conference papers

Dynamics of dense, charge-stabilized                                        with the results of the measurement of the diffusion coefficient in the
suspensions of colloidal silica studied by                                  zero density limit we determine the hydrodynamic interaction over
                                                                            the relevant wave-vector range free from any modeling of the static
correlation spectroscopy with coherent                                      or dynamic properties.
X-rays                                                                          Correlation spectroscopy determines the dynamic properties of
                                                                            matter by measuring the temporal correlations in the fluctuations of
                                                                            the intensity scattered by the sample. Correlations can be quantified
                                                                            via the normalized time correlation function

Gerhard Grübel, = Douglas L. Abernathy, = Dirk O. Riese,>                             g(Q,t) =  < I(Q,0) I(Q,t)> / <I>2 = 1 + A(Q) | f(Q,t) |2,    (1)
Willem L. Vos > and Gerard H. Wegdam >
aESRF, BP220, 38043 Grenoble, France                                        where I(Q,t) is the scattered intensity at wave-vector Q and the
bVan der Waals-Zeeman Institute, U. Amsterdam, 1018 Amsterdam,              angular brackets denote the average. A(Q) is a function depending on
The Netherlands                                                             the coherence properties of the source and typical values range
Email:gruebel@esrf.fr                                                       between 0.1 and 0.25 in the X-ray case. The function f(Q,t) denotes
                                                                            the normalized intermediate scattering function also known from
                                                                            quasielastic light or neutron scattering. The simplest functional form
The dynamics of concentrated, charge-stabilized colloidal silica            for a translational diffusion process is an exponential
suspensions was studied over a wide range of wave-vectors. The
short-time diffusion coefficient, D(Q), was measured for                                             f(Q,t) = exp ( -D(Q) Q2 t),                   (2)
concentrated suspensions up to their solidification points by photon
correlation spectroscopy with coherent X-rays and compared to free          where D(Q) is the diffusion coefficient. The exponential decay of
particle diffusion D , studied by Dynamic Light Scattering (DLS) in         temporal correlations is illustrated in Fig. 1 showing a DXS
                     0
the dilute case. Small angle X-ray scattering (SAXS) was used to            measurement (open circles) of the intermediate scattering function of
determine the static structure factor S(Q). D /D(Q) peaks for Q             a sample of colloidal silica suspended in a mixture of glycerol/water
                                                 0 
values corresponding to the maximum of the static structure factor          with a refractive index of 1.42 for visible light, making the sample
showing that the mostly likely density fluctuations decay the slowest.      opaque to the eye. The result of a DLS measurement from the same
The data allow one to estimate the diffusion coefficient D(Q) in the        sample is shown for comparison. The  DLS intermediate scattering
Q   0 and Q      limits. Thus, hydrodynamic functions can be                function (closed squares) relaxes faster as a result of multiple
derived free from any modeling of the static or dynamic properties.         scattering since the light wave scatters from more than one moving
The effects of hydrodynamic interactions on the diffusion coefficient       particle. It hence suffers a larger phase shift and the correlation is lost
in charge-stabilized suspensions are presented for volume fractions         faster than for single scattering. Although the characteristic square
0.075 <   < 0.28.                                                           root dependence of f(Q,t) on time, known from diffusive wave
                                                                            spectroscopy (Pine et al., 1990), is not yet observed one readily
                                                                            realizes that the correlation time is already affected. The DXS data
1.  Introduction                                                            are not subject to multiple scattering effects since the refractive index
Dynamic Light Scattering (DLS) with visible coherent light from a           for X-rays is always very close to one. DLS and DXS results are
laser source is a well established technique to investigate the             identical in the absence of multiple scattering in perfectly index-
dynamic properties of colloidal suspensions (Pusey, 1989). DLS, also        matched samples. A detailed discussion of multiple scattering effects
known as photon correlation spectroscopy, is based on a very                will be given in a forthcoming paper (Riese et al., 2000).
prominent feature, occuring whenever coherent light is scattered
from a random medium. There are strong spatial modulations of the
scattered intensity, called a speckle pattern, resulting from the
summing of the randomly phased electric fields from the individual
scatterers in the medium. If the spatial arrangement of scatterers
changes in time the corresponding speckle pattern will also change
and a measurement of the intensity fluctuations at a point in the far
field can reveal the dynamics of the system. Photon correlation spec-
troscopy in the visible is however subject to two main limitations : (i)
the occurence of multiple scattering in opaque systems (e.g. concen-
trated colloidal suspension) considerably complicates the interpreta-
tion of the experiments, (ii) the use of visible light prevents the
dynamics to be traced on length scales smaller than about 2000 Å.
Both limitations can be surmounted by correlation spectroscopy with
coherent X-rays (DXS) provided by a synchrotron source. DXS is a
novel technique that only recently has started to be applied to colloi-
dal systems (Dierker et al., 1995; Thurn-Albrecht et al., 1996; Tsui &      Figure 1
Mochrie, 1998) including free particle diffusion in a dilute suspen-        Intermediate scattering functions f(Q,t) measured by DXS (open circles) and
sion of colloidal silica (Grübel et al., 1999). We have studied the         DLS (closed squares) at Q=0.0009 Å-1 in an optically opaque sample of
static and dynamic properties of dense, charge-stabilized colloidal         colloidal silica (1820Å radius) suspended with a volume fraction of  =0.09
silica suspensions by scattering with coherent X-rays. Combined             in a glycerol/water solvent.

424 # 2000 International Union of Crystallography   Printed in Great Britain ± all rights reserved                J. Appl. Cryst. (2000). 33, 424±427



                                                                                                                             conference papers

In the dilute case (  << 1) the colloidal particles migrate driven by       Table 1
the thermal fluctuations of the solvent, with a diffusion coefficient       Summary of quantities obtained from static SAXS characterization.
given by the Stokes-Einstein relation,                                          Volume                  Average                              Size               Location of first
                       D(Q) = D  = kT / 6  R ,                      (3)         Fraction                radius                    polydispersity                   S(Q) peak
                                 0                         h                                             R[Å]                            R/R                          Q R
                                                                                                                                                                            o
                                                                                 0.075                   561                            0.017                         2.41
where D  is the (free particle) diffusion coefficient, k is the
           0                                                                      0.15                   555                            0.035                         2.47
Boltzmann constant, T the temperature,   the viscosity of the solvent             0.28                   481                            0.150                         2.79
and R  the hydrodynamic radius of the diffusing particle.
      h                                                                     with size  . A partially coherent X-ray beam was selected by placing
    At larger concentrations, interparticle interactions as well as         a d=20 µm collimating pinhole aperture 0.8m downstream of the
indirect, hydrodynamic interactions, mediated by the solvent become         mirror. The longitudinal coherence length     = ( /  ) = 1µm was
important. Then, the short-time (t<R2/D ) behaviour of the interme-                                                                                     l
                                                 0
diate scattering function can be described by a wave-vector depen-          determined by the bandpass   /  =1.4*10-4 of the Si(111) mono-
dent diffusion coefficient                                                  chromator. Coherent illumination of the sample volume requires that
                                                                            the maximum pathlength difference (d2sin22  + h2[1-cos2 ]2)1/2     l
                        D(Q) = D  H(Q) /S(Q),                       (4)     for rays scattered at an angle    in transmission through a sample
                                      0                                     with thickness h (=1mm). This is the case for Q < 0.15  Å-1  and is
where H(Q) is the hydrodynamic function and S(Q) the structure              sufficient for the present measurements. Experiments at even larger
factor (Hansen & McDonald, 1986). The slowing down of the                   wave-vectors are possible but might require a higher monochro-
diffusion around the peak in S(Q) is by now well established and is         maticity of the beam and the use of fast 2-D detectors. The samples
often referred to as the cage effect. However it is still an open           were positioned 0.13m downstream of the collimating aperture in an
question how the diffusion behaves in the small and large wave-             evacuated chamber connected to a drift tube. A guard slit was posi-
vector limits. For large Q one expects the colloidal particle to            tioned right in front of the sample capillaries to eliminate parasitic
perform a random walk inside the cavity formed by the surrounding           scattering from the collimating aperture. In order to detect intensity
particles. For hard spheres theory predicts H(Q    )    1. To date          fluctuations the detector aperture has to be comparable to the angular
there are no measurements to assess the influence of the direct and         size  /d of the speckles. An analysing aperture of 30 µm size was
hydrodynamic interactions on the diffusion process in this wave-            used in front of a scintillation counter at a distance of 1.4 m from the
vector regime.                                                              sample and correlation functions were recorded with a digital auto-
    For light scattering the determination of D(Q) and S(Q) over a          correlator. The SAXS data were taken in the same configuration.
large wave-vector range is a formidable task. The requirement of                DLS data were taken with a standard set-up including a diode
index matching to suppress multiple scattering has limited the DLS          pumped, frequency doubled Nd:YAG laser with an output of 100
measurements to hard-sphere and/or low density systems. A                   mW at  =5320 Å. Polarizers were inserted to observe just the VV
consistent picture has nevertheless emerged for hard-sphere fluids by       component of the scattering with a photomultiplier connected to the
a combination of experimental DLS data and hard-sphere cal-                 autocorrelator.
culations of the form and structure factors. The availability of                The colloidal silica particles were synthesized by polymerization
analytical expressions for the structure factor of the hard-sphere fluid    in a microemulsion or by Stöber synthesis. The dilute stock
within the Percus-Yevick approximation (Hansen & McDonald,                  suspension in pure ethanol was centrifuged to reduce the solvent
1986) and the calculations of hydrodynamic functions by Beenakker           content and resuspended in an ethanol-58 vol% benzylalcohol
and Mazur (1984) were essential for the interpretation of the data.         mixture with refractive index n=1.465 matching the refractive index
For charge stabilized colloidal systems the situation is more               of the silica particles. A concentrated master suspension (about 40
complicated. For these systems only numerical models for the                vol%) was prepared by centrifugation and a series of samples with
structure factor exist : the mean spherical approximation MSA               intermediate volume concentrations were prepared by diluting with
(Hayter & Penfold, 1981) or the rescaled RMSA version (Hansen &             known amounts of liquid. Samples from different batches with the
Hayter, 1982). Hydrodynamic effects in charged colloidal                    characteristics listed in table 1 were used.
suspensions have been calculated for the monodisperse case (Genz &
Klein, 1991) and for polydisperse systems (Nägele et al.,1993).             3.  Results and discussion
Experimental attempts to study interacting, charge-stabilized systems
are however scarce (Philipse & Vrij, 1988; Phalakornkul et al.,             Structural information in terms of size, shape, size polydispersity and
1996).                                                                      interparticle correlations were obtained by SAXS measurements in
                                                                            the concentrated and dilute suspensions. The scattering cross section
                                                                            may be described by
2.  Experimental details
The experiments were carried out at beamline ID10A (Grübel &                                   (d /d )/V = r 2 n  v 2 (  -  ) 2 |F(Q)| 2 S(Q)                                     (5)
                                                                                                                   o    p    p          c         s
Abernathy, 1997; Abernathy et al., 1998) of the European
Synchrotron Radiation Facility (ESRF) with X-rays of 8.2 keV                where r  is the Thompson radius, n  is the number concentration of
                                                                                    o                                              p
(1.51Å) energy, supplied by a single bounce Si(111) monochromator.          particles,    and    are the electronic densities of the colloidal particle
                                                                                          c        s
A vertically focussing Rh-coated Si mirror was set to a critical            and of the solvent respectively and v  is the particle volume. The
                                                                                                                                             p
energy just above 8.2 keV for harmonics rejection. The nominal              form factor F(Q) for a sphere of radius R is given by F(QR) =
transverse coherence lengths    = R  /2  were 144 µm (vertically)
                                 t         s                                3[(sin(QR)-QRcos(QR))/(QR)3] and size polydispersity  R/R was
and 13 µm (horizontally) at a distance R   = 44m from the source            taken into account by convolution with a Schultz distribution. The
                                                      s



J. Appl. Cryst. (2000). 33, 424±427                                                                                                                    G. GruČbel et al.         425



conference papers

                                                                                tudes that increase with increasing volume fraction. The positions of
                                                                                the D /D(Q) maxima mimic the behaviour of the static structure
                                                                                       0 
                                                                                factor and shift also to higher wave-vectors with increasing
                                                                                concentration, consistent with eq. (4). This confirms that the most
                                                                                likely density fluctuations, the ones measured at the structure factor
                                                                                peak, are the ones that decay the slowest.
                                                                                    The collective diffusion coefficient Dc=D(0) (i.e. the Q Ę  0
                                                                                limit of D(Q)) is finite and  D /Dc ranges between 0.2 and 0.7.  The
                                                                                                                   0 
                                                                                large-Q diffusion coefficient D( ) can be qualitatively estimated
                                                                                from the data for QR>3.5. It is smaller than the free particle diffusion
                                                                                coefficient (D( ) <D ) and D( ) decreases with increasing volume
                                                                                                         0
                                                                                fraction. This is expected with similar values for both charged and
                                                                                hard-sphere systems (Nägele et al.,1993). The prediction for the

Figure 2
Scattering intensity as a function of wave-vector transfer Q of a  =0.15
suspension of colloidal silica particles (R=555Å,  R/R=0.035) in
ethanol/benzylalcohol. The solid line is a fit to the data based on the form
factor of a sphere.

finite detector resolution was taken into account by convolution with
a box shaped resolution function. Data from a concentrated sample
( =0.15) are shown in Fig. 2. The solid line is a fit of eq. 5 to the
data based on the parameters for the dilute case (R=555Å,  R/R
=0.035) and assuming S(Q)=1. The structure factor is extracted by
dividing the  intensities I(Q) by their values for the respective dilute
samples. The resulting static structure factors S(Q) are shown in
Fig. 3 (open symbols) as a function of QR for three different volume
concentrations. The S(Q) curves oscillate in all cases to one at high Q
values (not shown). The positions of the structure factor maxima
shift to higher wave-vectors when the concentration is increased and
the respective peak values increase as expected, except for the high-
est concentration ( =0.28). This is caused by the higher size poly-
dispersity ( R/R =0.15) of this sample.
        The observed structure factors cannot be described within a hard-
sphere model. This is illustrated for the  =0.15 sample in Fig. 3
(middle) by the dashed line showing the calculated hard-sphere
structure factor for R=555Å,  R/R=0.035. It clearly fails to describe
the data. Not just the position, but also the shape distinctly differs
from the calculated hard-sphere result. Further modelling of the static
data was not attempted within the scope of this paper, but will be
discussed in a forthcoming publication.
        The dynamic data were all taken on index matched samples using
an ethanol-benzylalcohol solvent. Dilute suspensions were measured
by DLS to determine the free particle diffusion coefficient D . The
                                                                     0
concentrated suspension were measured with DXS. One sample (  =
0.15) was also studied by DLS, but the restricted Q range (Q<2.4.10-3
Å-1) does not allow to access the regime of the structure factor peak.
The DXS data were corrected for the known time structure of the
synchrotron beam and the diffusion coefficients were extracted by
fitting eq. (2) to the short-time regime of the correlation functions.
The normalized inverse diffusion coefficient D /D(Q) was deter-
                                                      0 
mined by dividing the dilute (DLS) data by the D(Q) values from the
concentrated samples. The results are plotted in Fig. 3 (solid sym-
bols) as a function of QR. Fig. 3 (middle) demonstrates that DLS and            Figure 3
DXS data perfectly superimpose for index-matched samples.  At                   Static structure factor S(Q) for suspensions of colloidal silica with volume
higher Q, a pronounced peak is observed that coincides in position              fractions  =0.075, 0.15 and 0.28 (open symbols). The dashed line (middle)
                                                                                represents the calculated hard-sphere structure factor. Normalized inverse
with the maximum of the static structure factor S(Q). Maxima in                 diffusion coefficient Do/D(Q) taken by DXS (closed circles) and DLS (closed
D /D(Q) are observed for all three concentrations with ampli-                   squares). The solid lines are guides to the eye.
  0 



426 G. GruČbel et al.                                                                                                    J. Appl. Cryst. (2000). 33, 424±427



                                                                                                                  conference papers

                                                                                 quantitatively and directly the hydrodynamic functions H(Q) over the
                                                                                 whole relevant wave-vector range without taking recourse to any
                                                                                 modeling of the static or dynamic properties.

                                                                                 References
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Figure 4                                                                            A.K. Freund, H.P. Freund & M.R. Howells, pp. 103-109. SPIE-
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to the eye.                                                                         Complex Systems,  Eighth Tohwa University  International  Sym-
                                                                                    posium, edited by M. Tokuyama & I. Oppenheim, pp. 158-159.
hard-sphere case is D / D( ) =(1-1.72  + 0.88 2)-1 = 1.14, 1.31 and
                           0                                                     Hansen, J.P. &  Hayter, J.B. (1982). Mol. Phys. 46, 651-656.
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                                                            0                       New York: Academic Press.
indicating that hydrodynamic interactions are relevant for the                   Hayter, J.B. & Penfold, J. (1981). Mol. Phys. 42, 109-118.
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                     0                                                           Phalakornkul, J.K., Gast, A.P., Pecora, R., Nägele, G., Ferrante, A.,
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that H(Q) < 1 for all samples and decreasing with increasing volume              Philipse, A.P. &  Vrij, A. (1988). J. Chem. Phys. 88, 6459-6470.
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               0
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J. Appl. Cryst. (2000). 33, 424±427                                                         Received 17 May 1999   Accepted 20 October 1999      427