Langmuir 1992,8, 2913-2920
2913
Structure of Binary Colloidal Mixtures of Charged and Uncharged Spherical Particles J. M. M6ndez-Alcaraqt B. D'Aguanno,t and R. Klein'pt Fakultiit ffirPhysik, Universitiit Konstanz, Postfach 5560, D-7750 Konstanz, FRG, and Center for Advanced Studies, Research and Development in Sardinia (CRS4),P.O. Box 488, 09100 Cagliari, Italy Received April 13, 1992. In Final Form: July 24, 1992
The microstructure and thermodynamic quantities of binary mixtures of hard spheres and Yukawa particles are investigatedby integral equation theories and Monte Carlo (MC) simulations. The OmeteinZemike equations for the partial pair correlation functions gi@) are solved with the Percus-Yevick, hypemetted-chainand Rogers-Young (RY) closure relations. $he resulta obtained with the RY scheme are in very good agreementwith MC data. Fromgij(r)the partial static structurefactorsS&), the number density structure factor S"(k), the measured structure factor S W ) , and the isothermal compressibility are calculated for various compositions. It is found that the replacement of small amounta of charged by uncharged particles has a profound effect on the main peak of S"(k), but it is of minor importance for the compressibility. The replacement of uncharged by charged particles, on the other hand, has strong effecta on the compressibility,whereas the structure changes only weakly. Recent experimental results on mixtures of polystyrene and silica particles are discussed. 1. Introduction
Structural properties of binary mixtures and, more generally, of polydispserse systems of spherical colloidal particles have been investigated recently by scattering experiments, computer simulations, and the integral equation methods of liquid state theory. These systems consistedeither of mixtures of hard spheres1+or of chargestabilized and it was shown that the integral equation theories were able to describe the structure of such systems very well. In most of the light and neutron scattering experiments only the total scattered intensity has been measured aa a function of scattering angle. Since the total intensity is given as a superposition of partial structure factors weighted by the scattering amplitudes of the various species of the mixtures, detailed information about the microstructure could not be directly obtained from these experiments. In order to determine more precisely the spatial orderingof, say, small and weakly charged particles around big and more strongly charged ones, it is necessary to use a contrast-matching method, which allows measurement of the correlations among one kind of (visible) particle in the presence of another species of (unvisible) particle. One example of such an experiment has been performed on a binary mixture of polystyrene and silica spheres by small-angle neutron ~cattering.~ A second new aspect of this experiment is that this system consists of a mixture of charged and uncharged + Univereitat
Konetanz.
Center for Advanced Studies, Research and Development in Sardinia. (1) van Beurten, P.; Vrij, A. J. Chem. Phys. 1981, 74, 2744. (2) Salacuse, J. J.; Stall, G. J. Chem. Phys. 1982, 77,3714. (3) Frenkel, D.; VM, R. J.; de Kruif, C. G.;Vrij, A. J. Chem.Phys. 1986, 84,4625. (4) de Kruif, C. G.; Briels, W. J.; May, R. P.; Vrij, A. Langmuir 1988, 4, 668. (5) DAguanno, B.; Klein, R. J. Chem. SOC.,Faraday Trans. 1991,87, 379. (6) Mhdez-Alcaraz, J. M.; DAguanno, B.; Klein, R. Physica A 1991, 178, 421. (7) Krause,R.;DAguanno,B.;MBndez-Alcaraz,J. M.;NAgele,G.; Klein, R.; Weber, R. J. Phye.: Condem. Matter 1991, 3, 4459. (8) D'Aguanno, B.; Krause, R.; MBndez-Alcaraz, J. M.; Klein, R. J. Phys.: Condens. Matter 1992,4,3077. (9) Hanley, H. J. M.; Straty, G . C.; Lindner, P. Phyaica A 1991, 174,
60.
spherical particles so that widely different types of potential interactions are determining the structural properties. Whereas the uncharged spheres interact among each other and with the charged particles through the short-rangedexcludedvolumeinteractions,the charged spheres feel each other over long distances because of the long-range character of the screened Coulomb potential. These mixtures are therefore qualitatively different from the previouslystudied mixtures of hard spheresor mixtures of charged particles. The purpose of this paper is to report results for thermodynamic properties and the microstructure of binary mixturesof charged and unchargedspherical particles obtained from integral equations theories and Monte Carlo simulations. In the next section we introduce the partial pair distribution function and partial static structure factors and establish the relations of these quantities to the experimentally accessible scattering intensity. The methods to obtain the pair distribution functionsfrom the model potentialsby solvingthe coupled Omstein-Zemike integral equations together with various closure relations are described in section 3. The results are presented in section 4, where it is shown that the effects of replacing one kind of particle by the other are quite different for the ordering in the mixtures and their thermodynamic properties. We will also compare our results to those of the above mentioned contrast-matching experiments. Section 6 contains a summary of the paper. 2. Description of the Structure
The microstructure of systems consisting of p species of suspended colloidal particles is described in terma of p @ + 1)/2 pair correlation functions gij(r), which are proportional to the probability to find a particle of species j at a distance r from a particle of species i. Once the pair correlation functions are known, one can calculate macroscopic thermodynamic quantities, such aa the pressure, the isothermal compressibility, and the internal energy, and, on the other hand, the microstructure, as it is measured in scattering experiments. In Fourier space the structure is described by the partial static structure factors Sij(k), which are defined as the correlation functions of the Fourier transforms Snki of the
0743-7463/92/2408-2913$03.00/00 1992 American Chemical Society
Mhndez-Alcaraz et al.
2914 Langmuir, Vol. 8, No.12,1992
fluctuations 6ni(r) = ni(r)- ni of the partial local densities ni(r) of the particles of species i
Here, N is the total number of suspended particles in the volume V, ni = Ni/ V the number density of species i, and the bracket denotesan ensembleaverage. Ni is the number of particles of species i. The partial structure factors are related to the pair correlation functions gij(r) by
Expression7 for the scatteringintensity can be rewritten in terms of the partial structure factorsll
-
Z(k) = Nf2(k)SM(k) where the measured structure factor
(9)
is a weighted superposition of the Sij(k) and where where xi = ni/n = Ni/N is the molar fraction of species i, n = N/V the total equilibrium density, and hij(k) denotes the Fourier transform of hij(r) = gij(r) - 1. It is furthermore of interest to consider the fluctuations of the total local density P
(3)
whose correlation defines the number density structure factor
P
(4) Another relevant quantity is the compressibilitystructure factorbJ0 S,(k) which is related to the response of the system to an external perturbation of wavevector k which compresses the system
-
P i=l
The reason for rewriting expression 7 as in eq 9 is that the latter is the natural extension to mixtures of the wellknown expression I(k) = N f ( k > S ( k )for the scattered intensity by a monodisperse system of N particles in the scattering volume and with f ( k ) being the form factor. If one considers the scattering of neutrons instead of photons, the above equations for the scattered intensity remain valid; the refractive indices have to be replaced by the densities of neutron scattering lengths. Since we will restrict ourselves in this paper to mixtures of particles with equal diameters and indices of refraction, it is convenient to point out that in this case the measured structure factor SM(k)reduces to S"(k). So we will see only the effects of the polydispersity due to the particle interactions and not to the particle scattering properties. 3. Theory and Computer Simulation 3.1. Ornstein-Zernike Equations and Their Closure Relations. The calculation of the microstructure for a system with given pair interaction potentials uij(r) is performed by solvingthe coupled Ornstein-Zemike (OZ) equations10 D
where the numerator is the determinant of the symmetric matrix of the structure factors Sij(k) and IS(k)lij is the cofactor of the (ij)element. The long wavelength limit of S,(k) is the normalized isothermal compressibility x of the mixture
x = nk,TKT = S,(k=O)
(6)
where KT is the isothermal compressibility. The microstructure of the mixture, as described bygij(r) in real space or Sij(k) in Fourier space, determines the angular-dependent static light scattering intensity P
Ni
which relate the total correlation functions hij(r) and the direct correlation functions cij(r),together with additional approximaterelations between hij(r),cij(r),and Uij(r),called closurerelations. Among the various closurerelations used in the literature we will be concernedhere with the PercusYevick (PY) relationslo cij(r)= e-@"ij@)[yij(r) + 11- yij(r)- 1 the hypernetted-chain (HNC) relationslo cij(r)= e-8uij(r)+yij(r)
- yij(r)- 1
(13) (14)
and the Rogers-Young (RY) relations12 Here, k is the scatteringvector, whosemagnitude is related to the scattering angle 0 by k = (47r/X) sin (W2) with X the wavelength of light in the system. The position of particle a of species i is denoted by rai,and fi(k) is the scattering amplitude. For spherical particles of diameter gi and a homogeneousdistribution of scattering material, we have
Here, Ri and fh are the indices of refraction of the particles of species i and the solvent, respectively, and j l ( k ~ i l 2 ) denotes the first-order spherical Bessel function. (10) Haneen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic: London, 1988.
where yij(r)
hij(r) - cij(r) and
fij(r) = 1 - e-W (16) As can be seen from eqs 15 and 16, the RY relations reduce to PY for aij = 0 and to HNC for aij OD, For values of aijbetween these limits, the RY closure relation is therefore mixing the two other closure schemes. However, the RY scheme necessitates additional conditionsto (11) Puaey, P. N.;Tough, R. J. A. In Dynamic Light Scattering; ed. Pecora, R., Ed.; Plenum: New York, 1985; p 85. (12) Rogers, F. J.; Young, D. A. Phys. Reo. A 1984, 30, 999.
Langmuir, Vol. 8, No.12, 1992 2916
Structure of Binary Colloidal Mixtures fix the values of These conditions are introduced in order to partially correct some deficiencies of the closure relations (13and 14)which do not contain free parameters. Since the closure relations are approximations, their use in calculating thermodynamic properties will, in general, lead to different results, when these properties are derived from different routes. For example, the isothermal compressibility can be calculated either from the compressibility equation of state P
or from the virial equation of state
mixtures of two types of hard spheres;one speciesconsists of uncharged particlesof diameter a, whereas the particles of the second species have diameter uy and carry a charge Q. These particles will be referred to as hard particles and Yukawa particles, respectively. The pair potential between hard particles is UHH(r) = O3 =O
uHy(r) = O3
(13) Zerah, G.; Haneen, J. P. J. Chem. Phys. 1986,84, 2336. (14) Rosenfeld, Y.;Ashcroft, N. W. Phys. Rev. 1979, A20, 1208. (15) Haneen, J. P.; Zerah, G. Phys. Lett. 1985, A108, 277.
r < UHy = ( b H
+ ay)/2
r l uHY
(21)
and the one between the Yukawa particles is taken as the repulsive part of the Derjaguin-Landau-Verey-Overbeek potential16
un(r) = = In the following, the isothermal compressibility, as calculated from eq 17 and eq 18, is denoted by xc and x,, respectively, and the pressure, as calculated from eq 19, is denoted by P,. If the OZ equations are solved either with the PY or the HNC closure and the results are used in eqs 17 and 19, the two isothermal compressibilitiesxc and xv will in general be different; there is some thermodynamicinconsistency. Using instead the RY closure, eqs 15 and 16 with aij = a for all i and j , one can fix the value of a by demandingxc = xv = x. This is the additional condition mentioned above. In this way thermodynamic consistency is, at least partially, restored. An additional advantage of the fact that the RY scheme mixes the PY and HNC closures arises from the observation that for charge particles the PY scheme overestimates and the HNC scheme underestimates the structure as compared to results obtained by Monte Carlo simulations. Therefore, the mixing of the two schemes, where the amount of mixing is determined from xC= x,, can be expected to result in an improved microstructure. The RY closure relations belong to a more general class of thermodynamicallyself-consistent closurescalled,after Zerah and Hansen,l3 HMSA closure relations. These relations interpolate, through the same mixing function of eq 16, between the soft mean spherical approximation (SMSA)13at small r and the HNC equations at large r. The generality of the HMSA relations comes from its capabilityof treating interaction potentials with attractive and repulsive parts. When the attractive part is not present, HMSA reduces to the RY closure. Between the thermodynamicallyself-consistent schemes we have also to mention the modified HNC (MHNC) cl0~ure.l~ Although the MHNC scheme gives the same quantitative agreement of the RY scheme when comparedto the "exact" simulation data? we do not use the MHNC closure since its extension to mixtures is not trivial.16 3.2. Model for Mixturesof Chargedand Uncharged Particles. Previous investigations of colloidal mixtures were restricted to either mixtures of charged or uncharged particles. Very good agreement with MC has been found if the uncharged particles are treated as hard spheres, and the charged ones as Yukawa particles. Here we consider
(20)
rlaH
the one between hard and Yukawa particles is of the same type =O
where the pressure P is given as
r < UH
r < uy
where K is the Debye-Hiickel screening parameter
Here, the index m denotes the various kinds of small ions (counterions and salt ions) of charge q m and number density nm. It should be noted that expression 22 for the interaction potential between the charged spheres in an effective potential which describes the small ions only through the screening parameter. There are various approaches to justify eq 22, which are also showing its limitations.17-23 In the next section we will present results for the pair distribution functions gij(r) and the structure factors SM(k)and S,(k) which are obtained in various approximations. It is known that the PY closure gives a rather good description for hard sphere systems, whereas the HNC scheme is better than PY for Yukawa systems. It is therefore suggested to calculate gHH(r) and gHY(r) with the PY closure and g w ( r ) using the HNC closure. This corresponds to CYHH = (YHY = 0 and a w = in the RY scheme and this method will be denoted as PY-HNC. These results will then be compared with those obtained from using the RY closure with one mixing parameter, CYHH= CYHY= ayy = a,where a is obtained by demanding xc = x,; this approximation will be denoted as RY1. As an alternative we will also treat the system by a twoparameter version of the RY scheme (RY2). Since UHH(r) and UHY(P) are both hard sphere interactions, whereas uw(r) is of long range, it is assumed that CYHH= (THY, but that a w can have a different value. Then two conditions (16) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (17) Medina-Noyola, M.; McQuarrie, D. A. J. Chem. Phys. 1980, 73, 6279. (18)Beresford-Smith, B.; Chan, D. Y. C.; Mitchell, D. J. J. Colloid Interface Sci. 1985,105, 216. (19) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincue, P. J. Chem. Phys. 1984,80,5776. (20) Belloni, L. J. Chem. Phys. 1986,85, 519. (21) Ruiz-Eetrada, H.; Medina-Noyola,M.; NHgele, G. Physica A 1990, 168,919. (22) Ronis, D. Phys. Rev. A 1991,44,3769. (23) LBwen, H.; Madden, P. A.; Hansen, J. P. Phys. Rev. Lett. 1992, 68, 1081.
2916 Langmuir, Vol. 8, No.12,1992
Mhdez-Alcaraz et al.
are needed. Among the various possibilities of splitting the compressibilities xc and xv up into hard sphere and Yukawa contributions, the following choice is made: Expressions 17 and 18 are written as &-' = 1 + ( X i l ) H xv-'
1 + (X;')H
+ (x:')y + (x;')y
(24)
g(r) I
3
(25)
where
2
1
0
and
1
11
12
13
II
15
r/U
Figure 1. MC, PY, HNC, and RY results for g(r) for a monodisperse system of hard spheres.
In eq 29, gHH(UH+) denotes the contact value of gHH(r) (r-c UH+), and similarlyforgHy(my+).The two conditions for (rHH = (YHy and for ayy are now obtained by demanding (Xc)H
= (Xv)H
(31)
and
= (XJy (32) It is clear from eqs 24 and 25 that the totalcompressibilities xc and xv are equal, if eqs 31 and 32 are simultaneously fulfilled. To calculate the various virial-type compressibilities in eqs 18 and 28, the pressures are first evaluated for the densities n and n f An, assuming that the a's have the same values for the three rather close values of the density. The derivatives of the pressure are then determined numerically. 3.3. Computer Simulation. The validity of the suggested schemes for the calculationof the structure and thermodynamicpropertiesof mixturesof hard and Yukawa particles is teetsd makingthe comparisonwith Monte Carlo (MC) results. The MC simulationsare carried out using a conventional Metropolis MC (NVT canonical ensemble) methodz4on a totalnumber of Nparticles. A value of N = 864 is chosen, except for the case xy = 0.9 for which extra runs were performed using N = 1372. Initially the particles are put randomly on the fcc lattice and the system is equilibrated performing 2000 MC cycles (1 cycle = N trial moves). The equilibration was checked by monitoring the values of the potential energy and of the instantaneous pressure (for the Yukawa particles) and of the translational order parameter. Long before the end of the equilibrationcycles the potential energy and the pressure ceased to show any systematicdrift. At the same time, the translational order parameter was oscillating about zero with amplitude 0(N-ll2). The pair correlation functions gij(r) are then evaluated by performing block averages over MC runs consisting typically of 4000 cycles. During the runs, the (XJy
(24) Allen, M. P.; Tildeeley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1990.
maximum allowed displacement of the particles is automatically readjusted in order to maintain the acceptance ratio around 0.5, and its value is chosen to be the same for both kind of particles. The Values Of gHH(r) and gHy(r) at contact cannot be directly evaluated. They are extracted using the extrapolation procedure of Freasier.24126All the parameterswhich characterizethe investigatedsystem and which are needed in the MC runs are listed in the following section which presents the results. 4. Results 4.1. Monodisperse Systems. In order to understand better the role of the mixing of PY and HNC as it is done by using the RY scheme and to illustrate the accuracy of the various schemes for hard spheres and for Yukawa particles, we first consider a monodisperse hard-sphere system and a monodisperse Yukawa system. Figure 1 shows results for g(r) for a rather concentrated suspension of hard spheres of volume fraction (a = 0.4 obtained from PY, HNC, and RY and compared to MC data. Although the differencesbetween the various results are not large and are only noticeable near contact, it is seen that in the neighborhood of r = u the PY results are below and the HNC results are above the MC data. Moreover, the PY scheme provides more accurate results thanHNC, which is in agreement with previous experience. The curve corresponding to the RY closure is very close to the MC results. Thermodynamic consistency was obtained for a* = au = 0.26, which shows that the RY scheme gives more weight to PY than to HNC. All further values of a which are quoted in what follows are always in units of u = UH = uy. This behavior is to be contrasted to the case of a Yukawa system with its long-range interaction. Figure 2 shows results for g(r) for a system of volume fraction (a = 0.2, diameter u = 50 nm, and charge Q = 1OOe- at temperature T = 300 K. For this system monovalent salt has been added at a normalized density n*dt = n d t d = 62.5. In contrast to the hard-sphere case, the P Y results are now overestimatingand the HNC results are underestimating the structure of g(r), although HNC is more accurate than PY. The agreement of RY and MC data is excellent. The thermodynamic consistency (xc = xv) gave a = 0.81, showing that the RY scheme contains more of HNC character than for hard spheres. (25) Freasier, B. C. Mol. Phys. 1980,39, 1273.
Structure of Binary Colloidal Mixtures
Langmuir, Vol. 8, No.12, 1992 2917
-- RY ( a = o 8 1 )
--- HPYN C
I\
I' '\,
0000
MC
u3 = 0.2
6=50nm
I
Q = 100e-
T =300K n'&,,: 62.5
BE
9 = 0.2
I
ol
O H =Oy = 50
Q =100e-
1
0.5
nm
X Y = 0.5
T=WK
nsalt=O -PY-HNC ...... R Y I (Cr=l.081 - - - RY2 ( a n = 0 0 1 5 , a y=1.23)
O Q 1
MC
0000
' I
1.5
2
2.5
3
1.5
3.5
2 r /UH
rlG
Figure 2. MC, PY, HNC, and RY results for g ( r ) for a monodisperse system of Yukawa particles.
2
Table I. P x / n h T for the Monodisperse System of Hard &heres in Figure 1. HS MC PY HNC RY FYnkBT 5.99 5.33(') 7.36(') 5.78(") 0 The results from the integral equation theories are obtained from the virial equation of state.
1.5
Table 11. P x / n h T and t P = / N h Tfor the Monodisperse System of Yukawa Particles in Figure 2. Y MC PY HNC RY Fa/nkBT 18.12 16.17m 18.91(') 18.13(') UgalNkeT 7.64 6.57(e) 8.08") 7.65") 0 Upper indices (v) and (e) refer to the virial and to the energy equation, respectively.
Concerning the ability of the various schemes to reproduce the correct thermodynamics, we compare in Tables I and I1 the excess pressures and excess internal energies for the two systems of the Figures 1 and 2, respectively. In view of the results for the pair distribution functions, it is no surprise to observe the very good agreement of RY and MC for the thermodynamics of the systems. 4.2. Mixtures of Hard and Yukawa Particles. We are now presenting the results for the structure (g.(r)), the angular distribution of scattered radiation (Sd(k)), and the compressibility structure factor (S,(k))for mixtures of charged and uncharged particles. Figure 3 shows the partial pair distribution functions for a system of total volume fraction cp = 0.2, diameters UH = uy = 50 nm, Q = 100e-,and molar fractions ZH = xy = 0.5, which contains no added salt. The results have been obtained by the PY-HNC method and the RY1 and RY2 schemes and are compared to MC data. Whereas and gHY(r) are fairly well reproduced by all three methods, the RY schemes give much better resulta forgyy(r)than PY-HNC. The thermodynamicconsistencywas obtained for the RY1 method with a = 1.08, which means that the HNC contribution is important. By use of conditions 31 and 32 for determining the parameters of the RY2 scheme, the values (YH E O~HH= CYHY= 0.015 and a y ayy = 1.23were found. As it is evident from Figure 3b for gyy(r),the RY2 results are not improvingthe slight disagreementnear the main maximum, they are instead slightly below the curve for RY1. We have therefore to conclude that the twoparameter RY scheme, for which the parameters are determined by the conditions in (31) and (32), is not an appropriate extension of the computationally much simpler RY1 closure relation. For this reason all further
I 3
2.5
0.5
1
1.5
2
2.5 3 r/GH
L
3.5
L.5
2.5 -;.2 >
d 1.5 1
t
0.5
01 1
1.5
2
2.5
3
3.51
r /On
Figure 3. MC, PY-HNC, RY1, and RY2 results for the pair distribution functions of a binary mixture of hard and Yukawa Particles: (a) gHH(r); (b) g d r ) ; (C) gHY(r). Table 111. P x / n h ? and LP./NhT for the Binary Mixture of Hard and Yukawa Particles in Figure 3. HS-Y MC PY-HNC RYl RY2 F'JnkBT 22.84 22.96(') 22.98(') 22.82(") Uea/NkBT 13.22 13.52(O) 13.29U 13.34@) Upper indices (v) and (e) refer to the virial and to the energy equation, respectively.
calculations have been performed with RY1. The thermodynamic quantities are well described by the three versions of the closure relations, as can be seen from Table 111. We are now changing the molar fraction xy of Yukawa particles in order to study the influence on the structure of adding charged particles to a system of uncharged hard
Mkndez-Alcaraz et al.
2918 Langnuir, Vol. 8, No. 12, 1992 3 1
I
. _
0
1
1.5
2 r/a,
- R Y I (CL=I.II)
I
xy=O3
2.5
3
0000
MC xy
0.6
7
OI
2
1.5
3
2.5
r /aH
1.5
1.0
0.5
0
"
0
"
"
"
I
IO
5
'
"
'
15
OH
o--' 0
/ , "
"
5
"
"
IO
"
"
15
k OH
Figure 4. MC and RY1 results for the structure of a binary mixture of hard and Yukawa particles with X Y = 0.3: (a) gHH(r), gHY(r), and g y d r ) ; (b) SM(k)(=SNN(k)) and S,(k). The other parameters are as in Figure 3.
Figure 5. MC and RY1 results for the structure of a binary mixture of hard and Yukawa particles with 3ey = 0.6. (a) gHH(r), g d r ) , and g y d r ) ; ,(b) SM(k) ( = S N N (and ~ ) )S,(k). The other parameters are as in Figure 3.
spheres and of adding hard spheres to a system of charged particles. Results are shown in Figures 4,5, and 6 for the partial pair distribution functions,the measured structure factor, and the compressibility structure factor for xy = 0.3, 0.6, and 0.9, respectively. The RY1 results for the gij(r) are for all cases in very good agreement with the MC data; the values of the mixing parameter were found as CY = 0.93, 1.11,and 1.22, respectively. The increase of CY is expected, since the number of Yukawa particles and therefore the HNC part of the mixing increases. Comparingthe behavior of gHH(r)in Figures 4a, 5a, and 6a, it is seen that the contact values gHH(UH+)decrease with increasing xy and that the second maximum of gHH(r) increases. At the same time,gHY(uHY+)is growing. From these observations it follows that the replacement of hard spheres by Yukawa particles changes the local order of the hard spheres. Since the Yukawa particles want to stay as far from each other as possible because of the long-range character of their repulsive interactions, the hard spheres have to go between the charged particles. As a result, the hard spheres will have mostly Yukawa particles as next neighbors (increase of gHY(UHY+) with increasing xy) and the most probable distance of hard spheres is no longer at UH but at the position of the second maximum of gHH(r), which is roughly at 2 ~ . This behavior of the partial pair distribution functions has profound consequences for the dependence of the on scatteringangle. What measuredstructure factorSM(k) is seen as a weak shoulder in Figure 4b at xy = 0.3 grows for xy = 0.6 in Figure 5b to produce a second peak of SM(k) and becomes the only peak at xy = 0.9 in Figure 6b. The
reason for this behavior is the presence of two characteristic length scaies in the mixture of charged and uncharged spheres. At low values of xy the order in the system is typical for hard sphere suspensions,in which the particle diameter is the characteristic length scale. At high values of XY the order is essentially determined by the Yukawa particles, for which the mean interparticle distance (proportional to ny-l4 is the characteristic length scale. The behavior seen in Figure 5b shows the crossover between the two length scales. Finally, it is interesting to compare SM(k)and the compressibility structure factor SJk), in particular in the long-wavelength limit, where according to eq 6 S,(k=O) determines the normalized isothermal compressibility x of the system. As shown in Figure 7,the behavior of x and of SM(0) = S"(0) is very different. The compressibility can be obtained from the experimentally accessible scattering intensity for k 0 only for monodisperse systems. For mixtures, where 0 < xy < 1,the quantities x and SM(0) have to be distinguished. It is seen from Figure 7 that the replacement of a few hard spheres by Yukawa particles has a profound effect on the compressibility,whereasthe replacement of a few Yukawa particles by hard spheres changes x only very little. This behavior is to be contrasted with the effects of such replacements on the ordering in the systems, as described by S'(k). Even at x y = 0.3the measured structure factor is not much different than at XY = 0, whereas x has dropped from 0.216 to 0.031. However,near xy = 1the effect of replacing Yukawa particles by hard spheres weakens the structure strongly. Therefore, we can conclude that for the deter-
-
Langmuir, Vol. 8, No. 12, 1992 2919
Structure of Binary Colloidal Mixtures
-R Y I (a-1.221 0000
MC X"
0
1.5
1
2
= 0.9
3
2.5
r /OH
6
L
8
k OY
Figure 8. Comparison between the experimentally measured partialstructure factorsSHH(k)and SYY(k) and thosedetermined with PY-HNC.
2.5 2.0
by contrast matching; if the neutron scattering length density of the solvent can be varied such that it becomes equal to that of one of the suspended particles, only the other kind of particles scatters the neutrons. Suppose that tis = iiy in eq 8,then eqs 9 and 10 lead to
1.5 1.0
0.5
0 OH
Figure 6. MC and RY1 results for the structure of a binary mixture of hard and Yukawa particles with xy = 0.9. (a) gHH(r), gHY(r), and gyy(r);.(b) SM(k)(=SNdk)) and S,(k). The other parameters are as in Figure 3. SM(k=O)
0000
0.2
I\
0.05
I
1\
0 '
0
'
' 0.2
'
"
0.L
'
0.6
'
I
0.8
- I
'
-r I
XY
Figure 7. RY1 results for the values of SM(k=O)and S,(k=O) as a function of xy. The other parameters are as in Figure 3.
mination of the local order, as represented by the peak value of the microscopic quantity S"(k) = SM(k),the hard spheres play a dominant role. For the isothermal compressibility, which is a macroscopic quantity, the Yukawa particles are the important ones. 4.3. Comparison with the Experiment. The total scattered intensity I(&, which is proportional to SM(k), eq 9, is a superposition of the partial structure factors Si,(k). Since Sij(k) is a Fourier transform of gij(r), the experimentaldeterminationof the partial structure factors provides a much more direct test of the microstructure than the measurement of SM(k). Thiscan be accomplished
so that the hard-sphere partial structure factor can be ~ can get Syy(k), determined. Similarly, for iis = i i one and from a third sample with ii, # i i and ~ iis # ily, it is possible to determine SHY(k). Preliminary results of such experiments on a mixture of polystyreneand silica spheres by the contrast-matching methods have been reported and the partial structure factors S H H (and ~ ) Syy(k) were obtained. Figure 8 shows these results together with our attempts to describe them in terms of the model. We assume that the only small ions in the suspension are counterions and we vary the value of the charge Q, which was not reported by Hanley et al.! until the peak height of Syy(k) agrees with the results obtained from the experiments. It is seen that the theoretical results do not reproduce the peak position of Syy(k) and that SH&) has very little structure. Those calculations have been performed for the reported values of the totalvolume fraction cp = 0.15 and the molar fraction x y = 0.64 of Yukawa particles. Since the peak position of Syy(k) is mainly determined by cp, we have increased cp to find agreement with the experimentally determined peak position. This is approximately accomplished for cp = 0.3. But even then, S H H (is~not ) well reproduced. In view of the very good agreement of the results of the integral equation theories with those of the Monte Carlo simulations, we conclude that the disagreement of the theory and the experiment is either due to the way, in which the determination of the partial structure factors from the measured scattered intensities is performed or it is due to the failure to model the mixtures of polystyrene and silica particlesas mixturesof hard spheresand Yukawa particles. More experimentalwork on these systemsseems to be necessary to clarify this situation. 5. Discussion and Conclusions
From the theoretical point of view it is quite satisfactory to observe the excellent agreement of the Ornstein-Zemike method together with the Rogers-Young closure relation with the Monte Carlo simulation results for the mixtures of charged and uncharged sphericalparticlesover the whole
2920 Langmuir, Vol. 8, No. 12, 1992
range of relative composition. The fact that the RY scheme leads to more reliable results than the Percus-Yevick and the hypernetted-chain closures is to be expected on the basis of previous experience. The inclusion of more than one mixing parameter is however no garantee for further improvement of the resulta, although only one of several possible choices to establish (partial) thermodynamic consistencyhas been made in the present study. Another choice has been used in an investigation of a different system without any significantchange of the results of the one-parameter version.26 With regard to the structure of the investigated mixtures, it is found that the replacement of a relatively small amount of charged by uncharged particles has a much larger effect than the replacementof a comparableamount of uncharged by charged spheres. The influence of such (26) Bernu, B.;Hansen, J. P.;Hiwatari, Y.;Pastore, G.Phys. Rev. A 1987,36,4891.
Mhdez-Alcaraz et al.
replacementon the thermodynamic properties is however quite different; they are primarily determined by the charged particles. The form of the measured structure factor SM(k), as the molar fraction of charged particles is increased,is the result of the interplay of two length scales; the particle diameters are the only relevant scale for hard sphere suspensions, whereas it is the mean interparticle distancewhich is the Characteristiclength scalefor Yukawa systems. As the composition of the mixtures is changed, the relative importance of the two lengths changes, and this is reflected in a characteristic fashion in the total structure factor.
Acknowledgment. We are grateful for the financial support of the Deutsche Forschungsgemeinschaft (SFB 306). J.M.M.-A. acknowledges a fellowship from the Deutacher Akademischer Austauschdienst (DAAD). Registry No. Polystyrene, 9003-53-6; silica, 7631-86-9.
