15993
J. Phys. Chem. 1995,99, 15993-16001
Structural and Thermodynamic Properties of Charged Silica Dispersions Jeanne Chang,*Jq' Pierre Lesieur?T* Michel Delsanti? Luc Belloni? Cecile Bonnet-Gonnet,Sand Bernard Cabme"' Service de Chimie Molbculaire, CEN Saclay, 91 191 Gifsur-Yvette Cgdex, France, Equipe Mixte CEA-RP, Rh6ne-Poulenc, 93308 Aubervilliers, France, and Universitb Paris Sui, LURE Bat. 290-0, 91405 Orsay Cedex, France Received: May 2, 1995@
Electrostatically stabilized aqueous dispersions of nanometric-sized silica particles have been characterized by light scattering, small angle X-ray scattering, and osmotic pressure measurements. All three studies yield similar values for the particle size, molecular weight, and particle surface charge. In addition, good agreement is found between directly measured osmotic pressure values and those calculated from scattering studies. Using the particle properties as determined from the three experiments, the osmotic pressure as a function of volume fraction is compared to liquid state theory models with no adjustable parameters. Finally, small angle X-ray scattering studies indicating the presence of long-range order in moderately to highly concentrated dispersions are reported.
Introduction The effective use of colloidal dispersions in numerous applications requires a fundamental understanding of the interparticle interactions. Depending on the storage, handling, and processing requisites of a system, it may be desirable to have either a dispersed or flocculated system. In the case of the former, a repulsive particle potential is necessary to maintain a stable system. Depending on the nature of the repulsive potential, the interactions between particles may be either longor short-range. We refer to systems in which the range of interaction is short as hard sphere systems. Much as already been learned about the nature of interparticle interactions in these systems through extensive theoretical'-* and experimental'^^-^ studies. Typical hard sphere systems are sterically stabilized systems and charged particle systems at high ionic strengths. In these systems particles do not interact until they approach close enough such that a very steep repulsive barrier is reached. Visually these dispersions appear to pass through a liquid-solid transition with increasing volume fraction. This transition has been shown to correspond to a disorder-order transition. However, as the repulsion is so steep, this transition from liquid-like to solidlike behavior occurs over a very narrow range of volume fraction. In dispersions of particles influenced by long-range interactions transitions are less abrupt. Particles repel one another at large distances of separation, and this repulsion increases with decreasing particle-particle separation. Until recently, studies have primarily concemed highly deionized dispersions of charged In such cases, the electric double layer surrounding the particles extends on the order of more than a particle diameter beyond the surface of the particle. Thus, particles may interact at volume fractions of as little as 0.01, while volume fractions greater than 0.20 are generally unobtainable. However, although the disorder-order transition in these systems is not as abrupt as in the case of hard sphere systems, the transition may only be observed at very low volume fractions. Service de Chimie Molkculaire. Equipe Mixte CEA-RP. 5 Universiti Paris Sud. Abstract published in Advance ACS Abstracts, September 1, 1995. @
0022-365419512099-15993$09.00/0
One means of examining the transition over a wide range of volume fractions is to utilize nanometric-sized charged part i c l e ~ . ' ~ Working ,'~ at moderate ionic strengths, one can obtain a system in which particles begin to interact when separated by a surface to surface distance on the order of a particle diameter. This can be considered to correspond to a volume fraction, 4, of 0.10, if one assumes particles are packed in a highly ordered face-centered-cubic (fcc) arrangement (4fCc =
0.74)
4 = 0.74(a/2a)3
(1)
where a represents the particle radius, and 2a half of the center to center distance between particles. As the volume fraction is further increased up to 0.74,these dispersions change from liquid-like to soft solid-like and ultimately become solid-like.I 4 Increasingly, systems of nanometric-sized particles are becoming of interest in numerous applications such as coatings, pharmaceutics, and high-performance materials. Furthermore, from an experimental standpoint, nanometric particles provide certain advantages. The difficulty in studying charge systems has largely stemmed from the fact that studies are typically carried out using monodisperse suspensions of polystyrene or silica particles measuring at least 40 nm in diameter. To investigate such systems of larger particles, light scattering experiments are generally desirable. However, due to the size and the index of refraction of these particles in water, multiple scattering effects can often arise at a volume fraction as little as 0.01. Using an aqueous system of nanometric silica particles, approximately 20 nm in diameter, we have been able to circumvent these problems and characterize the properties of a charged system over a wide range of volume fractions using light and small angle X-ray scattering. Furthermore, the small size of the particles has made direct measurement of the osmotic pressure feasible. Thus, a comparison between osmotic pressure values calculated using structure factor data and those measured directly is possible. Characteristic of both highly deionized as well as hard sphere systems is the appearance of ordered regions resembling crystallike structures. Much effort has been made in predicting the various phases, typically fluid, body-centered cubic, and facesntered cubic, as a function of volume fraction and verifying 0 1995 American Chemical Society
15994 J. Phys. Chem., Vol. 99,No. 43,1995 their presence in real Such understanding of the structure of dispersions is of great interest particularly as it likely concerns how such systems f l ~ w . ' ~ . ' ~ Unfortunately, -~O with regard to charged particle systems, highly deionized systems are not ideal for rheological measurements, and only a few studies have concerned the static structure of charge particle systems used in rheological studies.2' Therefore, it was of interest to examine the structure of our system, which had already undergone rheological characterization.I 4 In particular it should be noted that due to the size and relatively low index of refraction of silica with respect to water, our dispersions are transparent. Thus, no ordering was apparent from the outset, as is often the case in other dispersions which are iridescent. In what follows, light scattering, small angle X-ray scattering, and osmotic pressure measurements of dispersions of charged silica particles are used to yield a detailed description of the particle characteristics. Consistency between the three experimental techniques is examined, and results are discussed in light of existing thermodynamic models which describe the interparticle potentials. Finally, evidence indicating the presence of ordered structures in concentrated dispersions will be presented and discussed.
Experimental Section Materials. The synthesis of silica particles, which has been described elsewhere,I4consisted of reacting an aqueous sodium silicate solution with nitric acid. By this means, fully dense silica particles are produced, and the presence of multivalent ions such as Fe3+ or Sod2- may be avoided. At the end of the reaction, however, the ionic strength is relatively high (-0.05 M) and must be reduced in order to avoid flocculation during concentration of the sol. Thus the sol is washed using a tangential ultrafiltration module. In a manner described previo ~ s l y ,the ' ~ procedure involves removing water at a high ionic strength from the sol and replacing it with deionized water, thus reducing the ionic strength of the sol. Using this procedure, a batch of silica sol was washed until the water leaving the system reached an ionic strength of M. At this pointthe batch of silica was divided into two equal portions. One portion was then concentrated, while the other portion was washed until the water leaving the system reached an ionic strength of 6.5 x M and then concentrated. This procedure allowed for two identical systems of silica, with respect to particle size and distribution, to be studied at two different ionic strengths. The particle radius as determined by transmission electron microscopy ( E M ) was found to be 10.2 nm, with a standard deviation, 0,of 0.9 nm. Concentration of the sols was carried out in the same tangential ultrafiltration module. However, rather than replace the water leaving the system, it was simply collected, and the sols were eventually concentrated to a volume fraction of 0.12. It is important to note that the water leaving the system during the concentration process was collected, as it was used later in both diluting and concentrating the sol. T h i s procedure ensured that the equilibrium system conditions, e.g. ionic strength and pH, were always maintained. Dilute sols (volume fraction 0.12) were prepared by diluting the concentrated sol with the collected water. More concentrated sols were prepared by placing the sols in dialysis membrane bags (VISIUNG 12000 MW), which were then placed in aqueous solutions of polyethylene oxide (PEO) (MW 35000) at varying concentrations which applied an osmotic pressure.22 Note again that the aqueous solutions of PEO were prepared using the water collected during concentration of the sols. After concentrating, the concentration of the sols was
Chang et al. determined by weighing a sample before and after drying in an oven at 150 "C. Conversion of units from mass fraction to concentration in grams of SiOAiter, c, and volume fraction, 4, was made by assuming the density of Si02 to be 2.0 g/cm3. It is important to note that in all osmotic pressure, light scattering, and X-ray scattering measurements of dilute sols, a single preparation of each concentration of sol was made and used throughout. Scattering Experiments. Suspensions of particles were studied using a combination of light scattering and small angle X-ray scattering (SAXS). When an incident beam strikes a suspension of particles, the beam is scattered and the average intensity, Z&), scattered by the particles is measured at different scattering vectors q, which are directly related to the scattering angles 8 by
4n
8
q = - sin(?)
A
where A is the wavelength of the scattering radiation in the sample. Light scattering experiments were performed using the green line (514.5 nm) of an argon laser and a Brookhaven goniometer. The value of q investigated varied between 8.5 x and 3.2 x nm-I. A detailed description of the spectrometer has been given in ref 23. Sample cells were cleaned by rinsing with the filtrated water collected during concentration of the sol. Sols were then filtered through a 0.2 pm Millipore membrane into the light scattering cells. These airtight cells were then centrifuged for 15 min at 2000g. S A X S experiments were carried out at the Laboratoire pour 1'Utilisation du Rayonnement Electromagnttique (LURE) on beam line D22 using the DCI synchrotron radiation source. A pinhole collimation was used to obtain data, which were recorded with a linear detector of 512 cells. In addition, diffraction patterns were obtained on image plates. The scattering vector q varied from 0.05 to 0.4 nm-I, and the resolution dqlq was on the order of 0.02. A further discussion of the spectrometer is given in ref 24. All samples were placed in 1 mm thick cell holders surrounded by Kapton films. Since evaporation of water from sample cells was noticed, experiments were carried out on a sample no more than 24 h after it had been placed in a sample cell. Osmometry. The osmotic pressure of dilute suspensions (0.0005 I9 5 0.12) were determined with a Knauer membrane osmometer, using a 10 000 MW membrane, which measures reliably pressures between 2 and 4000 Pa. In the experiments carried out, a small volume of sol of known concentration was placed on one side of the membrane, while on the other side of the membrane was water collected during the concentration process of the sol. Due to the presence of particles on one side but not the other, a pressure difference, e.g. the osmotic pressure, was measured. Note that it was assumed that since the water used in diluting the samples (that coming from the concentrating of the sol) was also used as a reference in the osmotic pressure measurements, then any extraneous ions which might have been present were in equilibrium. Osmotic pressure measurements of concentrated dispersions (0.10 5 4 5 0.70) were made directly from samples prepared using the previously described dialysis membrane bags submerged in PEO solutions. As described elsewhere,I4the osmotic pressure of the dispersion was determined on the basis of the known osmotic pressure of the surrounding PEO solutions, with which the bags of dispersion were in thermodynamic equilibrium. Small samples of dispersions from each bag were then withdrawn, weighed, dried, and weighed again, in order to
J. Phys. Chem., Vol. 99, No. 43, 1995 15995
Properties of Charged Silica Dispersions determine a mass fraction, which was then converted to a volume fraction.
Gaussian distribution for e(x):
1 :La)l1
e(x) DC exp - - -
Theoretical Background Monodisperse System. The average intensity of a light beam scattered off a suspension of monodisperse spherical particles may be expressed as25
I&q) = KcMP(q)S(q)
(3)
where M is the molecular weight of the particle, K is an experimental constant, P(q) is the particle form factor, and S(q) is the structure factor of the particle suspension. P(q) =f2(qa), where a is the particle radius and
.xy) =
3[sin(y) - y cos(y)] Y3
+ * L4 - [ g ( r )
- l ] r sin qr dr
+
(4)
In the case of X-ray scattering experiments, the expression for the average intensity is similar to eq 3. The structure factor, S(q), for a system of interacting particles, is25
S(q) = 1
where a is the average size and (5 is the width of the distribution. In the presence of interactions, the discrete expression (9) will be used and the polydisperse suspension will be represented by a three-component mixture of sizes a - u, a, and a (5 and relative number densities 1,2, and 1. The equilibrium (pressure, compressibility) and structural (Sij(q))properties of such solutions can be obtained from classical liquid state t h e ~ r y . ? ~We .’~ will assume that the colloids i andj interact via the ion-averaged DLVO-like p~tential:~’
(5)
where g(r) is the pair distribution function, e = c/M is the particle number density, and r is the center to center distance between particles. For a suspension of monodisperse particles, S(q) is equal to the osmotic compressibility in the limit of q 0 and thus may be written as25
-
where n is the osmotic pressure. Rewriting this expression in terms of S(0) in order to determine n yields
The van der Waals contribution is negligible in our systems. 2, is the number charge of colloids i (we will assume that Z, ar2),K = (2e21/cockr)”2 is the screening constant, and I is the ionic strength which includes the salt ions (salinity = I,) and the counter ions which equilibrate the colloidal charges: 0~
This multicomponent model (MCM) is solved28 using the hypemetted chain (HNC) integral equation.25 The pressure is calculated from the virial equation, and the osmotic compressibility is deduced from the Sij(0).26.29 The main effect of the polydispersity (compared to the monodisperse case) is to increase the scattered intensity at low q. It is always possible to define an effective structure factor from the intensities in the presence and in the absence of interactions,
(7) or, upon replacing
e with the volume fraction, 4 = e(4na3/3),
x,
Thus, for a system of monodisperse particles, S(0) and n may be directly compared. Polydisperse System. In the case of a suspension of polydisperse spherical particles the scattered intensity is expressed as
where a,, M, = ai3, and c, = elM, are the radius, molecular weight, and weight concentration of species i, respectively. The S, are the partial structure factors. In the limit of systems of weakly interacting particles, S,, = 6, and thus
where g(x) dx represents the number of particles per unit volume having a radius between x and x dx. In the absence of interactions (dilute systems), the polydispersity will be treated with the integral form of eq 10 and
but, contrary to the monodisperse case, the compressibility, is not given by S,ff(O). For example, at 4 = 0.30 without added salt, 2, S,ff(O),and n/ekT are for a monodisperse case ( a = 10 nm) 0.0069,0.0069, and 97, respectively, and for a polydisperse case ( a = 10 nm, cr = 1 nm) 0.0068, 0.0095, and 85, respectively. Considering that interactions are strong in this concentrated salt-free system, one can see from these values that the polydispersity effects, although not negligible, are small. This reflects the relatively small degree of polydispersity present in our system. Molecular Weight Averages. Extrapolating the lightscattered intensity and the osmotic pressure at infinite dilution (no interaction) yields the weight average, M,, , and number average, Mn,molecular weights, respectively. Comparison of the two values gives information on the polydispersity of the system. M , is given by
+
where Zpand Ibenlene are the intensities scattered by the solution and a reference (benzene). The apparatus constant K‘ is
15996 J. Phys. Chem., Vol. 99, No. 43, 1995
Chang et al. 110' I
where n~ and Rg are the index of refraction and Rayleigh factor of benzene, respectively. The constant K' in our case is 2.45 x mol Llg2, having taken dnldc = 0.059 mL1g. The index of refraction, dnldc, was measured using a Zeiss Jena interferometer operating in the white light range. In the same limit of weak interactions, M,, is given by
- _ -1imM,,
. --%
4106-1
Jc 0
coNAkTc
Particle Characterization Particle Size and Polydispersity. Assuming that particle interactions were negligible in a 4 = 0.01 sol at a salinity of lo-* M, Ip(q)was calculated from eq 10 and compared to that experimentally obtained from SAXS. Comparing the fit, an average radius, a, of 10 nm with a standard deviation, o, of 1.3 nm was obtained from the value of the intensity at the position of the first minimum, qmln.It may be recalled that this value is consistent with that found from TEM measurements. Note, Pusey' defines the polydispersity, oP,as
oP= ola
(18)
and sets as a criterion op < 0.05 for a system which may be treated as a one-component system. Using this definition, op as determined by TEM is 0.09, which is consistent with that found from SAXS, namely, 0.13. As has been discussed however, and as will be indicated later, it seems that a higher degree of polydispersity may in fact be tolerated in charged particle systems. It should be noted that, as expected, the number average radii determined by SAXS and TEM were found to be smaller than the hydrodynamic radius obtained from quasi-elastic light scattering experiments. The difference between the radii is typically attributed to the polydispersity of the system. However, using the radius and Gaussian size distribution determined from TEM, an estimate of the hydrodynamic radius was calculated and found to be 10.6 nm. This value is notably less than the 11.8 nm radius found from quasi-elastic light scattering experiments of dilute sols at a salinity of M. The difference is thought to be due, at least in part, to a hydrated silica gel layer surrounding the particles. A particle with such a layer would assumingly be slightly more voluminous and therefore diffuse more slowly, as compared to a particle with no such layer. In such a case, light scattering experiments could distinguish between particles with and without layers. On the other hand, if the density of the layer was not sufficient to provide adequate contrast, TEM or SAXS studies might not show evidence of such layers. The presence of a silica gel layer on silica surfaces has recently been reviewed3' and, in the cases studied, was reported to be on the order of 1-2 nm in thickness. Molecular Weight. Using dilute sols (4 < 0.005) at a salinity of lo-* M, M , was determined from eq 15. As can be seen in Figure 1, Z~(Z~enzeneK'c) is essentially independent of q5 in this range of concentrations and gives M,, = 5.8 x lo6 glmol. Similarly, on the basis of eq 17 M , was determined using sols at 4 < 0.05 and at both salinities. Plotting nlc as a function of c and extrapolating to c 0 yields an M,, of 3.1 x lo6 glmol (Figure 2). This is rather low as compared to M , or to what one might expect on the basis of a simple calculation using the density of silica and the average particle radius. Although M,, is expected to be less than Muin the case where there is some
-
. . . .I . . . .: . . . . ; . . . . I. . . . I
0.001
0.002 0.003 volume fraction,
0.004
0.005
Figure 1. Molecular weight determination using eq 15 and intensity measurements of static light scattering experiments carried out at q = 2.30 x lo-? nm-'. Salinity is lo-? M.
t 3.-5 1 3.0
-.
-.
f
e
2.5
w
M
e
. "
a
0.5
0
20
40
60
80
100
concentration ( g l l )
Figure 2. Molecular weight determination from osmotic pressure measurements and eq 17. Salinity is 6.5 x M (circles) and lo-* M (triangles).
degree of polydispersity, this does not sufficiently describe the discrepancy in this case. Assuming a Gaussian distribution, the particle size polydispersity as determined from M , and M,, was calculated to be 0.35. Note, this value is significantly higher than that found directly from particle size measurements. It was considered that perhaps TEM photos may have given a false impression of a Gaussian distribution when in fact the distribution was more like a log normal, where a tail exists extending to smaller particles. Since osmotic pressure measurements favor smaller particles, this could explain why M,, was lower than expected. Using a log normal distribution, the particle size polydispersity was then recalculated from Mu and M,, and found to be 0.27. This being still higher than that calculated from particle size measurements, the difference was attributed to the uncertainty in the values of M , and M,,, which gave the impression of a much higher degree of polydispersity than actually existed. In an electrostatically stabilized system at a given volume fraction, particle interactions decrease with increasing ionic strength. Consequently, eq 17 becomes increasingly valid with increasing ionic strength. However, due to the diminished particle interactions, the osmotic pressure simultaneously decreases as well with increasing ionic strength. Thus, although sols at a salinity of lo-* M were more likely to satisfy the condition of weak interactions, as required in the use of eq 17, the small osmotic pressure values made reliable measurements difficult. As osmotic pressure measurements were accompanied by an error of f 2 Pa, a 10-50% degree of uncertainty was associated with osmotic pressure measurements of sols at the three lowest concentrations. As can be seen from Figure 2 , this in turn causes a large degree of uncertainty in the extrapolation for M,.
Properties of Charged Silica Dispersions
J , Phys. Chem., Vol. 99,No. 43, 1995 15997
7 0.01 0.03
1 0-2 iod
10.~ 10'~ io1 volume fraction, 4
1 o'
*
vi
. 1 I
4
--
a
3 --
.-,x
2-1
5 9
1-:
A
1
0
0.01
0.02
q
0.03
0.04
0.02
0.04
0.05
q (ti-')
Figure 3. S,rt(q=O) as obtained from light scattering (circles) and SAXS (triangles) for silica dispersions at a salinity of 6.5 x M. Comparison is made to that calculated using MCM (solid line) with a salinity of 1.5 x lo-' M and three components: particle radius = 10.5 1 nm, particle charge = (1 10 & 20)e, relative number density = 1. 2, 1.
-
0
0.05
(A-',
Figure 4. I p ( q ) as obtained from SAXS (circles) for silica dispersions at a salinity of 6.5 x and C#J = 0.04, 0.083, and 0.10 compared to those determined using the MCM (solid line) with the same parameters as given in Figure 3.
Surface Charge. The particle surface charge was determined by comparing Sefi(0),measured by light scattering and small angle X-ray scattering, and Zp(q) obtained from small angle X-ray experiments, to the Sefi(q=O) and Z&q) calculated from liquid state theory. S,fi(q) was calculated from eq 14 with I,,O(q) deduced from the scattering intensity of the most dilute system (this allowed one to work in arbitrary units). For each concentration, S,ff(4'0) was calculated by extrapolating the values of Seffiq)to q = 0. The experimental and theoretical MCM results are presented in Figures 3,4, and 5 . This fitting procedure yields an average surface particle charge of (150 f 30) e for the salinity of M. For the systems at the lower salinity, it was found that it was necessary to assume a salinity of 1.5 x M (instead of 6.5 x M) and a particle charge of (110 f 20)e, in order to be consistent with both the S,fi(q=O) data and the intensity spectra. It is unclear why a discrepancy exists between the experimental and theoretical salinities of this latter case. The experimental value was determined by measuring the conductivity of the collected water obtained during concentration of the dispersion of synthesized silica. The conductivity meter had been calibrated using NaOH, which considering the synthesis route, was assumed to be the major constituent in the collected water. However, it is possible that if a significant quantity of other ions, such as soluble silica species, existed, then an error may have been made in the determination of the ionic strength from conductivity measurements. In the case of the dispersions at a salinity of M, a small error in the ionic strength would
Figure 5. Same as Figure 4 for dispersions at a salinity of lo-* M. For the MCM (solid line) a salinity of IO-* M and three components, particle radius = 10.5 f 1 nm, particle charge = (150 & 30)e, and relative number density = 1, 2, 1, were used.
have less of an effect, since particle interactions decrease with increasing ionic strength, and thus the modeling of Zp(q) would be less affected by the choice of the ionic strength. The significance of the particle charge found by fitting scattering spectra is a point of much d i s c ~ s s i o n . ~Since ~.~~,~~ the DLVO expression (12) implicity assumes a linearization of the colloid-ions interactions, it is necessary to use a renormalized (postcondensation) charge which is smaller than the structural charge. Many models relate the renormalized and structural value^.^"^^ Here the approach of Alexander et al.34 based on the Poisson-Boltzmann cell model is used. Using their criteria, it is found that if 200e is taken to be the structural particle charge, then the effective particle charge for the linearized Poisson-Boltzmann is llOe at a salinity of 1.5 x M and 150e at a salinity of M. Silica is considered to have 0.5 ~ i t e s / n m *which , ~ ~ would imply, if all sites were charged, a surface charge of 600 sites/particles for our system. The large discrepancy between the number of total sites (600 sitedparticle) and the structural colloidal charge (200e)may indicate that only a fraction of the sites are in fact ionized.
Thermodynamics
-
Osmotic Pressure and the Structure Factor in the q 0 Limit. Despite the degree of polydispersity in the system and therefore lack of validity of eqs 6-8, the directly measured osmotic pressure was still compared to that calculated from structure factor data using eq 8 with S(q=O) replaced by Seff(4'0). This was of interest since the comparison represented in Figures 6 and 7 is between experiments of very different nature, without the benefit of any adjustable parameters. The agreement (Figures 6 and 7) seems excellent although admittedly the log-log representation masks somewhat the discrepancies. The consistency between the directly and indirectly measured osmotic pressures is encouraging, since it indicates that eq 8 may still be useful for systems that are not truly monodisperse. Pusey' suggests that for hard sphere systems it is necessary to consider the polydispersity in cases where it exceeds 0.05. However, the data here suggest that, at least for charged particle systems, a higher degree of polydispersity may in fact be tolerated. This idea has been presented before1.*' for charged systems at low concentrations; however, the results indicate that this may be true at higher concentrations as well. In electrostatically stabilized system, diffuse ionic clouds surround particles and typically interact long before particles come into direct contact. Thus, in such cases, it seems reasonable to assume that the effect of the polydispersity of the hard core radius on the validity of eq 8 would be less.
15998 J. Phys. Chem., Vol. 99, No. 43, 1995
Chang et al.
10.f
10-1F 1A.3
'
"
~
~
*
*
~
I
'"....J lbo
1' " ' ~ ~ * * l
1
10.2
I!-'
0
0.1
Figure 6. Experimentally measured osmotic pressure of silica dispersions at a salinity of 6.5 x IO-' M as compared to that calculated from eq 8 with the experimental S,fdq=O) values determined from static light scattering and SAXS experiments. I
I
0.2
0.3
0.4
0.5
volume fraction, $
volume fraction, I$
..c
I
Figure 8. Osmotic pressure of dispersions of silica at a salinity of 6.5 x IO-' M determined experimentally and compared to the MCM and PBC model. Parameters used in the MCM are the same as those used in Figure 4, except that 1.5 x M is assumed to be the salinity of the reservoir. For the PBC model a structural particle charge of 200e was used.
ight scattering
- -SAXS
Y
I
f'
e' /'
[ f 60
10'
40
. . . . ' -
10' 1o 3
20
1o 2
1 oo
1 0'
volume fraction,
I$
Figure 7. Same as Figure 6 for dispersions at a salinity of IO-? M.
0 0
0.1
0.2
0.4
0.3
volume fraction,
0.5
I$
It is expected that the osmotic pressure as calculated from Figure 9. Same as Figure 8 for dispersions at a salinity of IO-' M. Parameters used in the MCM are the same as those used in Figure 5, light scattering experiments should be slightly more consistent except that IO-? M is assumed to be the salinity of the reservoir. For as compared to SAXS experiments since smaller q values are the PBC model a structural particle charge of 200e was used. accessible in light scattering experiments, and thus S&q=O) may be more accurately determined. However, at higher edge of the For both models, salinities of 1.5 x concentrations, where light scattering experiments are no longer and M were used with a structural particle charge of 200e, straightforward, due to multiple scattering effects, SAXS studies which yielded an effective particle charge of llOe and 150e provide an alternative means of calculating the osmotic pressure. for the two salinities, respectively, as discussed earlier.44 Both light scattering and SAXS results, however, appear to As can be seen from Figures 8 and 9, neither model seems further confirm the uncertainty in the measured osmotic to come too close in describing the osmotic pressure data. In pressures at low concentrations. Note that while the comparison the case of dispersions at a salinity of lo-? M, the trend of the of light scattering and SAXS calculated osmotic pressures to data appears better represented, with the PBC model modeling the experimentally determined values is good at high volume the data a bit better than the MCM. This observation is fractions, the comparison is poorer in both cases at low volume consistent with what one would anticipate considering the fractions. Even taking into account errors associated with limiting factors of the respective models. With increasing comparing the experiments assuming a monodisperse system, volume fraction, the prediction from the MCM is expected to one would expect the discrepancies to be greater at the higher become worse since the DLVO expression (12) becomes less volume fractions. Thus, if one is to believe the osmotic pressure valid and the counterions, which increasingly contribute to the values at the higher volume fractions, then the convergence at osmotic pressure, are not dealt with explicitly. On the other lower volume fractions must be that indicated by light scattering hand, the PBC model treats the ions correctly; however, since and SAXS calculated values. As can be seen in both Figures each particle is fixed to the center of a cell, the interparticle 6 and 7, it appears at both salinities that the osmotic pressure force is always 0 (this leads to a vanishing second virial in the limit of 4 0 should be lower than predicted from direct coefficient). Finally, it must be noted that there was likely some measurements. If this is the case, then M, should be higher, uncertainty associated with osmotic pressure measurements, which would bring it closer in line with M , and yield a lower which are highlighted by the linear representation of n/@T in degree of polydispersity. Figures 8 and 9. Modeling of the Osmotic Pressure. As has been previously done with latex dispersion^,^^.^^ attempts at modeling the Structure and Crystallization osmotic pressure were made using the MCM, as described X-ray spectra obtained from concentrated dispersions at both earlier, and the Poisson-Boltzmann cell model (PBC).40,41 The PBC model considers each colloid to be situated at the center ionic strengths indicate the presence of long-range order at of a spherical cell and calculates the ionic profiles in the cell concentrations of 4 = 0.22-0.23 (Figures 10a and 1la). Further using the nonlinearlized Poisson-Boltzmann e q ~ a t i o n .The ~ ~ , ~ ~ evidence of ordering is seen in X-ray image plates taken of the pressure is determined from the total ionic concentration at the samples (Figures lob, and llb). As can be seen, there is a
-
Properties of Charged Silica Dispersions
J. Pliys. Cliem., Vol. 99, No. 43, 1995 15999
3.5 3 L.
2.5
U
2
-%i cn
1.5 1
0.5 0.01 0.02 0.03 0.04 0.05
0
0.06 0.07
q bi-9
0 0
0.01 0.02
0.03 0.04 0.05 0.06 0.07
q (A-9
b.
C.
Figure 10. (a) S,rl(y) as obtained from SAXS of a silica dispersion at a salinity of IO-' M and b, = 0.23. Peak positions corresponding to a bcc type structure are indicated. (b) Image plate photo corresponding to spectra of part (a). In this figure up to Figure 13 the shadow of the beam stop is seen lying across the rings. (c) Same image as in part (b). The intensity of the photo has been reduced in order to reveal the spots present in the dark inner ring of part (b).
distinctive speckle pattern in the image of the sample at a salinity of IO-* M and d, = 0.23, and one sees several faint rings in the image plate taken of a sample at a salinity of 6.5 x M and d, = 0.22. With increasing volume fraction, the spectra, unfortunately, do not yield any definitive structure, although image plate pictures of dispersions at a salinity of 6.5 x M and volume fractions between 0.25 and 0.50 do show a certain degree of anisotropy. At d, = 0.62 and a salinity of 6.5 x M and at d, = 0.54 and a salinity of IO-' M, the evidence from image plate pictures seems to strongly suggest that an ordered structure is present, as indicated by the six distinct spots (Figures 12 and 13). However, since no higher order peaks are found, it is not possible to determine the nature of the structure. The indication of any long-range order at all again suggests that a high degree of polydispersity may be tolerated in charged particle systems. On the basis of both simulated and experimental work, Pusey estimated that the degree of polydispersity must not exceed 0.07-0. I 1 in order to observe colloidal crystals. Furthermore, it was implied that the time to form a crystal should increase with increasing degree of polydispersity. All scattering experiments were carried out on samples that had been placed in sample holders less than 24 h in advance, and except for samples at d, > 0.50, all samples were squeezed from a dialysis bag into sample cells, a procedure which likely perturbed any pre-existing ordered regions. Peak positions of spectra of samples at d, = 0.22-0.23 suggest that all had a body-centered-cubic (bcc) structure. This was somewhat unanticipated, as face-centered-cubic structures (fcc) are more often found in charged colloidal systems and are the only structures proposed to exist in hard sphere system^.^'.^^ A number of author^^.^.^^ have studied highly deionized charged particle systems and compared the resulting
Figure 11. (a) S,fl(q) as obtained from SAXS of silica dispersions at a salinity of 6.5 x IO-' M and b, = 0.22. Peak positions corresponding to a bcc type structure are indicated. (b) Image plate photo corresponding to spectra of part (a).
structures to those predicted from a simulated phase diagram." Following a procedure similar to that of Monovoukas and Gast,h we attempted to compared our findings to those predicted from simulations. Using the parameters as determined earlier from the MCM modeling of the dilute dispersions, kT/v(g-I/') was plotted as a function of K Q - ~ ~ ' . As can be seen from Figure 14, the energetics of the system at a salinity of 6.5 x 10-4 M suggest the existence of a transition from a liquid to fcc structure. However, there is no suggestion of the existence of bcc structures for this system. Furthermore, the prediction of the transition between the liquid and fcc phase is found between two dispersions which were both clearly liquids, showing at best short-range order. In the case of dispersions at a salinity of IO-' M, the conditions of the dispersions studied puts them outside the phase diagram as drawn by Robbins et al.Is In fact, it does seem a bit strange to observe bcc structures at approximately the same volume fraction in systems at two ionic strengths which differ by an order of magnitude. As indicated above, the bcc structure is generally associated with systems where particles begin to interact via very long range interactions. This would imply the type of behavior noted at very low ionic strengths. However, it is difficult to believe that
16000 J. Phys. Chem., Vol. 99, No. 43, 1995
Chang et al. 0.8 0.7 0.6 h
2
0.5
0.2
0.1 0 0
2
4
6
a
10
KP-"3
Figure 14. Comparison of the phase diagram, kTlv(~-"~) vs as simulated by molecular dynamicsI5 to our findings for silica dispersions. Salinity = 6.5 x lop4M (open symbols); salinity = M (solid symbols). We classify our dispersions as showing no or only short-range order (circles), bcc order (squares), or an unidentifiable long-range order (triangles).
Figure 12. Image plate photo of a silica dispersion at a salinity of 6.5 x lop4M and q5 = 0.62.
while those at 0.25 < 6 < 0.50 were highly viscous, and finally those at # '0.50 were solid-like.'4 Thus, if crystals had been disturbed during preparation, it is expected that samples at # < 0.25 would have had an easier time reordering as compared to samples at 0.25 < # < 0.50 and that the order in samples at # > 0.50 would not be disturbed at all. Finally it is possible that crystals may have been present in samples at 0.25 < # < 0.50 but that they were too small to be measurable. However, unfortunately, the resolution of the X-ray detector was not adequate to enable an accurate measure of the size of the ordered regions. M and # = From the spectra obtained at a salinity of 0.23, it was assumed that the first peak corresponds to that of the closest packed plane in the bcc structure, e.g. (1 10). The position of the peak in terms of q is related to a distance, d, according to
q = 2x/d Thus, if the first peak corresponds to the (1 10) plane, d is the distance between (1 10) planes, d = dllo, and the characteristic length of the unit cell, I, is
dl10= l l h Figure 13. Image plate photo of a silica dispersion at a salinity of M and 4 = 0.54.
the ionic strength is actually lower than found, as typically in colloidal systems the problem is how to lower the ionic strength. Furthermore, it seems unlikely that placing the sample in the cell disrupted the ordered arrangement and coincidentally yielded an ordered bcc structure. On the other hand, the system at a salinity of M comes closer to resembling a hard sphere system, and thus perhaps the system is more sensitive to the degree of polydispersity. In this case, it may be that the degree of polydispersity is adequate to inhibit the formation of longrange order of the fcc type, yet not enough to prevent the formation of a bcc type structure. Finally, it should be noted that the original phase diagram as presented by Robbins15 was in fact for point particle charges and that others who have studied particle have located their systems on such a diagram generally with the aid of an adjustable parameter. As to why no evidence of crystal structure was found in samples at 0.25 < # < 0.50, it is hypothesized that during sample preparation crystals which were present were destroyed and did not have the opportunity to reform in the X-ray cell. It should be noted that samples at # < 0.25 flowed rather easily,
Taking the volume of a unit cell to be l3 and the number of particles per unit cell to be 2, as is in the case of a bcc structure, the molecular weight calculated from the known concentration Of the is =2 ~ 1 1 ~ c = 2Ml(2/r2d110)
From eqs 19-22 it was then found that = 5.8 lo6g",' which is consistent with that found earlier from light scattering experiments.
Conclusion A thorough characterization of an aqueous dispersion of silica particles over a wide range of volume fractions has been carried out using static scattering and osmotic pressure measurements. Despite indications that there was a fair degree of polydispersity in the system, excellent agreement was found between the directly measured osmotic pressure and that indirectly obtained from the scattering data at zero angle. The agreement between SAXS results and osmotic pressure measurements is particularly
Properties of Charged Silica Dispersions encouraging for studies of charged systems of large particles, where the low magnitude of the osmotic pressure may make direct measurements difficult and where multiple scattering effects are more likely to interefere with light scattering experiments. Using liquid state theories, it was then attempted to determine the average surface charge of the particles by modeling the scattering intensities. Often the determination is made by fitting only the main peak of Zp(q). Yet, since interparticle interactions have the greatest influence in the limit of the zero scattering vector, the fit in this limit should be much more stringent. Therefore, our success in modeling both S,f+(q=O) and the main peak of Z&q) with the same particle charge was encouraging. However, liquid state theories were less adequate in predicting the directly measured osmotic pressure. Finally, what we found to be one of the most exciting discoveries was the evidence of colloidal crystals. In systems of large particles, ordering is often easily distinguished by an iridescent appearance. This is feasible when the particles are large enough that the dispersions reflect light. However, all the dispersions under study here were transparent and thus appeared to resemble more a gel or glass. Thus, while there have been many reports of colloidal crystals of larger (diameter > 80 nm) particle^,^-^^.'^.'*^^^ we believe that this may be one of the first reports of colloidal crystals formed from smaller particles.
References and Notes (1) Pusey, P. N. Liquids, Freezing, and Glass Transition; Hansen, J. P., Levesque, D., and Zinn-Justin, J., Eds.; North Holland Press: Amsterdam, 1989: Chapter 10. (2) Genz, U.; D’Aguanno, B.; Mewis, J.; Klein, R. Langmuir 1994, 10, 2206-2212. (3) Livsey, I.; Ottewill, R. H. Colloid Polym. Sci. 1989, 267, 421428. (4) de Kruif, C. G.; Briels, W. J.; May, R. P.; Vrij, A. Langmuir 1988. 4, 668-676. ( 5 ) de Kruif, C. G.; Rouw, P. W.; Briels, W. J.; Duits, M. H. G.; Vrij, A,; May, R. P. Langmuir 1989, 5, 422-428. (6) Monovoukas. Y.: Gast. A. P. J . Colloid Interface Sci. 1989, I28 (2), 533-548. (71 Okubo. T. Lannmuir 1994, 10, 1695-1702. (8) Okubo, T. Colioid Polym. Sci. 1993, 271, 190-196. (9) Okubo, T. Phase Diagram of Ionic Colloidal Crystals, Schmitz. K. S., Ed.; ACS Symposium Series 548 Macro-Ion Characterization From Dilute Solutions to Complex Fluids; Washington, DC, 1994; pp 364-380. (10) Sengupta, S.; Sood, A. K. Phys. Rev. A 1991,44 (2), 1233-1236. (11) Hartl, W.; Versmold, H.; Wittig, U. Mol. Phys. 1983,50 (4), 815823. (12) Cebula, D. J.; Goodwin, J. W.: Jeffrey, G. C.; Ottewill, R. H.; Parentich, A.; Richardson, R. A. Faraday Discuss. Chem. SOC.1983, 76, 37-52. (13) Ottewill, R. H.; Richardson, R. A. Colloid Polym. Sci. 1982, 260, 708-7 19. (14) Persello, J.; Magnin, A,: Chang, J.; Piau, J. M.; Cabane, B. J . Rheol. 1994, 38 (6), 1-26.
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