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A corrected 3D Ewald calculation of the low effective temperature properties of the electrochemical interface
4 August 2000
Chemical Physics Letters 325 Ž2000. 675–677 www.elsevier.nlrlocatercplett
A corrected 3D Ewald calculation of the low effective temperature properties of the electrochemical interface Paul S. Crozier
a,1
, Richard L. Rowley
a,2
, Douglas Henderson
b,)
, Dezso¨ Boda
c,3
a
b c
Department of Chemical Engineering, Brigham Young UniÕersity, ProÕo, UT 84602-4100, USA Department of Chemistry and Biochemistry, Brigham Young UniÕersity, ProÕo, UT 84602-5700, USA Department of Physical Chemistry, UniÕersity of Veszprem, ´ P.O. Box 158, H-8201 Veszprem, ´ Hungary Received 3 March 2000
Abstract The corrected 3D Ewald method is used to verify charged sheets method results that show increasing double-layer capacitance with increasing temperature in the low effective temperature region. The restricted primitive model is used where ions are represented as charged hard spheres and the solvent is represented by a uniform dielectric constant. It is shown that the capacitance temperature plot for the test system exhibits increasing capacitance with increasing temperature in the low effective temperature region, which contradicts common theories of the electrochemical interface. For this system, the corrected 3D Ewald method results coincide well with the charged sheets method results. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Recently, Boda et al. w1x have studied low effective temperature anomalies in the properties of the electrochemical interface. They showed that under certain circumstances, a decreasing double-layer ŽDL. capacitance with decreasing temperature is observed. The restricted primitive model ŽRPM. was used where the ions were represented by charged hard spheres of equal diameter, d, and the solvent was represented by a uniform dielectric constant, ´ .
)
Corresponding author. Fax: q1-801-3785474; e-mail: [email protected] 1 E-mail: [email protected] 2 E-mail: [email protected] 3 E-mail: [email protected]
The correct representation of long-range Coulombic interactions is of the utmost importance in studying interfacial electrolyte systems. Crozier et al. w2x have recently compared the charged sheets ŽCS. method w3–7x of modeling these interactions with the corrected 3D Ewald method w8x ŽEW3DC., and found EW3DC to be more reliable than CS for the calculation of long-range Coulombic interactions. Due to this finding, it is worth repeating the calculations of Boda et al. where CS was used, to verify their accuracy. Of specific interest here is the observed decrease in DL capacitance with decreasing temperature in the low effective temperature region, as this is the anamoly for which a quantitative theoretical treatment is not yet available. This behavior contradicts predictions of the Gouy–Chapman theory and the mean spherical approximation, even though these
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 6 4 5 - X
676
P.S. Crozier et al.r Chemical Physics Letters 325 (2000) 675–677
theories predict behavior in agreement with simulation results at higher effective temperatures. The distinction between effective and absolute temperature is important. Systems of molten salt have high absolute temperatures but low effective, or reduced, temperatures where the reduced temperature is T ) s ´ d kTrq 2 , q is the ionic charge, and k is the Boltzmann constant. Molten salts may have an absolute temperature of thousands of degrees, but ´ f 1 for an effective temperature similar to that of an aqueous system at very low absolute temperature, because ´ s 78.5 for the aqueous system. Both systems can have low effective temperatures and exhibit similar behavior for the capacitance. For example, a system with divalent ions would exhibit behavior similar to a system with monovalent ions with a much lower absolute temperature. The effective temperature is the important variable.
2. Simulations We have performed molecular dynamics ŽMD. simulations similar to the Monte Carlo ŽMC. simulations of case c as reported by Boda et al. The EW3DC method was used rather than CS, and oppositely charged walls with an equal number on anions and cations was used, rather than similarly charged surfaces balanced by an unequal number of anions and cations. Oppositely charged surfaces should give
Fig. 1. Capacitance as a function of effective temperature. Boxes are the MC, CS simulation results by Boda et al., and 3s are the MD, EW3DC simulation results of this work. System parameters are s ) s 0.00765, r ) f 0.04, and ´ s 78.5.
Fig. 2. Density profile of ions along the z-axis Žperpendicular to the electrode, z ) s z r d . for T ) s16r25. The thick line represents the counterions, and the thin line represents the coions. System parameters are s ) s 0.00765, r ) f 0.04, and ´ s 78.5.
equivalent results to those produced using similarly charged surfaces as long as there is sufficient separation distance between the walls to allow neutral fluid formation in the central bulk fluid region of the simulation cell. Canonical ŽNVT. simulations were performed with a simulation cell of dimensions L3L3H, and H s 20 d. Box length, L, was allowed to vary between 15d and 17d to give the desired bulk density. In all cases, 100 anions and 100 cations were placed between the hard walls at z s 0 and z s H. As in the work of Boda et al., reduced variables are defined as s ) s s d 2rq, r ) s r d 3 , C ) s s )rf ) , and f ) s f drq where s is the wall surface charge density, r is the bulk density, C is the integral capacitance, f is the potential drop across the DL, and ) represents the reduced variable in each case. Parameters identical to those in case c of the work by Boda et al. were used: s ) s 0.00765, r ) f 0.04, ´ s 78.5. Simulations were performed at T ) s 1r25, 2r25, 3r25, 4r25, 8r25, and 16r25 ŽT s 20, 40, 60, 80, 160, 320 K for monovalent ions with d s 0.425 nm and ´ s 78.5.. Ten repetitions, each lasting 1 ns after an initial 50 ps equilibration period were performed at each of the six temperatures. Each of the repetitions used a unique and randomly selected starting configuration for ion placements and initial velocities. Density profiles were then taken as an average of the 10 repetitions in each case. The MD equations of motion were integrated using a fourth-order Gear predictor–corrector scheme, with a timestep size of 2.5 fs. As required
P.S. Crozier et al.r Chemical Physics Letters 325 (2000) 675–677
Fig. 3. Density profile of ions along the z-axis Žperpendicular to the electrode, z ) s z r d . for T ) s 2r25. The thick line represents the counterions, and the thin line represents the coions. System parameters are s ) s 0.00765, r ) f 0.04, and ´ s 78.5.
by RPM, hard sphere overlaps between timesteps were treated as elastic collisions, with the appropriate changes in ion positions and velocities. All force interactions, including the long-range EW3DC interactions were updated at each timestep.
3. Results As can be seen in Fig. 1, the results produced in this study using MD and EW3DC agree well with the results produced by Boda et al. using MC and CS. Discrepancies between the two data sets are smaller than their combined statistical error. The same increase in capacitance with increasing temperature in the low effective temperature region is observed. Why is there a change in slope in the C ) versus ) T plot when going from the low effective temperature region to the high effective temperature region Žsee Fig. 1.? It is caused by a change in the dominant effect of two competing effects. At high effective temperatures, thermal energy dominates, and the DL thickness increases with increasing temperature. A
677
thicker DL causes a higher potential and a lower capacitance. At sufficiently low effective temperatures, ion–ion interactions become more important as thermal energy becomes insufficient to separate ion pairs. The usual DL build-up at the electrode surface Žsee Fig. 2. is replaced by ‘drying’ at the electrode surface Žsee Fig. 3. as ion–ion interactions become increasingly dominant. The DL thickness increases with the decreasing temperature in the low effective temperature region as ion pairing increasingly prevails and causes increased drying at the electrode surface. This unusual behavior of decreasing capacitance with decreasing temperature has been confirmed to be the correct behavior of RPM under the specified conditions rather than simply an artefact of the calculation method. We have shown that for this system, CS, MC calculations yield the same results as EW3DC, MD calculations within statistical uncertainty. Our calculation also shows that the use of CS by Torrie and Valleau w3–5x is justified for the systems of interest to them. Assumptions inherent in the use of CS seem to be valid when used with RPM; however, we caution the use of CS in systems with discrete solvent molecules and ´ s 1, as shown in the work of Crozier et al. w2x.
References w1x D. Boda, D. Henderson, K.-Y. Chan, D.T. Wasan, Chem. Phys. Lett. 308 Ž1999. 473. w2x P.S. Crozier, R.L. Rowley, E. Spohr, D. Henderson, J. Chem. Phys. 112 Ž2000. 9253. w3x G.M. Torrie, J.P. Valleau, Chem. Phys. Lett. 65 Ž1979. 343. w4x G.M. Torrie, J.P. Valleau, J. Chem. Phys. 73 Ž1980. 5807. w5x G.M. Torrie, J.P. Valleau, G.N. Patey, J. Chem. Phys. 76 Ž1982. 4615. w6x D. Boda, K.-Y. Chan, D. Henderson, J. Chem. Phys. 109 Ž1998. 7362. w7x D. Boda, D. Henderson, R. Rowley, S. Sokołowski, J. Chem. Phys. 111 Ž1999. 9382.
Chemical Physics Letters 325 Ž2000. 675–677 www.elsevier.nlrlocatercplett
A corrected 3D Ewald calculation of the low effective temperature properties of the electrochemical interface Paul S. Crozier
a,1
, Richard L. Rowley
a,2
, Douglas Henderson
b,)
, Dezso¨ Boda
c,3
a
b c
Department of Chemical Engineering, Brigham Young UniÕersity, ProÕo, UT 84602-4100, USA Department of Chemistry and Biochemistry, Brigham Young UniÕersity, ProÕo, UT 84602-5700, USA Department of Physical Chemistry, UniÕersity of Veszprem, ´ P.O. Box 158, H-8201 Veszprem, ´ Hungary Received 3 March 2000
Abstract The corrected 3D Ewald method is used to verify charged sheets method results that show increasing double-layer capacitance with increasing temperature in the low effective temperature region. The restricted primitive model is used where ions are represented as charged hard spheres and the solvent is represented by a uniform dielectric constant. It is shown that the capacitance temperature plot for the test system exhibits increasing capacitance with increasing temperature in the low effective temperature region, which contradicts common theories of the electrochemical interface. For this system, the corrected 3D Ewald method results coincide well with the charged sheets method results. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Recently, Boda et al. w1x have studied low effective temperature anomalies in the properties of the electrochemical interface. They showed that under certain circumstances, a decreasing double-layer ŽDL. capacitance with decreasing temperature is observed. The restricted primitive model ŽRPM. was used where the ions were represented by charged hard spheres of equal diameter, d, and the solvent was represented by a uniform dielectric constant, ´ .
)
Corresponding author. Fax: q1-801-3785474; e-mail: [email protected] 1 E-mail: [email protected] 2 E-mail: [email protected] 3 E-mail: [email protected]
The correct representation of long-range Coulombic interactions is of the utmost importance in studying interfacial electrolyte systems. Crozier et al. w2x have recently compared the charged sheets ŽCS. method w3–7x of modeling these interactions with the corrected 3D Ewald method w8x ŽEW3DC., and found EW3DC to be more reliable than CS for the calculation of long-range Coulombic interactions. Due to this finding, it is worth repeating the calculations of Boda et al. where CS was used, to verify their accuracy. Of specific interest here is the observed decrease in DL capacitance with decreasing temperature in the low effective temperature region, as this is the anamoly for which a quantitative theoretical treatment is not yet available. This behavior contradicts predictions of the Gouy–Chapman theory and the mean spherical approximation, even though these
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 6 4 5 - X
676
P.S. Crozier et al.r Chemical Physics Letters 325 (2000) 675–677
theories predict behavior in agreement with simulation results at higher effective temperatures. The distinction between effective and absolute temperature is important. Systems of molten salt have high absolute temperatures but low effective, or reduced, temperatures where the reduced temperature is T ) s ´ d kTrq 2 , q is the ionic charge, and k is the Boltzmann constant. Molten salts may have an absolute temperature of thousands of degrees, but ´ f 1 for an effective temperature similar to that of an aqueous system at very low absolute temperature, because ´ s 78.5 for the aqueous system. Both systems can have low effective temperatures and exhibit similar behavior for the capacitance. For example, a system with divalent ions would exhibit behavior similar to a system with monovalent ions with a much lower absolute temperature. The effective temperature is the important variable.
2. Simulations We have performed molecular dynamics ŽMD. simulations similar to the Monte Carlo ŽMC. simulations of case c as reported by Boda et al. The EW3DC method was used rather than CS, and oppositely charged walls with an equal number on anions and cations was used, rather than similarly charged surfaces balanced by an unequal number of anions and cations. Oppositely charged surfaces should give
Fig. 1. Capacitance as a function of effective temperature. Boxes are the MC, CS simulation results by Boda et al., and 3s are the MD, EW3DC simulation results of this work. System parameters are s ) s 0.00765, r ) f 0.04, and ´ s 78.5.
Fig. 2. Density profile of ions along the z-axis Žperpendicular to the electrode, z ) s z r d . for T ) s16r25. The thick line represents the counterions, and the thin line represents the coions. System parameters are s ) s 0.00765, r ) f 0.04, and ´ s 78.5.
equivalent results to those produced using similarly charged surfaces as long as there is sufficient separation distance between the walls to allow neutral fluid formation in the central bulk fluid region of the simulation cell. Canonical ŽNVT. simulations were performed with a simulation cell of dimensions L3L3H, and H s 20 d. Box length, L, was allowed to vary between 15d and 17d to give the desired bulk density. In all cases, 100 anions and 100 cations were placed between the hard walls at z s 0 and z s H. As in the work of Boda et al., reduced variables are defined as s ) s s d 2rq, r ) s r d 3 , C ) s s )rf ) , and f ) s f drq where s is the wall surface charge density, r is the bulk density, C is the integral capacitance, f is the potential drop across the DL, and ) represents the reduced variable in each case. Parameters identical to those in case c of the work by Boda et al. were used: s ) s 0.00765, r ) f 0.04, ´ s 78.5. Simulations were performed at T ) s 1r25, 2r25, 3r25, 4r25, 8r25, and 16r25 ŽT s 20, 40, 60, 80, 160, 320 K for monovalent ions with d s 0.425 nm and ´ s 78.5.. Ten repetitions, each lasting 1 ns after an initial 50 ps equilibration period were performed at each of the six temperatures. Each of the repetitions used a unique and randomly selected starting configuration for ion placements and initial velocities. Density profiles were then taken as an average of the 10 repetitions in each case. The MD equations of motion were integrated using a fourth-order Gear predictor–corrector scheme, with a timestep size of 2.5 fs. As required
P.S. Crozier et al.r Chemical Physics Letters 325 (2000) 675–677
Fig. 3. Density profile of ions along the z-axis Žperpendicular to the electrode, z ) s z r d . for T ) s 2r25. The thick line represents the counterions, and the thin line represents the coions. System parameters are s ) s 0.00765, r ) f 0.04, and ´ s 78.5.
by RPM, hard sphere overlaps between timesteps were treated as elastic collisions, with the appropriate changes in ion positions and velocities. All force interactions, including the long-range EW3DC interactions were updated at each timestep.
3. Results As can be seen in Fig. 1, the results produced in this study using MD and EW3DC agree well with the results produced by Boda et al. using MC and CS. Discrepancies between the two data sets are smaller than their combined statistical error. The same increase in capacitance with increasing temperature in the low effective temperature region is observed. Why is there a change in slope in the C ) versus ) T plot when going from the low effective temperature region to the high effective temperature region Žsee Fig. 1.? It is caused by a change in the dominant effect of two competing effects. At high effective temperatures, thermal energy dominates, and the DL thickness increases with increasing temperature. A
677
thicker DL causes a higher potential and a lower capacitance. At sufficiently low effective temperatures, ion–ion interactions become more important as thermal energy becomes insufficient to separate ion pairs. The usual DL build-up at the electrode surface Žsee Fig. 2. is replaced by ‘drying’ at the electrode surface Žsee Fig. 3. as ion–ion interactions become increasingly dominant. The DL thickness increases with the decreasing temperature in the low effective temperature region as ion pairing increasingly prevails and causes increased drying at the electrode surface. This unusual behavior of decreasing capacitance with decreasing temperature has been confirmed to be the correct behavior of RPM under the specified conditions rather than simply an artefact of the calculation method. We have shown that for this system, CS, MC calculations yield the same results as EW3DC, MD calculations within statistical uncertainty. Our calculation also shows that the use of CS by Torrie and Valleau w3–5x is justified for the systems of interest to them. Assumptions inherent in the use of CS seem to be valid when used with RPM; however, we caution the use of CS in systems with discrete solvent molecules and ´ s 1, as shown in the work of Crozier et al. w2x.
References w1x D. Boda, D. Henderson, K.-Y. Chan, D.T. Wasan, Chem. Phys. Lett. 308 Ž1999. 473. w2x P.S. Crozier, R.L. Rowley, E. Spohr, D. Henderson, J. Chem. Phys. 112 Ž2000. 9253. w3x G.M. Torrie, J.P. Valleau, Chem. Phys. Lett. 65 Ž1979. 343. w4x G.M. Torrie, J.P. Valleau, J. Chem. Phys. 73 Ž1980. 5807. w5x G.M. Torrie, J.P. Valleau, G.N. Patey, J. Chem. Phys. 76 Ž1982. 4615. w6x D. Boda, K.-Y. Chan, D. Henderson, J. Chem. Phys. 109 Ž1998. 7362. w7x D. Boda, D. Henderson, R. Rowley, S. Sokołowski, J. Chem. Phys. 111 Ž1999. 9382.