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Questions and difficulties are presented pertaining to the quantitative characterization of the electric field in certain scenarios. Specific examples concerning electrolytes are explored. Newton’s third law is invoked and the concept of mobile charge density is presented in relation to free charge density and bound charge density. The notion of mobile charge density is utilized to develop a theory and model for the electric field coupled with electrolytic properties and transport. Validations, simulations, and implications of the model are presented and discussed, including: is it possible to extend Maxwell’s equations to a more generalized form?

Maxwell extended Ampere’s circuital law by adding a time derivative electric displacement term accounting for the polarization of material in a capacitor, thus balancing an incoming electric current [

Publications and reviews regarding electrochemical systems and the coupled electric field are plentiful. Prentice and Tobias [

In 1864 James Maxwell presented nine equations summarizing all known laws on electricity and magnetism [

and the continuity equation:

For a linear isotropic material the Electric Displacement Field is [

where, the second term on the right-hand-side is the polarization vector,

Newman and Thomas-Alyea state that “no quantitative characterization or measure of the difference of electrical state of two phases has yet been given when the phases are of different chemical composition” [

Poisson’s equation, describing the electric field, may be derived from Gauss’s law in Maxwell’s equations. Poisson’s equation describes the Laplacian of electric field as a function of charge density and permittivity. Therefore, to calculate an electric field using Poisson’s equation the charge density distribution must be known or predicted/modeled simultaneously with an additional equation. An equation, such as the Nernst-Planck equation, is capable of modeling flux and ionic current to predict a charge density distribution in electrolytes. An extensively published model for electrolytes couples these two equations and is commonly known as the Nernst-Planck-Poisson equation set, describing coupled transport and the electric field. However, in certain scenarios this method has the disadvantage of requiring very small time steps, sometimes approaching the order of approximately 10^{−12} seconds, as the resulting equations are considered to be mathematically stiff [

Due to difficulties in solving transport equations coupled with Poisson’s equation other theories have been presented upon which models may be founded. One widely used theory in the open literature is dilute solution theory [

Concentrated solution theory [

The objective of this research is to overcome difficulties in conventional theory describing electrolytes and the coupled electric field. Specifically, can a general definition of electric potential be developed that requires fewer assumptions than conventional theory and also requires fewer experimentally determined transport properties? A secondary objective of this theory is to investigate the relationship of this new theory with Maxwell’s equations and to determine if Maxwell’s equations may be further generalized.

This section will explore the application of coupled equations for transport and the electric field. Although examples are provided, the concepts presented are expected to extend beyond the case of electrolytes and beyond the specific equations utilized. In Section 2.1 and Section 2.2, equations are arranged such that phenomenological issues may be apparent. These issues are not incorporated into the final model presented in this paper. Instead, the author proposes these difficulties are inherent in conventional theory. The only conclusion drawn from Section 2.1 is that the Nernst-Planck equation depends upon a time step to predict an electric displacement and Poisson’s equation does not. Section 2.2 concludes that to create a parallel kinetic model a poor assumption regarding displacement is required. Section 2.3 explores similarities between kinetic models regarding displacements and conventional models for electric displacement. Section 2.4 utilizes these similarities to define a new form of mobile charge density and Section 2.5 develops a model for the electric field when the actions causing electric displacement are described by a transport equation. Section 2.6 presents the Inherent Charge Density model.

A very well published method for modeling charge in an electrolyte is the Nernst-Planck-Poisson equation set. This method couples the Nernst-Planck equation with Poisson’s equation for the electric field. Although the following investigation will be focused on the Nernst-Planck-Poisson equation set, it should be equally valid for any dynamic transport equation coupled with electrostatics.

The Nernst-Planck equation describes flux. In this description, the two phenomena creating flux are electro migration due to an electric field and diffusion due to a concentration field and are represented in the equation:

Poisson’s equation, developed from Gauss’s law, is:

When no reactions are occurring, conservation of species provides:

Taking the divergence of Equation (7), combining with Equation (9), multiplying both sides by

where,

Substituting the definition of electric displacement from Maxwell’s equations (Equation (3)) into Equations (8) and (11) gives a rearranged form of the Nernst-Planck-Poisson equation set:

When applying the Nernst-Planck-Poisson equation set, the transport equation must be integrated over a time step to create a charge density distribution utilized as the independent variable in Poisson’s equation to then calculate the resultant electric field. With this in mind, for a first time step where the composition of the electrolyte is initially homogeneous, isotropic and electroneutral, Equations (12) and (13) simplify to:

Interestingly, comparing these equations shows that both Equation (14) and (15) demonstrate an electric displacement proportional to the electric field; however, three differences are apparent:

1) The proportionalities are of different sign

2) The proportionality constants may be different

3) The electric displacement predicted by Equation (14) depends upon the integration time step,

Several arguments might be made to theoretically explain: 1) why the proportionalities are of different sign and 2) why the proportionality constants may be different. These items will be dealt with later in this paper. At this stage the discussion is limited to the fact that the transport equation depends upon integration over a time step and Gauss’s law does not. Therefore, for Equations (14) and (15) to obey Newton’s third law that every reaction has an equal and opposite reaction, the time constant must be specified. Specifically, where the electric field and conductivity are assumed constant over the integration time period:

The following example illustrates underlying issues with the Nernst-Planck- Poisson equation set and tests Equation (16). Consider a rectangular vessel containing 0.1 mole/m^{3} of AgNO_{3} and split into two cubic control volumes with 0.5 cm sides (shown in

these two control volumes for 5 seconds. Under these conditions the Nernst- Planck equation predicts a charge density of 266.62 C/m^{3} in each control volume after 5 seconds. According to Poisson’s equation, this charge density corresponds to a voltage of −9.614 × 10^{6} V across the system. Alternatively, if the time step is calculated according to Equation (16), then,

applied for the same situation over 5.201 × 10^{−7} seconds, 2.773 × 10^{−5} C/m^{3} of charge is predicted at each node. According to Poisson’s equation, this charge density corresponds to a voltage of −1.000 V. In this case, incorporating Equation (16) ensured the adherence of Newton’s third law.

This section illustrated how the charge density predicted by the Nernst-Planck equation depends on the time over which the resulting transport equation is integrated and Poisson’s equation does not. This time step issue must be true for any dynamic transport/flux equation coupled with a static equation and is not limited to the specific equations or scenario provided here. If the correct time step is not calculated and used for every an iteration of the Nernst-Planck-Poi- sson equation set, Newton’s third law may not be guaranteed.

The following example utilizes a poor assumption to develop models regarding displacement that may be similar to models for electric displacement outlined in Section 2.1. It will be shown how both equation sets depend similarly on a specific time period to obey Newton’s third law. Consider a cart moving horizontally at a constant velocity in a linearly resistive medium, shown in

The linear resistive force is given:

For the system to be at steady-state and constant velocity, the spring and drag

forces must be equal to some force,

For Equations (19) and (20), Newton’s third law is only guaranteed when:

This cart example demonstrates a poor assumption: that the value for displacement of the cart may be interchanged with value for displacement of the spring. A better method for modeling the cart would be to realize that although the same applied force is balanced with both the stretching of the spring and the velocity of the cart, the resulting displacements,

linearly resistive medium.

Multiple forms of charge density may be simultaneously apparent in an electrolyte. Do these forms actually exist, or are they simply manifestations of phenomenological models and numerical procedures? Does it matter? In the phenomenological model presented here, it is suggested that ions in an electrolyte may react to an electric field in the manner shown in

This section will utilize the concept that multiple forms of charge density may exist in an electrolyte; however, an electrolytic transport equation will predict mobile charge density created by the movement of ions and transport equations do not generally simulate the polarization of bulk molecules. A transport equation which considers flux due to electromigration, diffusion, and activity coefficients is given:

Additionally, a material balance considering reactions:

By combining Equations (23) and (24), multiplying both sides by

where,

Therefore, from

Equation (26) can be rearranged to provide the electric field resulting from transport calculated using Equation (25); in other words Equation (26) can be rearranged to calculate the electric field resulting from diffusion potential, activity coefficients, creation or consumption of ions, and mobile charge density.

The previous section presented a flux equation, coupled with the continuity equation, and an equation for the electric field developed by extending the phenomenological model of Maxwell. These two equations will now be rearranged and presented together as an equation set describing the transport (from combining Equation (23) and (24)) and coupled electric field, respectively,

(where

This equation set is an Inherent Charge Density model and it may be implemented multi-dimensionally using the following boundary conditions, for Equation (27) and (28), respectively, in the appropriate dimensions:

Validations and simulations already conducted for the Inherent Charge Density model are too lengthy for a single publication. Therefore, this section will present some new validations with a brief mention of other validations available in the open literature. Interestingly, the same governing equations and the same boundary conditions may be used for all scenarios. The first test of the Inherent Charge Density model and underlying theory is to examine whether the predicted electric field is dependent upon the time step over which the transport equation is applied. Therefore, Equations (27) and (28) are applied to the same problem as examined earlier in this paper, presented in ^{3} in each control volume after 5 seconds. Upon application of Equation (28) over a time of 5 seconds a value of −1.000 V is calculated for the electric field. This demonstrates that the strength of the response of the electrolyte, in terms of electric field, is of similar magnitude to the strength of the applied action and Newton’s third law is obeyed.

Additional validations and results from numerical simulations using the Inherent Charge Density Model have been presented. Kennell [

Kennell [

The Inherent Charge Density Model has also been applied to more complex scenarios, including a lithium-ion battery undergoing charging, with details available elsewhere [^{6} stable sequential iterations of the Inherent Charge Density model to create the data displayed in ^{−2}. A more complete examination of the dynamic electric

field coupled with ionic transport and non-homogeneous properties in the electrolyte and electrodes can be found elsewhere [

Investigations demonstrated that an electric current integrated over a time period causes an electric displacement. Divergence of this electric displacement may cause charge density. Gauss’s law assumes charge density is stationary and in balance with the electric field. Ohm’s law assumes charges are in uniform motion. By assuming Newton’s third law should be valid for a charge/ion, whether stationary or in motion, a transport equation was modified by changing the sign of the charge density term, creating an Inherent Charge Density Model. The Inherent Charge Density model was developed without assuming electroneutrality or electrostatics. The main weaknesses of the model pertain to the flux equation upon which the development was founded. However, the theory and process demonstrated here could be applied to alternate transport models. The Inherent Charge Density model was validated for a number of liquid-junctions, balanced ionic current/moving liquid-junction, and a lithium-ion battery. However, two questions remain to be answered:

1) Does the Inherent Charge Density Model conflict with Maxwell’s equations?

2) How can possible discrepancies be resolved?

It may appear that Equation (22) conflicts with the continuity equation from Maxwell’s equations. This is because Equation (28), in the absence of reactions, can be rewritten as:

Equation (31) appears to conflict with Maxwell’s continuity equation, Equation (5), due to the absence of a negative sign. However, it has been shown various times in this paper that a negative sign may be due to Newton’s third law, where a reaction should be equal but opposite to an action.

In the case when Maxwell extended Ampere’s circuital law by adding a term to account for the divergence in electric displacement caused by dynamic bound charge density in a solid capacitor, the electric current and bound charge density were situated at physically adjacent locations; the electric current was occurring in a conductor and the bound charge density was located in a capacitor. The conductor and capacitor were adjacent and connected.

similar model for electric displacements in a conductor adjacent to a capacitor. In

In an electrolyte, electric displacement may be caused by multiple phenomena and charge density may be balanced with the electric field via Ohm’s law and/or Gauss’s law. By assigning a third form of charge density, called mobile charge density, it becomes possible to distinguish between different forms of electric displacement and different phenomena. Kennell [

where, the total charge density is the sum of the free, bound, and mobile charge density:

It is expected that extending Maxwell’s equations to incorporate the charge density calculated from integrating an electric current over a time period will not detract from their general and wide applicability. Also, one of the important accomplishments of Maxwell’s original equations was that the continuity equation could be derived from the equations. It will now be shown that the continuity equation can still be derived from the extended equations show in Section 4.2. The divergence of Equation (33) gives:

The left-hand-side of Equation (39) is zero. Incorporating Equation (34) into Equation (39) gives:

Inserting Equations (36) and (38) into Equation (40) provides the continuity equation, where the sum of free and bound charge density represents the electric displacement occurring within a capacitor:

Therefore, the theory presented in this paper does not appear to be in conflict with Maxwell’s equations. Instead, by acknowledging the fact that electric current applied over a time period causes electric displacement, and incorporating Newton’s third law, a more general form of Maxwell’s equations has been presented that resolves possible discrepancies and/or difficulties.

It was shown that coupling Gauss’s law with a transport equation (such as in the Nernst-Planck-Poisson equation set) may violate Newton’s third law. This is because the electric field calculated by Gauss’s law depends upon the divergence of electric displacement, or charge density, which calculated by integrating the transport equation over a period of time. Not only must the time period be very small, but for Newton’s third law to hold the time step would have to be accurately calculated for each iteration, assuming a dynamic non-homogenous pha- se(s). To rectify this problem the assumption was made that Newton’s third law should be incorporated and all reactions should have an equal and opposite reaction, which resulting in the Inherent Charge Density model. Validations and simulations for the Inherent Charge Density Model demonstrated the model’s ability to model multi-dimensional multi-component electrolytes whilst considered the effects of concentration gradients, the electric field, charge density, activity coefficients, ionic currents, and balanced but physically displaced electrochemical reactions.

The Inherent Charge Density model was developed by assuming a valid transport equation and that Newton’s third law should be applied. This model was validated. The subsequent discussion presented a theory that has not yet been validated, but lays the foundation to view Maxwell’s equations from a different perspective. This theory suggests the electric displacement or charge density calculated from integrating an electric current over a period of time has a different proportionality constant (conductivity) with the electric field than the static charge density balanced in steady-state displacement with the electric field (permittivity). In other words, both Ohm’s law and Gauss’s law need not be simultaneously assumed to represent the same electric displacement caused by the same electric field. This modification does not conflict with Maxwell’s equations. Instead, this modification extends Maxwell’s equations to a more general form and it is expected that the modification will not change the generally wide applicability of these equations.

St. Peter’s College, Muenster, SK, provided publishing funding. The author would like to thank the following researchers for feedback and advice during the creation of this manuscript: Professor Richard Evitts, Dr. Jim Zacaruk, Professor Adam Bourassa, and Professor Allan Dolovich.

Kennell, G. (2017) Free, Bound, and Mobile Charge Density. Journal of Electromagnetic Analysis and Applications, 9, 73-89. https://doi.org/10.4236/jemaa.2017.95007

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