The Generalized Stability Model and Its Applications in Polymer Colloids

Abstract

This chapter reviews the generalized stability model developed in recent years Jia et al. (J. Colloid Interface Sci. 302:187–202, 2006) and its application to different polymer colloids. The most important feature of this model is that it accounts simultaneously for the interplay between different physicochemical processes, such as surfactant adsorption equilibrium on the particle surface, association equilibria of surface charges with counterions involved in the system, and DLVO and non-DLVO colloidal interactions. Through its application to different polymer colloids produced from industrial polymerization processes, we demonstrate that the generalized stability model is powerful and capable of describing the stability of the most complicated colloidal systems. After the model has been established, it can be used to analyze how the interplay of the relevant processes affects the stability of colloids, which is not often feasible through experimental investigation.

1 Introduction

Latexes produced by emulsion polymerization are typical colloids, stabilized through adsorption of ionic surfactants on the particle surface. The electrical charges generate an electrical double layer (EDL), leading to electrostatic repulsion, which, when prevailing over the short-range van der Waals attraction, results in a repulsive barrier that kinetically stabilizes the particles. Quantification and control of the stability of polymer colloids are of great importance in industrial practice. Thus, it is essential to develop a stability model that can describe the stability of industrial polymer colloids, accounting for the effect of operating variables (e.g., type and concentration of electrolyte, system pH, etc.) on the stability.

The dominant theoretical studies describing colloidal stability focus on how to correctly describe colloidal interactions, but the effect on colloidal stability of interplay between various physicochemical processes is seldom accounted for. The centerpiece in describing colloidal interactions is the DLVO (Deryaguin–Landau–Verwey–Overbeek) theory [1, 2], which models the competition between van der Waals attraction and EDL repulsion. Additional non-DLVO forces (e.g., long-range dispersion forces, short-range hydration forces, steric forces, capillary condensation) are known to be important under specific conditions [3,4,5,6,7], but, unlike DLVO forces, such non-DLVO forces are difficult to measure experimentally or predict theoretically, particularly for industrial polymer colloids.

In practical applications, for a chosen colloid, one often first measures the surface charge or zeta-potential under well-defined conditions, and then uses the measured surface charge or potential in the DLVO model (accounting for the ionic strength) to compute the interaction energy barrier or the Fuchs stability ratio, W. Although this method can model the stability of a specific colloid under specific conditions (i.e., ionic strength, ion types, particle concentration, etc.), it cannot be applied 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 valence in the DLVO model. This is because the above approach ignores the interplay between the various physicochemical processes (e.g., colloidal interactions, counterion association equilibrium, surfactant adsorption equilibrium). As a result of such interplay, changes in the ionic strength and type lead to changes in the counterion association equilibria, and thus to changes in the surface charge and potential. Consequently, there are changes in the ion and surfactant distributions (the Boltzmann effect), thus coupled back with further changes in counterion association. The interplay becomes even more complex for industrial polymer colloids, where the surfactant systems are often very complex.

To account for the interplay mentioned above, our group has developed a generalized stability model to describe colloidal stability [8], where different physicochemical processes, such as surfactant adsorption equilibrium, counterion association equilibria, and DLVO and non-DLVO colloidal interactions, have been integrated in a single model so that their coupled interplays can be simultaneously accounted for and correctly described. In this review, we first briefly describe the developed generalized stability model and then discuss its successful application in modeling the stability of different polymer colloids.

2 The Generalized Stability Model

The interactions between charged particles and electrolytes in solution lead to ion distributions around the particles, which are governed by the Poisson–Boltzmann equation. When the ionic strength of the system is changed, redistribution of the ionic species occurs, leading to changes in the association equilibria between the ionic surfactant adsorbed on the particle surface and the counterions in the disperse medium. Consequently, this leads to changes in the surface charge density and colloidal stability. Here, we briefly describe how the generalized stability model accounts simultaneously for different processes, based on the theories available in the literature [2, 9,10,11].

2.1 Surfactant Adsorption Equilibrium

Ionic surfactants (denoted by E in the following) are commonly used in emulsion polymerization. They are adsorbed on the particle surface and their dissociation forms charges on the surface, stabilizing the particle. The charges generated from the surfactant molecules are referred to as mobile charges because of the reversibility of surfactant adsorption. As an equilibrium process, surfactant adsorption depends on the surfactant concentration in the disperse medium. Different adsorption isotherms are proposed in the literature to describe surfactant adsorption [12, 13], depending on the nature of the particle surface and surfactant type. The Langmuir-type adsorption isotherm is simple and probably the most commonly used:

$$ {bC}_t^i=\frac{\Gamma /{\Gamma}_{\infty }}{1-\Gamma /{\Gamma}_{\infty }} $$

(1)

where Γ is the surfactant surface coverage, Γ_∞ at saturation, and b the adsorption constant. It is particularly worth noting that ${C}_t^i$ is the total surfactant concentration at the particle–liquid interface, which is different from the concentration in the bulk disperse medium, ${C}_t^b$.

On the particle surface there are also ionically dissociable polymer end groups, such as the sulfate head groups ($-{\mathrm{SO}}_4^{-}$) from the initiator and potassium persulfate (KPS). These charges are covalently bound to the surface; they are denoted by L and referred to as fixed charges in t.

2.2 Electrolyte Dissociation Equilibrium

Apart from the ionic surfactant, different electrolytes may exist in polymer colloids. Thus, their dissociation equilibria should be taken into account. It should be pointed out that for each electrolyte, there are two different dissociation equilibria: one at the particle–liquid interface and another in the bulk disperse medium. The properties of the particle surface are computed on the basis of the equilibria at the particle–liquid interface, instead of the properties in the bulk disperse medium. This is one of the essential features of the generalized stability model.

In the description of the dissociation equilibria, we use E⁻ and L⁻ to denote the surfactant and fixed charge anions, ${\mathrm{M}}_{\mathrm{m}}^{+}$ and ${\mathrm{M}}_{\mathrm{d}}^{2+}$ for the mono- and divalent cations, ${\mathrm{A}}_{\mathrm{m}}^{-}$ and ${\mathrm{A}}_{\mathrm{d}}^{2-}$ for the mono- and divalent anions, and H⁺ for the proton. The divalent cation ${\mathrm{M}}_{\mathrm{d}}^{2+}$ is assumed to combine with E⁻ and L⁻ only in the form of 1:1 complexes, M_dE⁺ and M_dL⁺. This assumption is reasonable at the interface for aliphatic surfactants [11], because 1:2 complexes can be formed only when the surfactant hydrocarbon chains are oriented perpendicular to each other, facing the metal ions with their functional groups. Except for cases of extremely low surface coverage, this is not a preferred conformation for surfactant molecules adsorbed on a particle surface [14]. The association of ${\mathrm{M}}_{\mathrm{d}}^{2+}$ with ${\mathrm{A}}_{\mathrm{d}}^{2-}$ is also included, but all the weak associations of ${\mathrm{M}}_{\mathrm{m}}^{+}$ with anions, ${\mathrm{A}}_{\mathrm{m}}^{-}$ and/or ${\mathrm{A}}_{\mathrm{d}}^{2-}$, are ignored. The associations of H⁺ with both ${\mathrm{A}}_{\mathrm{m}}^{-}$ and ${\mathrm{A}}_{\mathrm{d}}^{2-}$ are accounted for to correctly predict the system pH. These associations are particularly important when carboxyl groups are the main source of the surface charges, which are very sensitive to the system pH. For the association between H⁺ with ${\mathrm{A}}_{\mathrm{d}}^{2-}$, we consider only 1:1 association and the very weak 2:1 association is ignored. Note that all these assumptions are not strictly needed for the generalized stability model, and are only proposed treatments based on some general validity in the literature. In addition, we generally assume that the equilibrium constant K is independent of whether the association occurs at the interface or in the bulk disperse medium. The acidic surfactant, HE, is assumed to be water-soluble, regardless of whether it is associated or dissociated.

It should be pointed out that when both cationic surfactant and cationic fixed charges, E⁺ and L⁺, are used, one should consider their associations with anions in the disperse medium. However, in the specific case of cationic surfactant and anionic fixed charges, the applied surfactant first neutralizes the negative fixed charges. Beyond the charge compensation point, the particle surface progressively becomes positively charged [15]. In this case, such charge compensation has to be included in the electrolyte association equilibria.

2.3 Colloidal Interactions

The classical DLVO model accounting for electrostatic repulsive (U_R) and van der Waals attractive (U_A) potentials is considered here. As indicated by various studies [4, 6, 7, 16,17,18,19], in many polymer colloids there is often an additional short-range repulsive force that decays exponentially with distance. This force exhibits a specific electrolyte ion effect, which is related to the hydration strength of the ions, and this non-DLVO force is often referred to as “hydration force.” Thus, in addition to the DLVO interactions, we generally include this non-DLVO hydration force, referred to as the hydration interaction, U_hyd. Thus, the total interaction energy U is given as follows:

$$ U={U}_{\mathrm{A}}+{U}_{\mathrm{R}}+{U}_{\mathrm{hyd}} $$

(2)

where the van der Waals attraction, U_A, is computed with the Hamaker relation [1]:

$$ {U}_{\mathrm{A}}=-\frac{A_{\mathrm{H}}}{6}\left[\frac{2}{l^2-4}+\frac{2}{l^2}+\ln \left(1-\frac{4}{l^2}\right)\right] $$

(3)

where A_H is the Hamaker constant and l = r/a, where r is the center-to-center distance between two particles and a is the particle radius. For the electrostatic repulsion, U_R, we use the modified Hogg–Healy–Fuersteneau expression [20]:

$$ {U}_{\mathrm{R}}=\frac{4{\pi \varepsilon}_0{\varepsilon}_{\mathrm{r}}a{\psi}_0^2}{l}\ln \left\{1+\exp \left[-\kappa a\left(l-2\right)\right]\right\} $$

(4)

where ε₀ is the vacuum permittivity, ε_r the relative dielectric constant of the medium, and ψ₀ the surface potential. The Debye–Hückel parameter κ is given by:

$$ \kappa =\sqrt{\left({e}^2{N}_{\mathrm{A}}\sum \limits_j{z}_j^2{C}_j^b\right)/\left({\varepsilon}_0{\varepsilon}_{\mathrm{r}} kT\right)} $$

(5)

with e being the electron charge, N_A the Avogadro constant, ${C}_j^b$ and z_j the concentration and charge valence of the j-th ion in the bulk disperse medium, respectively; k is the Boltzmann constant and T the absolute temperature. The hydration force is modeled with an exponential decay function [9, 21]:

$$ {F}_{\mathrm{hyd}}={F}_0\exp \left(-h/{\delta}_0\right) $$

(6)

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. Applying the Deryaguin approximation leads to the corresponding hydration interaction energy between two spherical particles [22]:

$$ {U}_{\mathrm{hyd}}=\pi {aF}_0{\delta}_0^2\exp \left(-h/{\delta}_0\right) $$

(7)

Typical examples of the total interaction profiles in the presence of the hydration interaction are shown in Fig. 1 (cases 2, 3, and 4) and compared with the case in the absence of hydration (case 1). It can be seen that the hydration interaction not only increases the interaction barrier but also moves the primary minimum upward.

Fig. 1

Example of the typical effect of the presence of the hydration interaction on the total interaction energy [23]

With the above total interaction energy U, one can compute the Fuchs stability ratio, W, based on its definition [24]:

$$ W=2\underset{2}{\overset{\infty }{\int \limits }}\frac{\exp \left(U/ kT\right)}{Gl^2}\mathrm{d}l $$

(8)

where G is a hydrodynamic function accounting for additional resistance caused by squeezing of the fluid during the approach of a particle [25]:

$$ G=\frac{6{l}^2-20l+16}{6{l}^2-11l} $$

(9)

2.4 Computation of Key Quantities of the Generalized Stability Model

One of the most important quantities related to the processes described above is the total effective charge density on the particle surface, with which one can directly compute the surface potential needed in colloidal interactions. As mentioned previously, there are two types of charge, mobile and fixed. Of the adsorbed species on the particle surface, HE, M_mE, M_dE⁺, and E⁻, only the last two contribute to the surface charge. The net mobile charge density coming from the adsorbed surfactants, σ_0, E, is given by:

$$ {\sigma}_{0,\mathrm{E}}=F\left({C}_{{\mathrm{M}}_{\mathrm{d}}{\mathrm{E}}^{+}}^{\mathrm{S}}-{C}_{{\mathrm{E}}^{-}}^{\mathrm{S}}\right)\frac{V_{\mathrm{p}}}{A_{\mathrm{p}}}=\frac{aF}{3}\left({K}_{{\mathrm{M}}_{\mathrm{d}}\mathrm{E}}{C}_{{\mathrm{M}}_{\mathrm{d}}^{2+}}^{\mathrm{i}}-1\right){C}_{{\mathrm{E}}^{-}}^{\mathrm{S}} $$

(10)

where the superscripts s and i denote quantities on the particle surface and at the interface, respectively. F is the Faraday constant and V_p and A_p are the volume and surface area of a particle, respectively. Note that the sign of the mobile charge density given by Eq. 10 depends on the difference in the concentrations of M_dE⁺ and E⁻ on the surface. When the association between E⁻ and ${\mathrm{M}}_{\mathrm{d}}^{2+}$ is very strong, or when the ${\mathrm{M}}_{\mathrm{d}}^{2+}$ concentration in the bulk disperse medium is substantially high, the net mobile charge on the surface is positive and charge sign inversion occurs. The charge sign inversion is often used to explain the re-stabilization phenomenon observed at substantially high concentrations of divalent cation in the liquid phase [26,27,28,29].

For the net fixed charge, we can similarly write:

$$ {\sigma}_{0,\mathrm{L}}=\frac{aF}{3}\left({C}_{{\mathrm{M}}_{\mathrm{d}}{\mathrm{L}}^{+}}^{\mathrm{S}}-{C}_{{\mathrm{L}}^{-}}^{\mathrm{S}}\right)=\frac{aF}{3}\left({K}_{{\mathrm{M}}_{\mathrm{d}}\mathrm{L}}{C}_{{\mathrm{M}}_{\mathrm{d}}^{2+}}^{\mathrm{i}}-1\right){C}_{{\mathrm{L}}^{-}}^{\mathrm{S}} $$

(11)

Therefore, the total surface charge density, σ₀, is given by:

$$ {\sigma}_0={\sigma}_{0,\mathrm{E}}+{\sigma}_{0,\mathrm{L}} $$

(12)

It should be noted again that the above equations for computation of charge are valid only when both the mobile and fixed charges are negative. When the mobile and/or fixed charges are positive, the above equations must be modified accordingly.

As mentioned above, the distribution of all the ionic species in the system is described by the Poisson–Boltzmann equation. For simplification of the description, we treat it in the frame of the classical Gouy–Chapman theory [2, 11]. Thus, we obtain the following expression to correlate between the surface charge density, σ₀, the surface potential, ψ₀, and the ionic strength in the bulk disperse medium, ${C}_j^b$:

$$ {\sigma}_0=-{\left\{{R}_0\sum {C}_j^b\left[\exp \left(-\frac{z_je{\psi}_0}{kT}\right)-1\right]\right\}}^{1/2} $$

(13)

where R₀ = 2Fε₀ε_rkT/e.

The other important quantities to be properly computed are the mass balances of each species distributed in the different phases. Let us use C_j, 0 to represent the concentration of the j-th species initially added to the system. It is distributed on the particle surface and in the disperse medium at equilibrium according to:

$$ {C}_{j,0}=\phi {C}_j^{\mathrm{S}}+{N}_0\underset{0}{\overset{V_{\mathrm{l},\mathrm{p}}}{\int \limits }}{C}_j(x)\mathrm{d}V(x) $$

(14)

where ϕ is the particle volume fraction, N₀ the particle number concentration, and V_l,p the liquid volume that on average can be assigned to each particle (V_l,p = 1/N₀). The first term on the right-hand side of Eq. (14) is the mass on the particle surface, and the second is the mass distributed in the entire disperse medium, which can be divided into two regions, the diffuse layer near the particle surface and the bulk disperse medium:

$$ {\displaystyle \begin{array}{c}\underset{0}{\overset{V_{\mathrm{l},\mathrm{p}}}{\int \limits }}{C}_j(x)\mathrm{d}V(x)=\underset{0}{\overset{V_{\mathrm{d}}}{\int \limits }}{C}_j^d\mathrm{d}V+\underset{0}{\overset{V_{\mathrm{l},\mathrm{p}}}{\int \limits }}{C}_j^b\mathrm{d}V\\ {}=\underset{0}{\overset{V_{\mathrm{d}}}{\int \limits }}{C}_j^d\mathrm{d}V+\left({V}_{\mathrm{l},\mathrm{p}}-{V}_{\mathrm{d}}\right){C}_j^b\end{array}} $$

(15)

where V_d is the liquid volume occupied by the diffuse layer, and ${C}_j^d$ and ${C}_j^b$ the concentrations of the j-th component in the diffuse layer and in the bulk disperse medium, respectively. To solve the material balance, it is necessary to estimate V_d. For a moderate or thin EDL (~2 nm) compared with the particle radius (a > 30 nm), the contribution of the diffuse layer (i.e., the first term on the right-hand side of Eq. 15) to the total material balance is relatively small, and the concentration in the diffuse layer can be simply replaced by the concentration in the bulk disperse medium (i.e., ${C}_j^d\approx {C}_j^b$), which should not result in significant error in the material balance. In this way, Eq. 14 reduces to:

$$ {C}_{j,0}=\phi {C}_j^{\mathrm{S}}+\left(1-\phi \right){C}_j^b $$

(16)

Note that such an approximation is only for the purpose of the material balance computation and is not applied in the calculation of association equilibria and surface charge density.

Therefore, in summary, once the surfactant adsorption parameters in Eq. (1) and all the electrolyte association constants are known, the set of equations described in Sect. 2 can be solved simultaneously to obtain the surface potential or charge density, the concentrations of all species on the particle surface and in the bulk disperse medium, and finally the Fuchs stability ratio W.

3 Applications of the Generalized Stability Model

To demonstrate the reliability of the generalized stability model, we have applied it to various polymer colloids stabilized in one of three ways: by purely mobile charges, by purely fixed charges, or by both mobile and fixed charges. The unknown model parameters were estimated using a few values of the Fuchs stability ratio, W, as determined experimentally for various salt types and concentrations. Application of the model allows one to monitor the dynamics of surfactant partitioning between particle surface and disperse medium, analyze the variation in surface charge density and potential as a function of the electrolyte type and concentration, and predict the critical coagulant concentration (CCC) for fast coagulation, which can be defined as the minimum coagulant (salt) concentration at which no repulsive barrier exists between colloidal particles.

3.1 Fluorinated Elastomer Latex with Only Mobile Charges [8]

The first polymer colloid used to demonstrate the feasibility of the generalized stability model was a fluorinated elastomer latex, manufactured by Solvay (Italy) through emulsion polymerization. This colloid does not have fixed charges and is stabilized purely by the surfactant (E), a perfluoropolyether (PFPE)-based carboxylate. The particle radius is 60 nm, the surfactant concentration is 33.3 mol/m³ polymer, and the surfactant counterion is Na⁺. The original latex is 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.

3.1.1 Estimation of the Model Parameters

Application of the model involves knowing various parameters. Some can be found in the literature, but others have to be determined experimentally. The proposed approach is first to experimentally measure a few W values with different types of salts and then fit these data with the generalized stability model to obtain the unknown parameters. Figure 2 reproduces some W values measured using NH₄HSO₄, H₂SO₄, and MgSO₄. The details of how to measure W values can be found in the original paper [8].

Fig. 2

Values of the Fuchs stability ratio (W) of a fluorinated elastomer latex measured as a function of the salt concentration, for three types of salt, NH₄HSO₄, H₂SO₄, and MgSO₄, at ϕ = 5.0 × 10⁻³. The solid curves show fitting of the model [8]

Let us first consider the cases using H₂SO₄ and NH₄HSO₄, both of which involve generation of protons in the solution that can associate with the carboxyl groups of E. There are two parameters (K_HE and ${K}_{{\mathrm{HSO}}_4^{-}}$) for the associations of H⁺ with the surfactant E and the anion ${\mathrm{SO}}_4^{2-}$, respectively:

$$ {\mathrm{H}}^{+}+{\mathrm{E}}^{-}\rightleftarrows \mathrm{HE}\kern0.5em {K}_{\mathrm{H}\mathrm{E}}={C}_{\mathrm{H}\mathrm{E}}/\left({C}_{{\mathrm{H}}^{+}}+{C}_{{\mathrm{E}}^{-}}\right) $$

(17)

$$ {\mathrm{H}}^{+}+{\mathrm{SO}}_4^{2-}\rightleftarrows {\mathrm{H}\mathrm{SO}}_4^{-}\kern0.5em {K}_{{\mathrm{H}\mathrm{SO}}_4^{-}}={C}_{{\mathrm{H}\mathrm{SO}}_4^{-}}/\left({C}_{{\mathrm{H}}^{+}}+{C}_{{\mathrm{SO}}_4^{2-}}\right) $$

(18)

The ${K}_{{\mathrm{HSO}}_4^{-}}$ value can be found in the literature and is equal to 97.0 L/mol [30]. There is also the constant, ${K}_{{\mathrm{NH}}_4\mathrm{E}}$, for the association of ${\mathrm{NH}}_4^{+}$ with E:

$$ {\mathrm{NH}}_4^{+}+{\mathrm{E}}^{-}\rightleftarrows {\mathrm{NH}}_4\mathrm{E}\kern0.5em {K}_{{\mathrm{NH}}_4\mathrm{E}}={C}_{{\mathrm{NH}}_4\mathrm{E}}/\left({C}_{{\mathrm{NH}}_4^{+}}+{C}_{{\mathrm{E}}^{-}}\right) $$

(19)

The associations of ${\mathrm{NH}}_4^{+}$ with the other anions are known to be very weak and can be ignored [31]. Thus, for H₂SO₄ and NH₄HSO₄, two association parameters K_HE and ${K}_{{\mathrm{NH}}_4\mathrm{E}}$ need to be estimated.

For the adsorption of surfactant (E) on the particle surface, we consider the Langmuir isotherm (Eq. 1), which involves two parameters, Γ_∞ and b. The saturation coverage (Γ_∞) depends mainly on the affinity between the surfactant and the particle surface but not on the electrolyte. Thus, we take the value Γ_∞ = 5.5 × 10⁻⁶ mol/m², as reported elsewhere [32].

For colloidal interactions, we assume the presence of the hydration interaction, which involves two parameters, F₀ and δ₀ in Eq. (6). Literature information indicates that F₀ lies in the range between 1 × 10⁶ and 5 × 10⁸ N/m² and δ₀ in the range between 0.2 and 1.0 nm [33]. We consider here that δ₀ = 0.6 nm, which is approximately twice the size of a water molecule and, instead, F₀ is to be fitted. Another interaction parameter, the Hamaker constant, is assumed to be equal to that of a similar fluorinated polymer, PTFE, so that A_H = 3.0 × 10⁻²¹ J [34].

Therefore, there are four totally unknown parameters, K_HE, ${K}_{{\mathrm{NH}}_4\mathrm{E}}$, b, and F₀. To obtain these unknown parameters, a proper set of initial values are chosen for each parameter to solve the set of nonlinear algebraic equations, Eqs. 1–16. Then, the best fit is obtained to the measured W values in the cases of H₂SO₄ and NH₄HSO₄, using a proper optimization algorithm. The values of the parameters obtained in the cases of H₂SO₄ and NH₄HSO₄ are given in Table 1, and the good agreement between simulated and experimental W values is shown in Fig. 2.

Table 1 Values of parameters for the generalized stability model in the case of fluorinated elastomer latex [8]

We next consider the case of MgSO₄. The surfactant adsorption parameters obtained previously are still valid. Association of the surfactant with H⁺ is also present in this system because the latex is acidic, and the association constant K_HE obtained above can be used directly. Thus, there are only two parameters to be determined, the association equilibrium constant, ${K}_{{\mathrm{MgE}}^{+}}$, and the hydration constant, F₀. The result of the fitting is also shown in Fig. 2, and the values of the parameters are reported in Table 1. It is interesting that the association constant of Mg²⁺ is slightly smaller than that of ${\mathrm{NH}}_4^{+}$. This could arise from the structure of ${\mathrm{NH}}_4^{+}$, which is different from that of the metal cations.

3.1.2 Model Predictions

With all the parameters available in Table 1, one can now use the generalized stability model to predict the desired quantities for the given latex. Let us first apply it to predict the system pH under various conditions, particularly as a function of the salt type and concentration. Figure 3 compares predictions with experimentally measured pH values. It is evident that the model predictions agree excellently with the experiment results in all three cases. In the cases of H₂SO₄ and NH₄HSO₄, because they produce H⁺ both measured and predicted values of pH in Fig. 3 decrease as the concentration of H₂SO₄ or NH₄HSO₄ increases. On the other hand, in the case of MgSO₄, because the anion ${\mathrm{SO}}_4^{2-}$, at equilibrium, can consume H⁺ to form ${\mathrm{HSO}}_4^{-}$, the system pH increases slightly as the concentration of MgSO₄ increases. The capture of these slight variation trends confirms the reliability of the generalized stability model.

Fig. 3

Comparison between the measured (symbols) and predicted (solid curves) pH values for a fluorinated elastomer latex as a function of the salt concentration, for three types of salt, NH₄HSO₄, H₂SO₄, and MgSO₄, at particle volume fraction ϕ = 5.0 × 10⁻³ [8]

The second verification of model reliability is to predict the CCC value. This signifies using the generalized stability model, whose parameters were estimated at low salt concentrations, to extrapolate its application to substantially larger salt concentrations.

The CCC values are experimentally determined based on the fact that when diffusion-limited aggregation occurs at ϕ > 1 × 10⁻³, clearly visible large pieces of clusters are formed immediately [35, 36]. Therefore, a given amount of latex was added to a series of salt concentrations to reach ϕ = 5 × 10⁻³, and the time for the appearance of large clusters was recorded. The CCC value was obtained by extrapolation to the salt concentration at which the large clusters appear at time zero. Table 2 compares the measured CCC values of the three salts with the predicted values for three types of latexes. It should be mentioned that each predicted CCC value is reported with its significant error bar, because near the CCC the interaction energy barrier reduces asymptotically without a sharp change. It is clear that the model predictions are in good agreement with experimental results. Note that of the three latexes mentioned in Table 2, latex 1 is the one used above to define the generalized stability model; latexes 2 and 3 are similar to latex 1 but differ in the particle size and amount of surfactant. Thus, the model developed from latex 1 should be applicable to the other two cases. It is worth mentioning that since latexes 2 and 3 use less surfactant (i.e., smaller surface charge densities), both the measured and predicted CCC values are smaller than those of latex 1.

Table 2 Comparison between measured and predicted CCC values for three different salts used for fast coagulation of three latexes [8]

3.2 Styrene–Acrylate Copolymer Latex with Both Mobile and Fixed Charges [37]

The second polymer colloid used to verify the generalized stability model was a styrene-acrylate copolymer latex, manufactured by BASF SE (Ludwigshafen, Germany) through emulsion polymerization with a carboxylate surfactant. Unlike the previous latex, this latex possesses fixed charges, $-{\mathrm{SO}}_4^{-}$, on the particle surface as a result of the use of a persulfate initiator. Because it is stabilized by both mobile and fixed charges, we can compare its stability behavior to that investigated above with only mobile charges. The radius of primary particles is a = 52 nm, and the stability behavior is investigated at a fixed particle volume fraction, ϕ = 0.02.

3.2.1 Estimation of the Model Parameters

To estimate the model parameters, we used the measured CCC values as a function of the system pH, using a bivalent salt, MgSO₄. The data were obtained using a similar method to that described above and are reported in Fig. 4. Note that because the pH of the original latex was 8.3, the pH value is tuned by adding H₂SO₄.

Fig. 4

CCC values (symbols) for MgSO₄ as a function of the system pH for a styrene-acrylate copolymer latex at ϕ=0.02; the continuous curve is the model simulation [37]

Considering MgSO₄ and H₂SO₄ as well as the mobile and fixed charges, we have three anions in the system, ${\mathrm{SO}}_4^{2-}$ and −COO⁻ from the surfactant (denoted by E⁻), and $-{\mathrm{SO}}_4^{-}$ from the fixed charges (denoted by L⁻), and two cations, H⁺ and Mg²⁺. Thus, there are a total of six association equilibria. To reduce fitting parameters, we use association equilibrium constants for similar molecules but with shorter chain lengths. This does not introduce significant error because, for long carbon chain surfactants, the electronic effects are transferred through molecular bonds and are not felt beyond two to three carbon atoms [38]. Five of the six association constants were taken from the literature and are reported in Table 3 [30]. The only association constant of Mg²⁺ with fixed charge L⁻ ($-{\mathrm{SO}}_4^{-}$), ${K}_{{\mathrm{MgL}}^{+}}$, was used as a fitting parameter, because of unavailability in the literature. We again used the Langmuir isotherm for surfactant adsorption, and of the two parameters, Γ_∞ and b, we assumed that Γ_∞ is equal to that of stearic acid on butadiene-styrene polymer particles, a system very similar to the present one [39], while b was used as a fitting parameter. The Hamaker constant, A_H, was estimated using the Lifshitz theory [16]. The concentration of fixed charge groups C_L on the particle surface was used as a fitting parameter. Therefore, in total, there were three fitting parameters, ${K}_{{\mathrm{MgL}}^{+}}$, b, and C_L, which were determined by fitting the nine CCC values at different pH values, as shown in Fig. 4.

Table 3 Values of parameters for the generalized stability model in the case of styrene-acrylate copolymer latex [37]

It should be noted that for a small radius of the particles and the computed separation distance at the potential barrier larger than or close to 1 nm, we use the classical DLVO model for colloidal interactions and ignore any short-range non-DLVO forces. The simulated CCC values are compared with the experimental values in Fig. 4, and it is seen that not only the trend but also the absolute values are in good agreement. The values of the three fitted parameters, together with the values of the other parameters, as taken from the literature, are listed in Table 3.

3.2.2 Model Predictions and Applications

Both the experimental and simulated CCC data in Fig. 4 show two distinct regimes: one for pH > 4 and another for pH < 4. This corresponds to two types of surface charge, mobile −COO⁻ (E⁻) and fixed $-{\mathrm{SO}}_4^{-}$ (L⁻), as a result of their substantially different association constants with H⁺, as reported in Table 3. Let us now manifest the system behavior by applying the generalized stability model to detail the surface charge concentrations on the particle surface as a function of pH.

Let us first consider the case in the absence of MgSO₄. The computed concentrations of mobile and fixed charge groups on the particle surface as a function of pH are given in Fig. 5a, and the corresponding total surface charge density (σ₀) and potential (ψ₀) are shown in Fig. 5b. It can be seen that at the original pH of the latex (8.3), because E⁻ = E_total on the surface and L⁻ is at plateau, all the mobile and fixed charge groups on the surface are in dissociated form (E⁻ and L⁻). The σ₀ and ψ₀ values in Fig. 5b reach their maximum (in absolute value) corresponding to the highest colloidal stability of the latex. In this case, the total surfactant concentration (E_total) on the surface is only composed of the dissociated anions, E⁻. It is particularly important to mention that the E_total values on the surface and in the disperse medium are 8.16 × 10⁻³ mol/L and 2.54 × 10⁻⁴ mol/L, respectively, and do not follow the partitioning computed by the adsorption isotherm. The latter gives E_total values on the surface and in the disperse medium of 1.11 × 10⁻¹ and 4.2 × 10⁻⁴ mol/L, respectively. This arises because the ionized E⁻ follows the Boltzmann distribution, resulting in its concentration at the particle–liquid interface being smaller than that in the bulk disperse medium. Because, for the generalized stability model, the surfactant adsorption equilibrium is considered to establish at the particle–liquid interface, the Boltzmann distribution certainly changes the amount of surfactant adsorbed on the particle surface (in fact, it is reduced because the ionic surfactant species are dominated by E⁻). For pH > 7.0, no significant changes occur, as shown in Fig. 5.

Fig. 5

Concentrations of charge groups E⁻ and L⁻ on the particle surface (a), and surface charge density σ₀ and potential ψ₀ (b), as a function of pH for the styrene-acrylate copolymer latex at ϕ = 0.02 and ${C}_{{\mathrm{MgSO}}_4}=0$ [37]

When the system pH is less than seven, the protonation process starts. Because −COO⁻ (E⁻) has a much larger association constant with H⁺ than the fixed $-{\mathrm{SO}}_4^{-}$ (L⁻), its protonation initially dominates and follows that shown in Fig. 5a. The E⁻ concentration decreases sharply as pH decreases, while the fixed charge L⁻ concentration remains nearly constant. The decrease in σ₀ or ψ₀with decreasing pH in Fig. 5b is a result of a reduction in the mobile charges E⁻. In the pH range between 3 and 4, the system reaches another (almost) flat region, where the E⁻ protonation is close to completion but the H⁺ concentration is still insufficient to significantly protonate L⁻. When the pH is less than three, L⁻ starts to protonate and its concentration decreases as pH decreases, as shown in Fig. 5a. This leads to sharper decreases in σ₀ and ψ₀ with decreasing pH. In this region, basically all E is in the protonated form, HE, and because HE does not follow the Boltzmann distribution, the partitioning of HE on the surface and in the bulk disperse medium is exactly given by the surfactant adsorption isotherm. Therefore, E_total is equal to HE on the particle surface and reaches a plateau. Referring back to Fig. 4, showing the behavior of CCC values for MgSO₄, it is clear that for pH > 4, the change in CCC value with pH is related to protonation of the mobile charges, E⁻; for pH < 4, it is due to protonation of the fixed charges, L⁻.

For the presence of MgSO₄ at 0.01 mol/L, Fig. 6a shows the computed concentrations of the surfactant and fixed charge species on the particle surface as a function of pH. The total surface charge density (σ₀) and the contribution of each surface charge group are given in Fig. 6b.

Fig. 6

(a) Concentrations of the surfactant, E, species (solid curves) and the fixed charge, L, (broken curves) groups on the particle surface, and (b) total and species charge densities as a function of the system pH, for the styrene-acrylate copolymer latex at ϕ = 0.02 and ${C}_{{\mathrm{MgSO}}_4}=$0.01 mol/L [37]

Comparing Fig. 6a to Fig. 5a in the absence of Mg²⁺, we see that the concentration variations of the surfactant species are very similar, but now the new species, MgE⁺, is present on the surface as a result of E⁻–Mg²⁺ association. In the presence of Mg²⁺, the plateau value of the total surface charge density, σ₀ in Fig. 6c for pH > 7, is substantially smaller than that in Fig. 5b in the absence of Mg²⁺. As the system pH decreases to less than seven, the surface concentrations of both E⁻ and Mg²⁺ species decrease. For E⁻, this arises because of its protonation (as discussed above), whereas for Mg²⁺, we should consider that MgE⁺ is progressively substituted by HE as pH decreases, because −COO⁻ association with H⁺ is much stronger than with Mg²⁺.

For the species related to the fixed charge (L) groups, their concentration variations shown in Fig. 6a differ from those in Fig. 5a, as a result of the presence of Mg²⁺. The L⁻ concentration in Fig. 6a increases as pH decreases, and then decreases after reaching a local maximum at around pH ~ 3.5. However, as shown in Fig. 5a, there is no such local maximum in the L⁻ concentration in the absence of Mg²⁺. To explain this, we should recall that the total amount of fixed charge groups (L_total) is constant, which is a sum of L⁻ and MgL⁺ in Fig. 6a. As pH decreases by adding H₂SO₄, the ionic strength increases, leading to a decrease in the surface potential. The latter, through the effects of Boltzmann distribution, results in a shift of positive Mg²⁺ ions from the particle–liquid interface to the liquid phase, thus favoring the dissociation of MgL⁺ on the surface and freeing more L⁻. Consequently, the L⁻ concentration increases as pH decreases for pH > 3.5 (Fig. 6a). In the region of pH < 3.5, protonation of the L⁻ species becomes dominant, and it follows that the L⁻ concentration starts to decreases with decreasing pH.

3.3 Butylacrylate-Methylmethacrylate-Acrylic Acid Copolymer Latexes

The third type of polymer colloids used here for verifying the model feasibility is represented by two butylacrylate-mehthylmethacrylate-acrylic acid copolymer latexes, referred to as P1 and P2. They were supplied by BASF SE (Ludwigshafen, Germany), produced through emulsion polymerization with Na₂S₂O₈ as initiator and sodium dodecyl sulfate (SDS) as emulsifier. The radius of the particles is 80.0 and 81.0 nm, respectively, for P1 and P2. The main difference between them is the amount of acrylic acid (AA) used in the polymerization, which is 0% for P1 and 1% for P2. It follows that the particles of P1 possess only the fixed charges $-{\mathrm{SO}}_4^{-}$ and those of P2 contain both $-{\mathrm{SO}}_4^{-}$ and −COO⁻ (fixed) charges. The SDS surfactant was removed completely using ion exchange resins [40]. Therefore, after cleaning, unlike the latexes described in Sects. 3.1 and 3.2, the P1 and P2 latexes possess only fixed charges. It should be pointed out that P1 and P2 latexes (after removing the surfactant used during polymerization) represent the simplest type of polymer colloid. Similar but different systems have also been used to successfully verify the generalized stability model [41].

3.3.1 P1 Latex with Only Sulfate Groups

For the P1 latex, the W values measured in the presence of NaCl and H₂SO₄ are reproduced from the literature [42] and shown in Fig. 7a. The corresponding pH values are reported in Fig. 7b. Because P1 possesses only the fixed charges $-{\mathrm{SO}}_4^{-}$ (denoted by I) on the particle surface, it is the simplest latex considered. For P1, we have a total of seven parameters: A_H, ${C}_{\mathrm{I},\mathrm{T}}^{\mathrm{S}}$, K_HI, K_NaI, F_0,H, F_0,Na, and δ₀. Of these, values for A_H(1.0 × 10⁻²⁰ J) and δ₀ (0.40 nm) are taken from the literature. For the total concentration of $-{\mathrm{SO}}_4^{-}$ on the particle surface, ${C}_{\mathrm{I},\mathrm{T}}^{\mathrm{S}}$, we used the result obtained previously [37] (i.e., with respect to the total added initiator during the polymerization) that 33.1% $-{\mathrm{SO}}_4^{-}$ groups remain on the particle surface. From the polymerization recipe, this corresponds to ${C}_{\mathrm{I},\mathrm{T}}^{\mathrm{S}}$= 0.0235 mol/kg polymer. The remaining four parameters, K_HI, K_NaI, F_0,H, and F_0,Na, are used as fitting parameters. In addition, because H₂SO₄ was used to destabilize the latex and tune pH, there are two associations of ${\mathrm{SO}}_4^{2-}$in the bulk disperse medium with H⁺ and Na⁺. Equilibrium constants ${K}_{{\mathrm{HSO}}_4^{-}}$ and ${K}_{{\mathrm{NaSO}}_4^{-}}$ were taken directly from the literature [30], being equal to 97.0 and 5.0 L/mol, respectively.

Fig. 7

Comparison between measured (symbols) and simulated (solid curves) values for (a) the Fuchs stability ratio W and (b) pH values, for P1 latex at ϕ = 2.0 × 10⁻⁵, destabilized by NaCl and H₂SO₄

Following the same procedure as above, the simulated values of two sets of W values are shown in Fig. 7a, as well as the corresponding pH values in Fig. 7b. All the simulations are very satisfactory, but in the case of NaCl, the simulations show significantly sharper decrease in the W value with C_S than the experimental data. This could indicate that the hydration interaction increases as more Na⁺ ions are associated on the surface. The obtained values for the unknown parameters are listed in Table 4 under P1. The obtained value for K_HI is about one order of magnitude larger than that for K_NaI, which follows the same trend as ${K}_{{\mathrm{HSO}}_4^{-}}$ with respect to ${K}_{{\mathrm{NaSO}}_4^{-}}$. Both K_HI and K_NaI values are substantially smaller than those of ${K}_{{\mathrm{HSO}}_4^{-}}$ and ${K}_{{\mathrm{NaSO}}_4^{-}}$. The obtained K_HI value is equal to 33.1 L/mol, which is comparable to the association constant of propyl sulfonate (${\mathrm{C}}_3{\mathrm{H}}_7{\mathrm{SO}}_3^{-}$) with H⁺ [30], but smaller. In fact, the association constant of ${\mathrm{SO}}_4^{2-}$ with H⁺ is also smaller than that of ${\mathrm{SO}}_3^{2-}$ with H⁺. The estimated values for the hydration force constants, F_0,H and F_0,Na for P1, are well within the range reported in the literature (1 × 10⁶–5 × 10⁸ N/m²). Note that F_0,H and F_0,Na were considered to be independent of the salt concentration. The F_0,H value is smaller than the F_0,Na value, indicating that the hydration force in the presence of H₂SO₄ is smaller than that in the presence of NaCl. Such a trend is consistent with those reported in the literature [8, 43].

Table 4 Values of the parameters for P1 and P2 latexes obtained from fitting the measured W values

With estimated values for all the model parameters, the generalized stability model has been well established for P1 latex and can be applied to analysis of the system stability. As examples, let us calculate the evolutions of the surface charge density (σ₀) and potential (ψ₀) with salt concentration, which in most cases are difficult to determine experimentally. Figure 8a, b shows the computed σ₀ and ψ₀ values as functions of the salt (NaCl or H₂SO₄) concentration. In the case of H₂SO₄, both σ₀ and ψ₀ (absolute values) decrease as the H₂SO₄ concentration increases. For σ₀, this arises because of association of added H⁺ with the surface $-{\mathrm{SO}}_4^{-}$ charge groups. For ψ₀, the decrease is the result of two factors: (1) σ₀ reduction and (2) the screening effect of an increase in ionic strength.

Fig. 8

Surface charge density σ₀ and surface potential ψ₀ computed from the generalized stability model using the estimated parameters in Table 4, for P1 latex at ϕ = 2.0 × 10⁻⁵, destabilized by NaCl and H₂SO₄

For NaCl as destabilizer, the variation in surface charge density σ₀ with NaCl concentration C_S in Fig. 8a is rather peculiar. At low C_S, the absolute σ₀ value increases as C_S increases, instead of decreasing. Only when the σ₀ value reaches a local maximum at around C_S = 0.04 mol/L, does it start to decrease with C_S. It is worth pointing out that, without the generalized stability model, such behavior would be very difficult to observe experimentally. To explain the observed phenomenon, we should first consider that the surface charge results from the association equilibria between the surface $-{\mathrm{SO}}_4^{-}$ groups and cations Na⁺ and H⁺ at the particle–liquid interface. As the NaCl concentration increases, the increased Na⁺ ions at the interface in principle drive the equilibrium towards association, thus decreasing the surface charge; however, the computed surface charge increases, which means that the increased Na⁺ concentration in the bulk disperse medium leads to a decrease (rather than an increase) in the Na⁺ concentration at the particle–liquid interface. The occurrence of this phenomenon is related to the Boltzmann equation:

$$ {C}_j^i={C}_j^b\exp \left(-\frac{z_je{\psi}_0}{kT}\right) $$

(20)

Indeed, the concentration at the interface, ${C}_j^i$, increases as the concentration in the bulk disperse medium,${C}_j^b$, increases (from Eq. 20), but the increase in ${C}_j^b$ also leads to a decrease in the surface potential (ψ₀) as a result of the screening effect. Then, the exponent term in Eq. (20) decreases with ${C}_j^b$. Thus, whether ${C}_j^i$ increases or decreases with ${C}_j^b$ depends on the combined effect of ${C}_j^b$ and the exponent term. As can be seen in Fig. 8a, at NaCl concentrations smaller than about 0.04 mol/L the absolute ψ₀ value is very large, and variations in ψ₀ result in substantial changes in the value of the exponent term. Thus, the effect of the exponent term overwhelms that of ${C}_j^b$, leading to ${C}_j^i$ decreasing with ${C}_j^b$. It follows that σ₀ increases with the NaCl concentration. With further decrease in ψ₀, because its value is small, the effect of the exponent term becomes smaller than that of ${C}_j^b$, and ${C}_j^i$ increases with ${C}_j^b$, leading to a decrease in σ₀.

3.3.2 P2 Latex with 1% PAA

On the particle surface of P2 latex, apart from the fixed $-{\mathrm{SO}}_4^{-}$ groups, there are also carboxylic groups from PAA formed from 1% acrylic acid monomer, which when ionized (–COO⁻) contribute to the surface charge. The measured W values at different pH values in the presence of NaCl and H₂SO₄, respectively, are reproduced from previous work [42] and shown in Fig. 9a; the corresponding pH values are reported in Fig. 9b. The contributions of 1% PAA to the colloidal stability can be modeled by simulating the W and pH values using the generalized stability model. Except for 1% AA, the recipe for P2 is basically the same as that for P1. Thus, to reduce the fitting parameters, we assume that the total $-{\mathrm{SO}}_4^{-}$ concentration on the particle surface is identical for P1 and P2. Then, the values for the parameters ${C}_{\mathrm{I},\mathrm{T}}^{\mathrm{S}}$, K_HI and K_NaI evaluated for P1 (Table 4) can be directly used for P2. Because of the presence of −COO⁻ groups on the surface, whose properties depend strongly on pH, we expect that the hydration force constants, F_0 , H and F_0 , Na, also vary with pH. Thus, they are still set as fitting parameters.

Fig. 9

Comparison between measured (symbols) and simulated (solid curves) values of (a) the Fuchs stability ratio W and (b) pH values, for P2 latex at ϕ = 5.0 × 10⁻², destabilized by NaCl and H₂SO₄

There are three parameters related to the −COO⁻ groups: the total carboxylic group contraction on the surface, ${C}_{\mathrm{L},\mathrm{T}}^{\mathrm{S}}$, and two association constants of −COO⁻ with H⁺ and Na⁺, K_HL and K_NaL, respectively. The ${C}_{\mathrm{L},\mathrm{T}}^{\mathrm{S}}$ value has been determined by titration and is equal to 0.0866 mol/kgP, as given in Table 4 under P2. Therefore, for P2, we have a total of four fitting parameters, F_0 , H, F_0 , Na, K_HL, and K_NaL, to be determined from simulations of W and pH values. The simulation results are shown in Fig. 9 a, b, and we can see that the agreement between experiments and simulations is very satisfactory. The obtained values for all the unknown parameters are reported in Table 4 under P2.

Let us first analyze the simulation results for W and pH corresponding to H₂SO₄, which involve only two fitting parameters, K_HL and F_0 , H. It is seen in Fig. 9 that the three W and three pH values have been excellently simulated. However, the obtained association constant of −COO⁻ with H⁺ (K_HL in Table 4), is unreasonably large, equal to 4.0 × 10⁶ L/mol, which is almost two orders of magnitude larger than the typical value of 2 × 10⁴–6.3 × 10⁴ L/mol for the association of −COO⁻ with H⁺ when −COO⁻ is present at the end of a polymer chain [30, 37, 41]. Such an extremely large K_HL value indicates that the consumption of H⁺ ions is more than the association equilibrium requires. We believe that this is a consequence of the presence of PAA chains, which are in the collapsed state in the H₂SO₄ solution. The protonated carboxylic groups (−COOH) are buried in the collapsed layer and do not participate in the association equilibrium. Because this protonated “dead” −COOH state consumes a lot of H⁺ ions, one has to increase the K_HL value to compensate for the consumed H⁺ in order to fit the measured pH values. The hydration force constant F_0 , H in the presence of H₂SO₄ is much larger for P2 than for P1 (as shown in Table 4), signifying that even in the collapsed state at such low pH, the PAA brushes lead to an increase in surface hydrophilicity as a result of the polar nature of carboxylic acid.

Using NaCl as destabilizer, because the W values in Fig. 9 were measured at different pH values (tuned using H₂SO₄), the association of −COO⁻ with H⁺ is also involved. In the simulations, we directly applied the value for the association constant of −COO⁻ with H⁺, K_HL, obtained in the case where H₂SO₄ was used as destabilizer. Thus, we have only two fitting parameters, F_0 , Na and K_NaL. The obtained value for K_NaL, the association constant of −COO⁻ with Na⁺, is equal to 2.80 L/mol (Table 4). This value is comparable with the value of 2.65 L/mol estimated for perfluoropolyether-based carboxylic groups in another work [8]. It is interesting to see that the estimated value for the hydration force constant, F_0 , Na as shown in Fig. 10, increases as pH increases. This indicates that, as pH increases, more carboxylic groups are deprotonated and the surface becomes more polar (i.e., more hydrophilic). However, such an explanation is inconsistent with the results of previous work [41], where we simulated the W values of a colloidal system stabilized by fixed carboxylic groups, but not PAA (previously measured at pH 3–10 by Behrens et al. [44]). It was found that, using the generalized stability model accounting for only the DLVO colloidal interactions, we can describe the system stability behavior in the entire range of pH 3–10, without even introducing the hydration force. This means that carboxylic acid when deprotonated acts mainly as a charge group contributing to electrostatic repulsion, and that its contribution to the hydration force is not substantial. Thus, the increase in F_0 , Na with pH (Fig. 10) cannot be explained by the hydration force. Instead, we believe that this results from the steric force of the PAA chains, which, as pH increases, become more stretched and contribute more and more to particle stability. Because steric interactions are not included in our model, they are naturally lumped into the hydration force, leading to an increase in F_0 , Na with pH.

Fig. 10

Hydration force constant F_0 , Na for P2 latex as a function of pH, estimated from W simulations using NaCl as the destabilizer at ϕ = 5.0 × 10⁻²

Based on the above discussion, we can explain why the F_0 , H value at very low pH (<1) in Table 4 is larger than the F_0 , Na values at pH ~ 2 in Fig. 10 by considering that the protonated −COOH contributes to the hydration force, but the deprotonated −COO⁻ does not. At pH 2, the amount of −COOH is reduced and the hydration force is reduced. Of course, the increase in deprotonated −COO⁻ (as well as $\hbox{--} {\mathrm{SO}}_4^{-}$) groups contributes to electrostatic repulsion, which has a dominant effect on W. In fact, the W values in the case of NaCl at pH 1.87 and 2.58 are larger than those for H₂SO₄ at pH < 1 (Fig. 9a).

Using the established generalized stability model, we calculated the surface charge density and potential (σ₀ and ψ₀) as functions of the concentration of total added salts (H₂SO₄ + NaCl), C_s. The conditions corresponded to those for the five sets of experiments shown in Fig. 9a, b, and the results are shown in Fig. 11a, b. It is seen that for pH < 3, the variations in σ₀ and ψ₀ with C_s in Fig. 11b are very similar to those in Fig. 8a, b for P1 latex. This is because at low pH most of the −COO⁻ groups are protonated and the surface charge mainly comes from $\hbox{--} {\mathrm{SO}}_4^{-}$, which is practically the same for P1 and P2 latexes. As pH increases, the surface charge density and potential increase as a result of the increase in −COO⁻ groups.

Fig. 11

Surface charge density σ₀ and surface potential ψ₀ computed from the generalized stability model using the estimated parameters in Table 4, for P2 latex at ϕ = 5.0 × 10⁻², destabilized by NaCl and H₂SO₄

4 Concluding Remarks

We have reviewed the generalized stability model developed in recent years [8] and particularly its application to various polymer colloids. The stability of a colloidal system is not only determined by the DLVO colloidal interactions but is also affected by the interplay of other physicochemical processes such as surfactant adsorption equilibrium on the particle surface, association equilibria of the surface charges with counterions involved in the system, and non-DLVO colloidal interactions. The generalized stability model can account simultaneously for such complex interplay between different processes. The model has been successfully validated through its application to different polymer colloids produced from industrial polymerization processes, and its powerful capacity to describe the stability of complicated colloidal systems demonstrated.

Various processes are involved in the model and therefore the model parameters need to be defined. The values of some of the parameters are reported in the literature, and values for the remaining unknown parameters are required. We propose that their values be estimated through application of the model to fit ad hoc experiments, such as the value of the Fuchs stability ratio as a function of salt concentration, the CCC values of different types of salts, etc. Once all the parameters are defined, the established model can be used to analyze in detail how the interplay between different processes affects the stability of the system.

From application of the generalized stability model to different polymer colloids, we have demonstrated that the interplay between various physicochemical processes is substantial. Such interplay can result in very different (peaked or monotonic) tendencies of the surface charge density and different surfactant partitioning between the particle surface and the bulk disperse medium, with respect to changes in the ionic strength, counterion type, pH, etc. Therefore, the generalized stability model can be used as a reliable tool for understanding and defining suitable conditions for obtaining the desired stability during the production and application of polymer colloids for both industrial practice and academic purpose.