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Osmotic pressure of electrostatically stabilized magnetic liquids
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids 83 (1999) 203-215
,i
www.elsevier.nl/locate/molliq
Osmotic pressure of electrostatically stabilized magnetic liquids F. Cousin a and V.Cabuil b'* aCentre de Recherche sur la Mati6re Divis6e, Centre National de la Recherche Scientifique, 45071 Orl6ans Cedex 2, France ~aboratoire des Liquides Ioniques et Interfaces Charg6es, Equipe Collo~des Magn6tiques, Universit6 Paris 6, Case 63, 75252 Pads Cedex 05, France Osmotic pressure measurements are performed, either using a membrane-osmometer, or by the method of osmotic stress, in order to determine the equation of state of acidic and pH 7 magnetic liquids. Both systems are stabilized by electrostatic repulsions and the magnitude of interactions is estimated through the value of the second-virial coefficient. The effect of the ionic strength and of particle size is investigated. It appears that, even for high ionic strength, repulsions dominate the sum of interactions. © 1999 Elsevier Science B.V. All rights reserved. INTRODUCTION Colloidal dispersions exhibit phase transitions described with the same formalism as the one used to describe the phase transitions in molecular or atomic systems [1]. The colloid analogue of an atomic fluid's pressure is the osmotic pressure. The major advantage for the experimental study of the thermodynamic behavior of colloidal dispersions is that scattering_ techniques are available in order to investigate the nanoscopic structure of the system [2]. Osmotic pressure measurements can also be directly performed as a function of the volume t~action of colloid in order to get experimentally the equation of state of the system [3]. This latter can be calculated using statistical mechanics combined to an interparticle potential, which form depends on the nature of the colloid [4]. Indeed, colloidal dispersions are sometimes classified according to the form of the interparticle potential. For example, one speaks about an electrostatically stabilized colloid when interparticle repulsions are due to particles surface charges; dipolar fluids design dispersions of particles which interact through an anisotropic potential due to electric or magnetic dipolar Interactions, In a first approach, magnetic liquids (so-called "ferrofluids") can be classified in this last category, as they are constituted of magnetic monodomains (magnetic ferric oxide nanoparticles) dispersed in an oily or in an aqueous liquid carder [5]. In these systems, interparticle repulsions have either an electrostatic or a steric origin, according to the nature of the particles surface which can be coated by ionic species or by surfaetants. The possibility to " Correspondingauthor 0167-7322/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0167-7322(99) 00086-0
204
modify chemieaUy the surface of the particles and thus the nature of the repulsions combined to the possibility to monitor the dipolar magnetic interaction through the particles size makes magnetic fluids good candidates for an experimental study of the phase behavior of colloids [6,7]. Moreover, an important difference between atomic and colloidal systems concerns the latter's size distribution. In the case of magnetic fluids, the width of the size distribution can be monitored and "almost monodisperse" systems can be produced [8]. We shall present here a description of magnetic fluids as gas of particles. The equation of state of several aqueous systems will be experimentally determined and discussed with respect to relevant theories. 1. GENERAL CONSIDERATIONS ON I N T E R P A R T I C L E S I N T E R A C T I O N S We shall characterize a magnetic fluid by the nature and the intensity of the interparticle forces. Indeed, as soon as the interparticle potential is established, it is theoretically possible to deduce, using statistical mechanics or computer simulations, the equation of state of the fluid, and to predict its phase behavior. 1.1. Van der Waals interactions For a dispersion of spherical particles of diameter do, the reduced potential due to Van der Waals attractions can be written ."Uvdw(S)kT = -6--~- ~
1
+ s-5-+ In s-5--~-~_ 4 where A is the
Hamaker constant and s=2r/d0, for particles such as r >> do, r being the distance between the centers of two particles [9].
1.2. Magnetic dipolar interactions In a magnetic fluid, each particle is a magnetic monodomain, i.e. each particle has a permanent magnetic dipole which intensity ~ depends on the nature of the material_ constituting the particles, via its specific magnetization ms, and on the particles diameter do: = rest,d3/6. If/~l is the dipolar moment of the particle i and g is the vector joining the centers of the particles, the potential relative to dipolar interactions between two dipoles in zero magnetic field is anisotropic and is given by the formula : U~p(r) y and - 4-~ (2cos01 cosO2 -sinO I sinO 2 cos(p) where 0 i is the angle between /~l and kT (p the azimutal angle between both dipoles. ~t*m,2v~ The reduced parameter r = k---k' T~--r'(V, volume of a particle) characterizes the magnetic coupling between particles. If y/4rc<
48rt 2
magnetic coupling). If T/4r~>>l there are correlations between the dipoles and the potential becomes anisotropic. Nevertheless, it is possible to integrate over the space to obtain a mean value of the dipolar interaction : (u,,)=_r (high magnetic coupling) [10]. kT
2~
205 When a magnetic field is applied, the expansion of the potential around the most probable configuration leads to a mean potential decreasing as 1/r3.
1.3. Repulsions In magnetic fluids, according to the chemical nature of the particles surface, repulsions can be steric ones (particles coated by surfactants or polymers) or electrostatic ones (particles with surface charges). Steric stabilization is achieved when two particles cannot interpenetrate because of the steric hindrance imposed on the terminal chain segments. Here we shall only consider the case of electrostatic repulsions. Usually, the colloidal dispersions constituted of particles with surface charges, surrounded by their counterions and dispersed in a polar liquid, are described using the celebrated DLVO formalism [11] which takes into account the van der Waals attractions and the coulombic repulsions. The simpliest form for the repulsive potential is a Yukawa potential:
U,I (r) = Z'~2Ln- e-~(~-2°~r) ;with K | e2 X" 2~ u2 kT r (1 + ~ a s ) 2 = \go~kT " ~ ¢"cizi , )" L 8 is the Bjerrum length, Z ~ is the effective charge of the particle, a ~ the effective radius of a particle; 1/~: is the Debye length and c i is the concentration of the ion species i of charge z i. This expression can be used for low concentrations of particles and for weak interactions because the DLVO potential comes from the linearization of the Poisson-Boltzmann equation. The effective charge Z ~ involved in the DLVO formalism replace the structural charge Z (related to the nature of the surface and to the pH conditions) when the surface charge density of particles becomes so high that the counterions condense at the surface. The obtained effective charge can then be much lower than the structural charge Z [12]. When the concentration of colloids is higher in particles, the DLVO theory, based on infinite dilution, has to be replaced by a more suitable one : the "cell models" are analytical approaches which place macroions on a lattice and calculate ions density profile around the surface of the particles [13]. All these mean-field theories neglect ion-ion correlations in the system and are no longer valid when electrostatic interaction becomes very important. In such cases, another analytical approach, the so-called "primitive" treatment has to be considered : it takes into account the coulombic interactions between all ions (particles and salts) and the colloids and ions are considered as highly asymmetric electrolytes [14]. 1.4. Total interparticle potential In the low magnetic coupling case, an isotropic formalism can describe the system and allows to discuss the stability of the samples in terms of efficiency of the repulsions. Thus, when particles are eleetrostatically stabilized, the total interparticle potential can be written as the sum of the DLVO potential with the magnetic dipolar interactions (as a perturbation). Sometimes, even though far from the chemical reality of the systems, a Yukawa potential is useful to fit the experimental results (and compare the different behaviors). In the high magnetic coupling case, even in zero-field, dipolar interactions have to be taken into account [15].
206 When a magnetic field is applied, the diameter over which magnetic interactions become greater than isotropic ones is lowered, depending on the intensity of the applied magnetic field. 2. T H E E Q U A T I O N OF STATE OF M A G N E T I C FLUIDS Making the analogy with a gas, the osmotic pressure of a colloidal suspension as a function of the density p is the equation of state of the system and is oiten estimated using a virial expansion. For a dilute dispersion, the term in p3 (relative to three-body interactions) can be neglected and the equation of state of colloids can be written: H/kT = p + A2p 2 where A2 is the second virial coefficient, related to the total pair potential v(r) by: v(r) A 2 = j~ (1--e -~¥- )4~tr2dr • The sign o f A2 allows therefore to determine whether the interactions
are attractive or repulsive. Such a formalism assumes a symmetric and additive pair potential and can be used only for systems not far from ideality. For a more concentrated dispersion of colloidal particles, the relation between the interparticle potential and the pressure of the gas has to be established by means of the Omstein-Zemicke integral equation associated to a closure relation [4]. The equation of state of a magnetic fluid can be established either directly by making osmotic pressure measurements, either from the analysis of small angle scattering experiments. We shall describe here only the results obtained by osmotic pressure measurements. The SANS results will be the object of a further paper [16].
2.1. Description of magnetic fluids The magnetic fluids under consideration are dispersions of maghemite (y-Fe203) nanoparticles. They are usually polydisperse systems. Particles size distribution is described by a log-normal law
[6]:P(d)= ~ e x p ] _ d2~crd
r
1 _f i n d S _2 1/ , where d o is the mean diameter
L 2tr" ~, d° J J
and cr the standard deviation The particles can be obtained with a mean diameter ranging between 5 and 12 nm. A reasonable value for the Hamaker constant is A=10 "19 J [9]. The specific magnetization is ms = 3.1 105 A/m [ 17], which means a magnetic moment intensity ranging between 2. 10.2o A.m 2 and 2.8 10 "19 A.m 2 according to the particles diameter. When the diameter d o is smaller than 10nm and the volume fraction less than 10%, the system is in the low magnetic coupling cfse.
When uncoated, the particles have surface charges due to the specific acid-base behavior of the oxide surface [ 18] (cationic in acidic medium, anionic in alkaline medium). It is thus possible to vary the number of structural charges per particle modifying the value of the pH [5]. In the case of acidic magnetic fluids (pH 2), for maghemite particles of diameter 7 rim, the number of structural charges is estimated to be about 200 charges/particle [19]. A PB cell approach has allowed to estimate the number of effective charges which varies with the pH : 80 at pH 2, 25 at pH 3 and 8 at pH 4 [10].
207 Magnetic fluids, stable at pH 7, have also been synthesized in order to get biocompatible systems[20]. They are obtained by coating the ferric oxide particles by citrate species and they are anionic for all pH higher than 3. For these particles, the adsorption equilibrium of citrate species controls the number of structural charges which is about 240 charges / particle for 7 nm particles and 400 charges / particle for 9 nm particles, at pH 7 and at the plateau of the adsorption curve [21]. In this case the ionic strength of the medium is due to the unadsorbed citrate species. At pH 7, these species constitute a 3:t electrolyte (tri-sodium citrate), and ionic strength is indeed always rather high. In the present work, we shall assume that the formalism of the virial expansion is suitable for magnetic fluids at the condition that we consider low ionic strenth systems and low magnetic coupling conditions (particles diameter lower than 10 run and zero magnetic field).The sum of interparticle interactions for given experimental conditions will thus be estimated through the sign and value of A2. 2.2 Methods used to establish the equation of state of magnetic liquids Two differents techniques have been used in order to determine the osmotic pressure of magnetic fluids. 2.2.1 .Osmotic pressure measurements with a membrane osmometer Before the measurements, the ionic strength of the colloidal dispersion is adjusted to the desired value through dialysis. Equilibrium is followed by conductivity measurements and the bath is changed as many times as necessary to reach the required ionic strength. The measurements are performed using a membrane osmometer Knauer (model A0330). This apparatus is divided in two parts, one containing the colloidal dispersion and the other one (the so-called "reservoir") containing an aqueous solution at the same ionic strength. These two parts are separated with a membrane across which the ions are allowed to diffuse. The difference of pressure between the two parts is exactly the osmotic pressure due to colloidal particles in equilibrium and is measured by a sensitive pressure sensor. The apparatus is thermalized. It theoretically allows to measure osmotic pressures as small as 1 Pa but the pressure sensor is very sensitive to external conditions and systematic errors are made, especially for low pressures. 2.2.2. Osmotic stress This method allows to perform osmotic pressure measurements in a large range of osmotic pressures without modifying the chemical composition of the samples [22]. The sample is dialyzed against a bath which has the required ionic strength and pH. A Dextran polymer (Mw = 110000 g/tool) is added in the bath and imposes its own osmotic pressure which is neither dependent on temperature, nor dependent on ionic strength or pH. The osmotic pressure follows a phcnomenological law which is only dependent on the concentration of polymer [3]: loglO (ITdyn/cr~) = 1.826 + 1.715w °.297 (100w is the massic fraction of polymer in solution), and which is easily experimentally verified using the membrane osmometer. The reservoir is changed as many times as necessary to reach the equilibrium, which usually takes about three weeks. The new volume fraction of particles in the ferrofluid is then determined by chemical titration of iron or gravimetry.
208 Please note that this method is not very suitable for low osmotic pressures because the phenomenological osmotic law o f the stressing polymer appears to diverge at low volume fractions o f Dextran. Nevertheless, it allows to complete the measurements obtained with the membrane osmometer, especially in the range o f the high values of I-l. 2.3. Results Measurements have been performed for dispersions" of cationic particles (acidic magnetic fluid) and for dispersions o f citrate-coated particles (pH 7 magnetic fluid), in equilibrium with aqueous solutions o f several pH or ionic strength, For these colloids, stabilized by electrostatic repulsions, the main problem is to study the effect o f the ionic strength without modifying the surface charge density o f particles. That is indeed impossible. As a matter o f fact, in an acidic magnetic fluid, it as been established that the only one way to modify the ionic strength without inducing the flocculation of samples is to vary the nitric acid concentration [23]. This implies a modification o f the surface charge density and opposite effects on 17 as noted in [lO]. We shaLl discuss here the effect on I-I o f t w o parameters : the particles diameters and the
ionic strength. 2.3.1. Effect o f particles size 6000
1400
I = 5 1 0 "4 m o l / L
A
500O
1200
4000
1000
O. 30O0
4"
v
I = 10 "a m o l / L
.o
/~
/r
/"
800
0
600 200{]
4~
0
'1000
@
200
.o o
0 •
•
0
0
0 0
8O0 700
1
2
(%)
3
4
1 • (%) 2 200
I = 10"zmoi/L
175
600 .-,500
I •lO -I mogL
150
•
@
- ~ 125
m
~,~ 100 I~ 75
400
"-"
II,
•
~ 3oo 21111 lO0 0
/i
@
50
i
25 0 1
2
(%)
3
4
1
•
2
(%)
3
Figure 1. Osmotic pressure FI of acidic magnetic fluids as a function of the volume fraction • for size-sorted particles ~ 75A • 95A o 103A ) at different concentrationsof nitric acid. Measurements(dots) are performed using the membrane osmometer.The fine lines are only guides for the eyes.
209 Polydispersity has been for a long time a limitation t o the interpretation o f the physicochemical properties o f magnetic fluids. It is now possible to reduce the width o f the size distribution by a size-sorting process, and to obtain almost monodisperse samples [9]. The measurements concerning acidic magnetic fluid have been performed on such size-sorted samples. The characteristics o f the particles were do = 7fiA, ~ = 0.15; do = 95A, ~ -- 0.15 and do = 103A, o = 0.25 (d and o are deduced fxom the analysis of the shape of the magnetization curve [24]). Figure 1 plots the osmotic pressure of these monodisperse acidic magnetic fluids at different ionic strengths. The ionic strength are imposed by a nitric acid solution. 2.3.2 Effect of the ionic strength 5000
6OOO
/ / / @/
4000
75A.
/
~'3000
50O0
/
/
:
~ ,'
¢0 ~,~ 3000
Q.
2000
20OO
1000
. .#
0
2
3
(%)
¢
1
(b)
2
3
(%) I
5000
/ /
lO3A.
/
4000
/,e.
' /
1000
/
/
/,
600 <
400
..
I~
/
2000
#l
/
0
2
3
(d)
0
2
//
~"
,o <
~'
lOOO
200
1
/
~" 3000
I~. 800
(c)
/
0
1
1400 1200
/
1000
o,
(a)
9~k
4000
. ~ o~
4
6
8
10
. • .(%) a~ (%) Figure 2 : Effect ofiouic su-engmon me osmotic pressure of magnetic fluids. (a) Co) (c) : Acidic magnetic fluids, for three different particles size Measurements (dots) are performed using themembraueosmometer. #Iffi510-4mol/L #I=10-3moFL o I = 10.-2mol/I., • I = 10-1mol/L (d) : pH 7 magnetic fluids (polydisperse samples, I = 6[NaCit]). Measurements (dots) are performed usin~ the membrane-osmometerfor low pressures and by osmotic stress for high pressures.The fine lines are only guides for the eyes. * [Cit] = 10-3mol/L • [Cit]= 5.10-3mol/L <>[Cit]= 10-2mol/L
Figure 2 focus on the effect o f the ionic s~ength for acidic magnetic fluids (three particles sizes) and for citmted=coated particles, dispersed in water at pH 7 (do = 85A, a - 0.40). For these latter, the ionic strength is due to unadsorbed citrate species in solution. Samples for which [Cit] = 5 10"4 mol/L have been impossible to study because the absorption equilibrium of citrate ions on particles is shifted [21] for low concentration of citrate in solution. On the
210
other hand, for high ionic strength such as [Cit] = 10q mol/L, the uncertainty on the osmotic pressures measurements is too important to give significant results concerning the destabilization of low volume fraction samples. The osmotic pressure values reported on figure 2 (d) have been determined with the membrane osmometer for low volume fraction and by osmotic stress for high volume fraction, at three differents ionic strengths. 3 DISCUSSION The relevant result when characterizing a colloidal dispersion is the plot 17/O as a function of O in order to get its osmotic compressibility and thus an estimation of the strength of interparticle interactions. The initial slope of such a curve is proportional to Az. The limit ffI/O)¢--~0) allows theoretically to get the molecular weight of the colloid m but here the experimental uncertainties on the low pressures does not allow to derive this molecular weight. Please note that this error has no effect on the calculation of A2, which is the slope of the plot. • [Cal = 10-3 mol/L Q. i ! 800
e [CIt] = 5.10-3 moR. i o [Ot] = ~0-2 rnoVL i
) */
~,oo
0 ¸
; 4¢ J
/ 200
~
2
4
_- J 6 / . O ~ -~¸
6
8
10
(%) Figure 3 : Evidence of repulsions in a pH 7 dispersion of magneticparticles (do = 85A, ~ = 0.40). The fine lines are only guides for the eyes. For all the studied samples, the compressibility is linear until a given value of ~, which depends on the nature of the samples (acidic or pH 7 magnetic fluid) and on the ionic strength. In the case of pH 7 magnetic fluids (citrated-coated particles), it appears (figure 3) that, even for high ionic strengths ([Cit]=5 l0 "3 mol/L and [Cit]=10"z reel/L) the interactions in the system remain repulsive. It has been verified that a change of ionic strength does not modify the surface charge density provided the concentration of unadsorbed citrate ions is not lower than a given concentration of citrate in solution. It is surprising to have electrostatic repulsions for so high ionic strength (for [Cit]=10"2 moFL, 1--6 10.2 moVL) according to the DLVO formalism [25]. The same results was obtained by Dubois and al [16], from analysis of SANS spectra, and the authors invoke a so-called "electro-steric" repulsions, the absorbed citrate species inducing an additive repulsion of steric nature. They propose as an equation of state the following formula that summaries their results at varying I and T :
211
H = • =-~--~[1 + 16.5.(1 - (4.4 - 0.045T)I)] v, k-¥ where Vw is the volume of the particle, T the temperature and I the ionic trength. We compare on figure 4 our osmotic measu~ments with the osmotic pressure deduced from such a =
32q ~
formula. We estimated Vw taking into account the polydispersity, though (r 3) = {ro3)e¢~-)). 4O0
.-.¢~ 300
/
200
10o
0 1
2 •
3
(%)
Figure 4 : Osmotic pressure of pH 7 magnetic fluids at several ionic strengths • [Cit] = 10-3 mol/I., : osmotic pressure measured - [CitI = 10-3 ~ I / L : state equation f:mm~f [16] • [Cit] = 5 10-3 mol/L : osmotic pressure measured . . . . [Cit] = 5 10-3 r~l/L : state-equation from ref[16] o [Cit] = 10-2 n'~l/L : osmotic pressure measured . . . . . . . [Cit] = 10-2 mol/L : state-equation ffomref [16]
The experimental data are in reasonable agreement with the formula for the highest ionic strength, corresponding to the conditions used by Dubois but not for [Cit] = 10-3 mol/L. Results are slightly lower than the calculated ones at very low volume fraction, hut this can be due to a polydispersity effect. The equation has been established for size-sorted particles although the samples studied here are polydisperse and contain larger particles, which are more sensitive to Van der Waals and dipolar interactions and lead to a reduced osmotic pressure, even with an equivalent value of Vw. Of course, a decrease of ionic strength, i.e. a n increase of Debye length, stabilizes the ferrofluid. The second-osmotic virial coefficient is much higher for [Cit] = 10-3 mol/L than for [Cit] = 5 10-3 mol/L. For low ionic strength the 1-I/~ curve diverges sharply from a linear behavior at low volume fraction, because the repulsions occur on a large range and n-body interactions (with n > 3) are no longer negligible. Assuming that repulsions are sufficiently high at such ionic strength to approximate the interparticle potential as a pseudo hard-core one, we compare on figure 6 the measured equation of state with the Camahan-Stirling [26], which gives the exact values for the virial coefficients until A4 for a hard-core potential (we renormalize here the radius of the particles by an effective one, sum of the radius of particles and of the Debye length). The effective volume fractions calculated in the range of our experimental data are low enough to prevent forced interpenetration of renormalized particles. I-I 1+ ~ r + ~ 2 _ ~ 3
(p)kT
212
Figure 5 indicates that our system is more repulsive than an hard-core one. As a matter of fact, repulsions begin to increase at lower volume fraction, indicating the presence of additional repulsions on a range larger than a Debye length. 500
t
~,~.~,300
•
I~ 2OO 100 e •
2
4
6
8
10
(%) Figure 5: Osmotic pressure measurements of pH 7 magnetic fluids for [NaCit] = l0 "3 mol/L. The dots correspond to measurements and the line to the equation of state according to Camahan-Stirling formalism.
Figure 6 plots i-I/O as a function of • for acidic particles of different sizes, at different ionic strengths. For pH 2, pH 3 and pH 3.3, corresponding respectively to I = 10"2 moFL, I = 10"3 mol/L and I = 5 10.4 mol/L, the system is repulsive, even for biggest particles. At pH 1, experimental errors are non-negligible, and the calculation of A2 is not meaningfid. For the pH values higher than 1, it can be noted on figure 6 that the I-I/O curves are all linear, meaning that the equation of state of an acidic magnetic fluid can be reduced to a second-virial development. Their slope when divided by the volume of the particles, is proportional to A2 by a factor (the same for all measurements) that is neither dependent on ionic strength neither nor on particle size. Figure 7 plots A2 as a function of the pH for the three particles sizes. This diagram shows first of all that the sum of interactions becomes less and less repulsive when the particle size increases, whatever the ionic strength. It can be explained by the volume-dependence of the Van der Waals interactions and of the dipolar interactions which begin to have a significant magnitude for a particles diameter of 10 rim. The electrostatic repulsions seem also to be size-dependent: for the two smallest particles, the system becomes more repulsive when the pH increases indicating that repulsions increase because of the ionic strength effects. On the contrary, for the biggest particles, the system becomes less repulsive for pH > 3. This result on large particles is in agreement with Menager and al [I 0] who observe a maximum in the magnitude of electrostatic repulsions for acidic polydisperse samples at pH 3. This maximum is due to a competition between the pH induced reduction of the surface charge density and the increase of the Debye length. Figure 7 plots A2 as a function of the pH for the three particles sizes. This diagram shows first of all that the sum of interactions becomes less and less repulsive when the particle size increases, whatever the ionic strength. It can be explained by the volume-dependence of the Van der Waals interactions and of the dipolar interactions which begin to have a significant magnitude for a particles diameter of 10 nm.
213 1500
2500
j
2000
12.
1200
,sA / f
900
1500
v
1000
600
500
300
0
0 0
1
2
,
•
.
0
1
2
3
,~ (%)
3
1
2
,~ (%)
3
4
600 500
~,
400
12. "'"
300
200 100 0
4
(%) Figure 6 : Interactionsin a dispersionof acidicparticles.The slopesallo~vscalculationsof A2 4,pH3.3 4,pH3 o p H 2
The electrostatic repulsions seem also to be size-dependent: for the two smallest particles, the system becomes more repulsive when the pH increases indicating that repulsions increase because of the ionic strength effects. On the contrary, for the biggest particles, the system becomes less repulsive for pH > 3. This result on large particles is in agreement with Menager and al [10] who observe a maximum in the magnitude of electrostatic repulsions for acidic polydisperse samples at pH 3. This maximum is due to a competition between the pH induced reduction of the surface charge density and the increase of the Debye length. The size-dependence of electrostatic repulsions cannot be interpreted invoking the term e~(2a)
depending on the radius in the Yukawa potential ( ~ )
because, at large Debye length,
e~¢z°' --> l) the potential becomes independent of the particles radius. 0+~)' Therefore, this size-dependence of repulsions has to be related to a size dependence of the number of the effective charges. According to [12], for highly charged particles, the effective charge is particle size dependent and decreases when the radius of particles increases. For the value of pH under consideration (pH -> 3), the number of structural charge is not at its maximum value, but it is reasonable to consider that the size-dependence of effective charges will keep the same tendency [10]. Moreover, it can be assumed that the density of structural (r.a --> 0 and
214
charges decreases when the particles radius increases through a size-dependence of the acid constant Ka for the ferric hydroxide groups at the surface of particles [27]. 0,0015
o,oo
i- _1o
i
0,0005
0 1,5
2
2,5
pH
3
3,5
Figure 7 : A2 as a function of the pH for the dispersions of cationic particles of different diameters.
CONCLUSION The first result of this study is to underline the stability of aqueous magnetic fluids even when the ionic strength is important. In the case of pH 7 magnetic fluids, it is in agreement with SANS-experiments, but at the present time the high stability of these magnetic liquids is not completely understood, except invoking additional repulsion terms due to steric hindrance. The study of acidic magnetic fluids brings out one of the first systematic experimental results on the effect of particles size on osmotic pressure of colloidal systems. At a given pH, it underlines that an increase of particles size lead to a destabilization of magnetic fluids by increasing Van der Waals and dipolar attractions but also to a decrease of electrostatic repulsions through a decrease of the effective charge of the particles. The next step of the study is to establish the equation of state of such liquids a magnetic field. Acknowledgments : We would like to thank C. Carvalho for her experiments on acidic magnetic fluids and P. Levitz for his help in the realization of the osmotic stress. We are indebted to E. Dubois, C. Menager and P. Levitz for helpful discussions. REFERENCES 1. P.N. Pusey, Colloidal suspensions, in Liquids, Freezing and Glass Transitions, J.P. Hansen, D. Levesque and J.Zinn-Justin (eds), North-Holland (Amsterdam) (1991) 765-942. 2. O. GlaRer, Small-angle scaterring and light scaterdng, in Neutron, X-Ray and Light Scattering, North-Holland, Delta Series, Elsevier Science Publisher (I 991) 33. 3. A. Mourehid, A. Delville, J. Lambard, E. I_~collier and P. Levitz, Langmuir 11(1995) 1942. 4. J.P.Hansen and I.R. Mac Donald, Theory of simple liquids, Academic, New York, (1992). 5. J.C. Bacri, R. Perzynski, D. Salin, V. Cabuil and R. Massart, J.Magn. Magn. Mat., 85 (1990) 27.
215 6. J.-C. Bacri, R. Perzynski, D. Salin, V. Cabuil, R. Massart, J. Colloid Interface Sci. 132(1) (1988) 43. 7. E. Dubois, V. Cabuil, F. Bou6, J.C. Bacri, R. Perzynski, Prog. Colloid Polymer Sci., 104 (1997) 173. 8. Massart R, Dubois E., Cabull V., Hasmonay E., J.Magn Magn Mat 149 (1995) 1. S. Lefebure, E. Dubois,V. Cabuil, S. Neveu and R. Massart, J. Mater. Res., 13 (1998) 10. 9. P.C. Sholten, Colloid Chemistry of magnetic fluids, in Thermomechanics of the magnetic fluids, Berkovsky Ed., Hemisphere Publishing Corporation, Bristol, PA, 1 (1978). 10. C. M~nager, L. Belloni, V. Cabuil, M. Dubois, T. Gulik-Krzywicki and T. Zemb, Langmuir, 12 (14) (1996) 3519. 11. E. J. W. Verwey and J.T.G. Overbeek, Theory of the stability of Lyophobic colloids, Elsevier, Amsterdam (1948). 12. L. Belloni, Interacting monodisperse and polydisperse spheres, in Neutron, X-Ray and Light Scattering, North-Holland, Delta Series, Elsevier Science Publisher (1991) 135. 13. S. Alexander, P.M. Chaikin, P. Grant, G.J. Morales, P. Pincus and D. Hone, J. Chem. Phys., 80 (1984) 5776. 14. H. l_,5wen,J.P. Hansen and P.A. Madden, J. Chem. Phys. 98 (4) (1993) 3275 15. P.G. de Germes and P.A. Pincus, Phys. Kondens. Mat., 11(1970) 189. K. Sano and M. Doi, J. of the Physical Society of Japan, 52 (8) (1983) 2810. M.J. Stevens and G.S. Grest, P.R.L., 72 (23) (1994) 3686. 16. E. Dubois, V. Cabuil, F. Bou6, R. Perzynski, submitted to J. Chem. Phys (1999). 17. J.C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salin, J. Magn. Magn. Mat., 62 (1986) 36. 18. T. Hiemstra, W.H. Van Riemsdijk, G.H. Bolt, J. Coll. Interface. Sci. 133(1) (1989) 91. 19. T. Bouchami, Thesis, Paris VI University (1990). 20. J.C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salin, J.-N. Pons and J. Roger, Springer Verlag Berlin, (1986) 59. 21. E. Dubois, Thesis, Pads VI University (1997). 22. V.A. Parsegian, N. Fuller and R.P. Rand., Proc. Natl. Acad. Sci., 76(6) (1979) 2750. 23. V. Cabuil, E. Dubois, S. Neveu, J.-C. Bacri, E. Hasmonay and R. Perzynski,Progr. Colloid Polym. Sci. 98 (1995) 23. 24. S. Foner and E. J. Macniff, The Rev. Scien. Inst. 39 (1968) 171. 25. J.Israelachvili, Intermolecular and Surface Forces, Academic Press (1992). 26. J. Lyklema, Fundations of interface and colloide science, academic press, (1991). 27. J. Sonnenfed, J. Colloid Interface Sci.,155 (1993) 191.
LIQUIDS ELSEVIER
Journal of Molecular Liquids 83 (1999) 203-215
,i
www.elsevier.nl/locate/molliq
Osmotic pressure of electrostatically stabilized magnetic liquids F. Cousin a and V.Cabuil b'* aCentre de Recherche sur la Mati6re Divis6e, Centre National de la Recherche Scientifique, 45071 Orl6ans Cedex 2, France ~aboratoire des Liquides Ioniques et Interfaces Charg6es, Equipe Collo~des Magn6tiques, Universit6 Paris 6, Case 63, 75252 Pads Cedex 05, France Osmotic pressure measurements are performed, either using a membrane-osmometer, or by the method of osmotic stress, in order to determine the equation of state of acidic and pH 7 magnetic liquids. Both systems are stabilized by electrostatic repulsions and the magnitude of interactions is estimated through the value of the second-virial coefficient. The effect of the ionic strength and of particle size is investigated. It appears that, even for high ionic strength, repulsions dominate the sum of interactions. © 1999 Elsevier Science B.V. All rights reserved. INTRODUCTION Colloidal dispersions exhibit phase transitions described with the same formalism as the one used to describe the phase transitions in molecular or atomic systems [1]. The colloid analogue of an atomic fluid's pressure is the osmotic pressure. The major advantage for the experimental study of the thermodynamic behavior of colloidal dispersions is that scattering_ techniques are available in order to investigate the nanoscopic structure of the system [2]. Osmotic pressure measurements can also be directly performed as a function of the volume t~action of colloid in order to get experimentally the equation of state of the system [3]. This latter can be calculated using statistical mechanics combined to an interparticle potential, which form depends on the nature of the colloid [4]. Indeed, colloidal dispersions are sometimes classified according to the form of the interparticle potential. For example, one speaks about an electrostatically stabilized colloid when interparticle repulsions are due to particles surface charges; dipolar fluids design dispersions of particles which interact through an anisotropic potential due to electric or magnetic dipolar Interactions, In a first approach, magnetic liquids (so-called "ferrofluids") can be classified in this last category, as they are constituted of magnetic monodomains (magnetic ferric oxide nanoparticles) dispersed in an oily or in an aqueous liquid carder [5]. In these systems, interparticle repulsions have either an electrostatic or a steric origin, according to the nature of the particles surface which can be coated by ionic species or by surfaetants. The possibility to " Correspondingauthor 0167-7322/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0167-7322(99) 00086-0
204
modify chemieaUy the surface of the particles and thus the nature of the repulsions combined to the possibility to monitor the dipolar magnetic interaction through the particles size makes magnetic fluids good candidates for an experimental study of the phase behavior of colloids [6,7]. Moreover, an important difference between atomic and colloidal systems concerns the latter's size distribution. In the case of magnetic fluids, the width of the size distribution can be monitored and "almost monodisperse" systems can be produced [8]. We shall present here a description of magnetic fluids as gas of particles. The equation of state of several aqueous systems will be experimentally determined and discussed with respect to relevant theories. 1. GENERAL CONSIDERATIONS ON I N T E R P A R T I C L E S I N T E R A C T I O N S We shall characterize a magnetic fluid by the nature and the intensity of the interparticle forces. Indeed, as soon as the interparticle potential is established, it is theoretically possible to deduce, using statistical mechanics or computer simulations, the equation of state of the fluid, and to predict its phase behavior. 1.1. Van der Waals interactions For a dispersion of spherical particles of diameter do, the reduced potential due to Van der Waals attractions can be written ."Uvdw(S)kT = -6--~- ~
1
+ s-5-+ In s-5--~-~_ 4 where A is the
Hamaker constant and s=2r/d0, for particles such as r >> do, r being the distance between the centers of two particles [9].
1.2. Magnetic dipolar interactions In a magnetic fluid, each particle is a magnetic monodomain, i.e. each particle has a permanent magnetic dipole which intensity ~ depends on the nature of the material_ constituting the particles, via its specific magnetization ms, and on the particles diameter do: = rest,d3/6. If/~l is the dipolar moment of the particle i and g is the vector joining the centers of the particles, the potential relative to dipolar interactions between two dipoles in zero magnetic field is anisotropic and is given by the formula : U~p(r) y and - 4-~ (2cos01 cosO2 -sinO I sinO 2 cos(p) where 0 i is the angle between /~l and kT (p the azimutal angle between both dipoles. ~t*m,2v~ The reduced parameter r = k---k' T~--r'(V, volume of a particle) characterizes the magnetic coupling between particles. If y/4rc<
48rt 2
magnetic coupling). If T/4r~>>l there are correlations between the dipoles and the potential becomes anisotropic. Nevertheless, it is possible to integrate over the space to obtain a mean value of the dipolar interaction : (u,,)=_r (high magnetic coupling) [10]. kT
2~
205 When a magnetic field is applied, the expansion of the potential around the most probable configuration leads to a mean potential decreasing as 1/r3.
1.3. Repulsions In magnetic fluids, according to the chemical nature of the particles surface, repulsions can be steric ones (particles coated by surfactants or polymers) or electrostatic ones (particles with surface charges). Steric stabilization is achieved when two particles cannot interpenetrate because of the steric hindrance imposed on the terminal chain segments. Here we shall only consider the case of electrostatic repulsions. Usually, the colloidal dispersions constituted of particles with surface charges, surrounded by their counterions and dispersed in a polar liquid, are described using the celebrated DLVO formalism [11] which takes into account the van der Waals attractions and the coulombic repulsions. The simpliest form for the repulsive potential is a Yukawa potential:
U,I (r) = Z'~2Ln- e-~(~-2°~r) ;with K | e2 X" 2~ u2 kT r (1 + ~ a s ) 2 = \go~kT " ~ ¢"cizi , )" L 8 is the Bjerrum length, Z ~ is the effective charge of the particle, a ~ the effective radius of a particle; 1/~: is the Debye length and c i is the concentration of the ion species i of charge z i. This expression can be used for low concentrations of particles and for weak interactions because the DLVO potential comes from the linearization of the Poisson-Boltzmann equation. The effective charge Z ~ involved in the DLVO formalism replace the structural charge Z (related to the nature of the surface and to the pH conditions) when the surface charge density of particles becomes so high that the counterions condense at the surface. The obtained effective charge can then be much lower than the structural charge Z [12]. When the concentration of colloids is higher in particles, the DLVO theory, based on infinite dilution, has to be replaced by a more suitable one : the "cell models" are analytical approaches which place macroions on a lattice and calculate ions density profile around the surface of the particles [13]. All these mean-field theories neglect ion-ion correlations in the system and are no longer valid when electrostatic interaction becomes very important. In such cases, another analytical approach, the so-called "primitive" treatment has to be considered : it takes into account the coulombic interactions between all ions (particles and salts) and the colloids and ions are considered as highly asymmetric electrolytes [14]. 1.4. Total interparticle potential In the low magnetic coupling case, an isotropic formalism can describe the system and allows to discuss the stability of the samples in terms of efficiency of the repulsions. Thus, when particles are eleetrostatically stabilized, the total interparticle potential can be written as the sum of the DLVO potential with the magnetic dipolar interactions (as a perturbation). Sometimes, even though far from the chemical reality of the systems, a Yukawa potential is useful to fit the experimental results (and compare the different behaviors). In the high magnetic coupling case, even in zero-field, dipolar interactions have to be taken into account [15].
206 When a magnetic field is applied, the diameter over which magnetic interactions become greater than isotropic ones is lowered, depending on the intensity of the applied magnetic field. 2. T H E E Q U A T I O N OF STATE OF M A G N E T I C FLUIDS Making the analogy with a gas, the osmotic pressure of a colloidal suspension as a function of the density p is the equation of state of the system and is oiten estimated using a virial expansion. For a dilute dispersion, the term in p3 (relative to three-body interactions) can be neglected and the equation of state of colloids can be written: H/kT = p + A2p 2 where A2 is the second virial coefficient, related to the total pair potential v(r) by: v(r) A 2 = j~ (1--e -~¥- )4~tr2dr • The sign o f A2 allows therefore to determine whether the interactions
are attractive or repulsive. Such a formalism assumes a symmetric and additive pair potential and can be used only for systems not far from ideality. For a more concentrated dispersion of colloidal particles, the relation between the interparticle potential and the pressure of the gas has to be established by means of the Omstein-Zemicke integral equation associated to a closure relation [4]. The equation of state of a magnetic fluid can be established either directly by making osmotic pressure measurements, either from the analysis of small angle scattering experiments. We shall describe here only the results obtained by osmotic pressure measurements. The SANS results will be the object of a further paper [16].
2.1. Description of magnetic fluids The magnetic fluids under consideration are dispersions of maghemite (y-Fe203) nanoparticles. They are usually polydisperse systems. Particles size distribution is described by a log-normal law
[6]:P(d)= ~ e x p ] _ d2~crd
r
1 _f i n d S _2 1/ , where d o is the mean diameter
L 2tr" ~, d° J J
and cr the standard deviation The particles can be obtained with a mean diameter ranging between 5 and 12 nm. A reasonable value for the Hamaker constant is A=10 "19 J [9]. The specific magnetization is ms = 3.1 105 A/m [ 17], which means a magnetic moment intensity ranging between 2. 10.2o A.m 2 and 2.8 10 "19 A.m 2 according to the particles diameter. When the diameter d o is smaller than 10nm and the volume fraction less than 10%, the system is in the low magnetic coupling cfse.
When uncoated, the particles have surface charges due to the specific acid-base behavior of the oxide surface [ 18] (cationic in acidic medium, anionic in alkaline medium). It is thus possible to vary the number of structural charges per particle modifying the value of the pH [5]. In the case of acidic magnetic fluids (pH 2), for maghemite particles of diameter 7 rim, the number of structural charges is estimated to be about 200 charges/particle [19]. A PB cell approach has allowed to estimate the number of effective charges which varies with the pH : 80 at pH 2, 25 at pH 3 and 8 at pH 4 [10].
207 Magnetic fluids, stable at pH 7, have also been synthesized in order to get biocompatible systems[20]. They are obtained by coating the ferric oxide particles by citrate species and they are anionic for all pH higher than 3. For these particles, the adsorption equilibrium of citrate species controls the number of structural charges which is about 240 charges / particle for 7 nm particles and 400 charges / particle for 9 nm particles, at pH 7 and at the plateau of the adsorption curve [21]. In this case the ionic strength of the medium is due to the unadsorbed citrate species. At pH 7, these species constitute a 3:t electrolyte (tri-sodium citrate), and ionic strength is indeed always rather high. In the present work, we shall assume that the formalism of the virial expansion is suitable for magnetic fluids at the condition that we consider low ionic strenth systems and low magnetic coupling conditions (particles diameter lower than 10 run and zero magnetic field).The sum of interparticle interactions for given experimental conditions will thus be estimated through the sign and value of A2. 2.2 Methods used to establish the equation of state of magnetic liquids Two differents techniques have been used in order to determine the osmotic pressure of magnetic fluids. 2.2.1 .Osmotic pressure measurements with a membrane osmometer Before the measurements, the ionic strength of the colloidal dispersion is adjusted to the desired value through dialysis. Equilibrium is followed by conductivity measurements and the bath is changed as many times as necessary to reach the required ionic strength. The measurements are performed using a membrane osmometer Knauer (model A0330). This apparatus is divided in two parts, one containing the colloidal dispersion and the other one (the so-called "reservoir") containing an aqueous solution at the same ionic strength. These two parts are separated with a membrane across which the ions are allowed to diffuse. The difference of pressure between the two parts is exactly the osmotic pressure due to colloidal particles in equilibrium and is measured by a sensitive pressure sensor. The apparatus is thermalized. It theoretically allows to measure osmotic pressures as small as 1 Pa but the pressure sensor is very sensitive to external conditions and systematic errors are made, especially for low pressures. 2.2.2. Osmotic stress This method allows to perform osmotic pressure measurements in a large range of osmotic pressures without modifying the chemical composition of the samples [22]. The sample is dialyzed against a bath which has the required ionic strength and pH. A Dextran polymer (Mw = 110000 g/tool) is added in the bath and imposes its own osmotic pressure which is neither dependent on temperature, nor dependent on ionic strength or pH. The osmotic pressure follows a phcnomenological law which is only dependent on the concentration of polymer [3]: loglO (ITdyn/cr~) = 1.826 + 1.715w °.297 (100w is the massic fraction of polymer in solution), and which is easily experimentally verified using the membrane osmometer. The reservoir is changed as many times as necessary to reach the equilibrium, which usually takes about three weeks. The new volume fraction of particles in the ferrofluid is then determined by chemical titration of iron or gravimetry.
208 Please note that this method is not very suitable for low osmotic pressures because the phenomenological osmotic law o f the stressing polymer appears to diverge at low volume fractions o f Dextran. Nevertheless, it allows to complete the measurements obtained with the membrane osmometer, especially in the range o f the high values of I-l. 2.3. Results Measurements have been performed for dispersions" of cationic particles (acidic magnetic fluid) and for dispersions o f citrate-coated particles (pH 7 magnetic fluid), in equilibrium with aqueous solutions o f several pH or ionic strength, For these colloids, stabilized by electrostatic repulsions, the main problem is to study the effect o f the ionic strength without modifying the surface charge density o f particles. That is indeed impossible. As a matter o f fact, in an acidic magnetic fluid, it as been established that the only one way to modify the ionic strength without inducing the flocculation of samples is to vary the nitric acid concentration [23]. This implies a modification o f the surface charge density and opposite effects on 17 as noted in [lO]. We shaLl discuss here the effect on I-I o f t w o parameters : the particles diameters and the
ionic strength. 2.3.1. Effect o f particles size 6000
1400
I = 5 1 0 "4 m o l / L
A
500O
1200
4000
1000
O. 30O0
4"
v
I = 10 "a m o l / L
.o
/~
/r
/"
800
0
600 200{]
4~
0
'1000
@
200
.o o
0 •
•
0
0
0 0
8O0 700
1
2
(%)
3
4
1 • (%) 2 200
I = 10"zmoi/L
175
600 .-,500
I •lO -I mogL
150
•
@
- ~ 125
m
~,~ 100 I~ 75
400
"-"
II,
•
~ 3oo 21111 lO0 0
/i
@
50
i
25 0 1
2
(%)
3
4
1
•
2
(%)
3
Figure 1. Osmotic pressure FI of acidic magnetic fluids as a function of the volume fraction • for size-sorted particles ~ 75A • 95A o 103A ) at different concentrationsof nitric acid. Measurements(dots) are performed using the membrane osmometer.The fine lines are only guides for the eyes.
209 Polydispersity has been for a long time a limitation t o the interpretation o f the physicochemical properties o f magnetic fluids. It is now possible to reduce the width o f the size distribution by a size-sorting process, and to obtain almost monodisperse samples [9]. The measurements concerning acidic magnetic fluid have been performed on such size-sorted samples. The characteristics o f the particles were do = 7fiA, ~ = 0.15; do = 95A, ~ -- 0.15 and do = 103A, o = 0.25 (d and o are deduced fxom the analysis of the shape of the magnetization curve [24]). Figure 1 plots the osmotic pressure of these monodisperse acidic magnetic fluids at different ionic strengths. The ionic strength are imposed by a nitric acid solution. 2.3.2 Effect of the ionic strength 5000
6OOO
/ / / @/
4000
75A.
/
~'3000
50O0
/
/
:
~ ,'
¢0 ~,~ 3000
Q.
2000
20OO
1000
. .#
0
2
3
(%)
¢
1
(b)
2
3
(%) I
5000
/ /
lO3A.
/
4000
/,e.
' /
1000
/
/
/,
600 <
400
..
I~
/
2000
#l
/
0
2
3
(d)
0
2
//
~"
,o <
~'
lOOO
200
1
/
~" 3000
I~. 800
(c)
/
0
1
1400 1200
/
1000
o,
(a)
9~k
4000
. ~ o~
4
6
8
10
. • .(%) a~ (%) Figure 2 : Effect ofiouic su-engmon me osmotic pressure of magnetic fluids. (a) Co) (c) : Acidic magnetic fluids, for three different particles size Measurements (dots) are performed using themembraueosmometer. #Iffi510-4mol/L #I=10-3moFL o I = 10.-2mol/I., • I = 10-1mol/L (d) : pH 7 magnetic fluids (polydisperse samples, I = 6[NaCit]). Measurements (dots) are performed usin~ the membrane-osmometerfor low pressures and by osmotic stress for high pressures.The fine lines are only guides for the eyes. * [Cit] = 10-3mol/L • [Cit]= 5.10-3mol/L <>[Cit]= 10-2mol/L
Figure 2 focus on the effect o f the ionic s~ength for acidic magnetic fluids (three particles sizes) and for citmted=coated particles, dispersed in water at pH 7 (do = 85A, a - 0.40). For these latter, the ionic strength is due to unadsorbed citrate species in solution. Samples for which [Cit] = 5 10"4 mol/L have been impossible to study because the absorption equilibrium of citrate ions on particles is shifted [21] for low concentration of citrate in solution. On the
210
other hand, for high ionic strength such as [Cit] = 10q mol/L, the uncertainty on the osmotic pressures measurements is too important to give significant results concerning the destabilization of low volume fraction samples. The osmotic pressure values reported on figure 2 (d) have been determined with the membrane osmometer for low volume fraction and by osmotic stress for high volume fraction, at three differents ionic strengths. 3 DISCUSSION The relevant result when characterizing a colloidal dispersion is the plot 17/O as a function of O in order to get its osmotic compressibility and thus an estimation of the strength of interparticle interactions. The initial slope of such a curve is proportional to Az. The limit ffI/O)¢--~0) allows theoretically to get the molecular weight of the colloid m but here the experimental uncertainties on the low pressures does not allow to derive this molecular weight. Please note that this error has no effect on the calculation of A2, which is the slope of the plot. • [Cal = 10-3 mol/L Q. i ! 800
e [CIt] = 5.10-3 moR. i o [Ot] = ~0-2 rnoVL i
) */
~,oo
0 ¸
; 4¢ J
/ 200
~
2
4
_- J 6 / . O ~ -~¸
6
8
10
(%) Figure 3 : Evidence of repulsions in a pH 7 dispersion of magneticparticles (do = 85A, ~ = 0.40). The fine lines are only guides for the eyes. For all the studied samples, the compressibility is linear until a given value of ~, which depends on the nature of the samples (acidic or pH 7 magnetic fluid) and on the ionic strength. In the case of pH 7 magnetic fluids (citrated-coated particles), it appears (figure 3) that, even for high ionic strengths ([Cit]=5 l0 "3 mol/L and [Cit]=10"z reel/L) the interactions in the system remain repulsive. It has been verified that a change of ionic strength does not modify the surface charge density provided the concentration of unadsorbed citrate ions is not lower than a given concentration of citrate in solution. It is surprising to have electrostatic repulsions for so high ionic strength (for [Cit]=10"2 moFL, 1--6 10.2 moVL) according to the DLVO formalism [25]. The same results was obtained by Dubois and al [16], from analysis of SANS spectra, and the authors invoke a so-called "electro-steric" repulsions, the absorbed citrate species inducing an additive repulsion of steric nature. They propose as an equation of state the following formula that summaries their results at varying I and T :
211
H = • =-~--~[1 + 16.5.(1 - (4.4 - 0.045T)I)] v, k-¥ where Vw is the volume of the particle, T the temperature and I the ionic trength. We compare on figure 4 our osmotic measu~ments with the osmotic pressure deduced from such a =
32q ~
formula. We estimated Vw taking into account the polydispersity, though (r 3) = {ro3)e¢~-)). 4O0
.-.¢~ 300
/
200
10o
0 1
2 •
3
(%)
Figure 4 : Osmotic pressure of pH 7 magnetic fluids at several ionic strengths • [Cit] = 10-3 mol/I., : osmotic pressure measured - [CitI = 10-3 ~ I / L : state equation f:mm~f [16] • [Cit] = 5 10-3 mol/L : osmotic pressure measured . . . . [Cit] = 5 10-3 r~l/L : state-equation from ref[16] o [Cit] = 10-2 n'~l/L : osmotic pressure measured . . . . . . . [Cit] = 10-2 mol/L : state-equation ffomref [16]
The experimental data are in reasonable agreement with the formula for the highest ionic strength, corresponding to the conditions used by Dubois but not for [Cit] = 10-3 mol/L. Results are slightly lower than the calculated ones at very low volume fraction, hut this can be due to a polydispersity effect. The equation has been established for size-sorted particles although the samples studied here are polydisperse and contain larger particles, which are more sensitive to Van der Waals and dipolar interactions and lead to a reduced osmotic pressure, even with an equivalent value of Vw. Of course, a decrease of ionic strength, i.e. a n increase of Debye length, stabilizes the ferrofluid. The second-osmotic virial coefficient is much higher for [Cit] = 10-3 mol/L than for [Cit] = 5 10-3 mol/L. For low ionic strength the 1-I/~ curve diverges sharply from a linear behavior at low volume fraction, because the repulsions occur on a large range and n-body interactions (with n > 3) are no longer negligible. Assuming that repulsions are sufficiently high at such ionic strength to approximate the interparticle potential as a pseudo hard-core one, we compare on figure 6 the measured equation of state with the Camahan-Stirling [26], which gives the exact values for the virial coefficients until A4 for a hard-core potential (we renormalize here the radius of the particles by an effective one, sum of the radius of particles and of the Debye length). The effective volume fractions calculated in the range of our experimental data are low enough to prevent forced interpenetration of renormalized particles. I-I 1+ ~ r + ~ 2 _ ~ 3
(p)kT
212
Figure 5 indicates that our system is more repulsive than an hard-core one. As a matter of fact, repulsions begin to increase at lower volume fraction, indicating the presence of additional repulsions on a range larger than a Debye length. 500
t
~,~.~,300
•
I~ 2OO 100 e •
2
4
6
8
10
(%) Figure 5: Osmotic pressure measurements of pH 7 magnetic fluids for [NaCit] = l0 "3 mol/L. The dots correspond to measurements and the line to the equation of state according to Camahan-Stirling formalism.
Figure 6 plots i-I/O as a function of • for acidic particles of different sizes, at different ionic strengths. For pH 2, pH 3 and pH 3.3, corresponding respectively to I = 10"2 moFL, I = 10"3 mol/L and I = 5 10.4 mol/L, the system is repulsive, even for biggest particles. At pH 1, experimental errors are non-negligible, and the calculation of A2 is not meaningfid. For the pH values higher than 1, it can be noted on figure 6 that the I-I/O curves are all linear, meaning that the equation of state of an acidic magnetic fluid can be reduced to a second-virial development. Their slope when divided by the volume of the particles, is proportional to A2 by a factor (the same for all measurements) that is neither dependent on ionic strength neither nor on particle size. Figure 7 plots A2 as a function of the pH for the three particles sizes. This diagram shows first of all that the sum of interactions becomes less and less repulsive when the particle size increases, whatever the ionic strength. It can be explained by the volume-dependence of the Van der Waals interactions and of the dipolar interactions which begin to have a significant magnitude for a particles diameter of 10 rim. The electrostatic repulsions seem also to be size-dependent: for the two smallest particles, the system becomes more repulsive when the pH increases indicating that repulsions increase because of the ionic strength effects. On the contrary, for the biggest particles, the system becomes less repulsive for pH > 3. This result on large particles is in agreement with Menager and al [I 0] who observe a maximum in the magnitude of electrostatic repulsions for acidic polydisperse samples at pH 3. This maximum is due to a competition between the pH induced reduction of the surface charge density and the increase of the Debye length. Figure 7 plots A2 as a function of the pH for the three particles sizes. This diagram shows first of all that the sum of interactions becomes less and less repulsive when the particle size increases, whatever the ionic strength. It can be explained by the volume-dependence of the Van der Waals interactions and of the dipolar interactions which begin to have a significant magnitude for a particles diameter of 10 nm.
213 1500
2500
j
2000
12.
1200
,sA / f
900
1500
v
1000
600
500
300
0
0 0
1
2
,
•
.
0
1
2
3
,~ (%)
3
1
2
,~ (%)
3
4
600 500
~,
400
12. "'"
300
200 100 0
4
(%) Figure 6 : Interactionsin a dispersionof acidicparticles.The slopesallo~vscalculationsof A2 4,pH3.3 4,pH3 o p H 2
The electrostatic repulsions seem also to be size-dependent: for the two smallest particles, the system becomes more repulsive when the pH increases indicating that repulsions increase because of the ionic strength effects. On the contrary, for the biggest particles, the system becomes less repulsive for pH > 3. This result on large particles is in agreement with Menager and al [10] who observe a maximum in the magnitude of electrostatic repulsions for acidic polydisperse samples at pH 3. This maximum is due to a competition between the pH induced reduction of the surface charge density and the increase of the Debye length. The size-dependence of electrostatic repulsions cannot be interpreted invoking the term e~(2a)
depending on the radius in the Yukawa potential ( ~ )
because, at large Debye length,
e~¢z°' --> l) the potential becomes independent of the particles radius. 0+~)' Therefore, this size-dependence of repulsions has to be related to a size dependence of the number of the effective charges. According to [12], for highly charged particles, the effective charge is particle size dependent and decreases when the radius of particles increases. For the value of pH under consideration (pH -> 3), the number of structural charge is not at its maximum value, but it is reasonable to consider that the size-dependence of effective charges will keep the same tendency [10]. Moreover, it can be assumed that the density of structural (r.a --> 0 and
214
charges decreases when the particles radius increases through a size-dependence of the acid constant Ka for the ferric hydroxide groups at the surface of particles [27]. 0,0015
o,oo
i- _1o
i
0,0005
0 1,5
2
2,5
pH
3
3,5
Figure 7 : A2 as a function of the pH for the dispersions of cationic particles of different diameters.
CONCLUSION The first result of this study is to underline the stability of aqueous magnetic fluids even when the ionic strength is important. In the case of pH 7 magnetic fluids, it is in agreement with SANS-experiments, but at the present time the high stability of these magnetic liquids is not completely understood, except invoking additional repulsion terms due to steric hindrance. The study of acidic magnetic fluids brings out one of the first systematic experimental results on the effect of particles size on osmotic pressure of colloidal systems. At a given pH, it underlines that an increase of particles size lead to a destabilization of magnetic fluids by increasing Van der Waals and dipolar attractions but also to a decrease of electrostatic repulsions through a decrease of the effective charge of the particles. The next step of the study is to establish the equation of state of such liquids a magnetic field. Acknowledgments : We would like to thank C. Carvalho for her experiments on acidic magnetic fluids and P. Levitz for his help in the realization of the osmotic stress. We are indebted to E. Dubois, C. Menager and P. Levitz for helpful discussions. REFERENCES 1. P.N. Pusey, Colloidal suspensions, in Liquids, Freezing and Glass Transitions, J.P. Hansen, D. Levesque and J.Zinn-Justin (eds), North-Holland (Amsterdam) (1991) 765-942. 2. O. GlaRer, Small-angle scaterring and light scaterdng, in Neutron, X-Ray and Light Scattering, North-Holland, Delta Series, Elsevier Science Publisher (I 991) 33. 3. A. Mourehid, A. Delville, J. Lambard, E. I_~collier and P. Levitz, Langmuir 11(1995) 1942. 4. J.P.Hansen and I.R. Mac Donald, Theory of simple liquids, Academic, New York, (1992). 5. J.C. Bacri, R. Perzynski, D. Salin, V. Cabuil and R. Massart, J.Magn. Magn. Mat., 85 (1990) 27.
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