J. Phys. Chem. C 2007, 111, 7645-7654
7645
General Method for Simulation of 2D GISAXS Intensities for Any Nanostructured Film Using Discrete Fourier Transforms Michael P. Tate and Hugh W. Hillhouse* School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907 ReceiVed: September 18, 2006; In Final Form: March 9, 2007
A fast, flexible 2D GISAXS simulation method based on the distorted-wave Born approximation (DWBA) has been developed for nanostructured thin films using discrete Fourier transforms of a N × N × N matrix that represents the electron density. By expressing the electron density in matrix form, various models of electron density distributions can be easily simulated and compared to experimental data. In addition to modeling the effects of overall symmetry and orientation on the relative intensities of the Bragg peaks, this approach can be used to evaluate specific details of the nanostructure such as pore connectivity, domain size, domain shape, positional disorder, orientation disorder, and polydispersity. These effects are included in a natural way without making a decoupling approximation between structure factor and form factor. The range of reciprocal space simulated is set by the size of the matrix N and the scale factor β that sets the real-space length of each matrix element, given by 1/(2Nβ) < |s| < 1/(2β). However, the computation time of the 3D transform scales with N as N3log(N3). A matrix with N ) 200 and a real-space resolution of β ) 1 nm was sufficient to model the relevant features of self-assembled nanomaterials while remaining computationally inexpensive. Here, we describe the methodology, show simulations for several examples, and compare simulation to experimental 2D GISAXS patterns. Specific examples include simulated 2D GISAXS patterns for 2D hexagonal nanostructures (p6mm) where the pores are perpendicular to the substrate and for (110)oriented body-centered cubic nanostructures (Im3hm) based on the level surface approximation of the I-WP surface. For the latter, results show that systematic suppression of Bragg peaks occurs for specific values of the contour level and may be used to identify accessible phases. In addition, we compare simulated patterns to experimental 2D GISAXS synchrotron data for (111)-oriented rhombohedral (R3hm) nanostructured films. Curved arcs in the experimental data are identified by simulations to result from domain shape effects. Simulations show that the domains in the film are rhombus shaped, where the edges of the domain are coaligned with the (100) faces of the R3hm unit cell. The simulation code, entitled NANODIFT, is written in Mathematica and is available upon request.
1. Introduction The synthesis of ordered nanoporous metal oxide powders1-4 through self-assembly has launched a new field of nanomaterials with high surface area, highly ordered pore geometries, and monodisperse pore sizes from 2 to 20 nm. Further, the development of oriented thin-film morphologies5-10 has brought the potential for advances in areas such as low-k dielectrics,11,12 low refractive index films,13-15 hydrogen sensors,16 nanostructured magnetic materials,17 photomodulated mass-transport layers,18 nanostructured solar cells,19 and nanostructured thermoelectrics.20 However, further engineering films for each of these applications requires detailed characterization of the nanostructure. Typically, nanostructured films are characterized by powder X-ray diffraction (PXRD) and transmission electron microscopy (TEM). TEM is tedious and must be conducted ex situ in high vacuum, typically from films removed from the substrate. PXRD may be conducted on as-synthesized films under note natural environmental conditions but reveals only a 1D view of reciprocal space. Oriented phases are notoriously difficult to identify by this method, and recently, in addition, it was demonstrated that refraction effects are not negligible for many * To whom correspondence should be addressed. E-mail: hugh@ purdue.edu.
of the synthesized films.21 A more ideal method is to use 2D small-angle X-ray scattering (SAXS), which reveals a 2D slice of reciprocal space and thus enables determination of oriented nanostructures. SAXS of thin films is divided into two distinct regimes, (1) transmission SAXS (TSAXS), where the scattered intensity of interest is transmitted through the substrate (angles of incidence ranging from perpendicular to the substrate down to just a few degrees) and (2) grazing-incidence SAXS (GISAXS), where the scattered intensity of interest is either above the horizon of the edge of the substrate or is reflected off the substrate (angles of incidence are typically above the critical angle for total reflection and range from 0.15 up to 1°). TSAXS is limited to experiments on very thin substrates and results in long data collection times but has been successfully used to identify self-assembled mesophases.15,16,22-26 GISAXS data, on the other hand, may be collected in seconds, even on laboratory instruments. Further, due to the existence of many domains oriented in the plane of the substrate, reciprocal space for oriented self-assembled films typically consists of rings that are parallel to the substrate.22,23,27 As a result, the slice of reciprocal space seen in the GISAXS experiment reveals all features of reciprocal space in one experiment. As a result, GISAXS has been used frequently to examine self-assembled nanostructured films.15,22,23,27-46
10.1021/jp066111n CCC: $37.00 © 2007 American Chemical Society Published on Web 05/05/2007
7646 J. Phys. Chem. C, Vol. 111, No. 21, 2007 However, detailed analysis of the GISAXS data are difficult to analyze because the Born approximation (BA) fails due to multiple scattering at the film interfaces.47 This effect becomes more pronounced near the critical angle as refraction of the incident and exit waves becomes large.48,49 Additionally, there are extra peaks that arise from reflection at the film/substrate interface.42,43 In many case, the overall symmetry and orientation of the film can be identified by qualitative analysis of the symmetry of the Bragg peak pattern. However, this may result in misidentification due to the presence of artifacts from reflected beams and shifts due to refraction. Recently, we reported41 the development of simulation code (“NANOCELL”) that allows one to quantitatively compare peak positions from experimental 2D GISAXS patterns with those predicted by the DWBA. This code allows the accurate determination of space group, orientation, and lattice constants. However, there are many other film properties that affect the 2D GISAXS pattern, and a full intensity calculation is needed. The intensities can be calculated under the DWBA. In this theory, scattering at the interfaces is treated dynamically, while scattering from within a structure is treated as a first-order perturbation. Thus, the implementation of the theory is dependent on the geometry of the surfaces/interfaces. The theory was first developed more than 20 years ago49,50 and later expanded to include rough interfaces.51-54 Lazzari55 recently developed simulation code (“IsGISAXS”) for variousshaped geometric islands on a substrate. However, this geometry is not relevant for self-assembled nanostructured films which have relatively ideal interfaces but, yet, have intricate scattering patterns that arise from the nanostructure within the film. More recently, an analytical approach was developed by Lee42,43 to calculate the scattering from nanostructured thin films. However, this method is difficult to implement for many situations since the structure must be described analytically and the Fourier transform must have an analytic solution. To overcome the limitations of an analytical approach, we report the development of a general method using discrete Fourier transforms (DFT) within the DWBA formalism. This allows for easy simulation of the effects due to specific electron density profiles, allowing one to examine the effects of pore connectivity, wall thickness, and density gradients. In addition, the effects of domain shape, size, twining, and disorder may be examined easily. The method allows for the full intensity of 2D GISAXS patterns of arbitrary structures to be simulated without decomposing the scattering into form factor and structure factor contributions. 2. Methods The simulation of 2D GISAXS patterns begins with the construction of an N × N × N matrix that represents the electron density of real space, F(x,y,z). The amplitude in reciprocal space, A(sx,sy,sz), is then calculated by a fast Fourier transform algorithm. The intensity is then calculated by accounting for the DWBA. Schematically, the flow of input and output information is shown in Figure 1. 2.1. Construction of Real Space. First, a representative region of the film must be discretized. An N × N × N matrix is set up with the z axis normal to the substrate and the y axis co-aligned with the projection of the incident beam on the substrate. The size of this array cannot be arbitrarily large since the discrete Fourier transform must be calculated and the best fast Fourier transform (FFT) algorithms scale with
Tate and Hillhouse
Figure 1. A schematic of the calculation method.
N3log(N3). This very quickly becomes computationally too expensive to choose large N. A scale factor (β) is then set that relates each matrix element to real-space dimensions. This scale factor sets the minimum feature size in real space (β) and a maximum length scale for features that can be modeled equal to Nβ/2. Note that it also sets the high-s limit of reciprocal space (s < 1/(2β)). For the simulations here, N ) 200 and β ) 1 nm yielded sufficient balance between resolution and computation time. Population of the real-space 3D matrix with values that represent electron density was performed by simple if-then statements in a nested loop. For nanostructures composed of periodic placement of discrete objects, spheres or rods were placed based on the space group symmetry elements (2D hexagonal mesophase from p6mm placement of cylinders, bodycentered cubic mesophase from Im3hm placement of spheres, etc.). Using this technique, it is easy to construct nanostructures with different electron density profiles, domain shapes, and sizes by simply modifying the if-then conditions. Specific types of disorder can also easily be modeled, including polydispersity, static positional disorder, paracrystalline disorder, and faulting. It is also just as easy to build nanostructures built from continuous surfaces (such as the G surface with Ia3hd symmetry, the I-WP surface with Im3hm surface, etc.) by placing electron density on one side or within a certain distance of the surface. A review of the “level set equations” for surfaces based on various space group symmetries are reviewed by Wohlgemuth et al.56 In addition, structures may be “handmade” using the exponential scale. This versatile method of constructing 3D objects is described in depth by Jacob and Andersson.57 Once the 3D surface is defined, it may be oriented by a simple rotation
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J. Phys. Chem. C, Vol. 111, No. 21, 2007 7647
matrix transformation defined below as
tering amplitude from just the nanostructured thin film is given by41-43
RRotation ) 0 cos(ζ) sin(ζ) 0 cos(ψ) 0 -sin(ψ) 1 0 sin(χ) 1 0 ‚ 0 cos(χ) -sin(ζ) cos(ζ) 0 ‚ 0 0 -sin(χ) cos(ψ) 0 0 1 sin(ψ) 0 cos(ψ) (1)
(
)(
)(
)
where ζ is the rotation about the x axis, ψ is the rotation about the y axis, and χ is the rotation about the z axis. Similarly, to model the uniaxial contractions that typically occur in selfassembled films, another matrix transformation may be applied. For example, a uniaxial contraction along the z direction is modeled using
(
1 0 0 TTransform ) 0 1 0 0 0 1/(1 - f)
)
(2)
H(x,y,z,Nβ) ) 2π x 2π y 2π z π π π π π π 3 Nβ 3 3 Nβ 3 3 Nβ 3 cos2 cos2 cos2 2 2 2 (3)
(
) (
)
(
) (
)
∫
∫
∞
-∞
drye -i((kxf-kxi)rx+(kyf-kyi)ry) (TiTf
r ∫ dr V(F)e R R ∫ dr V(F)e r 0
TiRf i f
-d 0
-d
-i(-kzf-kzi)rz
z
-i(-kzf+kzi)rz
z
∫
+ RiTf
0
-d
∫
0
-d
-i(kzf-kzi)rz drzV(F)e + r
-i(kzf+kzi)rz drzV(F)e + r
)
(4)
where V(r) is the scattering potential of the perturbation, T and R are the transmission and reflection coefficient within the film, respectively, k is the direction of the wave vector, the subscripts i and f represent the incident and final waves inside the thin film, and d is the film thickness. Here, the scattering potential, V(r), is defined by
V(F r ) ) 4πr0(F(F r ) - 〈F〉) ) 4πr0η(F r)
where f is equal to the fraction contracted (0.10 for a 10% contraction). 2.2. Simulation of Reciprocal Space. A discrete Fourier transform (DFT) was applied to the N × N × N matrix of realspace electron density to calculate an N × N × N matrix that represents reciprocal space. The reciprocal space matrix is centered and covers reciprocal space from 1/(2Nβ) < |s| < 1/(2β). The DFT algorithm used is based on the fast Fourier transform method as implemented in Mathematica. For N ) 200, the computation time of the 3D DFT typically took 15 s on a 2.4 GHz desktop machine and scaled as N3log(N3). The use of the DFT does give rise to “aliasing artifacts” which result from the fact that the real-space array is assumed to repeat to infinity in every direction. These reciprocal-space artifacts are oscillations that result from the abrupt change of electron density at the edges of the array. However, apodization functions which taper the matrix values near the edges are commonly used to reduce aliasing. This is typically referred to as windowing, and many specific window functions have been reported, such as the Hann, Blackman-Harris, Hamming, and Gaussian windows.58 Here, the Hann window is used on select patterns as indicated through this report and is given by
(
ψ(F) s ) ∞ eik0r dr 4πr0 -∞ x
(
)
)
where x, y, and z are the 3D array coordinates in nanometers. The Hann window presented above is defined such that the maximum of the Hann function occurs at the center of the realspace array and the edges are zero (see Figure 2 for a 2D illustration). Note the reduction of artifacts (oscillating side lobes on the Bragg peaks). 2.3. Simulation of Detector Intensity using DWBA. The DWBA requires that the scattering from the nanostructure is weak compared with that from the interfaces. Thus, the total scattering potential may be separated into two parts, and a first-order perturbation technique may be applied to solve for the scattering of the nanostructure from within the thin film. Using the Parratt formalism,48 the scat-
(5)
While the x and y components of the scattering vector are the same for each term, the z component of the “incident” beam direction is different for each term due to refraction and reflection. Thus, the scattering vector (s) is defined as given below for each of the four terms above
kzf kzi 2π 2π kzf kzi szTerm3 ) + 2π 2π
szTerm1 )
sx )
kxf kxi 2π 2π
(6a)
sy )
kyf kyi 2π 2π
(6b)
kzf kzi - , 2π 2π kzf kzi szTerm4 ) + 2π 2π
szTerm2 ) -
(6c)
The intensity is then given by 2 * I(F s )ψ (F s )) s ) ) r0 (ψ(F
(7)
where / represents the complex conjugate, which yields
I(F s))
∫-∞∞ ∫-∞∞ ∫-d0 η(Fr )e-i2π(s r +s r +s ∞ ∞ 0 -i2π(s r +s r +s T2iR2f ∫-∞ ∫-∞ ∫-d η(F r )e ∞ ∞ 0 -i2π(s r +s r +s R2iT2f ∫-∞ ∫-∞ ∫-d η(F r )e ∞ ∞ 0 -i2π(s r +s r +s R2iR2f ∫-∞ ∫-∞ ∫-d η(F r )e
r02|T2iT2f
x x
x y
zTerm1rz)
drzdrxdry +
x x
x y
zTerm2rz)
drzdrxdry +
x x
y y
zTerm3rz)
drzdrxdry +
x x
y y
zTerm4rz)
drzdrxdr|2 (8)
Now, szTermi (where i represents each of the four terms) will contain an imaginary term whenever Ri < Rc or Rf < Rc. This occurs as a result of the fact that, below the critical angle, only an evanescent wave penetrates the surface of the film. Additionally, due to the time reciprocity theorem of light, this occurs for both the incident and exit waves. Rauscher54 showed for 3D periodic structures embedded within a substrate that this imaginary component is analytically separable. More recently, this was implemented
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for the case of a finite thickness thin film by Lee,43 which results in 2 I(F s ) ) r0
(
)
1 - e-Im(2πsz)d 2πIm(sz)
∫ ∫ ∫ η(Fr )e T R ∫ ∫ ∫ η(F r )e R T ∫ ∫ ∫ η(F r )e R R ∫ ∫ ∫ η(F r )e |TiTf i f
i f
i f
∞
0
-∞ ∞
-∞ ∞
-d 0
-∞ ∞
-∞ ∞
-d 0
-∞ ∞
-∞ ∞
-d 0
-∞
-∞
-d
∞
-i2π(sxrx+sxry+Re(szTerm1)rz-Im(szIM)|rz|)
drzdrxdry +
-i2π(sxrx+sxry+Re(szTerm2)rz-Im(szIM)|rz|)
-i2π(sxrx+syry+Re(szTerm3)rz-Im(szIM)|rz|)
drzdrxdry +
drzdrxdry +
-i2π(sxrx+syry+Re(szTerm4)rz-Im(|szIM)|rz|)
drzdrxdry |
Where Im(2πszIM) ) |Im(kzf)| + |Im(kzi)|. Note that when Ri gRc and Rf gRc, 1 - e-2Im(2πszIM)d/2Im(2πszIM) ) d and that e-Im(2πszIM)|rz| ) 1. Additionally, when Rf < Rc and Ri > Rc, the imaginary term inside the integral may be approximated as e-Im(2πszIM)|rz| ≈ 1.54 The intensity for all Rf when Ri > Rc may then be given as
(
)
1 - e-Im(2πsz)d 2πsz
∫-∞∞ ∫-∞∞ ∫-d0 η(Fr )e-i2π(s r +s r +Re(s ∞ ∞ 0 -i2π(s r +s r +Re(s TiRf ∫-∞ ∫-∞ ∫-d η(F r )e ∞ ∞ 0 -i2π(s r +s r +Re(s RiTf ∫-∞ ∫-∞ ∫-d η(F r )e ∞ ∞ 0 -i2π(s r +s r +Re(s RiRf ∫-∞ ∫-∞ ∫-d η(F r )e |TiTf
x x
x y
zTerm1)rz)
x x
x y
zTerm2)rz)
x x
y y
zTerm3)rz)
x x
y y
zTerm4)rz)
drzdrxdry +
drzdrxdry +
drzdrxdry +
drzdrxdry|2 (10)
Note that this expression differs significantly from that which would be obtained under the regular Born approximation 2 I(F s ) ) r0 ||
∫ ∫ ∫ η(Fr )e-i(Fk -Fk )Fr d3Fr ||2 f
i
(11)
However, the DWBA expression does contain four Fourier transforms that each look similar to those contained in the Born approximation, but each is modified by a transmission and reflection coefficient and is carried out with a different wave vector. The modification of the wave vector component in the z direction results from refraction at the air/film interface and causes a shift in the Rf direction for the scattering. Additionally, the reflection at the film/substrate interface creates a second set of peaks which are also shifted in the Rf direction. The x and y components are not effected, and thus, the peaks are not shifted in the 2θf direction for ideal flat interfaces. In order to calculate the observed intensity using eq 10, the 2D detector is discretized in terms of Rf and 2θf. Thus, the wave vectors entering and exiting the film may be expressed in terms of these parameters, along with Ri and Rc, as 2π F k f ) (sin(2θf)cos(Rf), cos(2θf)cos(Rf), xsin2(Rf) - sin2(Rc)) (12a) λ
(
2π F ki ) 0, cos(Ri), xsin2(Ri) - sin2(Rc) λ
)
(
)
1 - e-Im(2πszIM)d |T2iT2f A(sx,sy, Re(szTerm1)) + 2Im(2πszIM) T2iR2f A(sx,sy,Re(szTerm2)) + R2iT2f A(sx,sy,Re(szTerm3)) +
I(Rf,2θf) ) r02
R2iR2f A(sx,sy ,Re(szTerm4))|2 (13)
2
(9)
2 I(F s ) ) r0
space array, A(sx,sy,Re(szTermi)), for each of the four terms such that
The Fresnel transmission and reflection coefficients then are calculated41 at the given Rf and 2θf, and a correction for roughness of the film/substrate interface is included as a DebyeWaller-like factor47
R ) RFresnel exp-2σ kz
2 2
(14)
where Rf is the Fresnel reflection coefficient. These quantities are then inserted into eq 13 to yield the intensity at a given Rf and 2θf. This is repeated for all Rf and 2θf to simulate the intensity on the 2D detector for a single domain. For thin films with planar disorder, an additional step is required to sample all possible orientations of the nanostructure about reciprocal space. To do so, we assume that the scatterings from individual domains do not interfere with each other and the intensity may be superimposed on the detector. This is achieved by a matrix transformation that rotates the amplitude information about the z axis such that
(s′x , s′y , Re(sz)) ) RPlanarDisorder‚(sx , sy , Re(sz))
(
cos(ζ) sin(ζ) 0 RPlanarDisorder ) -sin(ζ) cos(ζ) 0 0 0 1
)
(15)
(16)
where ζ is the angle of rotation about the z axis. This rotation matrix is applied in small increments, and the resulting intensities are summed to achieve the final intensity as seen on the detector. For presentation purposes, a small Gaussian blur was applied to the final discretized detector. The code that implements the discrete Fourier transform method as described above on nanostructured films to calculate the 2D GISAXS patterns is referred to as NANODIFT. Full intensity simulations with NANODIFT can thus be used to complement NANOCELL41 calculations for lattice constant fitting and space group determination. 3. Results and Discussion
(12b)
where Rc is the critical angle of the air/film interface. The wave vector directions are then substituted into eqs 5a and b to define scattering vectors for each of the four terms in eq 9. The amplitude information is then retrieved from substitution of the Fourier-transform-like terms in eq 9 with the discrete reciprocal-
As an example of the NANODIFT code describe above, we simulate 2D GISAXS patterns from several nanostructures of interest. Electron density models based on the I-WP surface have been used to explain the structure of some selfassembled nanoporous materials that have Im3h m symmetry.59 When synthesized in oriented thin-film morphology, these structures tend to orient with their (110) planes
Method for Simulation of 2D GISAXS Intensities
J. Phys. Chem. C, Vol. 111, No. 21, 2007 7649
Figure 2. Aliasing effects and apodization. (a) A 2D array (1024 × 1024) of a hexagonal packing of spheres and the 2D reciprocal space (b) that results from DFT. Note the aliasing artifacts around each peak in part (b) that result from the abrupt change in electron density near the edge of the array. Part (c) shows the same array as that in part (a) after apodization. Part (d) shows the 2D reciprocal space that results from the DFT of part (c). Note the significant decrease in the aliasing artifacts.
parallel to the substrate. The level set equation approximation of the I-WP surface is given by
FI-WP(x,y,z,a) ) 2πx 2πy 2πy 2πz 2πz 2πx cos + cos cos + cos cos cos a a a a a a (15)
( ) ( )
( ) ( )
( ) ( )
The surface is then oriented with (110) parallel to the substrate by applying a rotation matrix with ζ ) 90, ψ ) 0°, and χ ) -45°. For simulation purposes, the lattice constant a is set to 17.95 nm. Now, the values of x, y, and z that yield a constant value of FI-WP define a surface. For each different value of FI-WP, a different surface is obtained. This constant is called the contour level. For values of x, y, and z where FI-WP is less than the chosen contour level, electron density is placed to represent the wall. For values of x, y, and z where FI-WP is less than the chosen contour level, the electron density is set to zero
to represent the pore. Figure 3 shows two oriented films constructed with different contour levels (0.6 and -0.2). Note the dramatic physical difference of the films. One has cagelike pores (contour level 0.6), while the other has open accessible pores (contour level -0.2). This method for the Im3hm nanostructure is similar to that reported elsewhere by us for the Ia3hd nanostructure.60 NANODIFT simulations of the 2D GISAXS patterns are shown for each structure (assuming the domain size is infinite) in Figure 3. The NANOCELL-predicted spot positions are overlaid on the NANODIFT intensity simulations. Examination of the patterns indicates that some peaks are strongly suppressed, even though they are allowed according to the Im3hm reflection conditions. Further, upon comparison of the intensities for each contour level, distinct regions of peak suppression are seen in one but not in the other. These regions are color coded to facilitate the comparison. If one examines the blue labeled spots,
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Figure 3. Simulations of a (110)-oriented I-WP-based nanostructure, where a ) 17.95 nm (parts a and c), and the resulting GISAXS patterns (parts b and d) for contour levels of 0.6 (a and b) and -0.2 (c and d). The GISAXS patterns were calculated for Ri ) 0.20°, Rc ) 0.16°, and the Rc substrate ) 0.22°. The circles and squares identify transmitted and reflected Bragg peaks, respectively, that are allowed according to the Im3hm reflection conditions and intense enough to be seen in the simulation. Those in blue are suppressed for the 0.6 contour, and those in green are suppressed for -0.2 contour. Peaks in red are not suppressed in either case.
Figure 4. (a) Reciprocal space of the nanostructure constructed from the bcc packing with the (110)-oriented nanostructure and (b) reciprocal space of the form factor for a sphere of the same radius as that constructed in part a (radius ) 5.4 nm). Notice that the form factor minima occur at the same location as the minima in part a. The form factor is not a perfect circle, as shown in (c), which is the analytical solution to the form factor due to the fact that the small sphere within the real-space array appears discrete (more like a polyhedral than a true sphere).
they are intense from the accessible structure but nearly completely suppressed in the cage-like structure. The opposite is true for the green labeled spots. The peak suppression may be understood by analogy to a Born approximation calculation where the intensity has been “decoupled” into the product of a form factor and structure factor, I(q) ) P(q)S(q). In general, the form factor describes the scattering from an object, while the structure factor accounts for the placement of the objects. Under this “decoupling approximation,” a minimum in the form factor can cause a
suppression of a peak in the structure factor. To illustrate the relevance to the GISAXS shown in Figure 3, we show a TSAXS pattern calculated using the Born approximation for a bcc packing of 5.4 nm spheres in Figure 4a. Unlike the I-WP surface, this structure may be easily decomposed into the form factor (that for 5.4 nm spheres) and a structure factor (Im3hm symmetry). In Figure 4b, a DFT simulation is shown for a single sphere. Note that the minimum in the Figure 4b pattern corresponds to where the peaks are suppressed in Figure 4a. This indicates that the suppression is a form factor effect. Note
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Figure 5. Simulated GISAXS patterns from a 2D hexagonal phase with channels oriented perpendicular to the surface. (a) Schematic of the thin film with a large domain in all three directions of a 2D hexagonal structure (a ) 8.6 nm) with the pores oriented perpendicular to the substrate. (b) The corresponding GISAXS simulation where Ri ) 0.20°, Rc ) 0.16°, and Rc substrate ) 0.22°. (c) Schematic of the same structure but with a small domain length in the plane of the film. (d) The GISAXS simulation of (c). (e) Schematic of the structure with small domains along the c axis of the pores but with large domains in the plane of the film and (f) the corresponding simulation. Note that, in each GISAXS simulation, the intensities have been scaled identically.
that the suppressed ring in Figure 4b is not circular. This arises from the fact that the 5.4 nm sphere is defined from the 1 nm square elements of the matrix (hence, it is not completely circular). This is an unavoidable consequence of the discrete method employed here. This is compared to the analytical form factor for 5.4 nm spheres shown in Figure 4c. The suppressions that are observed in the full GISAXS patterns in Figure 3 result from the same basic phenomena. However, each pattern of suppressed peaks forms an ellipsoid due to refraction effects. The effects of such suppressions may be seen experimentally in the (hh0) set of reflections since they are typically the most intense. Simulations show significant suppression of the (220) intensity in the accessible structure. Simulation of these effects opens the possibility to readily determine if there are significant differences in accessibility between films.
In addition to interpreting systematic absences, the simulation code also readily allows for interpretation of domain shape and size effects. As an example of this, we consider a 2D hexagonal system in which the pores are oriented perpendicular to the substrate. This structure is ideal for many applications and is sought after in many experimental efforts. Figure 5 illustrates three cases, (1) where a large domain of pores extends through the thickness of the film (Figure 5a), (2) where a small domain of pores extends through the thickness (Figure 5c), and (3) where a domain of pores persists perpendicular to the film thickness for only a short length (Figure 5e). In all cases, the domains are composed of a hexagonal packing of cylinders (p6mm symmetry) with a ) 8.6 nm oriented perpendicular to the substrate. The corresponding GISAXS patterns (Figure 5b, d, and f) that result are immediately distinguished from the 6-fold pattern of nanostructures with pores parallel to the surface (not
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Figure 6. An experimental 2D GISAXS pattern collected on APS (beamline 8-ID-E) from a (111)-oriented rhombohedral nanostructured film synthesized by EISA upon dip-coating at 40% RH. The lattice constants are a ) 11.1 nm and R ) 84°. Also, Ri ) 0.20°, Rc ) 0.16°, and the Rc substrate ) 0.22°. Note that the red circles and squares indicate the location of transmitted and reflected Bragg peak locations for the left half of the detector, as determined by NANOCELL.
presented). This results from the fact that the Bragg peaks in reciprocal space lie along the x axis only and thus would only produce points along the Rf ) 0°. However, Rf ) 0° is well below the critical angle of the film, and thus, these peaks have little to no intensity, as predicted by DWBA. The reflected Bragg peaks, however, are predicted to occur at Rf ) Rc, which will appear as intensity in the GISAXS simulations. Recent experimental GISAXS patterns collected from polymer thin films61
Tate and Hillhouse and mesostructured silica thin films62 agree qualitatively with the simulations presented here. In the first case, the intensity from the peaks is very narrow in both the Rf and 2θf directions, as is expected when the domain size is large. This is contrasted with Figure 5d where the c axis is still infinite but the a and b directions are very short. Notice the peak broadening only in the 2θf direction. This is expected because the domain is short in only two directions (x and y) rather than in all three directions. If the domain length along the channels is small (see Figure 5e), the result is strikingly different from that in Figure 5b or d. Here, significant peak broadening occurs in the Rf direction. The breadth of this peak is related in a reciprocal fashion to the persistence length of the cylinders in the direction perpendicular to the substrate. Interpretation of the peak widths, however, must be done with care as the first form factor minimum from the cylinder crosses both the diffuse scattering of all three Bragg peaks shown in Figure 5f in an ellipsoidal ring that diminishes the intensity. In another example, we compare NANODIFT simulations from rhombohedral symmetry nanostructured thin films27 to experimental data to reveal the domain structure in the synthesized films. Synchrotron GISAXS data collected from a rhombohedral nanostructured film are shown in Figure 6. The lattice constants, space group, and film orientation were determined using NANOCELL to be a ) 11.1 nm and R ) 84°, R3hm, and (111), respectively. The predicted peak positions from these parameters are shown as circles and squares. In addition to these sharp Bragg peaks, we note the very distinct set of diffuse lines and arcs. In order to explain and interpret
Figure 7. NANODFT simulations of GISAXS from rhombohedral films. (a) Schematic of infinite domains of a rhombohedral nanostructured film (a ) 11.1 nm and R ) 84°) and (b) the resulting GISAXS simulation where Ri ) 0.20°, Rc ) 0.16°, and the Rc substrate ) 0.22°. (c) A schematic of rhombus-shaped domains of a rhombohedral nanostructured film and (d) the resulting GISAXS simulation at the same conditions as those in part (c). Note that the red circles and squares indicate the location of transmitted and reflected Bragg peak locations for the left half of the detector, as determined by NANOCELL.
Method for Simulation of 2D GISAXS Intensities these diffuse features, we generated NANODIFT simulations with various domain shapes and sizes. To model an infinite domain size, a Hann window was applied to eliminate aliasing effects. However, as a result, each peak was broadened isotropically. The simulation is shown in Figure 7b, where the incident angle is 0.20°, the critical angle of the air/film is 0.16°, and the film/substrate interface is 0.22°. Note that the peak width is broader than the experimental pattern in Figure 6. This simply indicates that the instrumental broadening is low and the experimental data has a higher resolution than the simulation. Nevertheless, the main features of the experimental pattern may be understood as shown in Figure 7 parts (c) and (d). Here, we show simulation of a rhombus shaped domain that is constructed such that the edges of the domain break along the (100) faces of the rhombohedral unit cells. Notice how the NANODIFT simulation shows diffuse scattering of lines and arcs connecting Bragg peaks that match very well with the experimental pattern. This is in contrast to other domain shapes tested, such as a cubic-shaped domain (not presented here) that produces strong diffuse scattering in the cardinal directions. The most prominent lines extend from the (111) to the (110) peak, and the even more unique arcs exist between the (100) peak and the (110) peaks. There are other higher order lines and arcs that connect other Bragg peaks in the patterns, but the intensity is faint, as would be expected. Additionally, examination of the reflected peaks reveals that the same lines and arcs, albeit much weaker in intensity. This very characteristic pattern of diffuse features is produced only when the nanostructure has many rhombus-shaped domains that are highly (111)-oriented. This has important implications for the accessibility (or lack there of) of this phase. Since the domains are rhombus-shaped and (111)-oriented, most domains will not have large contacts with both the substrate and the surface. As a result, most nanopore pathways from the surface to the substrate must cross a domain boundary. However, it is not unreasonable to hypothesize that the interdomain connectivity of these materials is low, which would result in films that have poor accessibility to the substrate. 4. Conclusions We have developed a method based on discrete Fourier transforms to simulate GISAXS patterns from nanostructured films using the DWBA. This method removes the restriction that an analytical solution to the Fourier transform must be obtained and thus allows for the easy simulation of any type of nanostructure, as long as the morphology is that of a thin film with flat interfaces. Using this code, it was determined that selfassembled R3hm nanoporous silica films contained rhombusshaped domains that terminated along (100) faces of the rhombohedral unit cell. Acknowledgment. The authors wish to acknowledge financial support from National Science Foundation under the CAREER Award (0134255-CTS) and the use of the NSF-funded facility for In-situ X-ray Scattering from Nanomaterials and Catalysts (MRI Program Award 0321118-CTS) to collect GISAXS data. Additionally, the authors would like to thank BASF for providing the triblock copolymer templates used in the examples. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC0206CH11357. The authors wish the thanks Jin Wang and Xuefa Li at APS for their help.
J. Phys. Chem. C, Vol. 111, No. 21, 2007 7653 References and Notes (1) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M. E.; Kresge, C. T.; Schmitt, K. D.; Chu, C. T. W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. A New Family of Mesoporous Molecular-Sieves Prepared with Liquid-Crystal Templates. J. Am. Chem. Soc. 1992, 114, 10834-10843. (2) Monnier, A.; Schuth, F.; Huo, Q.; Kumar, D.; Margolese, D.; Maxwell, R. S.; Stucky, G. D.; Krishnamurty, M.; Petroff, P.; Firouzi, A.; Janicke, M.; Chmelka, B. F. Cooperative Formation of Inorganic-Organic Interfaces in the Synthesis of Silicate Mesostructures. Science 1993, 261, 1299-1303. (3) Huo, Q.; Margolese, D.; Cielsa, U.; Feng, P.; Gler, T. E.; Sieger, P.; Leon, R.; Petroff, P. M.; Schuth, F.; Stucky, G. D. Generalized Synthesis of Periodic Surfactant/Inorganic Composite Materials. Nature 1994, 368, 317-321. (4) Firouzi, A.; Kumar, D.; Bull, L. M.; Besier, T.; Sieger, P.; Huo, Q.; Walker, S. A.; Zasadzinski, J. A.; Glinka, C.; Nicol, J.; Margolese, D.; Stucky, G. D.; Chmelka, B. F. Cooperative Organization of InorganicSurfactant and Bio-Mimetic Assemblies. Science 1995, 267, 1138-1143. (5) Lu, Y. F.; Ganguli, R.; Drewien, C. A.; Anderson, M. T.; Brinker, C. J.; Gong, W. L.; Guo, Y. X.; Soyez, H.; Dunn, B.; Huang, M. H.; Zink, J. I. Continuous Formation of Supported Cubic and Hexagonal Mesoporous Films by Sol Gel Dip-Coating. Nature 1997, 389, 364-368. (6) Yang, H.; Kuperman, A.; Coombs, N.; Mamiche-Afara, S.; Ozin, G. A. Synthesis of Oriented Films of Mesoporous Silica on Mica. Nature 1996, 379, 703-705. (7) Trau, M.; Yao, N.; Kim, E.; Xia, Y.; Whitesides, G. M.; Aksay, I. A. Microscopic Patterning of Oriented Mesoscopic Silica Through Guided Growth. Nature 1997, 390, 674-676. (8) Hillhouse, H. W.; Okubo, T.; van Egmond, J. W.; Tsapatsis, M. Preparation of Supported Mesoporous Silica Layers in a Continuous Flow Cell. Chem. Mater. 1997, 9, 1505-1507. (9) Ogawa, M. Formation of Novel Oriented Transparent Films of Layered Silica-Surfactant Nanocomposites. J. Am. Chem. Soc. 1994, 116, 7941-7942. (10) Ogawa, M. A Simple Set-Gel Route for the Preparation of SilicaSurfactant Mesostructured Materials. Chem. Commun. 1996, 1149-1150. (11) Schuth, F.; Schmidt, W. Microporous and Mesoporous Materials. AdV. Mater. 2002, 14, 629-638. (12) Fan, H. Y.; Bentley, H. R.; Kathan, K. R.; Clem, P.; Lu, Y. F.; Brinker, C. J. Self-Assembled Aerogel-like Low Dielectric Constant Films. J. Non-Cryst. Solids 2001, 285, 79-83. (13) Schmidt, M.; Boettger, G.; Eich, M.; Morgenroth, W.; Huebner, U.; Boucher, R.; Meyer, H. G.; Konjhodzic, D.; Bretinger, H.; Marlow, F. Ultralow Refractive Index Substrates: A Base for Photonic Crystal Slab Waveguides. Appl. Phys. Lett. 2004, 85, 16-18. (14) Konjhodzic, D.; Bretinger, H.; Wilczok, U.; Dreier, A.; Ladenburger, A.; Schmidt, M.; Eich, M.; Marlow, F. Low-n Mesoporous Silica Films: Structure and Properties. Appl. Phys. A 2005, 81, 425-432. (15) Falcaro, P.; Grosso, D.; Amenitsch, H.; Innocenzi, P. Silica Orthorhombic Mesostructured Films with Low Refractive Index and High Thermal Stability. J. Phys. Chem. B 2004, 108, 10942-10948. (16) Urade, V. N.; Hillhouse, H. W. Synthesis of Thermally Stable Highly Ordered Nanoporous Tin Oxide Thin Films with a 3D Face-Centered Orthorhombic Nanostructure. J. Phys. Chem. B 2005, 109, 10538-10541. (17) Luo, H. M.; Wang, D. H.; He, J. B.; Lu, Y. F. Magnetic Cobalt Nanowire Thin Films. J. Phys. Chem. B 2005, 109, 1919-1922. (18) Liu, N. G.; Dunphy, D. R.; Atanassov, P.; Bunge, S. D.; Chen, Z.; Lopez, G. P.; Boyle, T. J.; Brinker, C. J. Photoregulation of Mass Transport through a Photoresponsive Azobenzene-Modified Nanoporous Membrane. Nano Lett. 2004, 4, 551-554. (19) Hou, K.; Tian, B. Z.; Li, F. Y.; Bian, Z. Q.; Zhao, D. Y.; Huang, C. H. Highly Crystallized Mesoporous TiO2 films and their Applications in Dye Sensitized Solar Cells. J. Mater. Chem. 2005, 15, 2414-2420. (20) Hillhouse, H. W.; Tuominen, M. T. Modeling the Thermoelectric Transport Properties of Nanowires Embedded in Oriented Microporous and Mesoporous Films. Microporous Mesoporous Mater. 2001, 47, 3950. (21) Tanaka, S.; Tate, M. P.; Nishiyama, N.; Ueyama, K.; Hillhouse, H. W. Structure of Mesoporous Silica Thin Films Prepared by Contacting PEO106-PPO70-PEO106 Films with Vaporized TEOS. Chem. Mater. 2006, 18, 5461-5466. (22) Grosso, D.; Balkenende, A. R.; Albouy, P. A.; Lavergne, M.; Mazerolles, L.; Babonneau, F. Highly Oriented 3D-Hexagonal Silica Thin Films Produced with Cetyltrimethylammonium Bromide. J. Mater. Chem. 2000, 10, 2085-2089. (23) Klotz, M.; Albouy, P. A.; Ayral, A.; Menager, C.; Grosso, D.; Van der Lee, A.; Cabuil, V.; Babonneau, F.; Guizard, C. The True Structure of Hexagonal Mesophase-Templated Silica Films as Revealed by X-ray Scattering: Effects of Thermal Treatments and of Nanoparticle Seeding. Chem. Mater. 2000, 12, 1721-1728.
7654 J. Phys. Chem. C, Vol. 111, No. 21, 2007 (24) Alberius, P. C. A.; Frindell, K. L.; Hayward, R. C.; Kramer, E. J.; Stucky, G. D.; Chmelka, B. F. General Predictive Syntheses of Cubic, Hexagonal, and Lamellar Silica and Titania Mesostructured Thin Films. Chem. Mater. 2002, 14, 3284-3294. (25) Hayward, R. C.; Alberius, P. C. A.; Kramer, E. J.; Chmelka, B. F. Thin Films of Bicontinuous Cubic Mesostructured Silica Templated by a Nonionic Surfactant. Langmuir 2004, 20, 5998-6004. (26) Crepaldi, E. L.; Soler-Illia, G.; Grosso, D.; Cagnol, F.; Ribot, F.; Sanchez, C. Controlled Formation of Highly Organized Mesoporous Titania Thin Films: From Mesostructured Hybrids to Mesoporous Nanoanatase TiO2. J. Am. Chem. Soc. 2003, 125, 9770-9786. (27) Eggiman, B. W.; Tate, M. P.; Hillhouse, H. W. The Rhombohedral Structure of Highly Ordered and Oriented Self-Assembled Nanoporous Silica Thin Films. Chem. Mater. 2006, 18, 723-730. (28) Besson, S.; Gacoin, T.; Jacquiod, C.; Ricolleau, C.; Babonneau, D.; Boilot, J. P. Structural Study of 3D-Hexagonal Mesoporous Spin-Coated Sol-Gel Films. J. Mater. Chem. 2000, 10, 1331-1336. (29) Besson, S.; Gacoin, T.; Ricolleau, C.; Jacquiod, C.; Boilot, J. P. Phase Diagram for Mesoporous CTAB-Silica Films Prepared under Dynamic Conditions. J. Mater. Chem. 2003, 13, 404-409. (30) Doshi, D. A.; Gibaud, A.; Goletto, V.; Lu, M. C.; Gerung, H.; Ocko, B.; Han, S. M.; Brinker, C. J. Peering into the Self-Assembly of Surfactant Templated Thin-Film Silica Mesophases. J. Am. Chem. Soc. 2003, 125, 11646-11655. (31) Doshi, D. A.; Gibaud, A.; Liu, N. G.; Sturmayr, D.; Malanoski, A. P.; Dunphy, D. R.; Chen, H. J.; Narayanan, S.; MacPhee, A.; Wang, J.; Reed, S. T.; Hurd, A. J.; van Swol, F.; Brinker, C. J. In-situ X-ray Scattering Study of Continuous Silica-surfactant Self-Assembly during Steady-State Dip Coating. J. Phys. Chem. B 2003, 107, 7683-7688. (32) Gibaud, A.; Doshi, D.; Ocko, B.; Goletto, V.; Brinker, C. J. TimeResolved In Situ Grazing Incidence Small Angle X-ray Scattering Experiment of Evaporation Induced Self-Assembly. In Nanotechnology in Mesostructured Materials; Elsevier Science BV: Amsterdam, The Netherlands, 2003, Vol. 146, pp 351-354. (33) Gibaud, A.; Dourdain, S.; Gang, O.; Ocko, B. M. In Situ Grazing Incidence Small-Angle X-ray Scattering Real-Time Monitoring of the Role of Humidity during the Structural Formation of Templated Silica Thin Films. Phys. ReV. B 2004, 70, 161403-1-161403-4. (34) Gibaud, A.; Grosso, D.; Smarsly, B.; Baptiste, A.; Bardeau, J. F.; Babonneau, F.; Doshi, D. A.; Chen, Z.; Brinker, C. J.; Sanchez, C. Evaporation-Controlled Self-Assembly of Silica Surfactant Mesophases. J. Phys. Chem. B 2003, 107, 6114-6118. (35) Grosso, D.; Babonneau, F.; Albouy, P. A.; Amenitsch, H.; Balkenende, A. R.; Brunet-Bruneau, A.; Rivory, J. An In Situ Study of Mesostructured CTAB-Silica Film Formation during Dip Coating using Time-Resolved SAXS and Interferometry Measurements. Chem. Mater. 2002, 14, 931-939. (36) Grosso, D.; Babonneau, F.; Soler-Illia, G.; Albouy, P. A.; Amenitsch, H. Phase Transformation during Cubic Mesostructured Silica Film Formation. Chem. Commun. 2002, 748-749. (37) Grosso, D.; Balkenende, A. R.; Albouy, P. A.; Ayral, A.; Amenitsch, H.; Babonneau, F. Two-Dimensional Hexagonal Mesoporous Silica Thin Films Prepared from Block Copolymers: Detailed Characterization and Formation Mechanism. Chem. Mater. 2001, 13, 1848-1856. (38) Falcaro, P.; Costacurta, S.; Mattei, G.; Amenitsch, H.; Marcelli, A.; Guidi, M. C.; Piccinini, M.; Nucara, A.; Malfatti, L.; Kidchob, T.; Innocenzi, P. Highly Ordered “Defect-Free” Self-Assembled Hybrid Films with a Tetragonal Mesostructure. J. Am. Chem. Soc. 2005, 127, 38383846. (39) Soler-Illia, G.; Crepaldi, E. L.; Grosso, D.; Durand, D.; Sanchez, C. Structural Control in Self-Standing Mesostructured Silica Oriented Membranes and Xerogels. Chem. Commun. 2002, 2298-2299. (40) Tate, M. P.; Eggiman, B. W.; Kowalski, J. D.; Hillhouse, H. W. Order and Orientation Control of Mesoporous Silica Films on Conducting Gold Substrates Formed by Dip-Coating and Self-Assembly: A Grazing Angle of Incidence Small-Angle X-ray Scattering and Field Emission Scanning Electron Microscopy Study. Langmuir 2005, 21, 10112-10118. (41) Tate, M. P.; Urade, V. N.; Kowalski, J. D.; Wei, T. C.; Hamilton, B. D.; Eggiman, B. W.; Hillhouse, H. W. Simulation and Interpretation of
Tate and Hillhouse 2D Diffraction Patterns from Self-Assembled Nanostructured Films at Arbitrary Angles of Incidence: From Grazing Incidence (above the Critical Angle) to Transmission Perpendicular to the Substrate. J. Phys. Chem. B 2006, 110, 9882-9892. (42) Lee, B.; Park, I.; Yoon, J.; Park, S.; Kim, J.; Kim, K. W.; Chang, T.; Ree, M. Structural Analysis of Block Copolymer Thin Films with Grazing Incidence Small-Angle X-ray Scattering. Macromolecules 2005, 38, 4311-4323. (43) Lee, B.; Yoon, J.; Oh, W.; Hwang, Y.; Heo, K.; Jin, K. S.; Kim, J.; Kim, K. W.; Ree, M. In-Situ Grazing Incidence Small-Angle X-ray Scattering Studies on Nanopore Evolution in Low-k Organosilicate Dielectric Thin Films. Macromolecules 2005, 38, 3395-3405. (44) Grosso, D.; Soler-Illia, G.; Babonneau, F.; Sanchez, C.; Albouy, P. A.; Brunet-Bruneau, A.; Balkenende, A. R. Highly Organized Mesoporous Titania Thin Films Showing Mono-Oriented 2D Hexagonal Channels. AdV. Mater. 2001, 13, 1085-1090. (45) Smarsly, B.; Gibaud, A.; Ruland, W.; Sturmayr, D.; Brinker, C. Quantitative SAXS Analysis of Oriented 2D Hexagonal Cylindrical Silica Mesostructures in Thin Films Obtained from Nonionic Surfactants. Langmuir 2005, 21, 3858-3866. (46) Cagnol, F.; Grosso, D.; Soler-Illia, G.; Crepaldi, E. L.; Babonneau, F.; Amenitsch, H.; Sanchez, C. Humidity-Controlled Mesostructuration in CTAB-Templated Silica Thin Film Processing. The Existence of a Modulable Steady State. J. Mater. Chem. 2003, 13, 61-66. (47) Schmidbauer, M. X-ray Diffuse Scattering from Self-Organized Mesoscopic Semiconductor Structures; Springer: New York, 2004, Vol. 199. (48) Parratt, L. G. Surface Studies of Solids by Total Reflection of X-rays. Phys. ReV. 1954, 95, 359-369. (49) Vineyard, G. Grazing-Incidence Diffraction and the Distorted-Wave Approximation for the Study of Surfaces. Phys. ReV. B 1982, 26, 41464159. (50) Wagner, H.; Dietrich, S. Critical Surface Scattering of X-rays and Neutrons at Grazing Angles. Phys. ReV. Lett. 1983, 51, 1469-1471. (51) Sinha, S. K.; Sirota, E. B.; Garoff, S.; Stanley, H. B. X-ray and Neutron-Scattering from Rough Surfaces. Phys. ReV. B 1988, 38, 22972311. (52) Holy, V.; Kubena, J.; Ohlidal, I.; Lischka, K.; Plotz, W. X-ray Reflection from Rough Layered Systems. Phys. ReV. B 1993, 47, 1589615903. (53) Holy, V.; Baumbach, T. Nonspecular X-ray Reflection from Rough Multilayers. Phys. ReV. B 1994, 49, 10668-10676. (54) Rauscher, M.; Salditt, T.; Spohn, H. Small-Angle X-ray Scattering under Grazing Incidence: The Cross Section in the Distorted-Wave Born Approximation. Phys. ReV. B 1995, 52, 16855-16863. (55) Lazzari, R. IsGISAXS: A Program for Grazing-Incidence SmallAngle X-ray Scattering Analysis of Supported Islands. J. Appl. Crystallogr. 2002, 35, 406-421. (56) Wohlgemuth, M.; Yufa, N.; Hoffman, J.; Thomas, E. Triply Periodic Bicontinuous Cubic Microdomain Morphologies by Symmetries. Macromolecules 2001, 34, 6083-6089. (57) Andersson, S.; Jacob, M. The Nature of Mathematics and the Mathematics of Nature, 1st ed.; Elsevier: New York, 1998. (58) Weisstein, E. W. CRC Concise Encyclopedia of Mathematics, 2nd ed.; Chapman & Hall/CRC: New York, 2002. (59) Anderson, M. W.; Ohsuna, T.; Sakamoto, Y.; Liu, Z.; Carlsson, A.; Terasaki, O. Modern Microscopy Methods for the Structural Study of Porous Materials. Chem. Commun. 2004, 907-916. (60) Urade, V. N.; Wei, T.-C.; Tate, M. P.; Kowalski, J. D.; Hillhouse, H. W. Nanofabrication of Double Gyroid Thin Films. Chem. Mater. 2007, 19, 768-777. (61) Cavicchi, K. A.; Berthiaume, K. J.; Russell, T. P. Solvent Annealing Thin Films of Poly(isoprene-b-lactide). Polymer 2005, 46, 11635-11639. (62) Koganti, V. R.; Dunphy, D.; Gowrishankar, V.; McGehee, M. D.; Li, X.; Wang, J.; Rankin, S. E. Generalized Coating Route to Silica and Titania Films with Orthogonally Tilted Cylindrical Nanopore Arrays. Nano Lett. 2006, 6, 2567-2570.
