Langmuir 2000, 16, 4045-4048
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Notes Effect of Polymer Adsorption on the Surface Tension and Flocculation of Colloidal Particles in Incompatible Solvents Nily Dan† Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19104 Received October 27, 1999. In Final Form: January 18, 2000
Introduction The transition from hydrocarbon-based suspensions to water-based ones is driven by environmental concerns. This transition is nontrivial in systems where the particles are hydrophobic and flocculate in aqueous media. One such example is the car paint industry, which currently utilizes hydrophobic pigments in nonaqueous solutions. To develop water-based paints, one must either identify a new class of hydrophilic pigments or develop a method for effectively solubilizing known hydrophobic ones.1,2 One method to reduce particle flocculation is by polymer adsorption. The effect of polymer surface layers on the interactions between colloidal particles has been studied extensively and is well understood (see, for example refs 3-8). The repulsive interactions between the adsorbed polymer layers are balanced, in these models, against such attractive particle-particle forces as van der Waals, thereby leading to stabilization of the suspension. Solubilization of particles in an incompatible liquid is not, however, solely due to the direct interactions between the polymer layers: In systems where the surface tension between the particle and the liquid is high, the adsorbed polymer reduces the effective surface tension. In this Note we investigate the effect of adsorbed, soluble polymer layers on the surface tension of colloidal particles that are immersed in an incompatible solvent. In the absence of polymer, the high surface tension between the particles and the liquid dominates over any other particleparticle interaction (except for hard core), as well as particle entropy. As a result, a dense flocculate will form. Polymer adsorption reduces the effective surface tension of the particles, thereby enabling solubilization of individual particles.9 † Phone: 215 735 7256. Fax: 215 735 1654. E-mail: dan@ che.udel.edu.
(1) Fujitani, T. Prog. Org. Coat. 1996, 27, 97. (2) Spinelli, H. J. Prog. Org. Coat. 1996, 27, 255. (3) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: London, 1989. (4) Halperin, A.; Tirrell, M.; Lodge, T. P. Adv. Polym. Sci. 1992, 100, 31. (5) Dan, N.; Tirrell, M. Macromolecules 1993, 26, 4310. (6) Biver, C.; Hariharan, R.; Mays, J.; Russel, W. B. Macromolecules 1997, 30, 1787 (7) Zhulina, E. B.; Borisov, O. V. Makromol. Chem. Macromol. Symp. 1991, 44, 275. (8) Eskilsson, K.; Ninham, B. W.; Tiberg, F.; Yaminsky, V. V. Langmuir 1999, 15, 3242. (9) Since our goal is to focus on the effect of adsorbed polymer on the particle-liquid surface tension, we do not account for the direct interactions between the adsorbed polymer layers.
The model used for the adsorbed polymer is a simple mean-field one.10,11 Despite its limitations, this type of model has been shown to produce qualitatively correct trends (see, for example, ref 4) while allowing us to examine all ratios of colloid radius to chain dimensions. Quite surprisingly, we find that there is no equilibrium between a population of solubilized particles and flocculated ones. Our model predicts that the system will be found in one of two states: either all particles are flocculated into a large aggregate or all particles are individually dispersed by an adsorbed polymer layer. In the latter case, the adsorbed chains are at equilibrium with some free chains in solution. The transition from one state (floc) to the other (solubilized particles) depends on the polymer adsorption energy, the particle surface tension, and the ratio of chain molecular weight to particle surface area. Model Let us examine a suspension of P spherical particles of radius R, immersed in a liquid. The surface tension between the liquid and the particles is defined by γ per unit area (all energies are given in units of kT, where k is the Boltzmann coefficient and T the temperature). We assume that the overall particle volume fraction in solution is relatively low, namely, that 4πR3P/3V is much smaller than unity, where V is the suspension volume. Also, we take the surface tension to be relatively high, so that 4πγ R2 is much larger than unity, dominating over both the colloids entropy (kT) or direct particle-particle interactions such as van der Waals. The particles can either be individually dispersed or flocculate into a large aggregate. Let us define the fraction of individually solubilized particles as x and assume that the flocculate is dense and more or less spherical so as to minimize the liquid-particle contact area. The free energy of the suspension, neglecting mixing entropy, is then
FP ) 4πR2γxP + 4πR2γ(1 - x)2/3P2/3
(1)
where the first term denotes the interfacial energy of the individual particles and the second that of the floc. As may be expected, when γ is positive (namely, the particleliquid interaction is unfavorable), the energy is minimized when all particles aggregate into a large flocculate and x ) 0. The mean-field mixing free energy of a dilute or semidilute polymer solution in an athermal solvent (where χ ) 1/2) is given by10,11
Fpol ≈ ν ln Φ +
1 νNΦ2 6
(2)
where Φ is the polymer volume fraction, ν is the number of chains in solution, and N is the chain molecular weight. (10) 10.Flory, P. J. Principles of polymer chemistry; Cornell University Press: Ithaca, NY, 1953. (11) 11.de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.
10.1021/la991417p CCC: $19.00 © 2000 American Chemical Society Published on Web 03/01/2000
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Notes
In semidilute solutions the entropic term, ln Φ, is small relative to the mixing term ΝΦ2 and may be neglected. The adsorption of polymer chains onto a surface is determined by a balance between the adsorption energy, the polymer-solvent mixing energy, and the loss of configurational entropy due to chain confinement on the surface. The adsorption energy is given by the number of adsorbed monomers times δ, the adsorption energy per monomer. The mixing entropy is given by eq 2, where the polymer volume fraction is now the volume fraction in the adsorbed layer. The loss of configurational entropy scales as the ratio between the chain dimensions in solution and on the surface, squared.10,11 To simplify the model, we make several assumptions: (i) The configurational entropy loss is small when compared to the adsorption energy. This requires that δ be large. (ii) The configurational entropy loss is small when compared to the mixing energy. This assumption must be tested for consistency once the polymer volume fraction in the adsorbed layer is calculated. (iii) The volume fraction in the adsorbed layer is uniform and does not depend on distance from the surface. Although it is well established that the polymer volume fraction in an adsorbed layer is not uniform,12,13 it has been shown that the error in the free energy of a solvated polymer layer due to this assumption is small and affects only numerical prefactors.4 The number of adsorbed monomers per particle, l, may be calculated by requiring that the volume fraction of monomers in the first layer near the surface (whose thickness is a, a monomer dimension) be equal to φ. The free energy of the particle-polymer complex is then written as
Fc ≈
1 nNφ2 - 4π(R/a)2δφ + 4πR2γ(1 - φ) 6
(3)
where n defines here the number of adsorbed chains per particle. The last term accounts for the reduction in the surface tension of the particle, since some of its surface is now occupied by the soluble polymer and is no longer in direct contact with the (incompatible) liquid. Minimization of the free energy of the particle-polymer complex yields φ* ) 12∆π(R/a)2/nN, where ∆ ≡ (δ + a2γ). As may be expected, the optimal value of the volume fraction in the adsorbed layer, φ*, increases with either the adhesion energy (δ) or the particle surface tension γ. The complex energy is then given by (neglecting, from here on, all constants of order unity)
Fc ≈ R2γ -
∆2π2(R/a)4 nN
(4a)
Converting this energy into a surface energy, we find that the reduced particle surface tension is now given by 2
γ* ≈ γ - (∆R/a) /nN
(4b)
As expected, polymer adsorption always reduces the effective particle-liquid surface tension. However, since the number of adsorbed chains, n, is not determined, we cannot relate, as yet, this reduction to system parameters. To calculate the number of adsorbed chains per particle, we must now consider the entire system, namely, a solution containing P particles and a volume fraction Φ0 of a given polymer. The number of chains in the system is then given by ν ) Φ0V/Na3. Three types of entities may be obtained: (12) de Gennes, P. G. Macromolecules 1981, 14, 1637. (13) Fleer G. J.; Leermakers F. A. M. Macromol. Symp. 1998, 126, 65-78.
(i) a particle flocculate, (ii) dispersed particle-polymer complexes (each containing n adsorbed chains), and (iii) free, unadsorbed chains. We assume that the free, unadsorbed polymer volume fraction (either before or after adsorption) falls into the semidilute regime. Defining the fraction of particles forming solubilized polymer-particle complexes (ii) as x and the fraction of adsorbed chains as y, the free energy (per particle) can then be written as
(
F ≈ Φ02FN(1 - y)3 + R2γ
(1 - x)2/3 P1/3
)
+x 2
2
x ∆ (R/a)4/ΝFy (5)
where F ) ν/P ) (Φ0V/a3NP) defines the ratio of polymer chains to particles. Note that n, the number of chains adsorbed per particle, is given then by yν/xp ) yΦ0V/xPNa.3 Results To obtain the system parameters, including the reduced particle surface tension and the fraction of dispersed particles and adsorbed chains, we need to minimize the system energy (eq 5) with respect to x and y. Examining eq 5 yields a somewhat surprising observation: The second derivative, ∂2F/∂x2, is always negative. This means that the energy is not minimized by any finite x value. As a result, the free energy of the entire system would be lowest in one of the end points of the allowed interval, i.e., either when x ) 0 (all particles flocculate) or when x ) 1 (all particles disperse). Which of these states is preferable will depend on system parameters.9 The free energy can be minimized, however, with respect to y, the fraction of adsorbed chains. This indicates that an equilibrium between adsorbed and free chains is possible
y* ) (1/2)(1 + x1 - ∆(R/a)2x/ Φ0FN)
(6)
This solution is valid only for y g 1/2. As a result, it does not apply when no particles are solubilized and x ) 0. Also, obviously this solution will break down for systems where ∆(R/a)2x/Φ0FN is larger than unity. This ties in to our assumption that the polymer solution is semidilute (namely, that NΦ02 and, naturally, NΦ0 are large) and that the number of particles is small (so that F ) ν/P is large). To determine whether the system favors a floc state or a dispersed state, we must compare the free energy of the two options. If F(x)0) > F(x)1), the particles will be solubilized. This corresponds to the condition that
Φ02FN + ∆2(R/a)2/NF > γa2
(7)
In this case, where x ) 1 (all particles are solubilized), the number of adsorbed chains per particle is given by
n* ) y*Φ0V/PNa3 ) (1/2)Φ0V/PNa3(1 + x1 ∆(R/a)2x/Φ0FN) (8a) and the reduced particle surface tension is
γ* ≈ γ - (∆R/a)2/n*N ) γ - (∆R/a)2Pa3/y*Φ0V
(8b)
These results are valid only if the loss of configurational entropy (which scales as Na2/L2, where L is the thickness of the adsorbed layer10,11) is negligible when compared to either the adsorption energy or the mixing energy. We check this, for example, for the limit where the particle
Notes
Langmuir, Vol. 16, No. 8, 2000 4047
radius is much larger than chain dimensions and derive the requirement that (R/a)2/N2/3 , F1/3. Since the radius of gyration of the free chains is aN1/2, this translates to a limit where F, the ratio of polymer chains to particles, is high and the polymer chains are in large excess. A similar constraint is obtained for the opposite limit when the particle radius is much smaller than the chain dimensions.
Discussion and Conclusions
Figure 1. A schematic of polymer adsorption on a colloidal particle. The particle radius is R. The polymer molecular weight is N, and a denotes a segment size. Many polymers can adsorb in a dense layer on a large particle; fewer chains will adsorb, in a less dense layer, onto a small particle.
Using a simple mean field model, we calculate the effect of an adsorbing polymer on the effective surface tension, and, thereby, stability and dispersion of a colloidal suspension in an incompatible liquid. Our calculation indicates that there cannot be an equilibrium between a population of dispersed and flocculated particles.9 Thus, we predict that (providing our assumptions apply) such systems will be found at one of two states: a large colloidal floc or a solution where all particles are solubilized by a fraction of the available polymer chains. It should be noted that, in the floc states, no polymer adsorbs on the particles. We find that, as expected, the effective surface tension between the (solubilized) particles and the liquid is reduced by polymer adsorption (eq 8b). The extent of reduction increases with the polymer adsorption energy. However, somewhat surprisingly, we see that adsorption of more chains (namely, increasing y* or Φ0) increases the surface tension, while increasing the number of particles decreases it. Increasing either the initial polymer volume fraction Φ0, the polymer-to-particle ratio, F, or the chain molecular weight will lead to particle dispersion. Solubilization also becomes more favorable as either the particle surface tension (γ) or the polymer adsorption energy (δ) increase. These trends are as may be expected; polymer adsorption, which leads to dispersion, will increase with the chain availability in solution of the strength of the adsorption energy. Increasing particle surface tension increases the penalty associated with bare particle solubilization. Quite surprisingly, however, we find that to solubilize particles one should choose a low molecular weight chain, rather than a high molecular weight one. Conversely, a given polymer suspension will solubilize larger particles more easily than smaller ones. This is due to the structure of the adsorbed polymer layer. Solubilization is more easily obtained when the density of the adsorbed layer is high, since that leaves only a small fraction of the particle surface to interact with the unfavorable liquid. However, high molecular weight chains adsorbing on small particles tend to form fluffy dilute layers, while short chains adsorbing on a relatively flat surface may form dense coverage. As may be expected, the fraction of adsorbed chains, y*, increases with increasing polymer availability, i.e., with either Φ0 or F. It should be noted, however, that the dependence of y* on these parameters is not linear. More interesting is the effect of the net adsorption energy on both the fraction of adsorbed chains and the density of the adsorbed layer. We see from Figure 2 that,
Figure 2. Dependence of the adsorbed layer volume fraction, φ* ) 12∆π(R/a)2/nN, and the fraction of adsorbed chains y* (eq 6) on the net adsorption energy and the ratio of particle area to chain length.
as may be expected, the density of the adsorbed layer increases with the strength of the adsorption energy; when ∆ increases, the chains adsorb in a flat configuration to maximize the number of monomer-particle contacts. However, a chain that adsorbs in a flat configuration reduces the number of sites available to other chains. As a result, while increasing ∆ increases the adsorbed layer density φ, it leads to a reduction in the fraction of adsorbing chains y*. The dependence of y* and φ on the ratio of particle area to chain length (R2/N) is similar. Long chains adsorbing on a small particle will adsorb in a loose configuration where φ is low. Increasing the particle area, or decreasing the chain length, will lead to adsorption in a more dense configurations and, hence, to an overall reduction in the fraction of adsorbed chains. Perhaps the most questionable assumption made here is neglecting the interactions between the adsorbed polymer layers. These, repulsive, interactions enhance solubility once some polymer adsorbs onto the particle, thereby reducing the concentration of polymer required for solubilization. However, while it could affect the kinetics and/or position of the transition, accounting for these interactions should not change our main conclusion, which is as follows: either all particles carry some adsorbed polymer and are individually solubilized or they have no adsorbed polymer and are flocculated. Similarly, if the floc geometry is nonspherical, we expect the transition point to shift since the floc energy (eq 1) will be higher. However, this shift should not affect our conclusion either.
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In conclusion, we present here a simple mean field model for the solubilization of particles via polymer adsorption. The model, despite its simplicity, both accounts for the particle-liquid surface tension and applies to all polymer to particle size ratios. It should be noted that our results regarding the properties of the adsorbed polymer layer are in agreement with the extensive literature discussing polymer adsorption on surfaces. However, what is unique about this work is the coupling between the polymer adsorption and the effective particle surface tension and, thus, solubilization. We predict that there will be no equilibrium between dispersed and flocculated particles, either all particles
Notes
will flocculate into a large aggregate or all will be solubilized. To obtain particle dispersion, one may increase the amount of either available polymer or the polymer adhesion energy. Most importantly, we find that short (and thus, cheap to synthesize) polymer chains are more effective as dispersants than long chains. While more sophisticated models may provide more quantitative predictions, we expect these qualitative trends to hold. Acknowledgment. Thanks for the support of NSF Grant CTS 9814398. LA991417P
