The Kuznetsov Tensor as a Foundation of the Electric Double Layer Theory at the Metal–Electrolyte Interface

1. General Formulation and Choice of Mathematical Framework

We consider an electrochemical system of the metal–electrolyte type in which an electric double layer (EDL) is formed, characterized by a spatially inhomogeneous distribution of charge, energy, and chemical potentials [1,2,3,4]. Unlike classical models based on a scalar electrostatic potential φ(r) [1,2,3,4], the present work adopts a tensor-based approach that allows one to account for interfacial anisotropy, partial charge transfer [10,15], and specific ion adsorption [5,6,19].

The fundamental object of the theory is the Kuznetsov tensor Kμν [18], interpreted as a generalized interfacial field tensor combining contributions from energy, momentum, charge, and adsorption effects [11]. The spacetime indices (μ,ν=0,1,2,3) are introduced formally, while the temporal component is treated in the quasistationary limit [18].

2. Definition of the Kuznetsov Tensor for the EDL

The Kuznetsov tensor for an electrochemical interface is defined as [11,18]:

$$K^{\mu v}=T^{\mu v}+{\lambda }_1{J}^{\mu }{J}^v+{\lambda }_2{\nabla }^{\mu }{q}^v+{\lambda }_3{\text{Σ}}^{\mu v},$$

where:

Tμν is the energy–momentum tensor of the coupled electron–ion subsystem [11];

J^μ is the four‐current of charge, accounting for fractional effective ionic charges [10,15];

q^v is the vector describing partial charge transfer between the ion and the metal surface [10,15];

∑^μv is the adsorption tensor associated with chemical bonding of ions to the surface [5,6,19]; λ_i are dimensionless coefficients determined through parameterization based on quantum‐chemical calculations [9–12].

This representation reflects the fact that electrostatic, chemical, and energetic effects in the EDL cannot be separated without loss of essential physical information [5,6,19].

3. Equilibrium Condition of the Electric Double Layer

The central assumption is that the stationary state of the EDL satisfies the tensorial balance condition [11,18]:

$${\nabla }_{\mu }{K}^{\mu v}=0.$$

This equation replaces the classical Poisson equation [1,2,3,4] and expresses not merely charge neutrality, but the compensation of energy, momentum, and chemical interaction fluxes in the vicinity of the interface [5,6,19]. Physically, this implies that EDL equilibrium is achieved not by vanishing gradients, but by their mutual balance [18].

In a one-dimensional approximation normal to the electrode surface (the (x)-direction), the condition reduces to

$$\frac{d}{dx}{K}^{xx}=0$$

indicating spatial constancy of the normal component of the interfacial field tensor.

4. Partial Charge Transfer and Fractional Charges

In classical electrochemistry, the ionic charge is assumed to be an integer (z e) [1,2,3,4]. However, quantum-chemical interaction with a metallic surface leads to redistribution of electronic density, yielding an effective ionic charge [10,15]:

q_eff = ze − Δe,

where Δe is the fraction of an electron transferred to the metal.

Within the tensor formalism, this is reflected by the modified charge current Jμ [10,15].

J^μ = q_effnu^μ

where n is the ionic concentration and u^μ is the mean ionic velocity. The quadratic contribution J^uJ^v in the Kuznetsov tensor produces a nonlinear dependence of the energetic and capacitive characteristics of the EDL on the degree of charge transfer [10,15], which is fundamentally inaccessible in linear scalar models [1–4].

5. Specific Adsorption as a Tensorial Effect

Specific ion adsorption at the electrode surface is described by the adsorption tensor Σμν, which accounts for the directionality of chemical bonds and local deformation of the electronic cloud of the metal. In the stationary regime,

$${\nabla }_{\mu }{\text{Σ}}^{\mu v}=0$$

indicating the presence of internal sources and sinks of momentum and energy that are compensated by other components of the Kuznetsov tensor [11,18]. This mechanism naturally explains ion competition in multicomponent electrolytes [14], where one ionic species displaces another from the interfacial region not due to purely electrostatic effects, but because of differences in their adsorption-related tensor contributions [5,6,19].

6. Relation to Classical Models

Under simplifying assumptions, such as isotropy of the medium [1,2,3,4], the Kuznetsov tensor reduces to the classical Poisson equation, and the Gouy–Chapman–Stern models are recovered [1,2,3,4].

$$\begin{array}{c}\text{Δ}e\to 0\\ {\text{Σ}}^{\mu v}\to 0\end{array}$$

isotropy of the medium, the Kuznetsov tensor reduces to

$$K^{\mu v}\approx \epsilon E^{\mu }{E}^v,$$

and the condition ${\nabla }_{\mu }{K}^{\mu v}=0$ becomes equivalent to the classical Poisson equation

$${\nabla }^2\phi =-\frac{\rho }{\epsilon }.$$

Thus, the Gouy–Chapman–Stern models are recovered as limiting cases of the tensor theory and are applicable only within a restricted class of systems.

7. Physical Interpretation and Discussion

The proposed description allows the EDL to be interpreted not as a set of geometrically separated layers, but as a self-consistent tensor field localized near the metal–electrolyte interface [18]. Within this interpretation, the EDL capacitance is determined not only by the potential distribution, but also by the structure of the Kuznetsov tensor, which includes contributions from adsorption [5,6,19] and fractional charge transfer [10,15].

This framework explains experimentally observed deviations from classical capacitance laws, asymmetry with respect to potential sign reversal, and nonlinear behavior of electrolyte mixtures [3,5,6,7,14]. Moreover, the tensor approach enables direct incorporation of quantum-chemical data [9,10,11,12] into continuum models without introducing empirical correction parameters.

8. Summary of the Discussion

The present analysis demonstrates that the use of the Kuznetsov tensor provides a rigorous mathematical description of the EDL consistent with the actual physics of electrochemical interfaces [18]. The proposed formalism overcomes conceptual limitations of classical models [1,2,3,4] and establishes a foundation for further development of multiscale theories in electrochemistry and electrocatalysis [5,6,7,19].