Dynamic Article LinksC<Soft Matter

Cite this: Soft Matter, 2012, 8, 5180

www.rsc.org/softmatter PAPER

Dow

nloa

ded

by F

lori

da S

tate

Uni

vers

ity o

n 12

Mar

ch 2

013

Publ

ishe

d on

26

Mar

ch 2

012

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C2S

M07

425G

View Article Online / Journal Homepage / Table of Contents for this issue

Decay of interfacial fluid ordering probed by X-ray reflectivity

Kim Nyg�ard†* and Oleg Konovalov

Received 20th December 2011, Accepted 14th March 2012

DOI: 10.1039/c2sm07425g

To what extent is the ordering of fluids at interfaces governed by the bulk structure of the fluid? In order

to address this question, we have studied a dense, charge-stabilized colloidal suspension at the fluid–air

interface and at two different solid–fluid interfaces using specular X-ray reflectivity. The experimental

data are well described by a simple model of a stratified fluid, with the wall–particle potential mainly

affecting the position of the first particle layer with respect to the interface. The decay of the fluid’s

density profile can for all the studied interfaces be described by a characteristic wavelength and a decay

length, both of which are independently determined from the bulk phase using small-angle X-ray

scattering. This latter finding is consistent with theoretical predictions and recent surface-force

experiments.

1 Introduction

Studies on the asymptotic decay of pair correlations in fluids

were pioneered by Fisher and Widom in the late 1960s.1 In their

seminal paper, they predicted a sharp line in the phase diagram of

simple one-component fluids, known as the Fisher–Widom line,

at which the asymptotic decay of the total pair correlation

function changes from purely exponential to oscillatory expo-

nential. Later it was realized that this also has strong implica-

tions for the density profiles of inhomogeneous fluids, whether at

a single interface or in confinement. In particular, for a fluid

exhibiting short-ranged (i.e., finite or exponentially decaying)

interaction potentials, the asymptotic decay of the density profile

is governed by the leading complex (conjugate) poles of the bulk

total pair correlation function.2 In other words, the decay of the

fluid’s density profile is determined by the decay of its bulk

structure.

The theoretically predicted connection between the fluid’s

density profile and its bulk pair correlations is a general

phenomenon, and is not limited to the aforementioned one-

component fluids exhibiting short-ranged interaction potentials.

For example, upon considering binary mixtures instead of one-

component fluids, this property manifests as the so-called

structural crossover; for mixtures with a sufficiently large size

asymmetry, there exists a crossover line in the phase diagram

separating which of the fluid components govern the decay of the

density profile.3 Recently, the structural crossover was directly

observed in the total pair correlation function of bulk binary

colloidal mixtures.4 Moreover, the analysis of the decay of

correlations in fluids in terms of leading poles has also been

European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble,France

† Present address: Department of Chemistry and Molecular Biology,University of Gothenburg, Sweden. E-mail: kim.nygard@chem.gu.se

5180 | Soft Matter, 2012, 8, 5180–5186

extended to fluids exhibiting additional long-ranged (i.e., power-

law) interactions, such as Coulomb or dispersion interactions.5

From an experimental point of view, studies on the decay of

interfacial fluid ordering are scarce. Recently, Klapp et al.

addressed this question in a series of papers6–8 by combining

surface-force experiments on charge-stabilized colloidal suspen-

sions using the colloidal-probe atomic-force microscope (CP-

AFM), small-angle X-ray and neutron scattering on the bulk

fluid, and theoretical modeling. In these studies, which were

carried out on fluids in narrow confinement rather than on fluids

at single interfaces, they obtained the first experimental evidence

for the decay of the fluid’s density profile being governed by the

decay of its bulk structure, as predicted by theory. Moreover,

they found the ‘‘asymptotic limit’’ of the density profile to be

valid already for confining slit widths of approximately twice the

particle diameter. This latter finding is surprising and therefore

warrants experimental verification using complementary

techniques.

In this paper, we complement the aforementioned surface-

force studies by probing the ordering of a colloidal suspension at

the fluid–air and at two different solid–fluid interfaces using

specular X-ray reflectivity (XRR). Three particular differences

between our approach and that of Klapp et al. should be

mentioned. First, we carry out experiments at single fluid inter-

faces rather than in narrow confinement, i.e., we carry out

experiments in a geometry where the asymptotic limit is well

defined. Second, we perform scattering experiments, and are

therefore probing the structure of the fluid instead of the ensuing

structural force. Third, we study the effect of different wall–

particle interaction potentials on the ordering of the fluid, rather

than the influence of different particle–particle interaction

potentials. The results of the present study can be summarized as

follows: (i) the XRR data can be described using a simple model

of a stratified fluid, with the wall–particle interaction potential

This journal is ª The Royal Society of Chemistry 2012

Dow

nloa

ded

by F

lori

da S

tate

Uni

vers

ity o

n 12

Mar

ch 2

013

Publ

ishe

d on

26

Mar

ch 2

012

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C2S

M07

425G

View Article Online

predominantly influencing the distance between the first particle

layer and the interface. (ii) The decay of the fluid’s density profile

can be described by a characteristic wavelength and a decay

length, both of which are independently obtained from the bulk

phase using small-angle X-ray scattering (SAXS). This latter

observation is consistent with theoretical predictions and

corroborates the above mentioned CP-AFM results.

The rest of the paper is organized as follows. In Section 2 we

provide a brief review of pair correlations and density profiles in

homogeneous and inhomogeneous fluids as well as their

connection with our experimental data. The experimental part is

described in Section 3. The results are presented and discussed in

Section 4, while the conclusions and an outlook are given in

Section 5.

2 Theory

2.1 Pair correlations

The pair correlations, and hence the local structure, of simple

fluids are governed by the Ornstein–Zernike (OZ) equation.9 For

uniform and isotropic one-component fluids, the OZ equation

reads

h(r) ¼ c(r) + n0Ðh(r0)c(|r � r0|)dr0, (1)

with h(r) ¼ g(r) � 1 denoting the total pair correlation function,

g(r) the pair distribution function, c(r) the direct correlation

function, and n0 the bulk number density. The distance between

two particles is denoted by r. Eqn (1) is exact, but a closure

approximation between h(r), c(r), and the particle–particle

interaction potential u(r) is needed to solve it. The reader is

referred to ref. 9 for a review of the most common closure

relations.

In the case of inhomogeneous fluids, the lack of translational

invariance normal to the interface induces anisotropy of the pair

correlations.10,11 This phenomenon has been known for a long

time within inhomogeneous liquid-state theory, both for fluids at

single interfaces12–14 and in confinement.15–18 The anisotropy of

the pair correlations has also recently been verified experimen-

tally by SAXS for hard-sphere19 and charged20 colloidal

suspensions in confinement. However, with increasing distance

from the interface this feature is less prominent, and hence the

number density profile n(z) can be approximated in a simple

manner using the so-called wall–particle or singlet OZ

equation,21,22

hwpðzÞ ¼ cwpðzÞ þ 2pn0

ðN�N

hwpðtÞdtðNjz�tj

scðsÞds; (2)

in which the anisotropy of the pair correlations is neglected. In

eqn (2), z denotes the distance normal to the interface, hwp(z) the

wall–particle total correlation function, and cwp(z) the wall–

particle direct correlation function. Similar as for eqn (1), the

singlet OZ equation needs to be complemented by a closure

approximation. From eqn (2), the number density profile of the

fluid is recovered using the relation n(z)¼ n0[hwp(z) + 1]. It should

be noted that, in contrast to eqn (1), the singlet OZ equation is

not formally exact and it has been reported to exhibit qualitative

discrepancies close to the interface in comparison with more

This journal is ª The Royal Society of Chemistry 2012

sophisticated theories.13 Nevertheless, for the purpose of dis-

cussing the asymptotic limit of n(z), it suffices to consider the

singlet OZ approximation.

The connection between h(r) and hwp(z) becomes particularly

obvious in reciprocal space, with

h(q) ¼ c(q)/[1 � n0c(q)] (3)

being the three-dimensional (3D) Fourier transform of eqn (1)

and

~hwp(q) ¼ ~cwp(q)/[1 � n0c(q)] (4)

the one-dimensional (1D) Fourier transform of eqn (2).2 Here,

and throughout this article, f (q) denotes the 3D Fourier trans-

form of the spherically symmetric function f(r), while ~f (q) depicts

the 1D Fourier transform of f(z). Notably, for short-ranged

wall–particle interaction potentials, ~hwp(q) is governed by the

same complex leading poles as the bulk quantity h(q). It follows

that, beyond the Fisher–Widom line,1 the asymptotic limit of the

number density profile is given by2

n(z) � n0 / Aexp(�a0 z)cos(a1z � Q). (5)

Here, the decay length x¼ a0�1 and the characteristic wavelength

L ¼ 2pa1�1 are bulk properties of the fluid,23 whereas the

parameters A and Q depend on the wall–particle interaction

potential. We emphasize that eqn (5) is not a result of the singlet

OZ approximation, but holds also within formally exact theo-

ries.24 Although the asymptotic limit of the number density

profile n(z) might at first seem a rather abstract quantity, recent

surface-force experiments on charge-stabilized colloidal suspen-

sions in confinement have suggested eqn (5) to hold already for

confining slit widths of approximately twice the particle diam-

eter.6–8 Moreover, theoretical studies in the bulk phase have

shown the leading-pole analysis to be valid at surprisingly short

distances of a few particle diameters.2,25

2.2 Small-angle X-ray scattering

SAXS probes electron density variations from nanometre to

mesoscopic length scales.26 In the case of bulk colloidal suspen-

sions composed of spherical particles, SAXS yields the total pair

correlation function as follows. Within the monodisperse

approximation, the scattered intensity is given by

I(q) f h|F(q)|2iS(q), (6)

with q¼ 2k sinq denoting the scattering vector modulus, k¼ 2p/l

the wave vector modulus, l the X-ray wavelength, 2q the scat-

tering angle, F(q) the particle form factor, and the brackets h.ithe averaging over a distribution of particle sizes. The interpar-

ticle interactions are contained in the static structure factor,

S(q) ¼ 1 + n0h(q), (7)

which provides a connection between the SAXS data and the

bulk pair correlations of the fluid.

In this study, the SAXS data is modeled as follows. The size

polydispersity of the spherical particles is included using the

Soft Matter, 2012, 8, 5180–5186 | 5181

Dow

nloa

ded

by F

lori

da S

tate

Uni

vers

ity o

n 12

Mar

ch 2

013

Publ

ishe

d on

26

Mar

ch 2

012

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C2S

M07

425G

View Article Online

Schultz distribution.27 The structure factor S(q), in turn, is

determined within the rescaled mean spherical approximation

(RMSA),28,29 with the repulsive particle–particle interaction

potential u(r) being of hard-core Yukawa-type,

buðrÞ ¼8<:

N if r\2R0;

Z2lB

�expðkR0Þð1þ kR0Þ

�2expð�krÞ

rif r. 2R0:

(8)

Here R0 is the average particle radius, Z the number of

elementary charges per particle, lB the Bjerrum length, k�1 the

Debye screening length, and b ¼ (kBT)�1 with kB Boltzmann’s

constant and T the absolute temperature.

Table 1 The complex refractive index nc ¼ 1 � d � ib for the materialsand incident X-ray energies h-u of the present XRR study.38 The densitiesr are also shown

Material r/g cm�3 h-u/keV d b

H2O 1.0 8.0 3.6 � 10�6 1.3 � 10�8

H2O 1.0 22.1 4.7 � 10�7 2.5 � 10�10

SiO2 2.0 8.0 6.6 � 10�6 8.6 � 10�8

SiO2 2.0 22.1 8.5 � 10�7 1.6 � 10�9

Si 2.3 22.1 9.8 � 10�7 3.2 � 10�9

2.3 Specular X-ray reflectivity

The density profile of the fluid normal to the interface can be

determined using XRR.30 Within the kinematic approximation,

XRR data are given by

R(q) f q2RF(q)|~re(q)|2. (9)

Here q denotes the scattering vector modulus,31 RF(q) the Fresnel

reflectivity of an ideal, sharp interface, and iq~re(q) the Fourier

transform of dre(z)/dz, with re(z) being the electron density

profile. The kinematic approximation fails close to the critical

angle for total reflection, and therefore we use the exact,

dynamical reflection formalism of Parratt32 throughout this

study. Nevertheless, the main result of eqn (9) remains, i.e.,

deviations of R(q)/RF(q) from unity are due to density variations

of the fluid normal to the interface.33

Since R(q) is the reflected intensity rather than the amplitude,

it can not be Fourier inverted directly to yield the density

profile. Instead, XRR data are typically analyzed by compar-

ison with model density profiles. Here, we model the number

density profile as a stratified fluid, following previous XRR

studies of ordering at liquid–air34 and solid–liquid interfaces.35

Although simple, this approach provides a straightforward

means to address the decay of n(z). Within this model, an

oscillatory n(z) close to and a constant n(z) far from the inter-

face are constructed using an array of Gaussian peaks, the root-

mean-square width of which diverge with increasing distance

from the interface. More specifically, we model the number

density profile as

nðzÞ ¼XNm¼0

n0dffiffiffiffiffiffi2p

psm

exp

"�ðz� z0 �mdÞ2

2s2m

#; (10)

with z0 denoting the position of the first particle layer with

respect to the interface and d the distance between adjacent

layers. Although the distance between the particle layers may

vary, as exemplified by a recent XRR study on a molecular

liquid confined between mica crystals,36 we decided to adopt

a constant interlayer distance d in order to minimize the

number of free parameters. The diverging peak width is given

by sm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis20 þmhsi2

q, with s0 and hsi being empirical

constants. In order to calculate the reflectivity R(q) corre-

sponding to the model, we transform the number density

profile n(z) into a silica concentration profile, C(z) ¼ (4pR30/3)

n(z)5P(z), with (4pR30/3)n(z) denoting the particle volume

5182 | Soft Matter, 2012, 8, 5180–5186

fraction profile, 5 the convolution operator, and P(z) the

projection of the particle shape.37 In essence, the convolu-

tion with P(z) takes into account that XRR is sensitive to

the electron density (or, equivalently, the refractive index)

profile, whereas n(z) is a distribution of particle centers.

Finally, we determine a complex refractive index profile

nc(z) ¼ 1 � d(z) �ib(z) from C(z) using tabulated optical

constants of the material constituents (see Table 1),38 which

is then used as an input for determining the model R(q)

using Parratt’s recursion formalism.32 Throughout this study,

we include the interface roughness sr in a standard manner

by describing the profile of the interface using an error

function.

3 Experimental details

3.1 Samples

As a model system, we used a commercially available, charge-

stabilized colloidal suspension of spherical SiO2 particles

dispersed in H2O (Ludox SM-30, Sigma-Aldrich). The colloidal

suspension was used as received. It had a nominal particle mass

fraction xM ¼ 0.30 and a pH of 10. We note that the same

brand of colloidal suspensions were used in the CP-AFM

experiments of Klapp et al.,6–8 although at lower particle

concentrations and pH.

As solid surfaces, we prepared two polished, single crystalline

Si wafers with different surface termination. First, a Si wafer with

a native oxide layer was used without further preparation. It had

a hydrophilic surface termination, with the contact angle of

water being y ¼ 31 � 1�. Second, a Si wafer was coated with

dimethyldichlorosilane [Si(CH3)2Cl2, Sigma-Aldrich], leading to

a weakly hydrophilic surface termination with the water contact

angle y ¼ 78 � 1�.Two points about the electron density contrasts of the surface

layers on the Si substrate need to be mentioned: (i) the electron

density of dimethyldichlorosilane in the liquid phase, re ¼ 320

nm�3, is close to that of liquid water, re ¼ 330 nm�3. Conse-

quently, we expect Si(CH3)2Cl2 and H2O to be practically

indistinguishable in the XRR experiment. (ii) The electron

densities of Si (re ¼ 700 nm�3) and the native oxide layer may be

very similar, depending on the porosity of the latter (a density of

r ¼ 2.3 g cm�3 for the latter would correspond to an electron

density re ¼ 690 nm�3). Since we observed signatures of neither

the dimethyldichlorosilane nor the native oxide layer in R(q) for

the probed q regime, we have not included these layers explicitly

in the XRR modeling.

This journal is ª The Royal Society of Chemistry 2012

Fig. 1 Bulk SAXS data. Only every fourth data point is shown for

clarity. The error bars of the experimental data are smaller than the

symbol size. The solid line depicts a model of polydisperse spherical

particles, with hard-core Yukawa-type particle–particle interactions

included within the RMSA.

Dow

nloa

ded

by F

lori

da S

tate

Uni

vers

ity o

n 12

Mar

ch 2

013

Publ

ishe

d on

26

Mar

ch 2

012

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C2S

M07

425G

View Article Online

3.2 Experiments

The SAXS experiment was carried out at beamline ID02 at the

European Synchrotron Radiation Facility (ESRF). Details

about the beamline have been reported elsewhere.39 The incident

X-ray beam had an energy of h-u ¼ 12.5 keV (wavelength l ¼1.00 �A) and the scattered X-rays were collected 1.3 m behind the

sample using the two-dimensional FReLoN CCD detector. The

preliminary data treatment (masking and radial integration) was

done using software available at the beamline. We corrected for

parasitic scattering by subtracting the scattering pattern obtained

from a water sample. The SAXS experiment was carried out at

room temperature, T ¼ 295 K.

The XRR experiments were carried out at beamline ID10B at

the ESRF. Details about the beamline can be found elsewhere.40

Wemade use of two different setups: (i) for XRR experiments on

the fluid–air interface, we used an incident X-ray beam with an

energy of h-u ¼ 8.0 keV (l ¼ 1.55 �A) and a beam size of 1.0 mm

(H) � 0.1 mm (V) at the sample position (H and V denote

horizontal and vertical, respectively). In order to tilt the

incoming beam with respect to the fluid surface, we used

a Ge(111) deflector crystal available at the beamline. (ii) In order

to penetrate through 25 mm of the bulk colloidal suspension, the

XRR experiments on the buried solid–fluid interfaces were

carried out using an incident X-ray energy of h-u ¼ 22.1 keV (l ¼0.56 �A). In this experiment, the beam size at the sample position

was 1.0 mm (H)� 12 mm (V), and the experiment was carried out

by rotating the sample with respect to the incident X-ray beam.

In both setups, the sample-to-detector distance was about

600 mm and we collected the specularly reflected X-rays and the

parasitic background scattering simultaneously using the linear

(1D) VANTEC-1 detector. All the XRR experiments were

carried out at room temperature, T ¼ 295 K.

In the case of the XRR data collected from solid–fluid inter-

faces, the subtraction of parasitic scattering warrants special

attention. In the present study, the X-ray beam traversed

approximately 25 mm in the bulk colloidal suspension, either

prior to or after being reflected from the interface. Consequently,

the main contribution to the parasitic background scattering is

SAXS from the bulk fluid, which exhibits a strong maximum in

S(q) in the same q range as the interesting features in R(q).

Gratifyingly, we have carefully modeled the parasitic scattering

in this q regime prior to subtraction, and found the intensity of

the parasitic scattering to be less than 5% of the intensity of the

specular reflection.

4 Results and discussion

4.1 Bulk structure

In Fig. 1, we present the SAXS data obtained from the bulk

colloidal suspension (collected up to qmax ¼ 4.5 nm�1). Also

shown is the best model of polydisperse spherical particles

interacting via hard-core Yukawa-type forces within the RMSA,

as determined by fitting the model of Section 2.2 to the experi-

mental data in the range 0.2 nm�1 # q # 2.0 nm�1. In the low-q

regime, the model fails to describe the experimental data, an

effect which we attribute to a minor fraction of aggregated

colloidal particles. For larger q, the experimental data and the

model are in reasonable agreement, the latter providing the

This journal is ª The Royal Society of Chemistry 2012

average particle radius R0 ¼ 5.15 � 0.04 nm, the particle size

polydispersity DR0/R0 ¼ 0.21 � 0.02, the Debye screening length

k�1¼ 3.40� 0.12 nm, and the particle volume fraction f¼ 0.18�0.01. The volume fraction corresponds to the number density

n0 ¼ 0.34D0�3, with D0 ¼ 2R0 denoting the average particle

diameter. The error bars reported here have been determined

from a 10% increase in the error metric

ð1=NÞPNi¼1

½IeðqiÞ � ImðqiÞ�2=I2e ðqiÞ, with Ie(q) and Im(q) denoting

the experimental and model SAXS data, respectively. We note

that the average particle size R0 and polydispersity DR0/R0 are in

good agreement with the values recently reported by Zeng et al.8

for the same brand of Ludox SM-30 colloids. The value of the

Debye screening length implies Z z 26 elementary charges per

particle. This is roughly a factor of four larger than Z ¼ 6

reported in ref. 8, an effect which we attribute to the following

two phenomena: (i) the colloidal suspension of the present study

has a larger pH compared to that of Zeng et al., leading to

a higher surface charge on the particles, and (ii) the RMSA

overestimates Z (see, e.g., ref. 41). Finally, the nominal particle

mass fraction xM ¼ 0.30 and the obtained bulk number density

n0 ¼ 0.34D0�3 imply an average density rSiO2

¼ 2.0 g cm�3 for our

colloidal SiO2. In the subsequent modeling of the XRR data, we

use these values for n0 and rSiO2as fixed parameters.

4.2 Interfacial density profiles

The XRR data collected from the fluid–air and the two solid–

fluid interfaces are presented in Fig. 2. Although not shown here,

all the reflectivity curves R(q) have been collected up to at least

qmax ¼ 2.5 nm�1. The XRR data from the solid–fluid interfaces

do not exhibit total reflection at small q, in contrast to the data

from the fluid–air interface. We attribute this unfortunate effect

to a minor bending of the solid substrates used in the present

experiment. Nevertheless, it does not affect the results of the

experiment, which are based on features in the XRR data at

larger q values.

Soft Matter, 2012, 8, 5180–5186 | 5183

Fig. 2 XRR data collected from the silica colloidal suspension at three

different interfaces: air, a weakly hydrophilic surface, and a hydrophilic

surface. The curves are vertically offset and only every fourth data point

is shown for clarity. The experimental error bars are smaller than the

symbol size. The solid lines depict the best fits using the stratified fluid

model of eqn (10).

Fig. 3 The silica concentration profiles C(z) obtained by fitting eqn (10)

to the XRR data in Fig. 2.

Dow

nloa

ded

by F

lori

da S

tate

Uni

vers

ity o

n 12

Mar

ch 2

013

Publ

ishe

d on

26

Mar

ch 2

012

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C2S

M07

425G

View Article Online

Also shown in Fig. 2 are the best fits to the experimental data

using the stratified fluid model of eqn (10). The XRR data from

the fluid–air and solid–fluid interfaces were modeled in the range

0.15 nm�1 # q # 3.5 nm�1 and 0.24 nm�1 # q # 2.5 nm�1,

respectively. In general, the agreement between the experimental

and the model R(q) is good, although the model exhibits weak

oscillations at large q which are not present in the experimental

data. We attribute this minor difference to size polydispersity of

the colloidal particles, which is expected to induce size segrega-

tion in the density profile42 and which is not properly taken into

account in the model. Indeed, such a weak large-q oscillation in

R(q) was observed in a previous XRR study on the fluid–air

interface of a colloidal suspension containing spherical particles

with a smaller size polydispersity of DR0/R0 ¼ 0.13.43

The best fit parameters of the modeling in Fig. 2 are presented

in Table 2. The values obtained for the distance d between

adjacent layers as well as the layer roughness s0 and hsi agreewell for all three models. This is also reflected in the corre-

sponding density profiles presented in Fig. 3, which coincide

Table 2 Fit parameters of eqn (10) to the XRR data of Fig. 2: the positiod between adjacent layers, empirical layer roughness s0 and hsi, and the inter

particle diameter D0. The error bars are determined from a 10% increase in

denoting the experimental and model reflectivity curve, respectively

Interface z0/D0 d/D0

Air 1.19 � 0.02 1.32 � 0.03Weakly hydrophilic 1.46 � 0.02 1.38 � 0.07Hydrophilic 1.11 � 0.02 1.39 � 0.03

5184 | Soft Matter, 2012, 8, 5180–5186

within the accuracy of the present XRR experiments. This

finding will be discussed in more detail in Section 4.3. Here, we

will instead focus on the other parameters of the model. First, we

note that the surface roughness sr obtained for the fluid–air

interface is in good agreement with the capillary-wave-induced

surface roughness of water, 3.24 � 0.05 �A.44 The solid–fluid

interfaces, in turn, exhibit a somewhat larger surface roughness.

Second, andmore interestingly, the distance z0 of the first particle

layer from the interface is found to depend on the wall–particle

potential. In order to explain the values of z0 quantitatively, we

would need to carry out extensive simulations or theoretical

calculations, taking into account both wall–particle and many-

body particle–particle interactions. Since this is beyond the scope

of the present study, we instead provide a brief, qualitative

discussion below.

Even for an uncharged interface, such as the fluid–air inter-

face, we expect two electrostatic contributions to the wall–

particle interaction potential:45 (i) a repulsion due to image

charges induced by the dielectric discontinuity at the interface

and (ii) a repulsion due to distortions close to the interface of the

electrical double layer surrounding the charged particle. Conse-

quently, we expect the charged colloidal particles to be repelled

from the fluid–air interface, in agreement with our observation.

This effect was also observed in a previous XRR study on the

ordering of a charge-stabilized colloidal suspension at the fluid–

air interface.43 In the case of the hydrophilic surface, we must

consider the following two effects:45 first, the repulsion due to

image charges will be weaker than that at the fluid–air interface,

because of a smaller jump in the dielectric constant across the

n z0 of the first particle layer with respect to the interface, the distanceface roughness sr. Except for sr, all values are given in units of the average

the error metric ð1=NÞPNi¼1

½ReðqiÞ � RmðqiÞ�2=R2eðqiÞ, with Re(q) and Rm(q)

s0/D0 hsi/D0 sr/�A

0.18 � 0.02 0.36 � 0.03 3.1 � 0.20.17 � 0.02 0.38 � 0.08 5.9 � 0.30.21 � 0.02 0.34 � 0.03 6.3 � 0.2

This journal is ª The Royal Society of Chemistry 2012

Fig. 4 Decay of the model density profiles of Table 2. The dashed line is

Dow

nloa

ded

by F

lori

da S

tate

Uni

vers

ity o

n 12

Mar

ch 2

013

Publ

ishe

d on

26

Mar

ch 2

012

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C2S

M07

425G

View Article Online

interface. Second, we expect an additional electrostatic wall–

particle repulsion of the Yukawa-type, due to the negative

surface charges on both the solid surface and the particles. These

two effects are competing, as evidenced by z0 being practically

equal for the fluid–air and hydrophilic surface–fluid interfaces.

Surprisingly, in the case of the weakly hydrophilic surface, for

which we expect weaker Yukawa-type repulsions than for the

hydrophilic surface, z0 reaches its maximum value. Such a large

z0 can not be explained only by steric repulsions due to the

dimethyldichlorosilane coating. We have verified these results for

the two solid–fluid interfaces using an effective hard-sphere hard-

wall model within the singlet OZ approximation of eqn (2) using

the Percus–Yevick closure,21 ruling out the possibility that the

behavior of z0 is specific to the model of eqn (10). For the

moment, the behavior of z0 in the case of the weakly hydrophilic

surface remains unexplained.

a linear fit to the data for the hydrophilic surface.

4.3 Decay of interfacial density profiles

Finally, we turn to the main topic of this paper, i.e., the decay of

density profiles at fluid interfaces. It should be mentioned that

this topic has also been touched upon in previous XRR experi-

ments. In particular, Yu et al. adopted a model akin to eqn (5), in

order to explain XRR data for a molecular liquid at a solid

surface.46 Recently, one of us also rationalized the decay of

interfacial ordering for confined reverse micelles using this

model.47 However, to our knowledge this is the first XRR (or,

more generally, X-ray scattering) study, in which the decay of

interfacial fluid ordering and its connection with the decay of

bulk pair correlations is explicitly addressed.

First, we discuss the characteristic wavelength L, which is

straightforwardly obtained from the model of eqn (10) as the

distance between adjacent layers, L ¼ d. This should be

compared to the average particle distance in the bulk phase, db.

The latter quantity is most readily determined as the distance

between adjacent peaks in the bulk total pair correlation function

h(r), which in turn is obtained from the model of Fig. 1 by

Fourier inverting eqn (7). Using this approach, we obtain db ¼1.34D0 for interparticle distances beyond the second peak in h(r),

in good agreement with the values of d given in Table 2. Alter-

natively, we can use d0b z 2p/q0 following Zeng et al.,8 with q0

denoting the position of the first peak in S(q). It should be noted,

however, that while the relation between the interparticle

distance and the first peak position in S(q) is exact when the

system exhibits long-range order, i.e., in the limit of Bragg

diffraction, it holds only approximately for disordered systems

such as fluids. Indeed, from the model of Fig. 1 we estimate d0b z

1.26D0. Finally, we note that db < 2(R0 + k�1) ¼ 1.66D0. This

latter observation corroborates the previous finding8 that the

spatial extent of the electrical double layers, �2(R0 + k�1), does

not provide a good estimate of the characteristic wavelengthL in

this system.

Next, we consider the decay length x, which is not as

straightforwardly determined from the model of eqn (10) as the

characteristic wavelength L. In order to determine x, we present

in Fig. 4 the peak values of log([n(z) � n0]/n0) for the three model

density profiles of Table 2. Also shown is a linear fit to the model

obtained for the hydrophilic surface, which (i) illustrates the

exponential decay of the stratified fluid model of eqn (10) and (ii)

This journal is ª The Royal Society of Chemistry 2012

provides an estimate of x. From Fig. 4, we determine the decay

lengths xa ¼ 0.99D0, xwh ¼ 0.99D0, and xh ¼ 1.27D0 for the data

collected at the interfaces between the fluid and air, the weakly

hydrophilic surface, and the hydrophilic surface, respectively.

The large scatter in these values is striking, considering the small

variation in s0 and hsi in Table 2. The scatter shows the sensi-

tivity of the decay length x, and hence the difficulty in experi-

mentally determining it for inhomogeneous fluids. Nonetheless,

the obtained values do provide an estimate of x, which we can

compare to the bulk counterpart xb. In order to determine the

bulk decay length xb from the SAXS data, we apply a similar

analysis as in Fig. 4 on the model h(r) of Fig. 1 (not shown). For

interparticle distances beyond the second peak in h(r), the

exponential decay of the bulk total pair correlation function can

be described by a single decay length, xb ¼ 0.79D0. We primarily

attribute the deviations between xb and x to the modeling of

Fig. 2. Nevertheless, xb is in reasonable agreement with the

inhomogeneous counterparts x presented above, considering the

difficulties in determining the latter values as mentioned above.

Here, a few important points need to be mentioned: (i) we have

so far in the discussion neglected the van der Waals contribution

to the particle–particle interaction potential u(r). It is long

ranged, and we therefore expect it to ultimately govern the

asymptotic decay of h(r) and n(z).5 However, the short-ranged

hard-core Yukawa-type contribution to u(r) will dominate at

short and intermediate distances, and it is this decay that we are

concerned with in the present paper. (ii) Based on theoretical

studies of bulk fluids, the contribution from at least two of the

slowest decaying poles are needed to accurately describe h(r) at

short distances.2,25 Moreover, we do not determine the charac-

teristic wavelength L and the decay length x from the slowest

decaying complex poles of eqn (3), but rather from simplified

modeling. Consequently, the present quantities should be

considered as approximations to (or effective counterparts of)

L ¼ 2pa1�1 and x ¼ a0

�1 introduced in Section 2.1.

5 Conclusions and outlook

In summary, we have carried out XRR experiments on a charge-

stabilized colloidal suspension at the fluid–air and at two

different solid–fluid interfaces, in order to address the question:

Soft Matter, 2012, 8, 5180–5186 | 5185

Dow

nloa

ded

by F

lori

da S

tate

Uni

vers

ity o

n 12

Mar

ch 2

013

Publ

ishe

d on

26

Mar

ch 2

012

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C2S

M07

425G

View Article Online

to what extent is the decay of interfacial fluid ordering deter-

mined by the bulk structure of the fluid? Our experiments provide

support for a bulk-like decay of the density profile already at

short distances from the interface. This finding is consistent with

theoretical predictions and corroborates recent surface-force

experiments on a similar colloidal suspension under confinement.

The above result raises the obvious question: how can we

reconcile the bulk-like decay of the fluid’s density profile close to

the interface with the anisotropy of its pair correlations alluded

to in Section 2.1? In order to answer this question, we would need

to go beyond the effective characteristic wavelength and decay

length probed in this study. Instead, we propose a theoretical

study of the fluid’s density profile n(z) and its characteristic

parameters L and x, while systematically varying the strength of

the anisotropy in the pair correlation functions. In practice, such

a study could be carried out, e.g., on an inhomogeneous hard-

sphere fluid as a function of particle concentration.

Acknowledgements

We thank Manuel Fern�andez and Theyencheri Narayanan for

assistance with the SAXS experiment and Angelo Accardo for

help with the contact angle measurements.

Notes and references

1 M. E. Fisher and B. Widom, J. Chem. Phys., 1969, 50, 3756–3772.

2 R. Evans, R. J. F. Leote de Carvalho, J. R. Henderson andD. C. Hoyle, J. Chem. Phys., 1994, 100, 591–603.

3 C. Grodon, M. Dijkstra, R. Evans and R. Roth, Mol. Phys., 2005,103, 3009–3023.

4 J. Baumgartl, R. P. A. Dullens, M. Dijkstra, R. Roth andC. Bechinger, Phys. Rev. Lett., 2007, 98, 198303.

5 R. J. F. Leote de Carvalho, R. Evans, D. C. Hoyle andJ. R. Henderson, J. Phys.: Condens. Matter, 1994, 6, 9275–9294.

6 S. H. L. Klapp, Y. Zeng, D. Qu and R. von Klitzing, Phys. Rev. Lett.,2008, 100, 118303.

7 S. H. L. Klapp, S. Gradner, Y. Zeng and R. vonKlitzing, SoftMatter,2010, 6, 2330–2336.

8 Y. Zeng, S. Grandner, C. L. P. Oliveira, A. F. Th€unemann, O. Paris,J. S. Pedersen, S. H. L. Klapp and R. von Klitzing, Soft Matter, 2011,7, 10899–10909.

9 J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids,Academic Press, Amsterdam, 3rd edn, 2006.

10 J. K. Percus, The Equilibrium Theory of Classical Fluids, New York,1964, pp. II.33–II.170.

11 S. Sokolowski, J. Chem. Phys., 1980, 73, 3507–3508.12 R. Kjellander and S. Mar�celja, Chem. Phys. Lett., 1984, 112, 49–53.13 M. Plischke and D. Henderson, J. Chem. Phys., 1986, 84, 2846–2852.14 B. G€otzelmann, A. Haase and S. Dietrich, Phys. Rev. E, 1996, 53,

3456–3467.15 R. Kjellander and S. Mar�celja, J. Chem. Phys., 1988, 88, 7138–7146.16 R. Kjellander and S. Sarman, J. Chem. Soc., Faraday Trans., 1991, 87,

1869–1881.

5186 | Soft Matter, 2012, 8, 5180–5186

17 B. G€otzelmann and S. Dietrich, Phys. Rev. E, 1997, 55, 2993–3005.18 D. Henderson, S. Sokolowski and D. Wasan, J. Stat. Phys., 1997, 89,

233–247.19 K. Nyg�ard, R. Kjellander, S. Sarman, S. Chodankar, E. Perret,

J. Buitenhuis and J. F. van der Veen, Phys. Rev. Lett., 2012, 108,037802.

20 K. Nyg�ard, D. K. Satapathy, J. Buitenhuis, E. Perret, O. Bunk,C. David and J. F. van der Veen, Europhys. Lett., 2009, 86, 66001.

21 D. Henderson, F. F. Abraham and J. A. Barker,Mol. Phys., 1976, 31,1291–1295.

22 Here, we have for brevity reproduced only one of several equivalentversions of the singlet OZ equation.

23 More specifically, a0 and a1 are obtained from the lowest lyingcomplex conjugate pole p ¼ �a1 + i a0 of eqn (3).

24 J. R. Henderson and Z. A. Sabeur, J. Chem. Phys., 1992, 97, 6750–6758.

25 J. Ulander and R. Kjellander, J. Chem. Phys., 1998, 109, 9508–9522.26 O. Glatter and O. Kratky, Small Angle X-ray Scattering, Academic

Press, London, 1982.27 M. Kotlarchyk and S.-H. Chen, J. Chem. Phys., 1983, 79, 2461–2469.28 J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109–118.29 J.-P. Hansen and J. B. Hayter, Mol. Phys., 1982, 46, 651–656.30 J. Daillant and A. Gibaud, X-ray and Neutron Reflectivity: Principles

and Applications, Springer, Berlin Heidelberg, 2009.31 In line with eqn (4), we denote the scattering vector modulus simply

with q. However, while the bulk S(q) is an isotropic quantity, R(q)is defined normal to the interface.

32 L. G. Parratt, Phys. Rev., 1954, 95, 359–369.33 The effects of capillary waves or roughness of the solid surface have

for simplicity been neglected in this discussion. However, they areincluded in the modeling of the XRR data.

34 O. M. Magnussen, B. M. Ocko, M. J. Regan, K. Penanen,P. S. Pershan and M. Deutsch, Phys. Rev. Lett., 1995, 74, 4444–4447.

35 W. J. Huisman, J. F. Peters, M. J. Zwanenburg, S. A. de Vries,T. E. Derry, D. Abernathy and J. F. van der Veen, Nature, 1997,390, 379–381.

36 E. Perret, K. Nyg�ard, D. K. Satapathy, T. E. Balmer, O. Bunk,M. Heuberger and J. F. van der Veen, J. Phys.: Condens. Matter,2010, 22, 235102.

37 M. C. Gerstenberg, J. S. Pedersen andG. S. Smith,Phys. Rev. E, 1998,58, 8028–8031.

38 B. L. Henke, E. M. Gullikson and J. C. Davis, At. Data Nucl. DataTables, 1993, 54, 181–342.

39 T. Narayanan, O. Diat and P. B€osecke, Nucl. Instrum. Methods Phys.Res., Sect. A, 2001, 467–468, 1005–1009.

40 D.-M. Smilgies, N. Boudet, B. Struth and O. Konovalov, J.Synchrotron Radiat., 2005, 12, 329–339.

41 M. Heinen, P. Holmqvist, A. J. Banchio and G. N€agele, J. Chem.Phys., 2011, 134, 044532.

42 I. Pagonabarraga, M. E. Cates and G. J. Ackland, Phys. Rev. Lett.,2000, 84, 911–914.

43 A. Madsen, O. Konovalov, A. Robert and G. Gr€ubel, Phys. Rev. E,2001, 64, 061406.

44 A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsenand J. Bohr, Phys. Rev. Lett., 1985, 54, 114–117.

45 H. H. von Gr€unberg and E. C. Mbamala, J. Phys.: Condens. Matter,2001, 13, 4801–4834.

46 C.-J. Yu, A. G. Richter, J. Kmetko, A. Datta and P. Dutta, Europhys.Lett., 2000, 50, 487–493.

47 K. Nyg�ard, D. K. Satapathy, E. Perret, C. Padeste, O. Bunk, C. Davidand J. F. van der Veen, Soft Matter, 2010, 6, 4536–4539.

This journal is ª The Royal Society of Chemistry 2012