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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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Chapterr 1
Introduction n
1.11 Background
Inn this thesis we present a method for the structural analysis of fluids in a con-finedd geometry. An everyday example of a confined fluid is a lubricant between twoo sliding surfaces. It is a well-known fact that the properties of a lubricating filmm strongly depend on its thickness, i.e., the degree of confinement. For exam-ple,, very thin lubricating films are known to induce stick-slip motion in which the slidingg object alternates between stick and slip [1]. During these transitions the lubricantt is believed to transform repeatedly from an ordered, solid-like, state to aa fluid state (see Fig. 1.1) and vice versa. In the solid-like state the adhesion betweenn the sliding surfaces may be large enough to cause damage to the surfaces. Fromm an economical viewpoint, this is highly unwanted. Besides being of techno-logicall relevance, confinement-induced ordering phenomena are interesting from a fundamentall point of view.
Lett us illustrate our point by first considering the arrangement of the fluid's constituentss at the interface with a single solid wall. Close to the wall, the fluid orderss in layers parallel to the wall. As a result, the density oscillates with increas-ingg distance from the wall around the value of the density corresponding to that off the bulk fluid. The period of the oscillations is roughly equal to the diameter off the liquid's constituents. At some distance from the wall, the amplitude of the oscillationss decays to zero. At this point, the fluid no longer 'feels' the presence of thee wall and it attains bulk properties.
(a) )
(b) )
Figuree 1.1 During stick-slip motion the lubricant alternates between (a) a solid-likelike state and (b) a liquid or fluidized state.
Inn a confined fluid, an additional solid-fluid interface is present. For large distancess between the two surfaces, the region which is influenced by the presence off a surface does not extend far enough into the fluid that it influences the ordering off the fluid at the opposite wall (see Fig. 1.2a). As the wall separation is reduced, thesee regions start to overlap, causing the molecules to 'feel' the influence of both walls.. In this case, the density variations induced by both walls may 'interfere' as iss illustrated in Figs. 1.2b-d. The precise shape of the resulting density profile is determinedd by the nature of the interaction between the fluid's constituents and betweenn the wall and the fluid. A detailed description of the effects of various interactionss is found in Ref. [2], along with references to other work.
Experimentally,, layering at the single solid-liquid interface has been observed inn various fluids. In the case of colloidal suspensions this was achieved by means of opticall microscopy (see Fig. 1.3). Recently, x-ray scattering studies have revealed layeringg on the molecular scale at the interface between liquid gallium and a di-amondd surface [3]. Studies of confined fluids, however, are more difficult because thee region of interest is less well accessible. In general, ordering phenomena on the molecularr scale have been studied with the "surface force apparatus" (SFA) [2]. Thee SFA enables confinement of a fluid between two atomically smooth surfaces whichh may be separated by distances as small as a few tenths of a nanometer. Onee of the surfaces is attached to a spring enabling measurement of the normal forcess acting on it, and in some designs an additional spring is attached for studies
Introduction Introduction 13 3 (a) )
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(b) ) (c) ) (d) )Figuree 1.2 (a) Structure of a confined fluid for large wall separations and the
correspondingcorresponding density profile p{x). (b)-(d) Structure models and corresponding densitydensity profiles for three wall separations. From left to right the distance decreases fromfrom approximately 4 to 3 particle diameters. The figures in the center correspond
toto a wall separation of 3.5 particle diameters. In the plots of the density, the dashed curvescurves show the density profiles for a single solid-liquid interface. The solid curve showsshows the sum of the dashed curves.
Figuree 1.3 Differential interference contrast microscopy image of the ordering of colloidalcolloidal spheres at a smooth hard wall of a glass capillary. The spheres are made of PMMA-PHSPMMA-PHS and have a diameter of 1 jim. They are dissolved in a refractive-index andand density-matched solvent. The volume fraction of the spheres is approximately
O.48,O.48, just below the freezing transition at 0.494- (By courtesy of W. K. Kegel)
off friction and lubrication. The normal forces may have an oscillatory character ass a function of the distance. For the detection of these oscillations extremely fiatt surfaces are required. As the surface separation decreases, the amplitude of thesee oscillations becomes larger. This indicates an increasing degree of order [2]. Theree is believed to be a strong correlation between these force oscillations and the constructivee and destructive 'interference' between the density profiles discussed previously. .
Mostt of the above results were confirmed by computer simulations. Only for colloidall systems which consist of particles with diameters in the range of optical wavelengths,, direct experimental verification was possible (see above). Recently, thee SFA was adapted so as to allow for studies of the structure of confined fluids byy means of x-ray scattering [4]. In this device the x-ray beam passed through thee confined fluid in the direction normal to the confining surfaces (see Fig. 1.4a). Inn such a scattering geometry, however, the scattered intensity is dominated by aa background originating from scattering by the confining walls. Moreover, the sensitivityy to the structure of the fluid is highest in the plane of the surfaces. In orderr to reduce the background and to determine the structure of the fluid across thee gap, we let the x-ray beam enter the device from the side (see Fig. 1.4b). In thiss way, the x rays may be confined to the gap while they propagate along the directionn parallel to the walls. Effectively, the gap acts as a planar waveguide for xx rays.
Introduction Introduction 15 5
Figuree 1.4 (a) Normal-incidence scattering geometry employed in the adapted surfacesurface force apparatus, (b) Our scattering geometry, in which the x-ray beam entersenters the device from the side.
Thee subject of the thesis is the exploitation of the waveguiding properties of thee confining geometry for the determination of the structure of a confined colloid. Forr a colloidal suspension of ~ 100 nm particles, confinement effects can already bee expected at gap sizes of a few hundred nanometers. This makes the system is welll suited for analysis by our waveguiding method.
1.22 Structure determination
XX rays are widely used for imaging opaque objects. On airports the content of our suitcasess is screened while in the hospital doctors use them to check the contents off our body. These applications are based on the fact that x rays penetrate matter deeplyy and that different materials have different absorption cross sections. Due too their small wavelength, x rays are also suitable for the determination of the microscopicc structure of matter.
Thee conventional imaging techniques mentioned above rely on absorption con-trastt and totally neglect the potential of phase contrast. In phase-contrast imaging onee illuminates the object (or part of it) with a (partially) coherent wave field. As thee wave field traverses the object, phase differences may arise between different partss of the wave front, which enhances the contrast in the image. Such phase differencess are due to variations in the refractive index of the object, which for
(a)) (b)
Figuree 1.5 Imaging of a locust using a) absorption contrast and b) phase contrast. FromFrom Ref. [5j.
x-rayy wavelengths is given by
nn = l-S + i/3. (1.1)
Here,, (3 = a\/4-ir incorporates the absorption of the x rays, with A the wavelength off the x rays and \i the absorption coefficient. This term affects the amplitude of thee waves. The term 6 = \2rene/2w, with re the classical radius of the electron
andd ne the electron density of the material, incorporates the refraction of waves
andd affects the phase of the x rays. For sufficiently small wavelengths (A < 0.1 nm),, (3 may be as small as 1CT9 whereas 8 is of the order of 1CT6. Thus in the x-rayy regime it is possible to observe phase contrast without having to deal with absorptionn contrast.
Ann advantage of being sensitive to phase contrast is that already the smallest variationss in the refractive index can be detected. For this reason phase-contrast x-rayy imaging has great potential for application in medicine and in biology, enabling onee to differentiate between different kinds of soft tissue (see Fig. 1.5). In confined fluids,, the variations in the refractive index due to layering effects, as discussed above,, may also be very small.
Inn our waveguiding geometry (see Fig. 1.6) the x rays are confined to the gap whereinn the fluid resides. This is achieved if the refractive index of the fluid is higherr than that of the walls, i.e., if the density of the fluid is lower than that of
Introduction Introduction 17 7
Figuree 1.6 Schematic of the waveguiding geometry (not to scale). The fluid within
thethe gap between the confining walls has ordered in six layers. The layering gives riserise to pronounced oscillations in the density in the direction perpendicular to the confiningconfining plates. Due to these variations the incoming waves are scattered into differentdifferent directions. As a result of the layering, the transmitted intensity distribu-tion,tion, which is measured in the far field, exhibits pronounced peaks at different exit anglesangles 6e.
thee walls. In this way we do not have to be concerned with background scattering fromfrom the confining walls. Whenever there are variations in the refractive index n(x,z),n(x,z), the incident wave front experiences phase changes and is refracted into otherr directions. The refracted waves interfere with all the other waves present inn between the walls, resulting in a strongly modified field distribution at the exit off the waveguide. For these experiments an x-ray source with a sufficiently high spatiall coherence is required. Such a source is available at, e.g., the European Synchrotronn Radiation Facility in Grenoble (France), where we have performed thee experiments discussed in this thesis.
Forr imaging an object, one generally places a position-sensitive detector behind it.. Depending on the distance R between the object and the detector, three regimes forr detection can be distinguished. In the 'contact' or 'near-contact' regime, in whichh the detector is placed directly behind the object, the observed contrast in the imagee is due to absorption. In the Fresnel diffraction regime for R < W2/\, where WW is the size of the object, phase contrast is present in the image, which depends
Wee deduced the refractive-index profile of a confined colloid from measured Fraunhoferr diffraction patterns, i.e., from far-field intensity distributions as a func-tionn of the exit angle 8e (see Fig. 1.6). The patterns were typically measured for
aa range of incidence angles 0*, which results in a measured intensity distribution I{@t,I{@t, #e)- A direct inversion of I(0j, 6e) into the refractive-index distribution n(x, z)
iss difficult because information on the phase of the field within the waveguide is lostt in the measurement (the well-known 'phase problem' [6]). Here we also note thatt the confined colloid is a thick phase object:
66maxmaxLL « l O x A (1.2)
forr a maximum refractive-index contrast within the fluid of <5max ~ 1 x 10~6 and
ann object thickness equal to the waveguide length L — 5 mm. The large opti-call thickness makes a direct inversion a 'tour de force'. We therefore have taken recoursee to the following model-dependent analysis. Assuming a model for the refractive-indexx distribution n(x, z) we calculate the field at the exit of the wave-guide.. From this, we derive a far-field diffraction pattern which we compare with thee measured diffraction patterns. If there is no match between the measurements andd the calculation we adjust the model and repeat the procedure until matching iss achieved (see chapter 6).
1.33 The thesis
Thee outline of the thesis is as follows. In chapter 2 the theory of propagation of x rayss through planar waveguides is discussed. In addition, a few numerical methods aree presented for the calculation of wave fields within planar waveguides having variouss refractive-index profiles. Chapter 3 describes the experimental setup of thee x-ray waveguide together with the requirements imposed on the x-ray source.
Introduction Introduction 19 9
Inn a first essential step, we have studied the coherent propagation of x rays in the absencee of a fluid. This is the subject of chapter 4. In chapter 5 we show that it is possiblee to focus x rays into an extremely fine line by tapering a planar waveguide. Inn chapter 6 we present a determination of the structure of a confined colloidal suspension,, using the waveguiding method discussed in the preceding chapters.
