4012
Langmuir 1994,10, 4012-4021
Osmotic Pressure of Latex Dispersions C. Bonnet-Gonnet,? L. Belloni,t and B. Cabane*ptJ Equipe mixte CEA-RP, Rhbne Poulenc, 93308 Aubervilliers, France, and Service de Chimie Moldculaire, CEN Saclay, 91191 Gif sur Yvette, France Received April 6, 1994. In Final Form: July 15, 1994@ The resistance of aqueous latex dispersions to removal of water has been measured through osmotic compression. For pure polystyrene particles stabilized with sulfate groups anchored at their surfaces,the measured pressures match the predictions of models for the interactions of charged spheres in water. For copolymerparticles (styrene-butyl acrylate)covered with acrylic acid sequenceslinkedto the core polymers, the measured pressures are considerablyhigher than the predictions. This strongresistance to dehydration originates from polyelectrolytes which are released by the particles or extend fiom their surfaces into the aqueous phase. Consequentlythe behavior of such dispersions is determined by the content ofthe aqueous phase that separates the particles.
Introduction Water-based dispersions of small polymeric particles are called latex dispersions.' Such dispersions are obtained through an emulsion polymerization process, where hydrophobic monomers are transferred to growing polymer chains in the particlesS2Hydrophilic species are often added during the synthesis; they form membranes around the particles and generate electrostatic repulsions between them, which stabilize the dispersion against aggregati~n.~ The constraints on this stability are severe, because the dispersions are made and stored at high volume fractions (4 L 0.5) where interparticle distances are short (on the order of 10nm) and collision frequencies are quite high. Even when electrostatic repulsions are adequate, instability may develop after a long period of time, or in high stress devices (pumps or filtration systems). It is found that the stability of latex dispersions varies considerably, according to the nature of the hydrophilic membranes. Therefore it would be most useful to have a method to measure this stability in the conditions of application. The usual way to measure the stability of colloidal dispersions is to dilute the dispersions, depress their stability by addingsalt, and measure the aggregation rate.* The result is expressed as a critical coagulation concentration, which is the salt concentration where the aggregation of particles is no longer inhibited by electrostatic barriers. This result measures the limit of electrostatic stability, but not the stability in the conditions of applications, where the volume fractions are high and there is no added salt. In such conditions, the resistance against aggregation is determinedby interactions between latex particles at short distances. Thus a more general goal would be to measure the interactions oflatex particles in this range of distances. Equipe mixte CEA-RP. Service de Chimie MolBculaire. Abstract published in Advance A C S Abstracts, September 15, 1994. (1)(a)Vanderhof,J.W.; Bradford, E. B.; Carrington,W. K J.Polym. Sci. Symp. 1873,41,166. (b) Kast, H. Makromol. Chem. Suppl. 1886, 10-2 1,447.(c)Zozel, A,;Heckmann,W.; Ley, G.; Mlchtle,W. Macromol. Symp. ISSO,35-36,423. (2)Van den Hul, H. J.,Vanderhoff, J. W. Br. Polym. J.1970,2,121. (3)(a) Greene, B. W., Sheetz, D. P., Filer, T. D. J.Colloid Interface Sci. 1870,32,90.(b) Greene, B. W. J. Colloid Interface Sci. 1873,43, 449.(c) Greene, B. W.: Nelson, A. R.: Keskev, _ .W. H. J. Phrs. Chem. 1980,84,1616. (4)Vervey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. T
*
@
The interactions between particles dispersed in water can be measured through three different methods. (1) Use two macroscopic surfaces of the same nature as the particle surfaces, and measure the force-to-distance relation with a surface forces apparatuss or an atomic force microscope.6 At present this is not practical, because it is difficult to stick common latex particles to the surfaces of the SFA or AFM. (2) Measure the spontaneous fluctuations of particle volume fraction, using the scattering of incident radiation (light, X-rays, or neutrons). However, latex dispersions are not transparent to light and do not provide good contrast for X-rays; neutrons are not readily available. (3) Use membrane exchanges to measure the relation of osmotic pressure to volume fraction in the dispersions. This method was pioneered by Hachisu,' Rand and Parsegian,6Nagy and H ~ r k a yand ,~ Parentich and Ottewill.lo It is always applicable and requires very little equipmentbut a fair amount of physical chemistry work; it measures a behavior which is very close to that observed in applications,i.e. the resistance to forced deswelling. In this paper we report the use of the osmotic stress method to measure interactions between particles in latex dispersions. Osmoticpressures and volume fractionshave been measured for different latex dispersions, and they have been compared with predictions based on models for electrostatic interactions of charged spheres. If the measured pressures do match the predictions, we may conclude that the dispersion is well described as a collection of hard spheres bearing electrical charges on their surfaces. If the measured pressures are considerably higher than the predictions, then we may conclude that the resistance to deswelling results from interactions between charges which are not localized on the particle surfaces. (6)(a) Israelachvili, J. N. J.Colloid Interface Sci. 1979,44,269. (b) Israelachvili, J.N.;Adams, G. E. J.Chem.Soc., Faraday Trans.1 1978, 74,976.(c) de Costello,B. A,; Kim,1. T.; Luckham, P. F.; Tadros, Th. F. Colloida Surfaces 1883,77,66. (6)Binnig,G.;Quate, C. S.;Gerber, Ch. Phys. Reu. Lett. 1886,56, 930. (7) Hachisu, S.,Kobayashi,Y. J. Colloid Interface Sci. 1874,46,470.
(8)(a) Parsegian, V. A,; Fuller, N. L.; Rand, R. P. Proc. Natl. Acad. Sci. U S A . 1878,76,2750.(b) Parsegian, V. A,; Rand, R. P.; Fuller, N. L.;Rau, D. C. In Methoda in Enzymology: Bwmembranes, Protons and Water, Structure and Translocation; Packer, L., Ed.; Academic Press: New York, 1986;Vol. 127,p 400. (9)(a)Nagy, M.,Horkay, F. Acta Chim. Acad. Sci. Hung. 1880,103, 387.(b)Rohrsetzer, S.;Kovacs, P.; Nagy, M. Colloid Polym. Sci. 1986, 264,812. (10)Goodwin, J. W.; Ottewill, R. H.; Parentich, A. Colloid Polym. Sci. 1980,268,1131.
0743-7463/94/2410-4012$04.50/00 1994 American Chemical Society
Langmuir, Vol. 10,No. 11, 1994 4013
Osmotic Pressure of Latex Dispersions Methods Osmotic stress methods are based on exchanges ofwater molecules between the sample and a reservoir. In nature, similar processes govern the passage of water and ions between cells and their environment. In the laboratory, the sample is pIaced in contact with a dialysis membrane through which it exchanges water, ions, and surfactants with a reservoir containinglarge amounts ofthese species. At equilibrium, the chemical potentials of all species able to cross the membrane are identical in the sample and in the reservoir; therefore it is possible to control the state of the sample through manipulation of these chemical potentials. This is particularly useful for colloidal systems where the adsorption of ions (H+, OH-) or molecules (surfactants) from the aqueous solution onto the particle surfaces determines the surface charge, and therefore the stability of the dispersions. The same equilibria make it possible to extract water from the dispersion by adding to the resemoir high molar mass species which bind water. At equilibrium, the chemical potentials of water on either side of the membrane are equal, and therefore the osmotic pressure of the sample equals that of the polymer in the reservoir. Through this procedure, dispersions may be concentrated at a set value of the osmotic pressure. The stressing polymer was dextran, chosen because it gives the same osmotic pressure regardless of ions and temperature.8bJ1The grade of dextran was T110, obtained from Fluka. The distribution of molar mass in this polymer was determined through size exclusion chromatography (SEC);the number average molar mass was Mn = 70 000, and the weight average was Mw = 150 000.The osmotic pressures of dextran solutions were determined with a Knauer membrane osmometer for pressures up to 4000 Pa (dextran concentrations between 0.1%and 5%). At higher pressures the values were from the intensity of scattered light, which gives the osmotic compressibility (dextran concentrations between 0.2% and 15%).12In the range where both methods overlapped, they gave the same pressures. The results were fitted by polynomial expressions for pressure ll (in Pa) versus concentration c in weight percent:
TI = 286c + 87c2 + 5c3
(1)
The linear term corresponds to a number average molar mass Mn = 87 000 g/mol, in good agreement with the results from SEC. The next term is related to the second virial coefficient, which corresponds to an exclusion diameter of 131 8 for each macromolecule; this is consistent with the hydrodynamic diameter obtained through quasielastic light scattering, which is 177 8. Finally, at high pressures, the values calculated from eq 1 may be compared with those measured by Rand and Parsegian,8b and by VBr6to~t.l~These authors used concentrated dextran solutions of molar mass 500 000. Their published values match those calculated from eq 1, even though the dextran samples are different. This is not unexpected, since the macromolecules in these concentrated solutions overlap t o the extent that the sizes of individual macromolecules become irrelevant. For these reason we conclude that expression 1represents well the (11) (a) Vink, H.Eur. Polym. J. 1971, 7, 1411. (b) Le Neveu, D. M.; Rand, R. P.;Parsegian,V. A.Nuture(London) 1976,259,601.(c)Prouty, M. S.;Schechter, A. N.; Parsegian, V. A. J.Mol. Biol. 1986,184, 517. (12) BonnebGonnet,C. Ph.D.Thesis,Universitk Pierre et Marie Curie Paris 6, 1993. (13) Vkrktout, F.;Delaye, M.; Tardieu, A. J. Mol. Biol. 1989,205, 713.
osmotic pressures of our dextran solutions in the whole range of concentrations, which was 0.1 to 40% in weight percent. The membranes used for osmotic stress experiments were dialysis bags which allowed exchange of water and ions but not of polymers or particles; this was obtained with a cutoff at a molar mass of 15 000 (Visking 8/32). Prior to the experiments, these bags were conditioned in water at the appropriate ionic strength and pH. Then latex dispersions were placed in the bags and immersed in the stressing solution of dextran. During the initial deswelling of the dispersions it was necessary to refill the bags with more latex dispersion (in order to obtain a sufficient amount of concentrated dispersion) and to exchange the stressing solutions (in order to avoid dilution with water from the dispersion);equilibrium was reached after 3 weeks. Then the content of each bag was extracted and the latex volume fraction was determined through drying at 120 "C. The reservoir was also analyzed, and its dextran concentration was measured through SEC or through total organic carbon analysis (TOC); from this concentration the osmotic pressure was calculated according to expression 1. Consequently, the osmotic pressure of the sample was known, and its latex volume fraction was measured; from these measurements one point of the equation of state of the dispersion was determined.
Materials Three types of latex dispersions have been studied, one containing pure polystyrene particles, and the other two containing copolymer particles. PS Latex. This dispersion was obtained from IDC (ref 815226). The particle cores are made of pure polystyrene, with glass transition temperature 110 "C; the Hamaker constant of pure polystyrene is 2kT.14 The particle diameters are monodisperse a t 61 nm. The core surfaces are largely hydrophobic, but they carry a few charged groups: sulfate groups which are the ends of the polystyrene chains, and sulfonate groups from styrene sulfonate monomers which are copolymerized with the styrene monomers. The total surface density of charged groups has been measured through potentiometric titration;ls it is 2 = 0.14 e/nm2. The aqueous phase ofthese dispersions is pure water. L1 Latex. This dispersion was synthetized through emulsion copolymerization of a mixture of hydrophobic and hydrophilic monomers.16 The particle cores are made of styrene-butyl acrylate copolymers, with glass transition temperature 25 "C; their Hamaker constant may be estimated as a linear combination of the Hamaker constants of polystyrene and poly(methy1methacrylate), which yields 2.5kT.14The particle diameters are nearly monodisperse a t 115 nm. Each particle core is protected from water by a membrane made of a copolymer of acrylic acid (AA,42%), butyl acrylate (BA, 35%),and styrene (S, 32%). The membrane polymers are copolymerized with the core polymers. The total amount of acrylic acid in the surface layer is 2%of the total latex mass. The total surface density of charged groups has been measured through potentiometric titration;16 it is 2 = 3.5 e/nm2a t pH 9 and 2 = 0.4 e/nm2at pH 3. The aqueous phase of these dispersions contains water, loose membrane polymers, and salt. L2 Latex. This dispersion is similar to L1,but the particle cores are made of styrene-butadiene copolymers, with glass transition temperature 0 "C; their Hamaker constant is 2.3kT.14 The particle diameters are nearly monodisperse at 170 nm. Each particle core is protected from water by a membrane made of monomers with carboxylic acid groups (37%)copolymerizedwith (14)Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (15) Van den Hul, H. J.; Vanderhoff, J. W. J. Electroanal. Chem. 1972,37, 161. (16) (a)Chevalier,Y.;Pichot, C.;Graillat, C.;Joanicot, M.;Wong, K.; Maquet, J.; Lindner, P.;Cabane, B. Colloid Polym. Sci. 1992,270,806. (b) Joanicot, M.; Wong, IC; Richard, J.; Maquet, J.; Cabane, B. Macromolecules 1993,26, 3168.
Bonnet-Gonnet et al.
4014 Langmuir, Vol. 10, No. 11, 1994 hydrophobic monomers (63%). The total amount of acid monomers in the surface layer is 1.4% of the total latex mass. The total surface density of charged groups is 2 = 4.3 dnm2a t pH 8.6 and 2 = 2.7 dnm2 at pH 5. The aqueous phase of these dispersions contains water, loose membrane polymers, and salt. All these dispersions were washed (Le. the aqueous phase was exchanged with pure water) through tangential ultrafiltration and then exchange of ions on ion exchange resins (Amberlite IRN 77 and IRN 78). Subsequently they were brought to the desired pH and ionic strengthlthrough dialysis against solutions of sodium hydroxide or sodium chloride. PolyacrylicAcid (PAA). Pure sodium polyacrylate solutions were also made in order to estimate the resistance of the particle membranes to osmotic compression. These solutions were made with sodium polyacrylate bought from Fluka (ref 81138). The distribution of molar mass in this polymer was determined through SEC;the number average was Mn = 110 000 g/mol and the weight average was Mw = 330 000 g/mol. In aqueous solutions, polyacrylate macromolecules expand t o large dimensions. Their average radius can be calculated from the viscosity laws measured for sodium polyacrylate in water.17J8 Above a certain concentration, denoted 0*,the macromolecules are forced to overlap and form a semidilute solution. The overlap concentration can be calculated from the dimensions of the macromolecules;the values are given below for the polyacrylate used in this study.
0*(g/g) p H = 9 -Z=lOmSM pH = 9 - Z = 0.1 M M pH = 3 - I = pH = 3 - Z = 0.1 M
0.0034 0.0082 0.0142 0.0291
In the present work, the polyacrylate concentration was always above the overlap concentration. Consequently, the macromolecules were fully interpenetrated, and their osmotic pressure was independent of molar mass.
Models The osmotic pressure of colloidal dispersions is measured as a resistance to a decrease in the volume available to the particles; this resistance originates from an increase of energy (interparticle forces) or from a loss of entropy (configurations) or from both. In order to calculate this pressure, it is necessary to have some information on the interactions and configurations of the particles; the required information is the pair distribution functiong(r) of interparticle distance r, and the pair interaction energy (or “potential”)u(r).19 If these functions were known, the pressure could be calculated from the virial equation:
where I7is the pressure, Q the number density ofparticles, k the Boltzmann constant, T the temperature, and u’(r) the interparticle force. However, g ( r ) and u(r) are not known; they must be calculated from the known quantities, which are the number density 8 , the surface charge Z of a particle, and the concentration of salt in the reservoir, expressed as a number density of ions e’s. At present, there is no model which calculates these functions accurately at all volume fractions ofthe dispersion; however, there are some models which are very good approximations at low volume fractions, where the particles do not get in each other’s (17)Encyclopedia of Polymer Science and Technology; Interscience Publishers, 1971; Vol. 14,p 736. (18)Van Kreveld,M. E.; Van den Hoed, N. J. Chromatogr. 1978,83,
111.
(19)Hansen, J. P.; MacDonald, I. R. Theory Academic Press: London, 1986.
of
Simple Liquids;
way, and others which are good at high volume fractions, where the particle positions are fixed. (1)Hard-SphereGas. In the limit of very low volume fractions, the osmotic pressure of a colloidal dispersion must follow the law of perfect gases, which is the first term of eq 2. For dispersions which are somewhat more concentrated, the loss of available configurations caused by the excluded volume around each particle can be taken into account through an expansion in powers of the volume fraction 4. A good approximation is provided by the Carnaham-Starling (CS) equation:
n 1+4+42-43 ijE= ( 1 - 413
(3)
In the case of charged particles which repel each other through overlap of their ionic clouds, the volume fraction & ~ f f used by the particles must include the volume of the = 0.6)are obtained if ionic clouds; good results (up to the extension of ionic clouds is chosen equal to the Debye screening length r 1 . 1 4 (2) One-Component Model. At still higher volume fractions, or with strongly repelling particles, it is necess a r y t o describe properly the interactions and the distances between the particles. In the “one-componentmodel”this is done by assuming that the particles interact directly with each other through an effective potential. This potential is calculated numerically accordingto the DLVO theory and the procedures described below.4 Once the interparticle potential is known, the resulting distribution of distances g(r) is calculated according to an integral equation (HNC) commonly used in the theory of simple 1iq~ids.l~ Finally, the pressure is calculated from the virial equation (2). This model works quite well at intermediate volume fractions but fails at high volume fractions, for three reasons. First, the DLVO pair potential is not calculated from the correct Poisson-Boltzmann (PB) equation but from a linearized version. At large distances this linearized equation gives the same solution as the correct equation if it uses an effective charge 2,~ which is lower than the true surface charge 2 of the particle; however, this procedure fails when the particles are at short distances. Second, the screening length used in DLVO theory is calculated from the concentration of passive salt in the reservoir; however, at high volume fraction, this does not match the concentration of salt in the sample. Third, even with a good pair potential, at high volume fractions the integral equations may not converge to the correct distribution of distances if the correlations are quite strong. These shortcomings originate from trying to describe the dispersion as a gas of particles and assuming that the ionic clouds are simply tied to the particles; then the pressure is proportional to the number density ofparticles, and the ions are included only through their effect on the interparticle potential. This simplifies the problem, as it reduces it to a calculation of interparticle distances and their fluctuations. However, a t high volume fractions, this description is inaccurate, for the reasons given above. In fact, when the interparticle distances are short, the resistance to a decrease in volume originates from the strong overlap of ionic clouds; then the pressure is proportional to the number density of ions, while the particles come in only through their effect on the ionic density profiles. (3) Poisson-Boltzmann Models. The next two models tackle this situation by calculating the pressure from the excess concentration of ions caused by the overlap of ionic clouds. For a dispersion containing counterions (in our case cations) at a local concentration
Osmotic Pressure of Latex Dispersions
Langmuir, Vol. 10,No. 11, 1994 4015
e+(r)and co-ions (in our case anions) at a local concentration e-(r),the osmotic pressure with respect to a reservoir containing salt at a concentration e’s iszo
n
= [e++ @-]E=() - 2e‘, kT
(4)
where E = 0 indicates that the concentrations of counterions and co-ions are measured at a point between particles where the electric field vanishes (closer to a particle, the concentrations may be higher or lower because of the nonvanishing electric field, but of course the chemical potentials are the same). (4) Poisson-Boltzmann Cell (PBC)Model. In this model the distribution of ions around each particle is calculated from the Poisson-Boltzmann (PB) equation with boundary conditions (the field vanishes) on a spherical shell of radius b around the particle.21 All counterions of the particle are within this boundary (the cell is electrically neutral); in addition, there are ions from passive salt, at a chemical potential which is determined by the reservoir. The model calculates the reduced electrostatic potential q(r)from the PB equation, and then the concentration profiles of counterions, g+(r),and of coions, e-(r),from the conditionthat their chemical potential equals that of passive salt which is in the reservoir at a concentration e‘,:
e+(r>= e’, exp[-v(r)I e-(r) = e’, exp[+q(r)l ( 5 ) Then the pressure may be calculated from eqs 4 and 5; it can be expressed as the difference, at the cell boundary ( r = b), between the arithmetic and geometric averages of e+ and e-:
n
- = [(e+(b> + e-(b)l - 2[e+(b)e-(b)lY2
kT
two dodecahedral particles; then the pressure of the dispersion is calculated from the virial equation (2)limited to first neighbors at the lattice positions. For a facecentered cubic lattice of particles, where a is the lattice parameter and u’(r)the force between neighboring par, virial ticles with a center to center distance ~ 1 4 2the equation yields
(7) For dodecahedral particles, the force is calculated as explained above from the pressure lIpexerted on opposing faces separated by a distance (a142 - D):
In this formula,the coefficientwhich takes the geometry of the particles into account is of order unity; therefore the osmotic pressure of the dispersion is nearly equal to the pressure between infinite planes at a separation determinedby the volume fraction of water and the particle size. Alternatively,ifthe particles remain spherical, the force between spherical surfaces can be calculated from an integration of the pressures exerted on planes located at different distances, followed by the use of the Derjaguin equation.* The virial equation is then written as in eq 7, but the force uYr) between neighboring spheres can now be expressed according to the potential up between opposing planes:
(6)
This model provides an excellent description of the electrostatic effects in the middle range of volume fractions. It is also the only proper way to calculate the concentration of salt in the dispersion. Its main shortcoming is the spherical shape of the cell, which does not match the unit cell in a very concentrated dispersion, where the neighboring particles come very close to the reference particle. In particular, this model does not predict the divergence ofthe pressure at a volume fraction Q = 0.74,because it does not prevent overlap ofneighboring particles. (6) Poisson-Boltzman Model with Planar Symmetry. When the particle surfaces are very close to each other, the repulsions are generated in thin layers of water which have a nearly lamellar geometry. Then it is more appropriateto use the PBC with planar symmetry, where the PB equation is solved in one dimension with boundaries at the particle surface and at the plane located halfway between the particles. As in the previous model, the pressure between planes is calculated from eq 4 or 6, but the number densities of ions are taken at the midplane where the electric field vanishes. This model is particularly appropriate for deformable particles which turn into dodecahedra under strong compression, as in the case of film-forming latex dispersions.16 The pressure exerted on a planar face can be multiplied by the area ofthe face to give the force between (20) Marcus, R. A. J. Chem. Phys. 1955,23, 1057. (21)(a) Katchalsky, A.;Alexandrowicz, 0. In Chemical Physics of ZonicSolutions;Conway,B.E.,Barradas, R. G., Eds.;Wiley: NewYork, 1966.(b)Bell, G.M.; Dunning, A. J. Trans.Faraday SOC.1970,66,500. (c) Belloni, L.;Drifford, M.; Turq, P. Chem. Phys. 1984,83, 147.
This model behaves well at high volume fractions; in particular, it reproduces the divergence of the pressure at 4 = 0.74 for dispersions of spherical particles and at 4 = 1for dispersions of deformable particles. It is not as good at low volume fractions, where the particles are far apart, because it no longer represents properly the shape of the unit cell. Attractive Forces. Finally, it may be necessary to include the effect of attractive forces. For Van der Waals attractions the interaction energy between particles of radius R and Hamaker constant H is4
In all the situations investigated, this attractive contribution was much smaller than the repulsive contribution from electrostatics; therefore it was neglected. It becomes significant only at high ionic strengths, comparable to the critical coagulation concentration of the dispersion, i.e. 0.08 M for the PS latex and > 1M for the other dispersions.
Results and Discussion PS Latex. Dispersions of the PS latex were compressed by osmotic stress at pH 7 and ionic strength 3 x M. The original dispersion, at a volume fraction Q = 0.06, was fluid and white; as the pressure was raised, it became iridescent and viscous; at a pressure of 1800Pa the volume fraction reached 0.214 and the dispersion turned to a soft solid. At this volume fraction the mean surface to surface separation is 31 nm; it is comparable with the Debye screening length (55 nm). Previous work on latex disper-
4016 Langmuir, Vol. 10, No. 11, 1994
Bonnet-Gonnet et al. 100000
T 8
10000
loo0
1
/ t a
I 100 -r
0.1
-I
~
0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
Figure 1. Osmotic pressures of PS latex dispersions (particle diameter 61 nm, surface charge density 0.14 dnma)in water M. Horizontal scale: volume at pH 7 and ionic strength 3 x fraction of latex in the dispersion. Vertical scale: osmotic pressure, log 10 scale. Squares: measured osmotic pressures and volume fractions. Full line: calculated values for a perfect gas of particles. Dashed line: values calculated through the CS equation for particleswithhard sphere repulsions at an effective radius Res = R K - ~ .
+
sions indicates that in such conditions the dispersions order as colloidalcrystals.22Thereforethe iridescent colors were the result of Bragg diffraction on reticular planes of particles within the crystallites. The experimental pressure us volume fiaction curve is shown in Figure 1. Over a large range ofvolume fractions (0.1-0.Q the pressures are between 1000and 10 000Pa. These pressures are 3 orders of magnitude higher than the pressures expected for a perfect gas at the same number concentration. This discrepancy may have two possible causes: an error in the number of objects contributing to the pressure or a highly nonideal behavior caused by the interactions between particles. For this latex, the number of particles is very well known, and we have found no other soluble species which could also contribute to the measured osmotic pressures. Therefore the discrepancy is more likely to come from interactions, which are expectedto be quite strong since they are almost unscreened. At very low volume fractions, the effect of such repulsions can be estimated through the CS formula (eq 3) applied to effective hard spheres with a radius equal to the particle radius plus the Debye screening length. The pressure of this hard-sphere gas is also shown in Figure 1;it rises extremely fast with volume fraction and crosses above the data beyond Q = 0.05;this is because these effective hard spheres use a large volume fraction; assuming that they cannot overlap overestimates the actual pressure. The pressures calculated for dispersions of charged spheres are shown in Figure 2. For this calculation, the parameters are the particle diameter, their surface charge, and the ionic strength of the reservoir. All these parameters are determined experimentally; therefore there are no adjustable parameters. At low volume fractions it is necessary to take into account the distribution ofdistances between interacting particles; this was done through the HNC equation used in the OCM model. The calculations (22) Monovoukas, Y.;Gast, A. 3. Colloid Interface Sci. 1989, 128,
633.
0
1
I
/
’ /
/
/
11‘’
/
0 I
0.1
I
0.2
0.3
0.4
0.5
0.6
0.7
Figure 2. Data as in Figure 1.Fits according to electrostatic models for interacting charged spheres. Dashed line: OCM model. Full line: PBC model.
converge for volume fractions up to Q = 0.2; the results are close to the data, but below. The validity of this procedure becomes questionable when the particles crystallize on a lattice. According to the iridescent colors of the samples, this occurs at Q = 0.05. Then it is more appropriate to use a lattice model such as the PBC model (eq 6). The calculated pressures reproduce the measured pressures accurately over the range ofvolume fractions 0.1-0.55 (Figure 2). At higher volume fractions the model underestimates the resistance to osmotic compression, because it does not describe the unit cell properly; as explained above, the calculated pressures diverge only at Q = 1instead of Q = 0.74. The good agreement between the measured pressures and those calculated by the PBC model indicates that the PS latex dispersions are well described as dispersions of charged spheres repelling through overlap of their ionic clouds. L1 Latex. Dispersionsofthe L1 latex were compressed by osmotic stress at 2 pH values: pH 3, where sulfate groups at the particle surfaces are ionized but carboxylates are not, and pH 9, where all acid groups are ionized. In addition, the ionic strength was varied between and 10-1 M, generating long-range or short-range screening of the surface charges. Unless specified otherwise, the temperature was below the glass transition temperature of the particle cores, and therefore the cores maintained a rigid spherical shape even under high pressure. The original latex dispersion at volume fraction 50% was white and fluid. Its osmotic pressure was between 1000and 10 000 Pa, dependingon ionic strength and pH: the pressure was higher at high pH (higher surface charge) and low ionic strength,as expected. When this dispersion was equilibrated at a lower pressure, it took up large amounts of water; at low pressures (100 Pa or less) the dispersion expanded so much that the dialysis bag was mechanicallystretched; then the pressure inside the bag was no longer equal to the pressure of the stressing solution. This artefact was eliminated by mounting the open bag at the tip of a glass capillary where the liquid dispersion could rise; then the pressure of the dispersion could be read directly from the level difference in an equilibrium against an aqueous solution with or without polymer (Figure 3). When the dispersion was equilibrated at a pressure higher than its original pressure, it lost water, the dispersion became unable to flow, and the resistance to
Osmotic Pressure of Latex Dispersions 10000
Langmuir, Vol. 10,No. 11, 1994 4017 1000000
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Figure 3. Swelling of latex dispersion L1 at pH 9 and ionic strength 7.5 x M. Diamonds: pressures and volume fractionsmeasured for dispersions placed in closed dialysis bags and immersed in dilute dextran solutions.Squares: same for dispersions in open dialysis bags. The measured pressure is the osmoticpressure ofthe dextran solution,outside the dialysis bag; it equals the osmotic pressure inside the bag minus a contribution from the stretching of the bag. compression became very high. The precise location of the crossover from fluid to solid was determined by the range of repulsions: a t low ionic strength and high pH the transition occurs when the counterion clouds overlap (separation = 2 Debye lengths); at high ionic strength and low pH, the transition occurs at the volume fraction where a hard sphere gas turns into a hard-sphere solid (4 = 0.49 to 4 = 0.55).19 Even with very high pressures (up to 4 atm) it was not possible to concentrate the dispersion to the maximum volume fraction of a face-centered cubicpacking (4 = 0.74); this resistance originated from the particle membranes, which are swollen with water and resist dehydration. Some typical compression curves are shown in Figure 4 (at pH 9) and Figure 5 (at pH 3). For all the curves there is a wide range of volume fractions (4 = 0.1 to 4 = 0.5) where the pressure varies smoothly and slowly; then, beyond 4 = 0.6,the pressure rises abruptly. We discuss these regions successively. Moderate Volume Fractions (4 = 0.1 to 4 = 0.5). In this regime the dispersion is expected to behave as a moderately concentrated gas, because the repulsions are short range (no more than 10 nm) and there is plenty of room available (the separation between particles is at least 50 nm). A calculation of pressure through the CS equation shows that the pressure from the particles is expected to vary between 1 and 500 Pa, depending on the effective volume fraction (Figures 4 and 5). The measured pressures are 3 orders of magnitude higher at pH 9, and 2 orders of magnitude higher a t pH 3. Therefore the pressure must originate from other species besides the particles. These species must be numerous, since they produce a high pressure, proportional to their number density, at a low volume fraction, since the slope of the compression curve is the same as that of a perfect gas, therefore of a small molar mass, but still large enough to be excluded by the pores of the dialysis bags. The molar mass of these species was determined as follows. The particles were separated by centrifugation, and the supernatant was examined through light scattering, size exclusionchromatography,and refractometry.
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Figure 4. Osmotic pressures of L1 latex dispersions at pH 9. Squares: measured osmoticpressuresand volume fractionsat ionic strength loe3M. Dashed line: values calculated through the CS equation for particles with hard-sphere repulsions at R,B = R K - ~ . Diamonds: measured values at ionic strength 10-l M. Semidashed line: corresponding values calculated through the CS equation. Full line: perfect gas of particles. Arrows indicate the location of the fluid-solid transition.
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Figure 6. As in Figure 4, but at pH 3. These studies indicate the presence of macromolecules with molar mass distribution centered at M = 3000 g/mol. Besides molar mass, some other characteristics of these macromolecules may be inferred from the shape of their compression curve. Indeed, the interactions between charged species in solution are reflected in their osmotic pressure, which may be calculated through the OCM model, as explained above. For this calculation, the number density of macromolecules must be known. In a first step, we take the number density obtained from the measured pressures at high dilutions. With this number density, all the compression curves can be reproduced through the OCM if the charge per macromolecule is 2 or 3 electron charges (Figures 6 and 7). Since these macromolecules must be made of the same monomers as the latex particles, it would then follow that each one would have 2 or 3 AA monomers, and about 30 BA monomers. However, macromolecules with this composition would not be soluble. On the other hand, macromolecules with more charges would give pressures which rise faster with concentration.
Bonnet-Gonnet et al.
4018 Langmuir, Vol. 10, No. 11, 1994 1000000 I I
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Figure 6. Comparison of osmotic pressures measured at low volume fractionsfor latex L l with pressures calculated for small macromolecules released by the particle membranes into the aqueous phase. Squares: measured osmotic pressures and volume fractions of the dispersion at pH 9 and ionic strength M. Full l i e : calculated osmotic pressures for a solution of macromolecules with molar mass M = 3000 g/mol, charge per macromolecule 2 = 3e, and number density 9000 times higher than that of latex particles. Dashed line: same for a perfect gas of uncharged macromolecules. I
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Figure 7. Same as Figure 6,but at pH 3 and ionic strength 10-1M. Fit (fullline) with M = 3000 g/mol,charge 2 = 2e, and number density 300 times higher than that of latex particles.
In order to use this information, we need to figure out how macromolecules which carry many charges (they are water soluble) do not repel very much (the pressure rises slowly with concentration). If these macromolecules carry sequences of charged AA monomers and sequences of uncharged BA monomers, then they may self-associate as other amphiphilic molecules do. At high concentrations, this association would be more extensive, thereby reducing the pressure. The origin of these soluble macromolecules can be understood by retracing the history of the dispersion. During the synthesis, AA monomers are copolymerized with hydrophobic monomers (mostly BA) to form a collection of macromolecules with different proportions of AA and BA monomers. Because the pH is kept low during
synthesis, all these macromolecules condense on the growing particles and form their membranes. After synthesis, the dispersion is washed, and all uncondensed monomers or oligomers are eliminated. However, the membranes may contain a fraction of low molar mass macromolecules which are soluble but physically trapped by other macromolecules; these species will be slowly released during long-term storage of the dispersion. This release will be accelerated by a dilution or by a rise in pH; indeed, from the compression curves (Figures 6 and 71, the number concentration of such soluble macromolecules appears to be much larger at pH 9 than at pH 3. The presence of such soluble macromolecules in the aqueous phase must have consequences for the properties of dilute latex dispersions. In addition to osmoticpressure (resistance to loss of water), vicosity and interfacial properties may also be modified. Thus, for latex dispersions with polymeric membranes, the properties of dilute dispersions are controlled by polymers released from the membranes. High Volume Fractions (4 > 0.5). The compression curves show a steep rise when the volume fraction reaches 4 = 0.5 (at high pH) or 4 = 0.6 (at low pH) (Figures 4 and 5). At the same point, the dispersions turn from fluid to solid. This abrupt change cannot result from the presence of released macromolecules, because their mass concentration in the aqueous phase is still small. At this point, however, the separations between particle surfaces are on the order of 12 nm. Moreover, the surface polymers are charged and thus may be extended into the aqueous phase; therefore the aqueous separation between surface polymers of neighboring particles may be quite short. Thus the steep rise in pressure corresponds to short-range interactions between charged polymers anchored on opposing surfaces. It may not be possible to predict the configurations of surface polymers extending into the aqueous phase separating particles: the simpler problem of polyelectrolytes grafted to a surfacehas been described theoretically,= but the present situation is complicated by the distribution of AA and BA sequences, which depends on the history of copolymerization. Therefore it may be useful to describe two extreme situations, where an accurate prediction of interactions and pressure can be made. (1)Assume that the surface polymers are so strongly anchored that all ionizable groups remain located at the particle surfaces. In this case the pressure may be calculated through the PB models. In conditions where water is a poor solvent for the surface polymers, e.g. at pH 3, the calculated pressure is of the same order of magnitude as the measured pressures (Figure 8). In conditions where water is a good solvent for the surface polymers, e.g. at pH 9 and low ionic strength,the measured pressures are much above the calculated pressures (Figure 91, indicating that the ionizable groups are not localized on the particle surfaces. (2) Assume that the charged groups of the sueace polymers are completely free to move in the aqueous solution between particles. Then the distribution of negative and positive charges in these aqueous layers is uniform, and their pressure can be calculated as that of a perfect gas of ions. The concentrations of ions of each type are calculated as above, by counting negative charges from the surface polymers, positive counterions, and passive salt in equilibrium from the reservoir (the PBC model is used beforehand to determine the concentration (23)(a)Van de Steeg, H. G. M.; Cohen Stuart, M. A.; de Kreizer,A.; Bijsterbosch,B. M. Langmuir lSS!2,8,2538. (b)Blaakmeer,J.;Btihmer, M. R.; Cohen Stuart,M. A.; Fleer, G. J. Macromolecules 1990,23,2301.
Osmotic Pressure of Latex Dispersions
Langmuir, Vol. 10, No. 11, 1994 4019
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Figure 8. Comparison of osmotic pressuresmeasured at high vnliima fvootinna f,-w lotnv T . l with
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interactingcharged particles. Squares: data tor tne dispersion M. Full line: calculated at pH 3 and ionic strength pressures from a model where all ionizable groupsof the particle membranes have been released into the aqueousphase. Dashed lines: calculated pressures from models where the ionizable groups are localized at the particle surfaces, and only counterions are into the aqueous phase; the models are PBC (semidashedline) and PB with planar geometry (dashed line). 1000000
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Figure 10. Comparison of osmotic pressures measured at high volume fractions for latex L1 with pressures measured for polyelectrolytes confined in the aqueous layers separatingthe particles. Squares: data for the dispersion at pH 9 and ionic M. Full line: osmotic pressuresof PAA solutions strength with the same amount of aqueous phase and an amount of PAA corresponding to the release of all the PAA from the particle membranes;the molar mass of PAAmacromoleculeswas chosen at 2 x lo5 g/mol. Diamonds: measured pressures and volume M. fractions of the dispersion at pH 3 and ionic strength Dashed line: osmotic pressuresof PAA solutionswith the same amount of aqueous phase.
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Figure 9. As in Figure 8, but at pH 9.
ofpassive salt). The resulting pressure is shownin Figures 8 and 9; it is the highest pressure that could be generated by the dispersion, since a reduction in volume would suppress a maximum number of degrees of freedom. At moderate volume fractions this pressure is much higher than the measured pressures; the discrepancy disappears in the limit of high volume fractions, where the confinement is so strongthat the distribution of charges is uniform anyway. As shown in Figures 8 and 9, the measured pressures are intermediate between these two extremes, indicating that the ionizable groups are neither completely localized nor completely free. As mentioned above, this is because the AA monomers are linked as macromolecules which
extend into the aqueous phase. Thus a better approximation would be to forget the anchoring and retain the polymer effect, by considering free PAA macromolecules confined in the aqueous phase separating the particles. This comparison is performed as follows. First, the concentration of passive salt in the aqueous phase is determined through a PBC calculation, as above. Then, the amount of PAA in the aqueous phase is chosen: we assumed that all PAA present in the membranes was released in the aqueous phase; accordingly the amount of PA4 was 2% of the amount of latex. Finally, the osmotic pressure of a PA4 solution at this concentration in an aqueous solution of matching pH and ionic strength was measured; the molar mass of PAA was chosen high enough so that its osmotic pressure was independent of molar mass (semidilute regime). The pressures of such PAA solutions are compared in Figure 10 with the pressures for latex dispersions. There is a range of volume fractions where the match is quite good, indicating that in this range the dominant interactions are between surface polymers extending into the aqueous phase separating the particles. At higher volume fractions, where there is a pressure "wall" in the compression curves of the latex dispersion, the curves from the PAA solutions continue to rise smoothly. This discrepancy is caused by the rigid shape of the latex particles, which forces the dense parts of the membranes to be in contact at a volume fraction where there is still a substantial amount of water in the dispersion. This comparison is instructive, because it determines precisely the extension ofthe three regimes for the osmotic pressures of such latex dispersions: moderate volume fractions, where the resistance to compression originates from soluble macromolecules released by the membranes; high volume fractions, where it originates from interac-
4020 Langmuir, Vol. 10, No. 11, 1994 1000000
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Figure 11. Comparison of osmotic pressures measured for latex L2 with pressures calculated for membrane polymers released or extended into the aqueousphase. Squares: data for the dispersion at pH 8.6 and ionic strength 10-1M. Dashed line: calculatedpressures for a solution of macromoleculeswith molar mass M = 3000 g/mol,charge per macromolecule2 = 2e, and number density 400 times higher than that of latex particles. Semidashed line: pressures of PAA solutions with an amount of PAA correspondingto the release into the aqueous phase of all the PAA sequences of the particle membranes. Arrow indicates the location of the fluid-solid transition. tions of macromolecules extending from the membranes into the aqueous phase; contact, where it results from the resistance to dehydration of the dense regions of the particle membranes. It is remarkable that, in all fluid states ofthe dispersion, the pressure ofthe dispersion is the pressure of an aqueous phase containing a certain amount of the polymers which make the particle surfaces. L2Latex. Dispersions of the L2 latex were compressed in the same conditions as those of L1 latex, but higher volume fractions were reached, up to 4 = 0.95. This was a consequence of the low glass transition temperature of the core polymers, which made the particles soft enough to deform under pressure. The compression curves are shown in Figures 11and 12. The fluid-solid transition is indicated as an arrow on each graph. At low ionic strength its location corresponds approximatelyto the volume fraction for which the counterion clouds overlap (separation of surfaces = 2 Debye lengths). At high ionic strength it corresponds to the fluid-solid transition of a hard-sphere gas. At volume fractions below the fluid-solid transition, the interactions between particles are largely screened, yet the pressure remains high even at large dilutions. As in the case of the L1 latex, this excess pressure originates from soluble macromolecules which are released by the particle membranes during storage of the dispersion. The concentration and interactions of these soluble macromolecules can be determined from a fit ofthe compression curve by the model pressure for the same type of soluble species as in the case of the L1 latex; this fit is shown in Figures 11 and 12. At volume fractions above the fluid-solid transition, the pressure rises much above the pressure from the soluble macromolecules (Figure 11).As in the case of the L1 latex, this rise results from repulsions between surface polymers which are extended into the aqueous layers separating particles. This contribution to the pressure can be approximated as the pressure from free PAA
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Figure 12. Data as in Figure 11but at ionic strength M. Dashed line: fit with macromoleculesof number density 12 000 times higher than that of latex particles. Semidashed line: osmotic pressures of PAA solutions with an amount of PAA corresponding to the release into the aqueous phase of all the PAA sequences of the particle membranes. macromolecules in this aqueous phase; this fit is also shown in Figures 11 and 12. The good match between pressures of the dispersion and pressures of the PAA solution, up to the highest volume fraction (6 = 0.951, indicates that the main contribution to osmotic pressure is the resistance to dehydration of the PAA sequences at the particle surfaces. A closer look at the compression curves in the high volume fraction range reveals that the pressure actually rises in two steps (Figure 11). We have observed this feature for all latex dispersions where the particle cores are soft,the membranesare made of hydrophilicpolymers, and the ionic strength is high. We have not observed it at low ionic strengths and high pH, where the pressure is dominatedby the soluble macromolecules released from the particle membranes. Hence, this feature reflects the direct interaction of soft particles with hydrophilic membranes. The first step occurs in a range of volume fractions where the particles must deform from spheres to dodecahedra; therefore it is likely that this step includes a mechanical resistance to deformation, while the second step must reflect only the resistance to dehydration of the swollen particle membranes in a fixed geometry. At low ionic strengths (Figure 12) this two-step behavior is harder to recognize because the resistance to deformationis small compared to the resistance to compression of highly extended PAA sequences. Conclusion In this work, the process of removing water from an aqueous dispersion has been studied. Initially, the dispersion was in a fluid state, where the particles were dispersed in a large volume of water. At the end of the transformation, the dispersion was in a solid state, where the particles had been forced into contact. At a macroscopic level, this transformation corresponds to a process where charged surfaces are displaced with respect to each other, which is a very general process in aqueous dispersions. It was expected that the behavior of different dispersions in this process should have some common, maybe universal, features. In order to characterize this behavior, different dispersions were concentrated through
Osmotic Pressure of Latex Dispersions osmotic stress, and their osmotic pressures were measured. It was found that the response of the dispersions to osmotic stress changed with the nature of the particles and with the content of aqueous phase in which they were dispersed. Two situations were examined: one where all ionizable groups were localized on the particle surfaces, and the aqueous phase contained only the counterions of these groups; in the other one the aqueous phase was loaded with polyelectrolytes which were either released from surfaces or extended from the surfaces. The first situation was obtained with “model” dispersions made from pure PS and stabilized with sulfate groups anchored to the particle surfaces. In this case, significant osmotic pressures (1000-10 000 Pa or0.01-0.1 atm)were obtained only at very low ionic strengths. These osmotic pressures were reproduced theoretically by a model which described the overlap ofthe counterion clouds surrounding the particles in concentrated dispersions. Since no adjustable parameters were used in this model, the agreement between measured and calculated pressures indicates that the overlap of ionic clouds is indeed the only cause of resistance to dehydration in these dispersions. The other situation was obtained with “industrial” dispersions made of S-BAcopolymers and stabilized with PAA sequences linked to the core polymers. In this case significant osmotic pressures were obtained even in the presence of passive salt in the aqueous phase. At low volume fractions of latex, these osmoticpressures originate from soluble macromolecules released from the particle membranes. These macromolecules are presumablyAABA copolymers which are normally associated with the membranes or with each other but dissolve in the aqueous phase at high dilution, high pH, and low ionic strength. At a high volume fraction of latex, other membrane polymers which extend from the particle surfaces into the aqueous phase become the dominant source of osmotic pressure, because the compression forces them to interpenetrate. This osmotic pressure was reproduced by the pressure of a semidilute PAA solution with a concentration corresponding to a complete swelling of the membrane PAA into the aqueous phase separating the particles. Since
Langmuir, Vol. 10, No. 11, 1994 4021 no adjustable parameters were used in this comparison, the agreement between PAA pressures and dispersion pressures indicates that the presence of polyelectrolyte sequencesin the aqueous layers is indeed the cause of the very strong resistance to dehydration of these dispersions. From these results, we can also draw two conclusions which have implications beyond the scope of this study: one that is specific to latex dispersions and another one which concerns the resistance to dehydration of aqueous dispersions. (1)Through osmotic pressure measurements, we have observed that latex particles are rarely “passive”objects. They can carry adsorbed amphiphilic species, including macromolecules, surfactants, and other molecules which are marginally water soluble; these molecules will be bound or released according to the solvent quality of the aqueous phase. They also carry grafted macromolecules, and again,depending on ionic conditions in the aqueous phase, these macromolecules will adsorb (collapse)on the particle surfaces or desorb and extend into the aqueous phase. Consequently, by modifying their environment, latex particles may yield dispersions that function differently from other particle dispersions. (2) For all dispersions, the osmotic pressure is basically proportional to the number of particles. With particles that are at least 100 nm in diameter, we have found that the osmotic pressure is never going to be very high, even with strong repulsions between them. A high resistance to dehydration can only be obtained if the dispersion has many more degrees of freedom that would be lost upon dehydration. This can be achieved if the aqueous phase contains large molecules or macromolecules which resist dehydration; we have seen that these molecules can be either dissolved or grafted. Because the number density of these macromolecules may be orders of magnitude higher than that of particles, the resistance to dehydration which they confer to the dispersion is also many orders of magnitude higher. Similarly, the ability of a dispersion to rehydrate spontaneously will be greatly increased if the continuous phase contains molecular or macromolecular species which have a strong hydration pressure.
