11960
J. Phys. Chem. B 2008, 112, 11960–11967
Electric Double Layer of Spherical Particles in Salt-Free Concentrated Suspensions: Water Dissociation and CO2 Influence Emilio Ruiz-Reina*,† and Fe´lix Carrique‡ Departamento de Fı´sica Aplicada II, UniVersidad de Ma´laga, Campus de El Ejido, 29071, Ma´laga, Spain, and Departamento de Fı´sica Aplicada I, UniVersidad de Ma´laga, Campus de Teatinos, 29071, Ma´laga, Spain ReceiVed: April 1, 2008; ReVised Manuscript ReceiVed: July 8, 2008
We present a model for the theoretical description of the electric double layer of realistic salt-free colloidal suspensions. This kind of systems consist of aqueous suspensions deionized maximally without any electrolyte added during the preparation, in which the only ions present can be (i) the added counterions that counterbalance the surface charge, (ii) the H+ and OH- ions from water dissociation, and (iii) the ions produced by the atmospheric CO2 contamination. Our theory is elaborated in the framework of the classical Poisson-Boltzmann theory, the spherical cell model approach, and the appropriate local equilibrium reactions, and it also includes an efficient mathematical treatment for dealing with the resulting integro-differential equations. We have applied it to the study of the surface electric potential in a wide range of volume fraction and surface charge density values in a variety of cases. The numerical results show that it is necessary to consider the water dissociation influence for volume fractions lower than approximately 10-2, whereas the atmospheric contamination, if the suspensions are open to the atmosphere, is important in the region of φ < 10-1. The present work sets the basis for theoretical models concernig the equilibrium phase diagram, electrokinetics, and rheology of such systems. Introduction
e2njcz2c -3ezc φ κ ) ) σ ε0εrkBT ε0εrkBTa 1 - φ 2
The study of suspensions of charged colloidal particles in a salt-free medium is nowadays increasing.1-10 The term saltfree does not mean that there are not ions present in the suspension because those ions coming from the charge process of the colloidal particles, which are known as “added counterions”, will always dissolve into the supporting medium. This fact implies that the electric double layers that surround the colloidal particles are constituted by one single ionic species. In contrast with the situation of added electrolyte, there is a singular relationship between the surface potential and the surface charge density in salt-free suspensions of spherical particles.3,4 While for low surface charge density this relation is roughly linear, above a critical value of the surface charge density, the surface potential increases very slowly. This phenomenon is called counterion condensation and can considerably affect the macroscopical physical behavior, such as the electrokinetics, the rheology, and so forth, of these suspensions. The study of the Poisson-Boltzmann condensation effect in a cell model scheme was introduced in the 80s by Alexander et al.11 More recent efforts12-15 have explored it to a large extent, whereas its performance has been also checked against numerical simulations.16 The present treatment goes beyond previous work in including realistic chemistry in exhaustively deionized concentrated suspensions, although surface group dissociation not touched on here has been treated by Palberg et al.17 The thickness of the electric double layer is characterized by the Debye length κ-1, which, in the case of a salt-free suspension with only added counterions, is given by18 * To whom correspondence should be addressed. E-mail:
[email protected]. † Departamento de Fı´sica Aplicada II. ‡ Departamento de Fı´sica Aplicada I.
(1)
where e is the elementary electric charge, kB is Boltzmann′s constant, T is the absolute temperature, εr is the relative permittivity of the suspending medium, ε0 is the vacuum permittivity, a is the particle radius, σ is the particle surface charge density, φ is the volume fraction of particles, and njc and zc are the mean number density in the liquid medium and the valence of counterions. On the other hand, the majority of the real systems studied or used in industrial applications are aqueous and usually in contact with the atmosphere. As a consequence, there will be other ionic species in the liquid coming from the water dissociation and CO2 contamination that provide a more complicated picture of the EDL. We will call this kind of systems “realistic salt-free suspensions”, in the sense that they are deionized maximally and there is no other salt added during the preparation, although there are actually various ionic species in it. Here, we present a description of the EDL for the case of concentrated suspensions that have no ions different than (i) those stemming from the colloidal particles, which we will call added counterions, that counterbalance their surface charge, (ii) the H+ and OH- ions from water dissociation, and (iii) the ions produced by the atmospheric CO2 contamination. In particular, we have studied the influence of the surface charge density, the particle volume fraction, and the type of added counterions on the electric double-layer distributions. The theory is valid for arbitrary surface charge density and very low to moderate particle concentrations, 0 < φ < 0.5. First, we describe briefly the spherical cell model approach used in our model and the case of “pure salt-free suspensions”, where the only ions present in the system are the added counterions. The treatment will next be extended to realistic
10.1021/jp8027885 CCC: $40.75 2008 American Chemical Society Published on Web 09/03/2008
Electric Double Layer in Salt-Free Suspensions
J. Phys. Chem. B, Vol. 112, No. 38, 2008 11961 Applying the spherical symmetry of the problem and combining eqs 3 and 4, we get
zcenc(r) 1 d 2 dΨ(r) d2Ψ(r) 2 dΨ(r) r ) + (5) ) 2 dr 2 dr r dr εrε0 r dr
(
)
As the system is in thermodynamic equilibrium, it is assumed that the ionic density follows a Boltzmann distribution
[
nc(r) ) bcexp -
zceΨ(r) kBT
]
(6)
where bc is an unknown constant representing the ionic concentration where the equilibrium electric potential is zero. The zero level of the electric potential is arbitrary, and we can choose that it is reached at the outer cell boundary
Ψ(b) ) 0
Figure 1. The cell model.
salt-free suspensions, with the water dissociation and the atmospheric CO2 contamination described by local equilibrium reactions which, coupled to the integro-differential PoissonBoltzmann equation, allow us to find the correct ionic distributions and electric potential around the particles. Theory The Cell Model. In order to take into account the overlapping of adjacent electric double layers, a cell model is used. The cell model concept has been successfully applied to develop theoretical models for different electrokinetic phenomena in moderately concentrated colloidal suspensions of charged particles, such as static electrophoresis and electrical conductivity,19-22 sedimentation velocity and potential,23,24 dynamic electrophoresis,10,25-27 complex conductivity and dielectric response,28 electroviscous effect,29-31 and electroacoustic phenomena,32,33 to mention just a few. An excellent review on the use of the spherical cell approach has been recently written by Zholkovskij et al.34 According to this model (Figure 1), each spherical particle of radius a is surrounded by a concentric shell of the liquid medium, having an outer radius b such that the particle/cell volume ratio in the cell is equal to the particle volume fraction throughout the entire suspension,35 that is
φ)
3
[
zceΨ(r) zce d2Ψ(r) 2 dΨ(r) + bcexp ) 2 r dr εrε0 kBT dr
F(r) εrε0
F(r) ) zcenc(r)
(3) (4)
Equation 3 is Poisson’s equation, where F(r) is the electric charge density given by eq 4.
]
(8)
The electroneutrality of the cell implies that
Q ) 4πa2σ ) -4π
∫ab r2F(r)dr
) -4πezcbc
(
∫ab r2exp
-
)
zceΨ(r) dr kBT
(9)
where Q is the total charge on the particle surface. From the last equation, we can obtain the unknown constant bc
bc ) ezc
-a2σ zceΨ(r) b 2 r exp dr a kBT
)
(
∫
[
(2)
The basic assumption of the cell model is that the suspension properties can be derived from the study of a unique cell. By its own nature, the cell model is only applicable when the suspension is homogeneous and isotropic. Electric Double Layer: Only Added Counterions. Let us consider now a charged spherical particle of radius a and surface charge density σ immersed in a salt-free medium, with only the presence of the added counterions that have a valence zc. The axes of the spherical coordinate system (r, θ, φ) are fixed at the center of the particle. In the absence of any external field, the particle is surrounded by a spherically symmetrical charge distribution. A complete solution of the problem would require knowledge of the electric potential Ψ(r) and the number density nc(r), which are related by
∇2Ψ(r) ) -
and, consequently, nc(b) ) bc. The combination of eqs 5 and 6 leads to the well-known spherical Poisson-Boltzmann equation for the equilibrium electric potential
and, using eq 8
( ba )
(7)
d Ψ(r) 2 dΨ(r) + ) r dr dr2
εrε0
]
zceΨ(r) kBT (11) zceΨ(r) b 2 dr r exp a kBT
a2σ exp -
2
(10)
∫
(
)
The last expression is an integro-differential equation for the electric potential Ψ(r), and we only need to add the initial conditions of eq 7 and
dΨ dr
|
r)b
)0
(12)
coming from the electroneutrality condition of the cell and Gauss theorem to define the problem completely. It can be solved iteratively by using eqs 8 and 10. We choose an initial guess for the bc parameter, say bc(0) ) 0, and perform the numerical integration of eq 8 from the initial point r ) b at the cell surface to the particle surface r ) a. We obtain the solution Ψ(0)(r) ) 0, and then, we insert it into eq 10 to obtain the new value bc(1) ) -3σa2/(ezc(b3 - a3)), which will give us Ψ(1)(r) using eq 8 again. The numerical iterative process is repeated until the relative variation of the electric potential at the particle surface is lower than a prescribed order of magnitude. The iterations
11962 J. Phys. Chem. B, Vol. 112, No. 38, 2008
Ruiz-Reina and Carrique
converges fastly to the solution of eq 11 in the present case, but we have found that it is not so rapid when there are other ionic species dissolved in the cell, as in the cases of water dissociation, CO2 contamination, or added salt. Fortunately, there is another way of solving the problem that can be also used in those other cases. Multiplying eq 8 by r2, integrating from a to b, and using eq 9, we find that
( |
σ ) εrε0
b2 dΨ a2 dr
r)b
-
dΨ dr
| )
(13)
r)a
By using eq 12, it reduces to
dΨ dr
|
r)a
) -
σ εrε0
(14)
which is a result that could also be found by applying Gauss theorem to the outer side of the particle surface r ) a. We will use nondimensional variables, which are defined by
x)
r a
˜ (r) ) eΨ(r) Ψ kBT
b˜c )
e2a2 b εrε0kBT c ea σ˜ ) σ (15) εrε0kBT
and eq 8 is written as
g(x) ≡
˜ (x) ˜ (x) ˜ d2Ψ 2 dΨ + ) -zcb˜ce-zcΨ(x) 2 x dx dx
(16)
where in the last expression we have defined the function g(x). If we differentiate it, after a little algebra, it is possible to eliminate the unknown parameter b˜c and find that
˜ ′(x) ) 0 g′(x) + zcg(x)Ψ
(17)
where the prime stands for differentiation with respect to x. In ˜ (x), eq 17 is rewritten as terms of the electric potential Ψ
˜ ′′′(x) + 2 Ψ ˜ ′(x) + ˜ ′′(x) - 2 Ψ Ψ x x2 ˜ ′(x) Ψ ˜ ′′(x) + 2 Ψ ˜ ′(x) ) 0 (18) zcΨ x
(
)
This is a nonlinear third-order differential equation that needs three boundary conditions to completely specify the solution. They are provided by eqs 7, 12, and 14, which now read
˜ (h) ) 0 Ψ
˜ ′(h) ) 0 Ψ φ-1/3.
˜ ′(1) ) -σ˜ Ψ
(19)
where h ) (b/a) ) The electric state of the particle is specified at its surface by the last condition in eq 19. It can be easily demonstrated that solving the problem defined by eqs 18 and 19 is mathematically equivalent to finding the solution to the problem specified by eqs 7, 11, and 12, that is, that any function satisfying eqs 18 and 19 also satisfies eqs 7, 11, and 12, and vice versa. We have avoided the iterative process because eq 18, subject to boundary conditions in eq 19, can be solved numerically in one single step to obtain the electric potential. We use the mathematical application MATLAB with its built-in routines for this purpose. The boundary value problem solver used is a finite difference code that implements the three-stage Lobatto-Illa formula. This is a collocation formula, and the collocation polynomial provides a C1-continuous solution that is fourth order accurate uniformly in the functions domain.36 Mesh selection and error control are based on the residual of the continuous
Figure 2. Nondimensional surface potential against the volume fraction for different surface charge density values with only zc ) 1 added counterions present.
solution. The relative tolerance, which applies to all components of the residual vector, has been taken as equal to 10-6. ˜ (x) with the system of After we find the electric potential Ψ eqs 18 and 19, it is possible to obtain the parameter b˜c using eq 10 and the ionic number density of added counterions from eq 6. The integration in eq 10 has to be done numerically. However, if we evaluate eq 16 at x ) h, we find that
˜ ′′(h) Ψ b˜c ) zc
(20)
which is a more useful expression for b˜c. In all of the figures, the temperature T and the relative electric permittivity of the suspending liquid εr have been chosen as 298.16 K and 78.55 respectively. In Figure 2, we show the nondimensional electric potential at the particle surface (x ) 1) against the volume fraction in the case of zc ) 1 added positive counterions. The different curves correspond to different negative surface charge density values. It can be noticed that the electric potential at the surface decreases with the particle concentration at fixed charge density in all cases. When the volume fraction is increased, the room available for the counterions inside of the cell diminishes, and consequently, the shielding of the particle surface charge is improved, giving a lower value of the surface electric potential. As was expected, the surface potential always augments with the surface charge; in Figure 3, we can observe clearly this trend for various fixed values of the volume fraction. It is possible to distinguish two different regimes. Initially, there is a fast and roughly linear increase of the surface potential with the surface charge density, which is followed by a much slower growth at higher surface charge density values. This phenomenon is known as the counterion condensation effect: the surface potential turns gradually insensitive to the surface charge density in the case of highly charged colloidal particles. Finally, in Figure 4, we see that the higher the valence of the added counterions, the lower the surface potential in all cases due to the upgraded shielding of the particle charge with increasing valence. Electric Double Layer: Added Counterions and Water Dissociation. Let us consider now that, in addition to the added counterions stemming from the charging process, there are also H+ and OH- ions coming from water dissociation in the liquid medium. This will always occur in aqueous suspensions. We
Electric Double Layer in Salt-Free Suspensions
J. Phys. Chem. B, Vol. 112, No. 38, 2008 11963 temperature, 298.16 K, and K˜w is a nondimensional quantity defined by
K˜w )
( )
NAe2a2 2 K εrε0kBT w
(23)
with NA as Avogadro’s constant. The electroneutrality of the cell implies that
σ ˜ ) -b˜H+
Figure 3. Nondimensional surface potential against the surface charge density for different volume fraction values with only zc ) 1 added counterions present.
∫1h x2e-Ψ(x)dx + b˜OH ∫1h x2eΨ(x)dx ˜
˜
-
(24)
and sets the electrical state of the particle surface. The system of eqs 21 and 22, subjected to the initial ˜ (h) ) 0 and Ψ ˜ ′(h) ) 0, completely defines the conditions Ψ problem and could be solved iteratively using eq 24 because there are two unknown parameters b˜H+ and b˜OH-. However, again, there is a better way to solve the problem. Differentiating eq 21
˜ ′(x)(A(x) + B(x)) g′(x) ) Ψ
(25)
and after a little algebra, we find that
˜ ′(x) ) 2Ψ ˜ ′(x)A(x) g′(x) - g(x)Ψ ˜ ′(x) ) 2Ψ ˜ ′(x)B(x) g′(x) + g(x)Ψ
(26)
Multiplying both expressions and using eq 22
˜ ′2(x)(g2(x) + 4K˜w) g′2(x) ) Ψ
(27)
we have eliminated the unknown parameters b˜H+ and b˜OH-. Finally, taking the square root, we obtain
˜ ′(x)√g2(x) + 4K ˜w g′(x) ) (Ψ
(28)
where we have to eliminate the negative sign to ensure consistency with eq 25. In terms of the electric potential, eq 28 becomes Figure 4. Nondimensional surface potential against the volume fraction for different added counterion valences zc. The surface charge density is σ ) -0.2 µC/cm2.
can distinguish between two cases, (a) when the added counterions are H+ or OH- ions and (b) when they are of a different ionic species. The distinction is important because in the (a) case, the added counterions will enter in the equilibrium reaction equation for water dissociation, whereas in the (b) case, they do not. Case a. In this case, the nondimensional Poisson-Boltzmann equation takes the form
˜ ′′(x) + 2 Ψ ˜ ′(x) ) -b˜H+e-Ψ˜ (x) + g(x) ≡ Ψ x ˜ b˜OH-eΨ(x) ) -A(x) + B(x) (21) where we have utilized zH+ ) 1 and zOH- ) -1, and we have defined A(x) ) b˜H+ e-Ψ˜ (x) and B(x) ) b˜OH- eΨ˜ (x), which are positive functions in the entire domain [1,h]. Equation 21 is coupled with the equilibrium mass-action equation for water dissociation, which we assume to hold at any point in the liquid medium
[H+][OH-] ) Kw ⇒ A(x)B(x) ) K˜w
(22)
where the square brackets stand for the molar concentration, Kw ) 10-14 mol2/L2 is the water dissociation constant at room
˜ ′′′(x) + 2 Ψ ˜ ′(x)˜ ′′(x) - 2 Ψ Ψ x x2 2 ˜ ′′(x) + 2 Ψ ˜ ′(x) ˜ ′(x) + 4K˜w ) 0 Ψ Ψ x
(
)
(29)
The three boundary conditions for eq 29 are the same as those in the previous section, eq 19. Now, we have transformed the iterative problem defined by the systems of eqs 21, 22, and 24, with two unknown parameters, to other one, consisting of a unique third-order differential eq 29 that can be solved numerically in one single step. ˜ (x), the paramOnce we have found the electric potential Ψ eters b˜H+ and b˜OH- can be obtained from the systems of equations
b˜H+b˜OH- ) K˜w
˜ ′′(h) ) -b˜H+ + b˜OHΨ
(30)
In Figure 5, we compare the only added counterions case with the water dissociation case. It can be observed that, for highly concentrated suspensions, the curves generated by taking into account the presence of water dissociation ions (dashed lines) coincide with the corresponding ones in the case of pure salt-free conditions (solid lines). This is a direct consequence of the increase in added counterion concentration with the volume fraction; at a volume fraction of φ g 10-3, the added counterions mask completely the influence of the water dissociation ions. However, for φ e 10-3, the situation is very different. When we approximate to the dilute limit, the electric potential at the particle surface becomes approximately constant,
11964 J. Phys. Chem. B, Vol. 112, No. 38, 2008
Ruiz-Reina and Carrique
˜ ′′(x) + 2 Ψ ˜ ′(x) ) -zcb˜ce-zcΨ˜ (x) - b˜H+e-Ψ˜ (x) + g(x) ≡ Ψ x ˜ b˜OH-eΨ(x) ) C(x) - A(x) + B(x) (31) ˜ (x). The last equation is coupled again where C(x) ) -zcb˜ce-zcΨ with eq 22. The added counterions balance the overall charge on the particle surface
b˜c )
-σ˜ zc
(32)
∫
˜ (x) h 2 -zcΨ xe dx 1
which is the nondimensional form of eq 10, whereas the number of H+ and OH- ions must be equal due to the electroneutrality of the cell
b˜H+ Figure 5. Nondimensional surface potential against the volume fraction for different surface charge density values. Solid lines: only added counterions H+. Dashed lines: added counterions H+ and water dissociation H+ and OH- ions.
∫1h x2e-Ψ(x)dx ) b˜OH ∫1h x2eΨ(x)dx ˜
˜
-
(33)
Now, we differentiate eq 31 and rearrange terms
˜ ′(x) ) Ψ ˜ ′(x)(A(x) + B(x)) g′(x) + zcC(x)Ψ
(34)
and, after a little algebra, we find that
˜ ′(x)) - (g(x) - C(x))Ψ ˜ ′(x) ) 2Ψ ˜ ′(x)A(x) (g′(x) + zcC(x)Ψ ˜ ′(x)) + (g(x) - C(x))Ψ ˜ ′(x) ) 2Ψ ˜ ′(x)B(x) (g′(x) + zcC(x)Ψ (35) Multiplying both expressions and using eq 22
˜ ′(x) ) Ψ ˜ ′(x)√(g(x) - C(x))2 + 4K˜w g′(x) + zcC(x)Ψ (36) where we have taken the positive sign of the square root for consistency with eq 34. In terms of the electric potential, the last equation becomes
Figure 6. Nondimensional surface potential against the surface charge density for different volume fraction values with added counterions and water dissociation ions present.
in contrast with the growth noticed in a pure salt-free system. This behavior is the result of the contribution of the H+ counterions stemming from the water dissociation. Now, the existent H+ counterions in the diffuse layer have two different sources, the added counterions from the particle charge process and those from the water dissociation equilibrium. In the high φ region, the first ones dominate, while in the low φ region, the second ones do. The augmented concentration of counterions inside of the cell also explains the noted decrease of the surface potential in comparison with the case of only added counterions. The surface potential against surface charge curves for fixed φ values are presented in Figure 6. For the higher particle concentrations, the curves entirely match with the corresponding ones in Figure 3. In all cases, the trends are the same as those found previously under the only added counterions condition. However, for low volume fractions, all of the curves tend to collapse into a single one, in agreement with the plateaus of Figure 5. Case b. In this case, the nondimensional Poisson-Boltzmann equation is
˜ ′′′(x) + 2 Ψ ˜ ′(x) - Ψ ˜ ′(x) ˜ ′′(x) - 2 Ψ Ψ x x2 2 ˜ ′′(x) + 2 Ψ ˜ ′(x) + zcb˜ce-zcΨ˜ (x) + 4K˜w + Ψ x
[(
)
˜
]
z2c b˜ce-zcΨ(x) ) 0 (37) The three boundary conditions for this equation are given by eq 19. In this case, the iterative problem cannot be eliminated completely because the parameter b˜c appears in eq 37. We have to choose an initial guess for the bc parameter, say b˜c(0) ) 0, and perform the numerical integration of eq 37 from x ) h to the particle surface x ) 1 using boundary conditions of eq 19. ˜ (0)(x), and then, we insert it into eq 32 We obtain the solution Ψ ˜ (1)(x) using to obtain the new value b˜c(1), which will give us Ψ eq 37 again. The numerical iterative process is repeated until the relative variation of the electric potential at the particle surface is lower than a prescribed order of magnitude. Although it has been necessary to follow an iterative process to obtain the solution to eqs 32 and 37 with boundary conditions of eq 19, this procedure is much better than the original and equivalent iteration problem defined by eqs 31, 32, and 33 with initial ˜ (h) ) 0 and Ψ ˜ ′(h) ) 0. The improved convergency conditions Ψ and superior numerical efficiency that are obtained when using eqs 32 and 37 lie in the facts that (i) all of the intermediate ˜ (n)(x) of the iterative method have the correct slope solutions Ψ ˜ ′(1) ) -σ˜ at the particle surface and (ii) we only have to Ψ calculate one numerical integration in eq 32. This is not true if we use the original scheme because in that case, the slope at
Electric Double Layer in Salt-Free Suspensions
J. Phys. Chem. B, Vol. 112, No. 38, 2008 11965
K˜1 )
NAe2a2 K εrε0kBT 1
K˜2 )
NAe2a2 K εrε0kBT 2 N˜H2CO3 )
NAe2a2 [H CO ] (39) εrε0kBT 2 3
where all of the values are taken in S.I. units. We can distinguish between two cases, (a) when the added counterions are coincident with one of the ionic species in the system (H+, OH-, HCO3-, or CO3 2- ions) and (b) when they are of a different ionic species. In the (a) case, the added counterions will enter in one of the equilibrium dissociation equations, whereas in the (b) case, they do not. Case a. In this case, the nondimensional Poisson-Boltzmann equation takes the form
Figure 7. Nondimensional surface potential against the volume fraction for different surface charge density values. Solid lines: only added counterions H+. Dashed lines: case (a), added counterions H+ and water dissociation H+ and OH- ions. Dotted lines: case (b), added counterions different than H+ with zc ) 1 and water dissociation H+ and OH- ions.
the particle surface is not determined and we have to perform two numerical integrations in every step. In Figure 7, we test the influence of univalent noncommon added counterions (dotted lines). There is an additional decrease of the surface potential and a more extended influence of water dissociation to higher volume fraction values in comparison with the case of a common counterion (dashed lines). The explanation lies in the fact that now the added counterions do not enter into the water dissociation reaction. In this case, the reaction H2O a H+ + OH- cannot reduce the number of added counterions by acting to the left to reach local equilibrium. The consequence is that the total number of counterions is greater than if they all were H+ ions, causing a better shielding of the particle charge and, therefore, lowering the electric potential at the particle surface. Electric Double Layer: Added Counterions, Water Dissociation, and CO2 Contamination. Let us consider now that, in addition to the added counterions and the H+ and OH- ions coming from water dissociation, there are also present ions stemming from the atmospheric CO2 contamination in the liquid medium. This will always occur in aqueous suspensions in contact with the atmosphere; the CO2 gas diffused into the suspension combines with water molecules to form carbonic acid H2CO3, and then, the following dissociation reactions take place
H2CO3 a H+ + HCO3HCO3
+
aH +
CO32-
(38)
with equilibrium dissociation constants K1 ) 4.47 × 10-7 mol/L and K2 ) 4.67 × 10-11 mol/L at room temperature (25 °C), respectively. The concentration of H2CO3 molecules in water can be calculated from the solubility and the partial pressure of CO2 in standard air. For a temperature of 25 °C and an atmospheric pressure of 101300 Pa, the concentration of carbonic acid is approximately [H2CO3] ) 1.08 × 10-5 mol/L, depending its particular value on the local environmental conditions. The dissociation constants and this concentration are nondimensionalized as
˜ ′′(x) + 2 Ψ ˜ ′(x) ) -A(x) + B(x) + F(x) + G(x) g(x) ≡ Ψ x (40) where we have set zH+ ) 1, zOH- ) -1, zHCO3- ) -1, zCO32) -2 and we have defined A(x) ) b˜H+e-Ψ˜ (x), B(x) ) b˜OH-eΨ˜ (x), F(x) ) b˜HCO3-eΨ˜ (x), and G(x) ) 2b˜CO32-e2Ψ˜ (x), which are positive functions in the entire domain [1,h]. Equation 40 is coupled with the equilibrium mass-action equations
[H+][HCO3-] A(x)F(x) ) K˜1 ) K1 ⇒ [H2CO3] N˜H2CO3 [H+][CO32-] [HCO3-]
(41) ) K2 ⇒
A(x)G(x) ) 2K˜2 F(x)
and with eq (22), which we assume to hold at any point in the liquid medium. Using eqs 22, 40, and 41, we can eliminate the functions B(x), F(x), and G(x) in terms of A(x)
g(x) ) -A(x) +
) -A(x) +
˜ 1K˜2N˜H CO K˜1N˜H2CO3 2K K˜w 2 3 + + 2 A(x) A(x) A (x) E˜2 E˜1 + 2 A(x) A (x)
(42)
where we have defined the constants E˜1 ) K˜w + K˜1N˜H2CO3 and E˜2 ) 2K˜1K˜2N˜H2CO3. Differentiating eq 42
(
˜ ′(x) A(x) + g′(x) ) Ψ
E˜1 2E˜2 + 2 A(x) A (x)
)
(43)
and, after some algebra, we can find the function A(x) from eqs 42 and 43
˜ ′(x) - 3E˜2(g′(x) - 2g(x)Ψ ˜ ′(x)) 2E˜21Ψ A(x) ) ˜ ′(x)) - 9E˜2Ψ ˜ ′(x) E˜1(g′(x) + g(x)Ψ
(44)
Inserting the last expression into eq 42 or 43, we obtain a rather cumbersome nonlinear third-order differential equation for the ˜ (x) that will not be written here. We can electric potential Ψ solve it numerically with the boundary conditions of eq 19 and get the electric potential profile in one single step, avoiding any iterative procedure again. However, it is better to make use of an approximated equation by employing the fact that the constant E˜1 is about five orders of magnitude greater than the constant E˜2, and the final term in the righthand side of eq 42 can be neglected. This is equivalent to the omission of the
11966 J. Phys. Chem. B, Vol. 112, No. 38, 2008
Figure 8. Nondimensional surface potential against the volume fraction for different surface charge density values. Solid lines: only added counterions H+. Dashed lines: added counterions H+ with water dissociation ions. Dotted lines: added counterions H+ with water dissociation and atmospheric contamination ions.
second proton dissociation in the chemical reactions described by eq 38. The problem is now the same as that in the water dissociation case but with the constant K˜w replaced by the constant E˜1. Recalling eq 29, the approximated equation for the ˜ (x) in this case reads electric potential Ψ
˜ ′′′(x) + 2 Ψ ˜ ′(x) ˜ ′′(x) - 2 Ψ Ψ x x2
(Ψ˜ ′′(x) + 2x Ψ˜ ′(x))
˜ ′(x) Ψ
2
+ 4E˜1 ) 0 (45)
We have found that the electric potential profiles provided by this equation differ from the exact one by less than 10-4% in the worst case studied. In Figure 8, we compare the only added counterions, the water dissociation, and the water dissociation plus CO2 contamination cases. Again, we can observe that for highly concentraded suspensions, all of the curves coincide due to the predomination of added counterions over the water dissociation or CO2 contamination ions. The influence of the water dissociation plus
Ruiz-Reina and Carrique
Figure 10. Nondimensional surface potential against the volume fraction for different surface charge density values. Solid lines: only added counterions H+. Dashed lines: case (a), added counterions H+ with water dissociation and atmospheric contamination ions. Dotted lines: case (b), added counterions different than H+ with zc ) 1 and water dissociation and atmospheric contamination ions.
CO2 contamination extends from the very dilute limit to φ ≈ 10-2, where the surface potential is seriously diminished in comparison to the other cases and presents a plateau against the volume fraction. Now, the H+ counterions inside of the cell arise from three different sources, (i) the charge process of the colloidal particle, (ii) the water dissociation equilibrium, and (iii) the proton dissociation from carbonic acid. Consequently, there is a great increase of the concentration of counterions that accounts for the marked reduction of the surface potential in the low volume fraction region. Figure 9 shows the surface potential against the surface charge curves for fixed φ values in the case of water dissociation plus CO2 contamination. Once more, the curves completely coincide with the ones shown in Figures 3 and 6 for the more concentrated suspensions, and the trends are similar. The collapse of the curves for low volume fractions is now much more important than that in the absence of atmospheric contamination. Case b. We will adopt from the beginning the approximation made before. In this case, the nondimensional Poisson-Boltzmann equation is
˜ ′′(x) + 2 Ψ ˜ ′(x) ) C(x) - A(x) + B(x) + F(x) g(x) ≡ Ψ x (46) ˜ (x), A(x) ) b˜H+e-Ψ˜ (x), B(x) ) b˜OH-eΨ˜ (x), where C(x) ) -zcb˜ce-zcΨ ˜ Ψ (x) and F(x) ) b˜HCO3 e . The problem is the same as that in the water dissociation case with added counterions not coincident with one of the ionic species in the system. Looking at eq 37, we find the new equation for the electric potential
˜ ′′′(x) + 2 Ψ ˜ ′(x) - Ψ ˜ ′(x) ˜ ′′(x) - 2 Ψ Ψ x x2 2 ˜ ′′(x) + 2 Ψ ˜ ′(x) + zcb˜ce-zcΨ˜ (x) + 4E˜1 + Ψ x
[(
)
˜
]
z2c b˜ce-zcΨ(x) ) 0 (47) Figure 9. Nondimensional surface potential against the surface charge density for different volume fraction values in the case of added counterions H+ with water dissociation and atmospheric contamination ions.
The three boundary conditions for this equation are given by eq 19. The system is solved using the iterative procedure described before.
Electric Double Layer in Salt-Free Suspensions The influence of univalent noncommon added counterions (dotted lines) is illustrated in Figure 10. We can observe an extra reduction of the surface potential and a wider influence of CO2 contamination that reaches higher volume fraction values in comparison with the case of a common counterion (dashed lines). The reason for this behavior is found again in the fact that the noncommon added counterions do not take part in the water and carbonic acid dissociation reactions. These reactions cannot reduce the number of added counterions in their way to local equilibrium. The resulting shielding of the particle charge is improved in comparison with the common added counterions case, and therefore, the surface electric potential is diminished. Conclusions We have developed a model for the theoretical description of the electric double layer of realistic salt-free concentrated suspensions. It is made in the framework of the classical Poisson-Boltzmann theory, the spherical cell model approach, and the local equilibrium reactions. The model takes properly into account the water dissociation and CO2 contamination by a coupling between the Poisson-Boltzmann equation and the dissociation equations. A robust and efficient mathematical treatment for dealing with the resulting integro-differential equations has been conceived and tested in all of the cases. The numerical results show that the influence of the water dissociation in aqueous salt-free suspensions has to be considered for volume fractions lower than 10-2, whereas the atmospheric contamination, if the suspensions are in contact with air, is significant in the region of φ < 10-1. They also demonstrate that it is important to distinguish whether the added counterions coincide or not with one of the ionic species arising from the dissociation of water and carbonic acid. We think that the present work establishes the starting point for the development of future theories about the electrokinetics, rheology, and phase behavior of realistic salt-free concentrated suspensions. Acknowledgment. Financial support for this work by Ministerio de Educacio´n y Ciencia, Spain, Project FIS2007-62737 (cofinanced with FEDER funds by the European Union), is gratefully acknowledged. References and Notes (1) Medebach, M.; Palberg, T. J. Chem. Phys. 2003, 119, 3360. (2) Medebach, M.; Palberg, T. Colloids Surf. A 2003, 122, 175.
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