J. Phys. Chem. C 2009, 113, 13241–13248
13241
Ion Diffusion in the Time-Dependent Potential of the Dynamic Electric Double Layer Hang Li,*,†,‡ Laosheng Wu,† Hualin Zhu, and Jie Hou‡ Department of EnVironmental Sciences, UniVersity of California, RiVerside, California 92507 and College of Resources and EnVironment, Southwest UniVersity, Chongqing 400716, People’s Republic of China ReceiVed: March 15, 2009; ReVised Manuscript ReceiVed: May 3, 2009
The description of ion diffusion in the electric field set up by the electric double layer (EDL) is an important issue in many scientific fields because of its close relevance to diffusion-controlled chemical kinetics and ion transport occurring in the environmental, biological, and other systems with multiphase chemical reactions and charged particle transports. When considering ion diffusion in the EDL, the change of ion density in space leads to the change of potential with time. Currently, the static distribution of electric potential in the EDL at equilibrium is described by the nonlinear Poisson-Boltzmann equation. However, describing a timedependent potential during a diffusion process in the EDL still remains a challenge. In this study, a dynamic Poisson-Boltzmann equation that describes the time-dependent potential was suggested for a slow diffusion problem. By combining the generalized nonsteady state diffusion equation (the linearized Fokker-Planck equation) with the time-dependent potential (the dynamic Poisson-Boltzmann equation), analytic solutions that can be expressed in algebraic form for describing the dynamic ion distribution with time-dependent potential and dynamic potential distribution in the EDL were obtained for the reflecting and adsorbing boundary conditions, respectively. The effective and simple approach for analytically solving the generalized linear equation of the complexity nonlinear diffusion in time-dependent potential advanced in the research could be potentially applied to solve other similar systems. 1. Introduction The electric double layer (EDL) emerges as a central feature for many problems in colloid science, electrochemistry, environmental science, and biophysics,1 because of its close relevance to diffusion-controlled chemical kinetics and ion transport occurring in the environmental, biological, and other systems with multiphase chemical reactions and charged particle transports. For a diffusion problem in the EDL, the potential change with time is the natural result of ion diffusion, since ion diffusion can result in change of charge density. For ion diffusion with a time-dependent potential, another equation that describes the change of potential with position and time is required to couple with the Fokker-Planck equation. Nevertheless, solving the Fokker-Planck equation still remains a challenge because of its inherent complexity in its classic form. Even with a potential independent of time, the exact analytic solutions of the equation can only be obtained for some specific cases;2-5 for other cases, the solutions were either numerical or approximated.6-12 When a time-dependent potential must be considered, the solution of the Fokker-Planck equation becomes extremely difficult. Samson et al. applied the Poisson equation directly to couple with the Fokker-Planck equation (or Nernst-Planck equation) to obtain a numerical solution.13 Their approach, however, could not be used to describe a genuine diffusion in a real colloidal suspension. Because of the electroneutrality rule, for a real charged colloidal system, the surface charges must be equilibrated with counterions forever. Therefore, before a new type of electrolyte can be added into the equilibrated colloidal * To whom correspondence may be addressed. E-mail: hli22002@ yahoo.com.cn. Phone: (086) 023-68250674. † University of California. ‡ Southwest University.
suspension, the surface charges must be equilibrated with the existing counterions in EDL (e.g., A+ for electrolyte A+X- in a negatively charged colloidal suspension). Now, we add a new type of electrolyte (B+X-) into the bulk solution of the suspension; the equilibrium is broken and the cation species B+ will diffuse from the bulk solution into EDL; and at the same time, the existing cation species A+ in EDL will diffuse out to the bulk solution. Therefore, for a genuine diffusion in a real colloidal suspension, the diffusion is a mutual diffusion or an exchange process between A+ and B+. Therefore, if the Poisson equation is to be used to describe the time-dependent potential, the time-dependent density distributions for both A+ and B+ must be known. However, in Samson’s approach, only the density distributions for B+ were considered and could be obtained by solving the Fokker-Planck equation. Chan and Hughes and Chan and Halle used the Smoluchowski (Fokker-Planck) Poisson-Boltzmann approach to describe ion diffusion near the charged surface14,15 Chan also used instantaneous relaxation approximation to solve the Fokker-Planck diffusion equation.4 In their approach, the Fokker-Planck diffusion equation was used to describe the diffusion of single ion that moves in the equilibrium potential field of the mean force, and the Poisson-Boltzmann equation was employed to describe the potential distribution of the electric field during the diffusion process. As a result, the solutions of single ion diffusion in the time-independent potential were obtained. However, if the genuine diffusion is not a single ion diffusion, then the equilibrium potential field of the mean force is incorrect; both A+, the having existed cation, and B+, the new added cation, do not obey the Boltzmann distribution in the diffusion process. Another disadvantage was that their solutions were expressed in complex integral forms. In order to obtain the final result, numerical calculations must be used.
10.1021/jp902302t CCC: $40.75 2009 American Chemical Society Published on Web 07/02/2009
13242
J. Phys. Chem. C, Vol. 113, No. 30, 2009
Li et al.
He et al. discussed a dynamic electric double layer model for describing the electrode/electrolyte interface by combining the Nernst-Planck equation and the Poisson equation and found that (1) the dynamic interfacial concentration distribution is similar to the Boltzmann distribution equation but contains the influence of current density and (2) the dynamic potential distribution equation containing the influence of current and double layer effects.16 The classic form of the Fokker-Planck equation, as applied to describing ion diffusion in a time-dependent potential, can be expressed as
∂ f (r, t) ) Di∇ • ∂t i
{[
] }
ZiF ∇φ(r, t) fi(r, t) RT
∇ +
(1)
where fi(r, t) is the concentration of ion species i at location r and time t; φ(r, t) is the electric potential in the electric field; Zi are the charges of the ion; F is the Faraday constant; Di is the diffusion coefficient of the ion; R is the gas constant; and T is the absolute temperature. To solve eq 1, it is required to know the mathematic form of φ(r, t) in advance. However, current approaches can not specify φ(r, t) in advance for an ion diffusion process. At present, we can only obtain the equilibrium distribution of φ(r) through the solution of the static Poisson-Boltzmann equation:
∇2φ(r) ) -
4πF ε
∑ Zifi0e-
ZiFφ (r) RT
(2)
i
where fi0 is the concentration of ion species i as |r| f ∞ and ε is the dielectric constant. Therefore, it is extremely difficult to describe ion distribution in a time-dependent potential by directly applying the classic Fokker-Planck equation. Li and Wu demonstrated that, as the diffusion velocity is slow, using the local equilibrium assumption, the nonlinear differential form of the Fokker-Planck equation (eq 1) is equivalent to the following linear equation:17
∂Ai(r, t) ) Di∇2Ai(r, t) ∂t
(3)
where Ai is apparent concentration (or apparent density), and it has the following form:
Ai(r, t) ) fi(r, t)eZiFφ(r,t)/RT
(4)
For a genuine diffusion in a real colloidal suspension, there are three factors to ensure the diffusion is a slow process: (1) usually in a real colloidal suspension, the electrostatic force is weak; seldom is the surface potential higher than (300 mV; (2) for liquid aqueous system, the viscous resistance force is strong for the diffusion ions; (3) for a genuine diffusion in a real colloidal suspension, the diffusion process is actually the mutual diffusion. Because of the electroneutrality rule, as one type of ionic species diffuses into the EDL, another type of ionic species that existed in the EDL will diffuse out. Because the moving direction of the existing ionic species ions in the EDL is opposite to the direction of the electrostatic attractive force from the particle surface, it will control the whole diffusion velocity and make the diffusion velocity slow. Therefore, the
Figure 1. A conceptual diagram of ion diffusion from an instantaneous source (T indicates the diffusion direction at the initial stage).
local equilibrium assumption for a genuine diffusion in a real colloidal system would be correct. It was shown that the solution of the diffusion problem was much simpler adopting eq 3 than eq 1. For eq 3, it is not required to know the mathematic form of φ(r, t) in advance for solving the equation. More importantly, this new equation (eq 3) implies that it is possible to characterize the time-dependent potential, φ(r, t). Just the same to the eq 1,4 the eq 3 can be used only if the following three assumptions are correct: (1) the relaxation of ion and solvent momenta and of solvent configuration is fast, compared with the time-scale for the change in ion configuration, (2) the interaction potentials among different ions change slowly over the ionic momentum correlation length, and (3) the solventmediated dynamic coupling between the ions and with the surface can be neglected. This study is to apply eq 3 to a time-dependent potential system to obtain the analytic expressions for describing ion diffusion for the reflecting and the adsorbing boundary conditions, respectively, and to calculate the time-dependent potentials of φ(x, t) by using those new analytic solutions. 2. Analytic Solutions of Ion Diffusion with Time-Dependent Potential 2.1. Instantaneous Source Diffusion for the Reflecting Boundary Condition. Figure 1 shows the conceptual diagram of ion diffusion from an instantaneous source. Considering the 1-dimensional case, the instantaneous source means that, at the initial time t ) 0, all of the ions stay on the plane at x ) x0, where the concentration of the ion is infinite. Thus, the initial conditions can be expressed as the Dirac’s delta function:
{
Ai(x, 0) ) δ(x, 0)
∞, 0,
x ) x0 x * x0
(5)
The reflecting boundary condition requires that the real flux ji(x, t) vanishes at the field source surface (x ) 0). Thus we have
ji(x, t)| x)0 ) ji(0, t) ) 0
(6)
According to the relationship developed by Li and Wu,17,18 the apparent flux Ji(x, t) is
Ji(x, t) ) ji(x, t)eZiFφ(x,t)/RT
(7)
Dynamic Electric Double Layer
J. Phys. Chem. C, Vol. 113, No. 30, 2009 13243
and
Ji(x, t) ) -Di
∂Ai(x, t) ∂x
(8)
Thus, the reflecting boundary condition can be expressed as
Ji(0, t) ) -Di
∂Ai(x, t) ) eZiFφ(0,t)/RTji(0, t) ) 0 ∂x x)0
(9) To satisfy the boundary condition (eq 9) and the initial conditions (eq 5), the solution of eq 3 can be expressed as4
Ai(x, t) )
c1 2√πDit
[e-(x - x0)2/4Dit + e-(x + x0)2/4Dit] (10)
where c1 is an integration constant. Using the apparent mass conservation equation,17 we can demonstrate that c1 is equal to the total apparent mass Mi0 (see Appendix A). Thus, eq 11 becomes
Ai(x, t) )
Mi0 2√πDit
[e-(x - x0)2/4Dit + e-(x + x0)2/4Dit] (11)
According to the definition of the apparent mass,17,18 at t ) 0, x ) x0, the relationship of the total apparent mass with the total real mass (m0) can be expressed as
Mi0 ) mi0eZiFφ(x0,0)/RT
mi0 2√πDit
eZiF/RT[φ(x0,0)-φ(x,t)][e-(x
- x0)2/4Dit
e-(x
+
+ x0)2/4Dit
] (13)
2.2. Instantaneous Source Diffusion for the Adsorbing Boundary Condition. For the adsorbing boundary condition, the concentration of the ion at the field source surface (x ) 0) remains zero; therefore
Ai(0, t) ) fi(0, t)eZiFφ(0,t)/RT ) 0
(14)
Applying the same initial conditions as for the reflecting boundary (eq 5), the solution of eq 3 that satisfies the adsorbing boundary condition is4
Ai(x, t) )
c2 2√πDit
[e-(x - x0)2/4Dit - e-(x + x0)2/4Dit]
where c2 is an integration constant.
Again, using the apparent mass conservation equation,17 we can demonstrate that c2 is equal to the total apparent mass (Mi0) at the initial time of diffusion for the adsorbing boundary condition (see Appendix B). Thus, eq 15 becomes
Ai(x, t) )
Mi0 2√πDit
[e-(x - x0)2/4Dit - e-(x + x0)2/4Dit] (16)
Introducing eqs 4 and 12 into eq 16, the concentration distribution of ion species i with time-dependent potential is
(12)
Introducing eqs 4 and 12 into eq 11, we obtain the concentration distribution of ion species i with time-dependent potential:
f(x, t)i )
Figure 2. The conceptual diagram of cation distribution in EDL in diffusion process.
(15)
fi(x, t) )
mi0 2√πDit
eZiF/RT[φ(x0,0)-φ(x,t)][e-(x
- x0)2/(4Dit)
e-(x
-
+ x0)2/4Dit
] (17)
According to eqs 13 and 17, if φ(x, t) is obtained, then fi(x, t) can be calculated. 2.3. Evaluation of φ(x, t) in the Dynamic EDL. In this section, we will discuss how to describe the φ(x, t) for a genuine diffusion in a real charged colloidal suspension. Because of the electroneutrality rule, for a real charged colloidal system, the surface charges must be equilibrated with counterions forever, for which the counterions distribution in EDL will obey the Boltzmann distribution law at equilibrium. Therefore, before a new type of electrolyte can be added into the equilibrated colloidal suspension, the distribution of the existing counterions (e.g., A+ for electrolyte A+X-) in EDL could be described by the Boltzmann distribution. Now, we add a new type of electrolyte (B+X-) into the bulk solution of the suspension; the equilibrium is broken and the cation species B+ will diffuse from the bulk solution into EDL; and at the same time, the existing cation species A+ in EDL will diffuse out to the bulk solution. Therefore, for a genuine diffusion in a real colloidal suspension, the diffusion is a mutual diffusion or an exchange process between A+ and B+ (Figure 2A). If the diffusion velocity is slow, the following deduction would be correct: the electroneutrality rule is still correct in the diffusion process; and the distribution of charge “+” (but not the cation species A+ or
13244
J. Phys. Chem. C, Vol. 113, No. 30, 2009
Li et al.
B+) in EDL must be the Boltzmann distribution, approximately. For the Poisson equation, merely the distribution of charge “+” in EDL is needed to be known; the knowledge about the distribution of A+ or B+ in EDL is actually not needed. The reason is that, for electrostatic effect, A+ and B+ are nondistinguishable; they are equally charged “+” (Figure 2B). Therefore, the diffusion problem can be approximately taken as the charge “+” diffusion in an initially having been equilibrated system with the same charge “+”. Obviously, for a slow diffusion process, at any moment of diffusion, if the electroneutrality is broken and if the distribution of “+” in EDL deviates from the Boltzmann distribution, the electrostatic force or the free energy in EDL must sharply increase, which will make the EDL recover to the electroneutrality and make the distribution of “+” recover to the Boltzmann distribution immediately. For a slow diffusion process, the EDL will have enough time to recover to the electroneutrality and the Boltzmann distribution for “+”. Therefore, for “+” (but not for A+ or B+), the EDL would be at equilibrium locally in the diffusion process, and “+” must obey an equilibrium distribution, the Boltzmann distribution, approximately. So, in EDL, there will be f+(x,t) ) fA(x,t) + fB(x,t) ≈ a exp[-Fφ(x,t)/RT]. Here, a would be an unknown imaginary constant, and in the diffusion process a * fA0(t) + fB0(t). On the basis of those analyses, the diffusion process can be illustrated by Figure 2. At equilibrium, the effective thickness of EDL, 1/κ, can be calculated by the following equation:
1 ) κ
εRT 1 8πF f Z2 2 i i0 i 2
(
∑
)
where fi0 is the concentration of ion species i in the bulk solution at equilibrium (including each counterion species and co-ion species in bulk solution). Because the values of ΣZi2fi0 at the final equilibrium of diffusion are larger than that at initial equilibrium of diffusion, the 1/κ(teq) will be smaller than the 1/κ0. In supposing the 1/κ continuously decrease from 1/κ0 to 1/κ(teq) in diffusion, the negative value of surface potential φ0 on the wall will continuously decrease from φ0(t ) 0) to φ0(t ) ∞) because of the increase of the charge (positive) density near the wall. By showing this, for a genuine diffusion in a real colloidal suspension, the potential must be time-dependent. When considering a symmetric electrolyte system in a diffusion process, based on the above analysis and the static Poisson-Boltzmann equation,19 the Poisson-Boltzmann equation at any given time t ) tj can be expressed as
d2φ(x, t ) tj)
)
2
dx
[
ZFφ(x, t ) tj) RT 2 κ (t ) tj) sin h ZF RT
]
(18)
with boundary conditions
{
φ(x ) 0, t ) 0) ) φ(0) φ(x ) 0, t ) tj) ) φ(tj)
where κ(t ) tj) is the Debye-Hückel parameter at a any given time t ) tj, and for nonequilibrium, it can not be obtained by
the equation for calculating Debye characteristic length; and Z ) Z+ ) -Z. Here, we call eq 18 the dynamic Poisson-Boltzmann equation. Subject to the boundary conditions, the solution for eq 18 is
{[
] }
ZFφ0(tj) -κ(tj)x 4RT e tan h-1 tan h ZF 4RT -κ(tj)x 2RT 1 + a(tj)e ) ln ZF 1 - a(t )e-κ(tj)x
φ(x, tj) )
(19)
j
where a(tj) ) tan h[ZFφ0(tj)/4RT]. Obviously, both Chan’s and Samson’s approach could not be used to describe a genuine diffusion in a real colloidal suspension. In Chan’s approach, it described single ion diffusion. The diffusion of a single ion could be taken as the diffusion in the equilibrium potential field of the mean force. However, for genuine diffusion, the potential would be time-dependent. For Samson’s approach, in the diffusion process, the charge density coming from the existing cation species A+ was neglected. For a genuine diffusion in a real colloidal suspension, the charge density would be F(x,t) ) ZF[fA(x,t) + fB(x,t)]. The fB(x,t) can be obtained by the Fokker-Planck equation, but usually the fA(x,t) will be difficult to obtain. In Samson’s approach, the fA(x,t) was neglected. 2.4. Final Expressions for Ion Diffusion in the TimeDependent Potential. (1) For the reflecting Boundary Condition. Introducing eq 19 into eq 13, we can obtain the final expression of the analytical solution for ion diffusion in the dynamic EDL for the reflecting boundary condition:
fi(x, tj) )
mi0 2√πDitj
[
γ0(0)
1 - a(tj)e-κ(tj)x 1 + a(tj)e-κ(tj)x
]
2Zi/Z
×
[e-(x - x0)2/4Ditj + e-(x + x0)2/4Ditj] (20)
where i represents a positive or negative ion of the electrolyte, and
ZFφ0(0) -κ(0)x0 e 4RT γ0(0) ) ZFφ0(0) -κ(0)x0 1 - tan h e 4RT 1 + tan h
in which φ0(0) and κ(0) are the surface potential and Debye-Hu¨ckel parameter at t ) 0, respectively. To calculate the a(tj) [or φ0(tj)] and κ(tj) in eq 20, let us consider a negatively charged field source surface where the counterions are positive and the co-ions are negative. At any given time t, for the reflecting boundary, the mass conservation for both counterions and co-ions leads to
∫0∞ fi(x, tj)dx ) mi0 Introducing eq 20 into eq 21, we have
(21)
Dynamic Electric Double Layer
∫0
∞
1 2√πDitj
[
γ0(0)
J. Phys. Chem. C, Vol. 113, No. 30, 2009 13245
1 - a(tj)e-κ(tj)x -κ(tj)x
1 + a(tj)e
]
2Zi/Z
e-(x
[e-(x - x0)2/4Ditj +
+ x0)2/4Ditj
]dx ) 1 (22)
For any given value of γ0(0), a(tj) [or φ0(tj)] and κ(tj) can be obtained, because eq 22 is applicable to both co-ions and counterions. (2) For the Adsorbing Boundary Condition. Similarly, introducing eq 19 into eq 17, we can get the final analytical solution for ion diffusion in the dynamic EDL for the adsorbing boundary condition:
mi0
fi(x, tj) )
2√πDitj
[
γ0(0)
1 - a(tj)e-κ(tj)x 1 + a(tj)e-κ(tj)x
]
2Zi/Z
×
Figure 3. Change of surface potentials with time in the diffusion process at T ) 298 K and Z ) 1, (line) reflecting boundary; (dashed line) adsorbing boundary. Numbers on the curves are the absolute values of φ0(0).
[e-(x - x0)2/4Ditj - e-(x + x0)2/4Ditj] (23)
Crank,20 and these approaches may be applicable to solve the linearlized Fokker-Planck equation.
Again, according to the mass conservation, the following holds for both counterion and co-ion at any given time:
3. Discussions on the Results 3.1. The Time-Dependent Potential in the Dynamic EDL. As was done by Chan,4 we used χ ) κ(tj)x as the dimensionless distance from the solid surface, τ ) κ2(tj)Dtj as the dimensionless time, and total mass m0 ) 1 as the dimensionless mass of the ion at the initial moment of diffusion. Therefore, the term f(x,tj) · [12/κ(tj)] ) f(x,tj)/κ(tj) ) f(χ,τ) represents the dimensionless concentration. We also assumed that the source plane of the diffusion is at position χ ) κ(0)x0 ) 1 and that the source electric field surface is negatively charged with constant charges. According to eq 20, the dimensionless forms of the ion distribution equations at any given time for the reflecting boundary condition are (1) for counterions
|
∫0∞ fi(x, tj)dx + ∫0t ji(x, tj) j
dtj ) mi0
(24)
x)0
The first integral in eq 24 represents the total existing quantity of the ion in space at any given time t; the second integral represents the total adsorbed quantity of the ion during t ) 0 to any given time t ) tj. Considering17,18
ji(x, tj)
|
x)0
) Die-ZFφ(0,tj)/RT
∂Ai(x, tj) ∂x
|
(25)
x)0
f(χ, τ) ) we have
∫0∞ fi(x, tj)dx + Di ∫0t e-ZFφ(0,t )/RT j
j
∂Ai(x, tj) ∂x
|
x)0
dtj ) mi0
[
1 2√πDitj
∫0t
[
γ0(0)
1 - a(tj)e-κ(tj)x 1 + a(tj)e-κ(tj)x
]
x0 2√
πDitj3
[
γ0(0)
1 - a(tj) 1 + a(tj)
]
+ 1)2/4τ
] (28)
ZFφ0(0) -1 e 4RT γ0(0) ) ZFφ0(0) -1 1 - tan h e 4RT 1 + tan h
×
2Zi/Z
e-(χ
+
where
2Zi/Z
[e-(x - x0)2/4Ditj - e-(x + x0)2/4Ditj]dx +
j
- 1)2/4τ
(26)
Introducing eqs 23 and 16 into eq 26, we obtain
∫0∞
]
1 1 - a(τ)e-χ 2[ -(χ e γ0(0) 1 + a(τ)e-χ 2√πτ
e-x0/4Ditjdtj ) 1 (27)
and (2) for co-ions are
2
Thus, for any given value of γ0(0), a(tj) [or φ0(tj)] and κ(tj) can be obtained from eq 27. By comparing this with the solutions of Chan,4 Chan and Hughes, or Chan and Halle, the new solution is much simpler; specifically, it can be expressed in algebraic form.14,15 The linearized Fokker-Planck equation has the same mathematic form as the linear Fickian diffusion equation. The approaches for solving the linear Fickian diffusion equation with other coordinates (cylindrical or spherical) were provided by
f(χ, τ) )
[
1 1 - a(τ)e-χ γ0(0) 1 + a(τ)e-χ 2√πτ
][ -2
e-(χ
- 1)2/4τ
e-(χ
+
+ 1)2/4τ
] (29)
From eq 22, the value of a(τ) [or φ0(τ)] in eqs 28 and 29 is the solution of the following integral equation:
∫0∞
[
]
1 1 - a(τ)e-χ 2[ -(χ γ0(0) e 1 + a(τ)e-χ 2√πτ
- 1)2/4τ
e-(χ
+
+ 1)2/4τ
]dχ ) 1 (30)
13246
J. Phys. Chem. C, Vol. 113, No. 30, 2009
Li et al.
The dimensionless forms of the ion distribution equations at any given time for the adsorbing boundary condition, according to eq 23, are (1) for counterions
f(χ, τ) )
[
]
1 1 - a(τ)e-χ 2[ -(χ γ0(0) e 1 + a(τ)e-χ 2√πτ
- 1)2/4τ
e-(χ
-
+ 1)2/4τ
] (31)
and (2) for co-ions are
f(χ, τ) )
[
1 1 - a(τ)e-χ γ0(0) 1 + a(τ)e-χ 2√πτ
][ -2
e-(χ
- 1)2/4τ
e-(χ
-
+ 1)2/4τ
] (32)
From eq 27, a(τ) [or φ0(τ)] is the solution of the following integral equation:
[
]
(χ+1) 1 1 - a(τ)e-χ 2 - (χ-1) e γ0(0) e 4τ 4τ dχ 1 + a(τ)e-χ 2√πτ τ 1 1 - a(τ) 2 - 1 + 0 e 4τ dτ ) 1 γ0(0) 1 + a(τ) 2√πτ3 (33)
∫0∞
∫
[
]
[
2
2
]
If the values of γ0(0) and τ are given, applying the Simpson rule, we can obtain a(τ) [or φ0(τ)] through the solutions of eqs 30 and 33 for the reflecting and adsorbing boundary conditions, respectively. As the φ0(τ) is obtained, the φ(x,t) can be easily calculated through solving eq 19. The curves in Figure 3 were obtained through solving eqs 30 and 33. The curves clearly show the change of surface
potentials with time during a diffusion process for the reflecting and adsorbing boundary conditions, respectively. It was observed that, at the initial stage of the diffusion, the surface potential (absolute value) decreases as time increases. The surface potential then bounces back as time increases for the reflecting boundary condition. This pattern of surface potential change is attributed to the fact that, at the initial stage, as ions diffuse from the plane χ ) 1 toward the field source surface, there are more counterions in the domain from χ ) 0 to 1 than the co-ions, which results in the decrease of the absolute value of the surface potential over time at the initial stage. As time increases to a certain point, the gathered counterions in the domain of χ ) 0 to 1 diffuse out to the infinite space of χ > 1, which in turn increases the surface potential because of the decrease of counterion density adjacent to the surface. In the adsorbing boundary case, the decrease of the surface potential at the very beginning is more rapid, but the duration of surface potential decreases before it bounces back and is much longer than that of the reflecting boundary case. Since many more counterions can be adsorbed on the surface than co-ions, the net effect is that the ion adsorption onto the surface is similar to the electro-neutralization reaction occurring on the surface. Therefore, the net charge density on the surface decreases during the adsorption process, leading to the rapid and longlasting decrease of the surface potential. In contrast, the net charge density on surface remains constant for the reflecting boundary condition, and the change of its surface potential is solely dependent on the change of the concentration of the counterions adjacent to the surface. The curves in Figure 3 also shows that, as the initial surface potential is very low, there will be φ0(τ) ≈ φ0(0) when the dimensionless time τ < 2. For this case, the obtained results for f(χ,τ) from our approach should be equal to the results from the Chan’s approach approximately. So, a comparison between
Figure 4. Comparison between the analytic solutions (curves) and Chan’s numeric solutions (dots) at T ) 298 K, Z ) 1 and ZFφ0(0)/RT ) 1.0. Numbers on the curves are dimensionless time. (a) Counterion distribution for reflecting boundary; (b) counterion distribution for adsorbing boundary; (c) co-ion distribution for reflecting boundary; and (d) co-ion distribution for adsorbing boundary.
Dynamic Electric Double Layer
J. Phys. Chem. C, Vol. 113, No. 30, 2009 13247
Figure 5. Comparison between the analytic solutions (curves) and Chan’s numeric solutions (dots) at T ) 298 K, Z ) 1 and ZFφ0(0)/RT) 2.0. Numbers on the curves are dimensionless time. (a) Counterion distribution for reflecting boundary; (b) counterion distribution for adsorbing boundary; (c) co-ion distribution for reflecting boundary; and (d) co-ion distribution for adsorbing boundary.
our approach and Chan’s approach can be made as the initial surface potential is low. 3.2. Comparison between the Analytic Solutions and Chan’s Numeric Results. According to the above discussions, as the initial surface potential is very low, the potential can be approximately taken as constant. For this case, the comparison between our approach and Chan’s approach could be made for f(χ,τ). Figure 4 and Figure 5 show the dimensionless concentration distributions of the counterions and co-ions as a function of time for the reflecting and adsorbing boundary conditions with the initial surface potential Fφ0(0)/RT ) eφ0(0)/kT ) 1 and 2, respectively. The figures indicate that our analytic solutions matched Chan’s numeric results well as the initial surface potential is very low.4 However, from Figure 5, it was observed that the difference between our solution and Chan’s solution increases when the initial surface potential is high. This may reflect the fact that Chan’s solutions neglected the timedependent nature of the potential in the EDL. The above discussions showed that the approach advanced in this paper could reasonably give an approximate description for both φ(x,t) and f(x,t) in a dynamic EDL system, and the application of the new approach is easy. Furthermore, the analytic solutions expressed in algebraic form could be obtained by the new approach. 4. Conclusions Based on the above discussions, we conclude the following: (1) The time-dependent potential in the electric field (electric double layer) can be described by the dynamic Poisson-Boltzmann equation. (2) Both time-dependent ion distribution and time-dependent potential distribution in the electric double layer can be quantitatively obtained by simultaneously applying the linearized nonlinear Fokker-Planck equation and the dynamic PoissonBoltzmann equation. (3) This paper provides an effective and simple approach for analytically solving the complexity nonlinear Fokker-Planck
equation with time-dependent potential in the one-dimensional case for genuine diffusion in real colloidal systems. Acknowledgment. This work was partially supported by the National Natural Science Foundation of China (Grant 40371061 and Grant 40671090). Appendix A Giving that the total apparent mass is Mi0, according to the apparent mass conservation equation,17 we have
∫0∞ Ai(x, t)dx
Mi0 )
(A1)
Introducing eq 12 into eq A1, we have
Mi0 ) )
∫0∞ c1
√π c1 √π c1
c1 2√πDit
[e-(x
- x0)2/4Dit
+ x0)2/4Dit
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
∫-x∞ /2√Dt e-( 2√Dt ) d 0
+ e-(x
∫x∞/2√Dt e-
x - x0
x + x0 2√Dt
2
2
d
x - x0
2√Dt x + x0
+
2√Dt x - x0 ) d + 2√Dt √π 2√Dt c1 ∞ - x + x0 2 x + x0 e 2√Dt d + √π 0 2√Dt x - x0 2 x - x c1 0 0 e- 2√Dt d -x0/2√Dt √π 2√Dt c1 x /2√Dt - x + x0 2 x + x0 0 e 2√Dt d √π 0 2√Dt c1 √π c1 √π ) · + · + 0 ) c1 √π 2 √π 2 0
∫
∞ e 0
∫
∫
∫
x - x0
]dx
2
(A2)
13248
J. Phys. Chem. C, Vol. 113, No. 30, 2009
Li et al. As t0 f ∞, eq B4 transforms to
Appendix B For the adsorbing boundary condition, the apparent mass flux at x ) 0 is17
Ji(0, t) )
dMi(0, t) ∂Ai(x, t) )D dt ∂x
|
(B1)
x)0
where Mi(0, t) is the apparent mass of the ion diffused through the field surface during the time from 0 to t, and it is equal to the total adsorbed apparent mass of the ion during the time period. From eq B1, we have
Mi(0, t0) ) Di
∫0
t0
∂Ai(x, t) ∂x
|
(B2)
dt
x)0
For the adsorbing boundary condition, according to the apparent mass conservation,17 the total apparent mass of the ion at t ) 0 is equal to the existing apparent mass in the space from x ) 0 to ∞ at any given time t0 plus the total adsorbed apparent mass on the surface during the time from t ) 0 to t0. Thus
Mi0 )
∫0∞ Ai(x, t0)dx + Di ∫0t
0
∂Ai(x, t) ∂x
|
dt
x)0
(B3)
Introducing eq 16 into eq B3, we have
Mi0 )
∫0∞
c2 2√πDit0
[e-(x
- x0)2/4Dit0
c2x0 2√πDi
- e-(x ·
+ x0)2/4Dit0
]dx +
∫0t t-3/2e-x /4D tdt 0
2 0
i
Mi0 )
c2x0
∫0∞ t-3/2e-x /4D tdt 2 0
2√πDi
(B5)
i
From eq B5, we have
Mi0 ) -
2c2
√π
( )
∫∞0 e-( 2√D t ) d x0
2
i
x0
2√Dit
)-
2c2
√π
( ) -
√π ) c2 2 (B6)
References and Notes (1) Åkesson, B.; Jo˝nsson, B.; Chan, D. Y. C. Mol. Phys. 1986, 57, 1105. (2) Hong, K. M.; Noolandi, J. J. Chem. Phys. 1978, 68, 5163. (3) Tachiya, M. J. Chem. Phys. 1979, 70, 238–241. (4) Chan, D. Y. C. J. Chem. Soc., Faraday Trans. 1987, 83, 2271– 2286. (5) Berezhkovskii, A. M.; D’yakov, Yu A. J. Chem. Phys. 1998, 109, 4182–4189. (6) Cohen, H.; Cooley, J. W. Biophys. J. 1965, 5, 145–162. (7) Clifford, P.; Green, N. J. B.; Pilling, M. J. J. Phys. Chem. 1984, 88, 4171–4176. (8) Benesi, A. J. J. Chem. Phys 1986, 85, 374. (9) Gitterman, M.; Weiss, G. H. Phys. ReV. E 1993, 47, 976. (10) Klik, I.; Yao, Y. D. Phys. ReV. E 2000, 62, 4469–4472. (11) Ansari, A. J. Chem. Phys. 2000, 112, 2516–2522. (12) Jones, R. B. J. Chem. Phys. 2003, 119, 1517. (13) Samson, E.; Marchand, J.; Robert, J. L.; Bournazel, J. P. Int. J. Numer. Methods Eng. 1999, 46, 2043–2060. (14) Chan, D. Y. C.; Hughes, B. D. J. Stat. Phys. 1988, 52, 383–394. (15) Chan, D. Y. C.; Halle, B. Biophys. J. 1984, 46, 387–407. (16) He, R.; Chen, S. L.; Yang, F.; Wu, B. L. J. Phys. Chem. B 2006, 110, 3262–3270. (17) Li, H.; Wu, L. New J. Phys. 2007, 9, 357. (18) Li, H.; Wu, L. J. Phys. Chem. B 2004, 108, 13821–13826. (19) Sposito, G. The Surface Chemistry of Soils; Oxford University Press: New York, 1984. (20) Crank, J. The Mathematics of Diffusion; Oxford University Press: New York, 2001.
(B4) JP902302T
