Colloidal aggregation in energy minima of ... - Semantic Scholar

JOURNAL OF CHEMICAL PHYSICS

VOLUME 110, NUMBER 11

15 MARCH 1999

Colloidal aggregation in energy minima of restricted depth ´ lvareza) J. A. Molina-Bolı´var, F. Galisteo-Gonza´lez, and R. Hidalgo-A Grupo de Fı´sica de Fluidos y Biocoloides, Departamento de Fı´sica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

~Received 15 May 1998; accepted 9 December 1998! Coagulation rates of bare and protein-covered colloidal particles show a different dependence on experimental conditions. While the rapid coagulation rate for the bare particles obeys the modified Smoluchowski theory and is independent of pH and the nature of the cation and the anion, the value for the coated particles is lower and depends on pH and ions’ nature. The variation in the Hamaker constant and the existence of a shallow primary minimum of the interparticle potential for the latex–protein complex, both attributed to the layer of water molecules and ions adsorbed on protein, may explain these results. Coagulation rates were measured with a low angle light scattering apparatus, and the experimental curves of stability fitted using Fuchs’ equation and the DLVO ~Derjaguin–Landau–Verwey–Overbeek! theory. In the case of covered particles, a modified expression of the van der Waals attraction was used. This attraction depends on the Hamaker constant for the protein in the vacuum, whose value was estimated from contact angle measurements. © 1999 American Institute of Physics. @S0021-9606~99!50711-3#

I. INTRODUCTION

mary minimum. Coagulation is usually assumed irreversible because the primary minimum is deep enough. There are, however, cases in which, by changing the ionic strength or the pH, the coagulated colloid redisperse.5–8 This phenomenon, known under the name of repeptization, can be explained by the existence of a distance of closest approach.9,10 The depth of the minimum can be restricted adsorbing a polymer on the particle surface, and then weak, reversible, and partial aggregation may occur. On the other hand, when aggregation occurs in a shallow minimum, the structure of the aggregates is more compact. It has been reported that the fractal dimension of an aggregate increases with decreasing the absolute magnitude of the minimum where coagulation occurs.11–14 Although a lot of work has been done on stabilization of colloidal systems by surfactants, only little attention has been paid to stabilization by amphoteric macromolecules like proteins. In the present paper we report on a detailed study of the influence of adsorbed proteins on the stability against coagulation of hydrophobic colloids ~the fundamental issue on particle enhanced immunoassays!. To do this, the stability domains and aggregation rates were determined using a lowangle scattering technique. Fragmentation experiments were carried out to obtain information relative to the internal stability of aggregates against breakup.

The stability and kinetics of coagulation of colloidal particles in liquids are of importance in many technological processes ~manufacturing of ceramic, magnetic, optic, electronic materials, and pharmaceutical and biomedical products!. The interaction energy between surfaces plays an important role in determining the behavior of the colloidal dispersion. Traditionally, colloidal stability has been described by the DLVO theory in which the total interaction energy between two particles is calculated as the sum of van der Waals attractive and electrical double layer repulsive forces.1,2 The adsorption of a macromolecule on the surface of the particle can affect this balance by significantly altering the interparticle electrostatic forces, by modifying the van der Waals attractive force or by producing significant additional interactions ~steric and hydration forces!. According to this theory, the total interaction energy versus distance curve should in principle have two minima and one maximum. This maximum represents an energy barrier opposing aggregation. The shape of the interaction profile has a direct influence on the kinetics of colloidal aggregation and on the structure of the aggregates. When aggregation occurs, the colliding particles overcome the maximum and come into direct contact in the infinitely deep primary minimum. But this situation is unlikely to occur since short-range effects, such as those caused by adsorbed layers of counterions ~in the Stern layer! and/or adsorbed solvent molecules, may keep particles from coming into physical contact, making the primary minimum of finite depth.3,4 The depth of the primary minimum at the distance of closest approach determines, therefore, the properties of colloidal dispersions in respect to the reversibility or irreversibility of coagulation in the pri-

II. EXPERIMENT A. Materials

All chemicals used were of analytical grade quality. Water was purified by reverse osmosis, followed by percolation through charcoal and a mixed bed of ion-exchange resins. The pH was controlled using different buffers ~acetate for pH 3–5 and phosphate for pH 6–7, constant ionic strength 2 mM!.

a!

Author to whom correspondence should be addressed; electronic mail: [email protected]

0021-9606/99/110(11)/5412/9/$15.00

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© 1999 American Institute of Physics

Downloaded 08 Mar 2005 to 150.214.68.115. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

´ lvarez Molina-Bolı´var, Galisteo-Gonza´lez, and Hidalgo-A

J. Chem. Phys., Vol. 110, No. 11, 15 March 1999

The latex was synthesized by means of a core-shell emulsion polymerization in a batch reactor. The core was a seed of polystyrene and the shell was obtained by copolymerization of styrene and chloromethylstyrene. Details are described elsewhere.15 The latex was cleaned by serum replacement. The particle size distribution was determined by transmission electron microscopy ~TEM! using an image analysis program ~Bolero, AQ systems!. The obtained diameter was (20165) nm and the polydispersity index 1.003, which indicates monodispersity. Surface charge, as determined by conductimetric titration, was (23.7 60.2) m C cm22, strong acid. F(ab 8 ) 2 antibody fragments from rabbit polyclonal IgG were kindly donated by Biokit S. A. ~Spain!. They were obtained by pepsin digestion of IgG, and purified by gel filtration followed by Protein A chromatography to remove undigested IgG. Purity was checked by SDS-PAGE, and the molecular weight was found to be 102 kDa. No IgG contamination was detected. The isoelectric points ~i.e.p.! of the F(ab 8 ) 2 preparations, measured by isoelectric focusing, were in the range 4.7–6.0.

B. Protein adsorption

The protein concentration in solution was determined, before and after adsorption at pH 7.2, by direct ultraviolet ~UV! spectrophotometry at 280 nm ~Spectronic 601, Milton Roy!. The total polymer area added was 0.4 m2 and the ionic strength of the medium 2 mM. Incubation was carried out in a thermostatic bath where the sample was gently shaken at 35 °C for 5 h. After incubation, the samples were separated from the supernatant by high-speed centrifugation and the supernatant filtered using a polyvinyldene difluoride filter ~Millipore, pore diameter 0.1 mm! before measuring the remaining protein concentration. Such a filter has an extremely low affinity for protein adsorption, so the filtration step does not interfere with the calculation of the adsorbed protein amount.

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where I u (0) is the initial intensity of light scattered at angle u, n s the number of primary particles, and k the rate constant. The scattered light intensity was followed at an angle of 4.5° during 100 s in a rectangular flow cell with 2-mm path length. Before use, the cell was thoroughly cleaned with chromic acid, rinsed with distilled water, and then dried using an infrared lamp. Equal quantities ~1 ml! of salt and latex were mixed and introduced in the cell by an automatic mixing device with a quite short dead time. The latex dispersions used for these aggregation experiments have to be sufficiently dilute to minimize multiple scattering effects, while still having an experimentally convenient coagulation time. The fresh suspensions of latex were sonicated for 2 min prior to experiments to break up any initial clusters. D. Photon correlation spectroscopy measurements

The mean hydrodynamic diameter of the latex and of the latex-F-(ab 8 ) 2 complexes was determined by photon correlation spectroscopy ~PCS!. The hydrodynamic layer thickness ~d! of the F(ab 8 ) 2 adsorbed on the polystyrene latex can be obtained from the difference between the mean diameter of the bare latex and the complex. Dynamic light scattering measurements were made using a Malvern 4700 system. The PCS measuring conditions were a temperature of 25 °C, scattering angle of 90°, wavelength of 488 nm ~Argon laser!, and particle concentration of ;109 ml21. E. Contact angle measurements

The sessile drop technique was used for contact angle measurement. A solution of F(ab 8 ) 2 is left to dry on glass ~microscope! slides for 1 or 2 days at ambient temperature. The advancing contact angle was measured directly for an a-bromonaphthalene drop placed on the substrate through a microsyringe. The contact angle was measured for five different drops with 3–4 mm drop base diameter at room temperature ~20 °C!. A few seconds were needed to obtain a stationary, not oscillating, drop. III. THEORY

C. Measurement of colloidal stability

The stability ratio ~W! is a criterion for the stability of a colloidal system,16–19 W5

kr , ks

~1!

where the rate constant k r describes rapid aggregation ~all collisions result in aggregation! and k s is the rate constant for the slow aggregation regime ~only a fraction of the collisions results in aggregation!. Thus, the inverse of the stability ratio provides a measure of the effectiveness of collisions leading to aggregation. In this work, the rate constant of coagulation was measured by low-angle light scattering, a technique developed by Lips and Willis.20 The total scattering intensity for a dispersion of identical primary particles with a time varying size distribution is given by21 I ~ t, u ! 5112kn s t, I u~ 0 !

~2!

The stability of a colloidal dispersion is determined by the total interaction potential close to the surface. According to the Derjaguin, Landau, Verwey, and Overbeek ~DLVO! theory,1,2 this total interaction potential, in the case of electrostatically stabilized colloids, is the sum of the repulsive electrostatic interaction energy (V E ) and the attractive London–van der Waals ~dispersion! energy (V A ) among two particles, V T 5V A 1V E .

~3!

This total interaction potential (V T ) is a function of particle diameter, the ionic strength of the aqueous phase, the Hamaker constant, and the diffuse potential of the particle. The van der Waals attraction takes the following nonsimplified form:22 V A 52

F

G

2a 2 A 2a 2 H ~ 4a1H ! 1 1ln , 6 H ~ 4a1H ! ~ 2a1H ! 2 ~ 2a1H ! 2

~4!


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where A is the Hamaker constant of particles interacting in the medium ~water!, a is the radius, and H is the distance between the particles. V E represents the repulsive interaction between the electrical double layers of the particles. According to the constant potential model for k a@1 and moderate potential, a reasonable expression for V E is23,24 V E 52 p a« « 0

S D 4kT g z ie

2

e 2kH,

~5!

where k is the Debye parameter and g,

g 5tanh

S D

ze c d , 4kT

~6!

being c d the Stern potential, z the valency of the ion, and a the particle radius. This equation is a good approximation for k H.1 and c d ,50 mV. In the original DLVO theory, the reference plane for attractive and repulsive energy coincide. However, Vincent et al.25,26 refined this idea by shifting the reference plane for the repulsive energy outward over a distance corresponding to the thickness ~D! of the Stern layer. Taking into account this correction, Eq. ~5! should be written as V E 52 p ~ a1D ! «« 0

S D 4kT g z ie

2

e 2 k ~ H22D ! .

~7!

This way, the total interaction energy as a function of distance between two particles presents a primary minimum, followed by a maximum, called the energy barrier, and another minimum, referred to as the secondary minimum. If through thermal agitation the particles have enough energy to overcome the energy barrier, the aggregation at the primary minimum occurs. However, if the secondary minimum is neither too shallow nor too deep, flocs are reversibly formed at the secondary minimum. Similarly, if the primary minimum is not too deep and the potential barrier is not too high ~or it can even be absent!, aggregates form reversibly at the primary minimum. Increasing the concentration of the electrolyte of the medium, the electrical double layer compresses and reduces the electrostatic repulsion between particles. The electrolyte concentration at which the energy barrier disappears, i.e., V T 50 and dV T /dH50, is known as critical coagulation concentration ~c.c.c.!.27 The stability ratio ~W!, previously introduced as the ratio between rapid and slow aggregation rates, depends on the total interaction energy. Using the modified Fuchs treatment28–30 we can write

E E

`

0

W5

`

0

b~ u ! ~ u12 !

exp 2

b~ u ! ~ u12 ! 2

exp

S D S D VT

k BT VA

k BT

described by a modified correction factor which takes electroviscous coupling into account. Higashitani et al.3 showed that this effect should influence the aggregation rates of particles smaller than ;100 nm in diameter. Since the particles used in this work have a diameter of (20165) nm, this approach will not be addressed. The stability factor can then be computed by numerical integration using Eq. ~8!. In this way, it is necessary to fit a theoretical curve to the experimental points of log W versus log@electrolyte# , using a pair of values of A and c d as fitting parameters. When polystyrene particles are coated with a protein, such as F(ab 8 ) 2 , the expression for the van der Waals attraction (V A ) has to be modified to include a Hamaker constant for the protein adsorbed. Assuming that the adsorbed layer is homogeneous, the following equation is obtained from the attraction energy:31,32 V A 52

1 ~ AA pp 2 AA i ! 2

~8!

du

where b is the hydrodynamic correction factor, and u is (H 22a)/a. When this correction is unnecessary, b 51. However, if an electrostatic interaction force exists, the hydrodynamic correction factor does not equal b (u), but must be

a 14 ~ AA aa 2 AA p p ! H12 d a1 d

3 ~ AA pp 2 AA i !

S

~ H1 d ! 21

d a

DG

,

~9!

where d is the thickness of the adsorbed layer, and A aa , A ll , and A pp are the Hamaker constants of water, particle, and protein, respectively. The layer thickness of F(ab 8 ) 2 adsorbed on the polystyrene latex ~d! can be determined by photon correlation spectroscopy, from the difference between the mean diameter of the bare latex and the complex. In the case of a protein coverage of 3 mg/m2, the value of d may be estimated as 5 61 nm. The literature values for A aa and A ll are 3.7 310220 and 6.5310220 J, respectively.33 These values are obtained via the Lifshitz approach, where spectroscopic data are needed. These necessary data are difficult to obtain in the case of a protein. An alternative way to determine the value of A pp is the van Oss’s surface thermodynamic approach.33,34 According to this approach, the surface energy of a solid, as well as the solid–liquid interfacial energy, may be separated in two components. The first is the apolar ~Lifshitz–van der Waals! component g LW, and the second is the polar ~Lewis acid–base! component g AB. It can be shown that there is a reliable proportionality between the Hamaker constant of a material (A ii ) and the apolar surface tension component g LW i ,

du ,

F

a1 d 1 ~ AA aa 2 AA pp ! 2 12 H

g LW i 5

Ai 24p l 20

,

~10!

where l 0 is a minimum equilibrium distance of 1.568 Å.33 Contact angles a measured with pure Lifshitz–van der LW Waals liquids yield the g LW S of the solid, once the g L of the 31 liquid is known, according to LW 1/2 11cos a 52 ~ g LW S /gL ! .

~11!




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TABLE I. Diffuse potential ( c d ) and Hamaker constant ~A! values ~at different ionic sizes! for latex calculated using DLVO theory.

FIG. 1. Theoretical dependence ~---! of log W on log@NaCl# @Eq. ~7!#. Experimental data ~d! for polystyrene latex at pH 5.5.

We have used this approach to obtain the Hamaker constant of the F(ab 8 ) 2 . A solution of F(ab 8 ) 2 is left to dry on glass ~microscope! slides for 1 or 2 days at ambient temperature. The contact angle obtained with a-bromonaphthalene ( g LW544.4mJ/m2! on the layer of F(ab 8 ) 2 was 13.8 °. Using Eqs. ~10! and ~11!, the value of the Hamaker constant for F(ab 8 ) 2 (A p p ) is estimated as 8310220 J. IV. RESULTS AND DISCUSSION A. Colloidal stability and interaction energy curves

Figure 1 shows the dependence of log W with log of the molar concentration of NaCl for the bare latex at pH 5.5 Log W decreases gradually with increasing NaCl concentration until a certain value is reached ~critical coagulation concentration, c.c.c.! and the curve remains parallel to the concentration axis. The latter part of the curve corresponds to the region of rapid aggregation, whereas the former part corresponds to the slow aggregation regime. Adding salt dispersion initiates aggregation by suppressing the double layer repulsion between particles. According to the DLVO, below the c.c.c., the thickness of the electrical double layer repulsion decreases with increasing salt concentration. The double layer is entirely suppressed, above the c.c.c., and the aggregation rate is independent of the salt concentration ~rapid aggregation!. Reerink and Overbeek35 have shown, considering several approximations, a linear relationship between log W and log electrolyte concentration. From this linear dependence an approximated value of the critical coagulation concentration can be obtained, which in the case of this latex is 365630 mM. The Hamaker constant and the diffuse potential of the latex were obtained by fitting the theoretical expression of W @Eq. ~8!# to the experimental values. The most relevant aspect of this fitting method is the fact that A and c d have different effects. c d affects the slope of the theoretical curve, whereas the Hamaker constant influences the plot intercept. To fit the experimental values of W it is necessary to know

Radius ion ~nm!

c d ~mV!

A31020 ~J!

0 0.095 0.36

216.42 226.94 226.27

0.2393 1.042 2.257

the ionic size, D, for the adsorbed ions of the Stern layer. Different situations of total or partial hydration of the counterion used in stability studies can be found in the literature.36–38 For this reason, both options, hydration and dehydration, were used in our calculations. Therefore, the distance between the surface of the particle and the center of the counterion (Na1) was considered to be 0.095 and 0.36 nm in the dehydrated and hydrated states, respectively.39 The A and c d values obtained from DLVO theory with and without considering these ionic sizes are shown in Table I. As can be seen, the fitted Hamaker value depends on the ion size used. These Hamaker constant values must be compared with the accepted value in the literature of 1.37310220 J for polystyrene in water.40 This value decreases if the retarded interaction of two polystyrene particles separated by water, with and without electrolyte, is considered. For example, at a separation of 2 nm the A value decreases until ;1 310220 J. The obtained Hamaker constant is closer to this value when the counterions are considered in a dehydrated situation ~see Table I!. The diffuse potential c d value has also been estimated with other techniques, such as zeta potential and conductometric measurements ~22164 and 22462 mV, respectively!. Since zeta potential depends on electrolyte concentration, mobility was measured using the closest possible concentration ~250 mM NaCl! to the c.c.c. which does not provoke aggregation. The other method implies the use of the Gouy–Chapman theory for the electrical double layer, where c d is obtained from the surface charge density. Both values are close to that obtained by fitting the stability plot. Using the Stern potential and Hamaker constant values obtained this way, estimates of the total potential energy of interaction curves were computed for the bare latex ~Fig. 2!. It can be observed that the increase in electrolyte concentration provokes a decrease in the height of the potential maximum. The latter, which prevents aggregation, finally disappears when the electrolyte concentration is similar to the experimental c.c.c. (365630mM! in agreement with the theoretical prediction. Figure 3 shows the stability curves for a latex-F(ab 8 ) 2 complex with a protein coverage of 3 mg/m2 at pH 2.3 and pH 6.8 ~plateau value of the adsorption isotherm 3.2 mg/m2!. Using the approximation of Reerink and Overbeek, the values of the experimental c.c.c. are 5465 at pH 6.8 and 366 640 mM at pH 2.3. The experimental values of the stability factor have been fitted to the theoretical expression @Eq. ~8!# using c d as fitting parameter, and Eq. ~9! with the previously cited values. The estimated values for c d are 16 at pH 6.8 and 124.5 mV at pH 2.3. Due to the amphoteric nature of the protein, the net charge of the complex depends to a large


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FIG. 2. Calculated total interaction potential ~V T in k B T units! versus distance for latex: ~1! 150, ~2! 186, ~3! 300, ~4! 360, and ~5! 409 mM. Hamaker constant ~A! 1.042•10220 J and Stern potential ( c d )226.94 mV.

FIG. 4. Calculated total interaction potential ~V T in k B T units! versus distance for latex-F(ab 8 ) 2 complex at pH 6.8 ~dashed line! and pH 2.3 ~solid line!: ~1! 186, ~2! 360, ~3! 409, ~4! 35, ~5! 55, and ~6! 70 mM.

extent on pH, becoming more negative the higher the pH. At pH 2.3, the net charge of the complex is positive, while negative at pH 6.8. In the case of pH 6.8, the existence of an anomalous stabilization mechanism at high electrolyte concentration is noteworthy. At low electrolyte concentrations, the colloidal stability proceeds as expected, decreasing with increasing ionic strength until a minimum stability is reached at the c.c.c. Nevertheless, above a certain value ~critical stabilization concentration, c.s.c.! a change in the trend is observed, and the coagulation diminishes with increasing ionic strength. This phenomenon is, of course, quite contrary to qualitative DLVO theory, since addition of electrolyte is generally expected to cause coagulation, not the opposite ~non-DLVO region!.

This phenomenon has been explained41,42 by the inclusion of hydration forces in the DLVO theory. Hydration forces are widely recognized for hydrophilic surfaces as strong stabilizers.43–45 The nature of water close to a surface can be very different from that of bulk water, especially in the case of proteins, where it is well established that water molecules strongly bind to their surface.46 The approach of two particles with hydrated surfaces will generally be hindered by an extra repulsive interaction, distinct from electrical double layer repulsion. This hydration repulsion arises essentially from the need for the surfaces to become dehydrated if true contact between particles is to occur. This involves work and hence an increase in the free energy of the system. At ionic strengths above the c.s.c., hydrated cations adsorb to hydrophilic proteins and they presumably retain some of their water of hydration.47 An overlap of the hydrated cations near the two mutually approaching particles creates the repulsive hydration force. Since the number of cations bound to a protein increases with salt concentration,48 it is expected that hydration forces increase with increasing ionic strength. At pH 2.3, the anomalous stabilization is not observed because the net charge of the complex is negative, and adsorbed anions do not provoke this effect.42 Figure 4 corresponds to the total potential energy of interaction calculated for the complex at pH 2.3 and 6.8. The obtained values of the Stern potential and Hamaker constant have been used to compute these interaction curves. For both pHs the potential maximum disappears for an electrolyte concentration similar to the experimental c.c.c. B. Fragmentation of aggregates

FIG. 3. Theoretical dependence ~---! of log W on log@NaCl# @Eq. ~7!#. Experimental data for latex-F(ab 8 ) 2 complex at pH 6.8 ~l! and pH 2.3 ~s!.

Fragmentation studies have been performed in order to obtain information about the structural stability of the aggregates. In a typical fragmentation experiment, equal volumes of latex and salt solution are mixed in a test tube at time




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FIG. 6. Time evolution of hydrodynamic diameter of aggregates before and after agitation for latex-F(ab 8 ) 2 complex at pH 2.3: ~l! 556 mM; ~s! 752 mM. The dashed lines indicate the moment when the agitation is carried out.

FIG. 5. ~a!. Time evolution of hydrodynamic diameter of aggregates before and after agitation for latex at 300 mM. The dashed line indicates the moment when the agitation is carried out. ~b!. Time evolution of hydrodynamic diameter of aggregates before and after agitation for latex at 370 mM. The dashed line indicates the moment when the agitation is carried out.

zero. The average diameter of the aggregates is determined as a function of time by photon correlation spectroscopy. At a certain time of the experiment, the cell is taken out of the device and shaken for 20 s, then inserted again. Figures 5~a! and 5~b! show the time evolution of the average diameter before and after agitation for bare latex aggregates at 300 and 370 mM NaCl, respectively, at pH 5.5. It can be observed that fragmentation of the aggregates does not occur in any case. But there are some differences in the behavior of the aggregating colloid after the agitation. For an electrolyte concentration lower than the c.c.c. @Fig. 5~a!#, the agitation does not influence the aggregation process. In this case, the interaction potential profile presents an energy barrier. If the primary minimum is deep, it is not expected that fragmentation occurs, since particles have to overcome the energy barrier of the primary minimum which, although not high enough to prevent coagulation in one direction, is high enough to prevent passage from the other side.

For an electrolyte concentration higher than the c.c.c. @Fig. 5~b!#, the agitation provokes an increase in the diameter of the aggregates. In this case the aggregation is controlled by the diffusion of the particles because the interaction potential profile does not present an energy barrier. Agitation can promote aggregation by increasing the frequency of particle collisions. But there is not fragmentation because the particles are aggregating in a deep primary minimum. Different electrolyte concentrations below and above c.c.c. ~186, 410, 480, and 556 mM! were tested with similar results. In the case of the latex-F(ab 8 ) 2 complex, Fig. 6 shows the time evolution of the average diameter before and after agitation at 556 and 752 mM NaCl at pH 2.3. As can be seen, the fragmentation of aggregates after agitation occurs for both electrolyte concentrations in exactly the same way. Similar fragmentation was observed for electrolyte concentrations above and below the c.c.c., and for pH 2.3 and 6.8. There is always fragmentation in the case of the complex, independent of the existence of energy barrier in the interaction potential profile. Therefore, in the case of the latex-F(ab 8 ) 2 complex, a break up of the aggregates occurs, whereas not in the bare latex. It suggests a weaker interaction between complex particles in the aggregate. The nature of the surface of the bare and the F(ab 8 ) 2 coated latex is quite different. The first is mainly hydrophobic, while the latter is hydrophilic due to the adsorbed protein layer. The structure of water around hydrophilic surfaces is highly ordered, and hydrated ions are easily adsorbed, as shown by the anomalous stabilization mechanism at high ionic strength previously described. Probably due to this hydration of the protein surface and/or the adsorption of solvated counterions, the primary minimum depth may become much lower than the calculated values from the superposition of van der Waals attraction and double layer repulsion forces. Moreover, Hamaker con-


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stant of this hydrated surface may well be diminished, with the consequent reduction in the attractive potential. For any or both of these reasons, interaction potential is low enough to allow fragmentation, although the existence of a distance of closest approach is the most accepted explanation in the literature for this kind of phenomena. Another factor giving rise to a shallow primary energy minimum is a porous or a rough particle surface. In this case the amount of contact area between two nominally touching particles is reduced, and may restrict the minimum separation distance attainable. Moreover, the van der Waals energies for rough particles are lower than for smooth ones.49 This reduction in the van der Waals energy, which is predominant at closest approach distance, provokes the existence of a shallow primary minimum. It is plausible then to suppose that the distance of closest approach for two coagulated particles in a bare latex aggregate is lower than that of a complex aggregate. As a consequence, the reduction in the primary minimum depth in the latter system may allow fragmentation of aggregates by agitation. C. Aggregation kinetics

In the classical theory of rapid coagulation developed by Smoluchowski,50 every collision between colliding particles is assumed to be effective. Rapid coagulation occurs when enough electrolyte is added to suppress electrostatic repulsion between interacting particles. Under these conditions, the rate of coagulation is determined by the diffusion of the particles. One of the main assumptions of the original Smoluchowski’s theory is that aggregation process is irreversible. As previously stated, however, the coagulation in the primary minimum is not necessarily irreversible. Retardation of coagulation kinetics may be caused by deaggregation if it occurs in a shallow primary minimum. The rapid coagulation constant (K R ) measured for the bare latex at pH 5.5 is 2.20310212 cm3 s21. This value is in good agreement ~within the standard deviation limits! with the interval of (3.2161.36)310212 cm3 s21 quoted by Sonntag and Strenge32 after summarizing the pertinent literature. On the other hand, the K R values for the latex-F(ab 8 ) 2 with a protein coverage of 3 mg/m2 at pH 6.8 and 2.3 are 1.60310212 and 1.96310212 cm3 s21, respectively. According to Eq. ~8!, rapid coagulation rates depend on the Hamaker constant of the colloid. If we assume that the Hamaker constant value for the protein-covered particle may be lower than that of bare particle due to hydration, the rapid aggregation rate should be also lower. But the observed decrease in the rapid coagulation constant when the latex particles are coated with protein could also indicate the possibility of deaggregation, due to the existence of a minimum distance of closest approach. In this case, the depth of the primary minimum becomes so shallow that particles can go over to deflocculate, and the effective coagulation rate is reduced. This hypothesis may be corroborated by studying the parameters that can affect the structure of this hydration layer ~pH and electrolyte ions nature!. To compare with a non-protein-covered surface, the influence of pH was tested on a carboxylic latex. The net

Molina-Bolı´var, Galisteo-Gonza´lez, and Hidalgo-A´lvarez TABLE II. Rapid coagulation constant (K R ) at different pHs for a carboxylic latex. pH

K R 31012 (cm3 s21)

4.1 5.3 6.7

2.2060.3 2.1760.3 2.1860.3

charge of this latex depends on pH due to the deprotonation of carboxylic groups. In Table II, the rapid coagulation constant values as a function of pH are presented for this latex. As can be seen, they are not dependent on the surface potential. Two latex-F(ab 8 ) 2 complexes with a protein coverage of 1.5 and 2.8 mg/m2 have been used to study the influence of pH on K R . The obtained values are shown in Table III. For both complexes the rapid coagulation constant decreases with increasing the pH. The highest K R values are obtained in both cases at pH 3.4, when the net charge of the complexes is positive, and in the stability diagrams stabilization does not appear by hydration forces ~nonspecific adsorption of anions!. Regardless, these K R values are lower than the rapid coagulation constant for the bare latex. The dependence of K R on the protein coating level is noteworthy. The decrease in the value is presumably due to the surface roughness provoked by a low coverage. This way, effective contact area is diminished, restricting the minimum separation distance.4,32 At pH 5.3, 6.2, and 7.5 the net charge of the complexes is negative, and more negative with increasing the pH. In these situations, there is specific adsorption of cations on the protein ~as the existence of stabilization by hydration forces indicates!, provoking an increase of the closest approach distance and therefore an increase in the deaggregation probability ~reduction in K R !. The minimum separation distance between coagulated particles should depend then on the counterions used in the aggregation process. The rapid coagulation constants for a latex-F(ab 8 ) 2 complex with a protein coverage of 2.6 mg/m2 at pH 6.2 are 1.17 and 1.4131012 cm3 s21 for Na1 and Cs1 salts, respectively. The hydrated radii of these cations are 0.36 and 0.33 nm.51 The coagulation rate decreases with increasing the diameter of the counterion, probably due to its influence in the primary minimum depth. The coagulation rate obtained for the bare latex at pH 5.5 with these cations, however, was the same, 2.21310212 cm3 s21.

TABLE III. Rapid coagulation constant (K R ) at different pHs for two latexF(ab 8 ) 2 complexes. Protein coverage ~mg/m2!

pH

K R 31012 (cm3 s21)

2.8 2.8 1.5 1.5 1.5

6.2 3.4 7.5 5.3 3.4

1.66 2.00 0.96 1.12 1.73



J. Chem. Phys., Vol. 110, No. 11, 15 March 1999 TABLE IV. Rapid coagulation constant (K R ) for the aggregation of a latexF(ab 8 ) 2 complex with 2.8 mg/m2 at pH 6.2 induced with different anions. Anion

K R 31012 (cm3 s21)

Cl2 NO2 3 ClO2 4

1.66 1.03 0.48

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stant of the complex on pH and cation and anion nature has been observed. For the bare particle this dependency does not exist. Finally, complex aggregates grown under rapid conditions are delicate, showing substantial fragmentation effects by agitating the sample. In the case of bare latex aggregates, fragmentation is not observed. ACKNOWLEDGMENTS

Finally, we have obtained the rapid coagulation constant for a latex-F(ab 8 ) 2 complex with 2.8 mg/m2 at pH 6.2 using salts with the same cation but different anions. The results are summarized in Table IV. As in the case of the cations, the coagulation rate decreases with increasing the size of the anion, whereas the rate for the bare latex is independent of the anion, with a constant value of 2.20310212 cm3 s21. The marked effect of these anions on the coagulation rate of the complex, apparently contradictory to the fact that anions do not provoke hydration stabilization when the net charge of the complex was positive is noteworthy. The reason for this is that anions do not adsorb specifically on the protein surface ~only to neutralize charge!, while cations do adsorb specifically ~by forces of nonelectrical nature!. Surrounding this excess of cations, however, anions must be situated to maintain electroneutrality. All of these results suggest that the latex-F(ab 8 ) 2 particles coagulate in a shallow primary minimum and therefore some deaggregation may occur, whereas the bare latex particles coagulate in a deep minimum.

V. CONCLUSIONS

This paper presents a comparison of the colloidal stability and characterization of aggregates between latex particles with and without protein on the surface. We have tried to explain the stability domains of bare and coated particles by including correction factors in the DLVO theory, such as the ionic size and the hydrodynamic effect. Theory and experiments are in good agreement. In the case of the complex, a complete expression for the van der Waals attraction between particles coated with protein has been introduced in the modified DLVO. This potential depends on the Hamaker constant of the protein, which has been determined by means of the van Oss’s surface thermodynamic approach. For this purpose, the contact angle of a-bromonaphthalene on a layer of F(ab 8 ) 2 was measured. The stability domains for the latex–protein complex show a non-DLVO region that has been explained by the existence of a repulsive hydration force. The rapid coagulation rate value for the bare latex is in agreement with irreversible Smoluchowski kinetics, whereas a reduction is observed for the complex. It is explained by the existence of a layer of water molecules, ions, and hydrated ions adsorbed on the latex–protein surface. This layer makes the primary minimum of the interparticle potential shallow and reduces the rate by deflocculation of coagulated particles ~reversibility in the aggregation process!, and may also reduce the effective Hamaker constant of the proteincovered particle. A dependence of the rapid coagulation con-

This work was supported by the Comisioń Interministerial de Ciencia y Tecnologıá CICYT, project MAT 96-103503-CO2. We thank Biokit S.A. ~Barcelona, Spain! for purifying, characterizing, and supplying the F(ab 8 ) 2 fragments. We are grateful to J. Sarobe and J. Forcada for providing the sample of polymer colloid used in this work. Also, we are grateful to A. Puertas and M. Romero for their computer programs, to A. Schmitt for the technical support in the nephelometer, and to Mustapha for the measurements of contact angle. And, also, we thank the referee for the significant suggestions. B. V. Derjaguin and L. Landau, Acta Physicochim. USSR 14, 633 ~1941!. E. J. W. Verwey and J. Th. G. Overbeek Theory of the Stability of Lyophobic Colloids ~Elsevier, Amsterdam, 1952!, Vols. 1 and 2. 3 K. Higashitani, M. Kondo, and S. Hatade, J. Colloid Interface Sci. 142, 204 ~1991!. 4 M. Elimelech, J. Gregory, X. Jia, and R. Williams, Particle Deposition and Aggregation ~Butterworth-Heinemann, London, 1995!. 5 E. Matijevic, J. Colloid Interface Sci. 58, 314 ~1977!. 6 S. Rohrsetzer and F. Csempesz, Colloid Polym. Sci. 257, 85 ~1979!. 7 S. Rohrsetzer, I. Paszli, F. Csempesz, and S. Ban, Colloid Polym. Sci. 270, 1243 ~1992!. 8 W. Wu, R. F. Giese, and C. J. van Oss, Colloids Surf., A 89, 253 ~1994!. 9 G. Frens and J. Th. G. Overbeek, J. Colloid Interface Sci. 38, 376 ~1972!. 10 G. Frens and J. Th. G. Overbeek J. Colloid Interface Sci. 36, 286 ~1971!. 11 P. Dimon, S. K. Sinka, D. A. Weitz, C. R. Safinya, G. S. Smith, W. A. Varady, and H. M. Lindsay, Phys. Rev. Lett. 57, 595 ~1986!. 12 W. Y. Shih, I. A. Aksay, and R. Kikuchi, Phys. Rev. A 36, 5015 ~1987!. 13 J. L. Wan, W. Y. Shih, M. Sarikaya, and I. A. Aksay, Phys. Rev. A 41, 3206 ~1990!. 14 P. W. Zhu and D. H. Napper, Phys. Rev. E 50, 1360 ~1994!. 15 J. Sarobe and J. Forcada, Colloid Polym. Sci. 8, 274 ~1996!. 16 J. T. G. Overbeek, Adv. Colloid Interface Sci. 16, 17 ~1982!. 17 J. E. Seebergh and J. C. Berg, Langmuir 10, 402 ~1994!. 18 F. J. Rubio-Hernańdez, J. Non-Equilib. Thermodyn. 21, 153 ~1996!. 19 ´ lvarez, A. Martıń, A. Fernańdez, D. Bastos, F. Martıńez, and R. Hidalgo-A F. J. de las Nieves, Adv. Colloid Interface Sci. 67, 1 ~1996!. 20 A. Lips and E. J. Willis, J. Chem. Soc., Faraday Trans. 1 67, 2979 ~1971!. 21 A. Lips and E. J. Willis, J. Chem. Soc., Faraday Trans. 1 69, 1226 ~1973!. 22 P. Hiemenz, Principles of Colloid and Surface Chemistry ~Marcel Dekker, New York, 1986!, p. 648. 23 J. Th. G. Overbeek, Adv. Colloid Interface Sci. 16, 17 ~1982!. 24 J. Th. G. Overbeek, Pure Appl. Chem. 52, 1151 ~1980!. 25 B. Vincent, B. H. Bijsterbosch, and J. Lyklema, J. Colloid Interface Sci. 37, 171 ~1970!. 26 E. Matijevic, K. G. Mathal, R. H. Ottewill, and M. Kerker, J. Phys. Chem. 65, 826 ~1961!. 27 N. Fuchs, Z. Phys. 89, 736 ~1934!. 28 D. W. J. Osmond, B. Vincent, and F. A. Waite, J. Colloid Interface Sci. 42, 262 ~1973!. 29 E. Barouch, J. Chem. Soc., Faraday Trans. 1 84, 3093 ~1988!. 30 E. P. Honing, G. I. Roebersen, and P. H. Wiersema, J. Colloid Interface Sci. 36, 97 ~1971!. 31 K. G. Mathai and R. H. Ottewill, Trans. Faraday Soc. 62, 759 ~1966!. 32 H. Sonntag and K. Strenge, Coagulation Kinetics and Structure Formation ~Plenum, New York, 1987!. 33 C. J. van Oss, Interfacial Forces in Aqueous Media ~Marcel Dekker, New York, 1994!. 1 2


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