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Ind. Eng. Chem. Res. 2004, 43, 7210-7219
Coagulation of Anionically Stabilized Polymer Particles Montserrat Fortuny,† Christian Graillat,‡ and Timothy F. McKenna*,‡ UNIT/ITP, Av. Murilo Dantas, 300 Farolaˆ ndia, CEP 49032-490 Aracaju SE, Brazil, and LCPP-CNRS/ESCPE, Baˆ t. F308, 43 Bd. du 11 Novembre 1918, B.P. 2077, 69616 Villeurbanne Cedex, France
The stability particles produced by emulsion polymerization and stabilized by the anionic surfactant sodium dodecyl sulfate, an electrosteric surfactant with a short ethylene oxide chain, or simply with sulfate end groups were studied by turbidity measurements. Results are compared to those of electrostatic stability models based on the Derjaguin-Landau-Verwey-Overbeek theory. The good agreement between experimental and theoretical data shows that the electrosteric surfactant could be described through electrostatic approaches and that the steric contributions could be neglected. This model was incorporated into a population balance model in order to describe the coagulation between particles in a stirred tank reactor in the absence of reaction. This model was validated with experiments in which coagulation between polymer particles was provoked by electrolyte addition. The decrease of the number of particles and the resulting monomodal particle size distribution were correctly described by the model, confirming the correct determination of coagulation rates as a function of the particle diameter and surfactant concentration. Introduction Emulsion polymerization is widely employed to produce latexes for coatings, inks, paints, and adhesives. The manufacture of a latex with desired end-use properties is a difficult task because of the heterogeneous nature of the reaction medium and the numerous mechanisms involved in the reaction. One of the main parameters influencing the final quality of the latexes is the particle size distribution (PSD). The range of particle sizes and the desired shape of the PSD can be achieved by tailoring the stability of the particles during both formation and growth of the polymer particles.1-7 To do so, it is necessary to quantify the role of the concentration and the types of surfactant and initiator species in providing the desired colloidal characteristics of the polymer particles. The influence of the surfactant and initiator species on PSD can be predicted through mathematical models that describe coagulation rates between polymer particles as a function of the particle size and operation conditions (in addition to particle creation and growth). For this, stabilization models which regard the interaction forces acting between particles, are incorporated into the well-known population balance equation approach. Such models have been used to describe emulsion and suspension polymerizations since Min and Ray8,9 proposed a comprehensive model based on the Brownian coagulation mechanism. Many of the models available in the literature that include coagulation phenomena are based on Deryaguin-Landau-VerweyOverbeek (DLVO) theory.10,11 For example, Richards et al.,12 Giannetti,13 Coen et al.,14 Zeaiter et al.,15 and Kiparissides et al.16 used this approach to model systems using electrostatic stabilization and assuming a Brownian mechanism for coalescence. In these works, reacting systems were studied where nucleation, par-
ticle growth, and coagulation can occur at the same time. While these studies showed good agreement with experimental data, it was necessary to use a number of model parameters that were highly specific to the stabilization of the polymerization in question. In addition, the simultaneous occurrence of these phenomena makes it difficult to correctly validate the coagulation approaches. To eliminate the influence of different events on the evolution of the PSD, Melis et al.17 used a nonreacting system for the study of coagulation of electrostatically stabilized poly(vinyl acetate) particles. This approach seems to be the most useful if we want to build a model for particle stabilization and validate it independently of the other phenomena. In addition, one of the difficulties often encountered is that a significant portion of the literature studies use only sodium dodecyl sulfate (SDS) as an electrostatic surfactant. However, other surfactants and types of charged surfactants (e.g., electrosteric surfactants) are commonly used in industrial recipes. These species comprise a hydrophilic “head” composed of a charged group and several ethylene oxide (EO) chains and a hydrophobic “tail” formed by an alkyl chain. The stabilization provided by ionic surfactants can be very different depending on the nature of the hydrophobic part of each surfactant molecule. Finally, the contribution of other active species to the system, and in particular that of the sulfate end groups produced by initiator decomposition and located at the particle surface, cannot always be neglected. The work presented in this paper is intended to overcome the lack of information on the stabilization of particles by surfactants other than SDS and will include the experimental and theoretical study of the stability imparted by SDS, a commercially available electrosteric surfactant, and sulfate end groups located on the surface polymer particles.
* To whom correspondence should be addressed. Tel.: 011 33 4 7243 1775. Fax: 011 33 4 7243 1768. E-mail: mckenna@ cpe.fr. † UNIT/ITP. ‡ LCPP-CNRS/ESCPE.
2. Experimental Section Latex Preparation. The composition of all latexes synthesized in this study was 80% butyl acrylate (BA) and 20% methyl methacrylate (MMA) by weight. Am-
10.1021/ie0342917 CCC: $27.50 © 2004 American Chemical Society Published on Web 04/17/2004
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7211 Table 1. Details of Recipes for Experiments Using SDS as the Surfactant BA (g) MMA (g) water (g) SDS (g) APS (g) Dp (nm) SDS2 SDS3 SDS4 SDS5
80 80 80 160
20 20 20 40
900 900 900 800
1.70 0.89 0.66 0.15
0.75 0.75 0.75 0.75
82 100 116 140
Table 2. Details of Recipes for the First Series of Experiments Using TA as the Surfactant TA4 TA5 TA6 TA7 TA8
BA (g)
MMA (g)
water (g)
TA (g)
APS (g)
Dp (nm)
80 80 80 160 250
20 20 20 40 60
900 900 900 800 700
1.60 0.83 0.56 0.15 0.45
0.26 0.26 0.26 0.27 0.26
70 90 102 135 141
monium persulfate (APS) was used as the initiator. These reagents, along with SDS, were obtained from Acros Organics (Geel, Belgium) and used as received. The electrosteric surfactant (TA) used in some of the experiments was Disponil FES 32 IS [a sodium salt of the sulfate of poly(glycol ether)]. This surfactant was supplied by Cognis (Meaux, France) and used as received. It was found to have a molecular weight of 464. It has a 12-carbon alkyl chain and four EO groups that impart a slightly steric character to this surfactant. It was found by Schneider et al.4 to have a critical micelle concentration of 0.2 g/L. Seed latexes were prepared in a batch in a 3-L jacketed glass vessel, equipped with a condenser and cooling jacket. The reactor temperature was controlled at (1 °C by means of adjustment of the temperature of the water bath. All polymerizations were performed at a reactor temperature setpoint of 70 °C. Tables 1-3 show recipes and average particle diameters (Dp) of the final latexes for syntheses carried out using SDS and TA as surfactants. The average particle sizes of the latexes were measured using quasi elastic light scattering (QELS) with either Malvern LoC or Malvern 4800 (multiple angle) purchased from Malvern Instruments (Lyon, France). The average particle sizes reported are an average of 10 measurements for each value. Measurement of the Surface Charge Density of Sulfate Groups. The latexes were diluted at solids contents of 2-3%, and surfactant and oligomers were removed by successive cleaning steps using a mixture of ion-exchange resins, Dowex MR-3 mixed-bed ionexchange resin (Aldrich, Geel, Belgium), with the technique described by Vanderhoff et al.18 The withdrawal of surfactant was monitored by measuring the conductivity, and it was assumed that the washing step was complete when the conductivity did not change from one sample to the next. The evolution of the conductivity of an aliquot of cleaned latex was followed while it was being titrated with NaOH in a solution at 9.9 × 10-4 mol‚dm-3. The amount of sulfate groups covalently adsorbed onto the surface particle was assumed to be equivalent to the number of NaOH molecules needed to reach the minimum of the conductivity curve (neutralization point). The surface charge density was then related to the total mass of the sulfate groups titrated (mA-) using the following equation:
σ)
m A-
e MW(A ) NpπDp2 -
(1)
where MW(A-) is the molecular weight of anionic species A-, e the charge of a mole of electrons, and Np the number of polymer particles. Equation 1 was also employed to calculate the surface charge density of surfactants on “unwashed” latexes. In this case, it was supposed that all surfactant added into the reactor was adsorbed onto the surface of particles and titration measurements were not performed. As we shall see below, this assumption appears to be valid, and given the structure of the anionic surfactant used in this work, it is reasonable. Turbidity Measurements. The stability of polymer latexes was evaluated using turbidity measurements, where the kinetics of coagulation was followed by measurement of the slope of the turbidity curves versus time. All measurements were performed at a constant temperature of 25 °C. The stability ratio (W) in homocoagulation processes is defined as the ratio of the rate of rapid to slow coagulation processes (eq 2), where τ is the turbidity and CE the electrolyte concentration. For
W)
(dτ/dt)0,CE>CCC
(2)
(dτ/dt)0,CE
electrolyte concentrations higher than the critical coagulation concentration (CCC), the electrostatic repulsive forces are completely canceled and rapid coagulation occurs as a result of the Brownian motion of the polymer particles; below this point, coagulation is slower. The use of eq 2 is based on the assumption that turbidity is a linear-dependent function of time. As will be demonstrated later, this assumption is only true for the initial moments of the coagulation process. During the initial stages of a homocoagulation process, the latex contains both coagulated and uncoagulated particles, with the former being formed by coagulation of two uncoagulated particles. Both of these particle types contribute to the turbidity of the polymer emulsion according to the Mie theory,19 which relates the turbidity to the concentration and the size of each particle type by eq 3, where κti, Ni, and ri are the optical n
τ)
κtiπNiri6 ∑ i)1
(3)
constant, the number of particles, and the particle radius of the ith-type particles, respectively. For rapid coagulation (during the early stages), population balances of each particle type including the coagulation rates are20 given by
dN1 ) -β11N12 dt
(4)
dN2 1 ) β11N12 dt 2
(5)
where N1 and N2 are noncoagulated and coagulated particle concentrations, respectively, and β11 is the coagulation rate between two noncoagulated particles. When eqs 3-5 are combined, the dynamic turbidity can be calculated from eq 6,
τ)
N0 1 κ πr 2 + κt2πr22N0β11t 1 + N0β11t t1 1 2
(
)
(6)
where N0 is the initial concentration of the polymer particles.
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Table 3. Details of Recipes for the Second Series of Experiments Using TA as the Surfactant Compato25 Compato20
BA (g)
MMA (g)
water (g)
TA (g)
APS (g)
Dp (nm)
170 600
47 150
727 2530
0.24 1.44
0.43 0.89
147 141
On the basis of the last equation, it can be seen that turbidity is a linear function of time solely for short times, when the following condition is achieved: β11N0t , 1. The linear evolution of turbidity can also be observed for slow coagulation processes. In this case, population balances are related to the coagulation rates and to the stability ratio, which accounts for the interaction between the polymer particles as represented by eq 7.
β11 2 dN1 )N dt W 1
(7)
The turbidity technique can provide accurate estimations of the stability of the polymer particles through measurements that are performed easily. This allows one to infer stability data of polymer latexes for wellknown experimental conditions. Turbidity measurements were performed in a turbidimeter that performs differential measurements of the intensity of scattered light from both reference and sample cells. During a typical turbidity measurement, 2.1 mL of diluted latex (2% solids content) were charged into both cells and 0.9 mL shots of electrolyte solution were added discontinuously. To perform turbidity measurements on the same volume and dilution in both the reference and sample cells, the volume of the reference cell was completed with 0.9 mL of deionized water. Online measurements of turbidity were conducted during the first 20 s after an electrolyte shot. This is the duration needed for the coagulation process to stop and the number of polymer particles to reach steady state. Coagulation Experiments in the Reactor. Coagulation kinetics were also studied with experiments conducted in a glass reactor with a polymer seed (again here in the absence of reaction). The reactor was initially charged with a seed latex, which was stirred throughout the experiment. Experiments were performed at a constant temperature of 25 °C. Coagulation of the polymer particles was provoked by successive additions of a known amount of an electrolyte solution at fixed time intervals (30 min). The electrolytes used in this work were NaCl and MgCl2. Samples were taken between two successive electrolyte injections to measure the average particle sizes. It was observed that the coagulation of the polymer particles is important during the early times after the electrolyte addition. As the number of polymer particles was reduced and the surface density of the particles was sufficiently increased, the rate of coagulation of the polymer particles diminished to attain low coagulation rates. 3. Model Development Electrostatic Stabilization Model. Interactions between particles determine the colloidal properties of polymer latexes. For latexes stabilized with ionic surfactants or/and charged groups on the surface of the particles (e.g., sulfate groups from initiators such as APS), the stabilization mechanism is based on the creation of electrostatic repulsive forces between polymer particles. The ionic species physically or chemically adsorb onto the particle surface, forming a charged layer near the surface. The surface charges are in equilibrium with
counterions in both the inner and diffuse regions of the electrical double layer. The inner part is called the Stern region, where counterions are strongly adsorbed onto ions located on the particle surface. The diffuse part is called the Gouy-Chapman region, where counterions are freely distributed because of the lower attractive force between surface ions and counterions. On the basis of this representation, Verwey and Oberbeek10 and Deryaguin and Landau11 established the DLVO theory applied to electrically charged surfaces submerged in a diluted solution of salts. According to this theory, the total potential energy of interaction (V) can be determined as the sum of the attraction energy (VA) and the repulsion energy (VR):
V ) VA + VR
(8)
Attractive forces between polymer particles are produced from the interaction between the temporary dipole on one molecule and the induced one on a neighboring one.21 This energy is proportional to the semiempirical Hamaker constant (A), which depends on the polymer properties, the molecular polarity, and the properties of the liquid phase where particles are dispersed. For polymer latexes, Blackey22 proposed values of the Hamaker constant between 0.2 and 1.7 × 10-20 J. The energy of attraction between two interacting particles whose radii are defined by ri and rj can be described by the following equation according to Hamaker:23
VA ) -
[
2rirj 2rirj A + 2 + 2 2 6 R - (r + r ) R - (ri - rj)2 i j ln
(
)]
R2 - (ri + rj)2
R2 - (ri - rj)2
(9)
where R is the center-to-center separation. The repulsive energy potential is determined as a function of the zeta potential (ζ) of each particle, the Debye-Hu¨ckel parameter (κ), and the distance between the surfaces of the two particles (L), as proposed by Hogg et al.:24
VR )
rirj(ζii + ζij)
4(rr + rj) 2ζiζj 1 + exp(-κL) ln + ln[1 - exp(-2κL)] i i 1 - exp(-κL) ζi + ζj
{
[
]
}
(10)
where the permittivity constant of the aqueous phase () is a function of the permittivity constant of vacuum (0) and water (r):
) 4π0r
(11)
The distance between the surfaces of the two particles is calculated as a function of the particle radius of each particle and the center-to-center separation by the following equation:
L ) R - (ri + rj)
(12)
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7213
The Debye-Hu¨ckel parameter is related to the ionic force (I) by the equation
κ)
(
)
8πNAIe2 kBT
0.5
(13)
where e is the electron charge, kB is the Boltzmann constant, and T is the temperature. The ionic force included in this equation is given by eq 14, n
I)
Cizi2 ∑ i)1
(14)
where Ci and zi are the concentration and the valence of the ionic species, respectively. The ζ potentials are determined as a function of the surface potential (ψ) and the Stern layer thickness (∆) by the following equations:10
ζ)
[ [
] ]
exp(λ4) + 1 2kBT ln z+e exp(λ4) - 1
λ4 ) κ∆ + ln
exp(λ5) + 1
exp(λ5) - 1
λ5 ) z+eψ/2kBT
(15)
(16) (17)
The surface potential can be determined as a function of the surface charge density (σ) as described by eqs 18 and 19. The appropriate description of this potential depends on the product κr. For values of this product lower than 1, spherical surfaces may be approximated to plate surfaces and use the approach proposed by Verwey and Overbeek10 as represented by eq 18. For values of the product higher than 1, spherical geometry has to be assumed, and then surface potentials are calculated by eq 19,10
4πrσ ψ) (1 + κr) ψ)
(18)
(
2kBT 2πreσ sinh-1 e κkBT
)
(19)
where r is the ratio of the polymer particles. Finally, the coagulation rate of two particles of size i and j (Bij) is related to Fuch’s stability ratio (Wij) as described by eq 21, which depends on the total potential of these particles as described by eq 20.10 The previous
Wij ) 2(ri + rj)
Bij ) Bij )
∫0
∞
exp
( ) V k BT
R2
dR
2kBT (ri + rj)2 3ηWij rirj
(20)
(21)
definition of Fuch’s stability applied to slow coagulation cases. Here, positive values of the total potential energy (negligible effect of attractive forces) lead to values of Fuch’s stability ratio higher than unity. On the other hand, values of Fuch’s stability ratio lower than 1 can also be found when the total potential energy is strongly
Figure 1. Surface particle density as a function of the particle size obtained for batch polymerizations with SDS and TA as surfactants.
influenced by the attractive forces, and the total energy can take on negative values. Note that experimental stability ratios are calculated through eq 2 and that values of this relation lower than unity cannot be obtained. Thus, to compare both experiments and model results on the same basis, the experimental data can be described through the definition of Fuch’s stability ratio proposed by Romero-Cano et al.25 represented as
∫0
∞
Wij )
exp
( ) V k BT
s2 VA exp k BT ∞
( )
∫0
s2
ds (22) ds
where s is defined by the following expression:
s)
L ri + rj
(23)
4. Results and Discussion Measurement of the Stability Ratio. The stabilization provided by sulfate end groups, SDS and TA, was studied for batch emulsion polymerizations (recipes are shown in Tables 1 and 2). It is important to point out that all of these species are anionic compounds that have a sulfate ion as the hydrophilic “head”. For TA, this hydrophilic part is also composed of four EO chains, which contribute to steric stabilization. SDS and TA surfactants are also composed of a hydrophobic “tail” formed by an alkyl chain of 12 carbons. The only difference between SDS and TA molecules is the EO chains, which can alter the stability provided by the two compounds. Obviously, this kind of stability is insignificant for sulfate radicals. Figure 1 shows the diameter of the final polymer particles as a function of the surface charge density (provided by the surfactants) for reactions performed with different amounts of TA or SDS. For both surfactants, similar trends can be observed for the evolution of the particle diameter as a function of the surface density charge (proportional to the surfactant coverage).
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Table 4. Values of CCC Determined by Turdidity Measurements for Seed Latex Compato25 and Three Stabilization Systems
a
expt no.
stabilization system
electrolyte
σsulfate, µC‚cm-2
σsurfactant, µC‚cm-2
CCC, mol‚dm-3
1 2 3 4 5 6 7 8
sulfate radicals SDS SDS SDS TA TA TA TA
NaCl NaCl NaCl NaCl NaCl NaCl NaCl MgCl2
0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57
0.00 0.80 5.12 11.64 0.71 1.27 5.00 5.00
0.06 0.15 0.71 1.47 0.21 0.48 a 0.54
No rapid coagulation was observed for NaCl concentrations up to 1.5 mol‚dm-3.
Figure 2. Stability ratio versus electrolyte concentration for seed latex Compato25 stabilized with SDS at three different concentrations (see Table 4).
Figure 3. Stability ratio versus electrolyte concentration for seed latex Compato25 without surfactant addition and stabilized with TA surfactant at three different concentrations (see Table 4).
Nevertheless, it can be seen that the surfactant type has an important effect on Dp, and different surface charge densities for each surfactant are needed to stabilize particles with equal Dp. TA leads to latexes with much smaller particles than latexes stabilized with SDS, and then it seems that electrosteric surfactant imparts stronger stability on the polymer particles. The degree of stability offered to the polymer particles by sulfate end groups, SDS and TA, was determined using turbidity measurements. All runs were performed using cleaned Compato25 latex as the polymer particles seed and NaCl or MgCl2 as electrolytes. In the case of the sulfate macroradicals, cleaned seed particles were used without surfactant addition. For SDS and TA surfactants, runs were performed for different surfactant concentrations. The samples were prepared by the addition of increasing amounts of a surfactant solution to the cleaned seed latex that was left to reach equilibrium for at least 30 min. At this point, particles were stabilized by both surfactant and sulfate macroradicals. Then the strength of the stability measured has the contribution of the two stabilization species. As we will see later, the stabilization imparted by the surfactant is much stronger than that offered by sulfate end groups, and for high surfactant concentrations, the sulfate macroradicals’ contribution to the stability can be neglected. Figures 2 and 3 show the experimentally measured stability ratio versus electrolyte concentration curves for both stabilization systems and for different surfactant concentrations. In addition to calculating the stability ratio, one can also determine the CCC of each latex sample. This concentration was obtained from the intercept with the abscissa of the log W-log C plots,
and the results are given in Table 4. Note that for experiment 8 (TA) the electrolyte used was MgCl2 because rapid coagulation was not observed when monovalent electrolytes were added to systems where the surface charge density was over 5 µC‚cm-2. To complete the experiment, it was necessary to switch over to a bivalent salt (otherwise, the experimental procedure remains identical). The stabilization of the polymer particles for each system can be compared through the obtained values of CCC. The lowest value of CCC was determined for sulfate end groups, where no surfactant was added. The addition of small amounts of surfactant to the cleaned seed latex produces an increase of CCC due to the improvement of the polymer particles stability. When both surfactants were compared, the highest stability ratio was obtained for the TA surfactant, which imparts stronger stability of the polymer particles than that offered by SDS surfactant systems. It can be clearly demonstrated that the degree of stability imparted by the sulfate groups to a latex also stabilized by the ionic surfactants is negligible for the cases studied here where either TA or SDS was used. Estimation of the Hamaker Constant. The Hamaker constant is the only adjustable parameter for the determination of the force balance between the polymer particles and the subsequent computation of the coagulation rates in the electrostatic stabilization model described above. For this reason, we can use the model and experimental data to fit reasonable values for A. Simulations of the electrostatic model were carried out in order to estimate the Hamaker constant for sulfate end groups, SDS and TA. Fuch’s stability ratio
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7215
Figure 4. Experimental and theoretical dependence of W on the [electrolyte] for seed Compato25 stabilized with SDS at three different concentrations.
TA, which corresponds to the stability system that also gives the highest CCC and the smallest particles (for a given surfactant concentration, in moles). On the other hand, it can be observed that experimental values of Fuch’s stability ratios are correctly described based on the electrostatic stability model, even for TA, which could be considered an electrosteric surfactant. The presence of the EO groups (with respect to SDS where there are not any but there is a molecule with the same alkyl chain and ionic species) enhances the stabilization of the polymer particles because of a different conformation of surfactant molecules on the polymer particles. These facts explain the good description of the stabilization of the electrosteric surfactant through the electrostatic stabilization model. In other words, it can be seen that the short EO chain on TA leads to a decrease in the Hamaker constant (with respect to SDS) and that the DLVO model accounts for the predominantly electrostatic nature of the stabilization. Modeling Evolution of PSD for Coagulation Experiments in the Reactor. On the basis of the estimates of the Hamaker constant, the electrostatic stabilization model was introduced to a population balance in order to determine the coagulation rate between two polymer particles as a function of their particle sizes. The population balances were used to describe the evolution of the PSD for nonreacting systems, where coagulation provoked by the addition of an electrolyte is the unique phenomenon that influences the density function. The population balance for this system is presented in eq 24,
∂f(mi,t) f(mi,t) mf β(mi,mj) f(mj,t) dmj + )∂t VT m0 1 mi-m0 β(mi-mj,mj) f(mi-mj,t) f(mj,t) dmj (24) 2VT m0
∫
Figure 5. Experimental and theoretical dependence of W on the [electrolyte] for seed Compato25 without surfactant addition and stabilized with TA at three different concentrations. Table 5. Parameter Values of the Electrostatic Stabilization Model Used in the Simulations parameter e kB ∆ 0 r η
1.60 × 10-19 26 1.38 × 10-23 26 1.40 × 10-10 23 8.85 × 10-12 27 8026 10-3 26
C J‚K-1 m C2‚N-1‚m-2 dimensionless kg‚m-1‚s-1
Table 6. Values of the Estimated Hamaker Constants for the Three Stabilization Systems stabilization system
A, J
sulfate macroradicals SDS TA
0.70 × 10-20 0.40 × 10-20 0.22 × 10-20
was calculated using eq 21, and theoretical results were compared to experimental data obtained from turbidity measurements. The values of the different parameters used in the simulations are shown in Table 5. Figures 4 and 5 show the experimental data compared to simulation results using the different estimates of the Hamaker constants resumed in Table 6. The Hamaker constant is related to the attractive forces between the polymer particles. Thus, low values of this constant correspond to a higher stability of polymer particles due to the decrease of the attractive forces. The lowest Hamaker constant was estimated for
∫
where f(mi,t) is the density function of the polymer particles of mass mi, VT is the reaction volume, and β(mi,mj) is the coagulation rate between two polymer particles of mass mi and mj. The density function was defined as the fraction of particles of each class expressed by the equation
f(m,t) dm )
N(m,t) dm NT(t)
(25)
where N(m,t) is the number of particles of mass m to m + dm at time t and NT(t) is the total number of particles at time t. The population balance is solved by introducing the coagulation rates given by eq 21, and Fuch’s stability ratio was computed using eq 20. To solve the population balance, the PSD is discretized into 200 mesh points (weight fractions) using regular central finite differences. Numerical integrations were performed using a Fortran code and DASSL subroutines.28 The initial PSD introduced in the model is that of the seed latex, which was represented as a normal distribution whose mean is the experimentally measured average of Dp, and the variance of the distribution was assumed to be 0.10 (typical for monodisperse distributions19,21,29) and similar to the experimental value given by QELS (multiple angle) equal to 0.08. The mathematical model for PSD was validated for coagulation experiments carried out in the reactor. These experiments were performed as described above,
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using Compato20 (the recipe is shown in Table 3), diluted to 9% solids content, as the seed latex (without a cleaning step) and a solution of NaCl (3.9 mol‚dm-3) as the electrolyte solution. A fixed volume of 0.6 mL was added into the reactor every 30 min. This length of the time step was chosen in order to ensure that samples reached low coagulation rates. During these experiments, it was necessary to verify whether there was formation of macroscopic coagulum in the reactor because this did occur and was thought to be a problem. The formation of lumps was mainly observed after the first electrolyte injections, when the coagulation rates are relatively high because of the relatively small sizes of the particles at this point. The injection of the electrolyte solution into the reactor can also lead to a sharp increase in the local electrolyte concentration, resulting in the excessive local flocculation of particles and thus the formation of coagulum. When it occurs, the coagulum grows rapidly during the initial experiment because of deadsorption of the polymer particles of the emulsion when the electrolyte concentration increases. To control the situation and eliminate the effect of the growth of the initial coagulum, when it was observed, the reactor was drained and the coagulum removed as many times as necessary until coagulum was no longer formed. Because the formation of coagulum can modify quantities such as the particle size and solids content, it was necessary to recharacterize the reactor contents after each cleaning step before the seed could be reintroduced into the reactor. Two runs were performed in order to determine the coagulation rates of the polymer particles of seed Compato20, to test the model, and to understand by how much the formation of coagulum perturbed the model predictions. An additional amount of TA (0.328 g) was added to the seed latex in order to increase the surface charge density to 3.5 µC‚cm-2 and thus reduce the coagulation rates. Macroscopic coagulum was observed to have formed during the six first electrolyte injections, but this was removed as described above, and by the end of the experiment, the latex contained less than 1% of scrap (with respect to the initial seed latex weight). In run 2, no additional surfactant was added and the initial surface density charge was 1.02 µC‚cm-2. In addition, this experiment was performed without removal of the coagulum formed during the initial electrolyte injections, and the final latex contained 30% scrap. Figures 6-9 show the evolution of Np and Dp for runs 1 and 2. Experimental data are compared to simulation results based on the population balance and electrostatic stabilization model using the Hamaker constant for TA estimated from the turbidity measurements. As discussed earlier, the influence of the sulfate macroradicals on particle stability can be neglected when an ionic surfactant such as TA or SDS also contributes to particle stabilization. In the case of run 1 (Figures 6 and 7), the simulations were begun only after the end of the sixth injection of electrolyte, once the formation of coagulum was largely reduced. Both experimental and simulation results describe the following: (i) a fast decrease of the number of particles immediately following the electrolyte addition and then a rapid dropoff in the coagulation rate; (ii) a decrease of the coagulation rate as the particle diameter increases. Note that these results were also observed for batch systems, where bigger and bigger particles were obtained and the surface charge density
Figure 6. Evolution of Np with time for successive electrolyte injections for seed Compato20 and a surface density charge of 3.5 µC‚cm-2.
Figure 7. Evolution of Dp with time for successive electrolyte injections for seed Compato20 and a surface density charge of 3.5 µC‚cm-2.
Figure 8. Evolution of Np with time for successive electrolyte injections for seed Compato20 and a surface density charge of 1.02 µC‚cm-2.
provided by the surfactant is insufficient to allow small particles to be stabilized. The agreement between the model and experiment for run 1 is satisfactory, and the simulations are clearly able
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7217 Table 7. Experimental and Theoretical Values of the Average Dp and the Variation Coefficient of the PSD as a Function of Time time (min)
avg Dp(exp), nm
expt polydispersity coefficient (from Malvern software)
avg Dp(theor), nm
theorl variation coefficient
180 209 239 329 359
177 195 216 245 250
0.08 0.07 0.04 0.05 0.06
175 201 218 243 259
0.101 0.109 0.105 0.107 0.105
Figure 9. Evolution of Dp with time for successive electrolyte injections for seed Compato20 and a surface density charge of 1.02 µC‚cm-2
Figure 10. Evolution of PSD with time for different electrolyte concentrations for Compato 20 and a surface density charge of 3.5 µC‚cm-2.
to describe the evolution of the average particle size and, implicity, to qualitatively describe the coagulation rates as a function of the electrolyte concentration and particle diameter and also to predict the correct number of polymer particles at the end of each electrolyte injection. Similar trends were obtained for both experimental and simulation results for run 2 but with less satisfactory agreement. The trends in the simulation appear to follow the trends in the data, but the final predictions are incorrect. For example, the model predicts larger particle sizes than can be found experimentally. In itself, this is not surprising. Recall that in run 2 the scrap formed during the initial injection was not removed from the reactor, which altered the stabilization conditions in a way that is not accounted for with the model presented here. At least part of the discrepancy between the model and data in this case is most likely due to poor mixing in the reactor. During the early stages of electrolyte addition, this leads to local “hot spots” of concentration that are poorly described by the model (that implicitly assumes perfect mixing). Once the coagulum has formed, it consumes the surfactant in an unpredictable manner and leads to slight differences between model predictions and the experimental data. However, even though the model does not correctly predict the particle size at the end of the experiment, the coagulation rate seems to be reasonably well predicted. After each electrolyte addition, the average particle sizes of the polymer particles and the breadth of the PSDs were measured using QELS29 to investigate the effect of the electrolyte-induced coagulation on the breadth of the PSDs. Experimental results showed that coagulation phenomena lead to monomodal distributions, and no increase of the breadth of PSDs was observed. As shown in Table 7, the experimental variation coefficients of the PSDs provided by the software of the apparatus are lower than 0.10, which is the value suggested by the manufacturer (Malvern Instruments Ltd.)
for monomodal distributions.19 These results agree well with simulation data, as can be observed in Figure 10, where the evolution of PSDs is shown for run 1. As can be observed in Table 7, the variation coefficient calculated for the model PSD does not increase with time. This indicates that the model is able to correctly predict the evolution of the PSD, the coagulation rates as a function of the electrolyte concentration, and the average particle diameter for the coagulation experiments. 4. Conclusions The stabilization of the polymer particles was investigated from experimental and theoretical points of view for three different stabilization species: sulfate macroradicals, SDS, and TA surfactants. First of all, TA can be considered to be an electrosteric surfactant, wherein the hydrophilic part of the chain is composed of a sulfate ion and four EO chains. This small steric contribution results in a lower minimal surface charge density for polymer particles synthesized using TA than those reached using SDS in batch polymerizations (i.e., given the same amount of monomer, initiator, and water, one can make more smaller particles with an equivalent amount of TA than is possible with SDS). Turbidity measurements were used to determine the stability imparted by each stabilizing species and to identify the relative influence of SDS, TA, and sulfate end groups on the stabilization of particles when several mechanisms act at the same time. The highest stability values were found for TA surfactant systems, and the lowest values were observed for polymer particles without surfactant addition, in which the stability was assured only by the sulfate end groups on the surface of the particles. As expected, the stability provided by the sulfate end groups is negligible compared to the contributions of either of the anionic surfactants. The experimental stabilities have been represented by an electrostatic stability model in order to estimate Ha-
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maker constants for each stabilization system. This constant is the only unknown parameter in the electrostatic model. The estimated Hamaker constants were 0.70 × 10-20 for sulfate end groups, 0.40 × 10-20 for SDS, and 0.22 × 10-20 for TA. The electrostatic stability model was then incorporated into a population balance in order to describe the evolution of the PSD during the addition of electrolytes for a nonreacting system. The model was validated for coagulation experiments carried out in a stirred tank reactor where the polymer particles were once again destabilized by electrolyte addition. The experimental PSD and coagulation rates of particles stabilized with TA were described by the model using population balances. Note that the model predictions showed much better agreement with experimental data in the absence of macroscopic coagulum, which is reasonable because it was not designed for this eventuality! Finally, it is interesting to note that although the steric contributions of the EO groups in the TA chains increased its ability to create and stabilize particles with respect to SDS, the presence of these groups did not change the fact that the stabilization behavior was well-described using an electrostatic model with only the Hamaker constant to be found for the system. Acknowledgment The authors are grateful to Atofina for financial support of this work. Nomenclature A ) Hamaker constant (J) CCC ) critical concentration coagulation (mol‚dm-3) Dp ) particle diameter (nm or m) e ) electron charge (C) f(m,t) dm ) particle fraction of particles of mass m to m + dm at time t I ) ionic force (mol‚dm-3) kB ) Boltzmann constant (J‚K-1) L ) distance between the surfaces of the two particles (m) mi ) mass of the ith-type particles/species (g) Ni ) particle concentration of the ith-type particles (dm-3) Np ) total number of particles NT ) total concentration of polymer particles (dm-3) R ) center-to-center separation (m) ri ) diameter of particle i in a two-body collision (m) T ) temperature (K) V ) total potential energy (J) VA ) attraction potential energy (J) VR ) repulsion potential energy (J) VT ) total volume reactor (dm3) W ) stability ratio βij ) coagulation rate between ith- and jth-type particles (m3‚s-1) ) aqueous-phase permittivity C2‚N-1‚m-2) 0 ) vacumm permittivity r ) water permittivity (C2‚N-1‚m-2) η ) aqueous-phase viscosity (kg. m-1.s-1) κ ) Debye-Hu¨ckel parameter (m-1) κti ) optical constant σ ) surface density charge (µC‚cm-2) τ ) turbidity (m-1) ψ ) surface potential (V) ζ ) zeta potential (V)
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Received for review December 8, 2003 Revised manuscript received February 6, 2004 Accepted February 13, 2004 IE0342917
