Research ArticleA Variational Approach to Electrostatics of PolarizableHeterogeneous Substances

Michael Grinfeld1 and Pavel Grinfeld2

1Aberdeen Proving Ground US Army Research Laboratory Aberdeen MD 21005-5066 USA2Drexel University Philadelphia PA 19104 USA

Correspondence should be addressed to Michael Grinfeld michaelgreenfield4civmailmil

Received 5 December 2014 Revised 2 April 2015 Accepted 8 April 2015

Academic Editor Giorgio Kaniadakis

Copyright copy 2015 M Grinfeld and P Grinfeld This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We discuss equilibrium conditions for heterogeneous substances subject to electrostatic or magnetostatic effects We demonstratethat the force-like aleph tensor alefsym119894119895 and the energy-like beth tensor ℶ119894119895 for polarizable deformable substances are divergence-freenabla119894alefsym119894119895= 0 and nabla

119894ℶ119894119895= 0 We introduce two additional tensors the divergence-free energy-like gimel tensor ℷ119894119895 for rigid dielectrics

and the general electrostatic gamma tensor Γ119894119895 which is not divergence-free Our approach is based on a logically consistentextension of the Gibbs energy principle that takes into account polarization effects While the model is mathematically rigorouswe caution against the assumption that it can reliably predict physical phenomena On the contrary clear models often lead toconclusions that are at odds with experiment and therefore should be treated as physical paradoxes that deserve the attention ofthe scientific community

1 Introduction

The goal of this paper is to present a logically consistentextension of the Gibbs variational approach [1] to elasticbodies with interfaces in the presence of electromagneticeffects Logical consistency and mathematical rigor in otherwords clarity do not always lead to physical theories thataccurately predict experimentally observable phenomena Infact Niels Bohr who stated that clarity is complimentary totruth may have thought that the clearer the model is theless likely it is to be a reflection of reality but neverthelessestablishing clarity is an essential step along the path towardsunderstanding This paper pursues clarity and thereforeposes the acute question of experimental verifiability

Gibbs suggested building an analysis of equilibrium ofheterogeneous substances by analogywith classical staticsHetransformed the principle of minimum energy by replacingmechanical energy with internal energy at fixed total entropyGibbsrsquo analysis incorporated phase transformations in het-erogeneous systems into a general variational frameworkGibbsmodeled phase transformations simply as an additionaldegree of freedom in his variational approach In the Gibbs

analysis the conditions of phase equilibrium arise as naturalboundary conditions (in the sense of variational calculus [2])corresponding to the additional degree of freedom

Simplicity was one of Gibbsrsquo primary objectives as hestated it in his own words [3] ldquoIf I have had any success inmathematical physics it is I think because I have been able tododge mathematical difficulties Anyone having these desireswill make these researches rdquo Perhaps foreseeing possiblemisinterpretations of the mathematical implications of hismethod Gibbs also wrote [3] ldquoA mathematician may sayanything he pleases but a physicist must be at least partiallysanerdquo

Let us now turn to the world of electromagnetism Oneof the major achievements of Maxwellrsquos theory [4] was thesuccessful introduction of the stress tensor originally foundin continuum mechanics to the concept of ether the agentof electrical and magnetic forces Historically Maxwellrsquostheory was not as readily accepted as one might imagine Onthe contrary several leading thinkers including Helmholtzrejected his theory either partially or completely In [5]Poincare emphasized that certain contradictions are inherentin Maxwellrsquos theory

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 659127 7 pageshttpdxdoiorg1011552015659127

2 Advances in Mathematical Physics

Maxwell himself pointed out a number of difficulties inhis theory Of relevance to this paper is his statement [4] ldquoIhave not been able to make the next step namely to accountbymechanical considerations for these stresses in dielectricsrdquoMany efforts have since been made to fix this shortcomingMany of those efforts are variational in nature since one of themost effective ways of coping with mathematical difficultiesand logical inconsistencies is to insist on a variational formu-lationAmong themany textbooks lectures andmonographson electromagnetism [6ndash16] there are many that discuss thevariational perspective and once again it is clear that there isno consensus on the right approach

One of the pioneers of variationalmethods in electromag-netismwasGibbs himself Gibbs studied the problem of equi-librium configuration of charges and discovered that (whatresearchers now call) the chemical potential of a chargedmaterial particle should be supplemented with an additionalterm 119902120593 (attributed by Gibbs to Gabriel Lippmann) where 119902is the electric charge of the particle and 120593 is the electrostaticpotential This is a very rough sketch of Gibbsrsquo vision Forinstance Gibbs himself has never used the term chemicalpotential and did not assign the corresponding quantity anyprofound meaning which was understood only much laterThe variational approach to polarizable substances was mostlikely pioneered by Korteweg [17] and Helmholtz [18]

Gibbs modeled heterogeneous systems or what he calledheterogeneous substances as macroscopic domains separatedby mathematical surfaces The difficulty in carrying overGibbsrsquo ideas to electromagnetism is that the analysis ofsingular interfaces in electrostatics and magnetostatics ismuch more challenging than it is in continuum mechanicsEven Lorentz chose to avoid the analysis of heterogeneoussystems stating in the preface to his classical treatise [19] thathe does not want to struggle with the boundary terms Manyof the difficulties that were faced (or should have been faced)by Lorentz can be overcome with the help of the calculus ofmoving surfaces

In this paper we make a new attempt at extending theGibbs variational framework to electrostatics Our approachis very simple and entirely straightforward conceptuallyContrary to many of the prior attempts ([12ndash14] to name justthree) we explicitly exclude the electric field and the electricdisplacement from the list of independent thermodynamicvariables Instead we account for polarization (ormagnetiza-tion) by adding a single term to the ldquotraditionalrdquo free energyfor a thermoelastic system The additional term representsthe potential energy accumulated in the electrostatic fieldover the entire space Different authors choose this termdifferently |E|2 E sdotD and so forth We choose the integrandin the simplest form |E|2 We build our approach on the exactnonlinear theory of continuum media and rely on Euleriancoordinates as the independent spatial variables

2 The Gibbs Thermodynamics in a Nutshell

According to the modern interpretation of Gibbs the chem-ical potential 120583 governs the equilibrium between the liquidand the vapor phases with respect to mass exchange between

them Equilibrium heterogeneous systems must satisfy anumber of conditions at the phase interface The first twoconditions those of thermal equilibrium (temperature 119879 iscontinuous across the interface (and of course spatially con-stant)) andmechanical equilibrium (pressure 119901 is continuousacross the interface) are satisfied by all equilibrium two-phase systems whether or not the phases are different statesof the same substance subject to a phase transformationLetting the brackets []+

minusdenote the jump discontinuity in the

enclosed quantity across the phase interface we write theseconditions as

[119879]+

minus= 0

[119901]+

minus= 0

(1)

Additionally when the interface is subject to a phase trans-formation the chemical potential 120583 is continuous across theinterface

[120583]+

minus= 0 (2)

This equation is interpreted as equilibrium with respect tomass exchange between the phases The chemical potential120583 is given by

120583 =120597120598 (120588)

120597120588 (3)

where 120588 is density and 120598(120588) is the free energy per unit volumeIn many physical systems equilibrium with respect to

mass exchange is attained over much longer time scalesthan thermal and mechanical equilibria The dynamics ofmass exchange in such systems is often well described by aquasi-static approximation which assumes that the systemmaintains thermal and mechanical equilibria throughoutevolution that is (1) are continuously satisfied while equi-librium equation (2) is replaced with the following equationfor the mass flux 119869

119869 = minus119870 [120583]+

minus (4)

where 119870 gt 0 is a kinematic quantity determined empiricallyor by some nonthermodynamic theory

3 A Variational Approach to Electrostatics ofHeterogeneous Systems

We will now briefly summarize a variational frameworkfor electrostatics of heterogeneous systems which was firstdescribed in [20ndash23] The presented model based on thechoice of the functional 119864 in (11) and the list of independentvariations is correct only in the mathematical sense thatis it is logically consistent Other authors [10 12 13 24]make different choices of energy functionals and sets ofindependent variations and arrive at different results

Our description uses the framework of tensor calculus[25] We refer the space to coordinates 119911119894 By conventionwe omit the superscript 119894 when the coordinate appears asan argument of a function We denote the covariant and

Advances in Mathematical Physics 3

S1

S2

S3

ΩqΩdminusΩd+

Figure 1 A heterogeneous system with distributed electric chargesand dipoles

contravariant ambient metric tensors by 119911119894119895and 119911119894119895 and the

ambient covariant derivative by nabla119894

Figure 1 illustrates the configuration of our system Sup-pose that the domain Ω

119889= Ω119889+

cup Ω119889minus

is occupied bysolid heterogeneous dielectric media with specific (per unitvolume) dipolemomentum119875

119894(119911)The domainΩ

119902is occupied

by a stationary electric charge distribution 119902(119911) The twosubdomains Ω

119889+and Ω

119889minusare occupied by two different

substances or two different phases of the same substanceThey are separated by the interface 119878

2

Suppose that 119880119894(119911) is the displacement field of the

material particles 120588(119911) is the actual mass density 120593(119911) is theelectrical potential

119864119894 (119911) equiv minusnabla119894120593 (119911) (5)

is the electrical field and

119863119894= 119864119894+ 4120587119875

119894 (6)

is the electric displacementFor the sake of simplicity we assume that the system

is kept under fixed absolute temperature 119879 and denote theelastic (internal) energy density 120595 of the dielectric substanceby

120595 (nabla119895119880119894 119875119896) (7)

Of course this elastic energy is actually the free energy den-sity of the system

The equilibrium of the system is governed by Poissonrsquosequation

nabla119894nabla119894120593 = 4120587119902 (8)

subject to the boundary conditions

[120593]+

minus= 0

119873119894[119863119894]+

minus= 0

(9)

across the interfaces (119873119894 is the unit normal) while at infinitythe electrical potential vanishes

1205931003816100381610038161003816infin = 0 (10)

The total energy 119864 of the system is given by the integral

119864 = int(120588120595 +1

8120587119864119894119864119894)119889Ω (11)

which extends over the entire spaceAccording to the principle of minimum energy we

associate equilibrium configurations with stationary pointsof the total energy 119864 In what follows we use the technique ofvariation of the energy functionals in the Eulerian descriptionpresented in detail in [21 22 26] Suggested procedures foranalyzing the equilibrium and stability conditions for two-phase heterogeneous systems can be found in [27ndash30]

We complete the description of the variational principleby presenting the list of quantities treated as the independentvariations

(i) virtual velocity 119891119894(119911) of the material particles(ii) virtual velocities119862

2and119862

3of the interfaces 119878

2and 1198783

(iii) variation 120575119875119894(119911) of the dipole momentum at the pointwith coordinates 119911119894

The geometry presented in Figure 1 was analyzed in [2128] which dealt with nucleation on stationary ions of liquidcondensate from the surrounding gaseous phase When thedomain Ω

119902is rigid the virtual velocities of the deformable

liquid phase should satisfy the boundary constraint

119873119894119891119894100381610038161003816100381610038161198781

= 0 (12)

4 The Bulk Equilibrium Equations ofDeformable Polarizable Substances

In this section we summarize the results and refer the readerto the relevant references for the corresponding derivations

Separating the independent variations in the volumeintegral of the first energy variation we arrive at the followingequilibrium equations [22 27]

minusnabla119894120577119894119896+ 120588120595119875119894nabla119896119875119894= 0

120588120595119875119894 = 119864119894

(13)

where 120595119875119894 = 120597120595120597119875

119894 the formal stress tensor 120577119898119896 is defined as

120577119894119895equiv 120588

120597120595

120597nabla119894119880119896

119860sdot119895

119896sdot (14)

and the tensor 119860119894sdotsdot119895is given by

119860119894sdot

sdot119895equiv 120575119894

119895minus nabla119895119880119894 (15)

Combining (13) we arrive at the equilibrium bulk equation

minusnabla119894120577119894119896+ 119864119894nabla119896119875119894= 0 (16)

Using the equations of electrostatics it can be shown that (16)can be rewritten as a statement of vanishing divergence

nabla119894(120577119894119895minus 119911119894119895(1

4120587119864119896119863119896minus

1

8120587119864119896119864119896) +

1

4120587119863119894119864119895) = 0 (17)

4 Advances in Mathematical Physics

For nonpolarizable substances the formal stress tensor120577119894119895 coincides with the Cauchy stress tensor in the Euleriandescription Relationship (17) generalizes to the celebratedKorteweg-Helmholtz relationship for liquid dielectrics [6 710ndash13 24] in the case of nonlinear electroelasticity

We can rewrite (17) as (see [22 23 27])

nabla119894alefsym119894119895= 0 (18)

where the aleph tensor alefsym119894119895 given by

alefsym119894119895equiv 120588

120597120595

120597nabla119894119880119896

119860sdot119895

119896sdot+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895

+1

4120587119863119894119864119895

(19)

can be thought of as the stress tensor of a polarizablesubstance We can rewrite the aleph tensor alefsym119894119895 as

alefsym119894119895equiv 120577119894119895+ Γ119894119895 (20)

where the electrostatic gamma tensor Γ119898119896 is given by

Γ119894119895equiv (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (21)

Equation (17) can be written in another insightful form

nabla119894120577119894119895= minusnabla119894Γ119894119895 (22)

In polarizable deformable substances neither one of thetensors 120577119894119895 or Γ119894119895 is divergence-free

The gamma tensor Γ119894119895 can be also considered as one ofthemanypossible generalizations of theMaxwell stress tensor119879119894119895

119879119894119895equiv minus

1

8120587119864119896119864119896119911119894119895+

1

4120587119864119894119864119895 (23)

since Γ119894119895 coincides with119879119894119895 when polarization vanishes Otherpossible generalizations of the Maxwell stress tensor

119879119894119895

1equiv minus

1

8120587119864119897119863119897119911119894119895+

1

4120587119863119894119864119895 (24a)

119879119894119895

2equiv minus

1

8120587119864119897119863119897119911119894119895+

1

4120587119863119895119864119894 (24b)

119879119894119895

3equiv minus

1

8120587119864119897119863119897119911119894119895+

1

8120587(119863119894119864119895+ 119863119895119864119894) (24c)

are perhaps more aesthetically appealing than the gammatensor Γ119894119895 We believe that the advantage of the gamma tensorover other possible generalizations is its variational origin andits ability to help address the issue of stability based on thecalculation of the second energy variation

One more useful tensor for polarizable materials is thebeth tensor ℶ119894sdot

sdot119895 or the tensor of electrochemical tensorial

potential It is defined by

ℶ119894sdot

sdot119895equiv (120588120595119911

119894119896minus alefsym119894119896+ Γ119894119896) 119861119896119895 (25)

where the tensor 119861119896119895

is the matrix inverse of 119860119896119895 definedin (15) As we show below the beth tensor ℶ119894sdot

sdot119895satisfies the

condition of zero divergence

nabla119894ℶ119894sdot

sdot119895= 0 (26)

similarly to the aleph tensor alefsym119894119895 The beth tensor ℶ119894sdotsdot119895can be

rewritten as

ℶ119894sdot

sdot119895= 120588119861119896119895120594119894119896 (27)

where 120594119894119896 is the Bowen symmetric tensorial chemical potential

120594119894119895= 120595119911119894119895minus1

120588120577119894119895= 120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895) (28)

The symmetric tensor 120594119894119895 should be distinguished fromthe typically asymmetric tensorial chemical tensor 120583119894119895

120583119894119895= 119911119896119894

∘119911119897119896120594119895119897 (29)

where 119911119898119894∘

is the contravariant metric tensor of the initialconfiguration

5 Conditions at the Interfaces

Boundary conditions depend on the various characteristicsof the interfaces Interfaces can differ by their mechanical orkinematic properties and whether or not they are subject tophase transformations We refer to interfaces that satisfy thekinematic constraint

[119880119894]+

minus= 0 (30)

as coherent interfaces The following condition for the alephstress tensor alefsym119894119895 is satisfied by equilibrium configurations atcoherent interfaces

119873119894[alefsym119894119895]+

minus= 0 (31)

If in addition to coherency the boundary is a phase interfacethe condition of phase equilibrium includes the beth tensorℶ119894119895

119873119894[ℶ119894119895]+

minus= 0 (32)

It makes sense then to call the beth tensor ℶ119894119895 the elec-trochemical tensorial potential for coherent interfaces indeformable substances because (32) is analogous to theequilibrium condition for the tensorial chemical potential

6 Nonfrictional Semicoherent Interfaces

By definition nonfrictional semicoherent interfaces are char-acterized by the possibility of relative slippage Nonfrictionalsemicoherent interfaces also may or may not be phaseinterfaces Regardless the following conditions ofmechanicalequilibrium must hold

119873119894120577119894119895

plusmn= minus119873

119895119901plusmn

119873119894119873119895[Γ119894119895]+

minus= [119901]+

minus

(33)

Advances in Mathematical Physics 5

At phase nonfrictional incoherent interfaces an additionalmass exchange equilibrium condition must be satisfied

119873119894119873119895[120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895)]

+

minus

= 0 (34)

7 Phase Interfaces in Rigid Dielectrics

When dealing with rigid solids all mechanical degrees offreedom disappear and the internal energy depends onlyon the polarization vector 119875119894 (and unless it is assumedto be constant temperature 119879) At the phase interface thecondition of phase equilibrium reads

119873119894119873119895[ℷ119894119895]+

minus= 0 (35)

where the gimel energy-like tensor ℷ119894119895 the electrostatic tenso-rial chemical potential for rigid dielectrics is defined by

ℷ119894119895equiv 120598 (119875) 119911

119894119895+ Γ119894119895 (36)

where 120598 equiv 120588120595 is the free energy density per unit volume (andwe once again suppress the index in119875119894 because it now appearsas an argument of a function) We refer to the gimel tensorℷ119894119895 as the electrostatic tensorial chemical potential because itplays the same role as the chemical potential 120583 in the classicalheterogeneous liquid-vapor system Contrary to the gammatensor Γ119894119895 the gimel tensor ℷ119894119895 is divergence-free

nabla119894ℷ119894119895= 0 (37)

One can analyze models in which the polarization vector119875119894 is fixed [20] Then 120595

plusmnare 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 twoidentities can be demonstrated similarly First let us rewritethe gimel tensor ℷ119894119895 as follows

ℷ119894119895= 120598 (119875) 119911

119894119895+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (38)

For the first term in (38) we have

nabla119894(120598 (119875) 119911

119894119895) = 119911119894119895 120597120598 (119875)

120597119875119896nabla119894119875119896 (39)

Using the thermodynamic identity

120597120598 (119875)

120597119875119894equiv 119864119894 (40)

we can rewrite (39) as

nabla119894(120598 (119875) 119911

119894119895) = 119864119894nabla119895119875119894 (41)

For the second term in (38) we have

nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896))

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896

(42)

which can be seen from the following chain of identities

2nd term = nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896)) (43a)

= nabla119896(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) (43b)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896119863119896minus 119864119896nabla119895119863119896) (43c)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896(119864119896+ 4120587119875

119896)

minus 119864119896nabla119895(119864119896+ 4120587119875

119896))

(43d)

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896 (43e)

For the third term in (38) we have

nabla119894(1

4120587119863119894119864119895) =

1

4120587119863119894nabla119894119864119895=

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (44)

Combining (41)ndash(44) we find

nabla119894ℷ119894119895= nabla119894120598 (119875) 119911

119894119895+ 119911119894119895nabla119894(1

8120587119864119897119864119897minus

1

4120587119864119897119863119897)

+1

4120587nabla119894(119863119894119864119895)

(45a)

= 119864119894nabla119895119875119894minus

1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894minus 119864119894nabla119895119875119894

+1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895

(45b)

= minus1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894+

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (45c)

Finally using the symmetric property nabla119894119864119895equiv nabla119895119864119894 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

119869 = minus119870119873119894119873119895[ℷ119894119895]+

minus (46)

The same approach can be applied to the case of an isolateddomain with fixed total volume yet subject to rearrangementIn this case the evolution equation should be slightly modi-fied to take into account surface diffusion Figure 2 illustratesan implementation of this approach in the two-dimensionalcase The quasi-static evolution of originally circular domainand fixed polarization vector leads to elongation in thedirection of polarization vector 119875

119894 and eventually to amorphological instability

6 Advances in Mathematical Physics

Figure 2 Onset of a morphological instability in a quasi-staticevolution of a domain filled with dipoles of fixed polarization

10 Conclusion

We discussed a phenomenological variational approach toelectrostatics and magnetostatics for heterogeneous systemswith phase transformations Although we focused on electro-statics almost all of the presented results are also valid formagnetostatics Our approach is an extension of the Gibbsvariational method as it was interpreted in [26]

The demand of having simultaneously a logically andphysically consistent theory remains to be the main driv-ing force of progress in thermodynamics The suggestedapproach leads to themathematically rigorous self-consistentresults Now it has to prove its viability in direct compar-ison with experiment That may prove to be difficult butreal progress is only possible when theory and experimentchallenge each other

Appendix

The summary of notations and variables is as follows (seeAbbreviations)

Abbreviations

119911119894 Eulerian coordinates in the ambient space119911119894119895 119911119894119895 Metrics tensors in the reference Eulerian

coordinates119911119894119895

∘ Metrics tensor of the coordinate system

generated by tracking back the coordinate119911119894 from the actual to the initialconfiguration [26]

nabla119894 The symbols of covariant differentiation

(based on the metrics 119911119894119895)

119902 119875119894 The electric charge density and

polarization (per unit volume)120593 119864119894 119863119894 The electrostatic potential field anddisplacement

Ω119902 Ω119889 Spatial domains occupied by free charges

and dipoles1198781 Interface separating the dielectric from the

distributed stationary electric charges1198782 Interface separating the different dielectric

phases

1198783 Interface separating the dielectric phase

from the surrounding vacuum119880119894 Displacements of material particles

119860119894sdot

sdot119895and 119861119894sdot

sdot119895 Mutually inverse geometric tensorsdefined in (15)

120588 Mass density119901 119879 120583 Pressure absolute temperature and

chemical potential of nonpolarizableone-component liquid phases

120583119894119895 120594119894119895 Asymmetric and Bowen chemical

potentials of nonpolarizable deformable(nonnecessarily liquid) media (forfurther details see [26])

120595 Free energy density per unit mass120577119894119895 Formal stress tensor defined in (14)119891119894 119862119894 119862119890 Admissible virtual velocities of the

material particles and interfacesalefsym119894119895 The aleph tensor a divergence-free

tensor defined in (19) the aleph tensorexhibits some of the properties of theclassical Cauchy stress tensor (inEulerian coordinates) and of theMaxwell stress tensor

ℶ119894119895 The beth tensor a divergence-free

tensor defined in (25) the beth tensorexhibits some of the properties of thescalar chemical potential ofnonpolarizable liquid and of thetensorial chemical potentials 120583119894119895 120594119894119896 ofnonpolarizable solids

Γ119894119895 The gamma tensor defined in (20) for

deformable media and in (21) forarbitrary polarizable media

ℷ119894119895 The gimel tensor which is defined in

(36) for rigid dielectrics and plays thesame role as the beth tensor ℶ119894119895 fordeformable dielectrics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Gibbs ldquoOn the equilibrium of heterogeneous substancesrdquoTransactions of the Connecticut Academy of Arts and Sciencesvol 3 pp 108ndash248 1876 vol 3 pp 343ndash524 1878

[2] I M Gelrsquofand and S V Fomin Calsulus of Variations Prentice-Hall Englewood Cliffs NJ USA 1963

[3] Josiah Willard Gibbs httpenwikiquoteorgwikiJosiah Wil-lard Gibbs

[4] J C Maxwell A Treatise on Electricity and Magnetism vol 1-2Dover Publications New York NY USA 1954

[5] H Poincare Lectures on Optics and Electromagnetism PrefaceCollected Papers of Poincare vol 3 Nauka Moscow Russia1974 (Russian)

Advances in Mathematical Physics 7

[6] Y I Frenkel Electrodynamics I General Theory of Electromag-netism ONTI Leningrad Russia 1934

[7] Y I Frenkelrsquo Electrodynamics Volume 2 Macroscopic Electrody-namics of Material Bodies ONTI Moscow Russia 1935

[8] I E Tamm Basics of the Theory of Electricity Nauka MoscowRussia 1989 (Russian)

[9] A Sommerfeld Electrodynamics Academic Press New YorkNY USA 1952

[10] J A Stratton Electromagnetic Theory McGraw-Hill New YorkNY USA 1941

[11] W K H Panofsky and M Phillips Classical Electricity andMagnetism Addison-Wesley Cambridge Mass USA 1950

[12] R A Toupin ldquoThe elastic dielectricrdquo Indiana University Math-ematics Journal vol 5 no 6 pp 849ndash915 1956

[13] L D Landau and E M Lifshitz Electrodynamics of ContinuousMedia Pergamon Press New York NY USA 1963

[14] I A Privorotskiı ldquoThermodynamic theory of ferromagneticgomainsinsrdquo Soviet Physics Uspekhi vol 15 no 5 pp 555ndash5741973

[15] L I Sedov and A G Tsypkin Fundamentals of MicroscopicTheories of Gravitation and Electromagnetism Nauka MoscowRussia 1989

[16] R E Rosensweig Ferrohydrodynamics Dover New York NYUSA 1985

[17] D J Korteweg ldquoUber die veranderung der form und desvolumens dielectrischer Korperunter Einwirkung elektrischerKrafterdquo Annalen der Physik und Chemie vol 245 no 1 pp 48ndash61 1880

[18] H Helmholtz ldquoUber die auf das Innere magnetisch oderdielectrisch polarisirter Korper wirkenden KrafterdquoAnnalen derPhysik vol 249 no 7 pp 385ndash406 1881

[19] H A LorentzTheTheory of Electrons and Its Applications to thePhenomena of Light and Radiant Heat Dover New York NYUSA 2011

[20] M A Grinfeld ldquoMorphology stability and evolution of dipoleaggregatesrdquo Proceedings of the Estonian Academy of SciencesEngineering vol 5 no 2 pp 131ndash141 1999

[21] P Grinfeld ldquoMorphological instability of liquid metallic nucleicondensing on charged inhomogeneitiesrdquo Physical Review Let-ters vol 87 no 9 Article ID 095701 4 pages 2001

[22] M Grinfeld and P Grinfeld ldquoTowards thermodynamics ofelastic electric conductorsrdquo Philosophical Magazine A vol 81no 5 pp 1341ndash1354 2001

[23] M A Grinfeld and P M Grinfeld ldquoThe exact conditions ofthermodynamic phase equilibrium in heterogeneous elasticsystems with dipolar interactionrdquo inNonlinearMechanics LMZubov Ed pp 47ndash51 Rostov University 2001

[24] M Abraham and R Becker The Classical Theory of Electricityand Magnetism Blackie amp Son 1932

[25] P Grinfeld Introduction to Tensor Analysis and the Calculus ofMoving Surfaces Springer New York NY USA 2014

[26] M Grinfeld Thermodynamic Methods in the Theory of Het-erogeneous Systems Interaction of Mechanics and MathematicsSeries Longman Scientific amp Technical Harlow UK 1991

[27] P Grinfeld and M Grinfeld ldquoThermodynamic aspects ofequilibrium shape and growth of crystalline films with elec-tromechanical interactionrdquo Ferroelectrics vol 342 no 1 pp 89ndash100 2006

[28] P Grinfeld ldquoMorphological instability of the dielectric thomsonnucleirdquo Physical Review B vol 81 no 18 Article ID 184110 2010

[29] P Grinfeld ldquoClausius-Clapeyron relations for an evaporatingsolid conductorrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 90 no 7-8 pp 633ndash640 2010

[30] P Grinfeld ldquoA proposed experiment for the verification ofThomsonrsquos nucleation theoryrdquo Ferroelectrics vol 413 no 1 pp65ndash72 2011

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Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

Maxwell himself pointed out a number of difficulties inhis theory Of relevance to this paper is his statement [4] ldquoIhave not been able to make the next step namely to accountbymechanical considerations for these stresses in dielectricsrdquoMany efforts have since been made to fix this shortcomingMany of those efforts are variational in nature since one of themost effective ways of coping with mathematical difficultiesand logical inconsistencies is to insist on a variational formu-lationAmong themany textbooks lectures andmonographson electromagnetism [6ndash16] there are many that discuss thevariational perspective and once again it is clear that there isno consensus on the right approach

One of the pioneers of variationalmethods in electromag-netismwasGibbs himself Gibbs studied the problem of equi-librium configuration of charges and discovered that (whatresearchers now call) the chemical potential of a chargedmaterial particle should be supplemented with an additionalterm 119902120593 (attributed by Gibbs to Gabriel Lippmann) where 119902is the electric charge of the particle and 120593 is the electrostaticpotential This is a very rough sketch of Gibbsrsquo vision Forinstance Gibbs himself has never used the term chemicalpotential and did not assign the corresponding quantity anyprofound meaning which was understood only much laterThe variational approach to polarizable substances was mostlikely pioneered by Korteweg [17] and Helmholtz [18]

Gibbs modeled heterogeneous systems or what he calledheterogeneous substances as macroscopic domains separatedby mathematical surfaces The difficulty in carrying overGibbsrsquo ideas to electromagnetism is that the analysis ofsingular interfaces in electrostatics and magnetostatics ismuch more challenging than it is in continuum mechanicsEven Lorentz chose to avoid the analysis of heterogeneoussystems stating in the preface to his classical treatise [19] thathe does not want to struggle with the boundary terms Manyof the difficulties that were faced (or should have been faced)by Lorentz can be overcome with the help of the calculus ofmoving surfaces

In this paper we make a new attempt at extending theGibbs variational framework to electrostatics Our approachis very simple and entirely straightforward conceptuallyContrary to many of the prior attempts ([12ndash14] to name justthree) we explicitly exclude the electric field and the electricdisplacement from the list of independent thermodynamicvariables Instead we account for polarization (ormagnetiza-tion) by adding a single term to the ldquotraditionalrdquo free energyfor a thermoelastic system The additional term representsthe potential energy accumulated in the electrostatic fieldover the entire space Different authors choose this termdifferently |E|2 E sdotD and so forth We choose the integrandin the simplest form |E|2 We build our approach on the exactnonlinear theory of continuum media and rely on Euleriancoordinates as the independent spatial variables

2 The Gibbs Thermodynamics in a Nutshell

According to the modern interpretation of Gibbs the chem-ical potential 120583 governs the equilibrium between the liquidand the vapor phases with respect to mass exchange between

them Equilibrium heterogeneous systems must satisfy anumber of conditions at the phase interface The first twoconditions those of thermal equilibrium (temperature 119879 iscontinuous across the interface (and of course spatially con-stant)) andmechanical equilibrium (pressure 119901 is continuousacross the interface) are satisfied by all equilibrium two-phase systems whether or not the phases are different statesof the same substance subject to a phase transformationLetting the brackets []+

minusdenote the jump discontinuity in the

enclosed quantity across the phase interface we write theseconditions as

[119879]+

minus= 0

[119901]+

minus= 0

(1)

Additionally when the interface is subject to a phase trans-formation the chemical potential 120583 is continuous across theinterface

[120583]+

minus= 0 (2)

This equation is interpreted as equilibrium with respect tomass exchange between the phases The chemical potential120583 is given by

120583 =120597120598 (120588)

120597120588 (3)

where 120588 is density and 120598(120588) is the free energy per unit volumeIn many physical systems equilibrium with respect to

mass exchange is attained over much longer time scalesthan thermal and mechanical equilibria The dynamics ofmass exchange in such systems is often well described by aquasi-static approximation which assumes that the systemmaintains thermal and mechanical equilibria throughoutevolution that is (1) are continuously satisfied while equi-librium equation (2) is replaced with the following equationfor the mass flux 119869

119869 = minus119870 [120583]+

minus (4)

where 119870 gt 0 is a kinematic quantity determined empiricallyor by some nonthermodynamic theory

3 A Variational Approach to Electrostatics ofHeterogeneous Systems

We will now briefly summarize a variational frameworkfor electrostatics of heterogeneous systems which was firstdescribed in [20ndash23] The presented model based on thechoice of the functional 119864 in (11) and the list of independentvariations is correct only in the mathematical sense thatis it is logically consistent Other authors [10 12 13 24]make different choices of energy functionals and sets ofindependent variations and arrive at different results

Our description uses the framework of tensor calculus[25] We refer the space to coordinates 119911119894 By conventionwe omit the superscript 119894 when the coordinate appears asan argument of a function We denote the covariant and

Advances in Mathematical Physics 3

S1

S2

S3

ΩqΩdminusΩd+

Figure 1 A heterogeneous system with distributed electric chargesand dipoles

contravariant ambient metric tensors by 119911119894119895and 119911119894119895 and the

ambient covariant derivative by nabla119894

Figure 1 illustrates the configuration of our system Sup-pose that the domain Ω

119889= Ω119889+

cup Ω119889minus

is occupied bysolid heterogeneous dielectric media with specific (per unitvolume) dipolemomentum119875

119894(119911)The domainΩ

119902is occupied

by a stationary electric charge distribution 119902(119911) The twosubdomains Ω

119889+and Ω

119889minusare occupied by two different

substances or two different phases of the same substanceThey are separated by the interface 119878

2

Suppose that 119880119894(119911) is the displacement field of the

material particles 120588(119911) is the actual mass density 120593(119911) is theelectrical potential

119864119894 (119911) equiv minusnabla119894120593 (119911) (5)

is the electrical field and

119863119894= 119864119894+ 4120587119875

119894 (6)

is the electric displacementFor the sake of simplicity we assume that the system

is kept under fixed absolute temperature 119879 and denote theelastic (internal) energy density 120595 of the dielectric substanceby

120595 (nabla119895119880119894 119875119896) (7)

Of course this elastic energy is actually the free energy den-sity of the system

The equilibrium of the system is governed by Poissonrsquosequation

nabla119894nabla119894120593 = 4120587119902 (8)

subject to the boundary conditions

[120593]+

minus= 0

119873119894[119863119894]+

minus= 0

(9)

across the interfaces (119873119894 is the unit normal) while at infinitythe electrical potential vanishes

1205931003816100381610038161003816infin = 0 (10)

The total energy 119864 of the system is given by the integral

119864 = int(120588120595 +1

8120587119864119894119864119894)119889Ω (11)

which extends over the entire spaceAccording to the principle of minimum energy we

associate equilibrium configurations with stationary pointsof the total energy 119864 In what follows we use the technique ofvariation of the energy functionals in the Eulerian descriptionpresented in detail in [21 22 26] Suggested procedures foranalyzing the equilibrium and stability conditions for two-phase heterogeneous systems can be found in [27ndash30]

We complete the description of the variational principleby presenting the list of quantities treated as the independentvariations

(i) virtual velocity 119891119894(119911) of the material particles(ii) virtual velocities119862

2and119862

3of the interfaces 119878

2and 1198783

(iii) variation 120575119875119894(119911) of the dipole momentum at the pointwith coordinates 119911119894

The geometry presented in Figure 1 was analyzed in [2128] which dealt with nucleation on stationary ions of liquidcondensate from the surrounding gaseous phase When thedomain Ω

119902is rigid the virtual velocities of the deformable

liquid phase should satisfy the boundary constraint

119873119894119891119894100381610038161003816100381610038161198781

= 0 (12)

4 The Bulk Equilibrium Equations ofDeformable Polarizable Substances

In this section we summarize the results and refer the readerto the relevant references for the corresponding derivations

Separating the independent variations in the volumeintegral of the first energy variation we arrive at the followingequilibrium equations [22 27]

minusnabla119894120577119894119896+ 120588120595119875119894nabla119896119875119894= 0

120588120595119875119894 = 119864119894

(13)

where 120595119875119894 = 120597120595120597119875

119894 the formal stress tensor 120577119898119896 is defined as

120577119894119895equiv 120588

120597120595

120597nabla119894119880119896

119860sdot119895

119896sdot (14)

and the tensor 119860119894sdotsdot119895is given by

119860119894sdot

sdot119895equiv 120575119894

119895minus nabla119895119880119894 (15)

Combining (13) we arrive at the equilibrium bulk equation

minusnabla119894120577119894119896+ 119864119894nabla119896119875119894= 0 (16)

Using the equations of electrostatics it can be shown that (16)can be rewritten as a statement of vanishing divergence

nabla119894(120577119894119895minus 119911119894119895(1

4120587119864119896119863119896minus

1

8120587119864119896119864119896) +

1

4120587119863119894119864119895) = 0 (17)

4 Advances in Mathematical Physics

For nonpolarizable substances the formal stress tensor120577119894119895 coincides with the Cauchy stress tensor in the Euleriandescription Relationship (17) generalizes to the celebratedKorteweg-Helmholtz relationship for liquid dielectrics [6 710ndash13 24] in the case of nonlinear electroelasticity

We can rewrite (17) as (see [22 23 27])

nabla119894alefsym119894119895= 0 (18)

where the aleph tensor alefsym119894119895 given by

alefsym119894119895equiv 120588

120597120595

120597nabla119894119880119896

119860sdot119895

119896sdot+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895

+1

4120587119863119894119864119895

(19)

can be thought of as the stress tensor of a polarizablesubstance We can rewrite the aleph tensor alefsym119894119895 as

alefsym119894119895equiv 120577119894119895+ Γ119894119895 (20)

where the electrostatic gamma tensor Γ119898119896 is given by

Γ119894119895equiv (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (21)

Equation (17) can be written in another insightful form

nabla119894120577119894119895= minusnabla119894Γ119894119895 (22)

In polarizable deformable substances neither one of thetensors 120577119894119895 or Γ119894119895 is divergence-free

The gamma tensor Γ119894119895 can be also considered as one ofthemanypossible generalizations of theMaxwell stress tensor119879119894119895

119879119894119895equiv minus

1

8120587119864119896119864119896119911119894119895+

1

4120587119864119894119864119895 (23)

since Γ119894119895 coincides with119879119894119895 when polarization vanishes Otherpossible generalizations of the Maxwell stress tensor

119879119894119895

1equiv minus

1

8120587119864119897119863119897119911119894119895+

1

4120587119863119894119864119895 (24a)

119879119894119895

2equiv minus

1

8120587119864119897119863119897119911119894119895+

1

4120587119863119895119864119894 (24b)

119879119894119895

3equiv minus

1

8120587119864119897119863119897119911119894119895+

1

8120587(119863119894119864119895+ 119863119895119864119894) (24c)

are perhaps more aesthetically appealing than the gammatensor Γ119894119895 We believe that the advantage of the gamma tensorover other possible generalizations is its variational origin andits ability to help address the issue of stability based on thecalculation of the second energy variation

One more useful tensor for polarizable materials is thebeth tensor ℶ119894sdot

sdot119895 or the tensor of electrochemical tensorial

potential It is defined by

ℶ119894sdot

sdot119895equiv (120588120595119911

119894119896minus alefsym119894119896+ Γ119894119896) 119861119896119895 (25)

where the tensor 119861119896119895

is the matrix inverse of 119860119896119895 definedin (15) As we show below the beth tensor ℶ119894sdot

sdot119895satisfies the

condition of zero divergence

nabla119894ℶ119894sdot

sdot119895= 0 (26)

similarly to the aleph tensor alefsym119894119895 The beth tensor ℶ119894sdotsdot119895can be

rewritten as

ℶ119894sdot

sdot119895= 120588119861119896119895120594119894119896 (27)

where 120594119894119896 is the Bowen symmetric tensorial chemical potential

120594119894119895= 120595119911119894119895minus1

120588120577119894119895= 120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895) (28)

The symmetric tensor 120594119894119895 should be distinguished fromthe typically asymmetric tensorial chemical tensor 120583119894119895

120583119894119895= 119911119896119894

∘119911119897119896120594119895119897 (29)

where 119911119898119894∘

is the contravariant metric tensor of the initialconfiguration

5 Conditions at the Interfaces

Boundary conditions depend on the various characteristicsof the interfaces Interfaces can differ by their mechanical orkinematic properties and whether or not they are subject tophase transformations We refer to interfaces that satisfy thekinematic constraint

[119880119894]+

minus= 0 (30)

as coherent interfaces The following condition for the alephstress tensor alefsym119894119895 is satisfied by equilibrium configurations atcoherent interfaces

119873119894[alefsym119894119895]+

minus= 0 (31)

If in addition to coherency the boundary is a phase interfacethe condition of phase equilibrium includes the beth tensorℶ119894119895

119873119894[ℶ119894119895]+

minus= 0 (32)

It makes sense then to call the beth tensor ℶ119894119895 the elec-trochemical tensorial potential for coherent interfaces indeformable substances because (32) is analogous to theequilibrium condition for the tensorial chemical potential

6 Nonfrictional Semicoherent Interfaces

By definition nonfrictional semicoherent interfaces are char-acterized by the possibility of relative slippage Nonfrictionalsemicoherent interfaces also may or may not be phaseinterfaces Regardless the following conditions ofmechanicalequilibrium must hold

119873119894120577119894119895

plusmn= minus119873

119895119901plusmn

119873119894119873119895[Γ119894119895]+

minus= [119901]+

minus

(33)

Advances in Mathematical Physics 5

At phase nonfrictional incoherent interfaces an additionalmass exchange equilibrium condition must be satisfied

119873119894119873119895[120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895)]

+

minus

= 0 (34)

7 Phase Interfaces in Rigid Dielectrics

When dealing with rigid solids all mechanical degrees offreedom disappear and the internal energy depends onlyon the polarization vector 119875119894 (and unless it is assumedto be constant temperature 119879) At the phase interface thecondition of phase equilibrium reads

119873119894119873119895[ℷ119894119895]+

minus= 0 (35)

where the gimel energy-like tensor ℷ119894119895 the electrostatic tenso-rial chemical potential for rigid dielectrics is defined by

ℷ119894119895equiv 120598 (119875) 119911

119894119895+ Γ119894119895 (36)

where 120598 equiv 120588120595 is the free energy density per unit volume (andwe once again suppress the index in119875119894 because it now appearsas an argument of a function) We refer to the gimel tensorℷ119894119895 as the electrostatic tensorial chemical potential because itplays the same role as the chemical potential 120583 in the classicalheterogeneous liquid-vapor system Contrary to the gammatensor Γ119894119895 the gimel tensor ℷ119894119895 is divergence-free

nabla119894ℷ119894119895= 0 (37)

One can analyze models in which the polarization vector119875119894 is fixed [20] Then 120595

plusmnare 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 twoidentities can be demonstrated similarly First let us rewritethe gimel tensor ℷ119894119895 as follows

ℷ119894119895= 120598 (119875) 119911

119894119895+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (38)

For the first term in (38) we have

nabla119894(120598 (119875) 119911

119894119895) = 119911119894119895 120597120598 (119875)

120597119875119896nabla119894119875119896 (39)

Using the thermodynamic identity

120597120598 (119875)

120597119875119894equiv 119864119894 (40)

we can rewrite (39) as

nabla119894(120598 (119875) 119911

119894119895) = 119864119894nabla119895119875119894 (41)

For the second term in (38) we have

nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896))

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896

(42)

which can be seen from the following chain of identities

2nd term = nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896)) (43a)

= nabla119896(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) (43b)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896119863119896minus 119864119896nabla119895119863119896) (43c)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896(119864119896+ 4120587119875

119896)

minus 119864119896nabla119895(119864119896+ 4120587119875

119896))

(43d)

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896 (43e)

For the third term in (38) we have

nabla119894(1

4120587119863119894119864119895) =

1

4120587119863119894nabla119894119864119895=

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (44)

Combining (41)ndash(44) we find

nabla119894ℷ119894119895= nabla119894120598 (119875) 119911

119894119895+ 119911119894119895nabla119894(1

8120587119864119897119864119897minus

1

4120587119864119897119863119897)

+1

4120587nabla119894(119863119894119864119895)

(45a)

= 119864119894nabla119895119875119894minus

1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894minus 119864119894nabla119895119875119894

+1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895

(45b)

= minus1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894+

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (45c)

Finally using the symmetric property nabla119894119864119895equiv nabla119895119864119894 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

119869 = minus119870119873119894119873119895[ℷ119894119895]+

minus (46)

The same approach can be applied to the case of an isolateddomain with fixed total volume yet subject to rearrangementIn this case the evolution equation should be slightly modi-fied to take into account surface diffusion Figure 2 illustratesan implementation of this approach in the two-dimensionalcase The quasi-static evolution of originally circular domainand fixed polarization vector leads to elongation in thedirection of polarization vector 119875

119894 and eventually to amorphological instability

6 Advances in Mathematical Physics

Figure 2 Onset of a morphological instability in a quasi-staticevolution of a domain filled with dipoles of fixed polarization

10 Conclusion

We discussed a phenomenological variational approach toelectrostatics and magnetostatics for heterogeneous systemswith phase transformations Although we focused on electro-statics almost all of the presented results are also valid formagnetostatics Our approach is an extension of the Gibbsvariational method as it was interpreted in [26]

The demand of having simultaneously a logically andphysically consistent theory remains to be the main driv-ing force of progress in thermodynamics The suggestedapproach leads to themathematically rigorous self-consistentresults Now it has to prove its viability in direct compar-ison with experiment That may prove to be difficult butreal progress is only possible when theory and experimentchallenge each other

Appendix

The summary of notations and variables is as follows (seeAbbreviations)

Abbreviations

119911119894 Eulerian coordinates in the ambient space119911119894119895 119911119894119895 Metrics tensors in the reference Eulerian

coordinates119911119894119895

∘ Metrics tensor of the coordinate system

generated by tracking back the coordinate119911119894 from the actual to the initialconfiguration [26]

nabla119894 The symbols of covariant differentiation

(based on the metrics 119911119894119895)

119902 119875119894 The electric charge density and

polarization (per unit volume)120593 119864119894 119863119894 The electrostatic potential field anddisplacement

Ω119902 Ω119889 Spatial domains occupied by free charges

and dipoles1198781 Interface separating the dielectric from the

distributed stationary electric charges1198782 Interface separating the different dielectric

phases

1198783 Interface separating the dielectric phase

from the surrounding vacuum119880119894 Displacements of material particles

119860119894sdot

sdot119895and 119861119894sdot

sdot119895 Mutually inverse geometric tensorsdefined in (15)

120588 Mass density119901 119879 120583 Pressure absolute temperature and

chemical potential of nonpolarizableone-component liquid phases

120583119894119895 120594119894119895 Asymmetric and Bowen chemical

potentials of nonpolarizable deformable(nonnecessarily liquid) media (forfurther details see [26])

120595 Free energy density per unit mass120577119894119895 Formal stress tensor defined in (14)119891119894 119862119894 119862119890 Admissible virtual velocities of the

material particles and interfacesalefsym119894119895 The aleph tensor a divergence-free

tensor defined in (19) the aleph tensorexhibits some of the properties of theclassical Cauchy stress tensor (inEulerian coordinates) and of theMaxwell stress tensor

ℶ119894119895 The beth tensor a divergence-free

tensor defined in (25) the beth tensorexhibits some of the properties of thescalar chemical potential ofnonpolarizable liquid and of thetensorial chemical potentials 120583119894119895 120594119894119896 ofnonpolarizable solids

Γ119894119895 The gamma tensor defined in (20) for

deformable media and in (21) forarbitrary polarizable media

ℷ119894119895 The gimel tensor which is defined in

(36) for rigid dielectrics and plays thesame role as the beth tensor ℶ119894119895 fordeformable dielectrics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Gibbs ldquoOn the equilibrium of heterogeneous substancesrdquoTransactions of the Connecticut Academy of Arts and Sciencesvol 3 pp 108ndash248 1876 vol 3 pp 343ndash524 1878

[2] I M Gelrsquofand and S V Fomin Calsulus of Variations Prentice-Hall Englewood Cliffs NJ USA 1963

[3] Josiah Willard Gibbs httpenwikiquoteorgwikiJosiah Wil-lard Gibbs

[4] J C Maxwell A Treatise on Electricity and Magnetism vol 1-2Dover Publications New York NY USA 1954

[5] H Poincare Lectures on Optics and Electromagnetism PrefaceCollected Papers of Poincare vol 3 Nauka Moscow Russia1974 (Russian)

Advances in Mathematical Physics 7

[6] Y I Frenkel Electrodynamics I General Theory of Electromag-netism ONTI Leningrad Russia 1934

[7] Y I Frenkelrsquo Electrodynamics Volume 2 Macroscopic Electrody-namics of Material Bodies ONTI Moscow Russia 1935

[8] I E Tamm Basics of the Theory of Electricity Nauka MoscowRussia 1989 (Russian)

[9] A Sommerfeld Electrodynamics Academic Press New YorkNY USA 1952

[10] J A Stratton Electromagnetic Theory McGraw-Hill New YorkNY USA 1941

[11] W K H Panofsky and M Phillips Classical Electricity andMagnetism Addison-Wesley Cambridge Mass USA 1950

[12] R A Toupin ldquoThe elastic dielectricrdquo Indiana University Math-ematics Journal vol 5 no 6 pp 849ndash915 1956

[13] L D Landau and E M Lifshitz Electrodynamics of ContinuousMedia Pergamon Press New York NY USA 1963

[14] I A Privorotskiı ldquoThermodynamic theory of ferromagneticgomainsinsrdquo Soviet Physics Uspekhi vol 15 no 5 pp 555ndash5741973

[15] L I Sedov and A G Tsypkin Fundamentals of MicroscopicTheories of Gravitation and Electromagnetism Nauka MoscowRussia 1989

[16] R E Rosensweig Ferrohydrodynamics Dover New York NYUSA 1985

[17] D J Korteweg ldquoUber die veranderung der form und desvolumens dielectrischer Korperunter Einwirkung elektrischerKrafterdquo Annalen der Physik und Chemie vol 245 no 1 pp 48ndash61 1880

[18] H Helmholtz ldquoUber die auf das Innere magnetisch oderdielectrisch polarisirter Korper wirkenden KrafterdquoAnnalen derPhysik vol 249 no 7 pp 385ndash406 1881

[19] H A LorentzTheTheory of Electrons and Its Applications to thePhenomena of Light and Radiant Heat Dover New York NYUSA 2011

[20] M A Grinfeld ldquoMorphology stability and evolution of dipoleaggregatesrdquo Proceedings of the Estonian Academy of SciencesEngineering vol 5 no 2 pp 131ndash141 1999

[21] P Grinfeld ldquoMorphological instability of liquid metallic nucleicondensing on charged inhomogeneitiesrdquo Physical Review Let-ters vol 87 no 9 Article ID 095701 4 pages 2001

[22] M Grinfeld and P Grinfeld ldquoTowards thermodynamics ofelastic electric conductorsrdquo Philosophical Magazine A vol 81no 5 pp 1341ndash1354 2001

[23] M A Grinfeld and P M Grinfeld ldquoThe exact conditions ofthermodynamic phase equilibrium in heterogeneous elasticsystems with dipolar interactionrdquo inNonlinearMechanics LMZubov Ed pp 47ndash51 Rostov University 2001

[24] M Abraham and R Becker The Classical Theory of Electricityand Magnetism Blackie amp Son 1932

[25] P Grinfeld Introduction to Tensor Analysis and the Calculus ofMoving Surfaces Springer New York NY USA 2014

[26] M Grinfeld Thermodynamic Methods in the Theory of Het-erogeneous Systems Interaction of Mechanics and MathematicsSeries Longman Scientific amp Technical Harlow UK 1991

[27] P Grinfeld and M Grinfeld ldquoThermodynamic aspects ofequilibrium shape and growth of crystalline films with elec-tromechanical interactionrdquo Ferroelectrics vol 342 no 1 pp 89ndash100 2006

[28] P Grinfeld ldquoMorphological instability of the dielectric thomsonnucleirdquo Physical Review B vol 81 no 18 Article ID 184110 2010

[29] P Grinfeld ldquoClausius-Clapeyron relations for an evaporatingsolid conductorrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 90 no 7-8 pp 633ndash640 2010

[30] P Grinfeld ldquoA proposed experiment for the verification ofThomsonrsquos nucleation theoryrdquo Ferroelectrics vol 413 no 1 pp65ndash72 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

S1

S2

S3

ΩqΩdminusΩd+

Figure 1 A heterogeneous system with distributed electric chargesand dipoles

contravariant ambient metric tensors by 119911119894119895and 119911119894119895 and the

ambient covariant derivative by nabla119894

Figure 1 illustrates the configuration of our system Sup-pose that the domain Ω

119889= Ω119889+

cup Ω119889minus

is occupied bysolid heterogeneous dielectric media with specific (per unitvolume) dipolemomentum119875

119894(119911)The domainΩ

119902is occupied

by a stationary electric charge distribution 119902(119911) The twosubdomains Ω

119889+and Ω

119889minusare occupied by two different

substances or two different phases of the same substanceThey are separated by the interface 119878

2

Suppose that 119880119894(119911) is the displacement field of the

material particles 120588(119911) is the actual mass density 120593(119911) is theelectrical potential

119864119894 (119911) equiv minusnabla119894120593 (119911) (5)

is the electrical field and

119863119894= 119864119894+ 4120587119875

119894 (6)

is the electric displacementFor the sake of simplicity we assume that the system

is kept under fixed absolute temperature 119879 and denote theelastic (internal) energy density 120595 of the dielectric substanceby

120595 (nabla119895119880119894 119875119896) (7)

Of course this elastic energy is actually the free energy den-sity of the system

The equilibrium of the system is governed by Poissonrsquosequation

nabla119894nabla119894120593 = 4120587119902 (8)

subject to the boundary conditions

[120593]+

minus= 0

119873119894[119863119894]+

minus= 0

(9)

across the interfaces (119873119894 is the unit normal) while at infinitythe electrical potential vanishes

1205931003816100381610038161003816infin = 0 (10)

The total energy 119864 of the system is given by the integral

119864 = int(120588120595 +1

8120587119864119894119864119894)119889Ω (11)

which extends over the entire spaceAccording to the principle of minimum energy we

associate equilibrium configurations with stationary pointsof the total energy 119864 In what follows we use the technique ofvariation of the energy functionals in the Eulerian descriptionpresented in detail in [21 22 26] Suggested procedures foranalyzing the equilibrium and stability conditions for two-phase heterogeneous systems can be found in [27ndash30]

We complete the description of the variational principleby presenting the list of quantities treated as the independentvariations

(i) virtual velocity 119891119894(119911) of the material particles(ii) virtual velocities119862

2and119862

3of the interfaces 119878

2and 1198783

(iii) variation 120575119875119894(119911) of the dipole momentum at the pointwith coordinates 119911119894

The geometry presented in Figure 1 was analyzed in [2128] which dealt with nucleation on stationary ions of liquidcondensate from the surrounding gaseous phase When thedomain Ω

119902is rigid the virtual velocities of the deformable

liquid phase should satisfy the boundary constraint

119873119894119891119894100381610038161003816100381610038161198781

= 0 (12)

4 The Bulk Equilibrium Equations ofDeformable Polarizable Substances

In this section we summarize the results and refer the readerto the relevant references for the corresponding derivations

Separating the independent variations in the volumeintegral of the first energy variation we arrive at the followingequilibrium equations [22 27]

minusnabla119894120577119894119896+ 120588120595119875119894nabla119896119875119894= 0

120588120595119875119894 = 119864119894

(13)

where 120595119875119894 = 120597120595120597119875

119894 the formal stress tensor 120577119898119896 is defined as

120577119894119895equiv 120588

120597120595

120597nabla119894119880119896

119860sdot119895

119896sdot (14)

and the tensor 119860119894sdotsdot119895is given by

119860119894sdot

sdot119895equiv 120575119894

119895minus nabla119895119880119894 (15)

Combining (13) we arrive at the equilibrium bulk equation

minusnabla119894120577119894119896+ 119864119894nabla119896119875119894= 0 (16)

Using the equations of electrostatics it can be shown that (16)can be rewritten as a statement of vanishing divergence

nabla119894(120577119894119895minus 119911119894119895(1

4120587119864119896119863119896minus

1

8120587119864119896119864119896) +

1

4120587119863119894119864119895) = 0 (17)

4 Advances in Mathematical Physics

For nonpolarizable substances the formal stress tensor120577119894119895 coincides with the Cauchy stress tensor in the Euleriandescription Relationship (17) generalizes to the celebratedKorteweg-Helmholtz relationship for liquid dielectrics [6 710ndash13 24] in the case of nonlinear electroelasticity

We can rewrite (17) as (see [22 23 27])

nabla119894alefsym119894119895= 0 (18)

where the aleph tensor alefsym119894119895 given by

alefsym119894119895equiv 120588

120597120595

120597nabla119894119880119896

119860sdot119895

119896sdot+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895

+1

4120587119863119894119864119895

(19)

can be thought of as the stress tensor of a polarizablesubstance We can rewrite the aleph tensor alefsym119894119895 as

alefsym119894119895equiv 120577119894119895+ Γ119894119895 (20)

where the electrostatic gamma tensor Γ119898119896 is given by

Γ119894119895equiv (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (21)

Equation (17) can be written in another insightful form

nabla119894120577119894119895= minusnabla119894Γ119894119895 (22)

In polarizable deformable substances neither one of thetensors 120577119894119895 or Γ119894119895 is divergence-free

The gamma tensor Γ119894119895 can be also considered as one ofthemanypossible generalizations of theMaxwell stress tensor119879119894119895

119879119894119895equiv minus

1

8120587119864119896119864119896119911119894119895+

1

4120587119864119894119864119895 (23)

since Γ119894119895 coincides with119879119894119895 when polarization vanishes Otherpossible generalizations of the Maxwell stress tensor

119879119894119895

1equiv minus

1

8120587119864119897119863119897119911119894119895+

1

4120587119863119894119864119895 (24a)

119879119894119895

2equiv minus

1

8120587119864119897119863119897119911119894119895+

1

4120587119863119895119864119894 (24b)

119879119894119895

3equiv minus

1

8120587119864119897119863119897119911119894119895+

1

8120587(119863119894119864119895+ 119863119895119864119894) (24c)

are perhaps more aesthetically appealing than the gammatensor Γ119894119895 We believe that the advantage of the gamma tensorover other possible generalizations is its variational origin andits ability to help address the issue of stability based on thecalculation of the second energy variation

One more useful tensor for polarizable materials is thebeth tensor ℶ119894sdot

sdot119895 or the tensor of electrochemical tensorial

potential It is defined by

ℶ119894sdot

sdot119895equiv (120588120595119911

119894119896minus alefsym119894119896+ Γ119894119896) 119861119896119895 (25)

where the tensor 119861119896119895

is the matrix inverse of 119860119896119895 definedin (15) As we show below the beth tensor ℶ119894sdot

sdot119895satisfies the

condition of zero divergence

nabla119894ℶ119894sdot

sdot119895= 0 (26)

similarly to the aleph tensor alefsym119894119895 The beth tensor ℶ119894sdotsdot119895can be

rewritten as

ℶ119894sdot

sdot119895= 120588119861119896119895120594119894119896 (27)

where 120594119894119896 is the Bowen symmetric tensorial chemical potential

120594119894119895= 120595119911119894119895minus1

120588120577119894119895= 120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895) (28)

The symmetric tensor 120594119894119895 should be distinguished fromthe typically asymmetric tensorial chemical tensor 120583119894119895

120583119894119895= 119911119896119894

∘119911119897119896120594119895119897 (29)

where 119911119898119894∘

is the contravariant metric tensor of the initialconfiguration

5 Conditions at the Interfaces

Boundary conditions depend on the various characteristicsof the interfaces Interfaces can differ by their mechanical orkinematic properties and whether or not they are subject tophase transformations We refer to interfaces that satisfy thekinematic constraint

[119880119894]+

minus= 0 (30)

as coherent interfaces The following condition for the alephstress tensor alefsym119894119895 is satisfied by equilibrium configurations atcoherent interfaces

119873119894[alefsym119894119895]+

minus= 0 (31)

If in addition to coherency the boundary is a phase interfacethe condition of phase equilibrium includes the beth tensorℶ119894119895

119873119894[ℶ119894119895]+

minus= 0 (32)

It makes sense then to call the beth tensor ℶ119894119895 the elec-trochemical tensorial potential for coherent interfaces indeformable substances because (32) is analogous to theequilibrium condition for the tensorial chemical potential

6 Nonfrictional Semicoherent Interfaces

By definition nonfrictional semicoherent interfaces are char-acterized by the possibility of relative slippage Nonfrictionalsemicoherent interfaces also may or may not be phaseinterfaces Regardless the following conditions ofmechanicalequilibrium must hold

119873119894120577119894119895

plusmn= minus119873

119895119901plusmn

119873119894119873119895[Γ119894119895]+

minus= [119901]+

minus

(33)

Advances in Mathematical Physics 5

At phase nonfrictional incoherent interfaces an additionalmass exchange equilibrium condition must be satisfied

119873119894119873119895[120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895)]

+

minus

= 0 (34)

7 Phase Interfaces in Rigid Dielectrics

When dealing with rigid solids all mechanical degrees offreedom disappear and the internal energy depends onlyon the polarization vector 119875119894 (and unless it is assumedto be constant temperature 119879) At the phase interface thecondition of phase equilibrium reads

119873119894119873119895[ℷ119894119895]+

minus= 0 (35)

where the gimel energy-like tensor ℷ119894119895 the electrostatic tenso-rial chemical potential for rigid dielectrics is defined by

ℷ119894119895equiv 120598 (119875) 119911

119894119895+ Γ119894119895 (36)

where 120598 equiv 120588120595 is the free energy density per unit volume (andwe once again suppress the index in119875119894 because it now appearsas an argument of a function) We refer to the gimel tensorℷ119894119895 as the electrostatic tensorial chemical potential because itplays the same role as the chemical potential 120583 in the classicalheterogeneous liquid-vapor system Contrary to the gammatensor Γ119894119895 the gimel tensor ℷ119894119895 is divergence-free

nabla119894ℷ119894119895= 0 (37)

One can analyze models in which the polarization vector119875119894 is fixed [20] Then 120595

plusmnare 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 twoidentities can be demonstrated similarly First let us rewritethe gimel tensor ℷ119894119895 as follows

ℷ119894119895= 120598 (119875) 119911

119894119895+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (38)

For the first term in (38) we have

nabla119894(120598 (119875) 119911

119894119895) = 119911119894119895 120597120598 (119875)

120597119875119896nabla119894119875119896 (39)

Using the thermodynamic identity

120597120598 (119875)

120597119875119894equiv 119864119894 (40)

we can rewrite (39) as

nabla119894(120598 (119875) 119911

119894119895) = 119864119894nabla119895119875119894 (41)

For the second term in (38) we have

nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896))

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896

(42)

which can be seen from the following chain of identities

2nd term = nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896)) (43a)

= nabla119896(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) (43b)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896119863119896minus 119864119896nabla119895119863119896) (43c)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896(119864119896+ 4120587119875

119896)

minus 119864119896nabla119895(119864119896+ 4120587119875

119896))

(43d)

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896 (43e)

For the third term in (38) we have

nabla119894(1

4120587119863119894119864119895) =

1

4120587119863119894nabla119894119864119895=

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (44)

Combining (41)ndash(44) we find

nabla119894ℷ119894119895= nabla119894120598 (119875) 119911

119894119895+ 119911119894119895nabla119894(1

8120587119864119897119864119897minus

1

4120587119864119897119863119897)

+1

4120587nabla119894(119863119894119864119895)

(45a)

= 119864119894nabla119895119875119894minus

1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894minus 119864119894nabla119895119875119894

+1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895

(45b)

= minus1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894+

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (45c)

Finally using the symmetric property nabla119894119864119895equiv nabla119895119864119894 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

119869 = minus119870119873119894119873119895[ℷ119894119895]+

minus (46)

The same approach can be applied to the case of an isolateddomain with fixed total volume yet subject to rearrangementIn this case the evolution equation should be slightly modi-fied to take into account surface diffusion Figure 2 illustratesan implementation of this approach in the two-dimensionalcase The quasi-static evolution of originally circular domainand fixed polarization vector leads to elongation in thedirection of polarization vector 119875

119894 and eventually to amorphological instability

6 Advances in Mathematical Physics

Figure 2 Onset of a morphological instability in a quasi-staticevolution of a domain filled with dipoles of fixed polarization

10 Conclusion

We discussed a phenomenological variational approach toelectrostatics and magnetostatics for heterogeneous systemswith phase transformations Although we focused on electro-statics almost all of the presented results are also valid formagnetostatics Our approach is an extension of the Gibbsvariational method as it was interpreted in [26]

The demand of having simultaneously a logically andphysically consistent theory remains to be the main driv-ing force of progress in thermodynamics The suggestedapproach leads to themathematically rigorous self-consistentresults Now it has to prove its viability in direct compar-ison with experiment That may prove to be difficult butreal progress is only possible when theory and experimentchallenge each other

Appendix

The summary of notations and variables is as follows (seeAbbreviations)

Abbreviations

119911119894 Eulerian coordinates in the ambient space119911119894119895 119911119894119895 Metrics tensors in the reference Eulerian

coordinates119911119894119895

∘ Metrics tensor of the coordinate system

generated by tracking back the coordinate119911119894 from the actual to the initialconfiguration [26]

nabla119894 The symbols of covariant differentiation

(based on the metrics 119911119894119895)

119902 119875119894 The electric charge density and

polarization (per unit volume)120593 119864119894 119863119894 The electrostatic potential field anddisplacement

Ω119902 Ω119889 Spatial domains occupied by free charges

and dipoles1198781 Interface separating the dielectric from the

distributed stationary electric charges1198782 Interface separating the different dielectric

phases

1198783 Interface separating the dielectric phase

from the surrounding vacuum119880119894 Displacements of material particles

119860119894sdot

sdot119895and 119861119894sdot

sdot119895 Mutually inverse geometric tensorsdefined in (15)

120588 Mass density119901 119879 120583 Pressure absolute temperature and

chemical potential of nonpolarizableone-component liquid phases

120583119894119895 120594119894119895 Asymmetric and Bowen chemical

potentials of nonpolarizable deformable(nonnecessarily liquid) media (forfurther details see [26])

120595 Free energy density per unit mass120577119894119895 Formal stress tensor defined in (14)119891119894 119862119894 119862119890 Admissible virtual velocities of the

material particles and interfacesalefsym119894119895 The aleph tensor a divergence-free

tensor defined in (19) the aleph tensorexhibits some of the properties of theclassical Cauchy stress tensor (inEulerian coordinates) and of theMaxwell stress tensor

ℶ119894119895 The beth tensor a divergence-free

tensor defined in (25) the beth tensorexhibits some of the properties of thescalar chemical potential ofnonpolarizable liquid and of thetensorial chemical potentials 120583119894119895 120594119894119896 ofnonpolarizable solids

Γ119894119895 The gamma tensor defined in (20) for

deformable media and in (21) forarbitrary polarizable media

ℷ119894119895 The gimel tensor which is defined in

(36) for rigid dielectrics and plays thesame role as the beth tensor ℶ119894119895 fordeformable dielectrics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Gibbs ldquoOn the equilibrium of heterogeneous substancesrdquoTransactions of the Connecticut Academy of Arts and Sciencesvol 3 pp 108ndash248 1876 vol 3 pp 343ndash524 1878

[2] I M Gelrsquofand and S V Fomin Calsulus of Variations Prentice-Hall Englewood Cliffs NJ USA 1963

[3] Josiah Willard Gibbs httpenwikiquoteorgwikiJosiah Wil-lard Gibbs

[4] J C Maxwell A Treatise on Electricity and Magnetism vol 1-2Dover Publications New York NY USA 1954

[5] H Poincare Lectures on Optics and Electromagnetism PrefaceCollected Papers of Poincare vol 3 Nauka Moscow Russia1974 (Russian)

Advances in Mathematical Physics 7

[6] Y I Frenkel Electrodynamics I General Theory of Electromag-netism ONTI Leningrad Russia 1934

[7] Y I Frenkelrsquo Electrodynamics Volume 2 Macroscopic Electrody-namics of Material Bodies ONTI Moscow Russia 1935

[8] I E Tamm Basics of the Theory of Electricity Nauka MoscowRussia 1989 (Russian)

[9] A Sommerfeld Electrodynamics Academic Press New YorkNY USA 1952

[10] J A Stratton Electromagnetic Theory McGraw-Hill New YorkNY USA 1941

[11] W K H Panofsky and M Phillips Classical Electricity andMagnetism Addison-Wesley Cambridge Mass USA 1950

[12] R A Toupin ldquoThe elastic dielectricrdquo Indiana University Math-ematics Journal vol 5 no 6 pp 849ndash915 1956

[13] L D Landau and E M Lifshitz Electrodynamics of ContinuousMedia Pergamon Press New York NY USA 1963

[14] I A Privorotskiı ldquoThermodynamic theory of ferromagneticgomainsinsrdquo Soviet Physics Uspekhi vol 15 no 5 pp 555ndash5741973

[15] L I Sedov and A G Tsypkin Fundamentals of MicroscopicTheories of Gravitation and Electromagnetism Nauka MoscowRussia 1989

[16] R E Rosensweig Ferrohydrodynamics Dover New York NYUSA 1985

[17] D J Korteweg ldquoUber die veranderung der form und desvolumens dielectrischer Korperunter Einwirkung elektrischerKrafterdquo Annalen der Physik und Chemie vol 245 no 1 pp 48ndash61 1880

[18] H Helmholtz ldquoUber die auf das Innere magnetisch oderdielectrisch polarisirter Korper wirkenden KrafterdquoAnnalen derPhysik vol 249 no 7 pp 385ndash406 1881

[19] H A LorentzTheTheory of Electrons and Its Applications to thePhenomena of Light and Radiant Heat Dover New York NYUSA 2011

[20] M A Grinfeld ldquoMorphology stability and evolution of dipoleaggregatesrdquo Proceedings of the Estonian Academy of SciencesEngineering vol 5 no 2 pp 131ndash141 1999

[21] P Grinfeld ldquoMorphological instability of liquid metallic nucleicondensing on charged inhomogeneitiesrdquo Physical Review Let-ters vol 87 no 9 Article ID 095701 4 pages 2001

[22] M Grinfeld and P Grinfeld ldquoTowards thermodynamics ofelastic electric conductorsrdquo Philosophical Magazine A vol 81no 5 pp 1341ndash1354 2001

[23] M A Grinfeld and P M Grinfeld ldquoThe exact conditions ofthermodynamic phase equilibrium in heterogeneous elasticsystems with dipolar interactionrdquo inNonlinearMechanics LMZubov Ed pp 47ndash51 Rostov University 2001

[24] M Abraham and R Becker The Classical Theory of Electricityand Magnetism Blackie amp Son 1932

[25] P Grinfeld Introduction to Tensor Analysis and the Calculus ofMoving Surfaces Springer New York NY USA 2014

[26] M Grinfeld Thermodynamic Methods in the Theory of Het-erogeneous Systems Interaction of Mechanics and MathematicsSeries Longman Scientific amp Technical Harlow UK 1991

[27] P Grinfeld and M Grinfeld ldquoThermodynamic aspects ofequilibrium shape and growth of crystalline films with elec-tromechanical interactionrdquo Ferroelectrics vol 342 no 1 pp 89ndash100 2006

[28] P Grinfeld ldquoMorphological instability of the dielectric thomsonnucleirdquo Physical Review B vol 81 no 18 Article ID 184110 2010

[29] P Grinfeld ldquoClausius-Clapeyron relations for an evaporatingsolid conductorrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 90 no 7-8 pp 633ndash640 2010

[30] P Grinfeld ldquoA proposed experiment for the verification ofThomsonrsquos nucleation theoryrdquo Ferroelectrics vol 413 no 1 pp65ndash72 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

For nonpolarizable substances the formal stress tensor120577119894119895 coincides with the Cauchy stress tensor in the Euleriandescription Relationship (17) generalizes to the celebratedKorteweg-Helmholtz relationship for liquid dielectrics [6 710ndash13 24] in the case of nonlinear electroelasticity

We can rewrite (17) as (see [22 23 27])

nabla119894alefsym119894119895= 0 (18)

where the aleph tensor alefsym119894119895 given by

alefsym119894119895equiv 120588

120597120595

120597nabla119894119880119896

119860sdot119895

119896sdot+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895

+1

4120587119863119894119864119895

(19)

can be thought of as the stress tensor of a polarizablesubstance We can rewrite the aleph tensor alefsym119894119895 as

alefsym119894119895equiv 120577119894119895+ Γ119894119895 (20)

where the electrostatic gamma tensor Γ119898119896 is given by

Γ119894119895equiv (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (21)

Equation (17) can be written in another insightful form

nabla119894120577119894119895= minusnabla119894Γ119894119895 (22)

In polarizable deformable substances neither one of thetensors 120577119894119895 or Γ119894119895 is divergence-free

The gamma tensor Γ119894119895 can be also considered as one ofthemanypossible generalizations of theMaxwell stress tensor119879119894119895

119879119894119895equiv minus

1

8120587119864119896119864119896119911119894119895+

1

4120587119864119894119864119895 (23)

since Γ119894119895 coincides with119879119894119895 when polarization vanishes Otherpossible generalizations of the Maxwell stress tensor

119879119894119895

1equiv minus

1

8120587119864119897119863119897119911119894119895+

1

4120587119863119894119864119895 (24a)

119879119894119895

2equiv minus

1

8120587119864119897119863119897119911119894119895+

1

4120587119863119895119864119894 (24b)

119879119894119895

3equiv minus

1

8120587119864119897119863119897119911119894119895+

1

8120587(119863119894119864119895+ 119863119895119864119894) (24c)

are perhaps more aesthetically appealing than the gammatensor Γ119894119895 We believe that the advantage of the gamma tensorover other possible generalizations is its variational origin andits ability to help address the issue of stability based on thecalculation of the second energy variation

One more useful tensor for polarizable materials is thebeth tensor ℶ119894sdot

sdot119895 or the tensor of electrochemical tensorial

potential It is defined by

ℶ119894sdot

sdot119895equiv (120588120595119911

119894119896minus alefsym119894119896+ Γ119894119896) 119861119896119895 (25)

where the tensor 119861119896119895

is the matrix inverse of 119860119896119895 definedin (15) As we show below the beth tensor ℶ119894sdot

sdot119895satisfies the

condition of zero divergence

nabla119894ℶ119894sdot

sdot119895= 0 (26)

similarly to the aleph tensor alefsym119894119895 The beth tensor ℶ119894sdotsdot119895can be

rewritten as

ℶ119894sdot

sdot119895= 120588119861119896119895120594119894119896 (27)

where 120594119894119896 is the Bowen symmetric tensorial chemical potential

120594119894119895= 120595119911119894119895minus1

120588120577119894119895= 120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895) (28)

The symmetric tensor 120594119894119895 should be distinguished fromthe typically asymmetric tensorial chemical tensor 120583119894119895

120583119894119895= 119911119896119894

∘119911119897119896120594119895119897 (29)

where 119911119898119894∘

is the contravariant metric tensor of the initialconfiguration

5 Conditions at the Interfaces

Boundary conditions depend on the various characteristicsof the interfaces Interfaces can differ by their mechanical orkinematic properties and whether or not they are subject tophase transformations We refer to interfaces that satisfy thekinematic constraint

[119880119894]+

minus= 0 (30)

as coherent interfaces The following condition for the alephstress tensor alefsym119894119895 is satisfied by equilibrium configurations atcoherent interfaces

119873119894[alefsym119894119895]+

minus= 0 (31)

If in addition to coherency the boundary is a phase interfacethe condition of phase equilibrium includes the beth tensorℶ119894119895

119873119894[ℶ119894119895]+

minus= 0 (32)

It makes sense then to call the beth tensor ℶ119894119895 the elec-trochemical tensorial potential for coherent interfaces indeformable substances because (32) is analogous to theequilibrium condition for the tensorial chemical potential

6 Nonfrictional Semicoherent Interfaces

By definition nonfrictional semicoherent interfaces are char-acterized by the possibility of relative slippage Nonfrictionalsemicoherent interfaces also may or may not be phaseinterfaces Regardless the following conditions ofmechanicalequilibrium must hold

119873119894120577119894119895

plusmn= minus119873

119895119901plusmn

119873119894119873119895[Γ119894119895]+

minus= [119901]+

minus

(33)

Advances in Mathematical Physics 5

At phase nonfrictional incoherent interfaces an additionalmass exchange equilibrium condition must be satisfied

119873119894119873119895[120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895)]

+

minus

= 0 (34)

7 Phase Interfaces in Rigid Dielectrics

When dealing with rigid solids all mechanical degrees offreedom disappear and the internal energy depends onlyon the polarization vector 119875119894 (and unless it is assumedto be constant temperature 119879) At the phase interface thecondition of phase equilibrium reads

119873119894119873119895[ℷ119894119895]+

minus= 0 (35)

where the gimel energy-like tensor ℷ119894119895 the electrostatic tenso-rial chemical potential for rigid dielectrics is defined by

ℷ119894119895equiv 120598 (119875) 119911

119894119895+ Γ119894119895 (36)

where 120598 equiv 120588120595 is the free energy density per unit volume (andwe once again suppress the index in119875119894 because it now appearsas an argument of a function) We refer to the gimel tensorℷ119894119895 as the electrostatic tensorial chemical potential because itplays the same role as the chemical potential 120583 in the classicalheterogeneous liquid-vapor system Contrary to the gammatensor Γ119894119895 the gimel tensor ℷ119894119895 is divergence-free

nabla119894ℷ119894119895= 0 (37)

One can analyze models in which the polarization vector119875119894 is fixed [20] Then 120595

plusmnare 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 twoidentities can be demonstrated similarly First let us rewritethe gimel tensor ℷ119894119895 as follows

ℷ119894119895= 120598 (119875) 119911

119894119895+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (38)

For the first term in (38) we have

nabla119894(120598 (119875) 119911

119894119895) = 119911119894119895 120597120598 (119875)

120597119875119896nabla119894119875119896 (39)

Using the thermodynamic identity

120597120598 (119875)

120597119875119894equiv 119864119894 (40)

we can rewrite (39) as

nabla119894(120598 (119875) 119911

119894119895) = 119864119894nabla119895119875119894 (41)

For the second term in (38) we have

nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896))

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896

(42)

which can be seen from the following chain of identities

2nd term = nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896)) (43a)

= nabla119896(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) (43b)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896119863119896minus 119864119896nabla119895119863119896) (43c)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896(119864119896+ 4120587119875

119896)

minus 119864119896nabla119895(119864119896+ 4120587119875

119896))

(43d)

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896 (43e)

For the third term in (38) we have

nabla119894(1

4120587119863119894119864119895) =

1

4120587119863119894nabla119894119864119895=

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (44)

Combining (41)ndash(44) we find

nabla119894ℷ119894119895= nabla119894120598 (119875) 119911

119894119895+ 119911119894119895nabla119894(1

8120587119864119897119864119897minus

1

4120587119864119897119863119897)

+1

4120587nabla119894(119863119894119864119895)

(45a)

= 119864119894nabla119895119875119894minus

1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894minus 119864119894nabla119895119875119894

+1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895

(45b)

= minus1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894+

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (45c)

Finally using the symmetric property nabla119894119864119895equiv nabla119895119864119894 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

119869 = minus119870119873119894119873119895[ℷ119894119895]+

minus (46)

The same approach can be applied to the case of an isolateddomain with fixed total volume yet subject to rearrangementIn this case the evolution equation should be slightly modi-fied to take into account surface diffusion Figure 2 illustratesan implementation of this approach in the two-dimensionalcase The quasi-static evolution of originally circular domainand fixed polarization vector leads to elongation in thedirection of polarization vector 119875

119894 and eventually to amorphological instability

6 Advances in Mathematical Physics

Figure 2 Onset of a morphological instability in a quasi-staticevolution of a domain filled with dipoles of fixed polarization

10 Conclusion

We discussed a phenomenological variational approach toelectrostatics and magnetostatics for heterogeneous systemswith phase transformations Although we focused on electro-statics almost all of the presented results are also valid formagnetostatics Our approach is an extension of the Gibbsvariational method as it was interpreted in [26]

The demand of having simultaneously a logically andphysically consistent theory remains to be the main driv-ing force of progress in thermodynamics The suggestedapproach leads to themathematically rigorous self-consistentresults Now it has to prove its viability in direct compar-ison with experiment That may prove to be difficult butreal progress is only possible when theory and experimentchallenge each other

Appendix

The summary of notations and variables is as follows (seeAbbreviations)

Abbreviations

119911119894 Eulerian coordinates in the ambient space119911119894119895 119911119894119895 Metrics tensors in the reference Eulerian

coordinates119911119894119895

∘ Metrics tensor of the coordinate system

generated by tracking back the coordinate119911119894 from the actual to the initialconfiguration [26]

nabla119894 The symbols of covariant differentiation

(based on the metrics 119911119894119895)

119902 119875119894 The electric charge density and

polarization (per unit volume)120593 119864119894 119863119894 The electrostatic potential field anddisplacement

Ω119902 Ω119889 Spatial domains occupied by free charges

and dipoles1198781 Interface separating the dielectric from the

distributed stationary electric charges1198782 Interface separating the different dielectric

phases

1198783 Interface separating the dielectric phase

from the surrounding vacuum119880119894 Displacements of material particles

119860119894sdot

sdot119895and 119861119894sdot

sdot119895 Mutually inverse geometric tensorsdefined in (15)

120588 Mass density119901 119879 120583 Pressure absolute temperature and

chemical potential of nonpolarizableone-component liquid phases

120583119894119895 120594119894119895 Asymmetric and Bowen chemical

potentials of nonpolarizable deformable(nonnecessarily liquid) media (forfurther details see [26])

120595 Free energy density per unit mass120577119894119895 Formal stress tensor defined in (14)119891119894 119862119894 119862119890 Admissible virtual velocities of the

material particles and interfacesalefsym119894119895 The aleph tensor a divergence-free

tensor defined in (19) the aleph tensorexhibits some of the properties of theclassical Cauchy stress tensor (inEulerian coordinates) and of theMaxwell stress tensor

ℶ119894119895 The beth tensor a divergence-free

tensor defined in (25) the beth tensorexhibits some of the properties of thescalar chemical potential ofnonpolarizable liquid and of thetensorial chemical potentials 120583119894119895 120594119894119896 ofnonpolarizable solids

Γ119894119895 The gamma tensor defined in (20) for

deformable media and in (21) forarbitrary polarizable media

ℷ119894119895 The gimel tensor which is defined in

(36) for rigid dielectrics and plays thesame role as the beth tensor ℶ119894119895 fordeformable dielectrics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Gibbs ldquoOn the equilibrium of heterogeneous substancesrdquoTransactions of the Connecticut Academy of Arts and Sciencesvol 3 pp 108ndash248 1876 vol 3 pp 343ndash524 1878

[2] I M Gelrsquofand and S V Fomin Calsulus of Variations Prentice-Hall Englewood Cliffs NJ USA 1963

[3] Josiah Willard Gibbs httpenwikiquoteorgwikiJosiah Wil-lard Gibbs

[4] J C Maxwell A Treatise on Electricity and Magnetism vol 1-2Dover Publications New York NY USA 1954

[5] H Poincare Lectures on Optics and Electromagnetism PrefaceCollected Papers of Poincare vol 3 Nauka Moscow Russia1974 (Russian)

Advances in Mathematical Physics 7

[6] Y I Frenkel Electrodynamics I General Theory of Electromag-netism ONTI Leningrad Russia 1934

[7] Y I Frenkelrsquo Electrodynamics Volume 2 Macroscopic Electrody-namics of Material Bodies ONTI Moscow Russia 1935

[8] I E Tamm Basics of the Theory of Electricity Nauka MoscowRussia 1989 (Russian)

[9] A Sommerfeld Electrodynamics Academic Press New YorkNY USA 1952

[10] J A Stratton Electromagnetic Theory McGraw-Hill New YorkNY USA 1941

[11] W K H Panofsky and M Phillips Classical Electricity andMagnetism Addison-Wesley Cambridge Mass USA 1950

[12] R A Toupin ldquoThe elastic dielectricrdquo Indiana University Math-ematics Journal vol 5 no 6 pp 849ndash915 1956

[13] L D Landau and E M Lifshitz Electrodynamics of ContinuousMedia Pergamon Press New York NY USA 1963

[14] I A Privorotskiı ldquoThermodynamic theory of ferromagneticgomainsinsrdquo Soviet Physics Uspekhi vol 15 no 5 pp 555ndash5741973

[15] L I Sedov and A G Tsypkin Fundamentals of MicroscopicTheories of Gravitation and Electromagnetism Nauka MoscowRussia 1989

[16] R E Rosensweig Ferrohydrodynamics Dover New York NYUSA 1985

[17] D J Korteweg ldquoUber die veranderung der form und desvolumens dielectrischer Korperunter Einwirkung elektrischerKrafterdquo Annalen der Physik und Chemie vol 245 no 1 pp 48ndash61 1880

[18] H Helmholtz ldquoUber die auf das Innere magnetisch oderdielectrisch polarisirter Korper wirkenden KrafterdquoAnnalen derPhysik vol 249 no 7 pp 385ndash406 1881

[19] H A LorentzTheTheory of Electrons and Its Applications to thePhenomena of Light and Radiant Heat Dover New York NYUSA 2011

[20] M A Grinfeld ldquoMorphology stability and evolution of dipoleaggregatesrdquo Proceedings of the Estonian Academy of SciencesEngineering vol 5 no 2 pp 131ndash141 1999

[21] P Grinfeld ldquoMorphological instability of liquid metallic nucleicondensing on charged inhomogeneitiesrdquo Physical Review Let-ters vol 87 no 9 Article ID 095701 4 pages 2001

[22] M Grinfeld and P Grinfeld ldquoTowards thermodynamics ofelastic electric conductorsrdquo Philosophical Magazine A vol 81no 5 pp 1341ndash1354 2001

[23] M A Grinfeld and P M Grinfeld ldquoThe exact conditions ofthermodynamic phase equilibrium in heterogeneous elasticsystems with dipolar interactionrdquo inNonlinearMechanics LMZubov Ed pp 47ndash51 Rostov University 2001

[24] M Abraham and R Becker The Classical Theory of Electricityand Magnetism Blackie amp Son 1932

[25] P Grinfeld Introduction to Tensor Analysis and the Calculus ofMoving Surfaces Springer New York NY USA 2014

[26] M Grinfeld Thermodynamic Methods in the Theory of Het-erogeneous Systems Interaction of Mechanics and MathematicsSeries Longman Scientific amp Technical Harlow UK 1991

[27] P Grinfeld and M Grinfeld ldquoThermodynamic aspects ofequilibrium shape and growth of crystalline films with elec-tromechanical interactionrdquo Ferroelectrics vol 342 no 1 pp 89ndash100 2006

[28] P Grinfeld ldquoMorphological instability of the dielectric thomsonnucleirdquo Physical Review B vol 81 no 18 Article ID 184110 2010

[29] P Grinfeld ldquoClausius-Clapeyron relations for an evaporatingsolid conductorrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 90 no 7-8 pp 633ndash640 2010

[30] P Grinfeld ldquoA proposed experiment for the verification ofThomsonrsquos nucleation theoryrdquo Ferroelectrics vol 413 no 1 pp65ndash72 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

At phase nonfrictional incoherent interfaces an additionalmass exchange equilibrium condition must be satisfied

119873119894119873119895[120595119911119894119895+1

120588(ℶ119894119895minus alefsym119894119895)]

+

minus

= 0 (34)

7 Phase Interfaces in Rigid Dielectrics

When dealing with rigid solids all mechanical degrees offreedom disappear and the internal energy depends onlyon the polarization vector 119875119894 (and unless it is assumedto be constant temperature 119879) At the phase interface thecondition of phase equilibrium reads

119873119894119873119895[ℷ119894119895]+

minus= 0 (35)

where the gimel energy-like tensor ℷ119894119895 the electrostatic tenso-rial chemical potential for rigid dielectrics is defined by

ℷ119894119895equiv 120598 (119875) 119911

119894119895+ Γ119894119895 (36)

where 120598 equiv 120588120595 is the free energy density per unit volume (andwe once again suppress the index in119875119894 because it now appearsas an argument of a function) We refer to the gimel tensorℷ119894119895 as the electrostatic tensorial chemical potential because itplays the same role as the chemical potential 120583 in the classicalheterogeneous liquid-vapor system Contrary to the gammatensor Γ119894119895 the gimel tensor ℷ119894119895 is divergence-free

nabla119894ℷ119894119895= 0 (37)

One can analyze models in which the polarization vector119875119894 is fixed [20] Then 120595

plusmnare 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 twoidentities can be demonstrated similarly First let us rewritethe gimel tensor ℷ119894119895 as follows

ℷ119894119895= 120598 (119875) 119911

119894119895+ (

1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) 119911119894119895+

1

4120587119863119894119864119895 (38)

For the first term in (38) we have

nabla119894(120598 (119875) 119911

119894119895) = 119911119894119895 120597120598 (119875)

120597119875119896nabla119894119875119896 (39)

Using the thermodynamic identity

120597120598 (119875)

120597119875119894equiv 119864119894 (40)

we can rewrite (39) as

nabla119894(120598 (119875) 119911

119894119895) = 119864119894nabla119895119875119894 (41)

For the second term in (38) we have

nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896))

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896

(42)

which can be seen from the following chain of identities

2nd term = nabla119894(119911119894119895(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896)) (43a)

= nabla119896(1

8120587119864119896119864119896minus

1

4120587119864119896119863119896) (43b)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896119863119896minus 119864119896nabla119895119863119896) (43c)

=1

4120587(119864119896nabla119895119864119896minus nabla119895119864119896(119864119896+ 4120587119875

119896)

minus 119864119896nabla119895(119864119896+ 4120587119875

119896))

(43d)

= minus1

4120587119864119896nabla119895119864119896minus 119875119896nabla119895119864119896minus 119864119896nabla119895119875119896 (43e)

For the third term in (38) we have

nabla119894(1

4120587119863119894119864119895) =

1

4120587119863119894nabla119894119864119895=

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (44)

Combining (41)ndash(44) we find

nabla119894ℷ119894119895= nabla119894120598 (119875) 119911

119894119895+ 119911119894119895nabla119894(1

8120587119864119897119864119897minus

1

4120587119864119897119863119897)

+1

4120587nabla119894(119863119894119864119895)

(45a)

= 119864119894nabla119895119875119894minus

1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894minus 119864119894nabla119895119875119894

+1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895

(45b)

= minus1

4120587119864119894nabla119895119864119894minus 119875119894nabla119895119864119894+

1

4120587119864119894nabla119894119864119895+ 119875119894nabla119894119864119895 (45c)

Finally using the symmetric property nabla119894119864119895equiv nabla119895119864119894 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

119869 = minus119870119873119894119873119895[ℷ119894119895]+

minus (46)

The same approach can be applied to the case of an isolateddomain with fixed total volume yet subject to rearrangementIn this case the evolution equation should be slightly modi-fied to take into account surface diffusion Figure 2 illustratesan implementation of this approach in the two-dimensionalcase The quasi-static evolution of originally circular domainand fixed polarization vector leads to elongation in thedirection of polarization vector 119875

119894 and eventually to amorphological instability

6 Advances in Mathematical Physics

Figure 2 Onset of a morphological instability in a quasi-staticevolution of a domain filled with dipoles of fixed polarization

10 Conclusion

We discussed a phenomenological variational approach toelectrostatics and magnetostatics for heterogeneous systemswith phase transformations Although we focused on electro-statics almost all of the presented results are also valid formagnetostatics Our approach is an extension of the Gibbsvariational method as it was interpreted in [26]

The demand of having simultaneously a logically andphysically consistent theory remains to be the main driv-ing force of progress in thermodynamics The suggestedapproach leads to themathematically rigorous self-consistentresults Now it has to prove its viability in direct compar-ison with experiment That may prove to be difficult butreal progress is only possible when theory and experimentchallenge each other

Appendix

The summary of notations and variables is as follows (seeAbbreviations)

Abbreviations

119911119894 Eulerian coordinates in the ambient space119911119894119895 119911119894119895 Metrics tensors in the reference Eulerian

coordinates119911119894119895

∘ Metrics tensor of the coordinate system

generated by tracking back the coordinate119911119894 from the actual to the initialconfiguration [26]

nabla119894 The symbols of covariant differentiation

(based on the metrics 119911119894119895)

119902 119875119894 The electric charge density and

polarization (per unit volume)120593 119864119894 119863119894 The electrostatic potential field anddisplacement

Ω119902 Ω119889 Spatial domains occupied by free charges

and dipoles1198781 Interface separating the dielectric from the

distributed stationary electric charges1198782 Interface separating the different dielectric

phases

1198783 Interface separating the dielectric phase

from the surrounding vacuum119880119894 Displacements of material particles

119860119894sdot

sdot119895and 119861119894sdot

sdot119895 Mutually inverse geometric tensorsdefined in (15)

120588 Mass density119901 119879 120583 Pressure absolute temperature and

chemical potential of nonpolarizableone-component liquid phases

120583119894119895 120594119894119895 Asymmetric and Bowen chemical

potentials of nonpolarizable deformable(nonnecessarily liquid) media (forfurther details see [26])

120595 Free energy density per unit mass120577119894119895 Formal stress tensor defined in (14)119891119894 119862119894 119862119890 Admissible virtual velocities of the

material particles and interfacesalefsym119894119895 The aleph tensor a divergence-free

tensor defined in (19) the aleph tensorexhibits some of the properties of theclassical Cauchy stress tensor (inEulerian coordinates) and of theMaxwell stress tensor

ℶ119894119895 The beth tensor a divergence-free

tensor defined in (25) the beth tensorexhibits some of the properties of thescalar chemical potential ofnonpolarizable liquid and of thetensorial chemical potentials 120583119894119895 120594119894119896 ofnonpolarizable solids

Γ119894119895 The gamma tensor defined in (20) for

deformable media and in (21) forarbitrary polarizable media

ℷ119894119895 The gimel tensor which is defined in

(36) for rigid dielectrics and plays thesame role as the beth tensor ℶ119894119895 fordeformable dielectrics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Gibbs ldquoOn the equilibrium of heterogeneous substancesrdquoTransactions of the Connecticut Academy of Arts and Sciencesvol 3 pp 108ndash248 1876 vol 3 pp 343ndash524 1878

[2] I M Gelrsquofand and S V Fomin Calsulus of Variations Prentice-Hall Englewood Cliffs NJ USA 1963

[3] Josiah Willard Gibbs httpenwikiquoteorgwikiJosiah Wil-lard Gibbs

[4] J C Maxwell A Treatise on Electricity and Magnetism vol 1-2Dover Publications New York NY USA 1954

[5] H Poincare Lectures on Optics and Electromagnetism PrefaceCollected Papers of Poincare vol 3 Nauka Moscow Russia1974 (Russian)

Advances in Mathematical Physics 7

[6] Y I Frenkel Electrodynamics I General Theory of Electromag-netism ONTI Leningrad Russia 1934

[7] Y I Frenkelrsquo Electrodynamics Volume 2 Macroscopic Electrody-namics of Material Bodies ONTI Moscow Russia 1935

[8] I E Tamm Basics of the Theory of Electricity Nauka MoscowRussia 1989 (Russian)

[9] A Sommerfeld Electrodynamics Academic Press New YorkNY USA 1952

[10] J A Stratton Electromagnetic Theory McGraw-Hill New YorkNY USA 1941

[11] W K H Panofsky and M Phillips Classical Electricity andMagnetism Addison-Wesley Cambridge Mass USA 1950

[12] R A Toupin ldquoThe elastic dielectricrdquo Indiana University Math-ematics Journal vol 5 no 6 pp 849ndash915 1956

[13] L D Landau and E M Lifshitz Electrodynamics of ContinuousMedia Pergamon Press New York NY USA 1963

[14] I A Privorotskiı ldquoThermodynamic theory of ferromagneticgomainsinsrdquo Soviet Physics Uspekhi vol 15 no 5 pp 555ndash5741973

[15] L I Sedov and A G Tsypkin Fundamentals of MicroscopicTheories of Gravitation and Electromagnetism Nauka MoscowRussia 1989

[16] R E Rosensweig Ferrohydrodynamics Dover New York NYUSA 1985

[17] D J Korteweg ldquoUber die veranderung der form und desvolumens dielectrischer Korperunter Einwirkung elektrischerKrafterdquo Annalen der Physik und Chemie vol 245 no 1 pp 48ndash61 1880

[18] H Helmholtz ldquoUber die auf das Innere magnetisch oderdielectrisch polarisirter Korper wirkenden KrafterdquoAnnalen derPhysik vol 249 no 7 pp 385ndash406 1881

[19] H A LorentzTheTheory of Electrons and Its Applications to thePhenomena of Light and Radiant Heat Dover New York NYUSA 2011

[20] M A Grinfeld ldquoMorphology stability and evolution of dipoleaggregatesrdquo Proceedings of the Estonian Academy of SciencesEngineering vol 5 no 2 pp 131ndash141 1999

[21] P Grinfeld ldquoMorphological instability of liquid metallic nucleicondensing on charged inhomogeneitiesrdquo Physical Review Let-ters vol 87 no 9 Article ID 095701 4 pages 2001

[22] M Grinfeld and P Grinfeld ldquoTowards thermodynamics ofelastic electric conductorsrdquo Philosophical Magazine A vol 81no 5 pp 1341ndash1354 2001

[23] M A Grinfeld and P M Grinfeld ldquoThe exact conditions ofthermodynamic phase equilibrium in heterogeneous elasticsystems with dipolar interactionrdquo inNonlinearMechanics LMZubov Ed pp 47ndash51 Rostov University 2001

[24] M Abraham and R Becker The Classical Theory of Electricityand Magnetism Blackie amp Son 1932

[25] P Grinfeld Introduction to Tensor Analysis and the Calculus ofMoving Surfaces Springer New York NY USA 2014

[26] M Grinfeld Thermodynamic Methods in the Theory of Het-erogeneous Systems Interaction of Mechanics and MathematicsSeries Longman Scientific amp Technical Harlow UK 1991

[27] P Grinfeld and M Grinfeld ldquoThermodynamic aspects ofequilibrium shape and growth of crystalline films with elec-tromechanical interactionrdquo Ferroelectrics vol 342 no 1 pp 89ndash100 2006

[28] P Grinfeld ldquoMorphological instability of the dielectric thomsonnucleirdquo Physical Review B vol 81 no 18 Article ID 184110 2010

[29] P Grinfeld ldquoClausius-Clapeyron relations for an evaporatingsolid conductorrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 90 no 7-8 pp 633ndash640 2010

[30] P Grinfeld ldquoA proposed experiment for the verification ofThomsonrsquos nucleation theoryrdquo Ferroelectrics vol 413 no 1 pp65ndash72 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

Figure 2 Onset of a morphological instability in a quasi-staticevolution of a domain filled with dipoles of fixed polarization

10 Conclusion

We discussed a phenomenological variational approach toelectrostatics and magnetostatics for heterogeneous systemswith phase transformations Although we focused on electro-statics almost all of the presented results are also valid formagnetostatics Our approach is an extension of the Gibbsvariational method as it was interpreted in [26]

The demand of having simultaneously a logically andphysically consistent theory remains to be the main driv-ing force of progress in thermodynamics The suggestedapproach leads to themathematically rigorous self-consistentresults Now it has to prove its viability in direct compar-ison with experiment That may prove to be difficult butreal progress is only possible when theory and experimentchallenge each other

Appendix

The summary of notations and variables is as follows (seeAbbreviations)

Abbreviations

119911119894 Eulerian coordinates in the ambient space119911119894119895 119911119894119895 Metrics tensors in the reference Eulerian

coordinates119911119894119895

∘ Metrics tensor of the coordinate system

generated by tracking back the coordinate119911119894 from the actual to the initialconfiguration [26]

nabla119894 The symbols of covariant differentiation

(based on the metrics 119911119894119895)

119902 119875119894 The electric charge density and

polarization (per unit volume)120593 119864119894 119863119894 The electrostatic potential field anddisplacement

Ω119902 Ω119889 Spatial domains occupied by free charges

and dipoles1198781 Interface separating the dielectric from the

distributed stationary electric charges1198782 Interface separating the different dielectric

phases

1198783 Interface separating the dielectric phase

from the surrounding vacuum119880119894 Displacements of material particles

119860119894sdot

sdot119895and 119861119894sdot

sdot119895 Mutually inverse geometric tensorsdefined in (15)

120588 Mass density119901 119879 120583 Pressure absolute temperature and

chemical potential of nonpolarizableone-component liquid phases

120583119894119895 120594119894119895 Asymmetric and Bowen chemical

potentials of nonpolarizable deformable(nonnecessarily liquid) media (forfurther details see [26])

120595 Free energy density per unit mass120577119894119895 Formal stress tensor defined in (14)119891119894 119862119894 119862119890 Admissible virtual velocities of the

material particles and interfacesalefsym119894119895 The aleph tensor a divergence-free

tensor defined in (19) the aleph tensorexhibits some of the properties of theclassical Cauchy stress tensor (inEulerian coordinates) and of theMaxwell stress tensor

ℶ119894119895 The beth tensor a divergence-free

tensor defined in (25) the beth tensorexhibits some of the properties of thescalar chemical potential ofnonpolarizable liquid and of thetensorial chemical potentials 120583119894119895 120594119894119896 ofnonpolarizable solids

Γ119894119895 The gamma tensor defined in (20) for

deformable media and in (21) forarbitrary polarizable media

ℷ119894119895 The gimel tensor which is defined in

(36) for rigid dielectrics and plays thesame role as the beth tensor ℶ119894119895 fordeformable dielectrics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Gibbs ldquoOn the equilibrium of heterogeneous substancesrdquoTransactions of the Connecticut Academy of Arts and Sciencesvol 3 pp 108ndash248 1876 vol 3 pp 343ndash524 1878

[2] I M Gelrsquofand and S V Fomin Calsulus of Variations Prentice-Hall Englewood Cliffs NJ USA 1963

[3] Josiah Willard Gibbs httpenwikiquoteorgwikiJosiah Wil-lard Gibbs

[4] J C Maxwell A Treatise on Electricity and Magnetism vol 1-2Dover Publications New York NY USA 1954

[5] H Poincare Lectures on Optics and Electromagnetism PrefaceCollected Papers of Poincare vol 3 Nauka Moscow Russia1974 (Russian)

Advances in Mathematical Physics 7

[6] Y I Frenkel Electrodynamics I General Theory of Electromag-netism ONTI Leningrad Russia 1934

[7] Y I Frenkelrsquo Electrodynamics Volume 2 Macroscopic Electrody-namics of Material Bodies ONTI Moscow Russia 1935

[8] I E Tamm Basics of the Theory of Electricity Nauka MoscowRussia 1989 (Russian)

[9] A Sommerfeld Electrodynamics Academic Press New YorkNY USA 1952

[10] J A Stratton Electromagnetic Theory McGraw-Hill New YorkNY USA 1941

[11] W K H Panofsky and M Phillips Classical Electricity andMagnetism Addison-Wesley Cambridge Mass USA 1950

[12] R A Toupin ldquoThe elastic dielectricrdquo Indiana University Math-ematics Journal vol 5 no 6 pp 849ndash915 1956

[13] L D Landau and E M Lifshitz Electrodynamics of ContinuousMedia Pergamon Press New York NY USA 1963

[14] I A Privorotskiı ldquoThermodynamic theory of ferromagneticgomainsinsrdquo Soviet Physics Uspekhi vol 15 no 5 pp 555ndash5741973

[15] L I Sedov and A G Tsypkin Fundamentals of MicroscopicTheories of Gravitation and Electromagnetism Nauka MoscowRussia 1989

[16] R E Rosensweig Ferrohydrodynamics Dover New York NYUSA 1985

[17] D J Korteweg ldquoUber die veranderung der form und desvolumens dielectrischer Korperunter Einwirkung elektrischerKrafterdquo Annalen der Physik und Chemie vol 245 no 1 pp 48ndash61 1880

[18] H Helmholtz ldquoUber die auf das Innere magnetisch oderdielectrisch polarisirter Korper wirkenden KrafterdquoAnnalen derPhysik vol 249 no 7 pp 385ndash406 1881

[19] H A LorentzTheTheory of Electrons and Its Applications to thePhenomena of Light and Radiant Heat Dover New York NYUSA 2011

[20] M A Grinfeld ldquoMorphology stability and evolution of dipoleaggregatesrdquo Proceedings of the Estonian Academy of SciencesEngineering vol 5 no 2 pp 131ndash141 1999

[21] P Grinfeld ldquoMorphological instability of liquid metallic nucleicondensing on charged inhomogeneitiesrdquo Physical Review Let-ters vol 87 no 9 Article ID 095701 4 pages 2001

[22] M Grinfeld and P Grinfeld ldquoTowards thermodynamics ofelastic electric conductorsrdquo Philosophical Magazine A vol 81no 5 pp 1341ndash1354 2001

[23] M A Grinfeld and P M Grinfeld ldquoThe exact conditions ofthermodynamic phase equilibrium in heterogeneous elasticsystems with dipolar interactionrdquo inNonlinearMechanics LMZubov Ed pp 47ndash51 Rostov University 2001

[24] M Abraham and R Becker The Classical Theory of Electricityand Magnetism Blackie amp Son 1932

[25] P Grinfeld Introduction to Tensor Analysis and the Calculus ofMoving Surfaces Springer New York NY USA 2014

[26] M Grinfeld Thermodynamic Methods in the Theory of Het-erogeneous Systems Interaction of Mechanics and MathematicsSeries Longman Scientific amp Technical Harlow UK 1991

[27] P Grinfeld and M Grinfeld ldquoThermodynamic aspects ofequilibrium shape and growth of crystalline films with elec-tromechanical interactionrdquo Ferroelectrics vol 342 no 1 pp 89ndash100 2006

[28] P Grinfeld ldquoMorphological instability of the dielectric thomsonnucleirdquo Physical Review B vol 81 no 18 Article ID 184110 2010

[29] P Grinfeld ldquoClausius-Clapeyron relations for an evaporatingsolid conductorrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 90 no 7-8 pp 633ndash640 2010

[30] P Grinfeld ldquoA proposed experiment for the verification ofThomsonrsquos nucleation theoryrdquo Ferroelectrics vol 413 no 1 pp65ndash72 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

[6] Y I Frenkel Electrodynamics I General Theory of Electromag-netism ONTI Leningrad Russia 1934

[7] Y I Frenkelrsquo Electrodynamics Volume 2 Macroscopic Electrody-namics of Material Bodies ONTI Moscow Russia 1935

[8] I E Tamm Basics of the Theory of Electricity Nauka MoscowRussia 1989 (Russian)

[9] A Sommerfeld Electrodynamics Academic Press New YorkNY USA 1952

[10] J A Stratton Electromagnetic Theory McGraw-Hill New YorkNY USA 1941

[11] W K H Panofsky and M Phillips Classical Electricity andMagnetism Addison-Wesley Cambridge Mass USA 1950

[12] R A Toupin ldquoThe elastic dielectricrdquo Indiana University Math-ematics Journal vol 5 no 6 pp 849ndash915 1956

[13] L D Landau and E M Lifshitz Electrodynamics of ContinuousMedia Pergamon Press New York NY USA 1963

[14] I A Privorotskiı ldquoThermodynamic theory of ferromagneticgomainsinsrdquo Soviet Physics Uspekhi vol 15 no 5 pp 555ndash5741973

[15] L I Sedov and A G Tsypkin Fundamentals of MicroscopicTheories of Gravitation and Electromagnetism Nauka MoscowRussia 1989

[16] R E Rosensweig Ferrohydrodynamics Dover New York NYUSA 1985

[17] D J Korteweg ldquoUber die veranderung der form und desvolumens dielectrischer Korperunter Einwirkung elektrischerKrafterdquo Annalen der Physik und Chemie vol 245 no 1 pp 48ndash61 1880

[18] H Helmholtz ldquoUber die auf das Innere magnetisch oderdielectrisch polarisirter Korper wirkenden KrafterdquoAnnalen derPhysik vol 249 no 7 pp 385ndash406 1881

[19] H A LorentzTheTheory of Electrons and Its Applications to thePhenomena of Light and Radiant Heat Dover New York NYUSA 2011

[20] M A Grinfeld ldquoMorphology stability and evolution of dipoleaggregatesrdquo Proceedings of the Estonian Academy of SciencesEngineering vol 5 no 2 pp 131ndash141 1999

[21] P Grinfeld ldquoMorphological instability of liquid metallic nucleicondensing on charged inhomogeneitiesrdquo Physical Review Let-ters vol 87 no 9 Article ID 095701 4 pages 2001

[22] M Grinfeld and P Grinfeld ldquoTowards thermodynamics ofelastic electric conductorsrdquo Philosophical Magazine A vol 81no 5 pp 1341ndash1354 2001

[23] M A Grinfeld and P M Grinfeld ldquoThe exact conditions ofthermodynamic phase equilibrium in heterogeneous elasticsystems with dipolar interactionrdquo inNonlinearMechanics LMZubov Ed pp 47ndash51 Rostov University 2001

[24] M Abraham and R Becker The Classical Theory of Electricityand Magnetism Blackie amp Son 1932

[25] P Grinfeld Introduction to Tensor Analysis and the Calculus ofMoving Surfaces Springer New York NY USA 2014

[26] M Grinfeld Thermodynamic Methods in the Theory of Het-erogeneous Systems Interaction of Mechanics and MathematicsSeries Longman Scientific amp Technical Harlow UK 1991

[27] P Grinfeld and M Grinfeld ldquoThermodynamic aspects ofequilibrium shape and growth of crystalline films with elec-tromechanical interactionrdquo Ferroelectrics vol 342 no 1 pp 89ndash100 2006

[28] P Grinfeld ldquoMorphological instability of the dielectric thomsonnucleirdquo Physical Review B vol 81 no 18 Article ID 184110 2010

[29] P Grinfeld ldquoClausius-Clapeyron relations for an evaporatingsolid conductorrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 90 no 7-8 pp 633ndash640 2010

[30] P Grinfeld ldquoA proposed experiment for the verification ofThomsonrsquos nucleation theoryrdquo Ferroelectrics vol 413 no 1 pp65ndash72 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of