# A Variational Approach to Electrostatics of Polarizable Heterogeneous Substances

Authors: Michael Grinfeld; Pavel Grinfeld
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

We discuss equilibrium conditions for heterogeneous substances subject
to electrostatic or magnetostatic effects. We demonstrate that the
force-like aleph tensorℵij and the energy-like beth tensor ℶij for polarizable
deformable substances are divergence-free: ∇iℵij=0 and ∇iℶij=0.
We introduce two additional tensors: the divergence-free energy-like gimel
tensor ℷij for rigid dielectrics and the general electrostatic gamma tensor Γij which is not divergence-free.
Our approach is based on a logically consistent extension of the Gibbs
energy principle that takes into account polarization effects. While the
model is mathematically rigorous, we caution against the assumption that
it can reliably predict physical phenomena. On the contrary, clear models
often lead to conclusions that are at odds with experiment and therefore
should be treated as physical paradoxes that deserve the attention of the
scientific community.


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## Body

## 1. Introduction

The goal of this paper is to present a logically consistent extension of the Gibbs variational approach [1] to elastic bodies with interfaces in the presence of electromagnetic effects. Logical consistency and mathematical rigor, in other words, clarity, do not always lead to physical theories that accurately predict experimentally observable phenomena. In fact, Niels Bohr, who stated that clarity is complimentary to truth, may have thought that the clearer the model is, the less likely it is to be a reflection of reality, but, nevertheless, establishing clarity is an essential step along the path towards understanding. This paper pursues clarity and, therefore, poses the acute question of experimental verifiability.Gibbs suggested building an analysis of equilibrium of heterogeneous substances by analogy with classical statics. He transformed the principle of minimum energy by replacing mechanical energy with internal energy at fixed total entropy. Gibbs’ analysis incorporated phase transformations in heterogeneous systems into a general variational framework. Gibbs modeled phase transformations simply as an additional degree of freedom in his variational approach. In the Gibbs analysis, the conditions of phase equilibrium arise asnatural boundary conditions (in the sense of variational calculus [2]) corresponding to the additional degree of freedom.Simplicity was one of Gibbs’ primary objectives as he stated it in his own words [3]: “If I have had any success in mathematical physics, it is, I think, because I have been able to dodge mathematical difficulties. Anyone having these desires will make these researches…” Perhaps foreseeing possible misinterpretations of the mathematical implications of his method, Gibbs also wrote [3], “A mathematician may say anything he pleases, but a physicist must be at least partially sane.”Let us now turn to the world of electromagnetism. One of the major achievements of Maxwell’s theory [4] was the successful introduction of the stress tensor, originally found in continuum mechanics, to the concept of ether, the agent of electrical and magnetic forces. Historically, Maxwell’s theory was not as readily accepted as one might imagine. On the contrary, several leading thinkers, including Helmholtz, rejected his theory either partially or completely. In [5], Poincaré emphasized that certain contradictions are inherent in Maxwell’s theory.Maxwell himself pointed out a number of difficulties in his theory. Of relevance to this paper is his statement [4]: “I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in dielectrics.” Many efforts have since been made to fix this shortcoming. Many of those efforts are variational in nature since one of the most effective ways of coping with mathematical difficulties and logical inconsistencies is to insist on a variational formulation. Among the many textbooks, lectures, and monographs on electromagnetism [6–16], there are many that discuss the variational perspective, and once again it is clear that there is no consensus on the right approach.One of the pioneers of variational methods in electromagnetism was Gibbs himself. Gibbs studied the problem of equilibrium configuration of charges and discovered that (what researchers now call) thechemical potential of a charged material particle should be supplemented with an additional term qφ (attributed by Gibbs to Gabriel Lippmann), where q is the electric charge of the particle and φ is the electrostatic potential. This is a very rough sketch of Gibbs’ vision. For instance, Gibbs himself has never used the termchemical potential and did not assign the corresponding quantity any profound meaning, which was understood only much later. The variational approach topolarizable substances was most likely pioneered by Korteweg [17] and Helmholtz [18].Gibbs modeled heterogeneous systems, or what he calledheterogeneous substances, as macroscopic domains separated by mathematical surfaces. The difficulty in carrying over Gibbs’ ideas to electromagnetism is that the analysis of singular interfaces in electrostatics and magnetostatics is much more challenging than it is in continuum mechanics. Even Lorentz chose to avoid the analysis of heterogeneous systems, stating in the preface to his classical treatise [19] that he does not want to struggle with the boundary terms. Many of the difficulties that were faced (or should have been faced) by Lorentz can be overcome with the help of the calculus of moving surfaces.In this paper we make a new attempt at extending the Gibbs variational framework to electrostatics. Our approach is very simple and entirely straightforward conceptually. Contrary to many of the prior attempts ([12–14], to name just three), we explicitly exclude the electric field and the electric displacement from the list of independent thermodynamic variables. Instead, we account for polarization (or magnetization) by adding a single term to the “traditional” free energy for a thermoelastic system. The additional term represents the potential energy accumulated in the electrostatic field over the entire space. Different authors choose this term differently: E2, E·D, and so forth. We choose the integrand in the simplest form E2. We build our approach on the exact nonlinear theory of continuum media and rely on Eulerian coordinates as the independent spatial variables.


## 2. The Gibbs Thermodynamics in a Nutshell

According to the modern interpretation of Gibbs, the chemical potentialμ governs the equilibrium between the liquid and the vapor phases with respect to mass exchange between them. Equilibrium heterogeneous systems must satisfy a number of conditions at the phase interface. The first two conditions, those of thermal equilibrium (temperature T is continuous across the interface (and, of course, spatially constant)) and mechanical equilibrium (pressure p is continuous across the interface), are satisfied by all equilibrium two-phase systems, whether or not the phases are different states of the same substance subject to a phase transformation. Letting the brackets []-+ denote the jump discontinuity in the enclosed quantity across the phase interface, we write these conditions as (1)T-+=0,p-+=0.Additionally, when the interface is subject to a phase transformation, the chemical potential μ is continuous across the interface(2)μ-+=0.This equation is interpreted as equilibrium with respect to mass exchange between the phases. The chemical potential μ is given by (3)μ=∂ϵρ∂ρ,where ρ is density and ϵρ is the free energy per unit volume.In many physical systems, equilibrium with respect to mass exchange is attained over much longer time scales than thermal and mechanical equilibria. The dynamics of mass exchange in such systems is often well described by a quasi-static approximation, which assumes that the system maintains thermal and mechanical equilibria throughout evolution; that is, (1) are continuously satisfied, while equilibrium equation (2) is replaced with the following equation for the mass flux J:(4)J=-Kμ-+,where K>0 is a kinematic quantity, determined empirically or by some nonthermodynamic theory.


## 3. A Variational Approach to Electrostatics of Heterogeneous Systems

We will now briefly summarize a variational framework for electrostatics of heterogeneous systems which was first described in [20–23]. The presented model, based on the choice of the functional E in (11) and the list of independent variations, is correct only in the mathematical sense; that is, it is logically consistent. Other authors [10, 12, 13, 24] make different choices of energy functionals and sets of independent variations and arrive at different results.Our description uses the framework of tensor calculus [25]. We refer the space to coordinates zi. By convention, we omit the superscript i when the coordinate appears as an argument of a function. We denote the covariant and contravariant ambient metric tensors by zij and zij and the ambient covariant derivative by ∇i.Figure1 illustrates the configuration of our system. Suppose that the domain Ωd=Ωd+∪Ωd- is occupied by solid heterogeneous dielectric media with specific (per unit volume) dipole momentum Piz. The domain Ωq is occupied by a stationary electric charge distribution qz. The two subdomains Ωd+ and Ωd- are occupied by two different substances or two different phases of the same substance. They are separated by the interface S2.Figure 1
A heterogeneous system with distributed electric charges and dipoles.Suppose thatUiz is the displacement field of the material particles, ρz is the actual mass density, φz is the electrical potential, (5)Eiz≡-∇iφzis the electrical field, and (6)Di=Ei+4πPiis the electric displacement.For the sake of simplicity, we assume that the system is kept under fixed absolute temperatureT and denote the elastic (internal) energy density ψ of the dielectric substance by (7)ψ∇jUi,Pk.Of course, this elastic energy is actually the free energy density of the system.The equilibrium of the system is governed by Poisson’s equation:(8)∇i∇iφ=4πq,subject to the boundary conditions(9)φ-+=0,NiDi-+=0,across the interfaces (Ni is the unit normal), while at infinity the electrical potential vanishes: (10)φ∞=0.The total energyE of the system is given by the integral (11)E=∫ρψ+18πEiEidΩ,which extends over the entire space.According to the principle of minimum energy, we associate equilibrium configurations with stationary points of the total energyE. In what follows, we use the technique of variation of the energy functionals in the Eulerian description presented in detail in [21, 22, 26]. Suggested procedures for analyzing the equilibrium and stability conditions for two-phase heterogeneous systems can be found in [27–30].We complete the description of the variational principle by presenting the list of quantities treated as the independent variations:(i)
virtual velocityfiz of the material particles,(ii)
virtual velocitiesC2 and C3 of the interfaces S2 and S3,(iii)
variationδPiz of the dipole momentum at the point with coordinates zi.The geometry presented in Figure1 was analyzed in [21, 28] which dealt with nucleation on stationary ions of liquid condensate from the surrounding gaseous phase. When the domain Ωq is rigid, the virtual velocities of the deformable liquid phase should satisfy the boundary constraint(12)NifiS1=0.


## 4. The Bulk Equilibrium Equations of Deformable Polarizable Substances

In this section, we summarize the results and refer the reader to the relevant references for the corresponding derivations.Separating the independent variations in the volume integral of the first energy variation, we arrive at the following equilibrium equations [22, 27]: (13)-∇iζik+ρψPi∇kPi=0,ρψPi=Ei,where ψPi=∂ψ/∂Pi, theformal stress tensorζmk is defined as (14)ζij≡ρ∂ψ∂∇iUkAk··j,and the tensor A·ji· is given by (15)A·ji·≡δji-∇jUi.Combining (13), we arrive at the equilibrium bulk equation (16)-∇iζik+Ei∇kPi=0.Using the equations of electrostatics, it can be shown that (16) can be rewritten as a statement of vanishing divergence:(17)∇iζij-zij14πEkDk-18πEkEk+14πDiEj=0.For nonpolarizable substances, the formal stress tensorζij coincides with the Cauchy stress tensor in the Eulerian description. Relationship (17) generalizes to the celebrated Korteweg-Helmholtz relationship for liquid dielectrics [6, 7, 10–13, 24] in the case of nonlinear electroelasticity.We can rewrite (17) as (see [22, 23, 27]) (18)∇iℵij=0,where the aleph tensor ℵij, given by (19)ℵij≡ρ∂ψ∂∇iUkAk··j+18πEkEk-14πEkDkzij+14πDiEj,can be thought of as the stress tensor of a polarizable substance. We can rewrite the aleph tensor ℵij as (20)ℵij≡ζij+Γij,where the electrostatic gamma tensor Γmk is given by (21)Γij≡18πEkEk-14πEkDkzij+14πDiEj.Equation (17) can be written in another insightful form:(22)∇iζij=-∇iΓij.In polarizable deformable substances, neither one of the tensors, ζij or Γij, is divergence-free.The gamma tensorΓij can be also considered as one of the many possible generalizations of the Maxwell stress tensor Tij:(23)Tij≡-18πEkEkzij+14πEiEj,since Γij coincides with Tij when polarization vanishes. Other possible generalizations of the Maxwell stress tensor, (24a)T1ij≡-18πElDlzij+14πDiEj,(24b)T2ij≡-18πElDlzij+14πDjEi,(24c)T3ij≡-18πElDlzij+18πDiEj+DjEi,are perhaps more aesthetically appealing than the gamma tensor Γij. We believe that the advantage of the gamma tensor over other possible generalizations is its variational origin and its ability to help address the issue of stability based on the calculation of the second energy variation.One more useful tensor for polarizable materials is the beth tensorℶ·ji·, or thetensor of electrochemical tensorial potential. It is defined by (25)ℶ·ji·≡ρψzik-ℵik+ΓikBkj,where the tensor Bkj is the matrix inverse of Akj defined in (15). As we show below, the beth tensor ℶ·ji· satisfies the condition of zero divergence(26)∇iℶ·ji·=0,similarly to the aleph tensor ℵij. The beth tensor ℶ·ji· can be rewritten as(27)ℶ·ji·=ρBkjχik,where χik is the Bowensymmetric tensorial chemical potential(28)χij=ψzij-1ρζij=ψzij+1ρℶij-ℵij.Thesymmetric tensor χij should be distinguished from the typicallyasymmetric tensorial chemical tensor μij:(29)μij=z∘kizlkχjl,where z∘mi is the contravariant metric tensor of theinitial configuration.


## 5. Conditions at the Interfaces

Boundary conditions depend on the various characteristics of the interfaces. Interfaces can differ by their mechanical, or kinematic, properties and whether or not they are subject to phase transformations. We refer to interfaces that satisfy the kinematic constraint(30)Ui-+=0,ascoherent interfaces. The following condition for the aleph stress tensor ℵij is satisfied by equilibrium configurations at coherent interfaces:(31)Niℵij-+=0.If, in addition to coherency, the boundary is a phase interface, the condition of phase equilibrium includes the beth tensor ℶij: (32)Niℶij-+=0.It makes sense, then, to call the beth tensor ℶij theelectrochemical tensorial potential for coherent interfaces in deformable substances because (32) is analogous to the equilibrium condition for the tensorial chemical potential.


## 6. Nonfrictional Semicoherent Interfaces

By definition,nonfrictional semicoherent interfaces are characterized by the possibility of relative slippage. Nonfrictional semicoherent interfaces also may or may not bephase interfaces. Regardless, the following conditions of mechanical equilibrium must hold: (33)Niζ±ij=-Njp±,NiNjΓij-+=p-+.Atphase nonfrictional incoherent interfaces, an additionalmass exchange equilibrium condition must be satisfied:(34)NiNjψzij+1ρℶij-ℵij-+=0.


## 7. Phase Interfaces in Rigid Dielectrics

When dealing with rigid solids, allmechanical degrees of freedom disappear and the internal energy depends only on the polarization vector Pi (and, unless it is assumed to be constant, temperature T). At the phase interface, the condition of phase equilibrium reads (35)NiNjℷij-+=0,where the gimel energy-like tensor ℷij, theelectrostatic tensorial chemical potential for rigid dielectrics, is defined by (36)ℷij≡ϵPzij+Γij,where ϵ≡ρψ is the free energy density per unit volume (and we once again suppress the index in Pi because it now appears as an argument of a function). We refer to the gimel tensor ℷij as theelectrostatic tensorial chemical potential because it plays the same role as the chemical potential μ in the classical heterogeneous liquid-vapor system. Contrary to the gamma tensor Γij, the gimel tensor ℷij is divergence-free:(37)∇iℷij=0.One can analyze models in which the polarization vectorPi is fixed [20]. Then ψ± are spatially constant but may still depend on temperature.


## 8. Divergence-Free Tensors in Electrostatics

We present a proof of the last of the three equations (18), (26), and (37) of vanishing divergence. The remaining two identities can be demonstrated similarly. First, let us rewrite the gimel tensor ℷij as follows:(38)ℷij=ϵPzij+18πEkEk-14πEkDkzij+14πDiEj.For the first term in (38), we have(39)∇iϵPzij=zij∂ϵP∂Pk∇iPk.Using the thermodynamic identity(40)∂ϵP∂Pi≡Ei,we can rewrite (39) as (41)∇iϵPzij=Ei∇jPi.For the second term in (38), we have(42)∇izij18πEkEk-14πEkDk=-14πEk∇jEk-Pk∇jEk-Ek∇jPk,which can be seen from the following chain of identities: (43a)2nd  term=∇izij18πEkEk-14πEkDk(43b)=∇k18πEkEk-14πEkDk(43c)=14πEk∇jEk-∇jEkDk-Ek∇jDk(43d)=14πEk∇jEk-∇jEkEk+4πPk-Ek∇jEk+4πPk(43e)=-14πEk∇jEk-Pk∇jEk-Ek∇jPk.For the third term in (38), we have (44)∇i14πDiEj=14πDi∇iEj=14πEi∇iEj+Pi∇iEj.Combining (41)–(44), we find (45a)∇iℷij=∇iϵPzij+zij∇i18πElEl-14πElDl+14π∇iDiEj(45b)=Ei∇jPi-14πEi∇jEi-Pi∇jEi-Ei∇jPi+14πEi∇iEj+Pi∇iEj(45c)=-14πEi∇jEi-Pi∇jEi+14πEi∇iEj+Pi∇iEj.Finally, using the symmetric property ∇iEj≡∇jEi, we arrive at identity (37).


## 9. Quasi-Static Evolution

A quasi-static evolution can be postulated by analogy with (4). In the case of nondeformable phases, it reads (46)J=-KNiNjℷij-+.The same approach can be applied to the case of an isolated domain with fixed total volume yet subject to rearrangement. In this case, the evolution equation should be slightly modified to take into account surface diffusion. Figure 2 illustrates an implementation of this approach in the two-dimensional case. The quasi-static evolution of originally circular domain and fixed polarization vector leads to elongation in the direction of polarization vector Pi and, eventually, to a morphological instability.Figure 2
Onset of a morphological instability in a quasi-static evolution of a domain filled with dipoles of fixed polarization.


## 10. Conclusion

We discussed a phenomenological variational approach to electrostatics and magnetostatics for heterogeneous systems with phase transformations. Although we focused on electrostatics, almost all of the presented results are also valid for magnetostatics. Our approach is an extension of the Gibbs variational method, as it was interpreted in [26].The demand of having simultaneously a logically and physically consistent theory remains to be the main driving force of progress in thermodynamics. The suggested approach leads to the mathematically rigorous self-consistent results. Now it has to prove its viability in direct comparison with experiment. That may prove to be difficult, but real progress is only possible when theory and experiment challenge each other.



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