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Theoretical Analysis of Factors Affecting the Formation and Stability of Multilayered Colloidal Dispersions D. Julian McClements Biopolymer and Colloids Research Laboratory, Department of Food Science, University of Massachusetts, Amherst, Massachusetts 01003 Received May 11, 2005. In Final Form: July 8, 2005 A mathematical analysis of the major factors influencing the formation and stability of colloidal dispersions containing spherical particles surrounded by multilayered polymeric interfacial membranes formed by the layer-by-layer electrostatic deposition technique is carried out. The mathematical model assumes that (i) the colloidal dispersion initially consists of a mixture of electrically charged monodisperse spherical particles and oppositely charged polymer molecules, (ii) the adsorption of polymer molecules to the particle surfaces is diffusion-limited, and (iii) the dominant particle-particle collision mechanism is Brownian motion. This approach was used to produce stability maps that highlight conditions under which bridging flocculation, multilayer formation, or depletion flocculation occurs. The stability maps are derived from calculations of the critical polymer concentrations required to (i) saturate the particle surfaces (CSat), (ii) ensure that polymer adsorption is faster than particle collisions (CAds), and (iii) promote depletion flocculation (CDep). In addition, the influence of interfacial properties on the stability of multilayer colloidal dispersions was assessed by calculating the colloidal interactions between the coated particles (i.e., van der Waals, electrostatic, steric, and depletion). These calculations indicated that the major factors are the interfacial charge and composition rather than the interfacial thickness. This article provides useful insights into the factors affecting the formation of stable multilayer colloidal dispersions.
Introduction Electrostatic layer-by-layer (LbL) deposition is a powerful method for manipulating the characteristics of interfacial membranes, such as their chemical composition, thickness, electrical charge, rheology, permeability, and environmental responsiveness.1-4 A solution containing a polyelectrolyte is brought into contact with an oppositely charged surface, which is usually either a macroscopic interface or a colloidal particle. The polyelectrolyte adsorbs to the surface through electrostatic attraction forming a multilayer interface: S-P1, where S refers to the surface and P1 refers to the first polyelectrolyte. Adsorption of a sufficiently high concentration of a highly charged polyelectrolyte causes reversal of the net charge on the interface (i.e., the net charge goes from positive to negative or vice versa). Consequently, it is possible to incorporate additional polyelectrolyte layers into the interface. In this case, the original polyelectrolyte solution is removed, and then a solution of a different polyelectrolyte (P2, with an opposite charge to that of the original polyelectrolyte) is brought into contact with the interface. This polyelectrolyte is adsorbed to the interface through electrostatic attraction leading to the formation of an S-P1-P2 interface with an opposite charge to that of the S-P1 interface. This process can be continued so that many different layers can be incorporated into the interfacial membrane: S-[P1-P2]n-P1-P2 or S-[P1P2]n-P1. The final outer layer normally determines the net charge of the interface.2 The thickness of the interfacial (1) Caruso, F.; Sukhorukov, G. In Multilayer Thin Films: Sequential Assembly of Nanocomposite Materials; Decher, G., Schlenoff, J. B., Eds.; Wiley-VCH: Weinheim, Germany, 2003; pp 331-362. (2) Decher, G. In Multilayer Thin Films: Sequential Assembly of Nanocomposite Materials; Decher, G., Schlenoff, J. B., Eds.; WileyVCH: Weinheim, Germany, 2003; pp 1-46. (3) Decher, G.; Schlenoff, J. B. Multilayer Thin Films: Sequential Assembly of Nanocomposite Materials; Wiley-VCH: Weinheim, Germany, 2003. (4) Panchagnula, V.; Jeon, J.; Rusling, J. F.; Dobrynin, A. V. Langmuir 2005, 21, 1118-1125.
membrane can be altered by choosing polyelectrolytes with different molecular weights and/or by varying the total number of layers formed. The overall electrical properties of the interface can be controlled by choosing polyelectrolytes with different electrical characteristics and/or by determining the polyelectrolyte that forms the outermost layer. The electrostatic LbL technique was originally used to modify the characteristics of electrically charged planar surfaces.2 More recently, it has been shown to be an extremely effective means of manipulating the interfacial characteristics of colloidal particles.3-6 In this case, polyelectrolytes are mixed with colloidal dispersions containing oppositely charged particles. This technology has been used to create colloidal dispersions with improved or novel physicochemical and functional properties. For example, LbL has been used to improve the stability of oil-in-water emulsions to environmental stresses, such as pH extremes, high mineral contents, thermal processing, freezing, and drying.7-15 In addition, it has also been used to engineer novel functional properties into colloidal dispersions, such as the encapsulation of active ingredi(5) Caruso, F. Top. Curr. Chem. 2003, 227, 145-168. (6) Antipov, A. A.; Sukhorukov, G. B. Adv. Colloid Interface Sci. 2004, 111, 49-61. (7) Aoki, T.; Decker, E. A.; McClements, D. J. Food Hydrocolloids 2004, 19, 209-220. (8) Dickinson, E. Food Hydrocolloids 2003, 17, 25-39. (9) Faldt, P.; Bergenstahl, B.; Claesson, P. M. Colloids Surf., A 1993, 71, 187-195. (10) Gu, Y. S.; Decker, E. A.; McClements, D. J. J. Agric. Food Chem. 2004, 52, 3626-3632. (11) Gu, Y. S.; Decker, E. A.; McClements, D. J. Food Hydrocolloids 2005, 19, 83-91. (12) Magdassi, S.; Bach, U.; Mumcuoglu, K. Y. J. Microencapsulation 1997, 14, 189-195. (13) Ogawa, S.; Decker, E. A.; McClements, D. J. J. Agric. Food Chem. 2004, 52, 3595-3600. (14) Ogawa, S.; Decker, E. A.; McClements, D. J. J. Agric. Food Chem. 2003, 51, 2806-2812. (15) Ogawa, S.; Decker, E. A.; McClements, D. J. J. Agric. Food Chem. 2003, 51, 5522-5527.
10.1021/la0512603 CCC: $30.25 © 2005 American Chemical Society Published on Web 09/09/2005
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Figure 1. Schematic diagram of the different events that occur when an electrically charged polyelectrolyte is added to a colloidal dispersion containing oppositely charged spherical particles.
ents, controlled or triggered release of active agents, controlled enzyme reactions, and so forth.1-5,6,16-19 One of the biggest problems reported in studies that have used the LbL technique to create multilayer-coated particles is the tendency for extensive flocculation of the particles to occur, even under conditions where they should be saturated with polyelectrolyte.1,13-15,20 It is sometimes possible to disrupt the flocs formed by application of mechanical agitation, such as sonication, blending, or homogenization.13 Alternatively, particle flocculation can be retarded by carefully controlling the preparation conditions,2,3 such as the characteristics of the colloidal particles (e.g., concentration, size, and charge), the characteristics of the polymer (e.g., concentration, molecular weight, charge density, conformation), the properties of the solvent (e.g., ionic strength, pH, and dielectric constant), and/or the mixing method (e.g., order of addition, mixing speed, flow profile). It would be highly desirable to have a better fundamental understanding of how these various parameters influence the formation of stable multilayer-coated particles. Two mechanisms have been proposed to account for the flocculation observed during the preparation of multilayercoated particles: (i) bridging flocculation occurs when particle collisions take place more rapidly than the time required for the polymers to saturate the particle surfaces completely,1 and (ii) depletion flocculation occurs when the free polymer concentration exceeds a particular value.21 One of the main objectives of this article is therefore to identify the major factors that influence bridging and depletion flocculation so as to develop effective strategies to prevent it from occurring during the formation of multilayer-coated particles. Stability Maps for Multilayer Formation Consider a system consisting of a suspension of charged spherical particles to which an increasing number of oppositely charged polyelectrolyte molecules is added (Figure 1). There will be an electrostatic attraction between the polyelectrolyte molecules and the bare surfaces of the (16) Riegler, H.; Essler, F. Langmuir 2002, 18, 6694-6698. (17) Antipov, A. A.; Sukhorukov, G. B.; Mohwald, H. Langmuir 2003, 19, 2444-2448. (18) Khopade, A. J.; Caruso, F. Langmuir 2003, 19, F6219-6225. (19) Dejugnat, C.; Sukhorukov, G. B. Langmuir 2004, 20, 72657269. (20) Kim, A. Y.; Berg, J. C. Langmuir 2002, 18, 3418-3422. (21) McClements, D. J. Food Emulsions: Principles, Practice and Techniques, 2nd ed.; CRC Press: Boca Raton, FL, 2004.
McClements
particles that will cause polyelectrolyte molecules to adsorb to the particle surfaces. Initially, we will assume that the system has been designed so that the particles are completely stable to aggregation when there are no polyelectrolyte molecules present (bare particles) and when the particles are completely saturated with polyelectrolyte molecules (provided there is no depletion attraction, see below). Practically, this could be achieved by ensuring that the magnitude of the electrical charge density on both the bare and saturated particles was high enough to generate a strong electrostatic repulsion between the particles. The stability of the particles to aggregation can then be divided into a number of different regimes depending on the polyelectrolyte concentration, C: (I) C ) 0. The particles are stable to aggregation because of the strong electrostatic repulsion between them. (II) 0 < C < CSat. When the concentration of polyelectrolyte in the system is insufficient to saturate the particle surfaces (CSat) completely, bridging flocculation occurs because of the sharing of individual polyelectrolyte molecules between more than one particle. (III) CSat < C < CDep. The particles should be stable to aggregation when their surfaces are completely saturated with polyelectrolyte, and there is not enough free polyelectrolyte present in the continuous phase to promote depletion flocculation (see IV). Under these circumstances, it should be possible to prepare stable systems consisting of particles completely surrounded by a polyelectrolyte layer. Nevertheless, it is still necessary to ensure that the time required for the particles to be saturated with polyelectrolyte (τAds) is appreciably shorter than the time between particle-particle collisions (τCol); otherwise, bridging flocculation will still occur. Typically, τAds falls below τCol when a certain polyelectrolyte critical concentration (CAds) is exceeded. (IV) C > CDep. When the concentration of free polyelectrolyte exceeds some critical value (CDep), depletion flocculation occurs because the attractive depletion forces are strong enough to overcome the various repulsive forces (e.g., electrostatic and steric). The purpose of this article is to give simple mathematical expressions for CSat, CDep, and CAds that will enable us to predict the major factors that influence the formation of stable multilayer-coated particles and therefore optimize their preparation. In addition, we will also carry out an analysis of the factors influencing the stability of multilayer colloidal dispersions once they are formed by analyzing the colloidal interactions between the multilayer-coated particles. Polyelectrolyte Saturation Concentration (CSat). If it is assumed that below the saturation concentration (C < CSat) all of the polyelectrolyte added to the system is adsorbed to the particle surfaces, which is usually a reasonable assumption given the strong electrostatic attractive forces involved, then the polyelectrolyte concentration required to reach saturation is given by21
CSat )
3φΓSat r32
(1)
where φ is the volume fraction of the particles, r32 () Σniri3/ Σniri2) is the volume-surface mean radius of the particles (in m), ΓSat is the surface load of the polyelectrolyte at saturation (in kg m-2), and CSat is the minimum concentration of polyelectrolyte in the whole system required to saturate the surfaces (in kg m-3). This equation shows that the polyelectrolyte concentration required to saturate the particle surfaces completely increases with increasing
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particle concentration, increasing surface load, and decreasing mean particle size. If it is assumed that the polyelectrolyte molecules have fairly similar conformations after adsorption to the particle surface as they have when dispersed in the continuous phase, then the surface load can be simply related to the effective radius (rPE) and molecular weight (M) of the polyelectrolyte in solution: ΓSat ≈ M/(NAπrPE2), where NA is Avogadro’s number. To avoid bridging flocculation when preparing multilayer particles, it is proposed that the overall polyelectrolyte concentration in the system must be at least greater than CSat. For example, about 0.6 kg m-3 (∼0.06 wt %) of a polyelectrolyte with a surface load of 1 mg m-2 is required to completely saturate the surfaces of a suspension containing 10 vol % of 1-µm-radius spherical particles. Polyelectrolyte Depletion Concentration (CDep). Once the surfaces of the particles have become completely saturated with polyelectrolyte, then any additional polyelectrolyte added to the system will remain free in the continuous phase and will therefore generate a depletion attraction between the particles (Figure 1). If this depletion attraction is strong enough to overcome any repulsive interactions, then the large particles will tend to flocculate. The following expression gives the strength of the depletion attraction (wDep,0) between two contacting spherical particles dispersed in a continuous phase containing nonadsorbed polyelectrolyte:21
(
) (
)
2πCfreeNA 2NACfreeν wDep,0 2 )1+ rPE2 r + rPE kT M M 3
(2)
Here, k is Boltzmann’s constant, T is the absolute temperature, Cfree () [C - CSat]/[1 - φ]) is the concentration of nonadsorbed polyelectrolyte in the continuous phase (in kg m-3), NA is Avogadro’s number, M is the molecular weight of the polyelectrolyte (in kg mol-1), v () 4πrPE3/3) is the effective molar volume of the polyelectrolyte in solution (in m3), r is the radius of the spherical particles, and rPE is the effective radius of the polyelectrolyte molecules in solution. The above equation shows that the strength of the depletion attraction increases with increasing concentration, increasing effective radius, and decreasing molecular weight of the polyelectrolyte. Nevertheless, it should be stressed that the molecular weight and effective radius of the polyelectrolyte are not independent variables: as M increases, rPE increases.22,23 The increase in the depletion attraction with increasing free polyelectrolyte concentration is shown in Figure 2 for spherical particles of different mean radii (0.2, 0.5, 1, and 2 µm), assuming that M ) 100 kDa and rPE ) 25 nm for the polyelectrolyte, which are in the range of experimentally determined values for polysaccharides, such as pectin.24 These predictions show that the depletion attraction is fairly strong, especially for large particles and high polyelectrolyte concentrations. Consequently, flocculation may occur in a colloidal dispersion when the free polyelectrolyte concentration is sufficiently high to generate a depletion attraction that overcomes the repulsive interactions between the particles (e.g., electrostatic and steric). To a first approximation, the percentage of particles that are flocculated because of the presence of the free polyelectrolyte can be estimated by assuming that the (22) McClements, D. J. Food Hydrocolloids 2000, 14, 173-179. (23) Grosberg, A. Y.; Khokhlov, A. R. Giant Molecules; Academic Press: San Diego, CA, 1997. (24) Fishman, M. L.; Chau, H. K.; Kolpak, F.; Brady, J. J. Agric. Food Chem. 2001, 49, 4494-4501.
Figure 2. Predictions of the strength of the depletion interaction between two spherical particles suspended in a continuous phase containing free polyelectrolyte.
Figure 3. Schematic diagram of the interaction potential assumed to calculate the critical free polyelectrolyte concentration required to promote depletion flocculation. The circles represent droplets in either a flocculated or stable state.
depletion attraction is constant from h ) 0 to 2rPE (wDep) but is zero at greater particle separations and that the only other interaction is an infinitely strong short-range repulsion (Figure 3). Hence, the particles partition between the flocculated state (low free energy) and the nonflocculated state (high free energy) according to the depth of the secondary minimum (wDep/kBT) and the relative volumes available to the flocculated state (θF) and the nonflocculated state (θNF):
F ) 100 ×
(
[
) ]
θNF wDep exp +1 θF kBT
-1
(3)
where
θF )
2rPE
2rPE ≈ L
x(4πr3/3φ) - 2(r + 2rPE) 3
θNF ) 1 - θF Here, F is the percentage of particles in the system that are flocculated (i.e., in the low free-energy state). The value of θF was calculated by assuming that the volume fraction that particles are restricted to when they are flocculated is approximately equal to the length of the depletion zone (2rPE) divided by the average distance between the surfaces of the particles (L):
x 3
L≈
4πr3 - 2(r + 2rPE) 3φ
In this analysis, it was assumed that the surface of the large particles was saturated with polyelectrolyte so that
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the effective radius of the multilayer particles was r + 2rPE. If it is assumed that 50% of the particles in the system are flocculated, then a critical value of the depletion attraction can be calculated:
( ) wDep kBT
[ ] θNF θF
) -ln
Crit
(4)
For a particular colloidal dispersion, it is possible to calculate the minimum amount of free polyelectrolyte (CDep) required to promote depletion flocculation (i.e., F ) 50%) by inserting the values of R into the equations for θNF and θF given above and calculating (wDep/kBT)Crit and then inserting this value into eq 2. The resulting quadratic equation can then be solved
CDep )
(
)
M -1 + x1 - 8vX NA 4v
( ) wDep kBT
τAds )
10ΓSat2 CPE2DPE
(9)
where CPE is the concentration of polyelectrolyte in the continuous phase (in kg m-3), DPE is the translation diffusion coefficient of the polyelectrolyte through the continuous phase (in m2 s-1), and ΓSat is the surface load of the polyelectrolyte at the interface (in kg m-2) (i.e., the mass of polyelectrolyte per unit surface area of particles). This characteristic adsorption time is the time required for the surface to be 90% saturated with the polyelectrolyte.26 The effective diffusion coefficient of the polyelectrolyte is given by
(5)
where
X)
the following expression26
kBT 6πηCrPE
(10)
τAds 60Γ2rPEφ ) τCol C 2r3
(11)
DPE ) Hence,
2 Crit2πrPE (r
1 + (2/3)rPE)
(6)
PE
Because some of the polyelectrolyte added to the system is adsorbed to the surfaces of the large particles, the total amount of polyelectrolyte in the overall system required to promote depletion flocculation is given by
CDep* ) (1 - φ) × CDep + CSat
(7)
To use the above equations to calculate CDep, it is necessary to know the molecular weight (M) and the effective radius (rPE) of the polyelectrolyte. These parameters can be determined using a variety of analytical techniques from measurements made on dilute solutions of polyelectrolytes, including dynamic light scattering, static light scattering, viscosity, chromatography, and sedimentation coefficient techniques.23-25 Nevertheless, the most appropriate physical parameter to represent the effective radius (e.g., hydrodynamic radius or radius of gyration) still has to be established. Polyelectrolyte Adsorption Concentration (CAds). Another important factor that influences the stability of the multilayer particles against aggregation during formation is the time taken for the surface of the large particles to be saturated (τSat) with polyelectrolyte relative to the time between collisions of the particles (τCol). If the surfaces of the particles are not completely saturated with polyelectrolyte when the particles encounter each other (τCol < τSat), then it is likely that the particles will aggregate with some polyelectrolyte acting as bridges between them (Figure 1). It is therefore useful to examine the relationship between τSat and τCol. If it is assumed that the dominant mechanism for the collisions of particles is Brownian motion, then the average time between collisions is26
τCol )
πr3ηC kBTφ
(8)
Here, ηC is the viscosity of the continuous phase. To a first approximation, the adsorption of spherical particles to a planar interface can be given by (25) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997. (26) Walstra, P. Physical Chemistry of Foods; Marcel Decker: New York, 2003.
The adsorption time decreases relative to the collision time (i.e., less expected particle aggregation) as CPE increases. It should be noted that the viscosity of the continuous phase does not appear in this expression, which means that it is not necessary to include the influence of the polyelectrolyte on the continuous-phase viscosity in the calculations. We can estimate a critical polyelectrolyte concentration (CAds) where the adsorption time is just equal to the collision time by letting τAds ) τCol:
CAds )
x
60Γ2rPEφ r3
(12)
If C > CAds, then we would expect the adsorption time to be faster than the collision time between particles and therefore little particle aggregation to occur. Overall Considerations Determining Stability. To produce a colloidal dispersion that is stable to flocculation, it is necessary to ensure that CSat < C < CDep* (i.e., that there is enough polyelectrolyte to completely saturate the surfaces of the particles but not too much to promote depletion flocculation). In addition, it is also important that the time required for the particles to be completely covered by polyelectrolyte is less than the time between particle-particle collisions: τAds/τCol < 1 (i.e., C > CAds). Using the various equations derived above for CSat, CDep* and CAds, it is possible to generate stability maps for the formation of multilayer particles without promoting bridging flocculation and depletion flocculation (see below). Numerical Calculation of Factors Affecting the Formation of Multilayer Systems. In this section, we examine some of the major factors that would be expected to influence the formation of stable multilayer colloidal dispersions, including polyelectrolyte concentration and size (C and rPE) and particle concentration and size (φ and r). Influence of Particle Concentration. A stability map showing the dependence of the critical saturation and depletion and adsorption concentrations on particle concentration is shown in Figure 4. In these calculations, it was assumed that the particles had a radius of 0.3 µm
Multilayered Colloidal Dispersions
Figure 4. Stability map showing the influence of droplet concentration on the critical polyelectrolyte concentrations for saturation, depletion, and adsorption. It was assumed that the droplets had a radius of 0.3 µm and the nonadsorbed polyelectrolyte had a molecular weight of 100 kDa and effective radius of 30 nm. Shaded area shows conditions where it should be possible to create a stable multilayer emulsion.
and that the polyelectrolyte had a molecular weight of 100 kDa and an effective radius of 30 nm. These polyelectrolyte characteristics are similar to those reported in the literature for a number of natural polymers, such as chitosan.27 This natural cationic polyelectrolyte was used in our previous studies to make up multilayer emulsions by adsorbing it to the surfaces of anionic droplets.13-15,28 The stability map indicates that at low polyelectrolyte concentrations the particles should be susceptible to bridging flocculation because there is insufficient polyelectrolyte present to completely saturate the surfaces of all the particles (C < CSat). Nevertheless, even when the polyelectrolyte concentration exceeds CSat, the particles may still be susceptible to bridging flocculation because the polyelectrolyte does not adsorb rapidly enough to saturate the particle surfaces before a particle-particle collision occurs (C < CAds). At relatively low particle concentrations (φ < 0.11), it should be possible to make stable multilayer-coated particles without flocculation occurring over a range of intermediate polyelectrolyte concentrations (CAds < C < CDep). However, if the polyelectrolyte concentration is increased further so that CDep is exceeded then the particles will be susceptible to depletion flocculation and it will not be possible to make a stable multilayer system. In addition, at relatively high particle concentrations (φ > 0.11), there is no range of polyelectrolyte concentrations where stable multilayercoated particles can be formed because depletion flocculation tends to occur at the polyelectrolyte concentration required to ensure rapid adsorption. In a recent study, we examined the stability of oil-inwater emulsions containing anionic SDS droplets after cationic chitosan had been added to the system.28 This study showed that extensive droplet flocculation occurred for C < CSat at all droplet concentrations and that it was not possible to make nonaggregated emulsions when the droplet concentration exceeded a particular value (φ > 0.05) at any polyelectrolyte concentration. The experimental data from this study therefore qualitatively supports the predictions of the stability map shown in Figure 4. Influence of Particle Size. A stability map showing the dependence of the critical saturation and depletion and adsorption concentrations on particle size is shown in Figure 5. In these calculations, it was assumed that the (27) Berth, G.; Dautzenberg, H. Carbohydr. Polym. 2002, 47, 39-51. (28) Mun, S.; Decker, E. A.; McClements, D. J. Langmuir 2005, 21, 6228-6234.
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Figure 5. Stability map showing the influence of droplet size on the critical polyelectrolyte concentrations for saturation, depletion, and adsorption. It was assumed that the droplets had a volume fraction of 0.03 (3 vol %) and the nonadsorbed polyelectrolyte had a molecular weight of 100 kDa and an effective radius of 30 nm.
Figure 6. Stability map showing the influence of the effective radius of the polyelectrolyte molecules on the critical polyelectrolyte concentrations for saturation, depletion, and adsorption. It was assumed that the droplets had a radius of 0.5 µm and a concentration of 5 vol % and that the nonadsorbed polyelectrolyte had a molecular weight of 100 kDa.
particle concentration was φ ) 0.03 (3 vol %) and that the polyelectrolyte had a molecular weight of 100 kDa and an effective radius of 30 nm. The stability map predicts that it should not be possible to make stable multilayer colloidal dispersions at relatively low particle radii (r < 0.15 µm), even above CSat, because the particles collide with each other before their surfaces are completely saturated with polyelectrolyte. However, at high particle sizes there should be a range of intermediate polyelectrolyte concentrations (CAds < C < CDep) where stable multilayercoated particles can be produced. Influence of Polyelectrolyte Characteristics. Stability maps showing the dependence of the critical saturation and depletion and adsorption concentrations on polyelectrolyte molecular weight and effective radius are shown in Figures 6 and 7. In these calculations, it was assumed that the particles had a radius of 0.5 µm and a volume fraction of 0.05. The stability map in Figure 6 assumes that the molecular weight of the polyelectrolyte stays constant (100 kDa) while the effective radius varies, which could occur if the polyelectrolyte conformation (e.g., sphere, coil, or rod) or solvent quality (e.g., pH, ionic strength, or temperature) was varied. For all polyelectrolyte effective radii, there is an intermediate range of polyelectrolyte concentrations where stable multilayer-coated particles could be formed (CAds < C < CDep). The stability map in Figure 7 assumes that the molecular weight of the polyelectrolyte varies while the effective radius remains constant (30 nm), which could occur if different kinds of polyelectrolytes were used (e.g., sphere, coil, or rod). The
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(
[
)
h 1 A (h) H ,1 + 12 131 2(r + ∂) h + 2∂ h+∂ r+∂ , 1 + 2A132(h) H , A232(h) H 2r 2r r
wVDW(h) ) -
(
)
(
)] (14)
where Aijk(h) refers to the Hamaker function, subscripts 1, 2, and 3 refer to the continuous phase and the particle and interfacial layers, respectively, h is the surface-tosurface separation between the outer regions of the adsorbed layers, δ is the thickness of the adsorbed layer, r is the particle radius, and H(x, y) is a function given by
y y + 2 + x + xy + x x + xy + x + y x2 + xy + x (15) 2 ln 2 x + xy + x + y
Figure 7. Stability map showing the influence of the molecular weight of the polyelectrolyte molecules on the critical polyelectrolyte concentrations for saturation, depletion, and adsorption. It was assumed that the droplets had a radius of 0.5 µm and a concentration of 5 vol % and that the nonadsorbed polyelectrolyte had an effective radius of 30 nm.
H(x, y) )
critical saturation and depletion and adsorption concentrations all increased linearly with increasing polyelectrolyte molecular weight so that the range of polyelectrolyte concentrations where stable multilayer-coated particles could be formed becomes wider as the conformation of the polyelectrolytes becomes more compact (i.e., the same effective polyelectrolyte radius for increasing molecular weight). In practice, the molecular weight and effective radius of polyelectrolytes cannot be changed independently because the effective radius will increase as the molecular weight increases.23
Mathematical formulas that can be used to calculate the Hamaker functions (A131, A232, and A132) are given in Appendix 1. Electrostatic Repulsion. The electrostatic repulsion between two similarly charged spherical particles suspended in a dielectric medium is given by the following equation:20,30,31
Interaction Potentials between Multilayer-Coated Particles
(13)
where wV(h), wE(h), wS(h), and wD(h) are the van der Waals, electrostatic, steric, and depletion interaction potentials, respectively. If the attractive forces between the multilayer particles dominate, then they will tend to flocculate, even if they are completely coated by a multilayer interface. However, if the repulsive forces dominate, then the particles will tend to remain separated, and it will be possible to produce a stable multilayer system. The sign, magnitude, and range of the individual colloidal interactions were predicted using the equations presented below. van Der Waals Attraction. To a first approximation, the van der Waals interaction between two coated spheres suspended in a dielectric medium that contains salt is given by the following equation29 (29) Vold, M. J. J. Colloid Sci. 1961, 16, 1-10.
(
wE(h) ) -2π0R(r + δ)Ψ2 ln[1 - e-κh]
)
(16)
Here, 0 is the permittivity of free space (8.854 × 10-12 J-1 m-1), R is the relative dielectric constant of the continuous phase, Ψ is the surface potential of the particles (in V), and κ-1 is the Debye screening length (in m). This equation assumes that the magnitude of the electrical potential on the particles is relatively small (0
(A1)
The influence of electrostatic screening and retardation can be taken into account using the following expression:
where
( )( )
i - j k - j 3 Aijk,v)0 ) kT 4 i + j k + j Aijk,v>0 )
the nonretarded frequency-dependent contribution.29 Here, k is Boltzmann’s constant (1.381 × 10-23 J K-1), T is the absolute temperature, h is Planck’s constant (6.626 × 10-34 J s), ε is the static relative dielectric constant, n is the refractive index, and ve is the major electronic absorption frequency in the ultraviolet region of the electromagnetic spectrum.
3hve 8x2
20,29
(A2)
Aijk ) [e-2κh]Aijk,v)0 +
[
(
1 - 5.32 × 107h ln 1 +
× (ni2 - nj2)(nk2 - nj2)
(ni2 + nj2)1/2(nk2 + nj2)1/2{(ni2 + nj2)1/2 + (nk2 + nj2)1/2} (A3) The Aijk,ν)0 term represents the nonscreened zerofrequency contribution, and the Aijk,ν>0 term represents
)]
1 Aijk,v>0 5.32 × 107h
The prefactors in the square brackets take into account the influence of electrostatic screening on Aν)0 and retardation on Aν>0, respectively. Here, κ-1 is the Debye screening length (see text). LA0512603
