Microscopic theory for electrocaloric effects in planar double layer systems
Rajeev Kumar, Jyoti P. Mahalik, Evgheni Strelcov, Alexander Tselev, Bradley S. Lokitz, Sergei.V. Kalinin, Bobby G. Sumpter
aa r X i v : . [ c ond - m a t . s o f t ] M a r Microscopic theory for electrocaloric effects in planar double layersystems
Rajeev Kumar,
1, 2, ∗ Jyoti P. Mahalik, Evgheni Strelcov, AlexanderTselev, Bradley S. Lokitz, Sergei.V. Kalinin, and Bobby G. Sumpter
1, 2 Center for Nanophase Materials Sciences,Oak Ridge National Laboratory, Oak Ridge, TN-37831 Computer Science and Mathematics Division,Oak Ridge National Laboratory, Oak Ridge, TN-37831 (Dated: October 6, 2018)We present a field theory approach to study changes in local temperature due to anapplied electric field (the electrocaloric effect) in electrolyte solutions. Steric effectsand a field-dependent dielectric function are found to be of paramount importancefor accurate estimations of the electrocaloric effect. Interestingly, electrolyte solu-tions are found to exhibit negative electrocaloric effects. Overall, our results pointtoward using fluids near room temperature with low heat capacity and high saltconcentration for enhanced electrocalorics.
There has been a renewed interest in developing caloric materials[1–4] and advancingtechnologies[1, 3] for various refrigeration applications. The caloric materials undergoreversible thermal changes under the influence of an applied field, which can be mag-netic, electric or mechanical in nature. These thermal changes due to magnetic field,electric field and mechanical stresses are known as the magnetocaloric, electrocaloric andmechanocaloric effects, respectively. Thermodynamic description of these changes was pro-vided by Thomson[5] and the changes are results of variations in entropy of the systemunder the influence of an applied field. The magnetocaloric effect is already used to reachtemperatures in the milliKelvin (mK) range and is in the stage of being commercialized forhousehold refrigeration. In contrast, search for novel materials that can achieve the so-calledcolossal or giant electrocaloric effect is currently a topic of extensive research. Historically,ferroelectric materials[1–4, 6], which are crystals with net polarization in the absence of anyexternal applied electric field, have been studied extensively for the electrocaloric effect andhave shown thermal changes as low as 0 .
003 K near room temperature and as high as 31K based on the operating temperature and applied electric field[3]. It is to be noted thatmost of the materials studied for the electrocaloric effects are in the solid state except somepolymeric films[1–3, 7], which are considered viscoelastic, and liquid crystalline fluids[8] thathave shown giant electrocaloric effects in thin film geometries.Thermodynamic description of the electrocaloric effect[6] in an adiabatic system relies onthe fact that changes in entropy resulting from application of an electric field must be zero.As entropy can be modified by varying either temperature ( T ) or difference in the surfacepotentials of the electrodes ( V s ), we consider infinitesimal changes in entropy (∆ S ) for asystem at an initial temperature of T = T and the potential difference V s = V undergoinginfinitesimal changes in the temperature (∆ T ) and the potential difference (∆ V s ) so that∆ S = (cid:20) ∂S∂T (cid:21) T = T ,V s = V ∆ T + (cid:20) ∂S∂V s (cid:21) T = T ,V s = V ∆ V s . (1)For adiabatic changes, ∆ S = 0 and noting that T (cid:2) ∂S∂T (cid:3) T = T ,V s = V = c v ( T = T , V s = V ),where c v is the volume heat capacity and depends on the initial temperature and the potentialdifference, we can write (cid:20) ∆ T ∆ V s (cid:21) T = T ,V s = V = − e/k B c v ( T = T , V s = V ) (cid:20) ∂S∂ψ s (cid:21) T = T ,ψ s = eV kBT (2)Here, we have defined ψ s = eV s /k B T , e is the charge of an electron and k B is the Boltzmannconstant so that e/k B = 1 . × K/V. It is to be noted that electrostriction[3] effectsleading to changes in the volume of the liquids are not taken into account here and form thebasis of multi-caloric materials exhibiting electrocaloric and mechanocaloric/elastocaloriceffects. This is an interesting direction for future research. Eq. 2 provides three insights.First, it is clear that the changes in temperature resulting from changes in the potentialdifference are inversely proportional to the volume heat capacity of the material. Hence,fluids with low heat capacity are preferable candidates for enhanced electrocalorics. Second,insight is obtained from the use of thermodynamic rules stating[10] that entropy must in-crease with an increase the temperature i.e., c v >
0. This implies that the dimensionlessquantity eV s /k B T , ratio of the electrostatic energy of a unit charge to the thermal energy,is the relevant variable. In particular, sign of changes in the temperature (i.e., increase ordecrease) with an increase in the surface potential depends on the changes in entropy withrespect to eV s /k B T . Third, the length scale of the region undergoing changes in temperatureis determined by the volume undergoing entropic changes.Larger entropic changes resulting from small changes in the potential difference are re-quired for enhancing the electrocaloric effect (cf. Eq. 2). As larger entropic changesare expected in liquids[7, 8] than solids in the presence of an external field, we have fo-cused on a theoretical description of the electrocaloric effect in electrolyte solutions. Weuse Eq. 2 and entropic changes computed using field theory[5, 6] to study the elec-trocaloric effect in planar double layer systems[1, 7, 14]. The free energy of the doublelayer can be constructed[8, 16, 17, 19, 20] with different approximations including variouseffects due to dielectric saturation[4, 21, 22], finite polarizability of ions[24–26], finite sizeof ions[9, 27, 29, 30], ion adsorption-desorption equilibrium[31] and image charges[14, 32].This allows systematic investigations into roles played by different factors in affecting theelectrocaloric effect and pave the way for rational design of enhanced electrocaloric fluids.Another motivation in studying such a system lies in the need for an improved theory for theelectrolyte solutions in strong external fields, where crowding and dielectric saturation effectsare important and a larger electrocaloric effect is observed for viscoelastic materials such aspolymer[7] films and liquid-crystalline solutions[8]. Furthermore, novel technologies[33, 34]for extracting energy by mixing fresh river water with saline ocean water can benefit froman improved theory for the electric double layer. These technologies are based on the well-known fact that an electric double-layer acts as a capacitor and salt concentration plays akey role in dictating its capacitance. Operating temperature has been shown to play a keyrole in affecting the energy that can be harvested[34] using these technologies.We use a microscopic field theory approach to study planar double layer systems (see theSupporting Information). In particular, we consider two parallel plates having surface area A each, separated by distance L and immersed in an electrolyte solution containing equalnumber density (= ρ c,b ) of positive and negative ions along with ρ s,b as the number density ofsolvent molecules. The plates are assumed to have uniform surface charge densities (numberof charges per unit area), σ and σ and the corresponding surface potentials are V , and V , (in units of Volts), respectively. Surface potentials and charge densities are related toeach other by electrostatic boundary conditions and depend on the mechanisms by whichthe plates acquire the surface charge. These relations can be formally derived by consider-ing different mechanisms for charging. We take molecular volumes of the solvent, positiveand negative ions to be v s , v + and v − , respectively. Noting that theoretical description ofpolarization under an external electric field and strong electric fields are pre-requisites fordeveloping theory for the electrocaloric effect, field dependent dielectric and steric effectsresulting from finite sizes of ions and solvent molecules are included in our model. In thiswork, we have built a minimal model that can capture the underlying physics based on treat-ing each solvent molecule as an electric dipole of length p s occupying molecular volume v s .Finite polarizability of ions and solvent molecules are not considered in this work. However,the current formalism can be extended to take into account the effects of polarizability. Wehave used the theory to study the electocaloric effects in non-overlapping double layers (i.e.,single double layer systems) so that V , = V and V , = 0 i.e., conditions of constant surfacepotentials are considered in this work so that the potential difference V s = V . Parametersare chosen to describe water molecules (such as the dipole moment p s ). Furthermore, inthese model calculations, we have considered symmetric ions and solvent molecules so that v s = v + = v − = a and ignored the asymmetry in sizes of the molecules. The size parameter a is chosen so that the density of pure water is reproduced i.e., 1 /a ≡ .Typical free energy changes (∆ F ) of the double layers (with respect to the electrolytesolution in the absence of applied surface potential ) are shown in Figure 1(a) for differ-ent values of eV /k B T and temperature ( T ) ranging from room temperature to near theboiling point of water. The free energy changes are negative for the entire parameter range,which is in qualitative agreement with the predictions of the standard Poisson-Boltzmann(PB) approach (i.e., ignoring field-dependent dielectric and steric effects) and the modi-fied Poisson-Boltzmann (MPB) approach (i.e., ignoring field-dependent dielectric effects)(cf. Eqs. 47 and 43, respectively, in the Supporting Information). Also, larger free energychanges are found with an increase in the temperature due to increased entropic contri-butions shown in Figure 1(b). Furthermore, an increase in the free energy changes withan increase in the surface potential is also in qualitative agreement with the PB and MPBapproaches. Corresponding entropic changes (∆ S = − ( ∂ ∆ F/∂T ) Ω , Ω = AL being the totalvolume) such as those shown in Figure 1(b) dictate the electrocaloric effect.As the free energy and entropy changes per unit area are computed using the field theory,we rewrite Eq. 2 to calculate the electrocaloric effect so that (cid:20) ∆ T ∆ V s (cid:21) T = T ,V s = V = − e/k B ¯ c v (cid:20) ∂ ∆ S/ ¯ Ak B ∂ψ s (cid:21) T = T ,ψ s = eV kBT (3)where ¯ A = A/a and ¯ c v = c v / ¯ Ak B is the rescaled heat capacity of the electrolyte solutionin the presence of applied electric field. Formally, it can be written as ¯ c v = ¯ c v ( T = T , V s =0) + ¯ c v ( T = T , V s = V ) so that ¯ c v ( T = T , V s = 0) = T h ∂S h / ¯ Ak B ∂T i T = T ,ψ s =0 is the rescaledheat capacity of the reference homogeneous electrolyte solution having S h as its entropyand ¯ c v ( T = T , V s = V ) = T h ∂ ∆ S/ ¯ Ak B ∂T i T = T ,ψ s = eV kBT accounts for additional contributions dueto the applied electric field. It is to be noted that in Figure 1(b), surface potentialsand temperature are varied simultaneously due to the variation of ψ = eV /k B T and thequantity ∂ ∆ S/ ¯ Ak B ∂T can be extracted from Figure 1(b) using the formal relation ∂ ∆ S/ ¯ Ak B ∂T = h ∂ ∆ S/ ¯ Ak B ∂T i ψ s = ψ − h ψ T ∂ ∆ S/ ¯ Ak B ∂ψ i T = T . In calculating the electrocaloric effect, we have taken¯ c v ( T = T , V s = 0) = 6 . L dl /a corresponding to molar heat capacities of water to be 3 . R (taken to be independent of temperature) and 3 / R for each type of ion treated as anideal gas[10] in the homogeneous phase, where L dl is the thickness of the double layer and R = k B N A is the universal gas constant so that N A is the Avogadro’s number. It is to benoted that L dl naturally sets the length scale of the region undergoing changes in temperatureas a result of the electrocaloric effect. For the numerical estimates, we have defined L dl /a as the distance from the electrode after which counterion and coion densities approach theirbulk values, ρ c,b ,From isothermal changes in the entropy in Figure 1(b), it is clear that entropy of thedouble layer increases with an increase in the surface potential (i.e., ∂ ∆ S/∂ψ s > . . T = 303 K (near room temperature). For this particular system, ∆ T / ∆ V s ∼ − . .
16 K is predicted. Itis found that magnitude of ∆
T / ∆ V s is dependent on the initial temperature and the saltconcentration in the bulk. In particular, the magnitude decreases with an increase in thetemperature and increases with an increase in the salt concentration. The decrease in the
300 310 320 330 340 350 360 370T (K)-2-1.5-1-0.50 a ∆ F / A k B T (a)
300 310 320 330 340 350 360 370T (K)00.511.522.53 a ∆ S / A k B (b) FIG. 1: (a) Changes in the free energy and (b) entropy as a function of applied surface potential( V ) and temperature for an electolyte solution containing 0 . eV /k B T (Volts)-0.4-0.3-0.2-0.10 ∆ T / ∆ V s ( K / V ) T = 303 KT = 313 KT = 323 KT = 333 KT = 343 KT = 353 KT = 363 K (a) (Volts)-0.8-0.6-0.4-0.20 ∆ T / ∆ V s ( K / V ) c s = 0.1 Mc s = 0.5 Mc s = 1.0 M (b) FIG. 2: (a) Effects of initial temperature ( T ) and (b) the bulk salt concentration (so that ρ c,b =0 . c s (nm) − and c s is in moles per litre (M)) on the electrocaloric effect. The left panelcorresponds to c s = 0 . T = 303 K. Parameter for doublelayer thickness L dl /a is found to be 3 , c s = 1 . , . . magnitude with an increase in the initial temperature is a direct outcome of an increase inthe heat capacity of the double layer with an increase in the applied surface potential, asevident from Figure 1(b). The increase in the magnitude of the electrocaloric effect with anincrease in the bulk salt concentration results from a decrease in thickness of the double layer( L dl ). Furthermore, larger free energy and entropic changes are found with an increase in thebulk salt concentration, as shown in Figure 1(a) in the Supporting Information. It shouldbe noted that qualitatively the same effects are predicted by the PB and MPB approaches,where the free energy and entropy changes increase as √ ρ c,b . However, quantitatively, the PBand MPB approaches digress from the numerical results due to errors made in predictingthe free energy changes. To demonstrate this point, we have shown a comparison of thefree energy changes for the same system, estimated using the PB, MPB and the numericalcalculations in Figure 4(a). It is found that the PB approach is off by factors of 10 − eV /k B T >
10. In contrast, the MPB approach corrects for some of the errors made inthe PB approach but it still deviates from the numerical results by a factor of 3. /k B T -10-9-8-7-6-5-4-3-2-10 a ∆ F / A k B T PBMPBNumerical (a) /k B T σ a c S = 0.1 Mc s = 0.5 Mc s = 1.0 M PB (Solid)MPB (Dashed)Numerical (Dash-dotted) (b)
FIG. 3: (a) Comparison of the free energy changes computed using the PB approach (i.e., ignoringfield-dependent dielectric and steric effects), the MPB approach (i.e., ignoring field-dependentdielectric effects) and the numerical calculations for c s = 0 . T = 303 K. (b) Computed surfacecharge density as a function of applied surface potential, estimated using the PB, the MPB andthe numerical calculations, for different bulk salt concentrations at T = 303 K. The solid linescorreponds to the analytical relation presented in the text. The relative importance of the field-dependent dielectric and steric effects in predictingstructure of the double layer and resulting changes in the free energy can be assessed bycomparing plots showing surface charge density as a function of the surface potential aspredicted by the PB, MPB and numerical approaches (Figure 4(b)). The surface chargedensity in the PB approach is given by σ = σ = κǫ h π | Z c | l Bo sinh h | Z c | eV k B T i where | Z c | is thevalency of ions (= 1 for monovalent ions), l Bo = e / πǫ k B T so that ǫ is the permittivity ofvacuum, ǫ h = 1 + 4 πl Bo p s ρ s,b / ρ s,b is the solvent density and κ = (8 πl Bo | Z c | ρ c,b /ǫ h ) / is the inverseDebye screening length. The PB and MPB approaches predict a monotonic increase in thesurface charge density with an increase in the surface potential, as shown in Figure 4(b),without showing any sign of saturation, leading to unphysical surface charge densities. Thenumerical calculations show agreement with the PB and MPB approaches for eV /k B T < − ∂σ/∂V ) hints at the breakdown of the one-dimensional uniform charge density model usedhere and plausible onset of in-plane charge density waves[35].In conclusion, we have presented a field theory approach for studying electrocaloric effectsin planar double layer systems. Two key ingredients of the theory are the consideration ofsteric effects and dipolar interactions resulting from polar solvent molecules. Althoughthe theory is general, in this work, we have presented calculations for aqueous solutionscontaining monovalent salt ions. It was shown that the electrocaloric effect in planar doublelayer systems is negative, i.e., the temperature of the double layer should decrease with anincrease in the applied surface potential. The magnitude of the electrocaloric effect dependson the initial temperature of the solution and the salt concentration. In particular, we showedthat the magnitude of the electrocaloric effect should decrease with increase in the initialtemperature and increase with an increase in the salt concentration. Due to the generalnature of the field theory approach[6] to tackle curved interfaces, polymers, multivalent ionsetc., our work opens up a new area of theoretical research focused on the rational designof electrocaloric fluids. Furthermore, we have shown that the field theory approach staysrobust for high surface potentials and the other approaches such as the PB and MPB are notreliable. This particular feature of the field theory is quite important for energy harvestingtechnologies based on electrochemical capacitors and supercapacitors.We acknowledge support from the Center for Nanophase Materials Sciences, which issponsored at Oak Ridge National Laboratory by the Scientific User Facilities Division, officeof Basic Energy Sciences, U.S. Department of Energy (DOE). ∗ Electronic address: [email protected][1] T. Correia, Q. Zhang,
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We consider two parallel plates separated by distance L and immersed in an electrolytesolution containing n s solvent molecules, n + positive and n − negative ions. The plates areassumed to have uniform surface charge densities (number of charges per unit area), σ and σ (in units of electronic charge, e ) and the corresponding surface potentials are V , and V , , respectively. It is to be noted that surface potentials and charge densities are related toeach other by electrostatic boundary conditions[1] and depend on the mechanisms by whichthe plates acquire the surface charge. These relations can be formally derived by consideringdifferent mechanisms for charging[1].Molecular volumes of the solvent, positive and negative ions are taken to be v s , v + and v − , respectively. We are interested in understanding the effects of dipolar interactions and2finite ion sizes on the thermodynamics of double layer. For such purposes, we seek a minimalmodel that can capture the underlying physics. In this work, we have studied a minimalmodel based on treating each solvent molecule as an electric dipole of length p s occupyingmolecular volume v s . Also, the positive and negative ions have molecular volumes of v + and v − , respectively. Finite polarizability[2, 3] of ions and solvent molecules are not consideredin this work. However, the current formalism can be extended to take into account theeffects of polarizability.The canonical partition function for such a system is written[4, 5] as Z = Z Y j = ± ,s n j ! n j Y α =1 d r j,α Z n s Y α =1 d u α exp h − ˆ H { r j,α , u α } i Y r δ X j = ± ,s ˆ ρ j ( r ) v j − ! (1)where r j,α is the position vector for the α th particle of type j and u α is the unit vectorquantifying orientation of α th solvent dipole. The Hamiltonian is written by taking intoaccount the contributions coming from ion-ion, ion-dipole and dipole-dipole interactions.Short range interactions between ions and solvent molecules are ignored in the minimalmodel studied here. ˆ ρ j ( r ) represents microscopic number density of the particles of type j at a certain location r defined asˆ ρ j ( r ) = n j X α =1 δ ( r − r j,α ) for j = s, + , − (2)The Hamiltonian for the ions and dipoles can be written as[4, 5]ˆ H = l Bo Z d r Z d r ′ h ˆ ρ e ( r ) − ∇ r . ˆ P ( r ) i h ˆ ρ e ( r ′ ) − ∇ r ′ . ˆ P ( r ′ ) i | r − r ′ | (3)where l Bo = e / πǫ o k B T is the Bjerrum length in vaccum and ˆ ρ e ( r ) is the charge density(in units of e ), given by ˆ ρ e ( r ) = P j = ± Z j ˆ ρ j ( r ) + σ δ ( x − x ) + σ δ ( x − x ), Z j being thevalency (with sign) of ions of type j and | x − x | = L is the distance between the plates.Also, ˆ P ( r ) is polarization density of dipoles (in units of e ) at location r , given byˆ P ( r ) = p s n s X α =1 δ ( r − r α ) u α (4) Field theory in the canonical ensemble
A field theory for the system described above can be constructed following a standardprotocol[6]. We start from the electrostatics contributions to the partition function. For3the electrostatics contribution to the partition function written in the form Eq. 3, we useHubbard-Stratonovich transformation[6] so thatexp h − ˆ H i = 1 N ψ Z D [ ψ ] exp (cid:20) − i Z d r n ˆ ρ e ( r ) − ∇ r . ˆ P ( r ) o ψ ( r ) + 18 πl Bo Z d r ψ ( r ) ∇ r ψ ( r )) (cid:21) (5)where N ψ is a normalization factor, given by N ψ = Z D [ ψ ] exp (cid:20) πl Bo Z d r ψ ( r ) ∇ r ψ ( r )) (cid:21) (6)Using this transformation and writing the local constraints (represented by delta functions)in terms of functional integrals using Y r δ X j = ± ,s ˆ ρ j ( r ) v j − ! = Z D [ η ] exp " − i Z d r η ( r ) ( X j = ± ,s ˆ ρ j ( r ) v j − ) (7)we can write the partition function given by Eq. 1 as Z = 1 N ψ Z D [ ψ ] Z D [ η ] exp (cid:20) − Hk B T (cid:21) (8)where Hk B T = − πl Bo Z d r ψ ( r ) ∇ r ψ ( r ) − i Z d r η ( r ) + σ Z d r k iψ ( r k , x )+ σ Z d r k iψ ( r k , x ) − X j = ± ,s { n j ln Q j { ψ, η } − ln n j ! } (9)and we have used the notation r = ( x, y, z ) ≡ ( x, r k ) so that r k denotes in-plane vectorparallel to the plates. Q j is the partition function for particles of type j , given by Q j = ± { ψ, η } = Z d r exp [ − iZ j ψ ( r ) − iv j η ( r )] (10) Q s { ψ, η } = Z d r Z d u exp [ − ip s u . ∇ r ψ ( r ) − iv s η ( r )] (11)In the following, we use the saddle-point approximation to estimate the functional inte-grals over ψ and η . An equivalent calculation in the grand canonical ensemble is presentedin the Appendix A.4 Saddle-point approximation: self-consistent equations, free energy and chemicalpotentials
The saddle point approximation with respect to η and ψ gives two non-linear equations.At the saddle-points, both η and ψ turn out to be purely imaginary. Writing iη ( r ) = η ⋆ ( r )and iψ ( r ) = ψ ⋆ ( r ) at the saddle point and defining densities of ions and solvent moleculesvia ρ j = ± ( r ) = n j Q j { ψ ⋆ , η ⋆ } exp [ − Z j ψ ⋆ ( r ) − v j η ⋆ ( r )] (12) ρ s ( r ) = 4 πn s Q s { ψ ⋆ , η ⋆ } exp [ − v s η ⋆ ( r )] sinh p s |∇ r ψ ⋆ ( r ) | p s |∇ r ψ ⋆ ( r ) | (13)the equations at the saddle point are given by X j = ± v j ρ j ( r ) + v s ρ s ( r ) = 1 (14) ∇ r · [ ǫ ( r ) ∇ r ψ ⋆ ( r )] = − πl Bo ρ e ( r ) (15)so that the local charge density ( ρ e ( r )) and dielectric function ( ǫ ( r )) are given by ρ e ( r ) = X j = ± Z j ρ j ( r ) + σ δ ( x − x ) + σ δ ( x − x ) (16) ǫ ( r ) = 1 + 4 πl Bo p s ρ s ( r ) L [ p s |∇ r ψ ⋆ ( r ) | ] p s |∇ r ψ ⋆ ( r ) | (17)where L ( x ) = coth( x ) − /x is the Langevin function. Corresponding Helmholtz free energy( F ) is given by the approximation F/k B T = − ln Z ≃ H ⋆ /k B T = F ⋆ /k B T so that (cf. Eq.9) F ⋆ k B T = 18 πl Bo Z d r ψ ⋆ ( r ) ∇ r ψ ⋆ ( r ) − Z d r η ⋆ ( r ) + σ Z d r k ψ ⋆ ( r k , x )+ σ Z d r k ψ ⋆ ( r k , x ) − X j = ± ,s { n j ln Q j { ψ ⋆ , η ⋆ } − ln n j ! } (18)Eq. 18 can be rewritten after eliminating n j using Eqs. 14 and 15. Furthermore, usingthe Stirling approximation ln n ! ≃ n ln n − n , Eq. 18 can be written as F ⋆ k B T = Z d r ρ e ( r ) ψ ⋆ ( r ) + 18 πl Bo Z d r ψ ⋆ ( r ) ∇ r ψ ⋆ ( r ) − Z d r ρ s ( r ) ln (cid:20) π sinh p s |∇ r ψ ⋆ ( r ) | p s |∇ r ψ ⋆ ( r ) | (cid:21) + X j = ± ,s Z d r ρ j ( r ) [ln ρ j ( r ) −
1] (19)5For study of opposing double layer systems in equilibrium with an electrolyte solution,chemical potential is determined by conditions in the solution far from the plates. In orderto fix the chemical potentials by specifying different conditions in the solution far from theplates, we rewrite the above equations in terms of chemical potenials. An approximationfor the chemical potentials ( µ j ) of different species can be derived from Eq. 18 using thethermodynamic relation µ j = ( ∂F/∂n j ) Ω ≃ ( ∂F ⋆ /∂n j ) Ω = µ ⋆j , Ω being the total volume.Using the Stirling approximation ln n ! ≃ n ln n − n , the chemical potentials within thesaddle-point approximation are given by µ ⋆j = ± ,s k B T = ln (cid:20) n j Q j { ψ ⋆ , η ⋆ } (cid:21) (20)Using Eq. 20, Eqs. 12 and 13 can be written as ρ j = ± ( r ) = exp (cid:20) µ ⋆j k B T − Z j ψ ⋆ ( r ) − v j η ⋆ ( r ) (cid:21) (21) ρ s ( r ) = 4 π exp (cid:20) µ ⋆s k B T − v s η ⋆ ( r ) (cid:21) sinh p s |∇ r ψ ⋆ ( r ) | p s |∇ r ψ ⋆ ( r ) | (22) Chemical part of the free energy: charging the electrodes and adsorption-desorptionelectrochemical equilibrium
The free energy (cf. Eq. 19) for the two opposing double layer system is obtained for a given surface charge density of the plates and has the charged plates at given surface poten-tials (in vacuum) as the reference frame. This can be easily seen by putting ρ j = ± ,s = 0 inEq. 19 so that F ⋆ /k B T { ρ j = ± ,s = 0 } = ( σ / R d r k ψ ⋆ ( r k , x ) + ( σ / R d r k ψ ⋆ ( r k , x ). This,in turn, means that Eq. 19 doesn’t include the work done (typically by an external source)in charging the two plates at a separation distance of L = | x − x | . This contribution[7, 8]to the free energy is F chem k B T = − Z d r k Z σ dσ ′ ψ ⋆ ( r k , x ) { σ ′ } − Z d r k Z σ dσ ′ ψ ⋆ ( r k , x ) { σ ′ } (23)Evaluation of the right hand side in Eq. 23 requires specification of the mechanisms bywhich the plates acquire their charge. In the following, we consider the specific case whenplates are kept at constant surface potentials.6 One dimensional model: plates at constant surface potentials with symmetrical ionsand solvent molecules
If the densities far from the plates are known to be ρ j,b corresponding to spatially uniform ψ ⋆ ( r ) = ψ ⋆b and η ⋆ ( r ) = η ⋆b then Eqs. 21 and 22 can be written as ρ j = ± ( r ) = ρ j,b exp [ − Z j { ψ ⋆ ( r ) − ψ ⋆b } − v j { η ⋆ ( r ) − η ⋆b } ] (24) ρ s ( r ) = ρ s,b exp [ − v s { η ⋆ ( r ) − η ⋆b } ] sinh p s |∇ r ψ ⋆ ( r ) | p s |∇ r ψ ⋆ ( r ) | (25)For two parallel plates, saddle point value of ψ varies only along the direction perpendic-ular to the charged surface (taken to be along x-axis) so that ψ ⋆ ( r ) ≡ ψ ⋆ ( x ) , η ⋆ ( r ) ≡ η ⋆ ( x ).Furthermore, considering the case of symmetric ions and solvent molecules so that v j = ± ,s = a and Z + = − Z − = | Z c | so that ρ j = ± ,b = ρ c,b , we can eliminate η ⋆ using Eqs. 14, 24 and25 and write Eq. 15 as ∂∂x (cid:20) ǫ ( x ) ∂ψ ⋆ ( x ) ∂x (cid:21) = − πl Bo ρ e ( x ) (26)where the local charge density ( ρ e ( x )) and dielectric function ( ǫ ( x )) are given by ρ e ( x ) = | Z c | [ ρ + ( x ) − ρ − ( x )] + σ δ ( x − x ) + σ δ ( x − x ) (27) ρ + ( x ) = ρ c,b exp [ −| Z c | { ψ ⋆ ( x ) − ψ ⋆b } ] f n ψ ⋆ ( x ) − ψ ⋆b , ∂ψ ⋆ ( x ) ∂x o (28) ρ − ( x ) = ρ c,b exp [ | Z c | { ψ ⋆ ( x ) − ψ ⋆b } ] f n ψ ⋆ ( x ) − ψ ⋆b , ∂ψ ⋆ ( x ) ∂x o (29) ǫ ( x ) = 1 + 4 πl Bo p s ρ s,b f n ψ ⋆ ( x ) − ψ ⋆b , ∂ψ ⋆ ( x ) ∂x o sinh p s | ∂ψ ⋆ ( x ) ∂x | p s | ∂ψ ⋆ ( x ) ∂x | L h p s | ∂ψ ⋆ ( x ) ∂x | i p s | ∂ψ ⋆ ( x ) ∂x | (30)so that f (cid:26) ψ ⋆ ( x ) − ψ ⋆b , ∂ψ ⋆ ( x ) ∂x (cid:27) = ρ s,b a sinh p s | ∂ψ ⋆ ( x ) ∂x | p s | ∂ψ ⋆ ( x ) ∂x | + 2 ρ c,b a cosh [ | Z c | { ψ ⋆ ( x ) − ψ ⋆b } ] (31)and [ ρ s,b + 2 ρ c,b ] a = 1. It is to be noted that solvent density is given by ρ s ( x ) = ρ s,b f n ψ ⋆ ( x ) − ψ ⋆b , ∂ψ ⋆ ( x ) ∂x o sinh p s | ∂ψ ⋆ ( x ) ∂x | p s | ∂ψ ⋆ ( x ) ∂x | (32)and satisfies the incompressibility constraint P j = ± ,s ρ j ( x ) a = 1.7 Free energy within saddle-point approximation : adiabatic changes
Changes in entropy (∆ S ) can be readily calculated from the corresponding free energychanges (∆ F ) and the thermodynamic relation ∆ S = − (cid:0) ∂ ∆ F∂T (cid:1) Ω . Free energy of the doublelayer system ( F ⋆dl ) is the sum of electrostatic contributions approximated by F ⋆ and thechemical part given by F ⋆chem . Superscript ⋆ implies the use of saddle-point approximation(mean-field like treatment) in estimating the free energy. In particular, assuming lateralhomogeneity, for plates (at known surface potentials) separated by distance L having surfacearea A each, F ⋆ and F ⋆chem are given by F ⋆ Ak B T = Z L dxρ e ( x ) ψ ⋆ ( x ) + 18 πl Bo Z L dxψ ⋆ ( x ) ∂ ψ ⋆ ( x ) ∂x − Z L dxρ s ( x ) ln " π sinh p s | ∂ψ ⋆ ( x ) ∂x | p s | ∂ψ ⋆ ( x ) ∂x | + X j = ± ,s Z L dxρ j ( x ) [ln ρ j ( x ) −
1] (33)and F ⋆chem Ak B T = − σ ψ ⋆ ( x ) − σ ψ ⋆ ( x ) (34)In order to compute the electrocaloric effect, free energy changes with respect to the systemin the absence of applied electric field are desirable. In the absence of applied electric field(i.e., when σ = σ = 0 and considered as the reference state), the same number of ions andsolvent molecules are homogeneously distributed in volume Ω = AL so that free energy ofthe reference state becomes F ⋆h ALk B T = (cid:20) F ⋆ + F ⋆chem ALk B T (cid:21) σ = σ =0 = 2 ρ c,b [ln ρ c,b −
1] + ρ s,b [ln ρ s,b − − ln 4 π ] (35)where, we have used the constraint A R L dxρ j,x = ρ j,b Ω for equating the number of ions andsolvent molecules in the absence and presence of applied electric field. Using these equations,the free energy changes (∆ F ⋆ ) due to the application of an electric field can be written as∆ F ⋆ Ak B T = F ⋆dl − F ⋆h Ak B T = F ⋆ − F ⋆h Ak B T − σ ψ ⋆ ( x ) − σ ψ ⋆ ( x ) (36) Spatially uniform dielectric : Poisson-Boltzmann (PB) and modifiedPoisson-Boltzmann (MPB) approaches
In the limits of small surface potentials so that ψ ⋆ ( x ) − ψ ⋆b → p s | ∂ψ ⋆ ( x ) ∂x | →
0, the dielectric function given by Eq. 30 becomes spatially8uniform so that ǫ ( x ) ≡ ǫ h = 1 + 4 π l Bo p s ρ s,b (37)Physically, this means that solvent density is spatially uniform in the limits of small surfacepotentials and weak coupling limit for dipoles so that ρ s ( x ) = ρ s,b as evident from Eq.32. It is to be noted that the quantity f is taken to be unity in these limits and leadsto the standard Poisson-Boltzmann results pioneered by Verwey and Overbeek[7]. Anothersomewhat recent development (so called modified Poisson-Boltzmann (MPB) approach[9])is to consider the case of uniform dielectric but include steric effects in the calculations ofcharge density by taking f (cid:26) ψ ⋆ ( x ) − ψ ⋆b , ∂ψ ⋆ ( x ) ∂x (cid:27) ≡ f MP B { ψ ⋆ ( x ) − ψ ⋆b } = 1 − α + α cosh [ | Z c | { ψ ⋆ ( x ) − ψ ⋆b } ](38), where α = 2 ρ c,b a is the packing fraction of ions in the bulk. Although it seems incon-sistent to ignore and retain functional dependence of a particular quantity such as f whileconsidering different physical quantities such as dielectric function and charge density, theMPB approach has been quite successful in predicting qualitative features of the doublelayer capacitance. Nevertheless, the MPB approach leads to semi-analytical predictions forthe electrostatic potential and the free energy, as described below.With the approximations described above, Eq. 26 can be readily integrated over x (aftermultiplying by ∂ψ ⋆ ( x ) ∂x on both sides). In particular, we obtain a self-consistent equation for ∂ψ ⋆ ( x ) ∂x (cid:20) ∂ψ ⋆ ( x ) ∂x (cid:21) = 4 πl Bo ǫ h a [ln f MP B { ψ ⋆ ( x ) − ψ ⋆b } − λ ] (39)where λ is an integration constant, which is determined below and the effects of surfacecharge densities ( σ , σ ) appear in the form of boundary conditions. Using Eq. 39 andequations at the saddle-point, the free energy changes of the double layer system, definedby Eq. 36, can be written as∆ F ⋆MP B Ak B T = F ⋆dl,MP B − F h Ak B T = − λLa − a Z ψ ⋆ ( x ) ψ ⋆ ( x ) dψ [ln f MP B { ψ ⋆ ( x ) − ψ ⋆b } − λ ] ∂ψ ⋆ ( x ) ∂x (40)where F dl,MP B = F ⋆MP B + F ⋆chem and F ⋆MP B is the approximation for Eq. 33 obtained usingEq. 39 and F ⋆chem is given by Eq. 34. We must point out that in obtaining Eq. 40, we9have retained functional dependence of the solvent density on f MP B through Eq. 32 andused the incompressibility constraint.In the following, we consider two cases of non-overlapping and overlapping double layersand eliminate λ from Eq. 40. In the case of non-overlapping double layers, ψ ⋆ ( x ) becomes anon-monotonic function of x with a minimum at x = x min . Integrating Eq. 39 over x withthe limits x and x , we obtain[8] Z ψ ⋆ ( x ) ψ ⋆ ( x min ) dψ ⋆ [ln f MP B { ψ ⋆ ( x ) − ψ ⋆b } − λ ] / + Z ψ ⋆ ( x ) ψ ⋆ ( x min ) dψ ⋆ [ln f MP B { ψ ⋆ ( x ) − ψ ⋆b } − λ ] / = (cid:20) πl Bo ǫ h a (cid:21) / L (41)Similarly, for the case of overlapping double layers so that ψ ⋆ ( x ) > ψ ⋆ ( x ), we obtain Z ψ ⋆ ( x ) ψ ⋆ ( x ) dψ ⋆ [ln f MP B { ψ ⋆ ( x ) − ψ ⋆b } − λ ] / = (cid:20) πl Bo ǫ h a (cid:21) / L (42)Eqs. 41 and 42 allows us to eliminate λ from Eq. 40 and write it as∆ F ⋆MP B Ak B T = − r ǫ h πl Bo a g { ψ ⋆ ( x ) , ψ ⋆ ( x ) } = − ρ c,b | Z c |√ κ α o g { ψ ⋆ ( x ) , ψ ⋆ ( x ) } (43)where we have defined κ = 8 πl Bo | Z c | ρ c,b /ǫ h . Also, g { ψ ⋆ ( x ) , ψ ⋆ ( x ) } = X k =1 , Z ψ ⋆ ( x k ) ψ ⋆ ( x min ) dψ q ln f MP B { ψ ⋆ ( x ) − ψ ⋆b } (44)for the non-overlapping double layers and g { ψ ⋆ ( x ) , ψ ⋆ ( x ) } = Z ψ ⋆ ( x ) ψ ⋆ ( x ) dψ q ln f MP B { ψ ⋆ ( x ) − ψ ⋆b } (45)in the case of overlapping double layers.Changes in entropy (∆ S ⋆MP B ) can be readily calculated using Eq. 43 and the thermody-namic relation ∆ S = − (cid:0) ∂ ∆ F∂T (cid:1) Ω so that∆ S ⋆MP B Ak B = − (cid:18) ∂ ∆ F ⋆MP B Ak B ∂T (cid:19) Ω = 6 ρ c,b | Z c |√ κ α o g (cid:20) T g ∂g∂T (cid:21) (46)where we have dropped explicit functional dependencies of g on ψ ⋆ ( x ) − ψ ⋆b for conveniencein writing. It is interesting to consider the limit of dilute solutions so that α → ψ ⋆ ( x min ) = ψ ⋆b and λ = 1 (due to the fact that ∂ψ ⋆ ( x ) /∂x = 0 at x = x min in Eq.39). This leads to ∆ F ⋆P B Ak B T = 4 ρ c,b κ X k =1 , [2 − | Z c | { ψ ⋆ ( x k ) − ψ ⋆b } )] (47)i.e., the total free energy change is the sum of changes in the individual double layers[8].This leads to entropic changes given by∆ S ⋆P B Ak B = − ρ c,b κ X k =1 , (cid:20) − (cid:18) | Z c | { ψ ⋆ ( x k ) − ψ ⋆b } (cid:19) − sech (cid:18) | Z c | { ψ ⋆ ( x k ) − ψ ⋆b } (cid:19) + (cid:26) − sech (cid:18) | Z c | { ψ ⋆ ( x k ) − ψ ⋆b } (cid:19)(cid:27) Tǫ h ∂ǫ h ∂T (cid:21) (48) NUMERICAL METHODS
We have solved the set of equations (Eqs. 26- 32) numerically after rewriting Eq. 26 inthe form ∂ψ ⋆ ( x ) ∂t = ∂ ψ ⋆ ( x ) ∂x + 1 ǫ ( x ) ∂ǫ ( x ) ∂x ∂ψ ⋆ ( x ) ∂x + 4 πl Bo ǫ ( x ) ρ e ( x ) (49)where t is a fictitious time. A steady state solution of Eq. 49 is obtained by using theextrapolated gear[5] scheme and using size of ions a to obtain dimensionless length variables.Time step of 0 . L/a = 20 −
40 (depending on thevalue of ρ c,b ) and 1024 grid points. Convergence of the numerical solution is checked bycomputing free energy changes between two consecutive time steps and the changes lessthan 0 . l Bo and the free energy changes (in units of Ak B T /a ) are computed using Eqs. 33,34, 35 and 36. In computing the electrocaloric effect, we have made use of the fact thatthe field variable ψ ⋆ ( x ) in the theory is the electrostatic potential (in units of k B T /e ) atlocation x . For example, ψ ⋆ (0) = eV , /k B T for the single double layer system studied inthis work. Numerical estimates for the surface charge densities we obtained by the relation σ = − [ ǫ ( x ) / πl Bo )( ∂ψ ⋆ ( x ) /∂x ] x =0 .1 /k B T -2-1.5-1-0.50 a ∆ F / A k B T c s = 0.1 Mc s = 0.5 Mc s = 1.0 M (a) ψ ★ ( x ) Numerical: eV /k B T = 1.0 MPB: eV /k B T = 1.0Numerical: eV /k B T = 10.0MPB: eV /k B T = 10.0 (b) FIG. 4: (a) Effects of the bulk salt concentration on the free energy changes (∆ F = ∆ F ⋆ ) of thedouble layer at T = 303 K. (b) Comparisons of electrostatic potential profiles ( ψ ⋆ ( x )) from theMPB approach and numerical calculations at c s = 0 . RESULTS: ANATOMY OF THE DOUBLE LAYER
Anatomy of the double layer is determined by the electrostatic potential profile. Asthe comparisons between the PB and MPB approaches are well-known[9], we only showcomparisons between the MPB and our numerical calculations in Figure 4(b) for low andhigh surface potentials. It is found that the MPB and numerical results are in excellentagreement at eV /k B T = 1 showing exponential decay appearing as linear on semi-logplot, as expected. In contrast, the electrostatic potential profiles differ near the surface (for x/a <
2) at eV /k B T = 10, which are responsible for differences in free energies predictedusing the MPB approach and the numerical calculations (cf. Figure 3(a) in the main text).The differences in the electrostatic potential near the surface show up in plots for surfacecharge density ( σ = σ ) as a function of applied surface potential (Figure 3(b) in the maintext). The structural changes resulting from an increase in the surface potentials are shown inFigure 5. In particular, an increase in surface potentials leads to an increase in the volumefraction of counterions (anions in this case) near the surface at the expense of excludingcoions and solvent molecules. However, further increase in the surface potential (e.g., seeplot for eV /k B T = 9 in Figure 5(c)) leads to increase in solvent volume fraction near thesurface at the expense of exclusion of counterions and coions. This is expected from theexpression for solvent volume fraction, Eq. 32, leading to higher volume fraction of solvent2 ε ( x ) eV /k B T = 1.0 eV /k B T = 5.0eV /k B T = 7.0eV /k B T = 9.0 (a) ψ ★ ( x ) eV /k B T = 1.0 eV /k B T = 5.0eV /k B T = 7.0eV /k B T = 9.0 (b) ρ ( x ) a eV /k B T = 1.0 eV /k B T = 5.0eV /k B T = 7.0eV /k B T = 9.0 SolventAnions (c) ρ + ( x ) a eV /k B T = 1.0 eV /k B T = 5.0eV /k B T = 7.0eV /k B T = 9.0 (d) FIG. 5: (a) The dielectric function, (b) electrostatic potential, (c) solvent and counterion (anion)densities, and (d) co-ion (cation) densities at different surface potentials are shown for bulk saltconcentration of 0 . T = 303 K. in regions having strong electric fields. Also, such an enrichment of solvent in regions ofstrong electric fields is in agreement with previous theoretical works[10, 11]. Furthermore,the electric field dependent sorption of water on the AFM tips has been used to modulatefriction at the nanoscale[12].3 APPENDIX A : Field theory for double layer systems in the grand canonicalensemble
For study of a double layer, grand canonical partition function can be constructed and isgiven by Γ = P j = ± ,s e µ j n j /k B T Z { n j } so thatΓ = 1 N ψ Z D [ ψ ] Z D [ η ] exp (cid:20) − H g { ψ, η } k B T (cid:21) (A-1)so that H g { ψ, η } k B T = − πl Bo Z d r ψ ( r ) ∇ r ψ ( r ) − i Z d r η ( r ) + σ Z d r k iψ ( r k , x )+ σ Z d r k iψ ( r k , x ) − X j = ± ,s e µ j /k B T Q j { ψ, η } (A-2)where we have used Eqs. 8 and 9 for the partition function in the canonical ensemble.The saddle point approximation with respect to η and ψ gives two non-linear equations.At the saddle-points, both η and ψ turn out to be purely imaginary. Writing iη ( r ) = η ⋆ ( r )and iψ ( r ) = ψ ⋆ ( r ) at the saddle point, the two equations are given by X j = ± v j ρ j ( r ) + v s ρ s ( r ) = 1 (A-3) ∇ r · [ ǫ ( r ) ∇ r ψ ⋆ ( r )] = − πl Bo ρ e ( r ) (A-4)where we have defined ρ j = ± ( r ) = exp (cid:20) µ j k B T − Z j ψ ⋆ ( r ) − v j η ⋆ ( r ) (cid:21) (A-5) ρ s ( r ) = 4 π exp (cid:20) µ s k B T − v s η ⋆ ( r ) (cid:21) sinh p s |∇ r ψ ⋆ ( r ) | p s |∇ r ψ ⋆ ( r ) | (A-6)so that ρ e ( r ) = P j = ± Z j ρ j ( r ) + σ δ ( x − x ) + σ δ ( x − x ) and the local dielectric functionis given by ǫ ( r ) = 1 + 4 πl Bo p s ρ s ( r ) L [ p s |∇ r ψ ⋆ ( r ) | ] p s |∇ r ψ ⋆ ( r ) | (A-7)where L ( x ) = coth( x ) − /x is the Langevin function. Corresponding approximation for theGibbs free energy is given by H ⋆g k B T = σ Z d r k ψ ⋆ ( r k , x ) + σ Z d r k ψ ⋆ ( r k , x ) + 18 πl Bo Z d r ψ ⋆ ( r ) ∇ r ψ ⋆ ( r ) − Z d r η ⋆ ( r ) − X j = ± ,s Z d r ρ j ( r ) (A-8)sing Eqs. A-3, A-4, A-5 and A-6, it can be shown that H ⋆g and F ⋆ given by Eq. 19 arerelated by F ⋆ k B T = H ⋆g k B T + X j = ± ,s µ j k B T Z d r ρ j ( r ) (A-9)in accordance with the thermodynamic relation that the Helmholtz free energy is the Gibbsfree energy plus chemical potential times the number of particles. REFERENCES ∗ Electronic address: [email protected][1] J.N. Israelachvili,
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