X-ray waveguiding studies of ordering phenomena in confined fluids - Chapter 6 Ordering phenomena in confined colloids

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X-ray waveguiding studies of ordering phenomena in confined fluids

Zwanenburg, M.J.

Publication date

2001

Link to publication

Citation for published version (APA):

Zwanenburg, M. J. (2001). X-ray waveguiding studies of ordering phenomena in confined

fluids.

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Orderingg phenomena in confined

colloids s

WeWe have determined the structure of a colloidal fluid confined in a gap between two wallswalls by making use of the waveguiding properties of the gap at x-ray wavelengths. TheThe method is based on an analysis of the coupling of waveguide modes induced by thethe density variations in the confined fluid. Studies on suspensions confined within gapsgaps of a few hundred nanometer showed strongly selective mode coupling effects, indicativeindicative of an ordering of the colloidal particles in layers parallel to the confining walls. walls.

6.11 Introduction

Recentt synchrotron x-ray scattering studies of liquids in contact with a single solidd wall have revealed that the molecules will become layered adjacent to the walll [3, 54]. An extension of such studies to liquids confined by two opposing wallss at close distance is highly relevant, given the frequent occurrence of confined liquidss in every-day life. An example is a thin lubricating film between two sliding surfaces.. It has been conjectured that the ordering within the fluid is particularly pronounced,, if an integer number of layers exactly fits within the gap [55]. This inn turn would cause the fluid to attain solidlike properties, adversely affecting the lubricationn [56].

Experimentall evidence for layering effects in confined fluids has mainly come

fromfrom studies performed with the surface force apparatus (SFA) [57]. Upon ap-plicationn of a normal force to one of the confining plates, the force was found to oscillatee with decreasing gap size, with one period corresponding to the removal off a single molecular layer. A recent synchrotron x-ray diffraction study showed smecticc ordering in liquid-crystal films confined within fim gaps [4, 58]. The latter experimentt was performed in a SFA which was modified so as to allow the beam too pass through the fluid along the direction normal to the confining surfaces.

Inn this chapter we present a novel coherent x-ray scattering method, which allowss for the determination of the fluid's density profile across the gap between twoo parallel plates. We direct the synchrotron x-ray beam into the gap along a directionn nearly parallel to the plates, using the system as a planar waveguide. As thee guided waves have their maximum amplitude within the fluid and a rapidly decayingg amplitude within the confining plates [53], the scattering contribution fromm the plates is minimized. This results in an unparalleled sensitivity to struc-turall changes within the fluid. Waveguiding is possible, provided the average x-ray refractivee index of the fluid is larger than that of the plate material. Most fluids off interest fulfil this requirement.

Thee method is based on an analysis of the coupling of waveguide modes induced byy the spatial variations in the electron density of the fluid. We demonstrate the methodd for a suspension of colloidal particles confined by flat plates of a few mm lengthh at a distance of a few particle diameters (300-600 nm).

6.22 Principle

Thee waveguiding method is illustrated in Fig. 6.1. The planar waveguide supports aa finite number of transverse electric (TE) modes. If the gap were empty, a given modee would propagate undisturbed through the waveguide. In the filled wave-guide,, however, the spatial variations in the electron density of the fluid give rise too scattering into other TE modes ('mode coupling'). The distribution of field am-plitudee over the modes as well as the interference between the mode amplitudes are observedd in Fraunhofer diffraction patterns of the field across the waveguide's exit plane.. From these, the density profile is determined through a model-dependent analysis. .

jW W

QQQQ Q

eoe e

^ > e e

t t

Figuree 6.1 Schematic of the waveguiding geometry. For illustration, the

wave-guideguide is filled with two layers of ordered fluid and a TE2 mode is drawn at the entrance.entrance. Angles and distances are not to scale.

withh wavenumber fco = 2ir/\ and with the electric field polarized perpendicular to thee plane of incidence, is incident onto the large bottom plate of the waveguide at ann angle Ö,, see Fig. 6.1. Interference with the reflected wave yields a standing wavee pattern above the bottom plate. For Ö; values equal to a mode angle 9m «

(m(m + l)w/koW, with W the gap width and m = 0,1, , a single TEm mode with modee number m passes through the entrance plane of the waveguide. At other

9i9i values, in between two consecutive mode angles, the field decomposes at the

entrancee into a linear combination of the neighboring TE modes as described in chapterr 4.

Nextt we consider the propagation of modes through the planar waveguide. The e.m.. field, i.e. ty(x,z) = Ey(x,z), within the waveguide satisfies the scalar wave equation n

V2^(x,, z) + n(x, z)2f$V(x, z) = 0, (6.1)

wheree V2 = d2/dx2 + d2/dz2 and n(x, z) is the spatially varying refractive index. Thee coordinates x and z are along the directions of confinement and propagation, respectively,, and their origin is taken at the bottom of the entrance plane. For xx rays, n(x, z) = 1 — a(x,z), with a(x,z) = A rene(x, z)/2ir being of the order

10"6.. Here, re is the classical electron radius and ne(x, z) the sought-after electron density.. Seeking for a solution of Eq. (6.1), we make the following Ansatz:

wheree {(j>m} are the orthogonal eigenmodes of the empty waveguide, (3m = fco cos 6m iss the corresponding propagation constant and mmax is the maximum allowed mode number.. The mode amplitudes have a standing-wave profile within the gap, and aree evanescent within the plates. As discussed in section 2.3, inserting Eq. (6.2) intoo Eq. (6.1) results in a set of coupled differential equations for the {cm}:

^ « ê ^ ^ r ^ ) ^ - ^ ,, (6.3)

withh the amplitude coupling coefficients given by

Tmk(z)Tmk(z) = ^ ƒ " <j>l(x) [n(x, zf - n0(x)2] 4>m{x)dx, (6.4) where e // oo o \^{x)\\^{x)\22dx.dx. (6.5) o o

Here,, n(x, z) is the refractive index of the confined medium and UQ{X) the stepped refractive-indexx profile of the empty waveguide (n0(x) = 1 for 0 < x < W and

no(x)no(x) — 1 - 2.57 x 10~6 within the confining silica plates). The starting values

Cm(0)Cm(0) follow from

11 f°°

c m ( 0 ) = p - // #»(*)*(*, 0)dx. (6-6)

*m*m J—oo

Givenn the field V(x,Q) at the entrance and the refractive index profile n(x,z), thee propagating field #(x, z) is found by numerically solving the set of Eqs. (6.3). Forr illustration, we have calculated the field amplitude ^{x,z) assuming a z-independentt layered profile

n(x)n(x) = 1 - a0 - af cos ( —— J (6.7) withinn the gap. Here, 1 — a0 is the spatially averaged refractive index and a/ the

modulationn amplitude for / layers. The incident field ^(x, 0) was chosen such that aa single TE2 mode was excited at the entrance. Figure 6.2a shows the field intensity |*J>(:r,, z)f within a gap of length L = 1.2 mm and width W — 615 nm, filled with fivefive layers having refractive-index parameters OQ — - a5 = 1.00 x 10- 6. The x-ray wavelengthh A was chosen to have the experimental value of 0.0931 nm. Clearly, the intensityy is redistributed over several modes and is concentrated in regions where

bothh the incident field amplitude and the refractive index are highest. Note, that forr symmetric systems in which the plane x = W/2 is a mirror plane, coupling onlyy occurs between modes of equal parity (even or odd). An ordering of the fluid mediumm into equidistant layers results in a further selectivity in the mode coupling. Substitutingg n(x) into Eq. (6.4) and taking

A,A, M / y/WMMn?) 0 < X < W

<P<Pmm\\xx)) = \ n „ , , (6.8) 00 x<OAz> W

wee obtain the following selection rules for a system of I layers:

rmf cc = ikoOoS^k -\ — ({>2l,m-k + <$2J,fc-m — f>2l,m+k+2 ~ t>2l,-m-k-2), (6-9)

wheree öij is the Kronecker-delta. The mode coupling strengths are of magnitude |T|| ~ k0ai/2 ~ 34 mm- 1. Transfer of amplitude between mode n and mode

nn + 2l takes place over a distance d ~ w/ (0n+2i — Pn) along the 2-direction, which rangess from tens of microns to a few millimeters depending on mode number andd mode distance. Hence, the relative magnitude of transferred amplitude over suchh distances is typically of the order of koai d ~ 1. We conclude that the modee coupling is strong, providing high structure sensitivity for systems with low refractive-indexx modulations.

Thee mode selectivity is most conveniently observed in a series of Praunhofer diffractionn patterns of the exit field ^f(x,L), each taken at a slightly increased valuee of di starting from zero. Whenever 9t equals 0n, with n = 0,1, , mode n iss excited at the entrance. Coupling to mode m within the waveguide then may resultt in a peak in the diffraction pattern at exit angle 6e = 0m. Figure 6.2b shows aa contour plot of the diffracted intensity I(9i,6e), calculated for the model system off Fig. 6.2a, with the use of Eq. (2.53). The peaks along the co-diagonals relate to thee second and third terms in Eq. (6.9), while the cross-diagonals originate from thee fourth term. The fifth term is always zero for positive mode numbers. The cross-diagonalss intersect the main diagonal at 0j = 9^i, with p = 1,2, . At thesee angles, the nodes of the incident mode lie precisely in between the layers.

6.33 Experimental

Thee waveguiding set-up has been described in chapter 3. In brief, the waveguide consistsconsists of two fused-silica plates which were coated with an aluminum layer (98 %

|V(x,L)| |

(a)) ...

0.33 0.6 0.9 zz (mm)

0.000 0.02 0.04 0.06 0.08 Incidencee angle 0; (degrees)

Figuree 6.2 ('oj Intensity distribution of the field within the waveguide, calculated

forfor a guiding medium having a refractive index profile n(x). (b) Linear contour plot ofof the calculated intensity I(8v9e), diffracted from the exit of the above waveguide.

reflection)) and subsequently a SiC>2 layer (650 nm). The liighly reflecting aluminum layerss form an optical interferometer which enables us to measure the gap width andd parallelism through use of the technique of Fringes of Equal Chromatic Order (FECO).. The gap size and tilt angle are controlled using a tripod of nanometer precisionn motors. The lower and upper fused-silica plates had a diameter of 25.4 mmm and 4.85 mm, respectively. The deposited SiC>2 layers had a surface roughness off ~1 nm (rms) as determined from atomic force microscopy measurements, which iss small enough not to affect the waveguiding properties.

Thee parallel-plate setup was mounted horizontally onto the diffractometer at thee undulator beamline ID10A of the European Synchrotron Radiation Facility (ESRF)) in Grenoble (France) [24]. A photon energy of 13.3 keV (A = 0.0931 nm) wass selected using the (111) reflection of a silicon monochromator. A mirror served too suppress higher harmonics from the undulator. The intensity of the beam of 0.11 mm diameter passing through a vertical gap of 500 nm was typically 3 x 108 photons/s.. The transverse coherence length of the beam is in the vertical plane £vv = 177 /zm and in the horizontal plane 4.5 fim. Since £v » W, the incident field iss fully coherent across the gap. In the horizontal plane, however, the beam has incoherentt properties.

6.44 Results and discussion

Forr our experiments we used a colloidal suspension of ~ 110 nm diameter SiC>2 spheress in dimethylformamide (DMF). The diameter of the spheres was deduced fromm AFM measurements (see Fig. 6.3). These were performed on the surface of thee residue which arose after a drop of the suspension dried out on a microscope slide.. The suspension was confined within gaps of various widths. Initially, the suspensionn was inserted in a gap of typically a few microns width, after which the gapp size was reduced to the desired value. We measured Fraunhofer diffraction patternss as a function of 0* for a gap of W = 655 nm filled with 10 vol. % suspen-sion.. The patterns were taken using a two-dimensional CCD camera (Sensicam, 6.77 /mi pixels, 12 bit) at 2.39 m distance from the waveguide. The angular resolu-tion,, as determined by the pixel size, was 0.2 millidegree. The measurements are presentedd in a contour plot of I(9i,9e), see Fig. 6.4a.

signif-Figuree 6.3 AFM image, from which the diameter of the SiC-2 particles was

deter-mined. deter-mined.

icantt mode coupling occurs. At angles larger than 0.065°, which is the critical anglee for total internal reflection of the DMF/Si02 interface, a much reduced in-tensityy is observed. The faint intensity along the diagonal for these angles is due too a part of the beam passing through the fused silica substrate which is reflected fromm the aluminum layer. A cross-diagonal intersects the diagonal at Ö, « 0.025°. Fromm 0j_i = 0.025° we deduce I = 6, which indicates the presence of ~ 6 layers inn between the plates. A cubic stacking of six layers of 110 nm particles does not fitfit within a gap of 655 nm width, while a stacking of hard spheres into a closed-packedd structure would result in a thickness less than the gap width (609 nm). Thiss suggests that the particles form a close-packed structure near the walls and aree less well-ordered in the center of the gap [59, 60]. Indeed, the cross-diagonal is nott as sharply defined as in Fig. 6.2b, indicating that the refractive-index profile containss more than a single Fourier component. Obviously, a 10 vol. % suspension cannott result in a complete filling of the waveguide by a close-packed structure and thee structure must be discontinuous along the z-direction. Evidence for this is the formationn of breaks in the FECO fringes upon closure of the gap to below ~ 1 /xm. Discontinuitiess in the refractive index profile along the z-direction also explain the asymmetryy with respect to the diagonal 6t = 6e in the contour plot of Fig. 6.4a.

(seee appendix A).

Inn a search for a fit to the data, we calculated diffraction patterns for various assumedd refractive-index profiles with the use of the finite-difference beam propa-gationn method as discussed in chapter 2. The major features in the I(9i,9e) plot, inn particular the cross-diagonal, are reproduced well for a profile of the form

6 6

n(x)n(x) = 1 - oo - J ^ a, cos{2lirx/W), (6.10) 1=4 1=4

withh a0 = 1.93 x 10"6-M3.1 x 10"9, a6 = -0.25 x 10~6-H3.00x 10~9, a4/o6 = 0.19, andd 0,5/0,$ = 0.85 (Figs. 6.4b) [61]. The plates are given a refractive index nSio2 = 11 — 2.57 x 10~6 — il.25 x 10"8, where the imaginary part here and above accounts forr absorption. The waveguide, of length L = 4.85 mm, was assumed to have the abovee profile n(x) only within the interval 1.81 < z < 3.85 mm, the remainder beingg filled with DMF only. The striking similarity between calculations and measurementss confirms the presence of six layers, with decaying order away from thee confining walls. The remaining discrepancies originate from uncertainty in the densityy distribution along the z-direction. We note here, that the measurements aree insensitive to the discreteness of the single particles along z, since the particle diameterr is much smaller than a typical scattering length (kai)~ .

Wee repeated the measurements in a gap of only W = 310 nm (Fig. 6.5a). Inn order to facilitate reduction of the gap, we diluted the suspension to 1 vol.

%.%. As expected, there are fewer guided modes, at larger angular spacing. A

cross-diagonall intersects the diagonal at 0, « 0.023°, from which we derive that twoo layers have formed. Fig. 6.5b shows a contour plot of I{9^9e) which was calculatedd for the profile

4 4

n(x)n(x) = 1 — oo — y^a/cos(2J7nr/W), (6-11) 1=2 1=2

whereinn ao = 1.99 x 10~6 + il.10 x 10"9, a2 = -0.36 x 10~6 + il.00 x 10"9, a3/a22 = 0.25, and a^fa-i = 0.13. The profile of n{x) was asymmetrically positioned betweenn 1.90 < z < 3.33 mm so as to reproduce the asymmetry in the data with respectt to the diagonal Qi = 9e. Again, there is reasonable agreement with the dataa [62].

0.000 0.02 0.04 0.06 0.08 Incidencee angle ^(degrees)

Figuree 6.4 Linear contour plot of the diffracted intensity 1(9^9^ (a) measured

forfor a waveguide having a 655 nm gap filled with a colloidal suspension (see text), andand (b) calculated for the profile n(x) = 1 — a(x), with parameter values as given inin the text.

0.08 8 0.06 6 0.04 4 <uu 0.02 i _ _ OJ J <D D ^^ 0.00 I » » _OJJ 0.08

(a) )

5 5

Incidencee angle ö^degrees)

Figuree 6.5 Linear contour plot of the diffracted intensity I(9v0e) (a) measured

forfor a waveguide having a 310 nm gap filled with a colloidal suspension (see text), andand (b) calculated for the profile n(x) = 1 — a{x), with parameter values as given inin the text.

6.55 Conclusions

Wee have determined density profiles in confined fluids using waveguiding of co-herentt x rays within the confined structure. In silica colloids, the proximity of thee walls is found to induce a strong layering effect. The ordering of the particles inn a closed-packed structure and the non-uniformity of the ordered regions in the planee of the gap strongly suggest that the confinement induces crystallisation at volumee densities much lower than the critical density for crystallisation of bulk colloidd [63].

Thee measured diffraction patterns are in essence phase-contrast images of the waveguide'ss filling. All interference takes place within the thick phase object. The detectorr in the far field then registers the intensity of the Fourier transform of the emergingg wavefield. The restriction that interference can only occur between dis-cretee modes of the waveguide greatly simplifies the structural analysis, especially if thee object's structure is translationally invariant over the length of the waveguide. Ourr waveguiding method can be modified to include studies of (molecular) fluids confinedd in much smaller gaps than reported here, provided that use is made of especiallyy tailored refractive index profiles in the confining walls and both guided andd radiative modes are detected [64].