- Email: [email protected]
Low temperature anomalies in the properties of the electrochemical interface
30 July 1999
Chemical Physics Letters 308 Ž1999. 473–478 www.elsevier.nlrlocatercplett
Low temperature anomalies in the properties of the electrochemical interface Dezso˝ Boda
a,1
, Douglas Henderson
a,)
, Kwong-Yu Chan b, Darsh T. Wasan
c
a
Department of Chemistry and Biochemistry, Brigham Young UniÕersity, ProÕo, UT 84602-5700, USA b Department of Chemistry, The UniÕersity of Hong Kong, Pokfulam Road, Hong Kong, China c Department of Chemical Engineering, Illinois Institute of Technology, Chicago, IL 60616-3793, USA Received 23 March 1999; in final form 27 May 1999
Abstract Some features of the adsorption isotherms and electrochemical capacitance of electrolytes in solvents at low temperatures are studied by means of computer simulations. The so-called restricted primitive model where the ions are represented as charged hard spheres of equal diameter and the solvent is represented by a uniform dielectric constant is used. It is found that at low temperatures the capacitance of double layers in dissolved electrolytes decreases with decreasing temperatures. This is similar to the behaviour of the capacitance of molten salt double layers but opposite to the behaviour of double layers in dissolved electrolytes at room temperature. Further, we find a maximum in the adsorption isotherm near the critical point of the electrolyte. This quantity may well be singular at the critical point. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Recently, we have used computer simulations to study electrochemical interfaces or double layers ŽDL.. Although our plan is to use realistic models for water, we have studied such interfaces using a dipolar hard sphere model for the solvent w1x. The advantages of using this, admittedly unrealistic, model for the solvent is that there have been some theoretical studies, the accuracy of which can be examined by simulations, that used this model for the solvent w2–5x.
) Corresponding author. Fax: q801 3785474; e-mail: [email protected] 1 Permanent address: Department of Physical Chemistry, University of Veszprem, ´ P.O. Box 158, H-8201 Veszprem, ´ Hungary.
Also, we have used the restricted primitive model ŽRPM. for the electrolyte where the ions are represented by charged hard spheres of equal size d and the solvent is represented by a uniform dielectric constant e . Torrie et al. w6–8x have studied the electrochemical interface with this model by simulation. Our studies complement the Torrie et al. work; we use higher densities w9x and, in this study, lower temperatures. In our previous simulations w9x, our purpose was to study molten salt DLs with the goal of determining whether the RPM could account for the fact that the capacitance of molten salt DLs increases with temperature. This contrasts with the opposite behaviour for DLs in dissolved electrolytes and the opposite prediction of all common theories of the electrochemical interface, such as the Gouy–Chapman theory – with ŽGCS. or without ŽGC. a Stern
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 6 4 3 - 0
474
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
layer – and the mean spherical approximation ŽMSA.. Through our simulations, we did find that the slope of the capacitance Ž C . versus temperature ŽT . was positive. However, equally interestingly, we found that the slope of the C versus T curve becomes negative at high temperatures and that at these higher temperatures the GCS theory and, especially, the MSA are in reasonable agreement with the simulation results. This suggests that the behaviour of molten salt DLs is a low-temperature effect. A first guess would have suggested that the behaviour of molten salt DLs was due to their high density or seemingly high temperatures. It may seem surprising to suggest that molten salts are low-temperature systems since they exist at thousands of degrees. However, if one reflects upon the fact that the effective Ža theorist would say reduced. temperature is T ) s e d 2 kTre 2 , where e is the electronic charge and k is the Boltzmann constant, it is the aqueous electrolyte that is the high reduced temperature system, since e s 78.5 for water while the reduced temperature of molten salts is low because e , 1 for molten salts. We speculated in Ref. w9x that DLs in dissolved electrolytes might exhibit this same changeover from negative to positive slope in a C versus T plot. One purpose of this study is to investigate this speculation. Henderson et al. w10x have suggested that the charge profiles and the adsorption isotherms of the ions are singular at the critical point of an electrolyte and that these quantities are large in magnitude in the vicinity of the critical point. The arguments of Henderson et al. are qualitative and involving grafting the compressibility onto the GC theory. However, the basic reasoning, involving the Ornstein– Zernike equation, seems plausible. The phenomena that Henderson et al. propose is analogous to the wetting of a surface by an adsorbed vapor. Hence, the other purpose of this study is to investigate the nature of the density and charge profiles in the vicinity of the critical point of an electrolyte.
2. Simulations Canonical ŽNVT. Monte Carlo simulations have been performed for the RPM in a rectangular simula-
tion cell of dimensions L = L = H at temperature T. Hard walls are placed at z s 0 and z s H that carry the same reduced surface charge s ) s y 12 Ž Nqy Ny .rŽ Lrd . 2 where Nq and Ny are the number of the cations and anions, respectively. A sufficiently large value of H is used so that a bulk electrolyte is formed in the center with equal averaged densities of cations and anions. Thus, the DL at each wall is independent of the DL at the other wall; the charge at each wall can be different, one can be uncharged or even oppositely charged. Hence, the choice of oppositely charged surfaces is quite feasible and should give equivalent results for the interfaces, provided a neutral bulk solution is formed in the center by using a sufficiently large value of H. Similarly charged surfaces represent the case of an electrolyte between two colloidal particles Žthrough the Derjaguin approximation.. There are contradictory data in the literature about the critical point of the RPM w12,13,11,14,15x. The critical temperature is about Tc) s d 2 kTcre 2 ; 0.05, while the critical density has been reported in a quite wide range: 0.01 - rc) s rc d 3 - 0.07. Therefore, simulations were performed in a temperature range 0.06 ( T ) ( 0.15 and density range 0.005 ( r ) ( 0.08 to study the temperature and density dependence of the capacitance and the adsorption. The reduced integral capacitance was calculated from C ) s s )rf ) where f ) s f dre is the potential drop across the DL and was obtained from the integration of the density profiles in the usual way. The Žreduced. adsorption was obtained from
Gs
H Ýr
) i
Ž x. yr ) d x ,
Ž 1.
i
where r i) Ž x . is the density profile of the cations or the anions and r ) is the total density in the middle of the cell, i.e. the bulk density. The number of particles was chosen to give a prescribed bulk density in the middle of the cell. This could not be done accurately in every case, so the density in the middle corresponds only approximately to the prescribed bulk density. However, the errors of C ) and G arising from the uncertainty of the bulk density are usually less than the order of magnitude of the size of the symbols in the figures,
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
and do not influence the basic behaviour of the curves. The length of the cell was H s 20 d, while the width and the number of particles varied in the ranges 14 d ( L ( 30 d and 130 - N - 560, respectively, depending on the density. For each state, 4 million configuration per particle were generated.
3. Results Fig. 1 shows the temperature dependence of the capacitance at four different densities that have been obtained from the simulations and from the GC, MSA, and GCS theories. The full circles represent integral capacitance data that should be close to the differential capacitances because we used low sur-
475
face charges Ž s ) s 0.005 for cases a, b, and d, and s ) s 0.00765 for the case c .. It can be seen that at low temperatures the capacitance increases with the increasing temperature. At a higher temperature the plot has a maximum, and at even higher temperatures it decreases in accordance with the theoretical predictions. At high temperatures the agreement between the MC and the MSA data is very good. The question arises that whether the same behaviour is valid for differential capacitances. For density r ) ; 0.04, we performed more simulations, and studied the capacitance as a function of the surface charge s ) . As is seen in Fig. 2, the capacitance decreases linearly as the electrode charge decreases at a given temperature. Moreover, the lower the temperature, the larger is the slope of the C ) Ž s ) .
Fig. 1. The capacitance as a function of the temperature at various densities. Full circles are the simulation results for the integral capacitance, and open circles are for the differential capacitance. The integral capacitances were obtained by using surface charge s ) s 0.00765 for the case c, and s ) s 0.005 for the cases a, b, and d. The solid, dot–dashed, and dashed lines represent the MSA, GC, and GCS results, respectively.
476
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
function. This makes the positive temperature dependence of the differential capacitance even more definite. On the basis of the method seen on Fig. 2, we have made estimates for the differential capacitances in the case of r ) ; 0.04. The open circles in Fig. 1 represent these estimates and show that the low-temperature behaviour of the capacitance is even more distinct. Therefore, we can say with a good certainty that the integral capacitances show the same behaviour as would be shown by the differential capacitances if the surface charge is low enough. Thus, we have shown that the RPM capacitance produces the same temperature dependence at low densities as it does at high densities Žmolten salts.. This implies that aqueous electrolytes should exhibit the same behaviour at low temperatures as do molten salts. And, indeed, the study of Hamelin et al. w16x showed that the DL capacitance of HClO4 P 5.5H 2 O at a gold electrode decreases with decreasing temperature at temperatures between 200 and 300 K. We are unable to compare our results with those of Hamelin et al. w16x because the value of e in frozen electrolytes is not known. An explanation of this peculiar behaviour can be the following. At high temperatures, the thickness of the DL increases, presumably because of the increased thermal energy. This leads to an increased
Fig. 2. The integral capacitance as a function of the surface charge at various temperatures for density r ) ; 0.04. The extrapolation to s ) s 0 gives the value of the differential capacitance.
Fig. 3. The integral capacitance Ž s ) s 0.005. as a function of the bulk density at temperatues T ) s 0.06 and 0.07.
potential and decreased capacitance with increasing temperature. At low temperatures, another effect becomes dominant. At low electrode charge, the ion– ion interactions become strong compared to the ion–electrode interaction, and the counterions tend to pull away from the electrode causing a thicker DL and a decreased capacitance with decreasing tempereture. At higher electrode charge, the ion–electrode interaction becomes dominant and the capacitance reverts to the usual temperature dependence Ža negative slope.. This behavior is seen in Fig. 2 for s ) ) 0.025. As far as the density dependence of the capacitance is concerned, nothing surprising was found. In accordance with the predictions of the theories, the capacitance increases nearly linearly with the increasing density as seen in Fig. 3. The temperature dependence of the adsorption was investigated for the the approximate bulk density r ) ; 0.04. According to theoretical considerations by Henderson et al. w10x mentioned in Section 1, the adsorption isotherms should be large in magnitude in the vicinity of the critical point. The results seen in Fig. 4 support their arguments. More accurately, the decrease of the temperature strengthens the effects observed at higher temperature. At zero surface charge Ž s ) s 0. the adsorption is negative Ždrying., and a decrease of the temperature makes it
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
more negative. This agrees with our arguments given above and makes sense because attractive forces act between the particles only and do not act between the particles and the wall. Of course, at high surface charge Ž s ) s 0.03. the adsorption becomes positive because of the strong attraction to the wall. Decreasing the temperature, the adsorption increases steeply. However, at an intermediate surface charge Ž s ) s 0.0125., where the adsorption is nearly zero Žthe two effects of the drying and the DL formation cancel each other., the decrease of the temperature does not seem to influence the adsorption. As discussed in Henderson et al. w10x, the existence of large adsorption or desorption near the critical point is related to the infinite compressibility of the fluid at its critical point. Infinite compressiblity implies infinite density fluctuations and long-range correlations. Small changes in pressure leads to very large changes in the concentration of the ions. Long-range correlations will allow the bulk fluid to adjust better with the presence of a surface or inhomogeneity and also implies a thicker interfacial region. The surface charge density gives an additional parameter over the case of simple fluid and the inhomogeneity can be enhanced in either direction, i.e. either adsorption, or drying Žor desorption. depending on the surface charge. In Fig. 5, a maximum in the adsorption occurs at a certain density. According to Henderson et al. w10x,
477
Fig. 5. The adsorption isotherms as functions of the density for various temperatures for surface charge s ) s 0.03.
this density must be close to the critical density since, according to their argument, the anomalous adsorption is infinite at the critical point of the bulk electrolyte. The maxima of the two temperatures simulated are located at about the same density, slightly less than 0.02. This should be very close to the critical density and falls within the range reported in the literature w11–15x for the critical density of the primitive model electrolyte. We are not able to conclude the critical temperature of the bulk electrolyte from results of this study. From Fig. 4, we can say the critical temperature is below 0.05 and probably not far from this value. The simulations at lower temperatures are possible but may face ergodicity problems, as well as problems of metastable states and phase separations. In any case, this is consistent with current estimates of the critical temperature.
Acknowledgements
Fig. 4. The adsorption as a function of the temperature for various surface charges. The density is r ) ; 0.04.
This work was supported in part by the National Science Foundation ŽGrants CHE96-01971 and CHE98-13729. and by the donors of the Petroleum Research Fund, administered by the American Chemical Society ŽGrant No. ACS-PRF 31573-AC9.. The authors thank Professor Wolfgang Schmickler for drawing their attention to the work of Hamelin et al.
478
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
References w1x D. Boda, K.Y. Chan, D. Henderson, J. Chem. Phys. 109 Ž1998. 7362. w2x S.L. Carnie, D. Henderson, J. Chem. Phys. 73 Ž1980. 2949. w3x L. Blum, D. Henderson, J. Chem. Phys. 74 Ž1981. 1902. w4x J.P. Badiali, M.L. Rosinberg, F. Vericat, L. Blum, J. Electroanal. Chem. 158 Ž1983. 253. w5x W. Schmickler, D. Henderson, J. Chem. Phys. 80 Ž1984. 3381. w6x G.M. Torrie, J.P. Valleau, J. Chem. Phys. 73 Ž1980. 5807. w7x G.M. Torrie, J.P. Valleau, G.N. Patey, J. Chem. Phys. 76 Ž1982. 4615.
w8x G.M. Torrie, J.P. Valleau, J. Phys. Chem. 86 Ž1982. 3252. w9x D. Boda, K.Y. Chan, D. Henderson, J. Chem. Phys. 110 Ž1999. 5346. w10x D. Henderson, J. Barojas, L. Blum, J. Phys. Chem. 87 Ž1983. 4544. w11x J.P. Valleau, J. Chem. Phys. 95 Ž1991. 584. w12x A.Z. Panagiotopoulos, Fluid Phase Equilibria 76 Ž1992. 97. w13x G. Orkoulas, A.Z. Panagiotopoulos, J. Chem. Phys. 101 Ž1994. 1452. w14x J.M. Caillol, D. Levesque, J.J. Weis, J. Chem. Phys. 107 Ž1997. 1565. w15x J.P. Valleau, G. Torrie, J. Chem. Phys. 108 Ž1998. 5169. w16x A. Hamelin, S. Rottgermann, W. Schmickler, J. Electroanal. Chem. 230 Ž1987. 281.
Chemical Physics Letters 308 Ž1999. 473–478 www.elsevier.nlrlocatercplett
Low temperature anomalies in the properties of the electrochemical interface Dezso˝ Boda
a,1
, Douglas Henderson
a,)
, Kwong-Yu Chan b, Darsh T. Wasan
c
a
Department of Chemistry and Biochemistry, Brigham Young UniÕersity, ProÕo, UT 84602-5700, USA b Department of Chemistry, The UniÕersity of Hong Kong, Pokfulam Road, Hong Kong, China c Department of Chemical Engineering, Illinois Institute of Technology, Chicago, IL 60616-3793, USA Received 23 March 1999; in final form 27 May 1999
Abstract Some features of the adsorption isotherms and electrochemical capacitance of electrolytes in solvents at low temperatures are studied by means of computer simulations. The so-called restricted primitive model where the ions are represented as charged hard spheres of equal diameter and the solvent is represented by a uniform dielectric constant is used. It is found that at low temperatures the capacitance of double layers in dissolved electrolytes decreases with decreasing temperatures. This is similar to the behaviour of the capacitance of molten salt double layers but opposite to the behaviour of double layers in dissolved electrolytes at room temperature. Further, we find a maximum in the adsorption isotherm near the critical point of the electrolyte. This quantity may well be singular at the critical point. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Recently, we have used computer simulations to study electrochemical interfaces or double layers ŽDL.. Although our plan is to use realistic models for water, we have studied such interfaces using a dipolar hard sphere model for the solvent w1x. The advantages of using this, admittedly unrealistic, model for the solvent is that there have been some theoretical studies, the accuracy of which can be examined by simulations, that used this model for the solvent w2–5x.
) Corresponding author. Fax: q801 3785474; e-mail: [email protected] 1 Permanent address: Department of Physical Chemistry, University of Veszprem, ´ P.O. Box 158, H-8201 Veszprem, ´ Hungary.
Also, we have used the restricted primitive model ŽRPM. for the electrolyte where the ions are represented by charged hard spheres of equal size d and the solvent is represented by a uniform dielectric constant e . Torrie et al. w6–8x have studied the electrochemical interface with this model by simulation. Our studies complement the Torrie et al. work; we use higher densities w9x and, in this study, lower temperatures. In our previous simulations w9x, our purpose was to study molten salt DLs with the goal of determining whether the RPM could account for the fact that the capacitance of molten salt DLs increases with temperature. This contrasts with the opposite behaviour for DLs in dissolved electrolytes and the opposite prediction of all common theories of the electrochemical interface, such as the Gouy–Chapman theory – with ŽGCS. or without ŽGC. a Stern
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 6 4 3 - 0
474
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
layer – and the mean spherical approximation ŽMSA.. Through our simulations, we did find that the slope of the capacitance Ž C . versus temperature ŽT . was positive. However, equally interestingly, we found that the slope of the C versus T curve becomes negative at high temperatures and that at these higher temperatures the GCS theory and, especially, the MSA are in reasonable agreement with the simulation results. This suggests that the behaviour of molten salt DLs is a low-temperature effect. A first guess would have suggested that the behaviour of molten salt DLs was due to their high density or seemingly high temperatures. It may seem surprising to suggest that molten salts are low-temperature systems since they exist at thousands of degrees. However, if one reflects upon the fact that the effective Ža theorist would say reduced. temperature is T ) s e d 2 kTre 2 , where e is the electronic charge and k is the Boltzmann constant, it is the aqueous electrolyte that is the high reduced temperature system, since e s 78.5 for water while the reduced temperature of molten salts is low because e , 1 for molten salts. We speculated in Ref. w9x that DLs in dissolved electrolytes might exhibit this same changeover from negative to positive slope in a C versus T plot. One purpose of this study is to investigate this speculation. Henderson et al. w10x have suggested that the charge profiles and the adsorption isotherms of the ions are singular at the critical point of an electrolyte and that these quantities are large in magnitude in the vicinity of the critical point. The arguments of Henderson et al. are qualitative and involving grafting the compressibility onto the GC theory. However, the basic reasoning, involving the Ornstein– Zernike equation, seems plausible. The phenomena that Henderson et al. propose is analogous to the wetting of a surface by an adsorbed vapor. Hence, the other purpose of this study is to investigate the nature of the density and charge profiles in the vicinity of the critical point of an electrolyte.
2. Simulations Canonical ŽNVT. Monte Carlo simulations have been performed for the RPM in a rectangular simula-
tion cell of dimensions L = L = H at temperature T. Hard walls are placed at z s 0 and z s H that carry the same reduced surface charge s ) s y 12 Ž Nqy Ny .rŽ Lrd . 2 where Nq and Ny are the number of the cations and anions, respectively. A sufficiently large value of H is used so that a bulk electrolyte is formed in the center with equal averaged densities of cations and anions. Thus, the DL at each wall is independent of the DL at the other wall; the charge at each wall can be different, one can be uncharged or even oppositely charged. Hence, the choice of oppositely charged surfaces is quite feasible and should give equivalent results for the interfaces, provided a neutral bulk solution is formed in the center by using a sufficiently large value of H. Similarly charged surfaces represent the case of an electrolyte between two colloidal particles Žthrough the Derjaguin approximation.. There are contradictory data in the literature about the critical point of the RPM w12,13,11,14,15x. The critical temperature is about Tc) s d 2 kTcre 2 ; 0.05, while the critical density has been reported in a quite wide range: 0.01 - rc) s rc d 3 - 0.07. Therefore, simulations were performed in a temperature range 0.06 ( T ) ( 0.15 and density range 0.005 ( r ) ( 0.08 to study the temperature and density dependence of the capacitance and the adsorption. The reduced integral capacitance was calculated from C ) s s )rf ) where f ) s f dre is the potential drop across the DL and was obtained from the integration of the density profiles in the usual way. The Žreduced. adsorption was obtained from
Gs
H Ýr
) i
Ž x. yr ) d x ,
Ž 1.
i
where r i) Ž x . is the density profile of the cations or the anions and r ) is the total density in the middle of the cell, i.e. the bulk density. The number of particles was chosen to give a prescribed bulk density in the middle of the cell. This could not be done accurately in every case, so the density in the middle corresponds only approximately to the prescribed bulk density. However, the errors of C ) and G arising from the uncertainty of the bulk density are usually less than the order of magnitude of the size of the symbols in the figures,
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
and do not influence the basic behaviour of the curves. The length of the cell was H s 20 d, while the width and the number of particles varied in the ranges 14 d ( L ( 30 d and 130 - N - 560, respectively, depending on the density. For each state, 4 million configuration per particle were generated.
3. Results Fig. 1 shows the temperature dependence of the capacitance at four different densities that have been obtained from the simulations and from the GC, MSA, and GCS theories. The full circles represent integral capacitance data that should be close to the differential capacitances because we used low sur-
475
face charges Ž s ) s 0.005 for cases a, b, and d, and s ) s 0.00765 for the case c .. It can be seen that at low temperatures the capacitance increases with the increasing temperature. At a higher temperature the plot has a maximum, and at even higher temperatures it decreases in accordance with the theoretical predictions. At high temperatures the agreement between the MC and the MSA data is very good. The question arises that whether the same behaviour is valid for differential capacitances. For density r ) ; 0.04, we performed more simulations, and studied the capacitance as a function of the surface charge s ) . As is seen in Fig. 2, the capacitance decreases linearly as the electrode charge decreases at a given temperature. Moreover, the lower the temperature, the larger is the slope of the C ) Ž s ) .
Fig. 1. The capacitance as a function of the temperature at various densities. Full circles are the simulation results for the integral capacitance, and open circles are for the differential capacitance. The integral capacitances were obtained by using surface charge s ) s 0.00765 for the case c, and s ) s 0.005 for the cases a, b, and d. The solid, dot–dashed, and dashed lines represent the MSA, GC, and GCS results, respectively.
476
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
function. This makes the positive temperature dependence of the differential capacitance even more definite. On the basis of the method seen on Fig. 2, we have made estimates for the differential capacitances in the case of r ) ; 0.04. The open circles in Fig. 1 represent these estimates and show that the low-temperature behaviour of the capacitance is even more distinct. Therefore, we can say with a good certainty that the integral capacitances show the same behaviour as would be shown by the differential capacitances if the surface charge is low enough. Thus, we have shown that the RPM capacitance produces the same temperature dependence at low densities as it does at high densities Žmolten salts.. This implies that aqueous electrolytes should exhibit the same behaviour at low temperatures as do molten salts. And, indeed, the study of Hamelin et al. w16x showed that the DL capacitance of HClO4 P 5.5H 2 O at a gold electrode decreases with decreasing temperature at temperatures between 200 and 300 K. We are unable to compare our results with those of Hamelin et al. w16x because the value of e in frozen electrolytes is not known. An explanation of this peculiar behaviour can be the following. At high temperatures, the thickness of the DL increases, presumably because of the increased thermal energy. This leads to an increased
Fig. 2. The integral capacitance as a function of the surface charge at various temperatures for density r ) ; 0.04. The extrapolation to s ) s 0 gives the value of the differential capacitance.
Fig. 3. The integral capacitance Ž s ) s 0.005. as a function of the bulk density at temperatues T ) s 0.06 and 0.07.
potential and decreased capacitance with increasing temperature. At low temperatures, another effect becomes dominant. At low electrode charge, the ion– ion interactions become strong compared to the ion–electrode interaction, and the counterions tend to pull away from the electrode causing a thicker DL and a decreased capacitance with decreasing tempereture. At higher electrode charge, the ion–electrode interaction becomes dominant and the capacitance reverts to the usual temperature dependence Ža negative slope.. This behavior is seen in Fig. 2 for s ) ) 0.025. As far as the density dependence of the capacitance is concerned, nothing surprising was found. In accordance with the predictions of the theories, the capacitance increases nearly linearly with the increasing density as seen in Fig. 3. The temperature dependence of the adsorption was investigated for the the approximate bulk density r ) ; 0.04. According to theoretical considerations by Henderson et al. w10x mentioned in Section 1, the adsorption isotherms should be large in magnitude in the vicinity of the critical point. The results seen in Fig. 4 support their arguments. More accurately, the decrease of the temperature strengthens the effects observed at higher temperature. At zero surface charge Ž s ) s 0. the adsorption is negative Ždrying., and a decrease of the temperature makes it
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
more negative. This agrees with our arguments given above and makes sense because attractive forces act between the particles only and do not act between the particles and the wall. Of course, at high surface charge Ž s ) s 0.03. the adsorption becomes positive because of the strong attraction to the wall. Decreasing the temperature, the adsorption increases steeply. However, at an intermediate surface charge Ž s ) s 0.0125., where the adsorption is nearly zero Žthe two effects of the drying and the DL formation cancel each other., the decrease of the temperature does not seem to influence the adsorption. As discussed in Henderson et al. w10x, the existence of large adsorption or desorption near the critical point is related to the infinite compressibility of the fluid at its critical point. Infinite compressiblity implies infinite density fluctuations and long-range correlations. Small changes in pressure leads to very large changes in the concentration of the ions. Long-range correlations will allow the bulk fluid to adjust better with the presence of a surface or inhomogeneity and also implies a thicker interfacial region. The surface charge density gives an additional parameter over the case of simple fluid and the inhomogeneity can be enhanced in either direction, i.e. either adsorption, or drying Žor desorption. depending on the surface charge. In Fig. 5, a maximum in the adsorption occurs at a certain density. According to Henderson et al. w10x,
477
Fig. 5. The adsorption isotherms as functions of the density for various temperatures for surface charge s ) s 0.03.
this density must be close to the critical density since, according to their argument, the anomalous adsorption is infinite at the critical point of the bulk electrolyte. The maxima of the two temperatures simulated are located at about the same density, slightly less than 0.02. This should be very close to the critical density and falls within the range reported in the literature w11–15x for the critical density of the primitive model electrolyte. We are not able to conclude the critical temperature of the bulk electrolyte from results of this study. From Fig. 4, we can say the critical temperature is below 0.05 and probably not far from this value. The simulations at lower temperatures are possible but may face ergodicity problems, as well as problems of metastable states and phase separations. In any case, this is consistent with current estimates of the critical temperature.
Acknowledgements
Fig. 4. The adsorption as a function of the temperature for various surface charges. The density is r ) ; 0.04.
This work was supported in part by the National Science Foundation ŽGrants CHE96-01971 and CHE98-13729. and by the donors of the Petroleum Research Fund, administered by the American Chemical Society ŽGrant No. ACS-PRF 31573-AC9.. The authors thank Professor Wolfgang Schmickler for drawing their attention to the work of Hamelin et al.
478
D. Boda et al.r Chemical Physics Letters 308 (1999) 473–478
References w1x D. Boda, K.Y. Chan, D. Henderson, J. Chem. Phys. 109 Ž1998. 7362. w2x S.L. Carnie, D. Henderson, J. Chem. Phys. 73 Ž1980. 2949. w3x L. Blum, D. Henderson, J. Chem. Phys. 74 Ž1981. 1902. w4x J.P. Badiali, M.L. Rosinberg, F. Vericat, L. Blum, J. Electroanal. Chem. 158 Ž1983. 253. w5x W. Schmickler, D. Henderson, J. Chem. Phys. 80 Ž1984. 3381. w6x G.M. Torrie, J.P. Valleau, J. Chem. Phys. 73 Ž1980. 5807. w7x G.M. Torrie, J.P. Valleau, G.N. Patey, J. Chem. Phys. 76 Ž1982. 4615.
w8x G.M. Torrie, J.P. Valleau, J. Phys. Chem. 86 Ž1982. 3252. w9x D. Boda, K.Y. Chan, D. Henderson, J. Chem. Phys. 110 Ž1999. 5346. w10x D. Henderson, J. Barojas, L. Blum, J. Phys. Chem. 87 Ž1983. 4544. w11x J.P. Valleau, J. Chem. Phys. 95 Ž1991. 584. w12x A.Z. Panagiotopoulos, Fluid Phase Equilibria 76 Ž1992. 97. w13x G. Orkoulas, A.Z. Panagiotopoulos, J. Chem. Phys. 101 Ž1994. 1452. w14x J.M. Caillol, D. Levesque, J.J. Weis, J. Chem. Phys. 107 Ž1997. 1565. w15x J.P. Valleau, G. Torrie, J. Chem. Phys. 108 Ž1998. 5169. w16x A. Hamelin, S. Rottgermann, W. Schmickler, J. Electroanal. Chem. 230 Ž1987. 281.