Scattering geometry

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 times the critical angles of the film material. 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. Largest scattering cross sections are achieved when the incident angle is inbetween the critical angles of the film and the substrate, however, multiple scattering effects have to be taken into account to properly model the data. If the incident angle is somewhat above the critical angle of the substrate, dynamic scattering effect are much reduced, and often the data can be modeled well within the quasi-kinematic approximation introduced by [Naudon]. In each of the latter two cases, a full penetration of the sample of several 100 nm is ensured.

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. The scattering geometry is thus relatively simple, and lends itself to study samples in in-situ environments [Renaud, Smilgies]. As the scattering intensity in the forward direction is high, real-time studies have become feasible [Renaud, Dourdain, Kim,  Smilgies2, Papadakis2, Paik, Zhang2].

In the scattering plane the GISAXS intensity distribution corresponds to a detector scan in Diffuse Reflectivity [Sinha]. The intensity distribution parallel to the surface plane corresponds to a line cut through the corresponding transmission SAXS pattern. The full GISAXS intensity map can be theoretically described within the framework of the Distorted-Wave Born-Approximation [Sinha, Rauscher, Lazzari, Lee1, Busch3, Tate, Stein].

GIWAXS is a related scattering technique probing atomic and molecular distances in crystal lattices. It is closely related to Grazing Incidence Diffraction, however, typically area detectors are used as in GISAXS. GIWAXS is typically used to probe the morphology of conjugated molecules and polymers[ Sirringhaus, Breiby, Chabinyc, He, Huang, Osaka]. Combined GISAXS and GIWAXS studies reveal the orientational order of crystalline blocks in polymers [Busch5, Sasaki, Darko] and orientational order of nonspherical nanocrystals on their superlattice sites [Bian, Choi, Choi2].

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 immiscible 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 incident angles and scattering angles are well above the critical angles of film and substrate. For incident angles between the critical angles of film and substrate, 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 immiscible 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, Potemkin], 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 rod-like 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 (aka "fingerprint patterns"). Such a system constitutes a 2D powder, similar to monolayers at the air-water interface [Als-Nielsen]. Another way of describing thin film samples is that they have uniaxial alignment. Closely related to such scattering patterns is fiber diffraction, and sometimes such images are also refered to as having "fiber texture" [Breiby]. While a fiber is the dual system to a thin film, it should be kept in mind that fiber diffration is a transmission experiment and well described within the kinematic approximation, while GISAXS works in reflection geometry, and reflection-refraction effect have to be included in a proper interpretation.
 
 

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 75 nm is strong and the ordering kinetics sufficiently slow, so that in-situ time-resolved measurements of the swelling of the film in solvent vapor on a timescale of tens of seconds 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 respect 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, Busch3, Busch4, Potemkin], where the morphology changes from parallel lamellae to perpendicular lamellae as a function of chain length. At 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. (Busch, Smilgies, Posselt, Papadakis, unpublished)

Thin films with other block copolymer morphologies than the lamellar phase have been analyzed as well: Du et al. characterized and modelled a monolayer of spherical voids in a matrix [Du] using the IsGISAXS code by [Lazzari] and applying Babinet's theorem. [Xu] and [Li] characterized standing hexagonal cylinders. The Ree group investigated lying hexagonally-packed cylinders, hexagonal perforated layers, and gyroid phases [Lee1, Park-I] and modeled the scattering for the cylinder phase within DWBA. In addition they characterized and modelled a film with randomly distributed spherical pores [Lee2].  The Hillhouse group independently  analyzed the gyroid phase [Tate, Urade]. Ed Kramer's group  performed a comprehensive study of thin film ordered spherical phases, from hexagonally packed monolayers to several tens of monolayers that form bcc-packed spheres with a (110) orientation, as expected from the bulk equilibrium phase [Stein]. Hence most of the regular bulk phases have been characterized, as they occur in thin film morphology, and preferential alignment with regard to the substrate surface could always be achieved. Finally [Zhang-Q] reported the preparation of a single gyroid phase.

In their study of silica surfactant mesophases the kinetics of the formation of various morphologies has been studied by [Gibaud]. [Wolff] studied the absorption of spherical micelles from the liquid onto a silicon substrate with grazing-incedence neutron scattering . Jin Wang, Sunil Sinha, and collaborators have shown, how nanoparticles trapped between two polymer surfaces diffuse laterally using resonance-enhanced GISAXS [Narayanan]. The Korgel group showed that monodisperse CuS nanodisks can form ordered columnar arrays on drop casting [Saunders]. Many more papers on nanoparticle self-assembly into two and three dimensional superlattices have been published recently, for example [Alexandrovic, Bian, Campolongo, Choi, Dunphy, Goodfellow, Hanrath, Heitsch, Smith, Zhang]

Hierarchical ordering has been reported by [Busch5] where a block copolymer in the cylindrical phase and with liquid crystalline side chains in the majority block showed ordering on the mesocopic scale (30 nm), the scale of the scale of the smectic layers (3 nm), and the molecular scale of the alkyl chain packing (0.5 nm). [Sasaki] studied thermal treatment of polyethylene films in-situ with simultaneous small- and wide-angle scattering. Simultaneous small and wide angle scattering was also the key to unravel the intricate relation of superlattice symmetry and on-site orientation of individual particles in PbS and PbSe nanocrystal assemblies [Bian, Choi, Choi2].
  
[Ree] has recently provided a comprehensive review on the multitude of scattering patterns observed in polymer films.

Preferential lateral ordering and patterning

All examples discussed so far were 2D powders, i.e. had a well-aligned axis perpendicular to the substrate and rotational averaging with respect to the surface normal, resulting in a rotationally homogeneous scattering intensity with respect to the azimuth angle f. However, by patterning the substrate, further ordering may be imposed on the film, and structures may show a preferential lateral 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 in photoresists by lithographic techniques representing artificial lamellar systems. In these cases the GISAXS intensity distribution depends now also on the azimuth angle f of the substrate. The Kowalewski group has developed the method of zone casting, which makes the preparation of laterally oriented block copolymer domains possible, and after calcination, the creation of oriented carbon nanogratings [Tang]. Nanogratings can also be prepared by using oriented block copolymer films as templates for reactive ion etching [Park-M], as shown in the example below. A quantitative description of such in-plane texture for the use with area detectors has been suggested by [Breiby] et al.

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 voids in a silica film on a wafer surface [Du], molecular sieves based on standing block copolymer cylinders with the cylinder material removed [Li] as well as quantum dot arrays [Metzger]. Below the calculated scattering intensity from a dilute layer of oblate elliptical nanoparticles on a wafer surface is shown in the quasikinematic approximation (left panel).
 
 

elliptical nanoparticles: dilute layer

oblate 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 Vineyard effect [Vineyard] occurs, resulting in a bright band of intensity at the critical angle. This is also referred to as the Yoneda peak [Yoneda]. Below the critical angle the scattering intensity falls quadratically off to zero. For thin films the Yoneda band extends between the critical angles of the film and the substrate, in particular if the former is smaller than the latter, as often encountered in organic thin films.

For a dense layer of nano-objects, 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.

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

Contributions to a GISAXS intensity map in the quasikinematic approximation.

GISAXS image from a Langmuir-Blodgett film of FePt nanocrystals (left) and simulation within the quasi-kinematic approximation (right). [Heitsch].

Laterally oriented nanostructures

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 in-situ growth studies by Renaud et al. in an all-vacuum, windowless small-angle scattering set-up [Renaud]. [Tang] created carbon nanogratings based on zone casting of a lamellar block copolymer and subsequent pyrolysis which removed one block and converted the other to carbon.
 
 

Dynamic scattering effects

Full scattering simulations

Indexation of complex scattering patterns

Self-organized nanoparticles synthesized by solution chemistry have attained better and better quality and monodispersity. Some spectacular results have been achieved for monolayers deposited on the air-water interface [Schultz] and on solid substrates  [Alexandrovic, Jiang, Heitsch]. Moreover, highly ordered and oriented unary [Saunders, Dunphy, Hanrath, Zhang] and binary [Smith] superlattices have be obtained. A key for these latter studies was careful tuning of deposition technique and annealing conditions.

Indexation schemes for such complex patterns have been described by [Breiby, Smilgies3, Tate, Hailey].

Other 3D nanostructures than nanoparticle assemblies and block copolymers have been studied with GISAXS as well: Nanotube forests can be grown using metal nanoparticles as nuclei and have been analyzed with GISAXS [Sendja]. Complex 3D nanostructured materials recently studied with GISAXS are block-copolymer templated nanoporous materials [Urade, Crossland] which are of interest for organic solar cells, catalyst scaffolds, and molecular sieves. Such structures a based on bicontinous phases such as the double gyroid and give rise to complex spot patterns.

Grazing-Incidence Wide-Angle Scattering (GIWAXS)

Bain transition in PbS nanocrystal superlattices as a function of drying dynamics. Structures covering the whole Bain path fcc > bct > bcc can be obtained by tweaking the drying kinetics. As structures approach bcc, nanocrystals display increasing relative orientation on their superlattice sites.

Orientational ordering of nanocrystals in oriented FCC(111) and BCC(110) superlattices [Choi]. In the FCC lattice nanocrystals behave like spheres and have random orientation on their lattice sites. In the BCC lattice formed by PbS nanocrystals with reduced ligand density GIWAXS data of the PbS atomic lattice reveals that particles are highly oriented.

Soft X-ray Scattering (GISoXS) and tender x-ray GISAXS

Microstructure characterization

• the radial widths of the spots
• within the Scherrer equation information on the average grain size of ordered material can be obtained, provided a careful determination of the resolution is done [Smilgies4].
• if the spot width increases with q (after a proper resolution correction !), further information about the type of disorder in the system can be obtained [Rivnay]
  
• the angular width of the spots
• the angular width of the spots or arcs in the GISAXS image is related to the degree of orientation of lamellae, cylinders, or lattices with regard to the substrate surface [Huang]. Orientation distribution functions for the so-called mosaicity can be extracted from the images following [Breiby] or [Baker2], for instance see [Darko] or [Baker]. 
• azimuthal scans reveal whether there is in-plane texture of the deposits [Tang, Breiby, Park-S]. Such studies are of particluar interest, if the deposition process or substrate modification break the isotropy of the substrate.
• in some cases orientational thin film polymorphs occur, when structures crystallize with two or more preferred planes parallel to the interface. In thin films, competing structures can be formed at the air-film and film-substrate interfaces [Choi2].
  
• the complete orientational effects can be described with the Orientation Distribution Function (ODF).

Spatially resolved studies

• [Roth] and [Muller-B] demonstrated that GISAXS studies can be performed with a spatial resolution down to a 5 microns. They studied the ordering in concentration gradients of self-organized gold clusters deposited on a polymer film.
• [Kuhlmann] showed that GISAXS can be combined with tomography, in order to characterize nanoparticle "coffee rings" resolved in 2D
  
• [Campolongo] devised a "lab-on-a-drop" experiment, to study the equilibrium configuration of Gibbs layers of DNA-coated gold nanoparticles. In this experiment the temperature, the nanoparticle concentration, the buffer salt concentration, and vapor pressure inside the sample cell are well defined. The Gibbs layer on the surface of the drop is in contact with a reservoir of nanoparticles and buffer salt in the interior of the drop. Vapor pressure is controlled by adding identical buffer solution to a reservoir below the drop. After a brief equilibration period the shape and size of the drop is stable. The sample cell consists mainly of aluminum and provides a homogeneous temperatures around the drop. Thus a small drop can be kept stable and fully equilibrated for hours. Finally, the particles in the Gibbs layer are free to move laterally due to the liquid substrate. They do not experience sticking or strain due to drying forces as in nanoparticles on a solid substrate. Hence these Gibbs layers constitute an ideally equilibrated system allowing detailed study of DNA-mediated particle interactions. In order to accomodate the curved surface of the drop, the usual GISAXS scattering geometry had to be modified. It was found that by keeping the x-ray beam parallel to the substrate ("parSAXS"), the apex of the drop could be studied with GISAXS, while the interior of the drop could be studied with transmission SAXS.
• [Li-R] used a 10 micron x-ray beam to study the microstructure of thin films of conjugated molecules in organic field-effect transistor structure with gate channels of 10-100 microns width. They found that the morphology on the gold source and drain electrodes is maintained on the silicon oxide gate electrode for tens of microns. However in wider channels a second orientational polymorph develops, that correlates with a much reduced mobility. Coating the device structure with a self-assembled monolayer helped to reduce the formation of the parasitic polymorph.
• [Böttiger] prepared a library of P3HT-PCBM bulk heterostructure film with varying composition by slot-die coating. They could read out the microstructure using a band transprt system with a barcode reader to identify individual composition. Such combinatorical studies show great promise for fine-tuning a large range of deposition parameters efficiently.
• [Smilgies6,Giri] demonstrated that the onset of crystallization of conjugated molecules can be studied in the meniscus region during shear coating with 15 micron spatial resolution and 10 msec time resolution.
• [Jacobs] performed detailed studies of laser spike annealed traces in a block copolymer film using microGISAXS with a resolution of 15 microns
  

Scanning a droplet of a solution of DNA-coated gold nanoparticles revealed formation of an ordered Gibbs layer at the apex of the drop [Campolongo].
The x-ray beam was parallel to the substrate (parSAXS).

In-situ and real-time studies

Instabilities during swelling of block copolymer lamellae [Papadakis2].	Kirkwood-Alder transition in dropcast PtCu nanoctahedra during drying under controlled vapor pressure [Zhang2].
Crystallization of nanocubes with competing structures at the substrate-solution interface and the air-solution interface [Choi2].	Self-organized Oswald ripening of 2nm gold nanocrystals during heating. Binary superlattices of large fused particles and the original small particles are formed [Goodfellow].

Block copolymers:

• [Papadakis]  identified the lowering of the glass transition and the order-disorder transition during solvent vapor precessing of block copolymers as determining factors whether a given polymer film can be vapor-annealed. They found that the degree of swelling or the solvent volume fraction in the block copolymer film should be chosen to lower the glass temperature as much as possible, while preventing the system to cross the order-disorder transition [Di].
  
• [Paik] found that block copolymer film morphology can be changed by swelling with a highly selective solvent. Using fast drying, this non-equilibrium structure can be quenched with a small vertical distortion due to drying. Treating such films with a non-selective solvent transformed the films back to their equilibrium configuration at high swelling ratio. This processing cycle could be repeated several times.
• [Zhang-J] observed the formation of a disordered PB top layer during swelling of vertical PS-PB block polymer lamellae in solvent vapor, and the formation of protrusions during successive controlled drying.
• [Chavis] found a full range of block copolymer nanostructures be annealing a block copolymer varying the mixsture solvents which were selective to one of the blocks. In addition a transition from BCC(110) to SC(100) was observed for the shorter one of two studied polymers.
  
• [Gu] observed the formation of a regular square pore lattice in PI-PS-P2VP triblock copolymers after doctor blading
  

• [Hanrath] found that evaporation kinetics has a profound effect on the degree of order and the degree of orientation attained in nanocrystal superlattices. By optimizing deposition conditions via slow drying of vapor annealing close to perfect FCC(111) and BCC(110) lattices could be obtained. Moreover, the exact packing of such objects was found to be dependent on nanocrystal shape [Bian] and the nanocrystal ligand density [Choi]. Using a combination of GISAXS and GIWAXS, first well-oriented superlattices were prepared and then the degree of orientation of individual nanocrystals on their lattice sites was probed. While nanocrystals should an uniform distribution in FCC(111), as to be expected for isotropic nanoparticle interaction, nanoparticles forming BCC(110) lattices were highly oriented. Lattices with an intermediate degree of on-site orientation were found to be tetragonal, covering the well-know Bain path between FCC and BCC. While BCC superlattices had been observed before, this work provided the first proof of the important on non-isotropic interactions in a BCC lattice.
  
• [Zhang2] observed a Kirkwood-Alder transition during the self-assembly of PtCu nanoctahedra to oriented BCC superlattices on a silicon wafer.
  
• [Senesi] engineered DNA-coated gold and silver nanoparticles in a way that they can do a strict layer-by-layer deposition of particles onto a gold film functionalized with complementary DNA.
• [Choi2] found competing structures of PbS nanocubes at the substrate solution (simple cubic 001) and the solution-air (rhombohedral 111) interfaces.
• [Goodfellow] observed complex fusion processes in gold nanoparticle superlattices upon heating that gave rise to ordered structures typically observed in binary superlattices.
• [Yu] found ordering upon heating in superlattices of small gold nanocrystals.
  
• [Ocier] observed reorganization of CdSe-CdS nanorods during solvent vapor uptake simultaneously with GISAXS, GIWAXS, and optical soectroscopy.
  
• [Weidman] obtained detailed simultaneous GISAXS/GIWAXS images on the Bain Transition in PbS superlattices during controled drying. The Bain transition is driven by the densification of the deposit. The FCC to BCC transition is accompanied by a rotation of the superlattice grains from FCC(111) to BCC(110).

• [Sanyal] studied the drying kinetics of P3HT-PCBM bulk heterojunctions after doctor blading.
  
• [Perlich] studied the formation of nanocrystal monolayers during dip coating.
  
• [Smilgies6] found that the crystallization of conjugated molecules at the edge of the meniscus can be studied with 15 micron spatial resolution and down to 10 msec time resolution during solution shearing.
  
• In-situ real-time studies of spin coating were performed by [Chou] and [Perez].
• [Wan] found intermediate phases in the crystallization of the small-molecule semiconductor C8-BTBT after hollow-pen writing.
• [Pozdin] observed thermal reorganization processes in a semiconducting conjugated polymer under inert atmosphere.
  

These discoveries were facilitated by the ability to study nanostructured materials in a well-controlled in-situ sample environment and in real time. It is to be expected that the GISAXS technique will unfold its full potential here.

Coherence Effects

Summary and Outlook

Mesoscopic systems can display a large range of ordering properties. Each of these has a well-defined signature in its GISAXS and/or GIWAXS 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 [Smilgies]. The GISAXS and GIWAXS scattering geometries  are straightforward and, in many cases, without the need for scanning, making GISAXS and GIWAXS very attractive to combine with elaborate in-situ chambers [Renaud]. GISAXS scattering intensities are high compared to grazing incidence diffraction, and in combination with the essentially static scattering geometry, make GISAXS an ideal technique to combine with real-time measurements. Typical CCD cameras acquire at 1 frame per 10 sec down to 1 frame per sec; commercially available pixel array detector can acquire data up to 100 frames per second.  While swelling kinetics in block copolymers happens on the time scale of minutes, and is well matched for the study of polymer kinetics, conjugated  molecules crsytallize from solution on a msec time scale. As area detectors are evolving, the msec time scale has become readily accessible with the Pilatus pixel array detector family, opening a new window in the self-organization kinetics of nanostructured materials.

All of these features make GISAXS and GIWAXS versatile tools to study shape and density correlations in nanoscopic systems in situ and in real time.

Links

A Final Comment

GISAXS and GIWAXS experiments, particaly for the charactrization of soft materials thin films, have recently experienced tremendous popularity and it has become hard to keep this tutorial up to date. Many new groups have gotten involved and are building up their own expertise. Please do not hesitate to point out papers describing new applications or technical break throughs that I may have missed. And please take a moment to fill out the feedback below. DS

References

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