Diffuse sorption modeling

Journal of Colloid and Interface Science 332 (2009) 54–59

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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis

Diffuse sorption modeling Sergey Pivovarov Institute of Experimental Mineralogy, Russian Academy of Sciences, 142432 Chernogolovka, Moscow District, Russia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 30 September 2008 Accepted 27 November 2008 Available online 20 January 2009 Keywords: Gouy–Chapman model Diffuse double layer Poisson–Boltzmann equation Ferrihydrite Surface charge Adsorption Ionic exchange Gaines–Thomas selectivity coefficient

This work presents a simple solution for the diffuse double layer model, applicable to calculation of surface speciation as well as to simulation of ionic adsorption within the diffuse layer of solution in arbitrary salt media. Based on Poisson–Boltzmann equation, the Gaines–Thomas selectivity coefficient for 0.5 /Q M ){M+ }/{Me2+ }0.5 , (Q is the equivalent fraction uni–bivalent exchange on clay, K GT (Me2+ /M+ ) = (Q Me + of cation in the exchange capacity, and {M } and {Me2+ } are the ionic activities in solution) may be calculated as [surface charge, μeq/m2 ]/0.61. The obtained solution of the Poisson–Boltzmann equation was applied to calculation of ionic exchange on clays and to simulation of the surface charge of ferrihydrite in 0.01–6 M NaCl solutions. In addition, a new model of acid–base properties was developed. This model is based on assumption that the net proton charge is not located on the mathematical surface plane but diffusely distributed within the subsurface layer of the lattice. It is shown that the obtained solution of the Poisson–Boltzmann equation makes such calculations possible, and that this approach is more efficient than the original diffuse double layer model. © 2008 Elsevier Inc. All rights reserved.

1. Introduction

into a mathematical model and applied to description of surface charge curves of ferrihydrite in NaCl solutions.

The diffuse double layer model (DDLM) is very complex for many applications, e.g., for simulation of ionic exchange. Of course, exchange phenomena are somehow related to DDLM. Indeed, numerical integration of the Poisson–Boltzmann equation is efficient for simulation of exchange equilibria [1]. However, clay science says “Excellent!” and uses exchange constants. Similarly, surface complexation modeling says “Wonderful!” and considers sorption onto “specific” and “exchange” sites in the modeling of specific sorption on clay minerals. Thus, there is a reason to formulate a simpler solution for DDLM, and this work presents such a solution. One may note that surface chemistry is one of the most eclectic fields of knowledge. Surface complexation theory considers surface potential, whereas electrochemistry considers interfacial potential. In accordance with the constant capacitance model, as well as the DDLM, the surface potential is equal to the interfacial potential. However, surface complexation theories are useless for calculation of pH from the potential of a glass electrode. Obviously, surface potential differs from interfacial potential, but the relations of these variables are not known. The diffuse distribution of net surface charge and the corresponding potential difference between the surface and bulk solid were first postulated by Grimley and Mott [2] in 1947. Indeed, this idea completely explains the difference between the surface and interfacial potential. However, this idea still has no applications. In the present paper, this idea is developed

E-mail address: [email protected]. 0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.11.074

© 2008

Elsevier Inc. All rights reserved.

2. Experimental All experiments were performed in a 70-ml plastic vessel, equipped with a magnet stirrer. The combined pH electrode and gas inlet were inserted into the holes in the cap of the vessel. The third hole, reserved for injections of solution, was plugged between the additions of reagents. The gas space of the vessel was purified with CO2 -free air (air was first dispersed at the bottom of a 3-L bottle with 0.1 M NaOH and then allowed to pass through the bottle with distilled water). All reagents were analytical grade. For the experiment, 29.5 ml of 0.1 M NaOH solution were gently added to 10 ml of 0.1 M Fe(NO3 )3 solution under constant mixing and then the suspension was allowed to settle (t = 22–25 ◦ C). The resulting pH was about 5.5–6. The next day, the solution was gently replaced twice with distilled water to adjust the ionic strength to (exactly) 0.01 m NaNO3 . The final concentration of ferrihydrite was about 0.025 M as Fe, and the total volume was about 50 ml. Titration experiments were performed with ferrihydrite aged for 2 days, at a temperature of 25 ◦ C, under constant mixing, and under CO2 -free air. Initial pH was about 4.5–5. The first point was equilibrated for 2 h in order to remove CO2 . After base and acid titration in 0.01 M NaNO3 , a known amount of NaCl was added to the suspension (all salt additions were performed at pH 4.5–5), the suspension was equilibrated for 1 h, and titration was repeated. Each point was equilibrated for 10–15 min.

S. Pivovarov / Journal of Colloid and Interface Science 332 (2009) 54–59

55

The application of the surface potential function P to calculation of adsorption is more convenient, since it allows reducing the system of equations to a more compact form. Also, the surface potential function P may be considered similar to the activity of a chemical component. For instance, Eqs. (1) and (2) may be rewritten as follows:



S + H+ ⇔ SH+ (+ P ),



◦ KH = [SH+ ] P / [S][H+ ] ,

−

−

◦

S + OH (+ P ) ⇔ SOH ,



−

−

(6)



K OH = [SOH ]/ P [S][OH ] .

(7)

However, this is a matter of taste. Traditionally, the system of Eqs. (1)–(4) is solved by the use of a function complementary to Eq. (4) (see Ref. [1] or rearrange Eq. (4)): P 0.5 = 0.5[SCh]/0.61I 0.5 +

Fig. 1. Surface charge of ferrihydrite in NaCl solutions (sign plus denotes that each solution contains also 0.01 M NaNO3 ). Curves calculated with use of 2pK diffuse double layer model (see Table 1).

Prior to the experiment, the pH electrode was calibrated via acid titration of a similar solution without ferrihydrite. Solutions of 0.1 M HCl and NaOH were used for titration. The pH values were calculated from [H+ ] with the mean activity coefficient of HCl, i.e., assuming γH (in NaCl) ≈ γH (in HCl of the same molarity).



0.5[SCh]/0.61I 0.5

2

0.5 +1 ,

(8)

where [SCh] = [SH+ ] − [SOH− ], μmol/m2 . However, solution of Eq. (8) is impossible at zero ionic strength due to its requiring division by zero. This leads to extremely difficult calculations at low ionic strengths. Besides, this solution is not applicable to arbitrary salt mixtures. A much easier method is to rearrange the charge–potential relationship (Eq. (4)) into a charge balance equation. Based on the Gouy–Chapman theory, the charge of the diffuse layer, [D], may be calculated from the equation (see Ref. [1]):



  [D],

μeq/m2 = 0.61

0.5 . [Ionz ]( P −z − 1)

(9)

3. Results and discussion The pH of zero charge point was obtained at about 8.0. In general, the modulus of surface charge monotonically increases with salt concentration. It is interesting that the difference between titration curves in 1 M and 6 M NaCl is much smaller than that for lower ionic strengths. This is because the effect of 6 M ionic strength approaches the effect of infinite ionic strength (within the 2pK DDLM approach, the model curve at infinite ionic strength coincides with that of dissolved dibasic acid). In an acidic field, surface charge approaches 0.2 mol/mol Fe. It should be noted that at 0.025 M Fe the free proton concentration is negligible up to pH 4. In the 6 M NaCl solutions, the saturation effect is distinctive even without subtraction of free protons. In Fig. 1, the experimental data are compared with curves calculated in accordance with the 2pK DDLM [3]:

  [SH ]/ [S][H+ ]γH = K H◦ /P ,   ◦ [SOH− ]/ [S][OH− ]γOH = K OH P, +

(1) (2)

T S = [S] + [SH+ ] + [SOH− ],

(3)

[SH+ ] − [SOH− ],

(4)

μmol/m2 = 0.61I 0.5 { P 0.5 − 1/ P 0.5 }.

Here S is a site (which may be also designed as >FeOH◦ , >MOH◦ , etc.), SH+ is an adsorbed proton, and SOH− is an adsorbed hydroxyl ion, γH and γOH are activity coefficients of protons ◦ and K ◦ and hydroxyl ions in solution. K H OH are constants; 0.61 2 0.5 [{μmol/m }/{mol/L} ] is the Gouy–Chapman constant (0.61 = 106 (103 × 2R T εε0 )0.5 / F , where 106 μmol per mol and 103 L/m3 are scaling factors; R = 8.314 J/mol K is the gas constant; T is temperature, K; ε is the dielectric constant of water, 78.47 at 25 ◦ C; ε0 is the dielectric constant of free space, 8.854 × 10−12 F/m; F = 96485 C/mol is the Faraday constant; I is the concentration of NaCl (ionic strength), moles per liter; and P is the surface potential function. The surface potential function may be considered as the electrostatic activity of the surface, since it is related to the surface potential by the equation

ψs = R T ln( P )/ F = 59.16 log( P ) mV (at 25 ◦ C).

(5)

For instance, in (Na, Mg)(Cl, SO4 ) solution, the charge of the diffuse layer may be calculated as

  [D],



μeq/m2 = 0.61 [Cl− ]( P − 1) + [Na+ ](1/ P − 1)

0.5 

. (10) + SO24− ( P 2 − 1) + [Mg2+ ](1/ P 2 − 1)

The sign of the charge of diffuse layer is positive at P < 1 and negative at P > 1. If the balance equation includes both positive and negative terms, its solution fails due to requiring division by zero. To exclude this difficulty, one may consider [D] as a difference in adsorption of cations and anions in the diffuse layer:

 [D] = [DNa+ ] + 2[DMg2+ ] − [DCl− ] − 2 DSO24− .

(11)

First, Eq. (10) may be rewritten as

[D] = [D]2 /[D].

(12)

Multiplying the numerator and denominator by P 0.5 /( P − 1), and using [Na+ ] + 2[Mg2+ ] = [Cl− ] + 2[SO24− ], one may obtain

 μeq/m2 = 0.61 [Na+ ] P −0.5 + [Mg2+ ]( P −0.5 + P −1.5 )

[D],



SO24−

I eff =

+



 0.5  , − [Cl− ] P 0.5 − SO24− ( P 0.5 + P 1.5 ) / I eff 2+

P + N + [Mg 2+

N = [Na ] + 2[Mg

]/ P ,

−



] = [Cl ] + 2

SO24−



(13) (14) (15)

.

Here I eff is effective ionic strength, and N is the normality of the solution. As may be seen from Eqs. (11) and (13), adsorption of ions in the diffuse layer may be calculated from

[DNa+ ], 2+

[DMg

−

],

0.5 μmol/m2 = 0.61[Na+ ] P −0.5 / I eff , 2

μmol/m = (1/2)0.61[Mg 2

−

0.5

2+

]( P

0.5 / I eff ,

−0.5

(16)

+P

−1.5

0.5 )/ I eff ,

[DCl ], μmol/m = 0.61[Cl ] P



 0.5 DSO24− , μmol/m2 = (1/2)0.61 SO24− ( P 0.5 + P 1.5 )/ I eff .

(17) (18) (19)

This method of solution of the Poisson–Boltzmann equation may be easily extended to electrolytes containing triply charged,

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S. Pivovarov / Journal of Colloid and Interface Science 332 (2009) 54–59

quadruply charged, etc. ions. In general case, adsorption of ions in the diffuse layer may be calculated in accordance with

Table 1 2pK diffuse double layer model for ferrihydrite in (Na, Mg)(Cl, SO4 ) solutions.

[DMez+ ],

Adsorption of proton and hydroxide ion S + H+ ⇔ SH+ (+P) − S + OH (+P) ⇔ SOH−

0.5 μeq/m2 = 0.61[Mez+ ] P −0.5 f me / I eff ,

[DAnz− ],

0.5 μeq/m2 = 0.61[Anz− ] P 0.5 f an / I eff ,

f me = 1 +



(20) (21)

P −i ,

(22)

P i,

(23)

i

f an = 1 +

 i

  [Anz− ] g an , [Mez+ ] g me +  g me = z + 2 ( z − i ) P −i ,

I eff = 0.5

i

g an = | z| + 2



(24) (25)



 | z| − i P i .

(26)

i

Adsorption of ions in diffuse layer 0 .5 [DNa+ ], μmol/m2 = 0.61[Na+ ] P −0.5 / I eff 0 .5 2+ [DMg ], μmol/m2 = (0.61/2)[Mg2+ ]( P −0.5 + P −1.5 )/ I eff 0 .5 [DCl− ], μmol/m2 = 0.61[Cl− ] P 0.5 / I eff 0 .5 [DSO24− ], μmol/m2 = (0.61/2)[SO24− ]( P 0.5 + P 1.5 )/ I eff I eff = [SO24− ] P + N + [Mg2+ ]/ P N = [Na+ ] + 2[Mg2+ ] = [Cl− ] + 2[SO24− ] Mass balance equations for surface species TS = [S] + [SH+ ] + [SOH− ] [SH+ ] + [DNa+ ] + 2[DMg2+ ] = [SOH− ] + [DCl− ] + 2[DSO24− ] Model parameters BET = 600 m2 /g FeOOHa MFeOOH = 89 g/mol TS = 0.2 mol/mol Fe = 0.2TFe mol/La

Here i is the sum from i = 1 to i = | z| − 1, or zero if | z| = 1. It should be noted that the value [DIon] is an “effective” but not a real one. For any ion, the value [DIon] is always positive, and this is convenient for calculations. However, the real adsorption of ions within the diffuse layer may be positive as well as negative. At the zero charge point, real adsorption of any ion in the diffuse layer is zero. Thus, as follows from Eqs. (16)–(19), real adsorption (i.e., enrichment or depletion of the diffuse layer with certain ion) may be calculated from [DIon]real = [DIon] − [DIon]neg ,

[DIon]neg ,

TIon + [DIon]neg = [DIon] + [Ion].

(29)

In the majority of cases, the effects caused by negative adsorption are negligible, and the difference between [DIon] and [DIon]real may be neglected. The system of equations that was used for solution of the 2pK diffuse double layer model is presented in Table 1. For practical calculations, reactions of protons and hydroxide ion adsorption (see the first two lines in Table 1) should be converted to





◦ ◦ ◦ log K ◦ = pHzpc = 0.5 log K H − log K OH + pK w , +

(30)

−

2S + H2 O ⇔ SH + SOH , ◦ ◦ ◦ log K self-ionization = log K H◦ + log K OH − pK w .

(31)

The algorithm for calculation of surface charge at a given pH, T Fe, [Na+ ], [Cl− ], [Mg2+ ], [SO24− ], and model parameters pHzpc = 8.11, K self-ionization = 10−1.64 , BET surface area 600 m2 /g FeOOH, and site density 3.75 μmol/m2 is as follows: Calculate α = 0.61 × 10−6 × 600 m2 /g × 89 g/mol × T Fe = 0.033TFe; T S = 3.75 × 10−6 × 600 m2 /g × 89 g/mol × T Fe = 0.2T Fe; N = [Na+ ] + 2[Mg2+ ]. Define [S], [SH+ ], and [SOH− ] (arbitrary). Cycle:

Values taken from Dzombak and Morel [3].

• calculate 

P = 10(pHzpc−pH) [SOH− ]/[SH+ ] I eff =



SO24−



0.5

;

2+

P + [Mg ]/ P + N ;  0.5  − 0.5 2−  0.5 ; [D ] = α [Cl ] P + SO4 ( P + P 1.5 ) / I eff  0.5  + −0.5 + 2+ −0.5 −1.5 + [Mg ]( P +P ) / I eff ; [D ] = α [Na ] P   +  +  + − − + [SH ]1 = [SOH ] + [D ] [SH ]/ [SH ] + [D ] ;   [S]1 = T S[S]/ [S] + [SH+ ] + [SOH− ] ; −

(28)

Here [DIon]real is real adsorption of the ion, [DIon]neg is maximum negative adsorption of the ion, [Ion] is its concentration in solution (molarity), and I is molar ionic strength. The contribution of negative adsorption to the charge of the diffuse layer is zero in all cases; i.e., z[DIonz ]neg = 0. Nevertheless, at very high solidto-water ratios, the composition of background electrolyte may be affected by the negative adsorption in accordance with the mass balance equation (all terms in moles per liter):

0.5SOH− + H+ ⇔ 0.5SH+ + 0.5H2 O(+ P ),

a

(27)

μmol/m2 = 0.61[Ion]/ I 0.5 .

log K ◦ 7.29a 5.07a

[SOH− ]1 = K self-ionization [S]2 /[SH+ ]; • compare [S] and [S]1 , [SH+ ] and [SH+ ]1 , [SOH− ] and [SOH]1 , • define new approaches for [SH+ ], [S], [SOH− ] (e.g. [S] = ([S] × [S]1 )0.5 ), • if necessary, repeat cycle.

Calculate [SH+ ] − [SOH− ], moles per liter. Practically, the total volume of calculations in accordance with Table 1 decreases by 20–100 times and more, as compared with traditional solution (Eqs. (1), (2), (3), (8)). Also, this method is applicable for arbitrary salt mixtures. Another advantage of this method is the possibility of calculating ionic adsorption within the diffuse layer. For instance, the replacement of Na+ by Mg2+ may be calculated with the Gaines–Thomas selectivity constant [4]:



2

K GT (Mg/Na)

  +2 2 = Q Mg / Q Na [Na ] /[Mg2+ ].

(32)

Here K GT (Mg/Na) is the constant of replacement of Na+ by Mg2+ (in designations of Gaines and Thomas), Q Mg and Q Na are equivalent fractions of magnesium and sodium in the exchange capacity ( Q Mg + Q Na = 1). Assuming Q Na ≈ [DNa+ ]/{[DNa+ ] + 2[DMg2+ ]}, and Q Mg ≈ 2[DMg2+ ]/{[DNa+ ] + 2[DMg2+ ]}, with Eqs. (16), (17) one may obtain the following equality:



2

K GT (Mg/Na)

≈ [Na+ ](1 + 1/ P ) + [Mg2+ ](1 + 1/ P )2 .

(33)

Now, by comparing Eqs. (33) and (10), one may obtain at a substantially negative surface charge ( P  1) K GT (Mg/Na) ≈ [ D , μeq/m2 ]/0.61.

(34)

Since the charge of the diffuse layer is equal by modulus to the surface charge, one may estimate the exchange constants K GT for some clay minerals. For example, the absolute values of permanent surface charge of ideal smectite and ideal illite are equal to 1.15 and 3 μeq/m2 [5]. Thus, the theoretical constants K GT (Mg/Na) for

S. Pivovarov / Journal of Colloid and Interface Science 332 (2009) 54–59

57

The algorithm of DDLM calculations of ionic exchange at given ionic activities {Na+ }, {Mg2+ }, and {ClO− 4 } and surface charge [SCh− , μeq/m2 ] is as follows. + Calculate I ac = 0.5({Na+ } + 4{Mg2+ } + {ClO− 4 }); define [DNa ] = − [SCh ]; I a = I ac . Cycle:

• calculate: 



P = 0.612 {Na+ }2 / I a [DNa+ ]2 , I a = {Na+ }/2 + {Mg2+ }(1 + 1/ P ) + {ClO− 4 }/2, − 0.5 , [DClO− 4 ] = 0.61{ClO4 }{ P / I a }

[DMg2+ ] = (1/2)0.61{Mg2+ }(1/ P 0.5 + 1/ P 1.5 )/ I a0.5 ,     + + 2+ ] , [DNa+ ]1 = [SCh− ] + [DClO− 4 ] [DNa ]/ [DNa ] + 2[DMg

Fig. 2. Diffuse double layer simulations (solid curves; Eqs. (35)–(38)) of experimental data on Mg/Na exchange onto illite (solid symbols; data from Sposito and LeVesque, [7]) and montmorillonite (open symbols; data from Sposito et al. [8]). Best fit values of surface charge 2.0 and 0.9 μeq/m2 , correspondingly. Dashed curves are Gaines– Thomas simulations calculated for the same values of surface charge in accordance with Eq. (34).

these minerals are 1.9 and 4.9. In general, these estimates are close to the experimental ones. There is a circumstance that should be taken into account. The Gaines–Thomas selectivity coefficient is constant if corrected for activity coefficients of ions in solution (e.g., see Ref. [6]). In general, Eq. (32) should contain the activities of sodium and magnesium in solution. Thus, a similar correction should be introduced into the DDLM. However, there is a difficulty. On theoretical grounds (in reality, this is a question!), equivalent activity of cations in solution may differ from equivalent activity of anions. Because of this, after correction for ionic strength, Eq. (15) loses validity. Nevertheless, the uncertainty caused by this fact is small, and one may consider activity-corrected effective ionic strength:





I a = γSO4 SO24− ( P + 1) + γCl [Cl− ]/2 + γNa [Na+ ]/2

+ γMg [Mg2+ ](1 + 1/ P ).

(35)

Using this variable, the adsorption of ions in the diffuse layer may be calculated from

[DNa+ ], [DMg2+ ],

μmol/m2 ≈ 0.61γNa [Na+ ] P −0.5 / I a0.5 ,

(36)

μmol/m2 ≈ (1/2)0.61γMg [Mg2+ ]( P −0.5 + P −1.5 )/ I a0.5 , (37)

[DCl− ], μmol/m2 ≈ 0.61γCl [Cl− ] P 0.5 / I a0.5 , (38)



 2− 2 − DSO4 , μmol/m2 ≈ (1/2)0.61γSO4 SO4 ( P 0.5 + P 1.5 )/ I a0.5 . (39) In Fig. 2, the experimental exchange curves [7,8] are compared with calculations (solid curves) performed in accordance with Eqs. (35)–(38). The best-fit values of the surface charge [SCh− ] of montmorillonite and illite are 0.9 and 2.0 μeq/m2 , correspondingly, i.e., approximately 1.5 times lower than for “ideal” ones. As can be seen, the selectivity of clay for bivalent metal really increases with surface charge. The dashed curves are Gaines–Thomas simulations performed with selectivity coefficients, calculated in accordance with Eq. (34) using the same values of surface charge (0.9 and 2 μeq/m2 ). As may be seen, the DDLM simulations do not coincide with the Gaines–Thomas ones (note that Eq. (34) is an approximate equality), but the difference decreases with surface charge.

• compare [DNa+ ]1 and [DNa+ ], • define new approach for [DNa+ ] as {[DNa+ ][DNa+ ]1 }0.5 , • if necessary, repeat cycle. Calculate: 0.5 [DNa+ ]real = [DNa+ ] − 0.61{Na+ }/ I ac , 0.5 , [DMg2+ ]real = [DMg2+ ] − 0.61{Mg2+ }/ I ac   2+ + 2+ Q Mg = 2[DMg ]real / [DNa ]real + 2[DMg ]real .

Now let us consider another application of the obtained solution of the Poisson–Boltzmann equation. As may be seen in Fig. 1, the 2pK DDL model of Dzombak and Morel [3] significantly overestimates the surface charge of ferrihydrite in an acidic field. Thus, the good fitting results of these authors [3] are due to the small range of surface charge covered by the experiments. The present experimental data show that the acid–base properties of amorphous ferric oxide are similar to these known for crystalline oxides. Thus, ferrihydrite is not an exclusion, for which the 2pK DDL model is well applicable. The slope of model curves may be adjusted with the constant capacitance model. This method is well known and applied in a huge number of publications. Let us consider another way. One may suggest that the oxide lattice contains the ions LH+ and LOH− . In the case of ferric hydroxide, these species may be con− sidered as (FeOOH)n Fe(OH)+ 2 and (FeOOH)n FeO2 . As is known from electrochemistry, for some special glasses and ceramics, the interfacial (Nernstian) potential is

ϕN = (2.3R T / F )(pHzpc − pH).

(40)

In the bulk lattice, [LH+ ]bulk = [LOH− ]bulk = [LI] (“lattice ionic strength”), whereas near the surface these species should be distributed in accordance with the Boltzmann law:

  [LH+ ] = [LI] exp F (ϕ N − ϕ )/ R T ,   [LOH− ] = [LI] exp − F (ϕ N − ϕ )/ R T .

(41) (42)

Thus, the surface charge may be calculated from the Gouy– Chapman equation:

  [SCh], μeq/m2 = 0.61[LI]e0.5 exp(Φs /2) − exp(−Φs /2) ,      Φs = F (ϕ N − ϕs )/ R T = ln [H+ ]/ [H+ ]zpc P .

(43) (44)

Here [LI]e = [LI](ε L /ε w ) is the effective ionic strength of the lattice, ε L is the dielectric constant of the lattice, ε w is the dielectric constant of water, and ϕs is the surface potential. Within the DDL approach, the density of ions in the diffuse layer (as moles per dm3 ) may increase up to infinity. If we consider the real ions, their concentrations should be realistic. For

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S. Pivovarov / Journal of Colloid and Interface Science 332 (2009) 54–59

example, the lattice of goethite contains 49 mol FeOOH/dm3 . Thus, the maximum charge density in the subsurface layer of the lattice is not greater. If the element of the lattice is (FeOOH)4 , the maximum charge density is about 12 moles per dm3 . Similar effects may be expected on the solution side. For instance, the number of molecules in water is 55.5 mol per dm3 , and this is about the maximum charge density for completely dehydrated counterions. If each counterion is hydrated with four water molecules, maximum charge density is about 11 moles per dm3 . Let us suggest that the neutral element of the ferrihydrite lattice (FeOOH)n dissociates in accordance with the scheme 2L ⇔ LH+ + LOH− .

(45)

Thus there is also the mass balance relation TL = [L] + [LH+ ] + [LOH− ]. The Poisson equation is

(46)





d2 ϕ /dx2 = −ρ /ε L ε0 = −{ F /ε L ε0 } [LH+ ] − [LOH− ] .

(47)

Fig. 3. Surface charge of ferrihydrite in NaCl solutions, simulated with use of Eqs. (58)–(64). Insert: the same data and curves in range of pH 0–8.

(48)

ionic strengths. The more convenient method is transformation of Eq. (57) into

From the Boltzmann equation and Eq. (46),

    [LH+ ]/[L] = [LI]/[L]bulk exp F (ϕ N − ϕ )/ R T ,     [LOH− ]/[L] = [LI]/[L]bulk exp − F (ϕ N − ϕ )/ R T , +

(49)

−

[L] = TL − [LH ] − [LOH ],

(50)

[L]bulk = TL − [LH+ ]bulk − [LOH− ]bulk = TL − 2[LI].

(51)

Using Eq. (46) and applying [LH+ ]/[LOH− ] = exp(2F (ϕ N − ϕ )/ R T ) (from Eqs. (48) and (49)), one may obtain

  [LH+ ] = TLexp(Φ)/ b + exp(Φ) + exp(−Φ) ,   [LOH− ] = TLexp(−Φ)/ b + exp(Φ) + exp(−Φ) .

(52) (53)

Here Φ = F (ϕ N − ϕ )/ R T and b = [L]bulk /[LI] = {TL − 2[LI]}/[LI]. Thus, from Eqs. (47), (52), (53),





d2 ϕ /dx2 = −{ F /ε L ε0 }TL exp(Φ) − exp(−Φ)

  / b + exp(Φ) + exp(−Φ) .

(54)

Multiplying both sides by 2dϕ (note that 2(dϕ /dx) d(dϕ /dx) = d(dϕ /dx)2 , and dΦ = d{ F (ϕ N − ϕ )/ R T } = −{ F / R T } dϕ ), one may obtain

(dϕ /dx)2 = a





+

  / b + exp(Φ) + exp(−Φ) dΦ    = a ln b + exp(Φ) + exp(−Φ) /{b + 2} .

(55)

Here a = {2R T /ε L ε0 }TL, and b = {TL − 2[LI]}/[LI]}. Since (ε L ε0 ) dϕ / dx at the interface is surface charge σ [C/m2 ], the latter may be calculated from





σ 2 = TL(2R T εL ε0 ) ln b + exp(Φs ) + exp(−Φs ) /(b + 2) . Applying known constants, and replacing exp(Φs ) P , one may obtain

 [SCh, μeq/m2 ]2 = 0.612 TLe ln b + 10(pHzpc−pH) / P   + P /10(pHzpc−pH) /(2 + b) .

(56)

by 10(pHzpc−pH) /

(57)

Here TLe = TL{ε L /ε w } is the maximum effective charge density of the lattice (moles per dm3 ), and TL is the real one. If TL is close to infinity, b is close to TL/[LI], and Eq. (57) may be reduced to Eq. (43). For practical calculations, one may use the function complementary to Eq. (57), but this method is not stable at low

2



(pHzpc−pH)

0.5

[SH , μmol/m ] = 0.61β [LI]e 10 /P ,   − 2 (pHzpc−pH) 0.5 [SOH , μmol/m ] = 0.61β [LI]e P /10 ,    0.5  β = a/ exp(a) − 1 at a < 0.1, β ≈ exp{−a/4} ,  2 a = [SCh, μeq/m2 ]/0.61 /TLe .

(58) (59) (60) (61) (62)

Here [SH+ ] and [SOH− ] are integer sums of [LH+ ]–[LI] and [LOH− ]–[LI] from the interface to the bulk lattice (plus the value μmol/m2 ), and β is of maximum negative adsorption, 0.61[LI]0.5 e the saturation factor, which ranges from 0 to 1. If to assume that the maximum concentration of ions within the diffuse layer of the solution is equal to that in the lattice, the saturation factor for counterions is equal to that for adsorbed protons and hydroxide ions. Thus, adsorption of sodium and chloride ions from NaCl solution may be calculated ({Na+ } and {Cl− } are ionic activities) from:

[DNa+ ], [DCl− ],

exp(Φ) − exp(−Φ)



[SCh] = [SH+ ] − [SOH− ],



μmol/m2 = 0.61β {Na+ }/ P



μmol/m2 = 0.61β {Cl− } P

0.5

0.5

.

,

(63) (64)

In Fig. 3, the experimental data are compared with model curves calculated in accordance with Eqs. (58)–(64). As may be seen, the effect of diffuse distribution of net proton charge is similar to that of incorporation of the constant capacitance model into DDLM: the model curves in Fig. 3 are more gentle than those in Fig. 1. This similarity is conditioned by the fact that the diffuse charge of the lattice is located, on the average, at some distance from the interface (which is head of diffuse layer in solution). The optimum value for maximum effective charge density, TLe = TL{ε L /ε w }, was estimated to be 9 moles per dm3 . In general, this value is more or less realistic. There is no significant difference, if saturation effects for the diffuse layer in solution are neglected, but maximum effective charge density drops to 6 moles per dm3 . As may be seen, the model curves look as if there is saturation of the surface, although the number of protonatable sites in the lattice is infinite. In accordance with the model, surface charge is not limited by some maximum value, but it increases slowly, as the concentration of ions at the interface (as moles per dm3 ) approaches the maximum value. Further increase of surface charge becomes slower and slower, as the saturated zone expands from the interface into the bulk lattice and bulk solution (see insert

S. Pivovarov / Journal of Colloid and Interface Science 332 (2009) 54–59

in Fig. 3). The applicable values of effective lattice ionic strength [LI]e = [LI]{εL /ε w } are these in the range 2–4.5 moles per dm3 . The model curves were calculated with the latter value (i.e., the lattice completely dissociates into LH+ and LOH− ). The algorithm for calculation of surface charge at a given pH, TFe, ionic activities {Na+ } and {Cl− }, and model parameters pHzpc 8.0, [LI]e = 4.5 M, [TL]e = 9 M, and BET surface area 600 m2 /g FeOOH is as follows: Calculate α = 0.61 × 10−6 × 600 m2 /g × 89 g/mol × TFe = 0.033TFe. Define [SH+ ], [SOH− ] (arbitrary), β = 1. Cycle:

• calculate 



P = 10(pHzpc−pH) [SOH− ]/[SH+ ] ,

 0.5 ; [DCl− ] = α β {Cl− } P   0.5 ; [DNa+ ] = α β {Na+ }/ P     − [SH+ ]1 = [SOH− ] + [DCl ] [SH+ ]/ [SH+ ] + [DNa+ ] ;   2 a = [SH+ ] − [SOH− ] /α /TLe , if a  0.1 then β1 = exp(−a/4),



0.5

if a > 0.1 then β1 = a/(exp(a) − 1) −

2

,

+

[SOH ]1 = (α β) [LI]e /[SH ]; • compare [SH+ ] and [SH+ ]1 , β and β1 , [SOH− ] and [SOH− ]1 , • define new approaches for [SH+ ], β , and [SOH− ] (e.g., β = (β × β1 )0.5 ), • if necessary, repeat cycle. Calculate [SH+ ] − [SOH− ], moles per liter. As may be seen, the present method of solution of the Poisson– Boltzmann equation allows consideration of the diffuse distribution of net proton charge. Even if this abstract mathematical model has no relation to reality at all, the ability to account for such effects seems to be important. Besides, there is one interesting moment. The effective lattice ionic strength, 4.5 M, corresponds to the effective Debye length, 1.4{ε L /ε w }0.5 Å, which is close to the radius of the oxygen ion. This means that the diffuse distribution of net

59

surface charge and corresponding changes of potential are limited, in the main, by one atomic layer. So, if other specifically adsorbed ions are located within the monolayer with a thickness of about atomic size (2–3 Å), the potential gradient should cause similar diffuse distribution of adsorbed species. 4. Summary Thus, it is possible to calculate adsorption of any kind of ion in a diffuse layer. Numerical integration of the Poisson–Boltzmann equation is not the only way. There are also some reasons to suspect diffuse distribution for all surface species. It is likely that the location of adsorbed species within the mathematical plane is a very rough approach. What is the difference between diffuse distribution within the monolayer and location within the mathematical plane? If all ions are located within the mathematical plane, the selectivity of the interface for multivalent ions (prior to univalent ones) is not dependent on the surface charge. In reality, selectivity of the interface for multivalent ions increases with surface charge (e.g., see data on 2− HCrO− 4 /SO4 exchange on ferrihydrite, Ref. [9]), and this is consistent with Eq. (34). This feature of specific adsorption may be simulated with a multilayer constant capacitance approach (e.g., [5]), or, almost the same, a charge distribution concept [10]. However, it is possible that the real cause is the diffuse character of specific adsorption. References [1] C.A.J. Appelo, D. Postma, Geochemistry, Groundwater and Pollution, second ed., A.A. Balkema Publishers, London, 2005. [2] T.B. Grimley, N.F. Mott, Discuss. Faraday Soc. 1 (1947) 3. [3] D.A. Dzombak, F.M.M. Morel, Surface Complexation Modeling: Hydrous Ferric Oxide, Wiley, New York, 1990. [4] G.L. Gaines, H.C. Thomas, J. Chem. Phys. 21 (1953) 714. [5] S. Pivovarov, in: A. Hubbard, P. Somasundaran (Eds.), Encyclopedia of Surface and Colloid Science, second ed., Taylor & Francis, New York, 2006, p. 4617. [6] W.J. Bond, I.R. Phillips, Soil Sci. Soc. Am. J. 54 (1990) 722. [7] G. Sposito, C.S. LeVesque, Soil Sci. Soc. Am. J. 49 (1985) 1153. [8] G. Sposito, K.H. Holtzclaw, L. Charlet, C. Jouany, A.L. Page, Soil Sci. Soc. Am. J. 47 (1983) 51. [9] J.M. Zachara, D.C. Girvin, R.L. Schmidt, C.T. Resch, Environ. Sci. Technol. 21 (1987) 589. [10] T. Hiemstra, W.H. van Riemsdijk, J. Colloid Interface Sci. 179 (1996) 488.