Effect of Steric, Double-Layer, and Depletion Interactions on the

Langmuir 2006, 22, 4541-4546

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Effect of Steric, Double-Layer, and Depletion Interactions on the Stability of Colloids in Systems Containing a Polymer and an Electrolyte Haohao Huang and Eli Ruckenstein* Department of Chemical and Biological Engineering, State UniVersity of New York at Buffalo, Buffalo, New York 14260 ReceiVed January 20, 2006. In Final Form: March 1, 2006 Experiments carried out by Stenkamp et al. [Stenkamp, V. S.; McGuiggan, P.; Berg, J. C. Langmuir 2001, 17, 637.] have shown that polystyrene latexes can be restabilized at sufficiently high electrolyte concentrations in the presence of an amphiphilic block copolymer [poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) (PEOPPO-PEO)] At even higher electrolyte concentrations, the systems can again be destabilized. The present paper attempts to explain the restabilization through the dominance of steric interactions and the destabilization through the dominance of depletion interactions. Because of salting out, as the concentration of electrolyte increases, the polymer molecules are increasingly precipitated onto the surface of the latex particles and, at sufficiently high electrolyte concentrations, form, in addition, aggregates. The precipitation onto the latex particles generates steric repulsion, which is responsible for the restabilization, whereas the formation of aggregates generates depletion interactions, which are responsible for destabilization.

1. Introduction The stability of a colloidal dispersion generally decreases as the electrolyte concentration increases.2 At low electrolyte concentrations, the electrostatic repulsion is responsible for the stability of the colloidal system. However, at sufficiently high electrolyte concentrations, the thickness of the double layer is significantly decreased, and the electrostatic repulsion no longer contributes to the stability of the system. The steric stabilization, which is imparted by polymer molecules grafted onto the colloidal particles, is extensively employed.3 Amphiphilic block copolymers are widely used as steric stabilizers. The solvent-incompatible moieties of the block copolymer provide anchors for the polymer molecules that are adsorbed onto the surface of the colloidal particles, and the solventcompatible (buoy) moieties extend into the solvent phase. When two particles with block copolymers on their surface approach each other, a steric repulsion is generated between the two particles as soon as the tips of the buoy moieties begin to contact, and this repulsion increases the stability of the colloidal system.4-6 Polymers can also induce aggregation due to either “depletion”7-11 or “bridging” interactions.12-15 * To whom correspondence should be addressed. Telephone: (716) 6452911 ext. 2214. E-mail: [email protected]. Fax: (716) 645-3822. (1) Stenkamp, V. S.; McGuiggan, P.; Berg, J. C. Langmuir 2001, 17, 637. (2) Verwey, E. J.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (3) Napper D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1983. (4) Alexander, S. J. Phys. 1977, 38, 983 (5) Roan, J. R.; Kawakatsu, T. J. Chem. Phys. 2002, 116, 7283. (6) Bell, N. S.; Sindel, J.; Aldinger, F.; Sigmund, W. M., J. Colloid Interface Sci. 2002, 254, 296. (7) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (8) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183. (9) Vrji. A. Pure Appl. Chem. 1976, 48, 471. (10) Joanny, J. F.; Leibler, L.; de Gennes, P. G. J. Polym. Sci., Polym. Phys. E. 1979, 17, 1073. (11) Rao, I. V.; Ruckenstein, E. J. Colloid Interface Sci. 1985, 108, 389. (12) Dickinson, E.; Eriksson, L. AdV. Colloid Interface Sci. 1991, 34, 1. (13) Podgornik, R. J. Phys. Chem. 1992, 96, 884. (14) Borukhov, I.; Andelman, D.; Orland, H. J. Phys. Chem. B 1999, 103, 5042. (15) Huang, H.; Ruckenstein, E. Langmuir 2006, 22, 3174.

Stenkamp et al.1 experimentally determined the stability ratio of polystyrene latexes as a function of electrolyte concentration for various electrolyte solutions in the presence of the amphiphilic block copolymer poly(ethylene oxide)-poly(propylene oxide)poly(ethylene oxide) (PEO-PPO-PEO). As the electrolyte concentration increases, the stability ratio of the colloidal system first decreases as a result of the decrease of the electrostatic stabilization. However, at sufficiently high electrolyte concentrations, the stability ratio increases with increasing electrolyte concentration. In some of the colloidal dispersions, Stenkamp et al. observed that the restabilization was followed by destabilization at higher electrolyte concentrations. Stenkamp et al. expressed the opinion that neither the additional adsorption of the block copolymer caused by decreased polymer solvency with increasing electrolyte concentration, nor ion binding, but rather the presence of micelles or aggregates is responsible for the restabilization. Another possible and more plausible explanation is that the steric interaction is responsible for the restabilization at sufficiently high electrolyte concentrations because of the salting out of the polymer onto the surface of the colloidal particles. The solubility of the polymer in water decreases as the electrolyte concentration increases. As a result, the hydrophobic PPO chains of the block copolymer are adsorbed upon the surface of the polystyrene particle, and the PEO chains stretch out, forming a brush that stabilizes the colloidal dispersion. Whereas, for not too large electrolyte concentrations, the polymer molecules precipitate upon the surface of the particles, at large concentrations, they generate, in addition, aggregates. The latter particles are responsible for the attractive depletion interactions and hence for the destabilization, which follows the maximum in the curve stability ratio against electrolyte concentration. The above explanation of the behavior of the complex dispersions investigated by Stenkamp et al. can be accurate only if, for not too low electrolyte concentrations, the concentration of the block copolymer is lower than the critical micelle concentration (CMC) and, for larger ones, greater than the CMC. The dependence of the CMC on the electrolyte concentration is not known for the present case. However, in pure water, at 25 °C, the CMC is very

10.1021/la0602057 CCC: $33.50 © 2006 American Chemical Society Published on Web 04/11/2006

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large (25.5 g/L for Pluronic L64 and 24.7 g/L for Pluronic L35 16), and at large electrolyte concentrations, intuition suggests that it is much smaller. Consequently, the proposed explanation might be accurate. In the present paper, calculations are carried out to explain the restabilization followed sometimes by destabilization, as the electrolyte concentration increases, observed by Stenkamp et al. In these calculations, double-layer, steric, and depletion interactions are taken into account. The steric interaction at various electrolyte concentrations is calculated using the scaling theory. The surface density of the polymer chain was evaluated by using the Sechenov equation for the polymer solubility as a function of electrolyte concentration.

the polymer chains present on the surface constrain the orientation of the water molecules nearby. As a result, in the region near the surface, the dielectric constant is lower than in the bulk. The region near the surface (region I) of thickness δ is considered to have a dielectric constant I different from that in the remaining region between plates (region II), II, because of the lower dielectric constant of the polymer and the constraint induced by the chains on the water molecules. For simplicity, the dielectric constants in both regions are assumed to be constant, with that in region II being equal to the dielectric constant of water (80) and that in region I being equal to 60. The electrical potential distribution between the two plates is given by the Poisson equations

2. Theoretical Framework

∂ 2Ψ F ) - I (for region I) ∂x2  0

(4)

∂2Ψ F ) - II (for region II) 2 ∂x  0

(5)

The colloidal stability can be expressed as the stability ratio, which is given by the expression17

W)

exp(Vtot/kT) ds/ s2

∫2

∞

∫2

∞

exp(VvdW/kT) ds s2

and

(1)

where W is the stability ratio, Vtot is the total interaction potential between two spherical particles, k is the Boltzmann constant, T is the absolute temperature, s is the ratio between the center distance of two colloidal particles and the radius of the particles, and VvdW is the van der Waals interaction potential. Because the block copolymer is uncharged, the electrostatic repulsion, the steric interaction, and the depletion repulsion are independent quantities and are hence additive. The total interaction potential Vtot is therefore given by the sum

Vtot ) Vd + Vsteric + Vdep + VvdW

(2)

where Vd is the double-layer potential between two spheres, Vsteric is the steric potential, Vdep is the depletion potential, and VvdW is the van der Waals interaction potential. When the radius of the spherical colloidal particles a is much larger than the shortest distance between the surfaces of two colloidal particles, the free energy between two identical spherical particles, due to double-layer, steric, and depletion interactions, can be calculated using the Derjaguin approximation:2

Vi ) πa

∫H∞ [Fi(H) - Fi(∞)]dH

i ) d, steric, dep

(3)

0

where Fi(H) is the free energy of interaction between two parallel plates, H is the distance between plates, and H0 is the shortest approach between the two spheres. The solubility of a block copolymer in water decreases as the concentration of electrolyte increases. When the concentration of the polymer is larger than its solubility, the polymer molecules precipitate onto the surface or form aggregates that remain dispersed into the colloidal system or deposit on the wall of the vessel that contains the colloidal dispersion. Let us start from two parallel plates, and then calculate the stability ratio of the system for spheres using the Derjaguin approximation. 2.1. Double-Layer Interaction. When the copolymer PEOPPO-PEO precipitates onto the surface of the particles, the PPO segments are attached to the surface, and the PEO segments extend into the solution. The dielectric constant near the surface of the plates is changed by the presence of the polymer chains because the latter have a lower dielectric constant. In addition, (16) Lopes, J. R.; Loh, W. Langmuir 1998, 14, 750. (17) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, England, 1986; Vol. 1.

where Ψ is the electrical potential, F is the local charge density, I and II are the dielectric constants in regions I and II, respectively, and 0 is the vacuum permittivity. At the boundary between the two regions, the electric potentials should be the same: I II ) Ψ|x)-d+δ Ψ|x)-d+δ

(6)

where 2d is the distance between the two plates, and x is measured from the middle of the two plates. If there is no charge accumulation, the Gauss law provides the equation

I

∂Ψ I ∂Ψ II ) II |x)-d+δ | ∂x x)-d+δ ∂x

(7)

where the superscripts I and II refer to regions I and II, respectively. The boundary condition at the surface has the form

dΨ σ )dx 0

(8)

where σ is the surface charge density. The symmetry condition provides an additional boundary condition,

∂Ψ | )0 ∂x x)0

(9)

Using boundary conditions 8 and 9, the Poisson equations 4 and 5 can be solved to calculate the distribution of the electrical potential for a given surface charge density. For two plates, the double-layer free energy Fd of the interaction between two plates is the sum of three contributions:18

Fd ) Fele + Fentropy + Fchem

(10)

where Fele, Fentropy, and Fchem are the electrostatic, entropic due to the small ions, and chemical contributions to the free energy, respectively. (18) Overbeek, J. T. G. Colloids Surf. 1990, 51, 61.

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The electrostatic contribution comes from the electrical potential energy of the charge in the field:

Fele )

∫-dd

I0 ∂Ψ 2 dx + 2 ∂x II d-δ  0 ∂Ψ 2 dx (11) -d+δ 2 ∂x

0 ∂Ψ 2 dx ) 2 2 ∂x

∫-d-d+δ

( )

( )

∫

( )

Because the two layers have different dielectric constants, the electrostatic contribution is the sum for those two layers. The entropic contribution to the free energy due to the electrolyte ions is given by18

Fentropy ) kT

∑i ∫-d ni ln d

() ni

n0i

- ni + n0i )dx

(12)

where ni is the number of ions of species i per unit volume, and n0i is its value in the bulk. The chemical contribution ∆Fchem is 0 at constant surface charge density. 2.2. Steric Interaction. The steric interaction force fsteric between two plates containing attached polymer chains can be evaluated on the basis of the scaling theory19

fsteric =

[( ) ( ) ]

kT hc D3 2d

9/4

-

2d hc

3/4

(13)

where D is the average distance between two attachment points, and hc/2 is the length of the buoy chain in the solvent phase. (It is assumed to be equal to δ.) This equation is valid only for 2d < hc. For 2d > hc, the steric force is zero. Therefore, the steric interaction free energy between two plates is given by

∫h2d fstericd(2d) ) -

Fsteric = -

c

[( ( ) )

4kThc 1 hc 5/4 1 + 2d D3 5 2d 7/4 1 1(14) 7 hc

( ( ) )]

The steric interaction energy increases as the average distance between the attachment points decreases. The electrolyte concentration affects the steric interaction through the distance D between the attachment points. To evaluate the distance between the attachment points at different electrolyte concentrations, we consider that the polymer begins to precipitate when the polymer concentration becomes larger than the solubility of the polymer in water. As the electrolyte concentration increases, the solubility of the polymer in the solution decreases. Therefore, the polymer chains dissolved in the solution are salted out onto the surface of the particles and/or form aggregates with increasing electrolyte concentration. As a result of the first effect, the steric interaction free energy increases. The second effect is responsible for depletion interactions. 2.3. Depletion Interaction. When the distance between two plates is smaller than the diameter l of the small particles (micelles), which, in the present case, are uncharged, the depletion force fdep acting between two plates is given by7,20,21

fdep ) -nmkT

(15)

(19) de Gennes P. G. AdV. Colloid Interface Sci. 1987, 27, 189. (20) Walz, J. Y.; Sharma, A. J. Colloid Interface Sci. 1994, 168, 485. (21) Huang, H.; Ruckenstein Langmuir 2004, 20, 5412

where nm is the number of aggregates (micelles) per unit volume. When the distance between the two plates is larger than the diameter of the small particles, the depletion force becomes zero because the small particles can be located in the gap. Consequently, the depletion free energy between two plates is given by

Fdep ) -

∫l 2d fdepd(2d) ) nmkT(2d - l)

(16)

To calculate the surface density of the polymer on the particles and the number of micelles at various electrolyte concentrations, the solubility of the polymer in the electrolyte solution is required. The Sechenov equation can be employed to calculate the solubility of a polymer in an electrolyte solution. This equation was initially proposed to calculate the solubility of a gas,22 but it is valid for a polymer solution as well.23 It has the form

log

C ) -kccsalt cw

(17)

where C is the solubility of the polymer in an electrolyte solution of concentration csalt, cw is the polymer solubility in pure water, and kc is the Sechenov constant, which depends on both the electrolyte and solute. Tadros and Vincent24 have observed that, by decreasing the polymer solvency, one can increase the adsorbed amount. When the polymer concentration is larger than its solubility, the excess polymer is assumed to precipitate on the surface of the particles and/or to form micelles. This assumption allows us to write for the polymer surface density η the expression

C0 - C + η0 η)R 4MNπa2

(18)

where N represents the colloidal particle concentration, 2 × 1012 particles/L in the experiments considered;1 C0 (g/L) is the initial concentration of the block copolymer in solution; η0 (mol/m2) is the surface density of polymer before the addition of the electrolyte; R is the fraction of precipitated polymer adsorbed onto the surface; and M is the molecular weight of the polymer. The average distance D between the attachment points decreases with increasing electrolyte concentration due to the precipitation caused by the salting out, and can be calculated from the surface density of the polymer:

D)

1

x2ηNA

(19)

where the factor 2 accounts for the two PEO blocks of the copolymer chain, and NA is Avogadro’s number. The ratio of the polymer deposited on the surface of the particles to that generating micelles was determined by fitting the experimental results. If the number of polymer chains in each micelle is m (which is taken as 50 in the present calculations1), the number density of the micelles is

nm ) (1 - R)NA

C0 - C mM

(20)

The size l of an aggregate can be easily calculated. For m ) 50, one obtains that l ≈ 100 Å. The above equation will be used to calculate the depletion free energy between particles. In some (22) Sechenov, M. Z. Phys. Chem. 1889, 4, 117. (23) Borukhov, I.; Leibler, L. Phys. ReV. E 2000, 62, R41. (24) Tadros, T. F.; Vincent, B. J. Phys. Chem. 1980, 84, 1575

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Figure 1. The stability ratio vs electrolyte concentration for various polymer lengths. The data are from ref 1. 0: L64 (M ) 2900); 4: L35 (M ) 1900). The initial polymer concentration is 60 × 10-3 g/L, a ) 110 nm, AH ) 1 × 10-21 J, hc ) 2 nm, σ ) -0.028 C/m2. (a) The electrolyte is LiCl. Parameters used in the calculation: for Pluronic L35, kc ) 5 × 10-7 M-1 and η0 ) 4.15 × 10-8 mol/m2; for Pluronic L64, kc ) 6 × 10-7 M-1 and η0 ) 5.1 × 10-8 mol/m2. (b) The electrolyte is NaCl. Parameters used in the calculation: for Pluronic L35, kc ) 1 × 10-6 M-1 and η0 ) 3.8 × 10-8 mol/m2; for Pluronic L64, kc ) 3 × 10-6 M-1 and η0 ) 5.1 × 10-8 mol/m2. (c) The electrolyte is KNO3. Parameters used in the calculation: for Pluronic L35, kc ) 1 × 10-6 M-1 and η0 ) 3.5 × 10-8 mol/m2; for Pluronic L64, kc ) 1 × 10-7 M-1 and η0 ) 5.1 × 10-8 mol/m2. The dotted lines represent the calculation results in a range for which no experimental data are available. (d) The electrolyte is KSCN. Parameters used in the calculation: for Pluronic L35, kc ) 5 × 10-7M-1 and η0 ) 4 × 10-8 mol/m2; for Pluronic L64, kc ) 1 × 10-7 M-1 and η0 ) 4.5 × 10-8 mol/m2.

cases, the stability ratio decreases with increasing electrolyte concentration after a minimum followed by a maximum. To explain this behavior, the depletion interaction was taken into account. In some cases, the stability ratio does not pass through a maximum after the minimum; the fraction of polymer that forms micelles is assumed in those cases to be zero. 2.4. van der Waals Interaction. The van der Waals interaction between two identical spherical particles of radius a has the form16

VvdW ) -

[

AH 2a2 2a2 + + 6 H0(4a + H0) (2a + H )2 0 ln

]

H0(4a + H0) (2a + H0)2

(21)

where AH is the Hamaker constant, and H0 is the closest distance between the surfaces of the two spheres. Adding all interaction free energies together, the stability ratio can be calculated for various electrolyte concentrations.

3. Fitting the Experimental Results In Stenkamp et al.’s experiments,1 the particles consisted of polystyrene latex containing sulfate functional groups. In the calculations, we assumed a constant surface charge density. Complex dispersions were generated by the addition of electrolyte solutions to polystyrene latex dispersions containing the amphiphilic block copolymer PEO-PPO-PEO. The stability of such a dispersion was quantified by determining the stability ratio through photon correlation spectroscopy of the incipient aggregation. The stability ratio first decreases and then increases with increasing electrolyte concentration. However, in some cases, at sufficiently high electrolyte concentrations, the stability ratio decreased again after passing through a maximum with increasing electrolyte concentration. The restabilization and destabilization concentrations depend on the nature of the electrolyte. In the present paper, we attempt to explain the restabilization and destabilization using the above theoretical framework. Our explanation implies that the excess polymer precipitates onto the surface of the particles when the polymer concentration becomes larger than its solubility, which decreases with increasing electrolyte concentration in the solution.

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Langmuir, Vol. 22, No. 10, 2006 4545

Figure 2. The stability ratio vs electrolyte concentration for various initial polymer concentrations. The data are from ref 1. Pluronic L64 is employed. a ) 110 nm, AH ) 1 × 10-21 J, hc ) 2 nm, σ ) -0.028 C/m2, l ) 10 nm. (a) The electrolyte is LiCl. Parameters used in the calculation: for C0 ) 60 × 10-3g/L, kc ) 6 × 10-7 M-1 and η0 ) 5.1 × 10-8 mol/m2; for C0 ) 0.6 × 10-3 g/L, kc ) 6 × 10-7 M-1 and η0 ) 4.8 × 10-8 mol/m2. (b) The electrolyte is KNO3. Parameters used in the calculation: for C0 ) 60 × 10-3 g/L, kc ) 1 × 10-7 M-1 and η0 ) 5.1 × 10-8 mol/m2; for C0 ) 0.6 × 10-3 g/L, kc ) 1 × 10-7 M-1 and η0 ) 4.2 × 10-8 mol/m2. The dotted lines represent the calculation results in a range for which no experimental data are available. (c) The electrolyte is KBr. Parameters used in the calculation: for C0 ) 0.01 × 10-3 g/L, kc ) 5 × 10-6 M-1 and η0 ) 1.2 × 10-8 mol/m2; for C0 ) 0.6 × 10-3g/L, kc ) 5 × 10-6 M-1 and η0 ) 4.9 × 10-8 mol/m2. (d) The electrolyte is KSCN. Parameters used in the calculation: for C0 ) 60 × 10-3g/L, kc ) 1 × 10-7 M-1 and η0 ) 4.5 × 10-8 mol/m2; for C0 ) 0.6 × 10-3g/L, kc ) 1 × 10-7 M-1 and η0 ) 4.2 × 10-8 mol/m2.

The effect of the molecular weight of the polymer is presented in Figure 1a-d. The increase in the molecular weight of the polymer increases the stability ratio. In some cases, restabilization occurs at sufficiently high electrolyte concentrations. The lowest molecular weight Pluronic L35 does not induce restabilization, while Pluronic L64 induces restabilization for the electrolytes LiCl, NaCl, and KNO3. For Pluronics L35 and L64 in KSCN, the stability ratio decreases to 1 at high electrolyte concentrations without passing through a minimum, or a minimum and a maximum. Another experimental aspect investigated by Stenkamp et al. was the effect of the concentration of the polymer initially dissolved in solution (Figure 2). In general, by decreasing the initial concentration of the polymer, the probability for restabilization to occur decreases. For LiCl and KNO3, a decrease from 60 × 10-3 to 0.6 × 10-3 g/L of the initial concentration of the polymer causes the loss of restabilization. KCl had a stronger propensity for restabilization and could restabilize the colloidal dispersion for a 0.6 × 10-3 g/L initial polymer concentration, but not for a 0.01 × 10-3 g/L concentration. The hydrodynamic diameter of the latex particles was also determined by Stenkamp et al. The results revealed that the average diameter of the restabilized latexes was close to that of

the polymer-coated latex particles in distilled water. However, this does not disprove that a significant additional adsorption is one of the possible causes of restabilization because the increase in the polymer coverage of the surface with increasing electrolyte concentration does not mean that the hydrodynamic diameter changed appreciably. Because the hydrophilic moiety of a polymer chain extends into the water phase, more adsorption of the hydrophobic moieties onto the particle surface due to the salting out will not lead to an appreciable increase in the thickness of the adsorption layer. On the contrary, the thickness of the adsorption layer has the tendency to decrease because of the poor solvency of the polymer at high electrolyte concentrations. The more than qualitative agreement between the experimental data and the calculations shows that the steric interaction can indeed contribute to restabilization at high electrolyte concentrations below the CMC. Of course, if the salting-out effect of the electrolyte is not large enough to precipitate enough polymer on the surface (small kc), the restabilization alone or restabilization followed by destabilization will not occur.

4. Conclusions The stability of the polystyrene latex in electrolyte solutions containing an amphiphilic block polymer first decreases because

4546 Langmuir, Vol. 22, No. 10, 2006

of the decrease in the electrostatic repulsion. However, at higher electrolyte concentrations, the polystyrene latex could be restabilized in the presence of amphiphilic block copolymers (Pluronics), which, because of salting out, precipitate onto the particles. This occurs because, with increasing surface coverage of the polymer, the steric interaction between two latex particles increases. When, because of salting out, aggregates are also

Huang and Ruckenstein

formed, an attractive depletion force is also generated, which decreases the stability ratio at very high electrolyte concentrations. The calculations suggest that the steric interaction and the depletion force can explain the restabilization followed sometimes by destabilization with increasing electrolyte concentration. LA0602057