Langmuir 2007, 23, 10323-10332
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Effect of Temperature and Surfactant Type on the Stability and Aggregation Behavior of Styrene-Acrylate Copolymer Colloids Zichen Jia, Hua Wu, Jianjun Xie,† and Massimo Morbidelli* Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland ReceiVed May 4, 2007. In Final Form: June 29, 2007 The colloidal stability, aggregation kinetics, and cluster structure of two styrene-acrylate copolymer latexes, stabilized with an aliphatic sulfonate and an aliphatic carboxylate surfactant, respectively, have been investigated experimentally in the temperature range between 283 and 323 K. The main objective of this study is to investigate the role of temperature and surfactant type on the aggregation kinetics and cluster structure. For this, the values of the Fuchs stability ratio and the time evolutions of the average radius of gyration, hydrodynamic radius, and structure factor of the clusters have been determined using static and dynamic light scattering techniques at different temperatures. It is found that although the two latexes exhibit a somewhat different dependence of the colloidal stability on temperature, all of the values of the average radius of gyration (or hydrodynamic radius) measured at different temperatures and surfactant types, which are plotted as a function of a properly defined dimensionless time, collapse to form a single master curve. Similarly, all of the measured average structure factors also collapse to form a single master curve when they are plotted as a function of the wavevector normalized using the average radius of gyration. These results indicate that, at least for the conditions investigated in this work, the aggregation mechanism and cluster structure are independent of temperature and surfactant type.
1. Introduction Studies of the response of a stagnant colloidal system to changes in the ionic strength in the disperse medium have been the main tools for investigating the stability behavior of colloidal systems. These include monitoring the kinetics and structure of cluster growth,1-14 measuring the critical coagulant concentration (ccc) for various coagulants,15-19 and determining the dependence of the Fuchs stability ratio on the coagulant concentration.18-21 * To whom correspondence should be addressed. E-mail: morbidelli@ chem.ethz.ch. Tel: +41 44 632 30 34. Fax: +41 44 632 10 82. † Current address: College of Material Science and Engineering, Central South University of Forestry and Technology, Changsha 410004, Hunan, PR China. (1) Family, F.; Meakin, P.; Vicsek, T. J. Chem. Phys. 1985, 83, 4144. (2) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Nature 1989, 339, 360. (3) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Phys. ReV. A 1990, 41, 2005. (4) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Klein, R.; Ball, R. C.; Meakin, P. J. Phys.: Condens. Matter 1990, 2, 3093. (5) Broide, M. L.; Cohen, R. J. J. Colloid Interface Sci. 1992, 153, 493. (6) Carpineti, M.; Giglio, M. Phys. ReV. Lett. 1992, 68, 3327. (7) Axford, S. D. T. J. Chem. Soc., Faraday Trans. 1997, 93, 303. (8) Odriozola, G.; Tirado-Miranda, M.; Schmitt, A.; Lopez, F. M.; CallejasFernandez, J.; Martinez-Garcia, R.; Hidalgo-Alvarez, R. J. Colloid Interface Sci. 2001, 240, 90. (9) Hanus, L. H.; Hartzler, R. U.; Wagner, N. J. Langmuir 2001, 17, 3136. (10) Lattuada, M.; Sandkuhler, P.; Wu, H.; Sefcik, J.; Morbidelli, M. AdV. Colloid Interface Sci. 2003, 103, 33. (11) Wu, H.; Lattuada, M.; Sandkuhler, P.; Sefcik, J.; Morbidelli, M. Langmuir 2003, 19, 10710. (12) Sandkuhler, P.; Sefcik, J.; Morbidelli, M. J. Phys. Chem. B 2004, 108, 20105. (13) Wu, H.; Xie, J.; Morbidelli, M. Biomacromolecules 2005, 6, 3189. (14) Sandkuhler, P.; Lattuada, M.; Wu, H.; Sefcik, J.; Morbidelli, M. AdV. Colloid Interface Sci. 2005, 113, 65. (15) Overbeek, J. T. G. In Colloid Science; Kruyt, H. R., Ed.; Elsevier: Amsterdam, 1952; Vol. 1. (16) Ottewill, R. H.; Shaw, J. N. Discuss. Faraday Soc. 1966, 42, 154. (17) Mabire, F.; Audebert, R.; Quivoron, C. J. Colloid Interface Sci. 1984, 97, 120. (18) Jia, Z.; Gauer, C.; Wu, H.; Morbidelli, M.; Chittofrati, A.; Apostolo, M. J. Colloid Interface Sci. 2006, 302, 187. (19) Jia, Z.; Wu, H.; Morbidelli, M. Ind. Eng. Chem. Res. 2007, 46, 5357. (20) Holthoff, H.; Egelhaaf, S. U.; Borkovec, M.; Schurtenberger, P.; Sticher, H. Langmuir 1996, 12, 5541.
In the fundamental studies of colloidal aggregations, similar to those of chemical reactions, the most important issue is to understand if different aggregation (reaction) processes follow the same aggregation (reaction) mechanism (i.e., the same cluster (product molecule) structure and the same aggregation (reaction) kinetics. An effective way to study the aggregation mechanism is to measure the time evolutions of both the radius of gyration and the hydrodynamic radius as well as the cluster structure factor using static and dynamic light scattering (SLS and DLS) techniques.10,12,14,22-25 Such measured quantities can then be compared with the corresponding values calculated using the Smoluchowski kinetic model, which is based on the following population balance equations (PBE)4-8,10,13,14,24,26-28
dNi dt
)
1i - 1
imax
Ki,jNiNj ∑ Ki-j,jNi-jNj - ∑ j)1
2 j)1
(1)
where Ni is the number concentration of clusters containing i primary particles at time t and Ki,j is the aggregation rate constant (or kernel) between two clusters containing i and j primary particles, respectively. This is a complex function of the mobility, structure, and other physicochemical properties of the clusters and can be generally expressed as7,10,24
Ki,j )
KB (i1/Df + j1/Df)(i-1/Df + j-1/Df) λ (ij) W 4
(2)
(21) Lopez-Leon, T.; Jodar-Reyes, A. B.; Bastos-Gonzalez, D.; Ortega-Vinuesa, J. L. J. Phys. Chem. B 2003, 107, 5696. (22) Bushell, G. C.; Yan, Y. D.; Woodfield, D.; Raper, J.; Amal, R. AdV. Colloid Interface Sci. 2002, 95, 1. (23) Lattuada, M.; Wu, H.; Sandkuhler, P.; Sefcik, J.; Morbidelli, M. Chem. Eng. Sci. 2004, 59, 1783. (24) Sandkuhler, P.; Sefcik, J.; Lattuada, M.; Wu, H.; Morbidelli, M. AIChE J. 2003, 49, 1542. (25) Sandkuhler, P.; Sefcik, J.; Morbidelli, M. Langmuir 2005, 21, 2062. (26) Gardner, K. H.; Theis, T. L. J. Colloid Interface Sci. 1996, 180, 162. (27) Schmitt, A.; Odriozola, G.; Moncho-Jorda, A.; Callejas-Fernandez, J.; Martinez-Garcia, R.; Hidalgo-Alvarez, R. Phys. ReV. E 2000, 62, 8335. (28) Ramkrishna, D. Population Balances; Academic Press: San Diego, 2000.
10.1021/la7013013 CCC: $37.00 © 2007 American Chemical Society Published on Web 08/22/2007
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where W is the Fuchs stability ratio, KB ) 8kT/3µ is the Smoluchowski aggregation rate constant between primary particles, and Df is the fractal dimension of the clusters. The last term, (ij)λ, accounts for the fact that the reactivity of clusters increases with the cluster mass in the reaction-limited cluster aggregation (RLCA) regime and the value of the exponent, λ, is generally in the range between 0 and 0.5.1,10,12,24,29 Axford, on the basis of the geometry of fractal clusters, derived a value of λ ) 0.5.7 In the diffusion-limited cluster aggregation (DLCA) regime, because the reactivity is independent of the cluster mass, the λ value in eq 2 becomes zero. The validity of eq 2 in describing the RLCA and DLCA processes has been further supported through a detailed modeling study of the aggregation process between two fractal clusters using a diffusion-reaction approach.30 The most important feature of the above kinetic approach is that if we introduce the following dimensionless time τ and cluster number concentration Xi10
tKBN1,0 τ) W
(3)
and
Xi )
Ni N1,0
(4)
where N1,0 is the initial number concentration of primary particles, then the PBE (eq 1) reduces in dimensionless form as follows
dXi
)
dτ
1i - 1
∑
2 j)1
imax
βi - j,jXi - jXj -
βi,jXiXj ∑ j)1
(5)
where βi,j is the dimensionless aggregation kernel and is defined as
βi,j )
(i1/Df + j1/Df)(i-1/D + j-1/D) λ (ij) 4
(6)
from eq 2. The dimensionless time τ and cluster number concentration Xi includes all parameters related to the aggregation conditions. In particular, τ accounts for the effect of colloidal stability (W), the initial number concentration (N1,0), and the temperature and viscosity (in KB) of the system on the aggregation rate. Because the cluster structure is defined by Df and the aggregation kinetics for the given second-order process is given only by βi,j, it can be observed from eqs 5 and 6 that if the values of Df and λ are given then the dimensionless cluster mass distribution Xi given by the solution of eqs 5 and 6 is a function of only the dimensionless time τ, independent of type of colloid and ionic strength (colloidal stability), particle volume fraction, temperature, and so forth. This implies that if the aggregation process follows the same mechanism (i.e., the same cluster structure and aggregation kinetics) then all of the experimental data of a given quantity measured under different aggregation conditions should collapse on a single master curve when they are plotted in terms of the dimensionless time τ. This feature has been confirmed for various colloidal systems.10,12,13 However, if the experimental data measured under different aggregation conditions cannot collapse on a single master curve, then there must be differences in the cluster structure and/or the aggregation kinetics with respect to the ones assumed in writing the PBE (eqs (29) Odriozola, G.; Moncho-Jorda, A.; Schmitt, A.; Callejas-Fernandez, J.; Martinez-Garcia, R.; Hidalgo-Alvarez, R. Europhys. Lett. 2001, 53, 797. (30) Lattuada, M.; Wu, H.; Sefcik, J.; Morbidelli, M. J. Phys. Chem. B 2006, 110, 6574.
Figure 1. Form factor P(q) of the primary particles of the L-S latex, obtained from SLS measurements (O) and compared to the fitting of the RDG expression (eq 7) with Rp ) 52 nm (-).
5 and 6). Therefore, such master curve analysis can readily reveal whether the aggregation process follows such a classical model. However, most of the studies reported so far in the literature about the master curves of the aggregation kinetics and cluster structures were based on variations in the ionic strength and initial particle number concentration.10,12,13 There are two parameters of highly practical importance, surfactant type and aggregation temperature, which have never been varied during such colloidal aggregation studies. The question has to be answered whether such master curves in the aggregation kinetics and cluster structure are still present for a given colloidal system when its stabilizer (surfactant) and the aggregation temperature are changed. To this aim, in this work we have selected a styrene-acrylate copolymer colloid to investigate experimentally the colloidal aggregation kinetics and the cluster structure by varying the surfactant type and the aggregation temperature under RLCA conditions. The values of the Fuchs stability ratio W and the cluster aggregation kinetics and structure have been measured for different surfactants at different temperatures to observe how the intrinsic stability of the colloidal system varies under different conditions. 2. Experiments 2.1. Colloidal System. The styrene-acrylate copolymer colloids used in this work were supplied by BASF AG (Ludwigshafen, Germany). For us to observe the effect of surfactant type on the aggregation kinetics and cluster structure, the supplier has manufactured the same polymer colloid using two different surfactants, an aliphatic sulfonate (L-S latex) and an aliphatic carboxylate (L-C latex) surfactant, respectively. The corresponding counterion is Na+ for the former and K+ for the latter. More details of the compositions of the two latexes can be found elsewhere.19 The form factor of the primary particles of the L-S latex, P(q) measured by SLS, is shown in Figure 1 (symbols). In the same Figure is also shown the fitting using the Rayleigh-Debye-Ganz (RDG) expression for the form factor of small spherical particles31
[
P(q) ) 9
]
sin(qRp) - qRp cos(qRp) (qRp)
3
2
(7)
where Rp is the radius of the primary particles, q is the magnitude of the wavevector defined as q)
4πn0 θ sin λ0 2
()
(8)
(31) Kerker, M. The Scattering of Light; Academic Press: New York, 1969.
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where θ is the scattering angle, λ0 is the wavelength of the incident light, and n0 is the refractive index of the disperse medium. It is found that the best fitting of the experimental form factor is obtained with Rp ) 52 nm in eq 7. Such an obtained Rp value is consistent with the DLS measurements, which give Rp ) 53 nm. The polydispersity, which is defined in the method of cumulants as 2µ2/ µ21 (where µ1 and µ2 are the first and second cumulants, respectively32), is equal to 0.02, indicating that the colloidal system is rather monodisperse. Moreover, the consistency between the Rp values determined by SLS and DLS indicate that the primary particles can be considered to be spherical. For the L-C latex, the form factor of the primary particles is practically the same as that of the L-S latex. The obtained radius of the primary particles Rp is also equal to 52 nm. The original latexes have been first diluted to the particle volume fraction, φ0 ) 5%, which are then used as the reference latexes. All of the colloidal systems investigated in the following are at a particle volume fraction of φ ) 0.5%, which are obtained by diluting such reference latexes with demineralized water. 2.2. Fuchs Stability Ratios at Different Temperatures. As mentioned in the Introduction, before investigating the effect of temperature on the aggregation kinetics and cluster structure, one needs first to quantify the effect of temperature on the intrinsic stability of the colloid (i.e., Fuchs stability ratio W). The approach used to determine Fuchs stability ratio W is based on the doublet formation kinetics monitored by the SLS technique. Details of this approach may be found elsewhere.10,33 It relies on the fact that in the initial stage of aggregation the system can be considered to contain only primary particles and doublets. Then, the following relation can be derived from the PBE (eq 1)10 x ) N1,0K1,1t 1-x
(9)
where K1,1 is the doublet formation rate constant and x () 1 N1/N1,0) is the conversion of the primary particles to doublets. Thus, by plotting x/(1 - x) against time t in the initial stage of aggregation, one can obtain a straight line whose slope (N1,0K1,1) gives the K1,1 value, which defines W according to the following relationship W)
KB K1,1
(10)
It is clear that to estimate K1,1 or W one essentially needs to determine the conversion value x at different times. This can be done from the average structure factor 〈S(q)〉 measured by SLS, which is given by the ratio between the scattered light intensity curve, I(q), and the form factor of primary particles, P(q). Note that, as mentioned above, because the measured P(q) curve can be well represented by the RDG expression, in order to obtain 〈S(q)〉 we have divided I(q) directly by eq 7 with Rp ) 52 nm. Then, because the 〈S(q)〉 curves in the initial stage of aggregation can be expressed as33 〈S(q)〉 ) 1 -
[
]
sin(2qRp) x 11+x 2qRp
(11)
when the measured 〈S(q)〉 values are plotted against [1 - sin(2qRp)/ (2qRp)] one obtains a straight line whose slope gives an estimate of the conversion of primary particles, x. The so-obtained x values plotted as a function of time, as indicated by eq 9, allow one to determine the doublet formation rate constant, K1,1, and consequently W. Therefore, to investigate the effect of temperature on W, we have determined the W values by monitoring the doublet formation kinetics at different temperatures but at fixed values of the particle volume fraction (φ ) 0.5%) and the electrolyte concentration. The electrolyte used in this study is MgSO4. For this electrolyte, several model (32) Koppel, D. E. J. Chem. Phys. 1972, 57, 4814. (33) Lattuada, M. Aggregation Kinetics and Structure of Gels and Aggregates in Colloidal Systems. Ph.D. Thesis, ETH Zurich, 2003.
parameters such as association constants of cations with the surfactants and with the surface fixed charges have been estimated in previous work19 and can be applied here to interpret the dependence of W on T. The MgSO4 concentration was fixed at 9 mmol/L for the L-S latex and at 6 mmol/L for the L-C latex. A smaller MgSO4 concentration was used for the L-C latex is because we found that at T ) 298 K the stability of the L-C latex is substantially lower than that of the L-S latex. To reduce the difference in the W values between the two latexes, we decided to use a smaller MgSO4 concentration for the L-C latex. In both cases, the used MgSO4 concentrations are substantially smaller than the critical coagulant concentration (ccc, which is 40 mmol/L for the L-S latex and 25 mmol/L for the L-C latex19) (i.e., all the aggregation experiments were performed under RLCA conditions). The aggregation temperature was varied in the range between 283 and 323 K. (Note that the Tg of the given polymer is 378 K.) Temperature control was realized by combining a cryostat (Lauda-RK20) with a thermostat (Julabo-5). The aggregating system was put in a closed vessel and introduced into the chamber of the thermostat, and to avoid evaporation, the vessel was opened only when a sample was taken. To initiate the aggregation process by mixing the latex with the salt solution, a proper procedure must be used in order to avoid local and temporary peaks in salt concentration, which may lead to very fast local aggregation, before mixing can make the system uniform. For this, the following procedure has been used:10 (1) dilute the salt solution to a concentration as close as possible to the final salt concentration desired in the colloidal dispersion and introduce it into the chamber of the thermostat to reach the desired temperature and (2) pour the salt solution into the vessel containing the required amount of the original latex and immediately put it into the chamber of the thermostat. Because the volume of the salt solution is generally much larger than that of the latex, the mixing induced by pouring the salt solution into the latex is sufficient to disperse the colloidal particles in the system uniformly. Samples were taken at various times during the aggregation and were immediately diluted in sample holders using Milli-Q water that was prefiltered using a 0.1 µm syringe filer (Acrodisc 32 mm, Pall, U.K.). The dilution ratio is at least 5-fold, depending on the particle volume fraction of the dispersion, and it is needed to quench the aggregation process as well as avoid multiple scattering during the light scattering measurements. The light scattering instrument used for this study is a BI-200SM (Brookhaven) with an argon ion laser (M95-2, Lexel) operating at a wavelength of λ0 ) 514.5 nm and a goniometer in the angle range from 15 to 150°. For each sample, at least three repetitions of SLS measurements were carried out, and their averages are reported in the following. 2.3. Time Evolutions of Average Cluster Sizes and Structures. The procedure for initiating a Brownian aggregation process to monitor the time evolutions of the cluster sizes and structures is the same as that for the determination of W described above. The only difference is that the aggregation process proceeds for a substantially longer time so as to have a clear picture of the aggregation kinetic behavior and significantly large clusters for which to observe their structures. To observe the aggregation kinetics and cluster structures better, we have determined the time evolutions of both the average radius of gyration and the hydrodynamic radius, 〈Rg〉 and 〈Rh〉 using SLS and DLS, respectively, which represent two different moments of the cluster distribution. Moreover, the time evolutions of the average structure factor 〈S(q)〉 at different temperatures, determined by SLS, allow us to estimate the fractal dimension of the clusters as a function of temperature and then the effect of temperature on the cluster structures.
3. Results and Discussion 3.1. Dependence of the Stability Ratio, W, on Temperature and Its Modeling. Experimental Results. Following the experimental procedure described above, we have monitored the doublet formation kinetics for the L-S particles at the same MgSO4 concentration (CMgSO4 ) 9mmol/L) and particle volume fraction
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Figure 3. Fuchs stability ratio W for the L-S and L-C latexes at φ ) 0.5% as a function of the system temperature T. (O) L-S, CMgSO4 ) 9 mmol/L; (b) L-C, CMgSO4 ) 6 mmol/L.
Figure 2. (a) Normalized average structure factors 〈S(q)〉 plotted as a function of [1 - sin(2qRp)/(2qRp)] according to eq 11 and (b) conversion of the primary particles to doublets x from part a, plotted against aggregation time, in order to estimate stability ratio W according to eqs 9 and 10. L-S latex. φ ) 0.5%, CMgSO4 ) 9 mmol/L, T ) 293 K.
(φ ) 0.5%) but at different temperatures. Figure 2a shows the measured average structure factors after normalization, 〈S(q)〉, as a function of the quantity [1 - sin(2qRp)/(2qRp)], following eq 11, for four selected aggregation times in the case of the L-S latex at T ) 293 K. It is seen that each 〈S(q)〉 curve can be well represented by a straight line, indicating that these systems are dominated by primary particles and doublets; therefore, the slope of the straight line gives the conversion of the primary particles to doublets, x, based on eq 11. The obtained x values are 3.87, 7.09, 16.4, and 23.6% for t ) 30, 60, 181, and 300 min, respectively. Using the obtained x values at different aggregation times, the plot of x/(1 - x) versus t can be made, as shown in Figure 2b. It is evident that the plot can be well represented by a straight line, and from its slope we obtain the doublet-formation rate constant K1,1 () 2.01 × 10-24 m3/s), based on eq 9, and the corresponding Fuchs stability ratio W () 5.33 × 106), based on eq 10. Following the same procedure described above, one can obtain plots of x/(1 - x) versus t at different temperatures and from these the dependence of W on the system temperature T. Figure 3 shows such obtained W values as a function of temperature for both the L-S (open symbols) and the L-C (solid symbols) latexes. It is seen that in both cases the W value decreases substantially as the system temperature increases. This indicates that the intrinsic stability of the latexes decreases with increasing system temperature. This is mostly related to variations in the surfactant adsorption equilibrium with respect to temperature because, considering that the adsorption process is exothermic, one can expect that as the temperature increases the adsorption equilibrium shifts to desorption from the particle surface to the bulk liquid phase. When the W-T data of the L-C latex are compared with those of the L-S latex, even though the measurements were made at a lower MgSO4 concentration (6 vs 9 mmol/L), the W values at
Figure 4. ln b as a function of 1/T for the L-S and the L-C latexes at φ ) 0.5%. (O) L-S, CMgSO4 ) 9 mmol/L; (b) L-C, CMgSO4 ) 6 mmol/L.
low temperatures are still smaller for the L-C latex than for the L-S latex. This result confirms the ccc values of the two latexes mentioned above, and the ccc is smaller for the L-C latex (25 mmol/L) than for the L-S latex (40 mmol/L). However, the W value decreases with temperature significantly more slowely for the L-C latex than for the L-S latex, indicating that with respect to the L-S latex the stability of the L-C latex is less sensitive to temperature variations. Modeling the Effect of Temperature on the Stability Ratio. In previous work,18 we have developed a generalized model for describing the stability of polymer colloids, which accounts simultaneously for the interactions among three important physicochemical processes: adsorption equilibrium of surfactants, association equilibria of the ionic surfactants with counterions, and colloidal interactions. This model has been successfully applied to interpret the stability behaviors of the L-S and L-C latexes at T ) 298 K in a recent paper.19 The model parameters of all three physicochemical processes at T ) 298 K obtained from previous work are reported in Table 1. Let us now apply the generalized stability model together with the estimated values of the model parameters to interpret the effect of system temperature T on W measured above. In particular, we assume that in the examined temperature range all of the parameters in Table 1 are independent of temperature, except for the adsorption parameters. Such an assumption is reasonable for the parameters involved in the colloidal interaction (DLVO) model (e.g., the Hamaker constant). Because the association reactions have generally low reaction enthalpies, their equilibrium constants can be assumed to be constant in a limited temperature range, such as the one considered in this work. Moreover, at the given particle volume fraction (φ ) 0.5%), both the L-S and L-C latexes are basic at pH 8.3, and under these conditions the protonation of the surfactant is so
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Table 1. Parameter Values Used in the Stability Model for Both L-S and L-C Latexes descriptiona
parameter
H+ + E- y\z HE
KHE, L/mol
33.9
KHL
H+ + L- y\z HL
KHL, L/mol
33.9
L-C latex
33.9
2.5
KMgL+
2.5
2.5
97.0
97.0
28.8
28.8
6.6 × 10-6 4.66 1.3 × 10-20
7.1 × 10-6 4.66 1.3 × 10-20
Mg2+ + E- y\z MgE+
KMgL+, L/mol
Mg2+ + L- y\z MgL+
KHSO4-, L/mol
H+ + SO2\z HSO4 y 4
KMgSO4, L/mol
\z MgSO4 Mg2+ + SO24 y saturation adsorption amount of fixed charge Hamaker constant
b b
3.5b c b
KHSO4
KMgSO
c
source
6.3 × 104
KMgE+
KMgE+, L/mol
Γ∞, mol/m2 Cf, mol/m3 polymer AH, J a
L-S latex
KHE
b
4
d c c
E- denotes the dissociated surfactant and L- denotes the dissociated fixed charge. a Reference 46. b Reference 19. c Reference 47.
Table 2. Surfactant Adsorption Constant b (L/mol) Estimated by Fitting the Stability Ratio W at Different Temperatures for the Two Latexes temperature, K
L-S
L-C
283 293 303 308 313 323
4.730 × 103 2.701 × 103 1.795 × 103 1.479 × 103 1.254 × 103
1.041 × 103 7.99 × 102 6.33 × 102 5.13 × 102
weak that it has a small effect on the colloidal stability. The surface coverage of the surfactants at saturation, Γ∞, is also assumed to be independent of temperature, and as a result, there is only one parameter, the adsorption constant b, that is considered to change with temperature. For a given temperature, the b value was evaluated by applying the generalized stability model to simulate the measured W values at the corresponding temperature, using b as a fitting parameter. Details of the fitting procedure can be found elsewhere.18,19 In Table 2 are reported the so-obtained b values at different temperatures for both L-S and L-C latexes. It is seen that b decreases as T increases, thus confirming that the surfactant adsorption is an exothermic process. It is known that adsorption constant b in the Langmuir isotherm can be expressed as a function of temperature as follows34
(
b ) A exp -
)
∆G°ads RT
(12)
where ∆G°ads is the standard Gibbs free energy of adsorption, R () 8.314 × 10-3 kJ/mol) is the gas constant and A is the prefactor. Then, one may estimate the ∆G°ads values for the L-S and L-C latexes from the slope of the plots, ln b versus 1/T, shown in Figure 4. The obtained ∆G°ads values are -32.5 and -18.5 kJ/ mol for the L-S and L-C latexes, respectively. These ∆G°ads values are very close to those of similar surfactants reported in the literature (e.g., -38.2 kJ/mol for sodium dodecyl sulfate adsorbed at the air/water interface35 and -23.5 kJ/mol for sodium laurate adsorbed at the air/water interface36). We can then conclude that the higher sensitivity of the stability of the L-S latex to temperature with respect to that of the L-C latex is due to its larger (in absolute value) adsorption free energy. 3.2. Aggregation Kinetics and Cluster Structure. Effect of Temperature. Let us first use the L-S latex to investigate the (34) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: New York, 2001. (35) Fainerman, V. B.; Miller, R.; Mo¨hwald, H. J. Phys. Chem. B 2002, 106, 809. (36) Zwierzykowski, W.; Konopacka-Lyskawa, D. Colloids Surf., A 1999, 160, 183.
effect of aggregation temperature T on the aggregation kinetics and cluster structures. The aggregation kinetics under RLCA conditions has been determined at the particle volume fraction of φ ) 0.5% using MgSO4 as the coagulant with CMgSO4 ) 9 mmol/L (i.e., under the same conditions used for the W measurements in Figure 3). Figure 5 shows the values of the average radius of gyration 〈Rg〉 and the average hydrodynamic radius 〈Rh〉 as a function of time determined by SLS and DLS, respectively, at five temperatures. It is seen that in all cases the values of both 〈Rg〉 and 〈Rh〉 increase with time and that at a given time their values increase as the aggregation temperature increases. This is consistent with the fact that the measured value of W decreases as temperature increases, as shown in Figure 3. However, the above-mentioned effect of W on the cluster growth rate is only a colloidal stability effect, but it cannot tell us everything about the aggregation kinetics at different temperatures. For this, we need to introduce a dimensionless time τ as defined by eq 3 to eliminate the effect of W on the cluster growth rate and replot the 〈Rg〉 and 〈Rh〉 values in Figure 5 as a function of τ, as shown in Figure 6. It is worth noting that in doing this we have to account for the temperature changes of the parameters involved in the definition of the dimensionless time, which, in addition to W, include the viscosity and the temperature itself appearing in the definition of parameter KB in eq 3. It is seen that when the curves of 〈Rg〉 and 〈Rh〉 at different temperatures in Figure 5 are plotted as a function of dimensionless time τ, all of the curves collapse to form a single master curve for 〈Rg〉 and 〈Rh〉, respectively. There are only a few experimental points with large 〈Rg〉 or 〈Rh〉 values, which failed to collapse on the master curve. This is mostly due to artifacts in the sampling procedure when the cluster size is very large, which may significantly alter the structure and size of the clusters. The collapse of all of the 〈Rg〉 and 〈Rh〉 curves in the range of T ) 283 and 313 K on a single master curve in Figure 6 indicates that the aggregation kinetics in the given temperature range is the same and the differences observed in Figure 5 are due to only the effects of temperature on the cluster growth rate, which include its effects on W and KB. Figure 7a shows the typical time evolution of the average structure factor determined by SLS, 〈S(q)〉, as a function of wavevector q corresponding to the case of T ) 293 K in Figure 5. Note that all of the 〈S(q)〉 curves in Figure 7a have been normalized such that 〈S(q)〉 f 1 as q f 0. It is known that when the growth of the clusters during aggregation follows the fractal
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Figure 5. Values of (a) the average radius of gyration 〈Rg〉 and (b) the average hydrodynamic radius 〈Rh〉 as a function of time determined by SLS and DLS, respectively, at various aggregation temperatures for the L-S latex. φ ) 0.5%, CMgSO4 ) 9 mmol/L.
Figure 6. Values of (a) the average radius of gyration 〈Rg〉 and (b) the average hydrodynamic radius 〈Rh〉, reported in Figure 5, as a function of the dimensionless time τ. L-S latex, φ ) 0.5%, CMgSO4 ) 9 mmol/L.
scaling, the structure of the clusters should exhibit a certain self-similarity. This requires a certain time, which under RLCA conditions can be particularly long, so that all aggregates develop their fractal structure and, for example, no more primary particles are present. In this case, if the average structure factors 〈S(q)〉
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Figure 7. (a) Average structure factors 〈S(q)〉 as a function of wavevector q at various aggregation times, corresponding to the aggregation system at T ) 293 K in Figure 5 and (b) the 〈S(q)〉 curves in part a plotted as a function of q〈Rg〉 for an aggregation time longer than 1000 min. L-S latex, φ ) 0.5%, CMgSO4 ) 9 mmol/L.
at different aggregation times are plotted as a function of the normalized wavevector q〈Rg〉, then all of the 〈S(q)〉 curves must collapse on a single master curve.13,37-39 In Figure 7b are shown the 〈S(q)〉 curves in Figure 7a corresponding to sufficiently long aggregation time (i.e., >1000 min in this case) but plotted as a function of q〈Rg〉. It is seen that the 〈S(q)〉 curves indeed collapse on a single curve. The same behavior also occurs for the time evolution of the average structure factors at the other aggregation temperatures (not shown) in Figure 5. We can therefore conclude that the growth of the clusters of the L-S latex follows the fractal scaling and the formed clusters are fractal objects. It is well known40,41 that for sufficiently large clusters the average structure factor, 〈S(q)〉, scales with q〈Rg〉 as 〈S(q)〉 ∝ (q〈Rg〉)-Df for q〈Rg〉 . 1. Then, the slope of the log 〈S(q)〉 versus log(q〈Rg〉) plot leads to an estimate of the fractal dimension, Df. For this, the 〈S(q)〉 curves at large aggregation times, determined in the range of T ) 293 - 313 K, have been plotted together as a function of q〈Rg〉 and are shown in Figure 8. Note that none of the 〈S(q)〉 curves at T ) 283 K are included in Figure 8 because at this temperature the aggregation rate is so slow that the 〈S(q)〉 curve measured even at the largest aggregation time is still dominated by the presence of primary particles. It is seen that the 〈S(q)〉 curves at different aggregation temperatures also collapse on a single master curve. This indicates that for the given temperature range T ) 283-313 K not only the aggregation kinetics, as demonstrated in Figure 6, but also the structures of the formed clusters are independent of the aggregation temper(37) Sciortino, F.; Tartaglia, P. Phys. ReV. Lett. 1995, 74, 282. (38) Sciortino, F.; Belloni, A.; Tartaglia, P. Phys. ReV. E 1995, 52, 4068. (39) Weijers, M.; Visschers, R. W.; Nicolai, T. Macromolecules 2002, 35, 4753. (40) Brown, W. Light Scattering: Principles and DeVelopment; Clarendon Press: Oxford, U.K., 1996. (41) Sorensen, C. M. Aerosol Sci. Technol. 2001, 35, 648.
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Figure 8. 〈S(q)〉 curves at large aggregation times for the four aggregation systems at different temperatures as a function of q〈Rg〉. L-S latex, φ ) 0.5%, CMgSO4 ) 9 mmol/L, T ) 293-313 K.
ature. Moreover, in Figure 8 there is a certain range of q〈Rg〉 . 1 where the 〈S(q)〉 master curve can be represented by a straight line in the log-log plane, further indicating that the formed clusters are fractal objects, and its slope () -Df) can be used to estimate the Df value. The decision for the slope is not easy because of the limited range of 〈S(q)〉, but the straight line with a slope of -2.15 covers the largest range of experimental points. Thus, we consider the value of Df ) 2.15. We will further verify such a value of the fractal dimension, Df, when the 〈S(q)〉 curves for the L-C latex are considered. A value of Df ) 2.15 indicates that the clusters of the L-S latex are typical of those obtained in RLCA processes.2 Another manifestation of the growth of clusters following the fractal scaling is to plot the time evolution of the 〈Rg〉 and 〈Rh〉 master curves in Figure 6 together, as shown in the Supporting Information (Figure S1). In this way, one can observe that initially 〈Rg〉 is smaller than 〈Rh〉, with the 〈Rg〉/〈Rh〉 ratio being very close to x3/5 (i.e., the Rg/Rh ratio of a sphere). Then, as the aggregation time increases, 〈Rg〉 increases with time faster than 〈Rh〉, and after a certain aggregation time, crossover occurs and 〈Rg〉 becomes larger than 〈Rh〉. This arises because 〈Rg〉 and 〈Rh〉 represent different moments of the CMD, and it is well known that for fractal clusters that the growth of 〈Rg〉 is always faster than that of 〈Rh〉, thus leading to the crossover.10 Effect of Surfactant Type. The effect of surfactant type on the aggregation kinetics and cluster structure can be investigated by comparing the aggregation kinetics and cluster structure of the L-C latex (with the carboxylate type surfactant) with those of the L-S latex (with the sulfonate type surfactant) discussed above. The aggregation kinetics and the cluster structure of the L-C latex have been investigated at φ ) 0.5% and using MgSO4 as the coagulant with CMgSO4 ) 6 mmol/L (i.e., under the same conditions used for the W measurements in Figure 3). The time evolutions of 〈Rg〉 and 〈Rh〉 at four temperature values are shown in Figure 9a,b, respectively. Similarly to the case of the L-S latex shown in Figure 5, because W decreases as T increases, as shown in Figure 3, the growth rate of both 〈Rg〉 and 〈Rh〉 increases as T increases. However, when the 〈Rg〉 and 〈Rh〉 values at different T values are plotted as a function of τ defined in eq 3, all the 〈Rg〉 or 〈Rh〉 curves collapse on a single master curve, as shown in Figure 10. This indicates that the aggregation processes of the L-C latex at different temperatures follow the same aggregation mechanism. Figure 11a shows the typical time evolution of the normalized average structure factor 〈S(q)〉 as a function of q, corresponding to the case of T ) 323 K for the L-C latex considered in Figure 9. When these 〈S(q)〉 curves at different aggregation times are
Figure 9. Values of (a) the average radius of gyration 〈Rg〉 and (b) the average hydrodynamic radius 〈Rh〉 as a function of time determined by SLS and DLS, respectively, at various aggregation temperatures for the L-C latex. φ ) 0.5%, CMgSO4 ) 6 mmol/L.
Figure 10. Values of the average radius of gyration 〈Rg〉 and the average hydrodynamic radius 〈Rh〉 at different temperatures, shown in Figure 9, plotted as a function of dimensionless time τ. L-C latex, φ ) 0.5%, CMgSO4 ) 6 mmol/L, T ) 293-323 K.
plotted as a function of the normalized wavevector q〈Rg〉, all of the 〈S(q)〉 curves collapse on a single master curve, as shown in Figure 11b. This confirms that the aggregation system follows fractal scaling and the formed clusters can be represented by fractal objects. For a better comparison, let us plot all of the kinetic data for the L-S and L-C latexes shown in Figures S1 and Figure 10 together, on the same plot as shown in Figure 12a. It is clearly seen that the kinetic data from the two latexes are practically overlapped and some scattering occurs only at large aggregation times, which as mentioned above is mostly related to artifacts in the sampling procedure for large clusters. Figure 12b shows the 〈S(q)〉 master curve as a function of q〈Rg〉, generated from all of the 〈S(q)〉 curves of both the L-S and L-C latexes at large aggregation times, indicating that the structures of the clusters formed in the two latexes are identical and independent of temperature, T, at least in the considered temperature range.
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Figure 11. (a) Average structure factor 〈S(q)〉 as a function of wave vector q at various aggregation times, corresponding to the aggregation system at T ) 323 K in Figure 9, and (b) 〈S(q)〉 curves in part a plotted as a function of q〈Rg〉 for aggregation times longer than 60 min. L-C latex, φ ) 0.5%, CMgSO4 ) 6 mmol/L.
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The above results indicate that although the colloidal stability behavior is somewhat different for the L-S and the L-C latexes, as observed from the measured temperature dependence of W in Figure 3, the aggregation kinetic behavior and the formed cluster structure under RLCA conditions are practically identical. Because the principal difference between the two latexes is the use of different surfactants, this means that for the given polymer particles and coagulant the RLCA kinetics and the cluster structure are not affected by the type of surfactant used in the range of temperature values from 283 to 323 K. PBE Simulation Results. Let us now use the Smoluchowski kinetic approach to simulate the experimentally observed aggregation kinetics and cluster structures. To do this, we first need to compute the time evolution of the cluster mass distribution (CMD) Ni from the PBE (eqs 1 and 2). Note that in the kernel (eq 2), the dependence of the physicochemical parameters on temperature has been considered. The W value is set to the one measured experimentally at the given temperature, the fractal dimension Df is set to 2.15, as estimated from Figure 12b, and the parameter KB has been recalculated at the various temperatures, also including the viscosity changes. The exponent of the product term, λ, is at this point unknown, and it has been estimated by the experimental data in Figure 12b. As mentioned above, its value is expected to be in the range between 0 and 0.5.1,10,12,24,29 The numerical procedure proposed by Kumar and Ramkrishna28,42 has been used to solve the PBE (eq 1). The algorithm consists of dividing the entire mass range using a geometric grid and writing the ordinary differential equations only for the pivots (i.e., for the masses corresponding to the boundaries between each interval). If the mass of a cluster produced by the aggregation falls within an interval, then it is split between the two pivots located at the boundary of the interval. The splitting factors are selected so that two (the zeroth and first order) moments of the original distribution are conserved. This procedure guaranties the high efficiency of the computations and allows one to simulate the evolution of very broad CMDs. To reconstruct the time evolutions of the average quantities 〈Rg〉, 〈Rh〉, and 〈S(q)〉 from the calculated CMDs, the following well-known relations have been used4,10,40,41 imax
Nii2Rg,i2 ∑ i)1
〈Rg〉2 )
(13)
imax
Nii2 ∑ i)1 imax
〈Rh〉 )
Nii2Si(q) ∑ i)1 imax
∑ i)1
Nii2Si(q)
(14)
eff Rh,i
imax
Figure 12. (a) 〈Rg〉 and 〈Rh〉 master curves for the aggregation kinetics of L-S and L-C latexes in Figures S1 and 10, respectively, plotted on the same graph. (b) 〈S(q)〉 master curve as a function of q〈Rg〉 for both the L-S and L-C latexes at large aggregation times. The solid curves are PBE simulations.
Moreover, the linear region of the 〈S(q)〉 master curve in Figure 12b becomes more clear, and its slope leads to the best estimate of the fractal dimension, Df ) 2.15.
〈S(q)〉 )
Nii2Si(q) ∑ i)1 imax
(15)
Nii2 ∑ i)1 where Rg,i, Reff h,i , and Si(q) are the radius of gyration, the effective hydrodynamic radius, and the structure factor of the cluster with (42) Kumar, S.; Ramkrishna, D. Chem. Eng. Sci. 1996, 51, 1311.
BehaVior of Styrene-Acrylate Copolymer Colloids
Figure 13. Comparison of the time evolutions of average structure factor 〈S(q)〉 (a) as a function of wave vector q and (b) as a function of q〈Rg〉 simulated by PBE (solid curves) with those measured by SLS experiments (symbols) shown in Figure 7a, corresponding to the L-S latex at T ) 293 K and CMgSO4 ) 9 mmol/L.
mass i, respectively. These structure properties of individual clusters are estimated from the clusters generated by Monte Carlo simulations, as described in detail elsewhere.43-45 The simulated time evolutions of the 〈Rg〉 and 〈Rh〉 master curves are shown in Figure 12a as a function of the dimensionless time and are compared with the experimental results. It is seen that the agreement between experiments and simulations is satisfactory, although the simulated 〈Rh〉 master curve is slightly below the experimental one. The λ value estimated from the data fitting is 0.45, which is well in the range of 0 to 0.5 often found in the literature. Moreover, the obtained λ value is very close to 0.5, as estimated on the basis of the geometry of fractal clusters7 and the diffusion-reaction model.30 We can then conclude that the aggregation kinetics of the L-S and L-C latexes follows the typical kinetics of RLCA processes. Typical time evolutions of the average structure factor 〈S(q)〉 obtained from the PBE simulations are shown in Figure 13a and compared with those measured by SLS for the case of the L-S latex at T ) 293 K. The agreement between simulations and experiments is generally satisfactory, but some deviations occur in the large-q region, where unlike the experimental ones, the simulated 〈S(q)〉 curves start to bend up slightly with increasing q. The bending up of the 〈S(q)〉 curves is reasonable because in this region the values of 1/q () 30-100 nm) are comparable to the radius of the primary particles (Rp ) 52 nm); therefore, we are approaching the scattering region of the primary particles. Such inconsistency between experiments and simulations in the (43) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 106. (44) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 96. (45) Lattuada, M.; Wu, H.; Morbidelli, M. Langmuir 2004, 20, 5630. (46) Martell, A. E.; Smith, R. M. Critical Stability Constants; Plenum: New York, 1977. (47) Maron, S. H.; Elder, M. E.; Ulevitch, I. N. J. Colloid Sci. 1954, 9, 89.
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large-q region may be due to some systematic error in the light scattering setup or something else, but no further explanation is given in this work. Moreover, when the 〈S(q)〉 data in Figure 13a are plotted as a function of q〈Rg〉, as shown in Figure 13b, both the measured and PBE simulated results indicate that as the aggregation time increases, the 〈S(q)〉 curves at different times progressively collapse on a single master curve. The failure of the collapsing for those 〈S(q)〉 curves at small aggregation times can be explained by recalling that the structure factor of the primary particles is S(q) ) 1 (i.e., a horizontal line). For an RLCA process, the cluster mass distribution is known to be very broad, and the number of primary particles can still be significant for substantially long aggregation time. Thus, those 〈S(q)〉 curves in Figure 13b that deviate from the master curve correspond to the situation where the number of primary particles is still substantial in the total cluster mass distribution (CMD), leading to a dominant effect of the primary particles on the average structure factor. Upon increasing the aggregation time, the effect of the primary particles on 〈S(q)〉 decreases, and the 〈S(q)〉 curve progressively merges onto the master curve. Figure 12b compares the simulated and experimental master curves of the average structure factor 〈S(q)〉 as a function of the normalized wavevector q〈Rg〉. Although the simulated master curve is slightly shifted to smaller q〈Rg〉 values (as a result of a slight underestimation of the 〈Rg〉 values), the shapes of the simulated and experimental master curves are very consistent.
4. Concluding Remarks Colloidal stability, aggregation kinetics, and cluster structure have been investigated experimentally in the temperature range from 283 to 323 K for two styrene-acrylate copolymer latexes made of the same polymer particles but stabilized with two different surfactants: an aliphatic sulfonate (L-S latex) and an aliphatic carboxylate (L-C latex). The main objective of this study is to investigate the roles of temperature and surfactant type in colloidal stability, aggregation kinetics, and cluster structure. Note that to avoid any softening effect of the particles the chosen temperature range is significantly below the glasstransition temperature (Tg ) 378 K) of the considered polymer. It is observed that for the given T range the value of W decreases as temperature increases for both latexes. However, W value decreases faster for the L-S than for the L-C latex (i.e., the stability of the L-S latex is more sensitive to temperature changes). This can be attributed to the fact that the adsorption enthalpy is larger for the L-S than for the L-C latex. The aggregation kinetics and the cluster structure of the two latexes have been investigated at different temperatures under reaction-limited cluster aggregation (RLCA) conditions. The growth rate of both the average radius of gyration and the hydrodynamic radius, 〈Rg〉 and 〈Rh〉, during aggregation as determined by SLS and DLS increases as temperature increases. However, when a proper dimensionless time is defined to account for the effect of temperature on W as well as on the other physicochemical properties of the disperse medium (e.g., KB, viscosity, etc.), all of the time evolutions of 〈Rg〉 and 〈Rh〉 measured for the two latexes at different temperatures collapse to form a master curve. This indicates that under the given conditions the aggregation mechanism of the considered polymer particles is independent of temperature and type of surfactant. Moreover, the curves of the average structure factor, 〈S(q)〉, measured by SLS at different temperatures and surfactants collapse to form a master curve when they are plotted as a function of q〈Rg〉, and the master curve in the range of q〈Rg〉 . 1 can be well represented by a straight line in the log-log plane. This
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means that the structures of all of the clusters formed under different conditions are self-similar and follow fractal scaling. The fractal dimension estimated from the linear part of 〈S(q)〉 is Df ) 2.15. Thus, the clusters formed by the considered polymer colloids correspond to the typical fractal objects generated by RLCA processes. It should be noted that the number of experiments and the investigated operation conditions are probably too limited to provide a full validation of the universal behavior of the system. Nevertheless, we keep the notion of a master curve, which is also based on the findings of previous studies10,12,13 where a large set of operating conditions have been investigated. Numerical simulations of the aggregation kinetics and the cluster structure factors have also been carried out using the Smoluchowski kinetic approach, which is based on the population balance equations (eq 1). During the simulations, the effect of temperature on W and the other physicochemical parameters in
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the kernel (eq 2) has been accounted for. The simulated master curves for 〈Rg〉, 〈Rh〉, and 〈S(q)〉 are in good agreement with those measured experimentally. The exponent of the product term, λ in eq 2, which is the only parameter used to fit the experimental data, is equal to 0.45, which is consistent with the typical λ values for the aggregation kinetics of RLCA processes. Acknowledgment. Financial support from BASF AG (Ludwigshafen, Germany) and the Swiss National Science Foundation (grant no. 200020-113805/1) is gratefully acknowledged. Supporting Information Available: Master curves of the time evolutions of the average radius of gyration 〈Rg〉 and the average hydrodynamic radius 〈Rh〉 shown in Figure 6a,b for the L-S latex, plotted as a function of dimensionless time τ. φ ) 0.5%; CMgSO4 ) 9 mmol/L; and T ) 283-313 K. This material is available free of charge via the Internet at http://pubs.acs.org. LA7013013
