A generalized model for the stability of polymer colloids

Journal of Colloid and Interface Science 302 (2006) 187–202 www.elsevier.com/locate/jcis

A generalized model for the stability of polymer colloids Zichen Jia a , Cornelius Gauer a , Hua Wu a , Massimo Morbidelli a,∗ , Alba Chittofrati b , Marco Apostolo b a Institute for Chemical and Bioengineering, ETH Zurich, Hönggerberg HCI, 8093 Zurich, Switzerland b Solvay Solexis SpA, Viale Lombardia 20, 20021 Bollate (MI), Italy

Received 4 May 2006; accepted 8 June 2006 Available online 25 July 2006

Abstract A generalized model has been proposed to describe the stability of polymer colloids stabilized with ionic surfactants by accounting simultaneously for the interactions among three important physicochemical processes: colloidal interactions, surfactant adsorption equilibrium, and association equilibria of surface charge groups with counterions at the particle–liquid interface. A few Fuchs stability ratio values, determined experimentally for various salt types and concentrations through measurements of the doublet formation kinetics, are used to estimate the model parameters, such as the surfactant adsorption and counterion association parameters. With the estimated model parameters, the generalized model allows one to monitor the dynamics of surfactant partitioning between the particle surface and the disperse medium, to analyze the variation of surface charge density and potential as a function of the electrolyte type and concentration, and to predict the critical coagulant concentration for fast coagulation. Three fluorinated polymer colloids, stabilized by perfluoropolyether-based carboxylate surfactant, have been used to demonstrate the feasibility of the proposed colloidal stability model. © 2006 Elsevier Inc. All rights reserved. Keywords: Colloidal stability; Fuchs stability ratio; Surface charge; Surfactant adsorption; Counterion association; Hydration force

1. Introduction Polymer latices produced by emulsion polymerization are typical colloidal suspensions with submicrometer particles. The kinetic stability of these systems is usually achieved through adsorption of ionic surfactants onto the particle surface, whose electrical charges create an electrical double layer (EDL), leading to electrostatic repulsive interactions between particles. The characterization and control of the stability of polymer colloids are of great industrial importance for both their manufacture and posttreatment (e.g., coagulation of the particles for separation of the polymer from the disperse medium). For this, it would be desirable to develop a model that can predict the effect of operating variables such as the type and concentration of electrolyte and the solution pH on the latex stability, i.e., on the Fuchs stability ratio, W .

* Corresponding author. Fax: +41 44 6321082.

E-mail address: [email protected] (M. Morbidelli). 0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.06.011

The state of the art in describing the stability of colloidal systems has been centered on how to correctly describe the colloidal interactions. The centerpiece of modeling the colloidal interactions has been the DLVO (Deryaguin–Landau–Verwey– Overbeek) theory [1,2], which accounts for the competing effects between van der Waals attraction and EDL repulsion. Additional interaction forces such as long-range dispersion forces, short-range hydration forces, and capillary condensation are known to be important under certain conditions [3–6], and need to be considered specifically. However, unlike the DLVO interactions, such non-DLVO interactions are very difficult to either measure experimentally or predict theoretically, particularly in the case of complex colloidal systems, as industrial polymer latices often are. Once a proper colloidal interaction model has been chosen, what one often proceeds to do in describing the stability of a colloidal system is as follows: first measure the surface charge or potential at properly defined conditions, and then use the measured surface charge or potential in the chosen colloidal interaction model, accounting for the ionic strength present in

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the system, to compute the interaction energy barrier or use the Fuchs stability integral to calculate the Fuchs stability ratio W . Such an approach may correctly characterize the stability of the colloidal system under the exactly given conditions (i.e., at the given ionic strength, ion types, particle concentration, etc.). However, the problem is that due to the needs of practical applications, one often has to extend such obtained results to describe the stability of the same colloid at different ionic strengths, ion types, and particle concentrations by simply changing the ionic strength and ion type (valency) in the colloidal interaction model. Obviously, this often fails because this procedure has ignored the interactions among various physicochemical processes (e.g., colloidal interactions, association equilibria of surface charge groups with counterions, surfactant adsorption equilibrium). In fact, different ionic strengths and different ions have different association behavior with the surface charge groups, leading to changes in the surface charge and potential. This consequently leads to changes in the distributions of ions and surfactants in the system (the Boltzmann effect), which are then coupled back again with the changes in the counterion association of the surface charge groups. Such problems become even more severe in the case of industrial colloidal systems, where the surfactant systems are often very complex and the surface charge is difficult to measure correctly. Based on the practical problems mentioned above in describing the stability of colloidal systems, we have in this work proposed to describe the colloidal stability by putting together all the important physicochemical processes (such as surfactant adsorption equilibrium, association equilibria of the surface charge groups with counterions, colloidal interaction model) in a single model so that their coupling interactions can be simultaneously accounted for and correctly described. Such a model is referred here to as a generalized model for describing the colloidal stability. It is evident that including so many physicochemical processes in one model leads to many model parameters such as surface charge or potential, surfactant adsorption parameters, and association equilibrium parameters of surface charge groups with counterions. To estimate these parameters, we have considered two aspects. First, if values of any parameters are reported in the literature and applicable, we use directly these values in our model. Second, for those parameters that cannot be found in the literature and also are difficult to measure experimentally, we estimate their values using a few Fuchs stability ratio values W , determined experimentally for various salt types and concentrations through measurement of the doublet formation kinetics. The estimated parameters are then compared with those reported in the literature for systems that are close to ours in order to assess their reliability. The paper is organized as follows. In Section 2 we describe all necessary details of each process, based on the literature sources, in order for the reader to understand how we treat each process and where the parameters come from. In the following two sections, as an illustrative example, we demonstrate the application of the model to three fluorinated latices for which limited information is available (for example, the surfactant adsorption isotherm is not known). It will be seen that with the

estimated parameters, the developed model is able to predict the colloidal stability behavior of the latices, which includes, for example, the stability ratio in the presence of different types and amounts of electrolytes, the extent of the double-layer compression under different operating conditions, and the critical coagulant concentration (ccc). 2. Model development As mentioned above, strong interactions exist between the charged colloidal particles and the electrolytes in solution, which lead to a distribution of the ionic species around the particle that is governed by the Poisson–Boltzmann equation. The change of the ionic strength in the bulk liquid phase causes the redistribution of the ionic species, which in turn changes the association and dissociation equilibria between the ionic surfactant adsorbed on the particle surface and the counterions in the liquid phase, and consequently also the surface charge density and then the colloidal stability. In the following, we briefly formulate a model for describing the equilibrium between the interface electrochemistry and the distribution of the ionic species in the liquid phase, accounting for the specifics of the polymer colloids, based on the theories and relevant treatments available in the literature [2,7–9]. 2.1. Sources of charges on the particle surface In polymer colloids produced by emulsion polymerization, ionic surfactants (denoted by E in the following) are often adsorbed on the particles, and their dissociation leads to the formation of negative charges on the surface. Since the surfactant adsorption is typically reversible, the charges deriving from the surfactant molecules are referred to as mobile charges. The amount of the surfactant molecules adsorbed on the particle surface depends on the adsorption equilibrium between the particle surface and the liquid dispersion. In this work, a Langmuir-type adsorption isotherm is used to describe the surfactant adsorption equilibrium, which may be written as [10] bCti =

Γ /Γ∞ , 1 − Γ /Γ∞

(1)

where Γ is the surface coverage of the surfactant, Γ∞ the surface coverage at saturation, and b the adsorption constant. Note that in the above adsorption isotherm (1) the total surfactant concentration at the particle–liquid interface, Cti , is used, instead of the concentration in the bulk liquid phase, Ctb . Furthermore, ionically dissociable polymer end-groups may exist on the particle surface, and these also contribute to the colloidal stability. A typical example is given by polymers produced using KPS (potassium persulfate) as initiator for the polymerization, which may result in sulfate head groups (–SO− 4 ) on the polymer chains, exposed on the particle surface. Since these charges (denoted by L) are covalently bound to the surface, they are referred to as fixed charges in the following.

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Table 1 The electrolyte association equilibria and the corresponding expressions of the equilibrium constants At the interface

In the liquid phase

H+ + E−  HE

KHE = C CHE H+ CE−

H+ + E−  HE

KHE = C CHE H+ CE−

− M+ m + E  Mm E

E KMm E = C Mm C M+ E−

− M+ m + E  Mm E

E KMm E = C Mm C M+ E−

− + M2+ d + E  Md E

CM E+ K M E+ = C d C d − M2+ E

− + M2+ d + E  Md E

C M E+ K M E+ = C d C d − M2+ E

H+ + L−  HL

KHL = C CHL H+ CL−

H+ + A − m  HAm

KHAm = C HACm H+ A−

− M+ m + L  Mm L

C L KMmL = C Mm + CL−

− H+ + A2− d  HAd

KHA− =

− + M2+ d + L  Md L

CM L+ K M L+ = C d C d − M2+ L

2− M2+ d + A d  M d Ad

K M d Ad = C

C

m

m

d

Mm

d

2.2. Dissociation equilibria of electrolytes in solution Besides the ionic surfactant, additional electrolytes are often present in a colloidal system. The relevant dissociation equilibria both at the particle–liquid interface and in the liquid phase are listed in Table 1, where E− and L− are the surfactant and 2+ fixed charge anions, M+ m and Md are the mono- and divalent 2− cations, A− m and Ad are the mono- and divalent anions, and H+ is the proton. Note that the divalent cation M2+ d is assumed to combine with E− and L− only in the form of 1:1 complexes, Md E+ and Md L+ , both at the interface and in the liquid phase. This is a reasonable assumption at the interface in the case of aliphatic surfactants, because, as pointed out by Bloch and Yun [9], the 1:2 complexes may be formed only when the surfactant hydrocarbon chains are oriented perpendicular to each other and facing the metal ions with their functional groups. Except for extremely low surface coverage, such a conformation is generally quite unfavored for surfactant molecules adsorbed on the particle surface [11]. 2− Moreover, the association of M2+ d with Ad is also consid2− − ered, while all the weak associations of M+ m with Am or Ad 2− are ignored, and those of H+ with both A− m and Ad are considered in order to correctly predict the pH of the system. The last is particularly important when carboxyl groups are the main source of the surface charge, which then is very sensitive to the system pH. For the association between H+ with A2− d , however, only the associate in the form of 1:1 is considered, while the very weak 2:1 association is ignored. Note that all these assumptions have been made because they have some general validity and apply to the particular experimental systems considered in this work. However, each of them can be changed without significantly altering the model. Expressions for the corresponding association equilibrium constants are listed in the second column of Table 1, where C represents the concentration of the component. In this work, we assume that for each reaction the equilibrium constant K is independent of whether the association reaction occurs at the interface or in the liquid phase, and we also neglect the effect of the possible surface aggregation of surfactant molecules. This is indeed a reasonable approximation in the case of the protonation of aliphatic surfactants [12,13]. In addition, it is assumed that the acidic sur-

C

C

d

m HA− d CH+ C 2− Ad CM A d d

C

d

C 2− M2+ Ad d

factant HE, whether associated or dissociated, is water-soluble. In other words, the possible precipitation of the surfactant is not accounted for. 2.3. Surface charge density Among all the species adsorbed on the particle surface, HE, Mm E, Md E+ , and E− , only the latter two contribute to the surface charge. Since Md E+ and E− have opposite charges, the net charge density on the particle surface arising from the adsorbed surfactants, σ0,E , is given by  Vp  s  aF  s σ0,E = F CM = KMd E C i 2+ − 1 CEs − , (2) + − CE− dE M Ap 3 d where the superscripts s and i denote quantities on the particle surface and at the interface, respectively, F is the Faraday constant, Vp and Ap are the volume and surface area of a particle, respectively, and a is the particle radius. It is worth mentioning that the sign of the charge density given by Eq. (2) depends on the difference in the concentrations between the charged groups, Md E+ and E− , on the surface. For example, we have positively charged particles when the association between E− and M2+ d is very strong or when the concentration of the cation M2+ d in the liquid phase is substantially high. The inversion of charge sign on the particle surface is often used to explain the restabilization phenomenon observed at substantially high concentrations of the divalent cation in the liquid phase [14–16]. Similarly, the contribution of the fixed charged groups on the particle surface is given by  aF   aF  s CMd L+ − CLs − = KMd L C i 2+ − 1 CLs − Md 3 3 and the total surface charge density σ0 is

σ0,L =

(3)

σ0 = σ0,E + σ0,L .

(4)

As mentioned above, the distribution of all the ionic species in the system is described by the Poisson–Boltzmann equation, which for convenience we apply in this work in the frame of the classical Gouy–Chapman theory [2,9]. This leads to the following relation between the surface charge density σ0 , the surface

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potential, and the bulk ionic strength (see Appendix A),      1/2 zj eψ0 Cjb exp − σ0 = − R0 (5) , −1 kT where R0 = 2F ε0 εr kT /e with ε0 the vacuum permittivity, εr the relative dielectric constant of the medium, e the electron charge, k the Boltzmann constant, and T the absolute temperature, while ψ0 is the surface potential, and Cjb and zj are the bulk concentration and charge valency of the j th ion (j applies 2+ 2− − − − + to H+ , M+ m , Md , Am , Ad , E , Md E , and HAd in Table 1), respectively.

Table 2 Species concentrations in the solid and liquid phases

2.4. Material balances The sum of the mass of each species distributed in the different phases has to be equal to the corresponding mass added to the system. Let us indicate Cj,0 as the initial concentration of the j th species added to the system. At equilibrium, this is distributed on the particle surface and in the liquid phase, so that Cj,0 = φCjs

Vl,p + N0 Cj (x) dV (x),

(6)

0

where φ is the particle volume fraction, N0 is the particle number concentration, and Vl,p is the liquid volume that on average can be assigned to each particle (Vl,p = 1/N0 ). The first term on the right-hand side of Eq. (6) represents the mass on the particle surface, while the second one is the mass distributed in the entire aqueous phase, which can be divided into two regions, the diffuse layer near the particle surface and the liquid bulk, i.e., Vl,p Vd Vl,p d Cj (x) dV (x) = Cj dV + Cjb dV 0

Vd

0

Vd =

Cjd dV + (Vl,p − Vd )Cjb ,

(7)

0

where Vd is the liquid volume occupied by the diffuse layer, while Cjd and Cjb are the concentrations of the j th component in the diffuse layer and in the bulk liquid phase, respectively. Thus, to solve the material balance, it is necessary to estimate the Vd value. In the present case of a moderate or thin EDL compared to the particle radius, since the contribution of the diffuse layer (i.e., the first term on the right-hand side of Eq. (7)) to the total material balance is relatively small, the concentration in the diffuse layer can be approximately replaced by the bulk phase concentration; i.e., Cjd ≈ Cjb . This should not lead to significant error in the material balance. In this case, Eq. (6) reduces to Cj,0 = φCjs + (1 − φ)Cjb .

(8)

It should be pointed out that the approximations proposed above are only for the purpose of material balance computations, and they are not involved in the calculation of the association equilibria and the surface charge density. For completeness, Table 2 gives the explicit expressions for Cjs and Cjb for each component j involved in Table 1.

Cjs

Cjb

s + Cs CHE HL s s CM + CM mE mL

b + Cb + Cb b CH + HE HAm + CHA− d b C b + + CM mE

s s CM + + C M L+ dE d

b C b 2+ + CM + dE M

s s s CEs − + CM + + CHE + CMm E dE

b b b CEb − + CM + + CHE + CMm E dE

–

b C b − + CHA

–

C b 2− + C b

s s s CLs − + CM + + CHL + CMm L dL

–

Mm d

m

Am Ad

HA− d

2.5. The colloidal interaction model To relate the particle surface charge to the aggregation kinetics, we use the classical DLVO model accounting for the electrostatic repulsive (UR ) and the van der Waals attractive (UA ) potentials. However, a large number of studies indicate that in many colloidal systems an additional short-range repulsive force, which decays exponentially with distance, is often present [6,17–21], especially at relatively high ionic strength. Moreover, this force shows the specific electrolyte ion effect, which is related to the hydration strength of the ions. Therefore, this non-DLVO force is often referred to as “hydration force.” The origin of this force is still under debate, and different models have been proposed in the literature to account for this short-range repulsive interaction in order to explain the experimental results, especially with respect to the ion specificity [4,22,23]. For instance, Manciu and Ruckenstein [6] have recently proposed a model that attributes the hydration force to the polarization of water molecules in the vicinity of the surface due to surface dipoles. This leads to an additional electric field, which affects the interaction potential. However, there are still a few parameters in the model that cannot be evaluated either experimentally or theoretically. Moreover, this short-range repulsive force can be different in different systems [24], and a model developed for a specific colloidal system is often difficult to apply to other systems. Thus, in our colloidal interaction model, in addition to the DLVO interactions, we also include the non-DLVO interaction coming from the hydration force, referred to as the hydration interaction, Uhyd , but in an empirical way. In particular, we consider that the total interparticle interaction energy U is given by the sum of the van der Waals, electrostatic, and hydration interactions, as follows: U = UA + UR + Uhyd .

(9)

The contribution of the van der Waals attraction, UA , is computed with the Hamaker relation [1],    2 AH 4 2 UA = − (10) + + ln 1 − , 6 l2 − 4 l2 l2 where AH is the Hamaker constant and l = r/a, where r is the center-to-center distance between two particles. For the electrostatic repulsion UR , we use the modified Hogg–Healy–

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Fuersteneau expression [25],

 4πε0 εr aψ02 ln 1 + exp −κa(l − 2) , (11) l where ψ0 is the surface potential and κ is the Debye–Hückel parameter, defined as   2  e NA j zj2 Cjb 1/2 κ= (12) , ε0 εr kT

UR =

where NA is the Avogadro constant. A simple exponential decay function is used to describe the hydration force between particles [7,26], Fhyd = F0 exp(−h/δ0 ),

(13)

where h = r − 2a is the surface-to-surface distance between particles, F0 is the hydration force constant, and δ0 is the characteristic decay length. The corresponding hydration interaction energy between two spherical particles can be obtained by applying the Deryaguin approximation [27], which leads to Uhyd = πaF0 δ02 exp(−h/δ0 ).

(14)

Finally, the relation between the above total interaction energy U and the measured Fuchs stability ratio W can be established based on the definition of W [28], ∞ W =2

exp(U/kT ) dl, Gl 2

(15)

2

where G is a hydrodynamic function accounting for the additional resistance caused by the squeezing of the fluid during the particle approaching [29]: 6l 2 − 20l + 16 . (16) 6l 2 − 11l Thus summarizing, the set of equations reported above can now be solved, once the surfactant adsorption parameters in Eq. (1) and all the association constants in Table 1 are known, to compute the surface potential or charge density, the concentrations of all species on the particle surface and in the bulk liquid phase, and finally the stability ratio, W .

G=

3. Experiments 3.1. The colloidal systems The polymer colloids used for demonstrating the feasibility of the proposed stability model are three types of fluorinated elastomer latices, T1, T2, and T3, manufactured by Solvay Solexis (Italy) through emulsion polymerization. The surfactant (E) used for the polymerization belongs to the family of perfluoropolyether (PFPE)-based carboxylates and has a molecular weight of 570. The adopted polymerization process does not generate any fixed charges on the particle surface. The three types of polymers have very similar chemical compositions and the main differences are in the size of the primary particles and in the amount of surfactant used during the polymerization, as

191

Table 3 Characteristics of the latices used in this work Name of latex

Particle radius (nm)

Surfactant concentration (mol/m3 Pol.)

Surfactant counterion

T1

60

33.3

Na+

T2

37.5

25.7

NH+ 4

T3

35

25.7

NH+ 4

reported in detail in Table 3. Note that the original latices are acidic due to the presence of a small amount of HF, whose concentration at the particle volume fraction φ = 5.0 × 10−3 is equal to 1.6 × 10−3 mol/L. The colloidal system T1 has been used to determine the Fuchs stability ratio W , while the remaining two systems, T2 and T3, are used only for validation of the model predictions. 3.2. Measurements of the Fuchs stability ratio W The approach based on doublet formation kinetics developed previously [30,31] has been used in this work to measure the Fuchs stability ratio W . In the initial stage of a Brownian aggregation process where the conversion of the primary particles to doublets is smaller than 20%, the rate of aggregation of primary particles can be approximated as follows [31], dN1 (17) = K1,1 N12 , dt where N1 is the number concentration of the primary particles and K1,1 is the doublet formation rate constant. Introducing the conversion of primary particles to doublets, x = 1 − N1 /N1,0 , into Eq. (17), where N1,0 is the initial number concentration of the primary particles, we obtain after integration

−

K1,1 =

x 1 . 1 − x N1,0 t

(18)

The K1,1 value defines the Fuchs stability ratio W according to the following relationship, W = KB /K1,1 ,

(19)

where KB (= 8kT /3μ) is the Smoluchowski rate constant and μ is the dynamic viscosity of the disperse medium. The remaining problem is to determine the conversion value x at each given time. This is done by taking samples at different aggregation times, diluting them immediately in demineralized water to quench the aggregation, and then performing static light scattering (SLS) measurements. In the very initial stage of the aggregation, where the system can be assumed to contain only the primary particles and doublets, the scattered light intensity I (q) measured by SLS is given as follows [32], I (q) =

N1 S1 (q)P (q) + 4N2 S2 (q)P (q) , N1 + 4N2

where q is the magnitude of the wave vector, defined as   θ 4πn0 sin . q= λ0 2

(20)

(21)

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Fig. 1. (a) I (q)/P1 (q) as a function of F (q) at different aggregation time; (b) The doublet formation rate constant K1,1 as a function of aggregation time t for the coalescence system. T1 latex; φ = 5.0 × 10−3 ; CNH4 HSO4 ,0 = 6.0 × 10−2 mol/L.

λ0 is the wavelength of the incident light and n0 is the refractive index of the disperse medium. N1 and N2 in Eq. (20) are the number concentrations of the primary particles and doublets, respectively. P (q) is the form factor and S1 (q) (= 1) and S2 (q) are the structure factors of the primary particles and doublets, respectively. In the present work, however, we have to consider that since the glass-transition temperature of the elastomer under examination is below −15 ◦ C, a doublet formed during the aggregation at room temperature coalesces quickly to become a spherical particle. In this case, we have S2 (q) = 1, and Eq. (20) may be rewritten as I (q) =

N1 P1 (q) + 4N2 P2 (q) , N1 + 4N2

(22)

where P1 (q) and P2 (q) are the form factors of the primary particle and the coalesced doublet, respectively. P1 (q) was determined from the original latex and can be accurately described by the RDG expression [32]  P1 (q) = 9

sin(aq) − (aq) cos(aq) (aq)3

2 .

(23)

For a coalescence system, where a doublet would √ coalesce to form a new spherical particle with radius R2 = 3 2a, the form factor of the doublets P2 (q) is also √ given by the RDG expression (23), where a is replaced by 3 2a: √ √ √  sin( 3 2aq) − ( 3 2aq) cos( 3 2aq) 2 . P2 (q) = 9 (24) √ ( 3 2aq)3 Substituting Eqs. (23) and (24) into Eq. (22), and assuming that N1 = (1 − x)N1,0 and N2 = xN1,0 /2, we have I (q) x =1− F (q), P1 (q) 1+x

(25)

where F (q) represents the q-dependent part of the form factor relation, √ √ √  1 sin( 3 2aq) − ( 3 2aq) cos( 3 2aq) 2 F (q) = 2 − (26) . 2 sin(aq) − (aq) cos(aq)

Therefore, when the measured I (q)/P1 (q) is plotted against F (q), one obtains a straight line, whose slope gives the conversion value x. Fig. 1a shows a typical plot of I (q)/P1 (q) vs F (q) in the initial stage of the aggregation process using NH4 HSO4 as the destabilizer, for the T1 latex at φ = 5.0 × 10−3 . These are straight lines corresponding to the times t = 1.5, 5, and 20 min, whose slopes lead to the estimation of x = 2.8, 7.7, and 16.6%, respectively. Such obtained conversions are used, based on Eq. (18), to compute the doublet formation rate constant K1,1 at different times t . The obtained results are shown in Fig. 1b. It is seen that the K1,1 value decreases with time. This is rather different from the doublet formation kinetics observed for rigid particles, of which the estimated K1,1 value is independent of time (e.g., the fluorinated polymer MFA colloids studied by Lattuada et al. [31]). The K1,1 value decreasing with time in Fig. 1b indicates that the aggregation to form doublets leads to an increase in the colloidal stability of the system. This arises because of the coalescence of two particles in a doublet to form a spherical particle, which reduces the total surface area available for the surfactant adsorption. The redistribution of the surfactants on the reduced total surface area of the particles and the new equilibrium with the liquid phase result in some gains in the particle surface charges, thus the colloidal stability. Then, in order to estimate the Fuchs stability ratio W representing the stability of the departure colloidal system, we need to extrapolate the results in Fig. 1b to t = 0. A second-order polynomial is found to well represent the trend of K1,1 vs t , as shown by the solid curve in Fig. 1b, which is then extrapolated to obtain K1,1 at t = 0, and thus the Fuchs stability ratio W according to Eq. (19). Following the same procedure, the stability ratio W has been measured at various concentrations of NH4 HSO4 , H2 SO4 , NaCl, and MgSO4 , as shown by the data in Figs. 2a to 2d, respectively. The use of the acid or acidic salts as destabilizer is due to the fact that the charges of the surfactants come from carboxyl groups, which are protonated at low pH, resulting in surface charge reduction. By comparing the data in Figs. 2a and 2c it can be observed that although both NH4 HSO4 and NaCl are monovalent salts, the former is more effective in destabiliz-

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193

Fig. 2. Values of the Fuchs stability ratio W of the T1 latex measured experimentally as a function of the salt concentration, using various salts: (a) NH4 HSO4 ; (b) H2 SO4 ; (c) NaCl; (d) MgSO4 . φ = 5.0 × 10−3 .

ing the system due to its capability of producing H+ from the dissociation of the HSO− 4 group. 4. Results and discussion 4.1. Estimation of the surfactant adsorption and association and the hydration parameters Using the W values measured with different types of salts and salt concentrations, we can now estimate the unknown parameters of the model that in the present case include: the adsorption and association equilibrium parameters of the surfactants with various cations as well as the hydration potential parameters. Let us first consider the cases of latex destabilization through the addition of H2 SO4 and NH4 HSO4 together, because they both generate protons in the solution, thus promoting the protonation of carboxyl groups on the particle surface. There are two association constants involved in the system for H+ , KHE and KHSO− , for the association of H+ with the surfactant E and 4

the anion SO2− 4 in the liquid phase, respectively. Note that as stated previously, we assume that the value of KHE in the liquid phase is the same as on the particle surface. The KHSO− value 4

can be obtained from the literature, as equal to 97.0 L/mol [33]. In the case of NH4 HSO4 , besides the H+ association, there is + also the association of NH+ 4 with E. The associations of NH4 with the other anions in the liquid phase are known to be very weak and are ignored. Thus, for the two cases of H2 SO4 and NH4 HSO4 , two association parameters need to be determined: KHE and KNH4 E . There are two parameters for the surfactant adsorption, Γ∞ and b, related to the Langmuir adsorption isotherm, Eq. (1). The surface coverage at saturation Γ∞ , which should depend mainly on the affinity between the surfactant and the particle surface and not on the electrolyte concentration and type, has been taken as equal to 5.5 × 10−6 mol/m2 , according to independent adsorption equilibria measurements [34]. Besides the adsorption and association parameters, there are two more parameters, the hydration force constant F0 and the characteristic decay length δ0 in Eq. (14) for the hydration interaction potential. The F0 value depends mainly on the type of the cations associated on the particle surface and often lies in the range between 106 and 5 × 108 N/m2 , while the δ0 value is weakly dependent on the type of the cations and is in the range between 0.2 and 1.0 nm [35]. The latter arises mostly because the decay length of the hydration force is related to the size

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Fig. 3. Comparisons between the measured (symbols) and simulated (solid curves) values of the Fuchs stability ratio W of the T1 latex as a function of the salt concentration, for different salt types: (a) (!) H2 SO4 and (P) NH4 HSO4 ; (b) NaCl; (c) MgSO4 . φ = 5.0 × 10−3 .

of the water molecules participating in the hydration interactions. Thus, in this work, the δ0 value is assumed to be constant and equal to 0.6 nm, which corresponds approximately to two times the size of a water molecule, independent of the type of cations associated on the particle surface. On the other hand, F0 is considered as a fitting parameter dependent upon the type of cations. Finally, the Hamaker constant has been assumed to be equal to that of a similar fluorinated polymer, PTFE, i.e., 3.0 × 10−21 J [36]. The general procedure for obtaining these adsorption, association, and hydration parameters is to first choose a set of values for these parameters to obtain the surface charge density or potential by solving the nonlinear algebraic equation system given by Eqs. (1)–(8). The obtained values of the surface potential ψ0 and Debye parameter κ, together with guessed values of F0 , are then applied to the colloidal interaction model to compute the W values from Eq. (15), with the salt types and concentrations used in the experiments. The computed W values are then compared with those measured by the experiments. Using a proper optimization procedure, the values of the adsorption, association, and hydration parameters have been estimated to best fit the W values measured at the different electrolyte concentrations Cs . The values obtained for the two electrolytes, H2 SO4 and NH4 HSO4 , are reported in Table 4. The agreement between simulated and experimental values of

Table 4 Values of the association equilibrium constants of the surfactant with various counterions, the hydration force parameters and the surfactant adsorption parameters used in the model Cations M H+

NH+ 4

Na+

Mg2+

29.4a (30) 97.0

11.2a –

2.65a –

7.8a 28.8

1.15a 0.6

1.25a 0.6

1.31a 0.6

1.36a 0.6

Association constantsb KME (L/mol) KMSO4 (L/mol) Hydration parameters F0 (106 N/m2 ) δ0 (nm) Adsorption parametersc Γ∞ (10−6 mol/m2 ) b (103 L/mol)

5.1 3.5a

(5.1) (4.0)

a Fitted values. b Value in the parentheses is reported in the literature [37]. c Values in the parentheses are reported in the literature [34].

W at the various considered concentrations Cs is satisfactory for both H2 SO4 and NH4 HSO4 as shown in Fig. 3a. It is worth noting that the obtained value of the association constant of H+ with the surfactant E, KHE in Table 4, is consistent with that measured by D’Aprile et al. through conductometric titration [37] and reported in the same table

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in parentheses. The obtained value of KHE is also comparable to the association constant of H+ with CF3 (CF2 )2 COO− , equal to 4.76 L/mol [38]. The larger value for KHE obtained in this work is reasonable because the longer chain of E leads to stronger binding with protons. As expected, due to the weaker + association of –COO− with NH+ 4 than with H , the predicted + association constant of NH4 , KNH4 E , is smaller than that of H+ . The obtained KNH4 E value is also comparable to that of another PFPE-based surfactant with smaller molar mass (= 480), given as 2.9 L/mol [39]. The value in this work is larger because of the larger molar mass (longer chain). The estimated values of + the hydration force constant F0 for NH+ 4 and H are well in the range reported in the literature as indicated above. Table 4 also compares the adsorption parameters predicted by the model with those measured experimentally [34]. The value of Γ∞ is identical, because, as mentioned above, the measured value has been adopted directly in the model. The value of the adsorption constant b for surfactant E obtained in this work is similar to that measured experimentally on a PTFE colloid under different conditions [34]. Some difference has to be expected because not only are the considered systems not identical, but in addition the adsorption isotherm considered in this work refers to the surfactant at the particle–liquid interface, while that in [34] refers to the surfactant in the bulk liquid phase. We now proceed to the experiments using the counterions Na+ and Mg2+ . Since the surfactant is the same as in the previous experiments, the corresponding adsorption parameters have not been changed. Note that the association of the surfactant with H+ is again present in these systems, because the latex is acidic. The corresponding association constant KHE has been maintained equal to the previous one. It follows that in the case of Na+ (or Mg2+ ), we have only two parameters to be determined, i.e., the association equilibrium constant KNaE (or KMgE+ ) and the hydration force constant F0 . The results of the fitting procedure are shown in Figs. 3b and 3c for the salts, NaCl and MgSO4 , respectively, and the obtained values of the corresponding parameters are summarized in Table 4. It is seen that the value of the Na+ association equilibrium constant is small with respect to the others, indicating a much weaker association with the surface charge groups. Mg2+ has a relatively larger value of the association constant than Na+ , which is straightforward because in principle the divalent ion has stronger association than monovalent ion. However, it is surprising that the association constant of Mg2+ is slightly smaller than that of NH+ 4 . Unfortunately, because of lack of data in the literature about the associations of NH+ 4 with other anions, such a result cannot be checked. This difference might be attributed to the structure of NH+ 4 , which is different from that of other metal cations. It is worth noting that the obtained values of the hydration force constants F0 in Table 4 are well within the typical range of the values reported in the literature, and follow the or+ der Mg2+ > Na+ > NH+ 4 > H , which is consistent with their hydration strength [40], except for H+ . This clearly indicates that for the given colloidal system, the hydration interaction depends on the type of involved electrolyte. In the case of H+ ,

195

although its hydration is known to be stronger than that of NH+ 4, its hydration force constant F0 obtained in this work is smaller than that of NH+ 4 . Similar results have been obtained previously by Pashley [18], who observed no hydration force between mica surfaces in the case of acid as electrolyte. This was ascribed to the different nature of the hydration of the H+ ion from that of a metal ion. Additional information provided by the model, which can be used to further test its reliability by comparison with the corresponding experimental data, is the system pH. Fig. 4 shows such a comparison. It is seen that the model results are in good agreement with the experiments in all cases. As expected, both measured and predicted values of pH in Fig. 4a decrease with increasing concentration of H2 SO4 or NH4 HSO4 . Since NaCl in Fig. 4b is a neutral salt and the associations of the Na+ and Cl− ions with the other species are very weak, both the measured and predicted values of pH do not change significantly with the salt concentration. In the case of MgSO4 , however, since its SO2− 4 ion can be weakly hydrolyzed to generate the OH− ion, the system pH slightly increases as the concentration of MgSO4 increases. Another test of the model reliability is the estimation of the critical coagulant concentration (ccc). This is a rather severe test of the model because its parameters, as described above, have been estimated at low salt concentrations, and we now need to extrapolate its results to substantially higher salt concentrations. For a given colloidal system, the ccc can be defined as the minimum salt concentration at which no repulsive barrier exists between the colloidal particles, so that the aggregation process is controlled entirely by diffusion, i.e., the so-called DLCA regime. In this work, however, such a definition cannot be used to compute the ccc with the model because the short-range repulsive hydration interaction, Uhyd , given by Eq. (14), implies an energy barrier that, although small, is always present, independent of the salt concentration. This means that at least in the frame of the particle interaction potential used in this work, in the presence of the hydration interaction, there exists a minimum interaction energy barrier that cannot be screened by increasing the salt concentration. In this case, we define the ccc, as the minimum salt concentration that leads to the minimum interaction energy barrier. This pragmatic approach is justified by the results shown in Fig. 5 representing the model-predicted interparticle interaction potentials for the T1 latex at the ccc, using H2 SO4 as coagulant. It is seen that in this case the classical DLVO (UA + UR ) interaction potential is practically overlapped with the van der Waals attraction potential (UA ), but due to the presence of the hydration interaction (Uhyd ), there is still a small maximum in the overall interaction potential (U ). Experimentally, the ccc values are determined using a simple approach based on the fact that when diffusion-limited aggregation occurs at a particle volume fraction larger than 1 × 10−3 , big and clearly visible clusters of gel are formed practically instantaneously [41,42]. Thus, in a typical experiment to measure the ccc, we add a given amount of latex to a series of salt concentrations to reach a particle volume fraction of 5 × 10−3 and observe the time for the appearance of big clusters. The ccc is then estimated by backward extrapolation to the salt concen-

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Fig. 4. Comparisons between the measured (symbols) and predicted (solid curves) pH of the T1 latex, as a function of the salt concentration, for different salt types: (a) (!) H2 SO4 and (P) NH4 HSO4 ; (b) NaCl; (c) MgSO4 . φ = 5.0 × 10−3 . Table 5 Comparisons of the values of the critical coagulant concentration ccc for fast coagulation measured experimentally with those estimated by the model for various latices using different electrolytes Latex

T1 T2 T3

Fig. 5. Interparticle interaction potentials as functions of the approaching distance computed from the colloidal interaction model for the T1 latex at ccc with H2 SO4 as coagulant. φ = 5.0 × 10−3 .

tration where the appearance of big clusters is instantaneous, i.e., at t = 0. The ccc values measured experimentally for various salts are compared with the predicted ones in the first row of Table 5 for the T1 latex. Note that each predicted ccc value is given by a narrow concentration range, instead of a specific

H2 SO4 (mol/L)

NH4 HSO4 (mol/L)

NaCl (mol/L)

MgSO4 (mol/L)

Exp.

Model

Exp.

Model

Exp.

Model

Exp.

Model

0.4 0.3 0.3

0.3–0.4 0.2–0.3 0.2–0.3

0.52 0.4 0.4

0.5–0.6 0.3–0.4 0.3–0.4

1.4 – –

1.3–1.5 – –

0.6 – –

0.4–0.5 – –

value, because near the ccc defined above, the interaction energy barrier reduces asymptotically without a sharp change. As can be seen, the model predictions are in good agreement with the experimental results, although the predicted ccc values are usually slightly smaller than the measured ones. As mentioned above, two other latices, T2 and T3 (Table 3) have also been considered in this work. Since the difference between these latices and the T1 latex is only in the particle size and amount of surfactant, the model developed above for the latex T1 should apply also in this case. The second and third rows of Table 5 compare the measured and predicted ccc values in the cases of H2 SO4 and NH4 HSO4 for T2 and T3, respectively.

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197

Fig. 6. The surfactant concentrations on the particle surface (a), the surfactant concentrations in the liquid phase (b), the surface charge density (c) and the surface potential and the Fuchs stability ratio (d) as a function of the added NaCl concentration CNaCl,0 . T1 latex; φ = 5.0 × 10−3 .

Again, the predictions are consistent with the experiments. It is particularly worth noting that since the colloidal systems T2 and T3 contain smaller amounts of surfactant than T1 (see Table 3), their surface charge densities are smaller, and therefore both the measured and predicted ccc values are smaller than those of the T1 latex. 4.2. Effects of salt type and concentration on the surfactant distribution and the surface charge density Besides the predictions of the ccc discussed above, the other important applications of the estimated values of the adsorption, association, and hydration parameters in industrial practice are to predict and analyze the dependencies of the surface charge density and the partitioning of the surfactant in different phases on the salt type and concentration. This information can help one to make decisions in the amount of surfactant used for the polymerization process, the use of the salt type and concentration for the latex coagulation, and the cleaning of the surfactant after the latex coagulation. Let us take the T1 latex at φ = 5 × 10−3 as an example to do such an analysis.

4.2.1. NaCl When NaCl is used as electrolyte, Figs. 6a and 6b show the surfactant concentrations on the particle surface and in the liquid phase as functions of the added NaCl concentration, CNaCl,0 , respectively. With increasing CNaCl,0 , the total amount of E (denoted as Et ) adsorbed on the particle surface increases, while the quantity decreases in the liquid phase. This arises because adding NaCl into the system leads to an increase in the ionic strength, which reduces the surface potential (its absolute value, the same in the following). Then, based on the Boltzmann equation (A.3), decreasing the surface potential leads to a shift of the anionic surfactant from the bulk liquid phase to the particle–liquid interface, while for the cations the opposite effects occur. Such variations require establishing new equilibria for both the surfactant adsorption and its association with cation, which obviously favor the adsorption of more surfactant from the liquid phase to the particle surface. This also leads to gains in the concentrations of the dissociated surfactant, CEs − on the particle surface, thus increasing the surface charge as shown in Fig. 6c. Such a trend will progressively diminish with further adding NaCl to the system, because of two reasons: the amount of the surfactant in the liquid phase that can be shifted

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Fig. 7. The surfactant concentrations on the particle surface (a), the surfactant concentrations in the liquid phase (b), the surface charge density (c) and the surface potential and the Fuchs stability ratio (d) as a function of the added H2 SO4 concentration CH2 SO4 ,0 . T1 latex; φ = 5.0 × 10−3 .

to the particle surface is limited and the associated surfactant in the liquid phase does not follow the Boltzmann distribution and has the same concentration at the interface (the side of liquid phase) as in the bulk liquid phase. As a result, with increasing the NaCl concentration, the association of the ionic surfactant continues but its partition between the particle surface and the liquid phase reaches progressively a plateau, as shown in Figs. 6a and 6b. Meanwhile, the dissociated surfactant on the particle surface, after reaching a local maximum, starts to decrease with increasing CNaCl,0 . This leads to a local maximum in the surface charge density σ0 (its absolute value, the same in the following) in Fig. 6c. In Fig. 6d are shown the predicted surface potential ψ0 and the Fuchs stability ratio W as functions of CNaCl,0 . It is seen that unlike the surface charge density σ0 , the surface potential ψ0 decreases monotonically as Cs,0 increases. This means that in the region where σ0 increases with CNaCl,0 in Fig. 6a, the screening effect of the added NaCl is still dominant. To better illustrate this point, let us consider Eq. (5), which for the symmetric electrolyte with zj = z+ = −z− = z and b = C b = C b can be simplified as C+ −

   1/2 zeψ0 sinh σ0 = 8ε0 εr kT NA C b 2kT or   σ0 zeψ0 ∝√ . sinh 2kT Cb Since C b increases linearly with Cs,0 , while σ0 in Fig. 6a increases only about 2 times when CNaCl,0 increases from 10−5 to 10−2 mol/L, it is straightforward to see from the above expres√ sion that the term σ0 / C b always decreases as C b increases. Because the hyperbolic sine is a monotonic function, it follows that the surface potential ψ0 decreases as the NaCl concentration increases. There is also the consequence that both the predicted and measured W values decrease as CNaCl,0 increases in Fig. 6d. 4.2.2. H2 SO4 and NH4 HSO4 In the case of using H2 SO4 as coagulant, the surfactant concentrations on the particle surface and in the liquid phase, the surface charge density and the surface potential and the Fuchs stability ratio as functions of the added H2 SO4 concentration, CH2 SO4 ,0 , are shown in Figs. 7a–7d, respectively. It is seen that,

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199

Fig. 8. The surfactant concentrations on the particle surface (a), the surfactant concentrations in the liquid phase (b), the surface charge density (c) and the surface potential and the Fuchs stability ratio (d) as a function of the added NH4 HSO4 concentration CNH4 HSO4 ,0 . T1 latex; φ = 5.0 × 10−3 .

similarly to the case of NaCl, the total amount of E adsorbed on the particle surface increases when CH2 SO4 ,0 increases, while the quantity in the liquid phase decreases. However, when Fig. 7 is compared with Fig. 6 for NaCl, all the curves in Fig. 7 are shifted to smaller electrolyte concentrations. The most significant results in Fig. 7 are twofold. First, due to significant protonation in the case of H2 SO4 , the surface charge density shown in Fig. 7c increases with CH2 SO4 ,0 to reach a local maximum, which is much smaller than that in Fig. 6c. Second, also due to the protonation, the dissociated surfactants both on the particle surface and in the liquid phase decrease more rapidly with increasing CH2 SO4 ,0 . This clearly shows that H2 SO4 is a more effective coagulant than NaCl for the given polymer latex. Moreover, it can be observed from Figs. 6 and 7 that no matter whether one uses H2 SO4 or NaCl as coagulant, above the ccc, the total amount of surfactant adsorbed on the particle surface is very similar. However, the form of the surfactant on the particle surface is different. In the case of H2 SO4 , the surfactant is in the protonated form (HE), while in the case of NaCl, it is mainly associated with the Na+ ions in the form of NaE and with a considerable amount of dissociated anion, E− .

In the case of using NH4 HSO4 as coagulant, the surface charge density, the surfactant concentrations on the particle surface and in the liquid phase, and the surface potential and Fuchs stability ratio as functions of the added NH4 HSO4 concentration, CNH4 HSO4 ,0 , are shown in Figs. 8a–8d, respectively. Since NH4 HSO4 is an acidic salt, results in Fig. 8 are very similar to those in Fig. 7 in the case of H2 SO4 , except that at a given salt concentration, the surface charge density is slightly larger for NH4 HSO4 . This arises because of two factors: (1) NH4 HSO4 is less acidic than H2 SO4 , leading to less protonation of the surface charge groups, and (2) the association of the NH+ 4 ion with the surfactant is relatively weak, which does not significantly change the surface charge density. These results indicate that NH4 HSO4 is less effective than H2 SO4 as coagulant for the given polymer latex, and in fact, the measured and predicted ccc values are larger for NH4 HSO4 than for H2 SO4 , as given in Table 5. 4.2.3. MgSO4 A similar analysis has also been carried out using MgSO4 as coagulant, and results are shown in Fig. 9. It is seen that the results in Fig. 9 are significantly different from those discussed above, mainly due to the divalent cation Mg2+ , which can as-

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Fig. 9. The surfactant concentrations on the particle surface (a), the surfactant concentrations in the liquid phase (b), the surface charge density (c) and the surface potential and the Fuchs stability ratio (d) as a function of the added MgSO4 concentration CMgSO4 ,0 . T1 latex; φ = 5.0 × 10−3 .

sociate with the monovalent carboxyl groups of the surfactant to form MgE+ and generate positive charge on the particle surface. Since the positive charge generated by the Mg2+ association increases monotonically with the added MgSO4 concentration, CMgSO4 ,0 , the net charge density in Fig. 9a quickly approaches zero with increasing CMgSO4 ,0 . Thus, in principle, one can expect that charge inversion would occur with further increasing CMgSO4 ,0 , which might lead to restabilization of the system. Such behavior is not investigated in this work, because this might occur at extremely large MgSO4 concentrations where the applicability of the model results through extrapolation is rather questionable. Moreover, at such large salt concentrations, the hydration interaction model, Eq. (14), needs to be further validated. 5. Concluding remarks A generalized model has been proposed to describe and to characterize the stability of polymer colloids and its dependence on the electrolyte type and concentration. It has simultaneously accounted for the interactions among three physicochemical processes: surfactant adsorption equilibrium at the particle–liquid interface, association equilibria of the surfactant

ions with the counterions involved in the system, and colloidal (e.g., DLVO) interactions. Values of the Fuchs stability ratio W determined experimentally at various salt types and concentrations, through measurements of the doublet formation kinetics, are used to estimate the model parameters that cannot be found in the literature. With the estimated model parameters, the proposed stability model allows one to analyze and describe the variations of the surface charge, potential, and surfactant distributions in the system as functions of the electrolyte type and concentration. Three industrial fluorinated polymer colloids have been used to demonstrate the applicability of the proposed methodology. For the three fluorinated polymer colloids, it is found that to properly describe their stability behavior, the colloidal interaction model should include not only the classical DLVO (van der Waals attractive and electrostatic repulsive) interactions but also the non-DLVO repulsive hydration interaction. In this way, a set of values for the surfactant adsorption and counterion association parameters and the hydration parameters can be found to well simulate the W values measured using different electrolyte types and concentrations. The obtained values of the hydration force constants for the salts NH4 HSO4 , NaCl, and MgSO4 used

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in this work follow the order Mg2+ > Na+ > NH+ 4 , consistent with their hydration strength. Moreover, we have applied the stability model with the estimated system parameters to predict the critical coagulant concentration (ccc) for fast coagulation, which is located at much higher electrolyte concentrations than those where the W values are measured. The predicted ccc values are in good agreement with those measured by experiments, supporting the reliability of the estimated system parameters and the proposed stability model. With the obtained system parameters, the proposed stability model has been applied to analyze the dynamics of the partition of the surfactant between the particle surface and the liquid phase and the variation of the surface charge density and potential as a function of the electrolyte type and concentration for the given fluorinated polymer colloid. The obtained information can help us to make decisions on the amount of surfactant used for the polymerization process, the use of the salt type and concentration for the latex coagulation, and the cleaning of the surfactant after the latex coagulation. Acknowledgment Financial support from the Swiss National Science Foundation (Grant 200020-101724) is gratefully acknowledged. Appendix A When the thickness of the electrical double layer (EDL) is relatively small with respect to the particle radius, the onedimensional Cartesian Poisson equation can be applied to a planar interface, ρ(x) d2 ψ =− , (A.1) ε0 εr dx 2 where x is the distance away from the surface, ρ is the charge density per unit volume,  ρ(x) = F (A.2) zj Cj (x), j

and the Boltzmann equation of the ionic species j is   zj eψ(x) . Cj (x) = Cjb exp − (A.3) kT Integrating Eq. (A.1) once with the boundary conditions ψ = 0 and dψ/dx = 0 at x = ∞, one obtains that x σd (x) 1 dψ −ρ(x) dx = , = dx ε0 εr ε0 εr

(A.4)

(A.5)

∞

where σd is the charge density per area of surface in the diffuse layer. On the other hand, the left-hand side of Eq. (A.1) can be written as 1/2 · d[(dψ/dx)2 ]/dψ . Then, after further rearrangement and integration of Eq. (A.1), we obtain ψ(x)  1/2  2 dψ   ρ(ψ ) dψ . = − dx ε0 εr 0

(A.6)

201

Comparing Eqs. (A.5) and (A.6), we have 

ψ(x) 

ρ(ψ  ) dψ 

σd (x) = −2ε0 εr

1/2 .

(A.7)

0

Substituting Eqs. (A.2) and (A.3) into Eq. (A.7), after integration, we obtain the relation between the local (area) charge density and potential:      1/2 zj eψ(x) σd (x) = R0 (A.8) Cjb exp − . −1 kT With the assumption of σ0 + σd|x=0 = 0 we obtain Eq. (5). References [1] J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1991. [2] R.J. Hunter, Foundations of Colloid Science, Oxford Univ. Press, New York, 2001. [3] B.W. Ninham, V. Yaminsky, Langmuir 13 (1997) 2097. [4] E. Ruckenstein, M. Manciu, Adv. Colloid Interface Sci. 105 (2003) 177. [5] W. Kunz, P.L. Nostro, B.W. Ninham, Curr. Opin. Colloid Interface Sci. 9 (2004) 1. [6] M. Manciu, E. Ruckenstein, Adv. Colloid Interface Sci. 112 (2004) 109. [7] S. Leikin, V.A. Parsegian, D.C. Rau, R.P. Rand, Annu. Rev. Phys. Chem. 44 (1993) 369. [8] D.C. Grahame, Chem. Rev. 41 (1947) 441. [9] J.M. Bloch, W. Yun, Phys. Rev. A 41 (1990) 844. [10] E.H. Lucassen-Reynders (Ed.), Anionic Surfactants: Physical Chemistry of Surfactant Action, Surfactant Science Series, vol. 11, Dekker, New York, 1981. [11] S. McLaughlin, N. Mulrine, T. Gresalfi, G. Vaio, A. McLaughlin, J. Gen. Physiol. 77 (1981) 445. [12] J.J. Betts, B.A. Pethica, Trans. Faraday Soc. 52 (1956) 1581. [13] A.A. Spector, J. Lipid Res. 16 (1975) 165. [14] M. Manciu, E. Ruckenstein, Langmuir 17 (2001) 7061. [15] E. Ruckenstein, H. Huang, Langmuir 19 (2003) 3049. [16] H. Huang, M. Manciu, E. Ruckenstein, Langmuir 21 (2005) 94. [17] J.N. Israelachvili, G.E. Adams, J. Chem. Soc. Faraday Trans. I 74 (1978) 975. [18] R.M. Pashley, J. Colloid Interface Sci. 80 (1981) 153. [19] R.M. Pashley, J. Colloid Interface Sci. 83 (1981) 531. [20] R.M. Pashley, J.N. Israelachvili, J. Colloid Interface Sci. 97 (1984) 446. [21] J. Faraudo, F. Bresme, Phys. Rev. Lett. 94 (2005) 077802. [22] M. Bostrom, D.R.M. Williams, B.W. Ninham, Phys. Rev. Lett. 87 (2001) 168103. [23] M. Manciu, E. Ruckenstein, Adv. Colloid Interface Sci. 105 (2003) 63. [24] J.J. Valle-Delgado, J.A. Molina-Bolivar, F.G. Gonzalez, M.J. Galvez-Ruiz, A. Feiler, M.W. Rutland, J. Chem. Phys. 123 (2005) 034708. [25] J.E. Sader, S.L. Carnie, D.Y.C. Chan, J. Colloid Interface Sci. 171 (1995) 46. [26] V. Runkana, P. Somasundaran, P.C. Kapur, AIChE J. 51 (2005) 1233. [27] B.V. Deryaguin, Kolloid-Z. 69 (1934) 155. [28] L.A. Spielman, J. Colloid Interface Sci. 33 (1970) 562. [29] E.P. Honig, G.J. Roebersen, P.H. Wiersema, J. Colloid Interface Sci. 36 (1971) 97. [30] M. Lattuada, Aggregation Kinetics and Structure of Gels and Aggregates in Colloidal Systems, Ph.D. thesis, ETH Zurich, 2003. [31] M. Lattuada, P. Sandkuhler, H. Wu, J. Sefcik, M. Morbidelli, Adv. Colloid Interface Sci. 103 (2003) 33. [32] M. Kerker, The Scattering of Light, Academic Press, New York, 1969. [33] A.E. Martell, R.M. Smith, Critical Stability Constants, Plenum, New York, 1977. [34] A. Chittofrati, Internal report, Solvay Solexis, 2005.

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