The Bridging Force between Colloidal Particles in a Polyelectrolyte

Nov 7, 2012 - In this paper, using a self-consistent field approach, a simple theory is developed which allows for the calculation of the bridging for...
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The Bridging Force between Colloidal Particles in a Polyelectrolyte Solution Haohao Huang*,†,§ and Eli Ruckenstein*,‡ †

School of Materials Science and Engineering, South China University of Technology, Guangzhou 510640, China Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260, United States

‡

ABSTRACT: The presence of a polyelectrolyte in a colloidal dispersion affects the interaction between colloidal particles through electrostatic and bridging interactions. In this paper, using a self-consistent field approach, a simple theory is developed which allows for the calculation of the bridging force between two plates and two colloidal particles. The present approach differs from the previous ones, since the contribution of the plate-solution interfacial tension to the free energy is taken into account in the calculation. The interfacial tension between solvent and plate depends on the nature of the particles and the concentration of the segments of the polymer at the surface. The surface−segment interaction has a significant effect on the segment concentration profile. When the segment−surface interaction is repulsive, the bridging force is weak because few polyelectrolyte chains are adsorbed onto the surface. When the segment−surface interaction is attractive, various segment concentration profiles could be identified. Depending upon the concentration of polyelectrolyte, the electrostatic plus bridging forces can be attractive or repulsive. The bridging force between two plates which is attractive has a longer range than the van der Waals interaction.

1. INTRODUCTION Polymers are often used to stabilize colloidal dispersions through steric repulsion among the particles.1 The steric interaction generated by grafted polymer molecules can be attributed to the elasticity of polymer chains and the entropic repulsion generated by the overlap of the polymer chains adsorbed on opposite particles.2 By the use of self-consistent field theories, the density functional theory, and scaling approaches, the conformation of a polymer grafted on a solid surface and the steric interaction could be predicted.3−8 Also, numerous experiments were performed to investigate the steric interaction.9−13 An individual polymer chain adsorbed on one particle surface can have the following three types of conformations: loops, trains, and tails. The fraction of each conformation was estimated by using a random walk treatment.14 However, when a polymer chain is simultaneously adsorbed onto two or more particles, the chains provide bridging interactions which link the particles together. As a result, an attractive force is generated which draws the particles toward each other. Therefore, in contrast to the steric force, the bridging one promotes, in general, the aggregation of colloidal particles and decreases the stability of the system. The bridging interaction was extensively examined experimentally,15−17 theoretically,18−21 and by simulations.22,23 Polyelectrolyte chains induce bridging interactions in a more complex manner than the neutral ones because they affect the electrical double layer,24,25 which, in turn, influences the © 2012 American Chemical Society

conformation of the chains and hence the bridging and the overall electrostatic interaction. The strength of the bridging force between two particles depends on the number of segments adsorbed on them. The adsorption of the segments of the chains onto the surface is dependent on the interaction between the segment and the surface. If the attraction between the segment and the surface is stronger than the affinity between the segment and the solvent molecules, the segment will be adsorbed onto the surface. In the opposite case, the segments will prefer the solvent to the interface. The interactions that promote adsorption are either nonelectrostatic (e.g., hydrophobic and van der Waals forces) or electrostatic. The net segment−surface interaction energies for polystyrene (PS) and polymethylmethacrylate (PMMA) adsorbed on oxidized silicon from carbon tetrachloride at 25 °C are ≈1.3kT (PS) and ≈4kT (PMMA) per segment, respectively.26 The adsorption profiles of polyelectrolyte on oppositely charged interfaces were examined on the basis of the Edwards equation for planar, cylindrical, and spherical geometries by Cherstvy and Winkler.27,28 In dilute solutions, the polymer chains can be considered isolated. However, the chains begin to interpenetrate in the semidilute concentration range. Because the solid surface interacts with the polymer segments, the segment concenReceived: October 1, 2012 Revised: November 7, 2012 Published: November 7, 2012 16300

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and 3 for dilute and semidilute solutions, respectively.31 In the present calculations, the solutions are considered dilute when the bulk volume fraction of segments is below 0.001. This volume fraction was estimated by de Gennes based on his scaling method to correspond to a chain length of 104 segments.30 The polymer contribution accounts for the polymer chain connectivity and the short-range excluded-volume interaction.29 It is provided for sufficiently long chains by the expression32,33

tration near the surface can be in dilute, semidilute, or concentrated concentration range even for dilute solutions. In the semidilute concentration range, the neighboring adsorbed segments also have interactions with each other. The change of the interfacial tension solution-plate induced by segment adsorption can have a nonlinear dependence on the segment concentration near the surface. In the present paper, the effect of the segment surface interaction on the bridging force between two parallel plates and two colloidal particles immersed in dilute and semidilute polymer solutions is examined. To account for this effect, the solid−solution surface tension is included in the calculation of the free energy. With the minimization of the free energy with respect to the segment concentration at the surface, a boundary condition is derived that allows for the examination of the effect of the segment−surface interaction on the interaction between two plates and two spherical particles.

⎡ a 2 ⎛ dφ ⎞ 2 1 ⎢ ⎜ ⎟ + v(φ4 − φb 4) ⎝ ⎠ 2 ⎣ 6 dx

+D /2

βFchain =

∫−D/2

⎤ − (φ 2 − φb 2)μp ⎥ dx ⎦

where a is the Kuhn segment length, β = 1/kT, k the Boltzmann constant, T the absolute temperature, φ the polymer order parameter [which is related to the local concentration cp(x) of segments through cp(x) = φ(x)2], v the excluded volume parameter, kTμp the chemical potential of the chains, and subscript b indicates the bulk. For good solvents, v is positive, whereas for poor solvents, it is negative. Under θ condition, v is zero. Because only good solvents are considered in the following calculations, a positive v is taken, independent of electrolyte concentration. The double-layer contribution includes the electrostatic free 2 energy of the solution (1/2)εε0∫ D/2 ‑D/2 [(dψ)/(dx)] dx, the D/2 chemical contribution of the surface 2σψs ≡ εε0∫ ‑D/2 {ψ[(d2ψ)/ (dx2)] + [(dψ)/(dx)]2} dx, and the ionic contribution (1/ 34 β)∫ D/2 ‑D/2∑i [ci ln(ci)/(ci,b) − ci + ci,b] dx.

2. THEORETICAL FRAMEWORK The system considered is depicted in Figure 1. It consists of two parallel plates immersed in an electrolyte/polyelectrolyte

+D /2

+

solution in contact with a bulk reservoir of electrolyte/ polyelectrolyte solution. The two plates possess constant electrical potential, ψs, and are located in the electrolyte/ polyelectrolyte solution at distances x = ± D/2. Within the mean field approximation, the free energy of the system with respect to that in the reservoir can be expressed as the sum of three contributions: the polymer contribution Fchain, the double layer contribution Fdl, and the surface tension contribution Fs

1 β

2 ⎡ d2ψ 1 ⎛ dψ ⎞ ⎤ ⎢ψ 2 + ⎜ ⎟ ⎥ d x 2 ⎝ dx ⎠ ⎦ ⎣ dx

⎛

+D /2

c ∫−D/2 ∑ ⎜⎜⎝ci ln c i

i,b

i

⎞ − ci + ci , b⎟⎟ dx ⎠

(4)

where ci is the concentration of ion i. In the present calculations, two kinds of small ions are present in solution: the ions of the salt and the counterions dissociated from the polyelectrolyte molecules which are assumed to be the same as one of the ions of the salt. The integral form −εε 0 2 2 2 ∫ D/2 ‑D/2 {ψ[(d ψ)/(dx )]+[(dψ)/(dx)] } dx was obtained using the Green theorem. Substituting the Poisson equation

d2ψ ρ =− 2 εε0 dx

(1)

The first two contributions were employed in the previous treatments,29 and the last term is new and is included here to account for the effect of the segment−surface interaction on the segment profile and bridging attraction. In addition there is an attractive van der Waals interaction which will be included later in the paper. The surface tension contribution is considered to have the form30,31

Fs = Fs0 + 2γV m /2φsm

∫−D/2

Fdl = −εε0

Figure 1. Two plates located at x = ± D/2 are immersed in a polyelectrolyte solution.

F = Fchain + Fdl + Fs

(3)

(5)

eq 4 becomes +D /2

Fdl =

∫−D/2 +

(2)

1 β

2 ⎡ 1 ⎛ dψ ⎞ ⎤ ⎢eψ (c + − c − + αφ 2) − ⎜ ⎟ ⎥ dx 2 ⎝ dx ⎠ ⎦ ⎣

⎛

+D /2

c ∫−D/2 ∑ ⎜⎜⎝ci ln c i i

i,b

⎞ − ci + ci , b⎟⎟ dx ⎠

(6)

where e is the electronic charge, α the average charge number of each segment, c+ the concentration of the cation, and c− the concentration of anion. Minimizing the free energy functional, Fchain + Fdl + Fs, with respect to ψ, φ, and φs, one obtains the following system of equations

where Fs0 is the interfacial tension between the solid and pure solvent, φs2 the segment concentration at the surface, V the volume of each segment, and γ the interaction between the surface and segments per unit area of one plate. The experiment has shown that the exponent m has the values 2 16301

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where ψ(0) is the electrical potential at x = 0, the middle of the two plates, and φ2(0) is the segment concentration at the middle, c is the bulk electrolyte concentration, and fdl is the double-layer interaction between the two plates. The first term in the second eq 14 is due to the segment−segment volume exclusion interaction, the second to the segment−surface interaction, and the third to the electrical double layer. For a good solvent (v > 0), the first term is negative. The second term is negative when the segment−surface interaction, γ, is positive. The double-layer contribution, fdl, is composed of three terms: the contribution of electrolyte ions, polyelectrolyte segments, and their counterions. The contribution of electrolyte ions to fdl is always positive, but the contributions of the segments and their counterions depend on the sign of the segment charge, segment concentration, and electrical potential. Because of the interdependence between the segment concentration and the electrical potential, the force does not depend only on the number of bridges.

d2ψ 1 [2ce sinh(βeψ ) − αeφ 2 + αeφb2 exp(βeψ )] = εε0 dx 2 (7)

a 2 d2φ = αβeψφ + v(φ3 − φφb 2) 6 dx 2

(8)

⎛ dφ ⎛ D ⎞⎟ D⎞ ⎜x = − = 3mβγa3m /2 − 2φm − 1⎜x = − ⎟ ⎝ ⎠ ⎝ dx 2 2⎠

(9)

Equation 7 constitutes a modified Poisson−Boltzmann equation in which the charges of the segments of the polymer and of their counterions and those of electrolyte are included. The last term in eq 7 accounts for the charges of the counterions. Equation 8 represents a self-consistent field equation for the local segment density of the polymer chains subjected to the electrical potential and to the volume-exclusion interactions. Equation 9 provides a boundary condition for eqs 7 and 8. For m = 2, the dilute concentration case, eq 9 reduces to that suggested by de Gennes.35,36 In previous theoretical investigations,18,29,37 either the zero segment concentration or the zero segment gradient concentration was employed as boundary conditions. They represent two limiting cases of eq 9: the infinite segment−surface repulsive interaction in the former case and the vanishing segment−surface interaction in the second. In contrast to the previous approaches, the boundary condition is derived in the present paper from the condition of minimum free energy. Additional boundary conditions are provided by the equation ⎛ D⎞ ψ ⎜x = ± ⎟ = ψs ⎝ 2⎠

3. NUMERICAL CALCULATIONS AND DISCUSSIONS 3.1. Effect of Segment−Surface Interaction on Segment Distribution Profile Near a Single Plate. To illustrate the effect of segment−surface interaction on adsorption, the segment distribution profiles near a single plate are first examined for various values of γ. If the plate and polymer are not charged, the segment distribution profile depends only on the interaction γ between the surface and the segment. The segment concentration near the surface is larger than that at a large distance from the plate when the segment−surface interaction is attractive (i.e., γ < 0); it is lower when that interaction is repulsive (i.e., γ > 0). The segment distribution profiles are plotted in Figure 2 for the semidilute case for various values of γ and in Figure 3 for

(10)

and by the symmetry conditions dφ (x = 0) = 0 dx

(11)

dψ (x = 0) = 0 dx

(12)

Equations 9−12 provide the boundary conditions for the self-consistent eqs 7 and 8. The numerical solution of eqs 7 and 8 provides the segment density and the electrical potential profiles. The force per unit area between plates can be calculated as the derivative of the free energy with respect to the distance between plates f=−

dF(D) dD

Figure 2. Segment distribution profiles near a single plate in semidilute solutions. The bulk segment concentration is 50 × 10−6 Å−3, v = 50 Å3, a = 5 Å, c = 0.01 M; 1024 × γ = −2 (blue), −0.6 (red), 0 (black), 1(green), and 2 (brown) J/Å2.

(13)

The van der Waals force between the two plates can be separately calculated, whereas the electrostatic and bridging forces are interdependent and provided together by eq 13

the dilute case for various values of γ. For γ > 0, as γ increases, the segment concentration at the surface decreases; for γ < 0, it increases as γ increases in absolute value. The segment−surface interaction has a stronger effect on the segment profile in the dilute than in the semidilute solutions. 3.2. Bridging Forces between Two Charged Plates and Two Particles. When both the surface−segment interaction, γ, and the segment concentration are small, eq 9 reduces to one of the commonly used boundary conditions

2γβa3m /2 m v [φ (0)] βf = − [φ 2(0) − φb 2]2 − 2 D ⎧ ⎡ eψ (0) ⎤ ⎫ − αβφ 2(0)eψ (0) + αφb2 ⎨exp⎢ ⎥ − 1⎬ ⎩ ⎣ kT ⎦ ⎭ + 2c{cosh[eψ (0)/kT ] − 1} 2γβa3m /2 m v [φ (0)] + fdl = − [φ 2(0) − φb2]2 − 2 D

(14) 16302

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Consequently, the repulsion becomes stronger when the surface charge density or potential increases. Because the bridging force is generated through the simultaneous adsorption of individual polymer chains on two or more particles, the number of polymer chains adsorbed changes the strength of the bridging force. With an increase in the bulk concentration of polyelectrolyte, the number of segments adsorbed increases. Moreover, the charges of the polyelectrolyte and of its couterions affect the electrical potential and, hence, the electrostatic interactions. If the segment−surface interaction is attractive, it promotes the accumulation of segments on the plates, and the accumulation increases the number of bridges which generate an attractive bridging force. The effect of the charge density of segments on the force is plotted in Figure 5 for the surface potential ψs=-kT/e. The

Figure 3. Segment distribution profiles near a single plate for dilute solutions. The bulk segment concentration is 1 × 10−6 Å−3. v = 50 Å3, a = 5 Å, c = 0.01 M; 1024 × γ= −0.6 (blue), −0.1 (red), 0 (black), 0.2 (green), and 1 (brown) J/Å2.

dφ (x = D/2) = 0 dx

(15)

In this case, the surface tension contribution to the free energy is negligible. The bridging force is then dependent upon the surface potential, the bulk concentration of segments, and their charge. The segment−surface interaction generates a concentration gradient near the surface. This interaction has a long-range effect on the segment distribution profile and the force between the plates (See Figure 4).

Figure 5. Effect of segment charge on the force between two particles. ψs=-kT/e, v = 50 Å3, a = 5 Å, φb2 = 1 × 10−6 Å−3, γ = −1 × 10−24 J/Å2, c = 0.01 M, α = 0.1 (black), 0.4 (red), 0.5 (green), 0.6 (purple), and 0.8 (blue).

charge of the segment is assumed to be generated by the complete dissociation of the charged segments of the polyelectrolyte. For a low positive segment charge, the electrostatic interaction with the surface is weak and few segments are adsorbed on the surface. In this case, the force is repulsive at short distances because of strong repulsive electrostatic interaction with little screening from the polyelectrolyte charges. With an increase in the charge of the segments when the surfaces have opposite charge, the force becomes increasingly more attractive. In Figure 6, the van der Waals force is compared to the force of each of the contributions involved in eq 14. Assuming large thicknesses of the plates, the van der Waals interaction has the form A Π=− (16) 6πD3 −21 The common value of the Hamaker constant, A = 5 × 10 J, is used. For the system considered, the van der Waals interaction is stronger at short distances than the sum of the bridging and electrostatic ones and is proportional to D−3. However, at large distances, the van der Waals interaction drastically decreases to a negligible value, whereas the bridging force has a longer range. The following expression for the van der Waals force between two particles of radius R was employed

Figure 4. Effect of surface−segment interaction on the force between two plates. ψs=-kT/e, α = 0.5, v = 50 Å3, a = 5 Å, φb2 = 1 × 10−6 Å−3, and c = 0.01 M; 1024 × γ = −20 (black), −5 (blue), 5 (red), 20 (green) J/Å2.

When the segment−surface interaction is attractive (hence negative), the surface will attract segments onto the plates. The overall force thus becomes more attractive with the increasing absolute value of the segment−surface interaction because a larger number of bridges form between the plates. If the segment−surface interaction is repulsive (positive), the segments are repulsed by the surfaces. As a result, the force becomes less attractive because few bridges are generated between plates. The force depends on both segment concentration and electrical potential profiles between the two plates. The surface charge density or potential can promote adsorption when it has the opposite sign to that of the segment charge. However, the repulsive double-layer force is also included in the total force. 16303

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in absolute magnitude increases from 1kT/e to 2kT/e. However, when the potential changes from 2kT/e to 3kT/e, the overall force (van der Waal force included) passes through zero, after which it becomes repulsive. The forces between two spherical particles for various segment−surface interactions are presented in Figure 8, which shows that the segment−surface interaction has a significant effect on the force between two spherical particles.

Figure 6. Comparison between van der Waals force and other contributions to the interactions: van de Waals force (blue), the first term in eq 14 (red), the second term (cyan), the third term (magenta), the fourth and fifth terms (black), and the electrostatic plus bridging force f (green). ψs = −kT/e, α = 0.3, v = 50 Å3, a = 5 Å, γ = −1 × 10−24 J/Å2, c = 0.001 M, φb2 = 1 × 10−6 Å−3, and A = 5 × 10−21 J.

Π′ = −

AR 6πd 2

(17)

where d is the shortest distance between the two spheres. The force between two spherical particles can be estimated using the Derjaguin approximation, x =∞

fs =

∫x=d

⎡ (x − d)2 ⎤ πf (x)d⎢R(x − d) − ⎥ 4 ⎣ ⎦

Figure 8. Effect of surface−segment interaction on the overall force between two particles. ψs = −1kT/e, α = 0.5, v = 50 Å3, a = 5 Å, φb2 = 1 × 10−6 Å−3, c = 0.01 M, R = 50 nm, A = 5 × 10−21 J, 1024 × γ = −20 (blue), −5 (red), and 20 (green) J/Å2.

(18)

The effect of surface potential on the overall force between two spherical particles is presented in Figure 7. It should be

5. CONCLUSIONS The segment−surface interaction free energy is included in the free energy of the system together with the polymer-excluded volume, connectivity, and electrostatic interaction contributions. The effect of the segment−surface interaction on the segment distribution is taken into account through a boundary condition for the self-consistent field equations for the segment concentration and electrostatic potential profiles. This boundary condition was derived via the minimum free energy with respect to the concentration of segments at the surface. When the interaction between the segments and plates is attractive, the force between the plates at a constant surface potential is attractive because a large number of segments are adsorbed on the surface. However, if the interaction is repulsive, the force between plates is less attractive and sometimes even repulsive because of fewer bridges between plates. The bridging interaction between two plates immersed in a polyelectrolyte solution has a longer range than the van der Waals interaction.

■

Figure 7. Effect of surface potential on the overall force between two spherical particles. α = 1, v = 50 Å3, a = 5 Å, φb2 = 1 × 10−6 Å−3, γ = −5 × 10−24 J/Å2, c = 0.001 M, R = 50 nm, A = 5 × 10−21 J, ψs = −kT/e (blue), −2kT/e (black), and −3kT/e (red).

AUTHOR INFORMATION

Corresponding Author

*H.H.: e-mail, [email protected], hhuang5@buffalo.edu. E.R.: e-mail, feaeliru@buffalo.edu.

noted that the overall force is the sum of several contributions, including the electrostatic and bridging forces. With an increase in the absolute value of the surface potential, more segments are attracted to the surface when they have opposite charges. As a result, the strength of the attractive bridging force increases. However, the increase in surface potential also leads to a stronger repulsive electrostatic interaction. Therefore, the force depends on the competition between these two forces. In the system considered in Figure 7, the overall force (van der Waal force included) becomes attractive when the surface potential

Present Address §

Department of Chemical and Biological Engineering, University at Buffalo, Buffalo, NY. Notes

The authors declare no competing financial interest.

■

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