An electrochemical sensor reaction-diffusion equation optimization method and optimization system

Abstract

The application relates to the technical field of electrochemical sensors, and discloses an electrochemical sensor reaction diffusion equation optimization method and an optimization system, which comprises the following steps: obtaining the relationship between reactant concentration and mass transfer process and reaction process and the relationship between reactant change rate and mass transfer rate and reaction rate based on the electrochemical reaction formula of a target sensor; obtaining the expression of the mass transfer rate and the expression of the reaction rate based on the electric field intensity in the reaction region of the double electric layer, the reactant concentration and the intermolecular interaction function; constructing the reaction diffusion equation of the target sensor according to the relationship between the reactant change rate and the mass transfer rate and the reaction rate, the expression of the mass transfer rate and the expression of the reaction rate; and processing the reaction diffusion equation through a dimensionless method to obtain the response current and the relationship between the response current and each parameter in the equation; the application solves the problems that the existing reaction diffusion equation has a small adaptive range and low description accuracy of a response signal.

Description

Technical Field

[0001] This invention relates to the field of electrochemical sensor technology, and in particular to a method and system for optimizing the reaction-diffusion equation of an electrochemical sensor. Background Technology

[0002] Electrochemical sensing is an analytical technique that obtains the concentration of a sample solution by detecting electron transfer generated by redox reactions. It is widely used in the detection of various substances due to its advantages such as simple operation, rapid response, high sensitivity, and strong selectivity. Various electrochemical sensors are devices that rely on electrochemical sensing technology for detection. Reaction-diffusion equations can be used to describe and explore electrochemical systems, thereby optimizing the performance of electrochemical sensors. Currently, while existing reaction-diffusion equation models are well applied to the H₂O₂ electrochemical system, they are difficult to extend to other enzyme or non-enzyme systems. The main reason is that existing reaction-diffusion equation models lack a description of the relationship between reaction sites and signals, and therefore cannot describe the impact of surface deactivation phenomena such as deposition and poisoning on the response signal. Therefore, existing reaction-diffusion equations suffer from limited applicability and relatively low accuracy in describing response signals. Summary of the Invention

[0003] This invention provides a method and system for optimizing the reaction-diffusion equation of an electrochemical sensor, in order to solve the problems of existing reaction-diffusion equations having a small applicable range and low accuracy in describing the response signal.

[0004] To achieve the above objectives, the present invention employs the following technical solution:

[0005] In a first aspect, the present invention provides a method for optimizing the reaction-diffusion equation of an electrochemical sensor, comprising:

[0006] Obtain the electrochemical reaction formula in the target sensor, and based on the electrochemical reaction formula, obtain the relationship between reactant concentration and mass transfer process and reaction process;

[0007] Based on the relationship between reactant concentration and mass transfer and reaction processes, the relationship between reactant change rate and mass transfer rate and reaction rate is obtained.

[0008] The expression for the mass transfer rate is obtained based on the Nernst-Planck equation, and the expression for the reaction rate is obtained based on the electric field strength, reactant concentration, and intermolecular interaction function in the reaction region within the electric double layer.

[0009] Based on the relationship between the reactant change rate and the mass transfer rate and reaction rate, the expression for the mass transfer rate, and the expression for the reaction rate, a reaction-diffusion equation for the target sensor is constructed.

[0010] The reaction-diffusion equation is processed by a dimensionless method to obtain a dimensionless target reaction-diffusion equation, which is then solved by the finite difference method to obtain the reactant concentration and the response current and its relationship with the parameters in the equation.

[0011] Optionally, obtaining the relationship between reactant concentration and the mass transfer and reaction processes based on electrochemical reaction equations includes:

[0012] In the reactive diffusion region of the solution within the target sensor, the change in reactant concentration arises from the increase in mass transfer and the decrease in reaction. Therefore, the expression for reactant concentration in relation to the mass transfer and reaction processes is:

[0013] ;

[0014] in, Indicates reactant concentration. This indicates that the mass transfer process increases the concentration of reactants. This indicates the concentration of reactants consumed during the reaction process.

[0015] Optionally, the process of obtaining the relationship between reactant change rate and mass transfer rate and reaction rate based on the relationship between reactant concentration and mass transfer process and reaction process includes:

[0016] Based on the reactant concentration being the difference between the concentration increased during the mass transfer process and the concentration consumed during the reaction process, the relationship between the reactant change rate and the mass transfer rate and reaction rate is obtained as the difference between the mass transfer rate and the reaction rate, expressed as:

[0017] ;

[0018] in, Indicates the mass transfer rate. Indicates the reaction rate.

[0019] Optionally, the expression for the mass transfer rate based on the Nernst-Planck equation includes:

[0020] The mass transfer process is determined by diffusion, convection, and electromigration using the Nernst-Planck equation. In an electrochemical sensing system where the target sensor solution remains quiescent and contains a significant amount of supporting electrolyte, the effects of convection and electromigration on the reactants are neglected, resulting in a diffusion-based mass transfer rate expression:

[0021] ;

[0022] in, Indicates the mass transfer rate. Indicates the diffusion coefficient of the reactants. Indicates the concentration of reactants.

[0023] Optionally, the expression for the reaction rate is:

[0024] ;

[0025] in, Indicates the reaction rate. Represents the reaction rate constant. Indicates the reaction order. Indicates the concentration of reactants.

[0026] Optionally, the expression for the reaction-diffusion equation is:

[0027] ;

[0028] in, Represents the reaction rate constant. Indicates the reaction order. Indicates reactant concentration. This represents the diffusion coefficient of the reactants.

[0029] Optionally, obtaining the target reaction-diffusion equation by performing partial differential calculations on the reaction-diffusion equation using a dimensionless method includes:

[0030] Based on the reaction-diffusion equation, a dimensionless method expression is set, and the dimensionless expression is substituted into the reaction-diffusion equation to obtain the dimensionless target reaction-diffusion equation. The finite difference method is then used to solve the equation and obtain the relationship between the reactant concentration and the parameters in the equation.

[0031] The dimensionless method expression includes: , , ;

[0032] in, Indicates the concentration of reactants in the solution. Indicates the width of the reaction diffusion region;

[0033] The target reaction diffusion equation is shown below:

[0034] ;

[0035] in, Indicates the reaction order. This indicates the dimensionless concentration of reactants. It represents the ratio coefficient between the reaction term and the diffusion term.

[0036] Secondly, embodiments of this application provide an electrochemical sensor reaction-diffusion equation optimization system, including a processor and a memory;

[0037] Memory, used to store computer programs for solving reaction-diffusion equations;

[0038] When a processor executes a program stored in memory, it implements any of the steps of the method described in the first aspect.

[0039] Beneficial effects:

[0040] This invention provides a method for optimizing the reaction-diffusion equation of an electrochemical sensor. It determines the relationship between reactant concentration and the mass transfer and reaction processes through the electrochemical reaction equation in the target sensor, thereby obtaining the relationship between the reactant change rate, mass transfer rate, and reaction rate. The expression for the mass transfer rate is obtained using the Nernst-Planck equation, and the expression for the reaction rate is derived based on the electric field strength, reactant concentration, and intermolecular interaction functions in the reaction region within the electric double layer. Based on the relationship between the reactant change rate and the mass transfer and reaction rates, the expression for the mass transfer rate, and the expression for the reaction rate, the reaction-diffusion equation of the target sensor is constructed. The dimensionless target reaction-diffusion equation is obtained by processing the reaction-diffusion equation using a dimensionless method, and then solved using the finite difference method to obtain the reactant concentration and the response current and its relationship with the parameters in the equation. This proposes a method to improve the stability of the electrochemical sensing system by controlling the k-value in the electrochemical system. In addition, the results also cover the study of reaction order and initial concentration difference, which can deepen the understanding of the electrochemical sensing process and help design electrochemical sensors with better performance. Attached Figure Description

[0041] Figure 1 This is a flowchart of a preferred embodiment of the electrochemical sensor reaction-diffusion equation optimization method of the present invention. Detailed Implementation

[0042] The technical solution of the present invention will be clearly and completely described below. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0043] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Similarly, the terms "an" or "a" and similar terms do not indicate a quantity limitation, but rather indicate the presence of at least one. The terms "connected" or "linked" and similar terms are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. "Up," "down," "left," "right," etc., are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship also changes accordingly.

[0044] Please see Figure 1 This application provides a method for optimizing the reaction-diffusion equation of an electrochemical sensor, including:

[0045] Obtain the electrochemical reaction formula in the target sensor, and based on the electrochemical reaction formula, obtain the relationship between reactant concentration and mass transfer process and reaction process;

[0046] Based on the relationship between reactant concentration and mass transfer and reaction processes, the relationship between reactant change rate and mass transfer rate and reaction rate is obtained.

[0047] The expression for the mass transfer rate is derived based on the Nernst-Planck equation, and the expression for the reaction rate is derived based on the electric field strength, reactant concentration, and intermolecular interaction function in the reaction region within the electric double layer.

[0048] Based on the relationship between the reactant change rate and the mass transfer rate and reaction rate, the expression for the mass transfer rate, and the expression for the reaction rate, a reaction-diffusion equation for the target sensor is constructed.

[0049] The reaction-diffusion equation is processed by a dimensionless method to obtain a dimensionless target reaction-diffusion equation, which is then solved by the finite difference method to obtain the reactant concentration and the response current and its relationship with the parameters in the equation.

[0050] In the above embodiments, the relationship between reactant concentration and mass transfer and reaction processes is determined by the electrochemical reaction equation in the target sensor, thereby obtaining the relationship between reactant change rate, mass transfer rate, and reaction rate. The expression for the mass transfer rate is obtained through the Nernst-Planck equation, and the expression for the reaction rate is obtained based on the electric field strength, reactant concentration, and intermolecular interaction function in the reaction region within the electric double layer. Based on the relationship between reactant change rate, mass transfer rate, and reaction rate, the expression for the mass transfer rate, and the expression for the reaction rate, the reaction-diffusion equation of the target sensor is constructed. The reaction-diffusion equation is processed using a dimensionless method to obtain a dimensionless target reaction-diffusion equation, which is then solved using the finite difference method to obtain the reactant concentration and the response current and its relationship with the parameters in the equation. Therefore, a method is proposed to improve the stability of the electrochemical sensing system by controlling the k-value in the electrochemical system.

[0051] In addition, the results also cover studies on reaction order and initial concentration, which can deepen our understanding of electrochemical sensing processes and help design electrochemical sensors with better performance.

[0052] Optionally, obtaining the relationship between reactant concentration and the mass transfer and reaction processes based on electrochemical reaction equations includes:

[0053] In the reactive diffusion region of the solution within the target sensor, the change in reactant concentration arises from the increase in mass transfer and the decrease in reaction. Therefore, the expression for reactant concentration in relation to the mass transfer and reaction processes is:

[0054] ;

[0055] in, Indicates reactant concentration. This indicates the increase in reactant concentration during the mass transfer process. This indicates the concentration of reactants consumed during the reaction process.

[0056] In the above embodiments, the concentration of the electroactive reactant in the reaction diffusion region... The change in the concentration of reactant A will depend on both mass transfer and reaction processes. Therefore, the change in the concentration of reactant A mainly stems from the increase in mass transfer and the consumption in the reaction process.

[0057] Optionally, the process of obtaining the relationship between reactant change rate and mass transfer rate and reaction rate based on the relationship between reactant concentration and mass transfer process and reaction process includes:

[0058] Based on the reactant concentration being the difference between the reactant concentration increased during the mass transfer process and the reactant concentration consumed during the reaction process, the relationship between the reactant change rate and the mass transfer rate and reaction rate is obtained as the difference between the mass transfer rate and the reaction rate, expressed as:

[0059] ;

[0060] in, Indicates the mass transfer rate. Indicates the reaction rate.

[0061] In the above embodiments, since the concentration change of reactant A mainly comes from the increase during the mass transfer process and the consumption during the reaction process, the concentration change rate of reactant A can be expressed as the difference between the mass transfer rate and the reaction rate, which can be obtained by the first derivative with respect to time.

[0062] Optionally, the expression for the mass transfer rate based on the Nernst-Planck equation includes:

[0063] The mass transfer process is determined by diffusion, convection, and electromigration using the Nernst-Planck equation. In an electrochemical sensing system where the target sensor solution remains quiescent and contains a significant amount of supporting electrolyte, the effects of convection and electromigration on the reactants are neglected, resulting in a diffusion-based mass transfer rate expression:

[0064] ;

[0065] in, Indicates the mass transfer rate. Indicates the diffusion coefficient of the reactants. Indicates the concentration of reactants.

[0066] In the above embodiments, as can be seen from the Nernst-Planck equation, the mass transfer process in electrochemical processes is determined by three aspects: diffusion, convection, and electromigration. When a solution remains static, the influence of the convection term in the internal electrochemical sensing system is small and can therefore be ignored. In addition, since a large amount of supporting electrolyte is often added to the electrochemical system to ensure the normal progress of the electrochemical reaction and reduce the influence of electromigration on the reactants, the influence of migration is ignored in this application. Finally, we only consider the influence of diffusion in the mass transfer process.

[0067] Optionally, the expression for the reaction rate is:

[0068] ;

[0069] in, Indicates the reaction rate. Represents the reaction rate constant. Indicates the reaction order. Indicates the concentration of reactants.

[0070] In the above embodiments, in the electrochemical reaction, the reaction rate depends on the electric field strength in the reaction region within the double layer, the reactant concentration, and the intermolecular interaction function. Since the electric field can provide the necessary energy for the reaction to occur, the reaction rate will be faster when the region where the reactant is located has a stronger electric field. When the electric field strength is constant, a fixed rate constant K0 will be obtained.

[0071] Furthermore, the intermolecular interaction function will include the interactions between reactant molecules as well as the molecular interactions between reactants, co-reactants, or catalysts. Therefore, the reaction rate in the reaction term can be represented by the expression in the above embodiments.

[0072] Optionally, the expression for the reaction-diffusion equation is:

[0073] ;

[0074] in, Represents the reaction rate constant. Indicates the reaction order. Indicates reactant concentration. This represents the diffusion coefficient of the reactants.

[0075] In the above embodiments, K0 and β are the reaction rate constant and reaction order, respectively, in the reaction rate expression (where β ∈ [0, 2]). For a given elementary reaction 2A → 2B n+ / n (Describing the interactions between reactant molecules), the reaction order should be 2; however, it should be noted that if competing interactions exist in the system (describing the molecular interactions between reactants / co-reactants or catalysts), some reactant molecules may not follow elementary reactions. In this case, the value of β will change. Therefore, we can use partial differential equations to describe the kinetic process of the electrochemical system, and thus obtain the expression of the reaction-diffusion equation in the electrochemical system through partial differential equations.

[0076] Optionally, obtaining the target reaction-diffusion equation by performing partial differential calculations on the reaction-diffusion equation using a dimensionless method includes:

[0077] Based on the reaction-diffusion equation, a dimensionless method expression is set, and the dimensionless expression is substituted into the reaction-diffusion equation to obtain the dimensionless target reaction-diffusion equation. The finite difference method is then used to solve the equation and obtain the relationship between the reactant concentration and the parameters in the equation.

[0078] The dimensionless method expression includes: , , ;

[0079] in, Indicates the concentration of reactants in the solution. Indicates the width of the reaction diffusion region;

[0080] The target reaction diffusion equation is shown below:

[0081] ;

[0082] in, Indicates the reaction order. This indicates the dimensionless concentration of reactants. It represents the ratio coefficient between the reaction term and the diffusion term.

[0083] In the above embodiments, in order to exhibit the characteristics of the above reaction-diffusion equation, the following dimensionless method is adopted:

[0084] , , ;

[0085] in, Indicates the concentration of reactants in the solution. The width of the reaction diffusion region (which depends on the specific electrode reaction and the electric field distribution in the reaction diffusion region) is represented by the dimensionless partial differential equation shown below:

[0086] ;

[0087] in, Indicates the reaction order. This indicates the dimensionless concentration of reactants. The ratio coefficient between the reaction term and the diffusion term is represented by the parameter k, which represents the ratio of the reaction term to the diffusion term within the scaffold. Its value quantitatively reflects the main kinetic process in the electrochemical system. For electrochemical systems with a large k value, the reaction process is dominant, while when the k value is small, the kinetic process mainly depends on the diffusion process.

[0088] Therefore, we can conclude that: ;

[0089] The boundary conditions described by the proposed model are shown in the following equations:

[0090] ;

[0091] Based on the principle of electrochemical sensing, the local current should be related to the concentration of remaining reactants and the diffusion flux at that location. Therefore, we assume a direct proportionality between the local current and the diffusion flux (with a coefficient of θ). Thus, the local current at any location (X0) will be given by the following equation:

[0092] ;

[0093] Where z, F, and A are the number of electrons transferred per mole of reaction, the Faraday constant, and the electrode area, respectively.

[0094] The conclusions obtained are all qualitative studies (such as the shape of the current-time curve), so we do not focus on the precise values ​​of coefficients such as θ.

[0095] Subsequently, let the total current be the sum of the local currents at various locations within the reaction diffusion region of the electric double layer, then we have,

[0096] ;

[0097] Here, I(X, T) represents the local current generated at different locations X at different times T. After the redox reaction generates freely moving electrons, these electrons are captured by the electrode, producing a corresponding electrochemical signal. However, we believe that not all generated electrons can be captured by the electrode. If the electrons are generated at a distance from the electrode, they are more difficult to capture. Therefore, we use a function P(X, T) to reflect the ease with which electrons are captured at different locations. In this study, to simplify the model, we will temporarily disregard the difference in the ease of electron capture due to distance and set P(X, T) as a constant.

[0098] This application also provides an electrochemical sensor reaction-diffusion equation optimization system, including a processor and a memory;

[0099] Memory, used to store computer programs for solving reaction-diffusion equations;

[0100] The processor, when executing a program stored in memory, implements any of the steps described in the method for optimizing the reaction-diffusion equation of an electrochemical sensor.

[0101] The above-described electrochemical sensor reaction-diffusion equation optimization system can realize various embodiments of the above-described electrochemical sensor reaction-diffusion equation optimization method and achieve the same beneficial effects, which will not be elaborated here.

[0102] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A system for optimizing the reaction-diffusion equation of an electrochemical sensor, characterized in that, The system includes a processor and a memory. The memory stores a computer program for solving the reaction-diffusion equation. When the processor executes the computer program stored in the memory, it performs the following method steps: Obtain the electrochemical reaction formula in the target sensor, and based on the electrochemical reaction formula, obtain the relationship between reactant concentration and mass transfer process and reaction process; Based on the relationship between reactant concentration and mass transfer process and reaction process, the relationship between reactant change rate and mass transfer rate and reaction rate is obtained; The expression for the mass transfer rate is obtained based on the Nernst-Planck equation, and the expression for the reaction rate is obtained based on the electric field strength, reactant concentration, and intermolecular interaction function in the reaction region within the electric double layer. Based on the relationship between the reactant change rate and the mass transfer rate and reaction rate, the expression for the mass transfer rate, and the expression for the reaction rate, a reaction-diffusion equation for the target sensor is constructed. The target reaction-diffusion equation is obtained by performing partial differentials on the reaction-diffusion equation using a dimensionless method.

2. The electrochemical sensor reaction-diffusion equation optimization system according to claim 1, characterized in that, The relationship between reactant concentration and mass transfer and reaction processes derived from electrochemical reaction equations includes: In the reactive diffusion region of the solution within the target sensor, the change in reactant concentration arises from the increase in mass transfer and the decrease in reaction. Therefore, the expression for reactant concentration in relation to the mass transfer and reaction processes is: ； in, Indicates reactant concentration. This indicates the increase in reactant concentration during the mass transfer process. This indicates the concentration of reactants consumed during the reaction process.

3. The electrochemical sensor reaction-diffusion equation optimization system according to claim 1, characterized in that, The relationship between reactant change rate and mass transfer rate and reaction rate, derived from the relationship between reactant concentration and mass transfer process and reaction process, includes: Based on the reactant concentration being the difference between the reactant concentration increased during the mass transfer process and the reactant concentration consumed during the reaction process, the relationship between the reactant change rate and the mass transfer rate and reaction rate is obtained as the difference between the mass transfer rate and the reaction rate, expressed as: ； in, Indicates the mass transfer rate. Indicates the reaction rate. The partial derivative sign is indicated by t, and time is represented by t. This indicates the increase in reactant concentration during the mass transfer process. This indicates the concentration of reactants consumed during the reaction process.

4. The electrochemical sensor reaction-diffusion equation optimization system according to claim 1, characterized in that, The expression for the mass transfer rate derived from the Nernst-Planck equation includes: The mass transfer process is determined by diffusion, convection, and electromigration using the Nernst-Planck equation. In an electrochemical sensing system where the target sensor solution remains quiescent and contains a significant amount of supporting electrolyte, the effects of convection and electromigration on the reactants are neglected, resulting in a diffusion-based mass transfer rate expression: ； in, Indicates the mass transfer rate. Indicates the diffusion coefficient of the reactants. Indicates the concentration of reactants.

5. The electrochemical sensor reaction-diffusion equation optimization system according to claim 1, characterized in that, The expression for the reaction rate is: ； in, Indicates the reaction rate. Represents the reaction rate constant. Indicates the reaction order. Indicates the concentration of reactants.

6. The electrochemical sensor reaction-diffusion equation optimization system according to claim 1, characterized in that, The expression for the reaction-diffusion equation is: ； in, Represents the reaction rate constant. Indicates the reaction order. Indicates reactant concentration. Indicates the diffusion coefficient of the reactants. The symbol represents the partial derivative, and t represents time.

7. The electrochemical sensor reaction-diffusion equation optimization system according to claim 1, characterized in that, The step of obtaining the target reaction-diffusion equation by performing partial differential calculation on the reaction-diffusion equation using a dimensionless method includes: Based on the reaction-diffusion equation, a dimensionless method expression is set, and the dimensionless expression is substituted into the reaction-diffusion equation to obtain the dimensionless target reaction-diffusion equation. The finite difference method is then used to solve the equation and obtain the relationship between the reactant concentration and the parameters in the equation. The dimensionless method expression includes: , , ; in, Indicates the concentration of reactants in the solution. Indicates the width of the reaction diffusion region; The target reaction diffusion equation is shown below: ； in, Indicates the reaction order. This indicates the dimensionless concentration of reactants. It represents the ratio coefficient between the reaction term and the diffusion term.

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