In order to make x-ray scattering surface sensitive, a grazing incidence angle a is chosen between about half the critical angle a_c and several critical angles. The actual choice depends on the system to be studied. For free-standing quantum dots, an incident angle below a_c may be chosen to make the scattering exclusively surface-sensitive [Metzger]. In order to probe the internal structure of a polymer thin film of 100 nm thickness, the incident angle should be above a_c of the film, to ensure a full penetration of the sample [Smilgies].

The area detector records the scattering intensity of scattered rays over a range of exit angles b and scattering angles y in the surface plane. A beam stop has to be set-up to block spill-over direct beam as well as the reflected beam and the intense diffuse scattering in the scattering plane.

In the scattering plane the GISAXS intensity distribution corresponds to a detector scan in Diffuse Reflectivity. The intensity distribution parallel to the surface plane corresponds to SAXS. The full GISAXS intensity map can be theoretically described within the framework of Distorted-Wave Born-Approximation [Rauscher, Lazzari].
 
 

Structure factor - lateral and normal density correlations

GISAXS provides information both about lateral and normal ordering at a surface or inside a thin film. This shall be illustrated with the example of lamellar films formed by symmetric polystyrene-polybutadiene block copolymers. In a block copolymer two inmiscible polymer chains are coupled by a chemical bond. If both chains occupy equal volumes, a lamellar phase is formed. In a thin film, i.e. if the thickness of the film is on the order of the lamellar period, the presence of two interfaces, air-film and film substrate, may induce preferential order in the film as compared to the bulk polymer which forms a 3D powder of micron-sized lamellar domains:

If interfacial energies are the dominant factor, i.e. if one of the block strongly favors the interface, parallel lamellae are formed. If the interfacial energies of the blocks are similar, interfacial entropy will determine the orientation of the blocks. In particular, chain stretching in the vicinity of the bond between the chains yields a perpendicular orientation of the lamellae, while the chain end effect favors a parallel orientation [Pickett]. As the entropic effects scale with the chain lengths, even a morphology change as a function of chain length is possible. This has been indeed observed for PS-PB, where parallel lamellae are observed for short chains and perpendicular lamellae for long chains [Papadakis]

What kind of scattering will result from these two extreme cases ?
 

If one of the blocks strongly favors one of the two interfaces, or even both, the lamellae will be parallel to the substrate. The classic example is PS-PMMA on a Si wafer covered with the native oxide [Anastasiadis]. The signature of parallel lamellae in GISAXS are stripes of intensity at regular spacings along the qz direction. In Langmuir-Blodgett films, such stripes in the diffuse reflectivity are referred to as Bragg sheets. The schematic shows the diffuse scattering only (the intense specular reflection from the surface is omitted). Strictly speaking, the sketched pattern is obtained within the validity of the Born-Approximation, i.e. if the scattering angles are well above the critical angle. For scattering angles close to the critical angle, scattering patterns may be more complicated, which can be explained within the framework of DBWA theory [Busch3]

If both blocks have similar interface energies, chain stretching at the interface comes into play. Chain stretching occurs at the link between the inmiscible blocks of the polymer. A nematic ordering of this stretched part parallel to the interface may become favorable giving rise to the formation of perpendicular lamellae. As both interactions scale differently with the degree of polymerization [Pickett], there can be a transition from parallel lamellae to vertical lamellae, as we have found for PS-PB [Busch]. The signature of perpendicular lamellae are correlation peaks parallel to the interface, with a streaklike shape normal to the surface, similar to the scattering rods in Grazing-Incidence Diffraction [Als-Nielsen].

 

Note that perpendicular lamellae still have the freedom to change direction parallel to the surface plane - in fact AFM pictures [Busch, Busch2] show that meandering lamellae are formed. Such a system constitutes a 2D powder, similar to monolayers at the air-water interface [Als-Nielsen].
 
 

AFM image of a diblock copolymer film displaying vertical lamellae [Busch2]

GISAXS from the polymer film shown on the left. [Smilgies]

The scattering from such a lamellar system with a period of about 800 Å is so strong, that in-situ time-resolved measurements of the swelling of the film in solvent vapor on a timescale of minutes were possible [Smilgies].
 

Often, though, the ordering is not quite so perfect:
 

In thick films the ordering induced at the interfaces may not prevail throughout the film, and the interior of the film may assume the 3D powder bulk structure. Rings or partial rings in the intensity maps can indicate anything from complete disorder of the lamellar domains to partial ordering, e.g. lamellae with a finite distribution of tilt angles with repect to the interface.

Not always does the kinetics of the film formation allow a complete ordering of the films. This is particularly important in a system like PS-PB [Busch, Papadakis], where the morphology changes from parallel lamellae to prependicular ones as a function of chain length. At these intermediate chain lengths there is only a small preference of one morphology over the other, and the system is hence slow to reach equilibrium. In this case a mixture of different structures is observed.

PS-PB sample with a chain length in the intermediate regime.
A mixture of  parallel, perpendicular, and unoriented lamellae
is observed.

Thin films with other block copolymer morphologies than the lamellar phase have been analyzed recently [Du, Xu, Li]. In their study of silica surfactant mesophases the kinetics of the formation of various morphologies has been studied by Gibaud et al. [Gibaud].
 
 

Preferential lateral ordering and patterning

All examples discussed so far were 2D powders, i.e. had rotational averaging with respect to the surface normal, resulting in a rotationally homogeneous scattering intensity. However, by patterning the substrate, further ordering may be imposed on the film, and structures may show a preferential orientation with respect to the substrate as seen in polymer blends where the surface had been prepared by alternating hydrophobic and hydrophilic stripes [Böltau].

Similarly, regular patterns can be prepared by lithographic techniques [Woll] representing an artificial lamellar system PMMA-air. In these cases the GISAXS intensity distribution depends now also on the azimuth angle f of the substrate.

Form factor

Another type of scattering is observed for nano-objects with a narrow size distribution and well-defined shape. Here the form factor dominates the scattering, in particular, if the nano-objects are randomly placed on the surface. Examples are monodisperse nanoparticles absorbed on a wafer surface [Du] and quantum dot arrays [Metzger]. Below the calculated scattering intensity from a dilute layer of elliptical nanoparticles on a wafer surface is shown in kinematical approximation (left panel).
 
 

elliptical nanoparticles: dilute layer

elliptical nanoparticles: dense layer  with in-plane correlations

The characteristic form factor oscillations are clearly to be seen in the parallel and the perpendicular direction. When the exit angle of the scattered beam is close to the critical angle, signal enhancement due to the Vinyard effect [Vinyard] occurs, resulting in a bright band of intensity at the critical angle. This is also refered to as the Yoneda peak. Below the critical angle the scattering intensity falls quadratically off to zero.

For a dense layer of nanoparticles, particles have more or less well-defined nearest-neighbor distances. This density correlation gives rise to a structure factor with characteristic modulations parallel to the surface, but constant in the perpendicular direction (right panel).  The characteristic intensity streaks are related to the scattering rods in Grazing-Incidence Diffraction [Als-Nielsen], and are modulated by the form factor.

In our simulation, all the structure was produced by the product of just three functions, the form factor F(q_y,q_z), the structure factor S(q_y), and the Vinyard factor T(q_z), where the q_y and q_z components of the scattering vector q are parallel and perpendicular to the substrate surface, respectively. This quasi-kinematic approach has been first introduced by Naudon for studying the formation of nanoparticles on surfaces and in buried layers [Naudon]. An exact approach has been given by [Rauscher] within the framework of distorted-wave Born approximation which was  further elaborated by [Lazzari].
 
 
 

structure factor S(qy)     form factor F(qy,qz) (log scale)   Vinyard factor T(qz)

Contributions to a GISAXS intensity map in the quasikinematic approximation.

Oriented nanoparticles

The highest degree of information on nanoparticles is obtained, if these have not only uniform shape, but also uniform orientation. The classic example are pyramid-shaped quantum dots on single crystalline surfaces [Metzger]. In the latter case the scattering intensity of a line scan taken parallel to the surface depends on the relative orientation of the pyramids to the beam as given by the azimuth angle f of the sample. Another spectacular example of very highly oriented monodisperse nanoparticles are the growth studies by Renaud et al. in an all-vacuum, windowless small-angle scattering set-up [Renaud].
 
 

Summary

Mesoscopic systems can display a large range of ordering properties. Each of these has a well-defined signature in its GISAXS intensity pattern.  Moreover, due to the penetration power of x-rays, not only surface structures, but also the internal structure of thin films and buried interfaces can be studied without any need of elaborate sample preparation, as needed for instance for cross-section transmission electron microscopy. Hard x-rays can penetrate air, vapor, and small amounts of liquid allowing samples to be studied in-situ and in real time [Smilgies]. All of these features make GISAXS a very versatile tool to study shape and/or density correlations in nanoscopic systems.

References
 

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[Woll]	unpublished. (follow the link to see a pretty picture)
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