Langmuir 2001, 17, 1841-1846
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Reversible Aggregation of Soft Particles A. Ferna´ndez-Nieves,† A. Ferna´ndez-Barbero,† B. Vincent,‡ and F. J. de las Nieves*,† Group of Complex Fluids Physics, Department of Applied Physics, University of Almerı´a, 04120 Almerı´a, Spain, and School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, U.K. Received September 20, 2000. In Final Form: November 30, 2000 In this work, the aggregation of microgel particles has been investigated, paying special attention to the structure of the clusters formed in the process. In particular, the aggregates’ fractal dimension was determined by static light scattering. The results indicate that the aggregates are more compact than expected for diffusive aggregation. A reversible aggregation mechanism is proposed on the basis of the competition between osmotic and elastic contributions arising from the soft character of the particles. Aggregation proceeds in an energy minimum of restricted depth, giving rise to the formation of more compact clusters than expected. Finally, the process reversibility is tested, confirming the secondary minimum controlled aggregation.
Introduction In the past decade, models have developed describing the structure that results from the union of subunits. The aggregation of colloidal particles is a good model for describing this phenomenon. Both theory and experiments have shown a universal behavior, independent of the particle nature, when the aggregation of clusters is diffusion-limited (DLCA) or reaction-limited (RLCA).1 DLCA occurs when every collision between clusters results in the formation of an irreversible bond. This regime gives rise to the formation of branched clusters, with a typical fractal dimension of 1.7-1.8. RLCA occurs when a small fraction of collisions leads to cluster formation. In this case, the aggregates are more compact than those formed in a DLCA process, with a fractal dimension of around 2.1. DLCA and RLCA are limited to certain ideal conditions. In particular, clusters have to be randomly distributed in space, with no position correlation at any time. Additionally, the aggregation must occur in a deep energy minimum that guarantees a Brownian path for every cluster. Intermediate aggregation modes between DLCA and RLCA have also been reported both, theoretically and experimentally.2,3 In addition, the formation of more compact structures than expected for RLCA processes has been encountered and explained in terms of reaction reversibility.4-6 In this case, the contact between particles is considered to be reversible so that they can loosen and re-form repeatedly after collision. This reversible mode of aggregation has been related to the finiteness of the energy well that holds the particles together.7,8 * To whom correspondence should be addressed (e-mail:
[email protected]). † University of Almerı´a. ‡ University of Bristol. (1) Carpineti, M.; Giglio, M. Adv. Colloid Interface Sci. 1993, 46, 73. (2) Kolb, M.; Botet, R.; Jullien, R. Phys. Rev. Lett. 1983, 51, 1123. (3) Asnaghi, D.; Carpineti, M.; Giglio, M.; Sozzi, M. Phys. Rev. A 1992, 45, 1018. (4) Aubert, C.; Cannell, D. S. Phys. Rev. Lett. 1986, 56, 738. (5) Dimon, P.; Sinha, S. K.; Weitz, D. A.; Safinya, C. R.; Smith, G. S.; Varaday, W. A.; Lindsay, H. M. Phys. Rev. Lett. 1986, 57, 595. (6) Jullien, R. CCACAA 1992, 65, 2, 215. (7) Shih, W. Y.; Aksay, I. A.; Kikuchi, R. Phys. Rev. A 1987, 36, 10, 5015.
In this work, the aggregation of mesoscopic gellike particles (i.e. soft particles) is studied. In particular, we have studied the structure of the aggregates formed under high salt conditions (far above the critical coagulation concentration of the colloidal system). The clusters present a more compact structure than expected for DLCA, which could be related to the soft character of these colloids, that are able to swell or deswell, depending on the environmental conditions. This feature not only modifies the particle structure and overall size but also gives rise to the appearance of new contributions to the total interaction potential between particles. Osmotic and elastic effects due to surface interpenetration are taken into account, yielding a finite energy minimum where aggregation takes place. This model supports the obtained cluster structures. Finally, the aggregation reversibility is experimentally tested, confirming the presented scenario. Monitoring Cluster Structure Cluster Morphology. It is well-known that colloidal clusters exhibit a fractal structure, characterized by a fractal dimension df, which is directly related to the cluster compactness. Cluster growth is such that its mass, M, increases slower than its volume. Mathematically, M(R) ∼ Rdf with df < 3 (in a three-dimensional space) and R the radius of the aggregate.9 This description implies that the pair correlation function of primary particles within a fractal aggregate scales as10
g(r) ∼ rdf-3
(1)
corresponding to a decay of the local density as the length scale increases. This power law is usually employed for obtaining the fractal dimension of growing mesoscopic clusters. Static Light Scattering. In a static light scattering experiment, the measured mean scattered intensity, I, (8) Lin, J.; Shih, W. Y.; Sarikaya, M.; Aksay, I. A. Phys. Rev. A 1990, 41, 6, 3206. (9) de Guzman, M.; Martı´n, M. A.; Mora´n, M.; Reyes, M. Estructuras fractales y sus aplicaciones; Labor SA: Madrid, 1993. (10) Vicsek, T. Fractal growth phenomena; World Scientific: Singapore, 1992.
10.1021/la001351u CCC: $20.00 © 2001 American Chemical Society Published on Web 02/13/2001
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may be expressed in the following factorized form:
I(q) ∼ P(q)S(q)
(2)
Here P(q) and S(q) are the form and structure factors, respectively, for the scattering wave vector q ) 4π/λ sin(θ/2), with λ the light wavelength in the solvent and θ the scattering angle. λ and θ set the characteristic observation length scale, by tuning the value of q. The form factor contains information concerning the interference of light coming from different volume elements within a particle, while the structure factor is determined by the relative positions of all particles. The structure factor is related to the pair correlation function through the following relationship:11
S(q) ∼
sin{qr} qr
∫0∞dr r2[g(r) - 1]
(3)
Inserting the pair correlation function for fractal structures (eq 1) and taking into account a finite structure size,12 the structure factor for a colloidal cluster may be expressed as
S(q) ∼ Γ(df - 1)
sin[(df - 1) arctan{qR}] qR[1 + (qR)2](df-1)/2
(4)
with R being the cluster radius. In the limit qR > 1, corresponding to a spatial length scale smaller than the cluster size, this equation predicts
S(q) ∼ q-df
(5)
establishing a power law relation between the structure factor and the scattering vector, which allows the cluster fractal dimension to be determined. This scaling law is valid while the q range does not correspond to a length scale smaller than the single particle dimensions (qb < 1, with b being the radius of the colloidal particle). For qb > 1, the scattered intensity is mainly determined by the particle form factor. For clusters formed by small hard spheres and in the visible light q range, I(q) ∼ S(q) since the form factor is approximately q independent. Under these experimental conditions, the structure factor is easily measured from the angular dependence of the scattered light. Experimental Details Experimental System. Microgel particles were employed as soft spheres. The synthesis of the system is described elsewhere.13 Spherical and monodisperse particles are based on poly(2vinylpyridine) (2VP), cross-linked with divinylbenzene (0.25% by weight). The initiator used in the synthesis was 2,2′-azobis(2-amidinopropane) dihydrochloride (V50, Wako). Two types of chemical groups are able to confer charge to the colloidal particles: (i) amidinium groups arising from the initiator, located essentially at the periphery of the particles; (ii) the constituent monomer 2VP, located within the particles. In this work, deswollen particles are employed with charge on their surface. The swelling of the microgel particles and its influence over their motion under external electric fields has been previously resported.14,15 (11) Dhont, J. K. G. An introduction to dynamics of colloids; Elsevier: Amsterdam, 1996. (12) Chen, S. H.; Texeira, J. Phys. Rev. Lett. 1986, 57, 2583. (13) Loxley, A.; Vincent, B. Colloid Polym. Sci. 1997, 275, 1108. (14) Ferna´ndez-Nieves, A.; Ferna´ndez-Barbero, A.; Vincent, B.; de las Nieves, F. J. Macromolecules 2000, 33, 6, 2114. (15) Ferna´ndez-Nieves, A.; Ferna´ndez-Barbero, A.; Vincent, B.; de las Nieves, F. J. J. Phys. C: Condens. Matter 2000, 12, 3605.
Figure 1. Hydrodynamic diameter versus salt concentration for three surface charge states corresponding to pH 5.8 (4), 6.8 (O), and 10.1 (]). The particle deswelling due to an increase in the χ parameter can be observed (region I) as well as the destabilization of the system caused by the screening of the electrostatic interaction between particles (region II). Instrumental Details and Experimental Conditions. Experiments were performed using a slightly modified Malvern Instruments 4700 System (U.K.) working with a 632.8 nm wavelength He-Ne laser. The q range is bound from 0.0023 to 0.026 nm-1, corresponding to scattering angles between 10 and 150°, respectively. Once the final structure of the clusters was totally established, the mean intensity showed an asymptotic time-independent behavior, as predicted theoretically, from which the fractal dimension was determined. The particle mean hydrodynamic diameter was measured using dynamic light scattering. Intensity autocorrelation functions were determined at a scattering angle of 90°, using a commercial equipment (Zetamaster-S, Malvern Instruments). Data analysis based on cumulant analysis was performed using homemade computer programs. All measurements were carried out under very dilute conditions, with the particle concentration being equal to 2 × 109 cm-3. This concentration was optimized in order to avoid multiple scattering. The medium pH and ionic concentration were adjusted with NaOH and NaCl, respectively. The temperature was constant and equal to (25.0 ( 0.1) °C. All experimental points correspond to the average of five different measurements.
Results Colloidal Stability. The aim of this section is to establish the ionic concentration region in which the microgel particles are electrostatically stabilized. The hydrodynamic diameter was measured as a function of ionic concentration, for three surface charge states. The results are presented in Figure 1, which shows two welldefined regions. In the first one, the effect of the ionic concentration translates in a particle diameter reduction. The addition of salt modifies the polymer/solvent interaction, favoring the polymer/polymer contacts compared to the polymer/solvent ones,16 thus increasing the Flory χ parameter. In the second region (above [NaCl] ∼ 15 mM), the mean hydrodynamic diameter increases due to destabilization of the colloidal dispersion. The electrostatic interactions responsible for the colloidal stability are screened out, giving rise to aggregation. The critical coagulation concentration was established at ∼50 mM, beyond which all aggregation experiments were performed. (16) Zhu, P. W.; Napper, D. H. Colloids Surf., A 1995, 98, 106.
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Figure 2. Intensity versus scattering vector, at different salt concentrations. [NaCl] ) 3 M (A), 2 M (B), 1 M (C), 0.5 M (D), and 0.25 M (E). Table 1. Cluster Fractal Dimension and Break Reflecting the Influence of the Microgel Particle Form Factor, for Different Salt Concentrations [NaCl] (M) 3 2 1 0.5 0.25
break (nm-1)
df
∼0.0119 ∼0.0052 ∼0.0044 ∼0.0042
1.819 ( 0.012 1.817 ( 0.016 1.88 ( 0.06 2.01 ( 0.06 2.13 ( 0.09
Cluster Structure. The fractal dimension of clusters formed by aggregation of soft particles was obtained by using eqs 2 and 5, at salt concentrations well above 50 mM. The I(q) curves are shown in Figure 2, and Table 1 summarizes the main features. The medium pH was set to ∼9, in all cases. The I(q) curves show a change in slope, with a q break separating two regions. The slope of the I(q) curves above the break is salt independent, supporting that in this region the intensity q dependence is dominated by the particle form factor. In addition, the q break shifts to lower q values as the salt concentration is decreased, this fact
being related to the particle size reduction induced by solvency effects (see Figure 1). Finally, in the low-q region, I(q) ∼ S(q), allowing the cluster fractal dimension to be determined. The fractal dimensions extracted from the I(q) relation at small scattering vectors increases from ∼1.8 to ∼2.1, as the ionic concentration decreases (Table 1). It is interesting to remark that df changes with salt concentration, since, at the NaCl concentrations studied, the electrostatic interaction is screened out. The microgel solvency reduction caused by the increase in the medium ionic concentration modifies the interaction potential between particles, thus affecting the growth process. Interpretation of Results The DLVO theory is the usual starting point for studying the colloidal stability of a dispersed system.17,18 In this (17) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (18) Derjaguin, B. V.; Landau, L. D. Acta Physicochim. 1941, 14, 633.
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theory, the interaction potential between a pair of particles is considered to be the sum of a long-range electrostatic contribution and a short-range attractive potential. The former arises from the presence of surface stabilizing charge, while the second contribution has its origin on the London-van der Waals dispersion forces. The addition of salt to the suspension medium translates into a screening of the electrostatic repulsion between particles, which permits the onset of aggregation. The particles stay together after contact in a deep primary minimum, due to van der Waals attraction. For spherical microgel particles the attraction potential is expressed as
(
[
])
H(4b + H) A 2b2 2b2 Va ) + ln + 6 H(4b + H) (2b + H)2 (2b + H)2
(6)
where A ) (xAm - xAw)2, with Am and Aw the Hamaker constants of the microgel particle and water, respectively. For microgel particles, Am is a function of the degree of swelling and is given by19
Am ) [xApφ + (1 - φ)xAw]2
(7)
where Ap is the Hamaker constant of the polymer (2VP in our case) and φ is the mean volume fraction of polymer within the microgel particle. For isotropic swelling or deswelling, this latter variable is related to the particle size by20
()
bo φ) b
3
(8)
where bo is the collapsed (hard sphere) radius. Once the critical salt value has been surpassed, any further increase in this variable should leave the aggregation mechanism unchanged. The fractal dimension of aggregates formed under these circumstances should be constant and equal to 1.7-1.8, which are typical values for diffusion limited aggregation. The structure of the clusters formed by the aggregation of microgel particles do not show this feature, indicating that the aggregation mechanism is being altered. The explanation for this unexpected behavior could have its origin in the soft character of the microgel particle, which permits interpenetration. As a consequence, the total interaction potential between gellike particles needs to be extended to account for this interpenetration. Vincent et al.21 studied this effect for core-shell systems, including an osmotic and an elastic contribution to the total interaction potential between particles. For the theoretical description of this effect, let H be the distance of separation between the surfaces of two colloidal particles (H ) r 2b) and δ the average interpenetration thickness. Depending on the values of H and δ, three situations can be distinguished: (i) noninterpenetrating domain; (ii) interpenetrating domain, δ < H < 2δ; (iii) interpenetrating and compression domain, H < δ. For H > 2δ, the polymeric chains are not in contact and, thus, the osmotic and elastic potentials, Vosm and Velas, respectively, are zero. In the interpenetrating domain, the microgel surfaces are in contact, causing an overlap region with a high density of polymer chains. When the (19) Snowden, M. J.; Marston, N. J.; Vincent, B. Colloid Polym. Sci. 1994, 272, 1273. (20) Hirotsu, H. Adv. Polym. Sci. 1993, 110, 1. (21) Vincent, B.; Edwards, J.; Emmett, S.; Jones, A. Colloids Surf. 1986, 18, 261.
interaction polymer/solvent is energetically more favorable than the polymer/polymer interaction (χ < 0.5), the contact between polymer chains leads to an increase of the system free energy, originating the appearance of a repulsive force between particles. For χ values above 0.5, the situation is the opposite: the contact between polymer chains causes a free energy decrease and, thus, an attraction between particles. In this domain, the osmotic potential is written as21
Vosm )
4πb 2 1 H Φ -χ δυ1 2 2
(
)(
)
(9)
where υ1 is the molecular volume of the solvent and Φ is the effective volume fraction of polymer in the interpenetration region. As can be seen, the value of χ controls the sign of this interaction potential. For 2VP, water is a bad solvent. χ ∼ 0.6 when the particle is in a deswollen state,14 implying that Vosm is attractive. If the two particles are closer than a distance equal to δ, the external polymer chains are forced to undergo elastic compression. Thermodynamically, this compression corresponds to a net loss of configurational entropy, giving rise to a repulsive force between particles. In this domain, the osmotic and elastic interaction potentials are given by21
Vosm ) Velas )
1 H H 4πb 2 1 - χ δ2 Φ - - ln υ1 2 2δ 4 δ
(
) [( )
( )]
2πb H H 3 - H/δ 2 FΦδ2 ln NAMw 2 δ 2 H 3 - H/δ ln +31+ 2 δ
[ ( ( (
)) ) (
(10)
)] (11)
where NA is the Avogadro number, F the polymer density, and Mw its molecular weight. The total interaction potential is obtained from the superimposition of all contributions: electrostatic; van der Waals attraction; osmotic; elastic. The increase in ionic concentration deswells the microgel particles (see Figure 1), implying an increase in Φ and a decrease in δ. The Flory parameter increases since the solvent becomes poorer. Additionally, for salt concentrations above the critical coagulation concentration, the electrostatic interaction is wiped out and plays no role. On the other hand, Va does not change appreciably within the experimental salt concentration range, since the microgel Hamaker constant calculated from equation 7 changes from 6.0 × 10-20 J at ∼0.25 M to 6.6 × 10-20 J at ∼3 M. For this estimation the particle size has been obtained through extrapolation of Figure 1 to the corresponding salt concentrations, considering that at 3 M the microgel particle has reached its collapsed size. Consequently, the interaction between particles should be controlled by the competition of osmotic and elastic interactions. The theoretical prediction for the interaction potential is presented in Figure 3. As can be seen, the competition between osmotic and elastic interactions yields the appearance of a secondary minimum, whose depth is controlled by the polymer volume fraction in the interpenetration region Φ (Vosm and Velas both depend on δ2; Vosm ∼ Φ2, while Velas ∼ Φ). An increase in salt concentration implies an increase in Φ; hence, the secondary minimum becomes deeper as the ionic concentration is raised. The interpretation of the results can now be tackled in terms of a reversible growth model, first introduced by
Reversible Aggregation of Soft Particles
Figure 3. Interaction potential curves between soft colloidal particles, illustrating the presence of a secondary minimum caused by competing osmotic and elastic interactions. The increase in salt concentration is modeled through an increase in χ. The solvency reduction implies a decrease in δ and an increase in Φ. χ ) 0.6: δ ) 4 nm; Φ ) 0.01. χ ) 0.7: δ ) 3 nm; Φ ) 0.015. χ ) 0.8: δ ) 2 nm; Φ ) 0.024. Ap ) 6.6 × 10-20 J and Aw ) 3.7 × 10-20 J.19
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Figure 5. Particle hydrodynamic diameter versus medium pH.
Figure 6. Average cluster hydrodynamic diameter as a function of time: aggregation (O) and stabilization (f) of the system.
depth is reduced, the number of particle neighbors n has to increase in order to reduce the probability for unbinding, thus yielding higher fractal dimensions. Figure 4. Effect of the χ parameter over the interaction potential curve.
Test to the Aggregation Reversibility
Shih, Aksay, and Kikuchi.7,8 It is basically a simulation model built by modifying the classical cluster-cluster aggregation model with a finite interparticle attraction energy, -Vmin. The aggregation reversibility is attained by introducing a probability for unbinding to occur: P ∼ exp[-nVmin/(kT)], where n is the number of closest neighbors of every single particle. Reducing Vmin results in increasing the probability for unbinding. In the limit Vmin f ∞, P f 0 and the predictions of the cluster-cluster aggregation model are recovered. Due to the finiteness of the attraction minimum, the cluster structure may be more compact in comparison with that arising from the cluster-cluster aggregation model, since the binding forces within the cluster are weaker. The deeper Vmin is, the smaller is the aggregate fractal dimension. The decrease in the fractal dimension with the increase in salt concentration (see Table 1) could be understood as an aggregation process taking place in a finite energy minimum, which is increasing in depth as the ionic concentration is raised. As the minimum becomes deeper, the probability for unbinding decreases and the cluster structure becomes more branched. As the energy minimum
The aggregation of colloidal particles in secondary minimums is characterized by the reversibility of the process. Experimentally, this feature can be tested by causing, first, the aggregation of the system, modifying afterward the values for the variables controlling the minimum depth. The experimental dependence of the fractal dimension on salt concentration, in the high salt limit, has been explained by considering that the aggregation process occurs in a minimum of finite depth, which is controlled by the balance between the osmotic and elastic interactions. The former is attractive since χ > 0.5, the latter being repulsive with independence of χ. As a consequence, a change in the Flory parameter from above to below 0.5 must eliminate the energy well, the net interaction between particles becoming totally repulsive (Figure 4). The aggregation reversibility for the microgel particles was tested by following the time evolution of the mean cluster hydrodynamic diameter during the aggregation process and after changing the Flory parameter. The aggregation was carried out at pH 4.9 and at [NaCl] ) 3 M in order to avoid any influence of electrostatic forces that could appear with the subsequent pH change. Once the particles aggregated, the medium pH was lowered to
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a value of 4.4. This pH value set the χ parameter of the polymer below 0.5 (water becomes a good solvent for 2VP), leaving unaltered the size of the particle (Figure 5) and thus the van der Waals interaction. The results are shown in Figure 6. After aggregation, the clusters broke down once the χ parameter was changed from above to below 0.5. Consequently, the aggregation of microgel particles is reversible taking place in a finite energy well.
colloidal system. The clusters have a higher fractal dimension than that typical for DLCA processes. This higher compactness results from the aggregation in energy minimums of finite depth, caused by the competition between osmotic and elastic particle interactions. These contributions to the total interaction potential arise due to the soft character of the microgel particles, which allows for surface interpenetration. Finally, the aggregation reversibility has been experimentally tested.
Conclusions
Acknowledgment. The financial support under Accio´n Integrada Hispano-Brita´nica HB 1998-0225 and Project MAT-2000-1550-C03-02 is greatly acknowledged.
The aggregation of soft particles has been studied, in particular, the cluster structure at ionic concentrations far above the critical coagulation concentration of the
LA001351U
