In order to make x-ray scattering surface sensitive, a grazing incidence angle a is chosen between about half the critical angle ac 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 ac 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
over a range of exit angles b
angles y in the surface
plane. A beam stop
has to be set up to
spill-over direct beam as well as the reflected beam and the intense
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,
In the scattering plane the GISAXS intensity distribution
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
map can be theoretically described within the framework of
Born-Approximation [Sinha, Rauscher,
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].
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
|If one of the blocks strongly favors one of the two
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
regular spacings along the qz direction. In
films, such stripes in the diffuse reflectivity are referred to as
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
parallel to the surface plane - in fact AFM pictures [Busch,
show that meandering lamellae are formed (aka "fingerprint patterns").
Such a system constitutes a
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
GISAXS from the polymer film
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
of the film in solvent vapor on a timescale of tens of seconds were
Often, though, the ordering is not quite so perfect:
|In thick films the ordering induced at the interfaces
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
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]. More recently [Zhang-Q] and collaborators described how to form single gyroid superlattices, 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.
their study of silica
mesophases the kinetics of the formation of various morphologies has
studied by [Gibaud].
[Wolff] studied the absorption of
micelles from the liquid onto a silicon substrate with
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
monodisperse CuS nanodisks can form ordered columnar arrays on drop
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
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.
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.
Basic scattering angles for a GISAXS experiment. If the
anisotropic in the film plane,
the GISAXS intensity map will depend on the sample azimuth f.
Scattering off an ordered array of Al nanowires, prepared by
reactive ion etching of an shear-oriented
block copolymer template [Angelescu, Pelletier]. When the sample is rotated by f, the scattering features become weaker.
Note that on the macroscopic scale, i.e. in the illuminated area of about 0.5 mm by 10 mm, the grating is not perfect.
Sample: Pelletier & Chaikin, Princeton. Scattering data: Smilgies & Gruner, CHESS (unpublished).
Another type of scattering is observed for nano-objects with a
size distribution and well-defined shape. Here the form factor
the scattering, in particular, if the nano-objects are randomly placed
on the surface. Examples are monodisperse voids in a silica film on a
surface [Du], molecular sieves
based on standing
block copolymer cylinders with the cylinder material removed [Li]
well as quantum dot arrays [Metzger].
Below the calculated scattering intensity from a dilute layer of oblate
nanoparticles on a wafer surface is shown in the quasikinematic
elliptical nanoparticles: dilute layer
oblate elliptical nanoparticles: dense layer
The characteristic form factor oscillations are clearly to be
the parallel and the perpendicular direction. When the exit angle of
scattered beam is close to the critical angle, signal enhancement due
the Vineyard effect [Vineyard]
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
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.
The highest degree of information on nanoparticles is
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
of a line scan taken parallel to the surface depends on the relative
of the pyramids to the beam as given by the azimuth angle f
of the sample. Another spectacular example of very highly oriented
nanoparticles are the in-situ growth studies by Renaud et al. in an
windowless small-angle scattering set-up [Renaud].
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.
Orientation-dependence of the scattering from quantum dots
shaped like three-sided pyramids [Metzger].
GISAXS from nanoscale gratings were
characterized in great detail by [Hofmann]
introduced a new way of obtaining grazing-incidence
transmission patterns which considerably simplifies the scattering
showed that laterally highly oriented hexagonal arrays of block
copolymer cylinders can be obtained on a miscut surface with regular
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,
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.
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.
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].
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
that the GISAXS
technique will unfold its full potential here.
Mesoscopic systems can display a large range of ordering
Each of these has a well-defined signature in its GISAXS and/or GIWAXS
Moreover, due to the penetration power of x-rays, not only surface
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.
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
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
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
become readily accessible with the Pilatus pixel array detector family,
new window in the self-organization kinetics of nanostructured
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.
(based on a talk given at the Physical Electronics Conference on Cornell Campus in June 2003)
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
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||Peter Busch, Sitaraman Krishnan, Marvin Paik, Gilman E. S. Toombes, Detlef-M. Smilgies, Sol M. Gruner, and Christopher K. Ober: "Surface induced tilt propagation in thin films of semifluorinated liquid-crystalline side-chain block copolymers", Macromolecules 40, 81-89 (2007).|
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