7200
Langmuir 2000, 16, 7200-7209
The Electrostatic Interaction of an Assemblage of Charges with a Charged Surface: The Charge-Regulation Effect Heng-Kwong Tsao* Department of Chemical Engineering, National Central University, Chung-li, Taiwan 32054, Republic of China Received November 22, 1999. In Final Form: June 13, 2000 The electrostatic interaction of a particle, modeled as an assemblage of point charges, with a charged plane is investigated on the basis of the Poisson-Boltzmann equation under the Debye-Hu¨ckel approximation. During interactions, the plate can be linearly regulated by the charged particle. The interaction energy and force are derived in an integral expression for a string-like particle and an ionpenetrable object. Analytical results can be obtained for a point charge and a fiber perpendicular to the plane. The interaction depends on three characteristic lengths, including the Debye length κ-1, the separation, and the regulation length λ. Though κλ . 1 corresponds to a plane of constant surface charge density, κλ , 1 does not necessarily represent that of constant surface potential. Two interesting results are observed. The interaction for κλ , 1 may change from repulsion to attraction, and eventually return to repulsion when the particle moves toward the like-charged plate. When the particle-plate distance is fixed, the interaction can change from repulsion to attraction upon varying the concentration of indifferent ions, such as adding salts or dilution. The effect of charge regulation on the change of both electric potential and charge density on the surface is also discussed.
1. Introduction The electrostatic interactions between charged particles and a charged plane are frequently encountered in practical situations, such as adsorption of polyelectrolytes on biological membranes. A charged particle sometimes can be considered as an assemblage of point charges. For instance, an ion-penetrable object, such as a cross-linked polyelectrolyte, can be regarded as a collection of point ions when dealing with electrostatic effects.1,2 For an ionimpenetrable particle, a point charge assumption is valid when the Debye length is large compared to the size of the particle. When the Debye length κ-1 is large compared with the smaller characteristic length d of the charged particle, i.e., κd , 1, the leading order behavior can also be demonstrated by an assemblage of point charge approximation. String-like particles, such as tobacco mosaic virus and DNA, can be modeled as a line of point charges, as a first approximation.3 Similarly, a particulate of oblate shape can be approximated as a disk of point ions when its short axis is small compared to the Debye length. When the particle is located near a charged wall, the primary effect of the electrostatic interaction can be illustrated by the point charge assumption when the particle-wall separation is large in comparison with the size of the particle. When the particle goes toward the plane, the surface of the plane are usually assumed to maintain a condition of constant surface potential or constant surface charge density. However, for a lot of systems, such as metal oxide and polymer latex, the electrostatic interaction may itself influence the degree of dissociation of ionizable groups on the surfaces. As a consequence, either the potential or the charge density on the surface is constant during interaction. The condition known as charge regulation is then more appropriate.4-8 * Fax: 886-3-427-6682. E-mail:
[email protected]. (1) Ohshima, H.; Kondo, T. J. Colloid. Interface Sci. 1993, 157, 504. (2) Chen, S. B. J. Colloid Interface Sci. 1998, 205, 354. (3) Tsao, H.-K. J. Colloid. Interface Sci. 1998, 202, 527. (4) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405.
The extent of dissociation of ionizable groups on the surface depends on the site-binding model, in which the specific type of ions, such as H+ or OH -, can bind with its own surface site. In general, there are single-site and two-site dissociation models. The latter includes the zwitterionic and amphoteric models. On the basis of the law of mass action and the Boltzmann distribution, the surface charge-surface potential relations are usually nonlinear. When the electric potential is small compared to the thermal energy (kBT), however, the chargepotential relation can be linearized to be consistent with the linearized Poisson-Boltzmann equation.5-10 Besides being a reasonable model for some types of charged particles, an assemblage of point charges approximation also provides a tractable method of mathematical treatment and possibilities for analytical solutions. In this paper I investigate the electrostatic interaction between a charge-regulation surface and a particle that is modeled as an assemblage of point charges. Upon linearization of the charge-potential relation on the surface, previous studies often adopt the expression that the surface charge density σ varies linearly with the surface potential ψ
σ ) σs - Cψ The constant C is positive and called the regulation capacitance of the surface.7 Here we rewrite the above relation as
ψ ) ψs + λ∇ψ‚n
(1)
(5) Hunter, R. J. Foundations of Colloid Science; Oxford: New York, 1992. (6) Krozel, J. W.; Saville, D. A. J. Colloid. Interface Sci. 1992, 150, 365. (7) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260. (8) Hsu, J.-P.; Liu, B.-T. Langmuir 1999, 15, 5219. (9) Tsao, H.-K. J. Colloid. Interface Sci. 1998, 205, 538. (10) Tsao, H.-K.; Sheng, Y.-J. Langmuir 1998, 14, 6793.
10.1021/la991520e CCC: $19.00 © 2000 American Chemical Society Published on Web 08/05/2000
Charge-Regulation Effect
Langmuir, Vol. 16, No. 18, 2000 7201
where n is the unit normal to the surface. The constant λ has units of length and can be regarded as the characteristic regulation length associated with the surface. Though the above two expressions are equivalent, one is able to sense the range of “potential regulation” due to the charged particle by the latter expression. The parameters in the linear charge-regulation model, i.e., ψs and λ, can be estimated in terms of the measurable quantities. As a consequence, there are at least three relevant length scales in our system, including the Debye length κ-1, the regulation length λ, and the distance of the particle to the plane R. The electrostatic interaction is therefore determined by two dimensionless groups, e.g., κR and κλ. Intuitively, the extent of charge regulation due to the charged particle can be revealed by the ratio κR/κλ and will be discussed later. The electrostatic interaction of a point charge with a planar surface can be described by the Poisson-Boltzmann (PB) equation under the Debye-Hu¨ckel approximation when the electric potential is less than the thermal energy. In the limiting cases of constant surface potential and surface charge density, the electric field can be solved by the image method.1,3 For an assemblage of point charges, the electric field can then be obtained by linear superposition of the contribution associated with each point charge due to the linearity of the linearized PB equation. However, under the condition of charge regulation, which is a more complicated boundary condition, the linearized PB equation has to be solved by the method of Green’s function and separation of variables. The paper is organized as follows. In the next section the electrostatic field of a fixed point ion interacting with a planar charge-regulation surface is derived on the basis of the linearized PB equation. To demonstrate the effect of charge regulation, the integration associated with the image-like contribution is carried out analytically along the line that passes through the point charge and is perpendicular to the plane. The electrostatic free energy of this system is then evaluated by the charging process. In Section 3, the results of a fixed point charge can be extended to N fixed point charges of arbitrary position distribution in an integral form. By generalizing the discrete charge distribution into a continuous distribution, the interaction energy for a string-like particle and any ion-penetrable object can also be obtained. In Section 4 we consider the effect of charge regulation on the electrostatic interaction for the case of a point ion and a charged rod perpendicular to the plane. Some interesting results that are quite different from those associated with constant surface potential and surface charge density, have been observed, in particular for κλ , 1. The effect of the potential regulation on the change of both surface potential and charge density on the surface is also discussed.
Figure 1. Schematic representation of the system containing a charged plate and fixed point charges. It includes a point charge Qi and its image Qi* with a separation Ri from the plane. The cylindrical coordinate (ri,zi) is associated with Qi.
r and 0 are the relative permittivity of the electrolyte solution and the permittivity of of a vaccum, respectively. During the interaction, the electric potential on the plane, i.e., z ) 0, is under the condition of charge regulation, instead of constant surface potential or constant surface charge density. A linear charge-regulation model is adopted. When eψ/kBT , 1, the linearized form is always obtained and given by eq 1. The constant λ g 0 is the characteristic length associated with the potential regulation. It controls the regulation property of the plane in that λ ) 0 reduces to a constant surface potential ψs and λ f ∞ corresponds to a constant surface charge density for a given separation R. The important features associated with λ will be discussed later. For any position P(x,y,z), the solution to eq 2 can be expressed as the sum of three contributions
ψ(P) ) ψp + ψq + ψq*
where ψp and ψq are, respectively, the undisturbed potentials produced by the plate and the fixed charge in the absence of interactions. The third term ψq* can be regarded as the potential due to the image ion with charge q*, which is required to satisfy the boundary condition. The choices of q* ) (q, respectively, are for constant surface potential and surface charge density.3 For the plane with charge regulation, however, q* varying from -q to +q depends on the position of the point charge. As a result, the image method is not applicable. Substituting eq 3 into eqs 1 and 2 and separating these equations into three groups of equations for ψp, ψq, and ψq* results in the following equations. For ψp,
∇2ψ ) κ2ψ -
q δ(Q) r0
∇2ψp ) κ2ψp
(4)
ψp ) ψs + λ∇ψp‚n, at z ) 0
(5)
subject to
For ψq,
∇2ψq ) κ2ψq -
2. A Fixed Charge First, let us consider a system of a charged plane and a fixed point charge with charge q immersed in an electrolyte solution. The charge is located at position Q and a distance R from the plane as shown in Figure 1. The electric potential ψ established in the system can be described by the linearized Poisson-Boltzmann equation if eψ/kBT , 1.
(2)
where e denotes the fundamental charge, 1.6 × 10-19 C.
(3)
q δ(Q) r0
(6)
Note that ψq need not satisfy any boundary condition. Then for ψq*, one must have
∇2ψq* ) κ2ψq*
(7)
ψq* - λ∇ψq* ) -(ψq - λ∇ψq)
(8)
with
The undisturbed potentials ψp and ψq can be easily obtained by solving the corresponding linear PoissonBoltzmann equations. For an isolated hard plate capable
7202
Langmuir, Vol. 16, No. 18, 2000
Tsao
of charge regulation, one has
ψp(z) )
ψs exp(-κz) 1 + κλ
(9)
For an isolated charge at Q in an unbound domain, the electric potential at P is given by
q exp(-κs˜ ) ψq ) 4πr0 s˜
(10)
where s˜ ) |P - Q|. The most complicated step is to solve eq 7 subject to eq 8. By adopting cylindrical coordinates, they become
( )
2
∂ ψq* 1 ∂ ∂ψq* r + ) κ2ψq* 2 r ∂r ∂r ∂z
(11)
(
)
∂ψq* ∂ψq ) - ψq - λ , at z ) 0 ∂z ∂z
(12)
where z ) 0 represents the plane and r ) 0 denotes the line passing through Q and perpendicular to the plane. Assume that ψq*(r,z) ) R(r)Z(z). Equation 11 then becomes
1 ∂2Z 1 1 ∂ ∂R r ) - κ2 ) ζ2 R r ∂r ∂r Z ∂z2
( )
(13)
where ζ is a constant. Using the method of the separation of variables, one obtains
Z(z) ) exp(-xκ2 + ζ2z)
(14)
R(r) ) aJ0(ζr) + bY0(ζr)
(15)
and
Here J0 and Y0 are the Bessel functions of the first and second kind, respectively. a and b are unknown constants dependent on ζ. Since the electric potential is finite at r ) 0, the coefficient b must be zero. As a consequence, eqs 14 and 15 can be combined to yield
ψq*(r,z) )
∫0∞C(ζ)J0(ζr) exp(-xκ2 + ζ2z) dζ
q 4πr0
(16)
where the unknown function C(ζ) has to be determined by the boundary condition. For the condition of constant surface potential, the electric potential ψq* on the plane is proportional to -exp-
(-xR2+r2)/xR2+r2. Therefore, one obtains the unknown function
C1(ζ) ) -
ζ
xκ2 + ζ2
exp(-xκ2 + ζ2R)
(17)
The above result is obtained by the inverse Laplace transform
{
J0(axt2 - k2) u(t - k) ) L-1
}
exp(-kxs + a ) 2
xs2 + a2
C2(ζ) ) -C1(ζ)
2
(18)
(19)
For the charge-regulation condition, the unknown function C3(ζ) has to satisfy eq 12 and one has
1 - λxκ2 + ζ2 C3(ζ) ) C1(ζ) 1 + λxκ2 + ζ2
(20)
Therefore, the electric potential ψq*, the image-like contribution, is solved and given by
ψq*(κr,κz;κR,κλ) ) ∞ 1 - κλ‚η qκ J (xη2 - 1‚κr) exp[-η(κR + 1 4πr0 1 + κλ‚η 0 κz)] dη (21)
∫(
and
ψq* - λ
where u(t - k) is the Heaviside function, u(t - k) ) 1 for t > k and zero otherwise. On the other hand, for constant surface charge density, the unknown function is given by
)
2.1. Asymptotic Limits κλ . 1 and κλ , 1. The effect of the boundary condition on the plane, i.e., κλ, is primarily displayed through ψq*. Unfortunately, the integral in eq 21 for ψq* cannot be carried out analytically for finite value of κλ. However, the general effect of κλ can still be demonstrated by considering the case r ) 0, at which the analytical expression can be obtained. For r ) 0, the complication due to J0 disappears and the integral can be performed.
ψq*(P) exp -κ(R + z) ) qκ/4πr0 κ(R + z) κ(R+z) exp 1 κλ E1 κ(R + z) 1 + 2 κλ κλ
[
(
)] (22)
where E1(z) is the exponential integral. Its asymptotic expression is E1 = -γ - ln z + z for z , 1 and E1 = [exp(-z)/z](1 - 1/z) for z . 1. Equation 22 can be simplified furthermore by considering the limiting values of the argument in E1. When κ(R + z)(1 + 1/κλ) . 1, one has
ψq*(P;r ) 0) 1 - κλ exp -(κR + κz) = qκ/4πr0 1 + κλ (κR + κz)
(23)
There are two asymptotic situations. For κλ . 1, eq 23 is valid for κ(R + z) . 1 and the electric potential ψq* can be represented by that associated with constant surface charge density with a small correction factor (1 - κλ)/(1 + κλ). For κλ , 1, eq 23 is valid for κ(R + z) . κλ and ψq* can be denoted by that associated with constant surface potential. On the other hand, when κ(R + z)(1 + 1/κλ) , 1, one obtains
ψq*(P; r ) 0) = qκ/4πr0 exp -κ(R + z) 1 + 2 exp (κR + κz) 1 + κ(R + z) 1 κ(R + z) 1 ln κ(R + z) 1 + κλ κλ κλ
{
)]
[
[
(
(
)]} (24)
Equation 24 is valid for κ(R + z) , 1 as κλ . 1 and for κ(R + z) , κλ as κλ , 1. Both limiting situations correspond
Charge-Regulation Effect
Langmuir, Vol. 16, No. 18, 2000 7203
to the condition of constant surface charge density since the second term in the bracket is O{[κ(R + z)/κλ]‚ ln[κ(R + z)(1 + 1/κλ)]} small. For κ(R + z)(1 + 1/κλ) ≈ O(1), the second term in eq 22 is O(κλ)-1 small if κλ . 1. As a result, one can conclude that as κλ . 1, ψq* behaves just like that associated with constant surface charge density. As κλ , 1, ψq* acts like that associated with constant surface potential for κ(R + z) . κλ. However, its behavior becomes such as that associated with constant charge density for κ(R + z) , κλ. Therefore, κλ , 1 is not a sufficient condition for constant surface potential. There always exists a region near the plane, i.e., κ(R + z) j O(κλ), so that ψq* varies from that of constant surface potential to that of constant surface charge density unless κλ ) 0 exactly. For r * 0, an approximate solution can also be obtained for κλ , 1 and κ(R + z) J κλ. By expanding eq 21 with respect to κλ in Taylor series, we obtained ∞
qκ ψq* ) 4πr0
∫1∞(-1 + 2∑(-κλη)i)J0(xη2 - 1‚κr) ‚ i)0
exp[-η(κR + κz)] dη (25)
For a plane with the linear charge-regulation condition, the free energy per unit area is given by7,8,11
fp )
φ0 )
∫1 J0(xη
2
Fp )
x(κR + κz)2 + (κr)2
∫(ψ - ψs) dA
(31)
A
By eliminating the contribution of the self-energy associated with the fixed charge and the charged plate themselves, the interaction energy W(R) is defined as the free energy at separation R subtracting that at infinite separation, W(R) ) F(R) - F(∞). It contains two contributions: W(R) ) Wq + Wp. The interaction energy Wq comes from the potential produced at Q by the plane and the image Q*.
(32)
Using eq 30 with z ) 0, one obtains the dimensionless energy
(26) W/q )
one has
Wq 1 1 ) e-κR + R‚Ξ(κR,κλ) qψs 2(1 + κλ) 2
(33)
where R ) qκ/4πr0ψs and
i
φi )
1 r0ψs 2 λ
q Wq(R) ) [ψp(Q) + ψq*(Q)] 2
- 1‚κr) ‚ exp[-η(κR + κz)] dη ) exp -x(κR + κz)2 + (κr)2
(30)
where the surface charge density σ is related to the potential by σ ) -r0∇ψ‚n. The first integral denotes the electrical work done in creating the double layer and the second integral represents the chemical part of the free energy due to the adsorption of potential-determining ions. Using eqs 1 and 22, one obtains
Since ∞
∫0σψdσ′ - ∫0σ(ψs - rλ0σ′) dσ′
∂ φ0 i
∂(κz)
)
∫1∞(-η)iJ0(xη2 - 1‚κr) ‚ exp[-η(κR + κz)] dη )
∂i exp -t t ∂(κz)i
[
]
(27)
where t ) κ|P - Q*| ) x(κR+κz)2+(κr)2. Q* is the image point. For example,
(28)
This result can also be obtained by solving eqs 11 and 12 based on a regular perturbation with respect to κλ. 2.2. Free Energy. The electrostatic free energy of the present system, F(R), includes both the free energy associated with the fixed charge, Fq, and the free energy associated with the charge regulation plane, Fp. For the fixed charge, the free energy can be evaluated by the charging process
Fq )
∫0qψdq′ )
qψ(Q) 2
)]
(34)
1 r0ψs 2 λ
∫0∞(ψq + ψq*)z)02πrdr
(35)
Since
1 r0ψs 2 λ
∫0∞ψq|z)02πrdr ) 41
Consequently, eq 25 can be expressed in terms of φi
∑
[ (
Apparently, for a specified κR, Ξ ) exp(-2κR)/2κR as κλ f ∞ and Ξ ) - exp(-2κR)/2κR as κλ ) 0. It should be noted that Ξ = exp(-2κR)/2κR if κR , κλ , 1. Wp is due to the potential produced at the plane by Q and Q*.
Wp(R) )
t+1 φ1 ) -κ(R + z) 3 e-t t
∞ qκ ψq*(P) ) {-φ0 + 2 (κλ)iφi} 4πr0 i)0
2κR 1 κλ E1 2κR 1 + κλ κλ
exp
exp(-2κR) -2 Ξ(κR) ) 2κR
(29)
qψs -κR e κλ
and
1 r0ψs 2 λ
∫0∞ψq*|z)02πrdr ) -41
qψs 2 - 1 e-κR κλ 1 + κλ
(
)
one has
W/p )
Wp 1 ) e-κR qψs 2(1 + κλ)
(36)
The integration of ψq* in eq 35 is shown in the Appendix. When κλ ) 0, the result for constant surface potential ψs (11) Chan, D. Y. C.; Mitchell, D. J. J. Colloid Interface Sci. 1983, 95, 193.
7204
Langmuir, Vol. 16, No. 18, 2000
Tsao
is recovered.12 On the other hand, as κλ f ∞, the result for constant surface charge density σ ) κr0ψs/(1 + κλ) is also regained.3,12 The interaction force, Π ) -dW/dR, can then be calculated
Π* ) )
Π(κR) κqψs
and
)
[
[ (
]
)]
3. N Fixed Charges of Arbitrary Position Distribution Consider N fixed point charges which are located at positions {Qi} with charges {qi}, where i ) 1,...,N. The electric potential, ψ, established in the present system is then governed by
qi
N
∇2ψ ) κ2ψ -
δ(Qi) ∑ i)1
(38)
r 0
The electric potential on the plane also obeys the charge regulation condition, eq 1. Similar to the solution for a point charge in Section 2, eq 38 can be solved by assuming
M(x1,x/2) )
λ
∫0 [ψ(i)q + ψq*(i)]z)02πridri ∑ i)1
1
ψs
∞
N
∑qie-κR 2 1 + κλi)1
i
(44)
∫1∞(-1 + 1 +2κλη)J0[xη2 - 1κ(x1 x/2)‚er] ‚ exp[-ηκ(x1 - x/2)‚ez] dη (45)
Obviously, M(x1,x/2) ) ( exp(-κ|x1 - x/2|)/|x1 - x/2| for κλ ) 0 and ∞, respectively. Equations 43 and 44 are expressions for discrete charge distribution. The generalization to continuous charge distribution is straightforward. Consider a string-like particle depicted by a space curve C
x(s) ) x(s)ex + y(s)ey + z(s)ez and the line charge density is γ(s). The interaction energy Wc can then be expressed by
Wc )
ψs γ(s) exp[-κz(s)] ds 1 + κλ C
∫
ψ ) ψp +
(i) [ψ(i) ∑ q + ψq*] i)1
(39)
The electric potential at P due to a fixed charge at Qi, ψq(i), is given by
qi exp(-κs˜ i) 4πr0 s˜ i
ψ(i) q )
(40)
where s˜ i ) |P - Qi|. The electric potential due to the image at Qi*, ψq*(i), satisfies
∇
2
(i) ψq*
)κ
2
(i) ψq*
(i) (i) (i) - λ∇ψq* ) - [ψ(i) ψq* q - λ∇ψq ]
(42)
(i) (κri,κzi;κRi,κλ) is simply given by eq The solution of ψq* 21. Here (ri,zi) denotes the cylindrical coordinate associated with the charge Q with the origin located on the plane as shown in Figure 1. The interaction energies Wq and Wp of the system are then given by
qi ψp(Qi) + Wq({Qi}) ) i)1 2 N
∑
N
∑ 2 1 + κλi)1
qie-κRi -
qi (j) ψq* (Qi) j)1 i)1 2 N N
∑∑
( )∑∑
1
κ
2 4πr0
(43)
N N
qiqjM(Qi,Q/j )
j)1 i)1
(12) Sader, J. E.; Chan, D. Y. C. J. Colloid Interface Sci. 1999, 218, 423.
)∫∫
γ(s)γ(s′)M(x,x′) dsds′ (46)
C C′
where s and s′ are, respectively, the distances from an arbitrarily defined beginning of the string C and its image C′ measured along the contours. x denotes the point in C and x′ the point in C′. For an ion-penetrable object with volume charge density F(x) and a volume V near a charged plate, the electrostatic interaction energy between them, Wv, can be written as
Wv )
ψs F(x) exp[-κz(x)] dV 1 + κλ V
∫
(
)∫∫
1 κ 2 4πr0
(41)
subject to the boundary condition at z ) 0
(
κ 1 2 4πr0
N
ψs
N
where
2κR e 1 R 2κR + 1 -2κR κλ 2 e-2κR e +4 E1 2κR 1 + 2 2 2 2(κR) κλ κλ κR (κλ)
1
2
(37)
1 e-κR + 1 + κλ
)
1 r0ψs
Wp({Qi}) )
F(x)F(x′)H(x,x′) dVdV′ (47)
V V′
where V′ represents the volume of the image and z is the separation of the element dV located at x from the plate. x denotes the point in V and x′ the point in V′. 3.1. A Charged Fiber. Some elongated, colloidal particles and charged biopolymers, such as DNA, have shapes similar to fibers with large aspect ratio of length L to diameter d. When the Debye length κ-1 is large compared with the diameter d, the charged fiber can be treated as a line of point charges. According to eq 46, the interaction energy is given by
ψs L/2 γ(s) exp[-κ(R + sp)‚ez] ds 1 + κλ -L/2 L/2 L/2 κ 1 γ(s)γ(s′)M(R + sp,-R + s′p′) dsds′ 2 4πr0 -L/2 -L/2 (48)
Wf )
(
)∫
∫
∫
where R is the position vector from the plate to the center of the fiber, and p and p′ are the orientations of the fiber and its image, respectively. If the fiber with net charge Q is perpendicular to the plate and the line charge density is uniform γ0 ) Q/L, the interaction energy in eq 48 can
Charge-Regulation Effect
Langmuir, Vol. 16, No. 18, 2000 7205
be further simplified
( )
κL 2ψsQ sinh 2 -κR 1 κQ2 e Wf ) I 1 + κλ κL 2 4πr0
(49)
I is an integral,
I)
contribution due to I. Accordingly, the interaction energy for a long fiber, κL . 1, behaves like that of constant surface potential for κλ , 1 and like that of constant surface charge density for κλ . 1. It seems that this result is different from that for a short fiber, κL , 1, as κλ , 1. However, the interaction force for a long fiber is still unlike that associated with constant surface potential as κλ , 1 and will be discussed later.
∫01∫01M(t,t′)dtdt′ )
{
4. Results and Discussion
exp -κ[2b + L(t + t′)] + κ[2b + L(t + t′)] κ[2b+L(t+t′)] exp 1 κλ E1 κ[2b + L(t + t′)] 1 + 2 κλ κλ
∫01∫01
-
{
(
}
)} dtdt′ (50)
where b ) R - L/2 is the shortest distance between the fiber and the plane. Using the following two equations
∫x∞E1(x)dx ) -xE1(x) + e-x
The interaction of a charged particle, modeled as an assembly of point charges, with a plane capable of charge regulation is investigated on the basis of linearized Poisson-Boltzmann equation. The interaction energies are derived and can be expressed in integral form. For the cases of a point charge and a fiber perpendicular to the plane, analytical expressions can be obtained. In this section, the effect of charge regulation on the interaction are discussed for both a point charge and a rod. The parameters in the linearized charge-potential relation can be determined by the model associated with chemical equilibrium taking place on the surface. For example, consider a single-site dissociation model -
AH ) A + H ; Ka )
and
∫abeβxE1(Rx)dx ) β1{eβxE1(Rx) - E1{(R - β)x]}ba I0 ) (κL)2I ) {H[κ(2R - L)] + H[κ(2R + L)] 2H(2κR)} -2κλ{G[κ(2R - L)] + G[κ(2R + L)] 2G(2κR)} (51)
H(x) ) e
- xE1(x)
and
[(
G(x) ) -ex/κλE1 1 +
1 x + E1(x) κλ
)]
Since
E1(x + ) = E1(x) -
exp(-x) 1 exp(-x) 1 + 1 + 2 + x 2 x x O(3) for , 1
(
)
eq 51 can be reduced to the result of a point charge approximation, eq 34, when κL , 1. Some other limiting cases are obtained from eq 51 by using the relations H(x) = exp(-x)/x as x . 1, H(x) f 1 as x , 1, G(x) = [exp(x)/x]/(1 + κλ) as x . 1, and G(x) f ln(1 + 1/κλ) as x , 1.
(i) I0 ) H(2κb) - 2κλG(2κb), if κL f ∞ (ii) I0 ) 1 - 2κλ ln(1 + 1/κλ), if κb ) 0 and κL f ∞ (iii) I0 ) 1 + H(2κL) - 2H(κL) - 2κλ[ln(1 + 1/κλ) + G(2κL) - 2G(κL)], if κb ) 0 κb ) 0 corresponds to the situation that one end of the fiber is in contact with the plate. Although there is a singularity at t ) 0 and t′ ) 0 in M(t,t′), the integral I is convergent. Case (ii) denotes the upper bound of the
[AH]
kBT 1 + [H+]/Ka ψs ) e [H+]/K a
and
λ)
where
[A-][H+]s
where s refers to a point on the surface. For eψ/kT , 1, eq 1 can be obtained with
the integration in eq 50 can then be carried out
-x
+
r0kBT (1 + [H+]/Ka)2 nae2
[H+]/Ka
Here na denotes the number of ionized group per unit area and the regulation length decays with increasing na. If Ka ) 10-7, [H+] ) 10-6 M, and na ) 0.01 - 0.2 nm-2, ψs ) -28.3 mV and λ ) 34 - 1.7 nm at 298 K. The electric field established in the system containing a point charge and a charged plane can be described by eqs 9, 10, and 21. When the size of a particle a is much smaller than both the Debye length and the particle-wall distance, i.e., κa , 1 and a , R, a point charge approximation is valid. In addition, the analytical result also provides us with the important features of the system. In a 1-1 electrolyte solution of about 0.1 mM, which corresponds to a Debye length κ-1 ≈ 30 nm, a particle of a ) 2 nm, such as for proteins, located at R J 10 nm satisfies this condition. There are three characteristic lengths in this system, κ-1, R, and λ. The magnitude of the electrostatic interaction always depends on κR. The degree of potential (charge) regulation is determined by both κR and R/λ. Note that λ represents the characteristic length of potential-regulation. Hence, when the distance of the charged particle is large compared to λ, the effect of potential regulation is pretty weak. The surface potential is uniform, (1 + κλ)-1, as κR f ∞. On the other hand, when R/λ j O(1), the potential on the plate is significantly regulated by the charged particle. As shown in Figure 2(a), the deviation of the surface potential at r ) 0, nondimensionalized by ψs, from (1 + κλ)-1 is increased as κλ increases for a fixed separation. The extent of regulation also grows with decreasing distance for a given κλ.
7206
Langmuir, Vol. 16, No. 18, 2000
Tsao
Figure 2. (a) The variation of the dimensionless surface potential at r ) 0 with the dimensionless separation at R ) 1 for various values of κλ. (b) The variation of the dimensionless surface charge density at r ) 0 with the dimensionless separation at R ) 1 for various values of κλ.
The variation of the surface charge density with κR and κλ can be demonstrated by considering the point r ) 0.
σ* ) )
σ(r ) 0) r0κψs
{
R exp -κR 1 κR -2 11 + κλ κλ κR κλ
[ (
exp κR 1 +
1 1 E κR 1 + κλ 1 κλ
)] [ (
)]}
When the particle is far away from the plane, i.e., κR . 1, the second term on the right-hand side of the foregoing equation is O[exp(-κR)/κR] small. As a result, the surface charge density at r ) 0, nondimensionalized by r0κψs, is σ = (1 + κλ)-1 > 0, which is due to the plane itself. However, σ* decreases and becomes negative as the particle approaches the plane as shown in Figure 2(b). When κλ ) 0, the adsorption/desorption of potential-determining ions has to take place substantially in order to maintain the condition of constant surface potential. Nevertheless, the extent of surface charge density alteration can be reduced as κλ increases. Equations 33 and 36 provide a concise expression for the interaction energy between a charge regulation plane and a point charge. The image-like contribution is depicted
Figure 3. (a) The variation of the dimensionless interaction energy with the dimensionless separation at R ) 1 for various values of κλ. (b) The variation of the dimensionless interaction force with the dimensionless distance at R ) 1 for various values of κλ.
by the term 1/2RΞ(κR,κλ). For κλ . 1, the charge regulation effect yields an O[(κλ)-1] correction for κR . 1 and an O[(κR ln κR)/κλ] correction for κR , 1. Hence, Ξ = [exp(-2κR)/2κR][1 + O(κλ)-1] and the interaction energy behaves like that of constant charge density with σ ) r0κψs/(1 + κλ). Though the surface potential is strongly regulated as κR j O(κλ), the surface charge density is only slightly affected. As depicted in Figure 3(a) for κλ ) 1000, the interaction is in general weak and rises rapidly only as κR , 1 because of the image contribution. For κλ , 1, the situation is a little complicated. When κR . κλ, the charge regulation effect gives a 2[exp(-2κR)/2κR](1 + κλ)-1 correction. Thus, Ξ = -[exp(-2κR)/2κR][1 + O(κλ)] and the interaction energy behaves like that of constant surface potential with ψ ) ψs. However, when κR ln κR , κλ, the correction becomes O[(κR ln κR)/κλ] small. As a result, the interaction energy behaves more like that associated with constant charge density. In fact, the charge density on the plane does not remain unchanged and varies as ≈-(2R/κλ)(κR)-1 at r ) 0. The surface potential goes as ≈ψs(1 + R/2κR), which is the same as the surface potential for the condition of constant surface charge density (σ ) r0κψs). The results indicates that although κλ . 1 is corresponds to the condition of constant surface charge density, κλ , 1 does not necessarily represent the condition of constant surface potential. For a given value of κλ , 1, as the particle is close enough to satisfy κR ln κR , κλ, the near constant surface potential condition is no longer
Charge-Regulation Effect
present. Only when κλ ) 0, which denotes a singular point mathematically, is the constant surface potential condition always guaranteed. This conclusion can also be proved by the failure of the regular perturbation method in solving this problem. It is worth mentioning that for a plate with constant surface potential which corresponds to λ ) 0, the interaction eventually becomes attractive when the separation is small enough and the magnitude grows with decreasing separation. To maintain a constant surface potential of the plate, the potential determining ions in the electrolyte solution have to bind/dissociate from the surface accordingly. In reality, unlike the condition of constant surface charge density, it would be difficult to achieve this condition. When λ is less than several Ångstroms, however, the plate can be modeled as having a constant surface potential because the particle cannot move closer to satisfy the condition κR , κλ , 1. Figures 3(a) and (b) illustrate the effect of both κR and κλ on the interaction energy W* and Π*. As depicted in Figure 3(a) for κλ ) 0.1, one can clearly see that when the regulation length is small enough, i.e., κλ , 1, the interaction energy essentially reaches a maximum as the particle approaches the plane. This points out the possibility that the interaction can change from repulsion to attraction (or vice versa) as the separation decreases. However, as shown in Figure 3(b) for κλ ) 0.1, the attraction eventually changes back to repulsion as the distance continues to decrease. This is consistent with the conclusion that when κR , κλ, the interaction behaves like that of constant charge density, of which the interaction is always repulsive. When κλ is large enough, e.g., κλ g 0.186 for R ) 1, the possibility of attraction disappears. Nevertheless, as shown in Figure 3(b) for κλ ) 0.186, the repulsion can become weaker first and then increase when the separation decreases. If the value of κλ increases further, i.e., κλ g 0.265, the repulsion decreases monotonically with increasing distance. From the viewpoint of the image contribution, the image possesses charge in opposite sign to the fixed charge q for κR J κλ and attraction can be observed when the image contribution is greater than that of the plane. However, the sign becomes the same as q as κR , κλ and strong repulsion appears. In the foregoing paragraph, the effect of the concentration of indifferent electrolytes is not discussed. From another point of view, one can consider a situation that the relative distance is fixed and the concentration of the electrolyte is varied, for instance by dilution. As shown in Figure 4 for λ ) 1 nm and q/ψs ) 0.3 e/mV, interesting results can be observed. The 1-1 electrolyte concentration ranges from 10-4 to 10-2 M. Upon dilution, the interaction force at R ) 5.0 nm may change from repulsion to attraction. At R ) 7.5 nm, the repulsion increases first and then decreases. At R ) 15 nm, the repulsion increases monotonically by dilution. A charged rodlike particle with diameter d ≈ 2 nm and length L ) 20-80 nm, such as DNA and polypeptides, in a 0.2 mM 1-1 electrolyte solution satisfies the conditions that κd , 1 and L/d . 1. κL is about 1-5. When the fiber is perpendicular to the plane, the interaction energy can be calculated analytically and is given by eqs 49 and 50. The parameter Λ ) γ0/4πr0ψs ≈ 0.1-5 for Q/L ≈ 0.1-2 e/nm and ψs ≈ 5-20 mV. To illustrate the effect of charge regulation, we consider the limiting case that the length of the fiber is large, i.e., κL f ∞, and the line charge density is uniform and fixed, γ0. Since the contribution from the element on the fiber located at z is at least O[exp(-κz)] small if κz . 1 , the
Langmuir, Vol. 16, No. 18, 2000 7207
Figure 4. The interaction forces vary with the concentration of indifferent electrolytes, which is expressed in terms of κ, for λ ) 1 nm and q/ψs ) 0.3 e/mV at different location of a point charge.
interaction is finite even though κL f ∞. Actually, the error by assuming κL f ∞ is O[exp(-κL)/κL] small if κL J 4. The interaction energy and force are given, respectively, by
W/f )
Wf 1 1 ) e-κb - Λ[H(2κb) - 2κλG(κb)] ψsγ0/κ 1 + κλ 2
and
Π/f )
exp(-κb) Π ) ψsγ0 1 + κλ
{
[(
Λ E1(2κb) - 2e2κb/κλE1 1 +
) ]} (52)
1 2κb κλ
Note that as κb f 0, the interaction energy is finite, (1 + κλ)-1 - Λ/2, but the interaction force is infinite, (1 + κλ)-1 - Λ ln[2κb(1 + 1/κλ)] > 0 for κλ , 1. The interaction energy for the conditions of constant surface potential (-) and constant surface charge density (+) are, respectively, W/f ) exp(-κb) ( Λ/2H(2κb).3 Similarly, the interaction forces are Π/f ) exp(-κb) ( ΛE1(2κb). From the viewpoint of interaction energy, the result of charge regulation resembles that of a constant surface potential for κλ , 1. However, the interaction force deviates from that significantly, particularly when κb , κλ. Figure 5 shows the variation of the interaction forces with κb with Λ ) 1.5 for different values of κλ. It is similar to Figure 3(b) for a point charge qualitatively. When the fiber moves toward the plate with κλ , 1 from the distance κb > 1, the interaction force is repulsive and increases to a maximum. Then it decreases and becomes an attractive force. Nonetheless, as κb , κλ, the attraction stops increasing and eventually turns into repulsion, which rises very rapidly. However, when κλ g (κλ)c, e.g., ≈0.15, the interaction is purely repulsion no matter what value κb has. When κλ is large enough, e.g., ≈0.27, the repulsion grows monotonically with decreasing distance. The variation of the interaction force with the length of the fiber is depicted in Figure 6 with κb ) 0.1 and Λ ) 1.5 for various values of κλ. The result shows that even for the same shortest separation κb, the interaction force may change from repulsion to attraction with increasing κL, depending on the charge regulation length κλ. For κL
7208
Langmuir, Vol. 16, No. 18, 2000
Tsao
Figure 5. The interaction forces vary with the shortest fiberplate distance at Λ ) 1.5 and κL . 1 for various values of κλ.
Figure 6. The interaction forces vary with the length of the fiber at Λ ) 1.5 and κb ) 0.1 for various values of κλ.
, 1, the interaction force can be expressed as
Π* )
exp(-κb) (κL) + 1 + κλ
{
(2κb κλ ) E
exp
exp(-2κb) δ (κL)2 -2 2 2κb
κλ
[2κb(1 + κλ1 )]
1
}
Since the total charge Q is proportional to the length κL for a given line charge density, the interaction force approaches zero as κL f 0. Because the second term is O(κL) smaller than the first term, the interaction is always repulsive if κL is small enough. However, for κλ , 1, the second term may dominate and the interaction becomes attractive as κL J 4κb/Λ. In the case of κλ ) 0.15, the interaction can vary from repulsion to weak attraction, and finally become repulsive with increasing κL. The limiting value of Π* associated with κL f ∞ can be either attraction or repulsion and has been shown in Figure 5 and eq 52. The interaction between a charge regulation plate and an assemblage of point charges of arbitrary shape have been obtained in eqs 46 and 47. Since the charges in the assemblage is kept constant during interactions, the present study should be regarded as the interaction between a particle of constant charge and a plate of charge
regulation. This is unlike the interaction between two particles with charge regulation surfaces.6-8 Consequently, the qualitative behavior of the former can be quite different from that of the latter. Some peculiar results associated with a point charge due to the charge regulation effect are also displayed in the case of a fiber; therefore we believe that those properties, particularly for κλ , 1, are general for any ion-penetrable object. Although κλ . 1 denotes a plate with constant surface charge density, κλ , 1 does not necessarily represent a plate with constant surface potential, except κλ ) 0 exactly. For a particle of charge density σp interacting with a plate of charge density σ, Grant and Saville13 have shown that the interaction energy can be split into three terms, which are proportional to σσp, σp2, and σ2, respectively. The σp2 term denotes the interaction between the particle and its image. The σ2 term is always repulsive and accounts for the disturbance associated with the distribution of ions and the electric field due to an uncharged particle. On the other hand, for a point ion of charge q interacting with a plate of constant charge density σ, our previous result3 obtains the first two terms only, i.e., σq and q2. Therefore, the size of a charged entity would give rise to repulsion that is proportional to σ2. When the σp2 (or q2) term or the σ2 term dominates, the interaction is repulsive. For a surface with constant surface potential ψs, based on a similar argument, the interaction energy also includes terms proportional to σpψs, -σp2, and ψs2. When the -σpψs term dominates, the interaction is attractive even for a particle interacting with a like-charged plate. When the regulation length is short, i.e., κλ , 1, the point charges located in distances less than λ will exhibit unusual behavior completely different from that of constant surface potential. In this paper, the particle size effect is neglected and thus the repulsion is not caused by the σ2 (or ψs2) term. For κR , κλ , 1, the short-range repulsion originates from the q2 term associated with a plate of constant charge density. The interesting behavior for a point charge may diminish when the size of an ionimpenetrable particle is considered. In addition, the nonlinear behavior of the potential-charge regulation relation and the Poisson-Boltzmann equation may become significant. Those effects are worth being investigated in the future. Acknowledgment. This research is supported by National Council of Science of Taiwan under Grant No. NSC 89-2214-E-008-012. Appendix In this Appendix, I derive eq 36. Using eq 28, I am going to perform the following integration for Wp.
J)
∫0∞ψq*|z)0rdr ) qκ
∫0
4πr0
∞
[
∞
-φ0 + 2
∑ (κλ)
n)0
i
∂iφ0
∂(κz)i
]
rdr (A1)
z)0
The integration of φ0 can be easily carried out.
∫0∞φ0|z)0‚(κr) d (κr) ) e-κR
(A2)
The next step is to carry out the integration of [∂iφ0/∂(13) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35.
Charge-Regulation Effect
Langmuir, Vol. 16, No. 18, 2000 7209
(κz)i]z)0 for i ) 1, 2, .... Since
∂kφ0
The integration in eq A1 can then be obtained as ∞ qκ J)[-e-κR + 2e-κR (-κλ)n] 4πr0 n)0
k
∞ ∂ φ (κr) d (κr) ∫0 ∂(κz)k(κr) d (κr) ) ∂(κz) k∫0 0 ∞
∑
2 qκ -κR e -1 + )4πr0 1 + κλ
[
one has
∂iφ0
∫0 ∂(κz)i|z)0‚(κr) d (κr) ) (-1)ie-κR, for i ) 1, 2, ..., ∞
(A3)
LA991520E
]
(A4)
