STEADY STATE AND PSEUDO-TRANSIENT ELECTRIC POTENTIAL USING THE POISSON- BOLTZMANN EQUATION

Transcription

1 Brazlan Journal of Chemcal Engneerng ISSN - Prnted n Brazl Vol., No., pp. 9 -, January - March, dx.do.org/.9/-.s STEADY STATE AND PSEUDO-TRANSIENT ELECTRIC POTENTIAL USING THE POISSON- BOLTZMANN EQUATION L. C. dos Santos,, F. W. Tavares,*, A. R. Secch, E. C. Bscaa Jr. and V. R. R. Ahón Petrobras, Centro de Pesqusas e Desenvolvmento (CENPES), Av. Horáco Macedo 9, CEP: 9-9, Ro de Janero - RJ, Brazl. E-mal: letcacota@hotmal.com COPPE, Programa de Engenhara Químca, Unversdade Federal do Ro de Janero, Av. Horáco Macedo, CEP: 9-97, Ro de Janero - RJ, Brazl. Phone: + () () -7 * E-mal: tavares@eq.ufrj.br ; arge@peq.coppe.ufrj.br; evarsto@peq.coppe.ufrj.br Escola de Químca, Unversdade Federal do Ro de Janero, Av. Horáco Macedo, CEP: 9-97, Ro de Janero - RJ, Brazl. Departamento de Engenhara Químca e de Petróleo, Unversdade Federal Flumnense, Rua Passo da Pátra,, CEP: -, Nteró - RJ, Brazl. E-mal: ruz@vm.uff.br (Submtted: January, ; Revsed: Aprl 9, ; Accepted: Aprl 9, ) Abstract - A method for analyss of the electrc potental profle n salne solutons was developed for systems wth one or two nfnte flat plates. A modfed Posson-Boltzmann equaton, takng nto account nonelectrostatc nteractons between ons and surfaces, was used. To solve the stated problem n the steady-state approach the fnte-dfference method was used. For the formulated pseudo-transent problem, we solved the set of ordnary dfferental equatons generated from the algebrac equatons of the statonary case. A case study was also carred out n relaton to temperature, soluton concentraton, surface charge and salt-type. The results were valdated by the statonary problem soluton, whch had also been used to verfy the onc specfcty for dfferent salts. The pseudo-transent approach allowed a better understandng of the dynamc behavor of the on-concentraton profle and other propertes due to the surface charge varaton. Keywords: ; Posson-Boltzmann; Fnte Dfference. INTRODUCTION At the nterface of the dsperse and the dspersant phases of a collodal system there are characterstc surface phenomena, lke adsorpton effects and an electrc double layer that are very mportant to determne the physcochemcal propertes of the whole system (Lma et al., 8). In the classcal approach, the Posson-Boltzmann (PB) equaton does not take nto account the non-electrostatc nteractons present between ons and surfaces. However, the modfed PB equaton used n ths study enables the onc specfcty to be descrbed, as verfed n several collodal systems. Even though the classcal form of the Posson- Boltzmann equaton presents lmtatons, an nnumerable number of applcaton are found n the lterature. A good ntroducton and dervaton can be found n Israelachvl (99). Concernng applcaton of the PB equaton, excellent revews are reported by Davs and McCammon (99) and by Hong and Ncholls (99). In partcular, Shestakov et al. () solved the nonlnear Posson-Boltzmann equaton usng pseudo-transent contnuaton and the fnte element *To whom correspondence should be addressed

2 9 L. C. dos Santos, F. W. Tavares, A. R. Secch, E. C. Bscaa Jr. and V. R. R. Ahón method to show the behavor of ons close to electrodes wth changnged potental. Although a very good numercal technque was used, Shestakov et al. () reported results that are lmted for general electrolytes. The authors dd not nclude onc specfcty (Hofmester effect). Here, we nclude the dsperson nteracton between each on and the electrode wall (from Lfshtz theory as descrbed n Nnham and Yamnsky, 997) to take nto account the dfference between salt types. The Hamaker potentals are obtaned elsewhere (Tavares et al.,, Lva et al., 7). The PB equaton n -D form s a second order non-lnear ordnary dfferental equaton wth Drchlet and/or Neumann boundares condtons. An analytcal soluton for ths equaton s only avalable for partcular cases lke when the system s composed of a sngle plate and the classcal PB approach for symmetrcal electrolyte soluton s used (Lma et al., 7). In ths study, the PB equaton was solved for the classc and modfed forms, for one or two flat plate systems, usng the fnte-dfference method and a numerc approach, whch s detaled n the second and thrd secton of ths paper. A pseudo-transent form of ths equaton s descrbed n the fourth secton. Fnally, results and conclusons are presented n the last sectons. POISSON-BOLTZMANN EQUATION In a lqud medum wth electrc charges, the basc form of the Laplace equaton gves place to the Posson equaton, shown n Equaton (), whch relates the vector feld dvergence to the charge densty, (Equaton ()). w x x () ( x) e zc( x) () n whch c s the concentraton of on, e s the elementary charge, z s the valence of on, the vacuum permttvty, and w the water delectrc constant (Lma et al., 7). From the chemcal potental of each on n soluton, the Boltzmann dstrbuton (Equaton ()) of the ons can be obtaned. E( x) Eo c( x) c exp kbt () where c s the concentraton of on n the bulk soluton, E s the reference state potental energy for on, and E s the potental energy of on defned as the sum of the electrostatc potentals plus the dsperson nteractons ( U ) between the on and the surface (non-electrostatc potentals). Consderng that all potentals between ons and macro partcles n an aqueous soluton go asymptotcally to zero n the bulk phase ( x E, ), Equaton () becomes: ze ( x) U( x) c( x) coexp kbt () Substtutng Equaton () n (), gves (Equaton ()): ze ( x) U( x) xe zc oexp () kbt Substtutng Equaton () nto Equaton (), gves the second-order non-lnear modfed PB equaton: ze ( x) U( x) w xezc oexp () kt B NUMERICAL SOLUTION OF THE POISSON- BOLTZMANN EQUATION Ths secton presents a pseudo-transent approach to calculate the electrcal potental profle usng a modfed Posson-Boltzmann equaton n dfferent condtons. The profle trends presented n the Results secton, based on the data calculated from ths approach, n some cases are all well known; however, these trends are confrmed and presented n a dfferent way usng D fgures. Two knds of geometry have been studed. The problem domans for each case are: ron x for one flat plate ron x L r on for two parallel plates n whch r on s the onc radus (here all ons have the same sze), x s the ndependent varable and L s the dstance between the two flat plates. The Boundary Condtons The -D form of the non-lnear PB equaton requres two boundary condtons. In ths study two knds of system are dscussed: ) Systems wth a charged surface, such as protens not at ther soelec- Brazlan Journal of Chemcal Engneerng

3 Steady State and Pseudo-Transent Usng the Posson-Boltzmann Equaton 9 trc pont and, ) Systems wth a non-charged surface, such as the ar/water nterface. For an nfnte flat plate, the frst boundary condton (BC), vald for both systems (Equaton (7)), admts that the electrc feld goes asymptotcally to zero n the bulk phase ( x ). Applyng Gauss law for charged surfaces, the second BC (Equaton (8)) comes from the electrc feld generated by the surface charge densty, (Morera et al., 7): x lm (7) x x d dx xron w (8) Equaton (8) s also used for non-charged surfaces, at whch. In the case of two parallel nfnte flat plates, the frst BC (Equaton (9)) for both systems admts that the electrc feld profle has a symmetry condton n the md-pont of the doman ( x L /). x d dx xl/ (9) The second BC s also represented by Equaton (8). One Infnte Flat Plate Dmensonless Form For both geometres studed, the correspondng model equatons were rewrtten n dmensonless form n order to avod scalng problems durng the numercal resoluton. Defnng the new ndependent varable y: y exp( ) yexp( kr on ) () k The Debye-Length ( k ) s defned by: e z c o wkt B () In whch k B s the Boltzmann constant and T s the temperature. The new dependent varable (dmensonless electrc potental) s defned as: y e y () kt B and the onc strength of the soluton, s gven by: I z co () The dsperson nteracton between each on and the flat surface, n the Hamaker approach, s gven by (Israellachvl, 99): / B B B U x H Hk k T kt ktx ln y () n whch H s the dsperson coeffcent, estmated here by the Lfshtz theory (for van der Waals nteractons) (Israellachvl, 99). Its dmensonless * form, H s defned by: H * B H k () k T For ths geometry, the modfed dmensonless form of the PB equaton s gven by: d( y) d ( y) y y dy dy * H zc exp z y I ln( y) () And the two dmensonless BC are represented by Equatons (7) and (8): y (7) y d y k y dy e I yexp( kr on ) Two Parallel Flat Plates Dmensonless Form (8) In the case of two parallel flat plates, a smlar procedure was performed; however, the ndependent varable was defned as: y kr y k L r (9) on For ths geometry, the modfed dmensonless form of the PB equaton s gven by: on Brazlan Journal of Chemcal Engneerng Vol., No., pp. 9 -, January - March,

4 9 L. C. dos Santos, F. W. Tavares, A. R. Secch, E. C. Bscaa Jr. and V. R. R. Ahón d ( y) H * zc exp z y () dy I y And the two dmensonless BC are represented by Equatons () and (): y d dy yk L d y k dy e I ykr on The Fnte-Dfference Method () () As already mentoned, for the examples studed here, there are no analytcal solutons. Therefore, the fnte-dfference method wth second-order approxmatons and n equally-spaced dscretzaton ntervals was used to solve the problem. The equatons used to calculate the dervatve at the doman endponts were generated from lnear nterpolaton n y y y and yn y yn, respectvely. a) One Infnte Flat Plate Steady State Condton Applyng the fnte-dfference method n Equatons ()-(8) ( j,,..., n ) gves: () j * exp H zc z j ln y j * exp H zc z j ln y I j j j j j j j () k n n n () Ine b) One Infnte Flat Plate Pseudo-Transent Condton To verfy the effects on the electrc potental profles, caused by changes n surface charge, a pseudotransent model of the modfed PB was proposed. The pseudo-transent form of the modfed PB equaton s an extenson form of Shestakov et al. (). Furthermore, verfyng these changes, t was possble to corroborate the results found wth the steady-state problem soluton. The pseudo-transent problem was formulated by defnng a dmensonless potental varaton wth respect to a dmensonless tme, t (Shestakov et al., ). The system of n+ algebrac equatons, generated n the prevous tem, s turned nto a system wth two algebrac equatons defned by the boundary condtons (Equatons () and ()), and n- dfferental Equatons () for the nternal ponts ( j,,..., n ). d j j dt * exp H zc z j ln y j * exp H zc z j ln y I j j j j j j j c) Two Parallel Plates Steady-State Condton () Applyng the fnte-dfference method n Equatons ()-() ( j,,..., n ) gves: ky (7) ei j j j y * H zc exp z j y j zc * H exp zj j y j (8) n n n (9) d) Two Parallel Plates Pseudo-Transent Condton In a smlar way, usng () t, the steady-state system of n+ algebrac equatons s redefned by Equatons (7) and (9), and n- dfferental Equatons () for the nternal ponts ( j,,..., n ). Brazlan Journal of Chemcal Engneerng

5 Steady State and Pseudo-Transent Usng the Posson-Boltzmann Equaton 97 d dt j j j j y * H zc exp z j y j * H zc expzj j y j I () the electrc potental value n the lmt of x converged asymptotcally to zero. To valdate the mplemented algorthm, we compared our results wth those presented n the recent lterature for NaCl, consderng the dsperson nteracton (Morera et al., 7). The base case was generated for NaCl solutons ( M at 98.K). One Infnte Flat Plate Steady State Condton To solve tem (b) and tem (d), the ntal condtons ( j ) are obtaned from the soluton of the statonary problem for dscharged surfaces. To solve the proposed problem, a computatonal code was wrtten n MATLAB, usng nternal solvers lke fsolve and ode. RESULTS AND DISCUSSION To establsh the mesh sze, an analyss of the convergence of the electrc potental on the surface as a functon of n was performed. It was verfed that, for above ntervals, the dfference between the surface electrc potentals was less than and These results were generated for NaCl solutons ( molar at 98. K). Fgure shows the electrc potental profle generated by a dscharged surface, Fgure (a) and (b) shows the electrc potental profles for surfaces wth postve and negatve charges, respectvely. From these results, t s possble to say that the modfed PB equaton accounts for the nfluence of the non-electrostatc potental of each on, that s, the onc specfcty gven by the dynamc reorentaton of the electronc cloud due to a nearby surface. Ths becomes evdent n the electrc potental value observed on the surface, whch s not zero even when the surface s not charged. The latter result would not be obtaned from the soluton of the classcal PB equaton. (mv)... (mv) x(nm) Fgure : Electrc potental profle for a dscharged surface (Surface Charge Densty = C/m )... (mv) x(nm) x(nm) (a) (b) Fgure : (a) Electrc potental profle for a postvely charged surface (Surface Charge Densty =. C/m ). (b) Electrc potental profle for a negatvely charged surface (Surface Charge Densty =-. C/m ). Brazlan Journal of Chemcal Engneerng Vol., No., pp. 9 -, January - March,

6 98 L. C. dos Santos, F. W. Tavares, A. R. Secch, E. C. Bscaa Jr. and V. R. R. Ahón soluton temperature, soluton concentraton, surface charge and the salt type. These results were obtaned by evaluatng the steady-state model for dfferent parameter values. Regardng the soluton temperature, two opposte effects can be observed, a negatve correlaton (the potental decreases as the temperature ncreases) when there s charge on the surface (Fgure ), and a postve one (Fgure 7) when the surface s dscharged. We show calculatons for a very large range of temperature (from to K). At very hgh pressure (.e., 9 Pa), water s an ncompressble lqud n ths temperature range. Because the calculatons are carred out n the McMllan-Mayer framework, results are ndependent of pressure. Other mportant pont s about the delectrc constant. The delectrc constant decreases when temperature ncreases. However, we assume that the product o wk BT s ndependent of temperature. One Infnte Flat Plate Pseudo-Transent Condton Results obtaned from the dmensonless pseudotransent model for one nfnte flat plate are shown n Fgures and ( molar NaCl solutons at 98. K). Based on these results t s possble to verfy changes n the electrc potental profle caused by changes n the surface charge. It s mportant to emphasze that the electrc potental on the surface s not zero, even when the surface has no charge (see Fgure ). Ths s not true when usng the classcal PB equaton, as can be seen n Fgure. One Infnte Flat Plate Case Study Fgures - show the electrc potental profles obtaned by perturbatons n the model parameters: t t...8. Fgure : Dmensonless electrc potental profle for ncreasng surface charge ( ntal= C/m and fnal=. C/m). Fgure : Dmensonless electrc potental profle for decreasng surface charge ( ntal=. C/m and fnal= C/m) t.. Fgure : Dmensonless electrc potental profle for decreasng surface charge ( ntal=. C/m and fnal= C/m), consderng only the electrostatc potental (classcal PB equaton). Brazlan Journal of Chemcal Engneerng

7 Steady State and Pseudo-Transent Usng the Posson-Boltzmann Equaton Temperature (K) Fgure : Dmensonless electrc potental profle for temperatures between and K wth charged surface (Surface Charge Densty =. C/m ) Temperature (K) Fgure 7: Dmensonless electrc potental profle for temperatures between and K for a dscharged surface (Surface Charge Densty = C/m ) Concentraton (M) Fgure 8: Dmensonless electrc potental profle for concentratons between and molar wth a charged surface (Surface Charge Densty =. C/m ). Fgures - 9 were obtaned for NaCl. The base case was done at 98 K and M. The results n Fgures (a) and (b) were obtaned by evaluatng the steady-state model for dfferent salts (NaCl, KCl, BaCl and CaCl ) and concentraton M. It s noteworthy that the on specfcty shown (Hofmester effect) n Fgure (a) loses ts nfluence n cases harge Surface Densty Charge (C/m) Densty (C/m ) Fgure 9: Dmensonless electrc potental profle for surface charge denstes between: -. and. C/m. where the surface charge s hgh. For these cases, electrostatc effects outwegh the others and the valence of the ons n soluton becomes more relevant (Fgure (b)). In concluson, n Fgure (b), t s not possble do dstngush NaCl and KCl. In addton, results obtaned for BaCl and CaCl are pratcaly the same NaCl KCl BaCl CaCl... NaCl KCl BaCl CaCl (a) (b) Fgure : (a) Dmensonless electrc potental profles for dfferent salts (Surface Charge Densty = C/m ); (b) Dmensonless electrc potental profles for dfferent salts (Surface Charge Densty =. C/m ). Brazlan Journal of Chemcal Engneerng Vol., No., pp. 9 -, January - March,

8 L. C. dos Santos, F. W. Tavares, A. R. Secch, E. C. Bscaa Jr. and V. R. R. Ahón pont s not necessarly zero and t ncreases (n magntude) as the plates come closer to each other. Two Parallel Plates Steady-State Condton These results were generated for NaCl solutons ( molar at 98. K). Fgures (a) and (b) show the results generated for the geometry wth two charged parallel flat plates, where the nfluence of the dstance ( L ) between the two plates on the electrc potental profle can be seen. A crtcal pont was observed half way between the two plates; however, the potental value at ths Two Parallel Plates Pseudo-Transent Condton The results obtaned from the dmensonless pseudo-transent model for ths geometry are shown n Fgures and. Based on these results, t s possble to verfy changes n the electrc potental profle due to changes n the surface charge. 7 (mv) (mv) 7.. x(nm)... x(nm). (a) (b) Fgure : (a) Electrc potental profle wth L. nm (Surface Charge Densty =. C/m); (b) Electrc potental profle wth L. nm (Surface Charge Densty =. C/m).. Electrc Potental t.. Fgure : Effect of nverson of the surface charge densty ( ) on the dmensonless electrc potental profle wth L nm ( ntal=+. C/m and fnal=-. C/m) t Fgure : Effect of nverson of the charge on the electrc potental profle wth L nm ( ntal= -. C/m and fnal=. C/m). Brazlan Journal of Chemcal Engneerng

9 Steady State and Pseudo-Transent Usng the Posson-Boltzmann Equaton Two Parallel Flat Plates Case Study Fgures -7 show the profles obtaned by changes n: soluton temperature, soluton concentraton, surface charge and type of salt used. Fgures - were obtaned for NaCl. The base case was done at 98 K and M. The results n Fgure 7 were obtaned by evaluatng the steady-state model for dfferent salts (NaCl, KCl, BaCl and CaCl ) and concentraton M. In a smlar fashon, Fgures,, and 7 show the electrc potental profles obtaned by changes n soluton temperature, soluton concentraton, surface charge, and type of salt used. The results show that the electrc potental s not necessarly zero n the mddle of the doman, only the crtcal pont condton s establshed by the problem boundary condtons. As shown n Fgure 7, n cases where the surface charge s hgh, the nfluence of the electrostatc effects ncreases and the behavor of the physcal propertes s only modfed by the valence of the ons present n the soluton. Once more, n Fgure 7, t s not possble to dstngush NaCl and KCl. Also, the results obtaned for BaCl and CaCl are the same Temperature (K) Fgure : Dmensonless electrc potental profle wth temperatures between and K, wth charged surface, and L nm (Surface Charge Densty =. C/m ) Concentraton (M). Fgure : Dmensonless electrc potental profle for concentratons between. and. molar, wth charged surface and L nm (Surface Charge Densty =. C/m ) Surface Charge Densty (C/m ) Fgure : Dmensonless electrc potental profle for surface charge densty between -. and. C/m ( L nm ) NaCl KCl CaCl BaCl -. 8 Fgure 7: Dmensonless electrc potental profle for dfferent salts (Surface Charge Densty =. C/m ). Brazlan Journal of Chemcal Engneerng Vol., No., pp. 9 -, January - March,

10 L. C. dos Santos, F. W. Tavares, A. R. Secch, E. C. Bscaa Jr. and V. R. R. Ahón CONCLUSIONS A modfed Posson-Boltzmann equaton, takng nto account non-electrostatc nteractons between ons and surfaces was used to descrbe salt concentratons close to one or two nfnte flat plates. To descrbe pseudo-transent behavor, a set of ordnary dfferental equatons generated from algebrac equatons and wrtten n dmensonless varables was solved. Ths procedure permtted obtanng the dynamc behavor of the on-concentraton profle and other propertes due to the surface charge varaton. The proposed method to solve the pseudotransent Posson-Boltzmann equaton that accounted for salt type and dvalent counterons can be used to descrbe electrochemcal devces, such as electrodes wth dfferent surface-charge frequency. Senstvty analyss was successfully carred out to verfy the potental and on concentratons close to the electrode n response to temperature, soluton concentraton, salt type, and surface charge. NOTATION c on concentraton c concentraton of on n the reference state (bulk phase) E potental energy E potental energy n the reference state o (bulk phase) ez charge of each on H dsperson potental (Hamaker constant) * H dmensonless dsperson parameter I onc force n the bulk phase Counter k Debye-Length k B Boltzmann constant L dstance between two flat plates r on radus of the on T temperature t dmensonless tme U dsperson potental x poston coordnate, ndependent varable y dmensonless ndependent varable n number of dscretzaton ntervals Z valence surface y nterval sze Greek Letters surface charge densty delectrc constant charge volumetrc densty electrc potental dmensonless electrc potental REFERENCES Davs, M. E. and McCammon, J. A., Electrostatcs n bomolecular structure and dynamcs. Chem. Rev., 9, 9 (99). Hong, B. and Ncholls, A., Classcal electrostatcs n bology and chemstry. Scence, 8, (99). Israellchvl, J., Intermolecular and Surface Forces. Second Edton. Academc Press, London (99). Lma, E. R. A., Tavares, F. W., Bscaa Jr., E. C., Fnte volume soluton of the modfed Posson Boltzmann equaton for two collodal partcles. Physcal Chemstry Chemcal Physcs, v. 9, pp. 7-8 (7). Lma, E. R. A., Hornek, D., Netz, R. R., Bscaa Jr., E. C., Tavares, F. W., Kunz, W., Boström, M., Specfc on adsorpton and surface forces n collod scence. Journal of Physcal Chemstry B, v., pp. 8-8 (8). Maslyah, J. H., Bhattacharjee, S., Electroknetc and Collod Transport Phenomena. John Wley & Sons, Inc., Hoboken, New Jersey (). Morera, L. A., Bostrom, M., Nnham, B. W., Bscaa Jr., E. C., Tavares, F. W., Effect of the on-proten dsperson nteractons on the proten-surface and proten-proten nteractons. J. Braz. Chem. Soc, v. 8, No., pp. - (7). Nnham, B. W. and Yamnsky, V., Ion bndng and on specfcty The Hofmester effect, Onsager and Lfschtz theores. Langmur,, 97-8 (997). Tavares, F. W., Bratko, D., Blanch, H. and Prausntz, J. M., Ion-specfc effects n the collod-collod or proten-proten potental of mean force: Role of salt-macroon van der Waals nteractons. J. Phys. Chem., B 8, 98-9 (). Shestakov, A. I., Mlovch, J. L., Noy, A., Soluton of the nonlnear Posson-Boltzmann equaton usng pseudo-transent contnuaton and the fnte element. J. Collod Interface Sc., 7(), -79 (). Brazlan Journal of Chemcal Engneerng