© Copyright 2003 American Chemical Society
FEBRUARY 4, 2003 VOLUME 19, NUMBER 3
Letters Amphiphile-Induced Stabilization of Hydrophobic Colloidal Particles in Aqueous Solutions S. Tikku and N. Dan* Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19104 Received February 27, 2002. In Final Form: July 1, 2002 We present a simple model for the solublization of hydrophobic colloidal particles, such as lipophilic drugs, in aqueous solutions through the use of bilayer-forming amphiphiles (surfactants or lipids). We predict that when the amphiphiles are in excess, there is a sharp transition from flocculated particles to dispersed ones. The system state (flocculated or dispersed particles) depends on the compressibility of the amphihphiles and their interaction energy with the particles. When the amphiphiles are not in excess, the volume fraction of solubilized particles is predicted to increase linearly with amphiphile concentration, with a slope that depends on amphiphile characteristics. These predictions are found to be in agreement with experimental observations.
Introduction Interest in the stabilization of hydrophobic colloidal particle suspensions is driven by two, seemingly disparate, needs. One is due to the transition in industrial processes from environmentally harmful solvents to aqueous solutions, necessitating thereby a method to disperse hydrophobic particles (e.g., pigments) in water. The other need arises from the frequently inefficient performance of new drugs in vivo, which is linked in many cases to ineffective delivery of hydrophobic drugs into the affected tissue. Particle flocculation is generally driven by two forces: attractive van der Waals interactions, tending to destabilize all colloidal suspensions,1 and the surface tension between the particle and the surrounding media, which dominates in incompatible systems. Thus, the stabilization of hydrophobic particles in aqueous solutions requires a reduction in the particle/solvent surface tension, which may be obtained through the use of surface-active agents. Amphiphilic molecules such as surfactants and lipids are attractive surface modifiers due to their availability and biocompatiblity. Indeed, recent studies examined the effect of surfactants on the stability of hydrophobic particles suspensions2,3 and on the dissolution of lipophilic drugs.4-7 (1) Russel, W.B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989.
Theoretical discussions of suspension stabilization tend to focus on van der Waals forces between particles immersed in a compatible solvent.1 Few papers to date discuss the parameters controlling the amphiphileinduced stabilization of particles in an incompatible medium. In a previous study8 we examined the effect of polymer adsorption on the surface tension of colloidal particle suspensions in an incompatible liquid, finding that there practically is no equilibrium between colloidal flocculates and dispersed particles; either all particles flocculate or all are solubilized. This sharp transition is due to the fact that colloidal entropy, which plays a significant role in colloidal suspensions in good solvents, is negligible when compared to the surface tension between the particles and a poor solvent. In this study we examine stabilization of hydrophobic particles by small molecule amphiphiles, focusing on (2) Yonezawa, T.; Gotoh, Y.; Toshima, N. React. Polm. 1994, 23, 4351. (3) Ivanova, A. S. Kinet. Catal. 2001, 42, 354-365. (4) Benita, S.; Levy, M. Y. J. Pharm. Sci. 1993, 82, 1069-1079. (5) Davis, S. S. Interdiscip. Sci. Rev. 2000, 25, 175-183. (6) Mehnert, W.; Mader, K. Adv. Drug Delivery Rev. 2001, 47, 165196. (7) Perkins, W. R.; Ahmed, I., Li, X..; Hirsh, D. J.; Masters, G. R.; Fecko, C. J.; Lee, JK.; Ali, S.; Nguyen, J.; Schupsky, J.; Herbert, C.; Janoff, A. S.; Mayhew, E. Int. J. Pharm. 2000, 200, 27-39. (8) Dan, N. Langmuir 2000, 16 (8), 4045-4048.
10.1021/la025672c CCC: $25.00 © 2003 American Chemical Society Published on Web 01/07/2003
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Figure 1. A sketch of our system. The system may contain any of the three types of phases: particle flocculate, particles solubilized by an amphiphile shell, and amphiphile bilayers. The particle radius is given by R, and the bilayer thickness is 2L*. In the case of small aprticles where R < L*, the particle is incorporated into the lamellar phase, while in the case of large particles (R > L*) the particles are individually solubilized. L denotes the thickness of the amphiphile layer surrounding the particle.
lamellae-forming surfactants or lipids since those are the most relevant to biomedical applications. Small stabilized particles may be incorporated into the lamellar phase while larger particles are individually stabilized by an amphiphile monolayer (see Figure 1). As in the case of polymer-stabilized suspensions,8 we find that there is a sharp transition from flocculates to stabilized particles. The critical particle volume fraction, above which the particles flocculate, decreases significantly with the thickness of the amphiphile lamellae and increases with particle radius. We also find that the strength of the interaction energy required to stabilize a suspension increases and then decreases with particle size. As a result, in the limit of large particles, a suspension may be stable even if the specific interaction between the amphiphile and the particle is unfavorable. Model The system examined consists of a dilute suspension of spherical (hydrophobic drug) colloidal particles of radius R, immersed in an incompatible liquid (water). The particle volume fraction is given by φ. We assume that the surface tension, γ, between the particles and the liquid is relatively high, dominating over both the long-range van der Waals interactions9 and colloidal entropy.10 The particles may be dispersed or flocculated into aggregates. For simplicity, we assume that if flocculation occurs, only one large aggregate forms whose geometry is spherical, thereby minimizing liquid-particle contact area (see Figure 1). The free energy of the suspension, per unit volume, is then1,9
F ) γyφ/R + γ(1 - y)2/3φ 2/3V-1/3
(1)
where y is the fraction of individually solubilized particles and V is the suspension volume. As a result, yφV/R3 is the number of solubilized particles per unit volume and the radius of the particle aggregate is given by (1 - y)1/3φ1/3V1/3. All energies are given in units of kT, where k is the (9) Israelachvili, J.Intermolecular and Surface Forces; Academic Press: New York, 1992. (10) The surface tension of an oil/water interface is of order 0.05 J/m2 (see, for example, ref 9). Thus, the surface energy of a 10 nm hydrophobic particle in water is of order 10-17 J. Entropic energy at room temperature is (10-21 ln φ) J, where φ is the particle volume fraction. The volume fraction at which entropic energy of a nanoscale particle is similar to that of the surface tension is therefore of order 10-22.
Boltzmann coefficient and T the temperature. For clarity we neglect all geometrical coefficients of order unity (e.g., π). As may be expected, if γ is positive (namely, the particle liquid interaction are unfavorable) the free energy of the particle suspension is minimized when all particles aggregate into a large flocculate10 and y ≈ 0. The addition of amphiphilic molecules (lipids or surfactants) may lead to coexistence between an amphiphilic microphase, particle-amphiphile complexes (namely, stabilized particles), and particle flocculates, as sketched in Figure 1. For simplicity, we assume that the critical aggregation concentration for the amphiphiles is very low so that we can neglect the presence of unassociated molecules and focus on bilayer-forming species, since those are the most relevant to biomedical uses. The energy of the amphiphilic molecules is set by a balance between head-head repulsion and tail packing.9 We assume that the area per molecule, Σ, is set by the head-head interactions, so that it remains constant regardless of the amphiphile phase geometry.11 The tail energy is given by a compressibility (B) times the perturbation from the favored phase state, i.e., the lamellar in our case. Thus, the free energy of an amphiphileparticle complex, per amphiphilic molecule, is given by
Fc ) B(L - L*)2 - δ
(2)
where L is the thickness of the amphiphile layer surrounding the particle and δ is the interaction energy, per molecule, between the amphiphile and the particle (see Figure 1). Using the notation here indicates that a positive δ value is associated with an attractive interaction between the particle and the amphiphilic tails. To calculate the energy of amphihpile deformation when adsorbed onto a particle, we must be able to relate L, the thickness of the layer surrounding the particle, to the particle radius R. To that end we may use geometrical considerations, assuming that the volume of the amphiphile molecule ΣL* is fixed. The number of amphiphiles in the shell surrounding the particle is given by (R + L)2/Σ (see Figure 1), so that their volume is equal to ΣL*(R + L)2/Σ. This volume must be equal to the volume of the amphiphile shell in the complex, so that
L*(R + L)2 ) (R + L)3 - R3
(3)
Thus, eq 3 relates the amphiphile layer thickness in the layer surrounding the particle, L, to the particle radius R and the lamellar phase thickness L*. Note that we make no assumptions regarding the relative dimensions of R and L*, so that the model applies to both small particles and large ones (see Figure 1). We compare the free energy between the case where no particles are stabilized (i.e., where all amphiphiles are in lamellar phases and all particles are flocculated in a large aggregate) and that where some of the particles are stabilized by complexing with amphiphiles and the others are aggregated. Assuming that Σ, the interfacial area per amphiphile is fixed,11 the number of amphiphiles per colloid-amphiphile complex is given by (R + L)2/Σ in the (11) This assumption is in agreement with the classic derivation by Tanford (Tanford, C. The hydrophobic effect; Wiley-Interscience: New York, 1973). It is also in agreement with Nagarajan, who has recently shown (Nagarajan, R. Langmuir 2002, 18, 31-38) that the tail can play either an explicit role through modification of the area per amphiphile or an implicit role without changing the packing parameter. We adopt the latter approach, whereby we assume that Σ, the area per amphiphile, is fixed but taking into account the effect of the tail packing on the aggregate free energy.
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case of large particles (see Figure 1), or some fraction thereof in the case of small particles. The number of colloid-amphiphile complexes, namely, solubilized particles, per unit volume is given by yφ/R3. Thus, the energy of the amphiphiles in the particle complexes relative to that of the unperturbed lamellae is given by (per unit volume)
∆Fs ) yφ(1 + L/R)2 [B(L - L*)2 - δ]/RΣ
(4.a)
while the energy of the unsolubilized, flocculated particles is given by
∆Fc ) γφ2/3V-1/3[(1 - y)2/3 - 1]
Figure 2. The critical particle volume fraction (φc) as a function of particle radius R, as defined by eq 7, for two values of bilayer thickness differing by 25%. If φ < φc (namely, the area underneath the curve), all particles will be solubilized. If φ > φc, all particles will flocculate. φc and R are given in arbitrary units.
(4.b)
In principle, we should also account for the entropy of the dispersed particles (when compared to the flocculated state), which is given by the number of free particles times the log of their volume fraction (including the effect of the amphiphile layer)
∆Ss ) yφ/R3 ln[yφ(1 + L/R)3]
(4.c)
For y ) 0 (no solubilized particles) the entropy is zero, increasing with the number of dissolved particles. However, unless φ is extremely small, the solubilized particle entropy is low when compared to ∆Fs.12 As a result, we neglect the contribution of the mixing entropy and focus on the interactions. The system free energy is therefore given by
∆FT ≈ ∆Fs + ∆Fc
(4.d)
R, φ ,VL*, and L(R,L*) are fixed system parameters. The only free variable is y, the fraction of solubilized particles. Results The equilibrium state is set by minimization of the system energy (eq 4) with respect to the stabilized particle fraction y. However, the second derivative of FT with respect to y is given by
∂2FT/∂y2 ) -(φV)2/3γ/(1 - y)4/3
(5)
which is always negative, thereby indicating that there is no true minimum in the system free energy as a function of the fraction of stabilized particles. As a result, the system will favor either the limit where all particles are stabilized (y ) 1) or the limit where no particles are stabilized and y ) 0. This is in agreement with a similar analysis of particle stabilization through polymer adsorption.8 The condition for obtaining a stable dispersion where all particles are is that the energy of y ) 1 be lower than that of y ) 0, namely
γφ2/3V-1/3 - φ(R + L)2[B(L - L*)2 - δ]/ΣR2 > 0 (6) In systems where the particle chemistry is similar to that of amphiphilic tails so that δ ≈ 0, eq 6 reduces to a balance between the particle surface tension (which favors stabilization) and the amphiphile deformation. As a result, (12) As shown in ref 10, the entropy per individual particle is of order (10-21 ln φ) J. The compression modulus B for lipid bilayers is of order 40-150 mJ/m2. (See: Gennis, R, B. Biomembranes; Springer-Verlag: Berlin, 1989. Evans, E.; Needham, D. J. Phys. Chem. 1987, 91, 42194228.) As a result, the colloidal volume fraction at which the colloid entropy becomes significant when compared to the membrane deformation energy, as given by eq 2, is of order 10-3-10-10, depending on the degree of deformation (1 - L*/L)2.
Figure 3. The critical particle-amphiphile interaction energy (δc) as a function of particle radius R, as defined by eq 8. If the interaction energy δ is more favorable (namely, larger than) δc, all particles will be solubilized.
the critical particle volume fraction under which stabilization will occur is given by
Vφc/R3 ) (γΣ/BL*2)3/(1 - L/L*)6(1 + L/R)6
(7)
where Vφc/R3 is the number of particles in the system. As may be expected, the critical limit for particle stabilization increases with increasing particle surface tension (γ) and decreases with increasing bilayer thickness (L*) or bilayer compressibility (B). However, quite surprisingly, there is no dependence of φc on the amphiphile concentration. In Figure 2 we examine the effect of the particle radius, R, on the critical particle volume fraction; for volume fractions below the critical one, all particles will be individually stabilized, while above it they would aggregate into a large flocculate. We see that φc increases rapidly with the particle radius in the limit where R is small, reaching a saturation value in the limit of large particles. The saturation limit decreases significantly with the amphiphilic bilayer thickness, scaling as (γΣ/BL*2)3. It is interesting to note that there is a finite critical volume fraction even in the limit where R is small; thus, even particles whose radius is smaller than the amphiphile bilayer thickness will undergo a transition from a stable solubilized suspension to a floc, as a function of their volume fraction. In systems where there is a favorable interaction between the amphiphilic tails and the particles (i.e., an “adsorption energy”), we find that the critical adsorption energy is given by
δc ) B(L - L*)2 - γΣR3/φ1/3V1/3(L + R)2
(8)
If the interaction energy is larger than the critical value, the particles will be stabilized. In Figure 3 we plot the effect of the particle radius on δc. We see that the critical interaction increases and then decreases with particle size. In the limit of large particles δc is negative, indicating that in such systems particles may be stabilized even if the interaction energy between the particle and the amphiphile is unfavorable. What if the amphiphiles are not in excess, so that the maximal possible fraction of stabilized particles cannot be unity? The free energy expression (eq 4) holds, but ymax
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) cΣR/φ, set by the particle geometry and the (molar) concentration of amphiphiles c. Since there is no minimum in the free energy as a function of y, we must compare the two states: y ) 0 and y ) yma. The condition for solubilization becomes then
1 < cV1/3(R + L)2[B(L - L*)2 - δ]/γφ2/3R2[1 (1 - cRΣ/φ)2/3] (9) which holds for systems where the perturbation energy of the amphiphiles [B(L - L*)2 - δ] is small relative to the particle surface tension. As a result, we predict that for a given particle solution (as characterized by φ and R), the number of stabilized, solubilized particles will increase linearly with amphiphile concentration c and a slope that depends on the amphiphile properties (as given through Σ). Discussion and Conclusions We present here a simple, mean field model for the solubilization and stabilization of hydrophobic colloidal particles against flocculation via lamellae-forming amphiphilic molecules. We find that there is practically no coexistence between flocculated and solubilized particles. One state or the other will be favored, depending on system parameters such as the particle size, the interaction energy between the particle and the amphiphile, and the particle volume fraction. In systems where the amphiphiles are not in excess, we predict that the fraction of stabilized particles will increase linearly with amphiphile concentration and a slope that depends on the amphiphile type. This prediction is in agreement with the experiments of Ahmed,13 who examined the effect of surfactant type and concentration on the solubilization of hydrophobic drug
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particles. Ahmed13 finds that the concentration of stabilized particles increased linearly with surfactant concentration for all types of surfactants, with a slope that varies from one surfactant to another. Our model also predicts that the interaction between the hydrophobic particle and the amphiphilic molecules plays a significant role in determining the stabilization of a suspension; the critical interaction energy required for the formation of stable particles (eq 8 and Figure 3) varies as a function of the amphiphile characteristics and the particle size. Indeed, Luner et al.14 have shown that there is a link between the wetting of hydrophobic drugs by surfactant solutions (a function of the surfactantsurface interactions) and particle solubilization. In conclusion, we present here a simple model for the solublization of hydrophobic aprticles in aqueous solutions via bilayer-forming amphiphiles. We predict that when the amphiphiles are in excess there will be practically no coexistence between solubilized particles and flocculated ones, but only one or the other. When the amphiphiles are not in excess, the volume fraction of solubilized particles will increase linearly with amphiphile concentration. We also find that, even when the amphiphile are in excess, particles may form a flocculate (rather than be solubilized) if their volume fraction is high or if the interaction energy between the amphiphile and the particle is higher than a critical value. Acknowledgment. This research was supported by the National Science Foundation Grant number 0049076. LA025672C (13) Ahmed, M. O. Eur. J. Pharm. Biopharm. 2001, 51, 221-226. (14) Luner, P. E.; Babu, S. R.; Mehta, S. C. Int. J. Pharm. 1996, 128, 29-44.
