obtained by the finite difference method is substituted into the diffusion coefficient calculation formula D(C) to calculate the estimated value D of the diffusion coefficient est ; Calculate D through the relative error calculation formula est and D pre for the relative error; Compare the relative errors of the traditional solution method and the improved solution method to evaluate the two within different ranges, especially outside the control range of the traditional solution method, that is, in the region where tD / L 2 is close to 1 or greater than 1, and analyze whether the improved solution method can improve the calculation accuracy.

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