Ann. Univ. Sofia, Fac. Chem. Pharm. 16 (14) 59-64 [arxiv 141.118] A novel protocol for linearization of the Poion-Boltzmann equation Roumen Tekov Department of Phyical Chemitry, Univerity of Sofia, 1164 Sofia, Bulgaria A new protocol for linearization of the Poion-Boltzmann equation i propoed and the reultant electrotatic equation coincide formally with the Debye-Hückel equation, the olution of which i well known for many electrotatic problem. The protocol i examined on the example of electrotatically tabilized nano-bubble and it i hown that table nano-bubble could be preent in aqueou olution of anionic urfactant near the critical temperature, if the urface potential i contant. At contant urface charge non nano-bubble could exit. The theory of the electric double layer [1] date back to the claical work of Helmholtz, Gouy, Chapman, Stern, Debye and Hückel. Due to the theoretical complication in dene ytem [] the application i mainly retricted to dilute ionic olution, where the electrotatic potential i decribed via the Poion-Boltzmann equation. The interaction between electric double layer ha been alo tudied intenively a an important component of the particle and colloidal force [3]. Significant attention ha been paid recently to highly-charged Coulomb mixture, where many pecific phenomena take place [4] among them mono-pecie electric double layer [5]. General theorie of charged fluid are developed [6-9], which account for ion correlation going beyond the Poion-Boltzmann theory. An intereting effect here i the nonelectrotatic interaction between the ion in an electric double layer expected to become important in concentrated olution []. The aim of the preent paper i to develop a new protocol for linearization of the Poion-Boltzmann equation, which could be ueful for olving eaily electrotatic problem in colloidal ytem. A an example, the protocol i applied for decription of electrotatically tabilized nano-bubble [1]. In electrotatic of dipere ytem one need regularly to olve the Poion equation [1] (1) governing the electrotatic potential. The charge denity in the olution ( ) i modelled often via the Boltzmann ditribution. Due to the mathematical complexity, the nonlinear Poion-Boltzmann equation i olvable numerically only. However, Eq. (1) i analytically olved in practice for many colloidal ytem in the frame of the Debye-Hückel linear approximation
D () where D ( ) / i the reciprocal Debye length. Thi equation i derived via a tandard linearization of Eq. (1) for low potential. In the preent paper a new protocol for linearization i propoed, which eem to be more general and ueful for arbitrary potential. Introducing a contant electrotatic potential, being typical for the conidered dipere ytem, one can approximate Eq. (1) a follow (3) where ( ) / i a new creening parameter. In fact, Eq. (3) could be applied to an arbitrary charge denity ditribution and the Boltzmann ditribution i imply an example here. Note that ( ) D and thu the Debye-Hückel approximation i a particular cae of Eq. (3). The advantage of thi new protocol a compared to the tandard linearization cheme ( ) ( ) ( ) ( ) i that Eq. (3) coincide formally with Eq. (), which i olved elewhere. Thu, the olution of Eq. (3) i well known and, for intance, it read to exp[ ( R r)]( R / r) exp( x) coh( z) / coh( h/ ) (4) near a pherical particle with radiu R, a flat urface and in a ymmetric film with thickne h, repectively. Thee olution certainly differ from the exact olution of the nonlinear Eq. (1) but varying the parameter one can alway find a good approximation. For example, if one i looking primarily for the correct urface potential it i reaonable to accept, while in the cae of interet on the overall dependence of the electric potential the choice / eem more plauible. Finally, a mean-field approximation i alo poible, which conit in replacement of by in the final olution (4). Thu, for example, the mean-field potential on a flat urface of 1:1 electrolyte will be the olution of the following trancendental equation exp[ ( k T / e)inh( e / k T) x] B B D D B B For thi particular cae the exact olution of the nonlinear Poion-Boltzmann equation i known, x arctanh[exp( e / k T)] arctanh[exp( e / k T)]. It i plotted in Fig. 1 together with the mean-field approximation in the form x ln( / ) / ( k T / e)inh( e / k T) D B B and the Debye-Hückel approximate olution x ln( / ). A i een, the mean-field olu- D
tion almot follow the exact olution near the urface, while at long ditance it tend aymptotically to the Debye-Hückel approximation. Thu, the juxtapoition i very good at larger potential, which are important for the pecific electric propertie on the boundary interface. Hence, the preent new protocol for linearization of the Poion-Boltzmann equation eem to be quite good and capture the eential phyic of the charged ytem. Fig. 1 The dependence of the dimenionle ditance x on the dimenionle potential / D for 1:1 electrolyte at a high urface potential 5 k T / e : the exact olution (olid line), the mean-field approximation (dahed line) and the Debye-Hückel approximation (dotted line) B In order to get advantage of the new protocol for linearization of the Poion-Boltzmann equation, let u conider a nano-bubble with a radiu R immered in aqueou electrolyte olution. It i aumed that ion can adorb at the bubble interface. Since there are no ion in the ga phae, the electrotatic potential inide the nano-bubble i contant. In the liquid atifie the Poion-Boltzmann equation (1), which, written in pherical coordinate, read (5) r( r r ) / r ( ) A wa mentioned, Eq. (5) can be analytically olved for low potential but in the preent cae we
are looking for trong electrotatic effect due to huge urface potential. Thu, the Debye- Hückel approximation i not applicable. Fortunately, in thi cae the new protocol, valid for electric double layer being very condened near the urface, i reaonable. Hence, cloe to the urface the electrotatic potential i of the order of and the Poion-Boltzmann equation (5) can be linearized in the form ( r ) / r (6) r r where ( ) /. Thi equation i exact at the bubble urface. In the bulk Eq. (6) i not precie but the defect i not eential ince the potential inide the olution i very low a compared to the urface potential due to trong electrotatic creening. A wa mentioned, the advantage of Eq. (6) i that it analytical olution i known exp[ ( R r)]( R / r) (7) The urface potential on the bubble/water interface i related to the urface charge denity q via the relation [1] q ( ) ( 1/ R) (8) r rr where the lat expreion i obtained by employing Eq. (7). The preence of the electrotatic potential in the liquid generate additional preure on the bubble via the Maxwell tre tenor [1]. Hence, the normal and tangential component of the preure tenor in the liquid read P p / ( ) / N L r P p / ( ) / (9) T L r repectively, where p L i the contant molecular preure far away from the bubble. The econd term here repreent the omotic preure due to the difference in the ionic concentration. One can eaily check that thee preure component atify the equilibrium condition. Integrating the latter yield the exce electrotatic force acting on the r ( r PN ) rpt nano-bubble N, L L T R R ( P p ) ( p P ) rdr ( R 1/ ) (1)
where P, P ( R) i the preure on the bubble urface. A i een, thi exce force i N N negative, thu leading to tabilization of the nano-bubble. On the other hand, the normal preure balance on the bubble urface read p P / R (11) G N, where p G i the ga preure inide the bubble and i the contant urface tenion on a flat liquid/ga interface at zero urface charge. Note that the electrotatic effect on the urface tenion are already accounted for in Eq. (1). Due to the urrounding atmophere, the aqueou olution i in equilibrium with a bulk ga phae a well. Therefore, pg pl and Eq. (11) accomplihed by Eq. (1) lead to an expreion for the equilibrium radiu of the bubble R / 4( / ) (1) In the colloidal cience there are two well-known interfacial model [11]: the contant urface charge denity and the contant urface potential. In the firt cae q q and, hence, according to Eq. (8) the urface potential equal to q / ( 1/ R). Since we are looking for nano-bubble with very mall radii it follow from thi relation that the urface potential i alo very low. Hence, D and Eq. (1) can be rewritten in the form (1 1/ DR) / (1 1/ DR) q / D (13) Thi equation ha a poitive olution for R only if tenion on a flat liquid/ga urface equal to q D. However, ince the urface q /, the inequality above i never atified. Therefore, the concluion i that no table nano-bubble i poible at contant urface charge denity. On the contrary, at contant urface potential Eq. (1) read D R / 4( / ) / 4 (14) Hence, the nano-bubble radiu i larger a higher the urface potential and lower the urface tenion are. A tandard way to try to achieve uch condition i addition ionic urfactant. However, traditional ionic urfactant cannot lower the urface tenion enough to tabilize the nano-bubble. If we accept the zeta-potential of pure water 65 mv [1] a the value of
urface potential, a nano-bubble with radiu R 1 nm would correpond to 1 mn/m according to Eq. (14). Obviouly, to reach thi very low value of urface tenion one ha to play with temperature. Therefore, table nano-bubble with radii of everal nanometer are expected to be preent in water near the critical temperature, epecially at preence of anionic urfactant. [1] H. Ohhima, Theory of Colloid and Interfacial Electric Phenomena. London, Academic, 6 [] R. Tekov, Ann. Univ. Sofia, Fac. Chem., 11, 1/13, 177 [3] J.-P. Hanen, H. Löwen, Annu. Rev. Phy. Chem.,, 51, 9 [4] Y. Levin, Rep. Prog. Phy.,, 65, 1577 [5] R. Tekov, O.I. Vinogradova, J. Chem. Phy., 7, 16, 9491 [6] T. Narayanan, K.S. Pitzer, Phy. Rev. Lett., 1994, 73, 3 [7] H. Weingärtner, W. Schröer, Adv. Chem. Phy., 1, 116, 1 [8] R. Ramirez, R. Kjellander, J. Chem. Phy., 3, 119, 1138 [9] Y. Levin, Braz. J. Phy., 4, 34, 1158 [1] O.I. Vinogradova, N.F. Bunkin, N.V. Churaev, O.A. Kieleva, A.V. Lobeyev, B.W. Ninham, J. Colloid Interface Sci., 1995, 173, 443 [11] D. McCormack, S.L. Carnie, D.Y.C. Chan, J. Colloid Interface Sci., 1995, 169, 177 [1] A. Graciaa, G. Morel, P. Saulner, J. Lachaie, R.S. Schechter, J. Colloid Interface Sci., 1995, 17, 131