Simulation and Interpretation of 2D Diffraction Patterns from Self

9882

J. Phys. Chem. B 2006, 110, 9882-9892

Simulation and Interpretation of 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 Michael P. Tate, Vikrant N. Urade, Jonathon D. Kowalski, Ta-chen Wei, Benjamin D. Hamilton, Brian W. Eggiman, and Hugh W. Hillhouse* School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907 ReceiVed: NoVember 15, 2005; In Final Form: February 21, 2006

A method to calculate the location of all Bragg diffraction peaks from nanostructured thin films for arbitrary angles of incidence from just above the critical angle to transmission perpendicular to the film is reported. At grazing angles, the positions are calculated using the distorted wave Born approximation (DWBA), whereas for larger angles where the diffracted beams are transmitted though the substrate, the Born approximation (BA) is used. This method has been incorporated into simulation code (called NANOCELL) and may be used to overlay simulated spot patterns directly onto two-dimensional (2D) grazing angle of incidence smallangle X-ray scattering (GISAXS) patterns and 2D SAXS patterns. The GISAXS simulations are limited to the case where the angle of incidence is greater than the critical angle (Ri > Rc) and the diffraction occurs above the critical angle (Rf > Rc). For cases of surfactant self-assembled films, the limitations are not restrictive because, typically, the critical angle is around 0.2° but the largest d spacings occur around 0.8° 2θ. For these materials, one finds that the DWBA predicts that the spot positions from the transmitted main beam deviate only slightly from the BA and only for diffraction peaks close the critical angle. Additional diffraction peaks from the reflected main beam are observed in GISAXS geometry but are much less intense. Using these simulations, 2D spot patterns may be used to identify space group, identify the orientation, and quantitatively fit the lattice constants for SAXS data from any angle of incidence. Characteristic patterns for 2D GISAXS and 2D low-angle transmission SAXS patterns are generated for the most common thin film structures, and as a result, GISAXS and SAXS patterns that were previously difficult to interpret are now relatively straightforward. The simulation code (NANOCELL) is written in Mathematica and is available from the author upon request.

1. Introduction Surfactant or block copolymer self-assembly may be used to generate a wide variety of highly ordered periodic structures. The symmetry of the periodic structure depends sensitively on the interfacial curvature, which can be chosen during synthesis by the choice of surfactant, cosurfactant, swelling agent, or the size of any coassembled oligomers. Here, we will focus on materials synthesized by cooperative self-assembly of surfactants and inorganic species.1-3 After self-assembly of the nanostructure and condensation of the inorganic species, removal of the surfactant yields a highly ordered nanoporous structure. Using the technique of evaporation-induced self-assembly (EISA),4-7 dip-coating and spin-coating have been used to fabricate many highly ordered and oriented nanostructured films including: (1) the two-dimensional (2D) hexagonal phase with hexagonal plane group p6mm (which distorts to a rectangular plane group c2mm upon drying),4,8,9 (2) a three-dimensional (3D) hexagonal phase with space group P63/mmc,10-12 (3) a body-centered cubic phase with space group Im3hm (which distorts to an orthorhombic Fmmm space group upon drying),13,14 (4) a primitive cubic phase with space group Pm3hn,10,15 (5) a body-centered tetragonal phase with space group I4/mmm,16 (6) a gyroid-based cubic phase with space group Ia3hd that distorts upon drying,17 and * To whom correspondence [email protected].

should

be

addressed.

Email:

most recently (7) a rhombohedral phase with space group R3hm.18 For a recent review of the EISA and explanations of the structures produced, see Grosso et al.4-7 The most common tools used to identify structure are “powder” X-ray diffraction (PXRD) and transmission electron microscopy (TEM). However, typically there are very few resolvable peaks in the PXRD data from nanostructured thin films due to the one-dimensional (1D) scan of reciprocal space in traditional 1D diffractometers. As a result, typical diffraction pattern indexing programs such as DICVOL,19-21 ITO,22 or TREOR23 are of little use in determining the crystal class and lattice constants of films (or even nanostructured powders, which also tend to have few resolvable peaks). To overcome this challenge, phase identification has relied heavily on the aid of TEM and electron diffraction,24 but these techniques provide only microscopic structural information and cannot be conducted in situ. In contrast, X-ray scattering methods provide structural information averaged over macroscopic regions, are nondestructive, and are capable of being applied in situ during synthesis, post-treatments, or application. To aid in the analysis of X-ray scattering patterns from these self-assembled nanomaterials, several special programs have been developed. Using a traditional crystallographic approach, Solovyov25 developed a special code to perform Rietveld refinement on powder X-ray diffraction data, which has been used to answer questions about the electron density distribution

10.1021/jp0566008 CCC: $33.50 © 2006 American Chemical Society Published on Web 04/28/2006

Self-Assembled Nanostructured Films in mesoporous powder samples. More recently, Fo¨rster et al.26 developed a general formalism using the decoupling approximation in which form factors and structure factors are calculated for many different shapes and lattices. The model includes effects of disorder and is very useful for understanding systematic absences in nanomaterials, particularly in identifying cases where diffraction peaks are strongly suppressed by minima in the form factor. However, neither code handles oriented structures such as those that are observed in self-assembled nanostructured films. Typically, films are highly oriented in the plane of the substrate but have many domains that sample different orientations about an axis perpendicular to the substrate (this type of disorder is referred to as “planar disorder” throughout the manuscript). As a result, the reciprocal space consists of rings oriented parallel to the substrate as opposed to spots.27 This structure of reciprocal space was recently imaged directly by collecting small-angle X-ray scattering (SAXS) patterns over the full range of incident angles from grazing to perpendicular transmission.18 Due to this geometry of reciprocal space, 2D scattering patterns collected at small angles of incidence to the film surface (but larger than the critical angle for total external reflection from the film) allow the Ewald sphere to slice though the rings. Thus, this single experiment then can reveal most of the features of reciprocal space. This technique is called 2D grazing angle of incidence small-angle X-ray scattering (GISAXS). At grazing angles, the effective sample volume for scattering becomes large, even for films that are only a few hundred nanometers thick. This results in strong scattering patterns from a small amount of material. However, the analysis of these patterns can be complicated by dynamical scattering effects (primarily reflection and refraction at the air-film and film-substrate interfaces). As a result of these effects, the intensity observed on the detector is not exactly a slice of reciprocal space. The diffuse scattered intensity and diffraction may be altered. Peak positions may shift, and additional peaks may appear on the detector. Analysis of these effects are typically handled within the framework of the distorted wave Born approximation (DWBA) as developed for GISAXS.28-33 Recently, Lazzari developed a simulation code that is available (called IsGISAXS) that uses the DWBA to simulate diffuse scattering GISAXS patterns.34 However, this code is restricted to islands distributed on a substrate. More recently, Lee et al.35 developed the theory to use DWBA for nanostructured films with perfect interfaces. However, GISAXS is not without its limitations as scattering at angles below the critical angle are strongly attenuated and regions of reciprocal space that correspond to diffraction planes that are perpendicular to the substrate are obscured. Also, the distribution of orientations in the plane cannot be obtained from a single GISAXS pattern. As a result, the structure may not be apparent from GISAXS data alone, and data may need to be collected at other angles. If data is collected at a spectrum of angles of increasing incidence, all of the reciprocal space is probed, as opposed to a single slice. Depending on the thickness of the substrate, this may or may not be possible. In principle, the space group, orientation, and lattice constants can be determined from these data. However, currently there are no software tools available to simulate SAXS patterns from oriented nanostructured films at arbitrary angles of incidence. Here, we report the development of theory and code (NANOCELL) to model diffraction spot patterns from ordered and oriented periodic nanostructured films for any angle of incidence above the critical angle of the film. At grazing angles, we use

J. Phys. Chem. B, Vol. 110, No. 20, 2006 9883 the DWBA approximation, simplified to calculate diffraction peak positions only. At higher angles of incidence, the diffraction peak positions are simulated within the normal Born approximation (BA). NANOCELL is then used to simulate and interpret the 2D scattering patterns typically observed from selfassembled nanostructured films. NANOCELL, written in Mathematica, is available from the authors for free for academic use. 2. Methods Employed 2.1. Calculation of Bragg Peak Locations. For our purposes, we are interested in determining the symmetry, orientation, and lattice constants of a nanostructured film, and thus, we are primarily concerned with only diffraction peak positions. The starting point for analysis of most X-ray scattering data is to apply the first BA, which assumes single scattering from the electron density distribution of the sample and assumes that the direction and amplitude of the main incoming and outgoing waves are equal to those in a vacuum. However, there can be significant changes in direction and intensity of these waves when the incident beam impinges on an interface at grazing angles due to dynamical effects (multiple scattering, reflection, refraction, etc.). These effects become increasingly significant as the grazing angle approaches the critical angle for the film material and the BA breaks down. Also, because we are working with nanostructured materials where the angle of diffraction is also very small, the effects of refraction and reflection at the interfaces must also be considered for the diffracted waves. As a result of these effects, the 2D pattern that appears on the detector is not exactly a slice of reciprocal space, and thus, the simulations of the 2D patterns are given in terms of exit angles, rather than scattering vector components. These reflection and refraction effects are most often handled by a first-order perturbation theory called the DWBA. The essence of this theory is that scattering at the film interfaces is treated dynamically, whereas the scattering from the internal structure of the film is treated kinematically (single scattering). Specifically, the scattering potential (V) is expressed as a superposition:

V ) V1 + V2

(1)

where V1 is a potential that yields an exact solution of the stationary wave equation and V2 is a potential that is a small perturbation. For X-ray scattering, the scattering potential is proportional to the electron density contrast, and thus, the DWBA assumes that one can express the total contrast as a superposition of two separate contrast distributions where one yields an exact solution to the stationary wave equation and the other results in much less-intense scattering. For the case of flat nanostructured film on a thick substrate, V1 is chosen to represent a homogeneous film on a thick substrate where the interfaces are flat and sharp. The stationary wave equation V1 then is:

(∇2 + k02 + V1)ψ ) 0

(2)

where k0 is the magnitude of the wave vector and ψ is the wave field. For the definition of V1 given above, Parratt formalism36 may be used to express the exact solution in terms of the Fresnel coefficients:

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ψ1i ) ei(kx1irx + ky1iry) (eikz1irz + R1ie-ikz1irz) for z > 0

(3a)

ψ2i ) ei(kx2irx + ky2iry)(T2ieikz2irz + R2ie-ikz2irz) for 0 > z > -d (3b) ψ3i ) ei(kx3irx + ky3iry)(T3ie-ikz3irz)

for z < -d

(3c)

where d, Ri, Ti, and ki are the film thickness, Fresnel reflectivity coefficient, transmission coefficient, and wave vectors, respectively, for each layer, i, within the sample. Equation 3 describes the refraction and reflection corrected wave fields in each segment (vacuum, inside the film, and inside the substrate). Also, the time-reversed wave is also a solution. For the region in the film, it is given by:

ψ2f ) ei(kx2frx + ky2fry)(T 2f* eikz2frz + R 2f* e-ikz2f ) *

* rz

(4)

where kz2f is the scattered wave vector inside the film and T2f and R2f are the transmission and reflection coefficients of the scattered wave, respectively. The asterisk * indicates the complex conjugate. The subscripts “i” and “f” denote terms for the incident and scattered waves throughout the manuscript, respectively. Now, with this exact solution for V1, we use firstorder perturbation theory to calculate the scattering from V2:

eik0r / ψ2f V2ψ2id3r 4πr0 where V2 for may be written as:37,38 V2 ) 4πr0F(r) ψs )

∫

(5)

(6)

where F(r) is the deviation in electron density distribution that describes the nanostructure. Now, the scattered intensity from the nanostructure can be calculated by:

I(q) ) r02|ψs|2

(7)

Substituting eqs 3b, 4, and 6 into eq 5, and then the result into eq 7, we arrive at a 4-channel DWBA expression where the intensity for scattering from the nanostructure is:

I(q) ) r20 |

∫∫e-i((k

x2f

- kx2i) + (ky2f - ky2i)ry)

∫ T2iR2f∫F(r)e-i(-k - k )r drz + R2iT2f ∫F(r)e-i(k + k )r drz + R2iR2f ∫F(r)e-i(-k + k )r drz) drxdry|2 (T2iT2f F(r)e-i(kz2f - kz2i)rz drz + z2f

z2i z

z2f

z2i z

z2f

z2i z

(8)

(see Figure 1). As expected, calculation of this intensity requires knowing the electron density (which is typically not known). However, for the determination of space group, film orientation, and lattice constants, the detailed electron density distribution is not required. For this, we are only interested in the directions of the beams. Thus, we simplify eq 8, discarding information about peak intensity or diffuse scattering while retaining the positions of the Bragg peaks. First, we neglect the cross terms after taking the absolute square. This gives the intensity as the sum of the squares of each term as opposed to the square of the sum. This simplification has been discussed previously.31,34,35 However, here (because we are discarding the intensity information), the assumption is much less restrictive and is expected to hold in general. The second simplification is to neglect the transmission and reflection coefficients. The result is four terms that each appear similar to the BA except with directions for the incident and exit wave vectors modified by reflection and refraction. Each term is shown schematically in Figure 2. Note

Figure 1. Scattering geometry for GISAXS: (a) Overview of refraction and reflection events; (b) sketch showing the exit angles and Cartesian coordinate frame of reference used (the substrate frame); (c) sketch showing the orientation of the substrate relative to the detector (shown here with substrate oriented vertically, but data is also collected with the substrate horizontal depending on the instrument used).

that for terms 1 and 2, the only difference is the sign of kz2f (the diffracted beam points toward the substrate instead of the film surface). Likewise, this is also the only difference between terms 3 and 4. Now, the direction of each wave inside the film, both incident and diffracted, may be calculated using the standard laws of refraction and reflection.28 When the wave vectors are written according to Figure 1b, the law of refraction is written as:

n1 cos(R1) ) n2 cos(R2)

(9)

where ni is the refractive index for each medium and Ri is the incident angle for the medium. The critical angle may then be expressed (for cases where n1 > n2) as cos(Rc) ) n2/n1. (Note: that Rc is the critical angle for the vacuum-film interface and n1 ) 1 for vacuum and air). When rewritten in terms of the z-component of the wave vector, the law of refraction may be expressed as:

kz2 ) xkz12 - kzc2

(10)

where kz2 is the z-component of the transmitted wave in the film, kz1 is the z-component of the incident wave in a vacuum, and kzc is the z-component of the incident wave in a vacuum when the incident angle is equal to the critical angle. Again,

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( ( ( (

)) ))

kz2f ) k0 sin cos-1

n1 cos(Rf) n2

kz2i ) k0 sin cos-1

n1 cos(Ri) n2

(13)

Equations 13 are substituted into eq 11a, and then by using cos(Rc) ) n2 (for n1 ) 1), qz may be written as:

( ( )) cos(Rf)

qz ) k0 sin cos-1

cos(Rc)

( ( ))

- k0 sin cos-1

cos(Ri)

cos(Rc)

(14)

Then, upon rearrangement, simplification, and substituting q ) 2πs, we have:

x ( x

cos Rf ) cos Rc

1 - λsz +

1-

cos2(Ri)

)

2

(15a)

cos2(Rc)

Using the procedure defined above for term 1, equations for Rf from each additional term in eq 8 may be determined. For term 2, this becomes:

cos Rf ) cos Rc For term 3:

this holds true for both the incident and scattered wave vectors. Similarly, the law of reflection is

(10)

or when written in terms of the wave vector, kzr ) - kzwhere kzr is the z-component of the reflected wave and kz is the incident wave. This law holds true for both the vacuum-film and filmsubstrate interfaces. Now, the exit angles for each term in eq 8 may be solved by examining the definition of each scattering vector (q) within the thin film. These terms may be written as qj ≡ k2f - k2i, where the magnitude of k is defined as k0 ) 2π/λ. Note, here qj represents the jth term of eq 8, not the layer index (as is used for k, T, and R). Now, the z-component of qj is then written as

q1z ≡ kz2f - kz2i

(11a)

q2z ≡ -kz2f - kz2i

(11b)

q3z ≡ kz2f + kz2i

(11c)

q4z ≡ -kz2f + kz2i

(11d)

By definition, the values of kz2f and kz2i are:

kz2f ) k0 sin(R2f) kz2i ) k0 sin(R2i) Then, using eq 9 for R2f and R2i, these become:

1 - -λsz -

(12)

For term 4: cos Rf ) cos Rc

1-

x ( x x ( x

cos Rf ) cos Rc

Figure 2. Schematic representation of each term in eq 8.

Rr ) -Ri

x ( x 1 - λsz -

1 - -λsz +

1-

1-

cos2(Ri) cos2(Rc)

)

2

) )

cos2(Ri)

2

cos2(Rc) cos2(Ri) cos2(Rc)

(15b)

(15c)

2

(15d)

These solutions define the exit angles (Rf) relative to the plane of the substrate for diffracted beams originating in the films, as shown in each term of Figure 2 and eq 8. Then, using a similar calculation, 2θf is defined by:

tan(2θf) )

λsx λsy - cos(Ri)

(16)

The much simpler form of eq 16 compared with eq 15 results from the fact that there is no change in the x and y components of the scattering vector upon refraction and reflection with a flat ideal interface. A similar approach was used by Lee35 to develop equations for Rf and 2θf that apply to the case where films have planar disorder. However, here eqs 15 and 16 are expressed in a more general form that may be used for films with any type of disorder (single domain, finite number of multiple domains, planar disorder, etc.). We see that in general for any sz, eq 15b reduces to 15a and eq 15d reduces to 15c. Thus, we see that for the calculation of Rf, terms 1 and 2 are identical and terms 3 and 4 are identical. Thus, the diffraction can be understood as originating from two terms, one from the transmitted and one from the reflected. However, care must be taken for the single-crystal case because the calculation of Rf and 2θf for the reflected beam can yield positions that (when combined with the transmitted beam diffraction positions)

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Figure 3. Sketch showing real space and reciprocal space for: (a) the substrate in transmission geometry (showing a case where the rings do not intersect the Ewald sphere); and (b) the substrate rotated almost 90° to a grazing angle of incidence geometry where the reciprocal space rings intersect the Ewald sphere, each producing two diffracted beams.

resemble the scattering pattern for the same unit cell and orientation but with planar disorder. For experiments at higher angles of incidence (using thin substrates) where the diffracted beams are transmitted through the thickness of the substrate, refraction effects become negligible and the BA with the unmodified direction of the incident beam becomes a good approximation for the diffraction pattern. Note that at an angle of incidence of 2°, refraction at the air-film interface will cause a deviation in the incident beam direction by less than 0.01° for most materials. For these cases, the angles given by the BA are:

sin(Rf) ) λsz + sin(Ri) tan(2θf) )

λsx λsy - cos(Ri)

(17) (18)

Thus, the position on the detector where intensity will be observed from scattering resulting from a point in reciprocal space with given sx, sy, sz and incident angle may be calculate with eqs 15-18. 2.2. Determination of the Scattering Vectors that Yield Diffraction at Any Given Angle of Incidence. Only reciprocal

lattice points that lie on the Ewald sphere will yield diffraction. For any given structure and orientation of the substrate, the reciprocal lattice points or rings that satisfy this criterion will be different (see Figure 3). In this section, we develop the equations to determine which points and rings yield diffraction for an arbitrary orientation of the substrate in the laboratory frame. For each set of sx, sy, sz below, the equations in section 2.1 may then be used to determine where the intensity will appear on the detector. Note that the position of the reciprocal lattice relative to the Ewald sphere changes with reflection and refraction. As discussed above, we take the direction of the beam in the film to determine the correct orientation of the reciprocal lattice. We define a Cartesian coordinate laboratory reference frame where the +z direction points toward the source and the +y is in the upward vertical direction (shown in Figure 3). Note that this Cartesian coordinate system (the lab frame) is different from the “substrate frame of reference” that is used in section 2.1 and shown in Figure 1b. To transform between the coordinate systems, a rotation matrix is applied. First a, b, and c (the real space lattice vectors) are defined in the lab frame coordinate system for a given set of constants. Next, the vectors are rotated

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within the lab frame to describe a given film orientation relative to the substrate. Then, these vectors are rotated to describe any rotation of the substrate in the lab frame. Once the real space vectors are set (as described above), the reciprocal space vectors are calculated in the lab frame. Because the Ewald sphere is defined in the lab frame, it is now easy to check which points of the reciprocal lattice lie on the surface of the Ewald sphere (within ∆s) using the following inequality:

(

) (

) (

) ()

kx2i 2 ky2i 2 kz2i 2 12 sx + sy + sz e ∆s (19) 2π 2π 2π λ

where ∆s accounts for the finite size of the Bragg peak in reciprocal space and any mosaicity of the domains. The components of s are calculated for each hkl by:

s ) ha* + kb* + lc*

(20)

where a*, b*, and c* are the reciprocal space lattice vectors and h, k, and l are the Miller indices of the Bragg peaks. For films that are single crystalline or have only a few domain orientations, each reciprocal space point for each orientation is handled separately. This scattering vector (s), which is defined in the lab frame, is rotated to the substrate coordinate frame and used to calculate the exit angles according to eqs 15-18. Now, for films with planar disorder, reciprocal space consists of rings. Here we calculate the intersection of the ring with the Ewald sphere by solving a set if three simultaneous equations. First, we define a vector (s′) in the substrate frame for each ring. s′z is the distance from center of the ring to the origin of reciprocal space. The radial component of the ring is given by s′r ) x(s′x2 + s′y2) , where s′x and s′y are the x- and y-components of the scattering vector, as illustrated in Figure 3a. Because s′ is defined in the substrate frame, the values of s′z and s′r do not change upon rotation of the substrate. This is in contrast to sx, sy , and sz which do change with rotation of the substrate as they define the current intersection of the Ewald sphere with the reciprocal space ring. Now, as shown in Figure 3, most rings do not intersect the Ewald sphere unless the substrate is rotated by a minimum angle, ζmin, about the lab frame y-axis (where ζ ) 90° - Ri). The value of ζmin (which is different for each reciprocal space ring) is given by:

(

ζmin

) ()

xs′r2 + s′z2

π ) - sin-1 2

2(1/λ)

- tan

-1

s′r s′z

(21)

for the case where the reciprocal space ring is completely inside the Ewald sphere. This equation takes a slightly different form when ring is outside the Ewald sphere:

ζmin ) sin

-1

(

2(1/λ)

ζmin ) sin

ζmin ) sin

(

-1

) ()

xs′r2 + s′z2 -1

( ) s′r2

2(1/λ)

for s′z > 0

for s′z ) 0

) ()

- xs′r2 + s′z2 2(1/λ)

s′r s′z

- tan-1

+ tan-1

s′r s′z

(22a)

(22b)

for s′z < 0 (22c)

Once the substrate is rotated by at least ζmin, the reciprocal lattice rings will intersect the Ewald sphere at two different points. The intersection of the reciprocal lattice ring with the

Ewald sphere defines a scattering vector that satisfies the Laue equation. This scattering vector is calculated by solving the equation for the Ewald sphere (eq 19 where ∆s ) 0), the equation for the rotated reciprocal space ring (eq 23), and the equation for the plane of the ring (eq 24). All three equations are solved simultaneously for any given value of ζ > ζmin to determine sx, sy, and sz.

sx2 + sy2 + sz2 - s′r2 + s′z2 + s′z(sx sin ζ - sz cos ζ) ) 0 (23) - sx sin ζ + sz cos ζ ) s′z

(24)

Again, depending on the angle of incidence, the scattering vector is substituted into either eqs 15 and 16 or eqs 17 and 18 to solve for the exit angles. These calculations are repeated for every (hkl) allowed by the systematic absences to get the 2D diffraction pattern. Further, rings that do not intersect the Ewald sphere directly are checked to determine if they are within ∆s of the Ewald sphere. If so, they are included on the detector. 2.3. Range of Incident Angles Over Which Calculations Are Accurate. Equations 15 and 16 accurately describes the exit angles for GISAXS geometries over a range of incident angles, but it is not without its restrictions. These limitations arise from the use of the law of refraction to predict the z-component of the wave vector inside the film. The law of refraction accurately predicts the regime of total external reflection for a wave that strikes a medium with lesser refractive index (and, as a result, predicts the presence of a critical angle, Rc). Consequently, if Ri e Rc, then no beam penetrates the film and all predicted angles are imaginary. Thus, eq 15 is accurate only when Ri > Rc. This limitation is not at all restrictive. Because we are interested in detecting Bragg diffraction from planes parallel to the substrate, we must set Ri > Rc to observe these. There is also an analogous restriction on the exit angle (Rf). Recall that Rc is defined with respect to the plane of an interface and the incident angle of the radiation on that interface. Because the time-reversal solution must also be satisfied, eq 15 will only be valid for Rf > Rc. This may readily be seen by rearrangement of eq 15:

x

1-

cos2 Rf cos2 Rc

) (λsz (

x

1-

cos2 Ri cos2 Rc

(25)

For Rf < Rc, the left-hand side becomes imaginary. Thus, eq 15 is valid for cases where both Ri > Rc and Rf > Rc. However, the exit angle is different for each (hkl) and depends on the magnitude of the z-component of the scattering vector and upon whether increasing Ri, Rf > Rc is satisfied for increasingly fewer (hkl). Once the angle of incidence is greater than the diffraction angle for the main diffraction peaks, the GISAXS geometry loses its utility. However, provided the substrate is thin enough and after sufficient rotation of the substrate, the diffracted beams will be transmitted though the film-substrate interface, the substrate, and the parallel substratevacuum interface on the backside. The refraction effects here are small and become increasing negligible as Ri becomes large. Under this geometry, the BA works very well and can use used to model the diffraction peak positions up to angles of incidence perpendicular to the substrate using eqs 17 and 18. One can also observe contributions from beams that exit out the lateral edge of the substrate (interface perpendicular to the interface on the backside). However, these deviations from BA are small. It should also be noted that for the case of surfactant assembled materials, the diffraction angle for the largest d spacing is

9888 J. Phys. Chem. B, Vol. 110, No. 20, 2006

Figure 4. GISAXS patterns collected from synchrotron source (Ri ) 0.23°) for a rhombohedral nanoporous thin film (R3hm symmetry with a ) 114 Å and R ) 87° and with the [111] perpendicular to the substrate) with overlays showing simulated spot patterns for: (a) the BA, (b) DWBA including refraction/reflection effects (transmitted beam), and (c) DWBA including refraction/reflection effects (reflected beam). The substrate is oriented with the substrate normal pointing vertical.

typically in the range of 0.8-3°. Because the critical angle is typically 0.1-0.6°, eq 16 is generally valid for most peaks from surfactant-assembled materials. 3. Results and Discussion 3.1. Quantitative Fit of Diffraction Peak Positions in GISAXS. To illustrate that the corrected peak positions match experimental data, a nanostructured thin film with rhombohedral symmetry (synthesized as previously reported by Eggiman et al.18) was examined with high-intensity synchrotron radiation at the advanced photon source (APS) on beamline 8-ID-E (see Figure 4). For this beamline at the APS, the substrate is horizontal, and thus, the scattering patterns are rotated 90° counterclockwise about the incident beam direction. Thus, the substrate normal is positioned along the y-axis, rather than the x-axis. Figure 4a shows the BA overlay of a nanostructure thin film (R3hm symmetry with [111] oriented perpendicular to the substrate) with a ) 114 Å and R ) 87°. The data were collected at Ri ) 0.23°, and the same lattice constants were used for overlays in each part of Figure 4 (a-c). It can be seen visually

Tate et al. in Figure 4a that the BA simulation becomes more inaccurate as Rf nears the critical angle, Rc. Note that the critical angle is easily identified by the diffuse intensity maximum along a constant Rf, in this case Rc ) 0.17°. This maximum in intensity is predicted by the DWBA when all the amplitudes and phases of each term with their transmission and reflection coefficients are included in the calculation.34 Additionally, there are several peaks not predicted at all by the BA. These peaks, which arise from diffraction of the transmitted then reflected incident beam, may prevent identification of the space group or be mistakenly identified as belonging to multiple phases. Even if these additional spots on the detector are recognized as “artifacts”, the use of the BA to calculate the lattice constants will lead to inaccuracies. In Figure 4b, the effects of refraction of the incident and exit beams are included in the simulation (beams from terms 1 and 2 in eq 8 and Figure 2). It is observed that the predicted diffraction spots near Rc are accurately predicted. Then, by adding diffraction spots from the reflected solution (terms 3 and 4 in eq 8 and Figure 2), the additional spots (not predicted by the BA) are accurately simulated (shown in Figure 4c). Note the correct prediction of the strong spots along 2θf ) 0°. When all of these effects are included, it becomes possible to correctly identify the structure and orientation and accurately calculate the lattice constants from GISAXS data collected from nanostructure films. 3.2. Comparison of Scattering Data from Common Experimental Geometries. Consider first the differences between a typical theta-theta Bragg-Brentano 1D diffractometer and a pinhole SAXS camera with a 2D detector. In a 1D diffraction pattern (as illustrated in Figure 5a,d), the intensity along a single radial line through reciprocal space is detected. This experimental setup shows all the main features of reciprocal space only if it contains a large number of crystals oriented in a large number of directions due to the fact that, in this case, reciprocal space is composed of concentric shells. However, when there are few domains or the orientation distribution of domains is not random, this 1D technique fails to show all the features of reciprocal space and it becomes difficult to determine the structure. The single Bragg peak illustrated for the 1D detector corresponds to a single spot along the x-axis in the 2D GISAXS pattern to its right. With only a single Bragg peak to identify the structure, the task becomes impossible because a single Bragg peak may be fitted to any hypothesized structure. For the case of materials with many domains that are each oriented with respect to the substrate (termed “planar disorder”), reciprocal space consist of rings that are parallel to the substrate (as shown in Figure 2). For this case, a single GISAXS scan cuts though almost all of the rings (leaving out only rings with large |s| but low sz due to the curvature of the Ewald sphere). For example, the (-110) and (-211) peaks in Figure 5e, which are often hard to observe in GISAXS data because they are below the critical angle, are readily observed when the sample is rotated to an angle of incidence of 90° (see Figure 5f). This is the result of the fact that the whole ring now lies on or very near the Ewald sphere. Observing these peaks in transmission can be a key piece of data in determining the structure and orientation of samples. Note that Figure 5b,e does not account for the effects of refraction and reflection within the thin film or attenuation due to absorption by the substrate or extended beam stops typically used in GISAXS. As such, the simulated patterns are slices of reciprocal space rather than a simulation of the spots as they would appear on the detector.

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Figure 5. (a-c) Comparison of expected diffraction patterns collected from the same single-crystal structure in three different geometries. (d-f) Comparison for the same structure but with planar disorder. In each case, the structure is an R3hm mesostructured thin film with lattice constants a ) 112 Å, R ) 86° and orientated such that the [111] is perpendicular to substrate. For the single crystal, the c lattice vector is oriented in the lab frame yz-plane (in transmission geometry). Additionally, the labeled peaks are highlighted red for clarity. Note that in parts (b) and (e), the substrate normal is horizontal.

Data collected at intermediate angles can also be very useful in verifying the structure and answering questions about the presence or absence of multiple phases. An example of experimental data collected at several angles of incidence and the corresponding NANOCELL simulation overlays are shown in Figure 6 for a orthorhombic nanostructured SnO2 film (synthesized as reported by Urade et al.39). This symmetry is shown to result from a uniaxially compressed Im3hm with planar disorder and the (110) oriented parallel to the substrate. Upon compression, the Im3hm symmetry is broken and the resulting structure has Fmmm symmetry with the [010] perpendicular to the substrate.14 The overlay of the simulated and experimental data confirmed this structure and orientation for the SnO2 shown in Figure 6. 3.3. Characteristic Patterns from GISAXS or Low-Angle Transmission from Self-Assembled Nanostructured Films. Ordered nanostructured thin films synthesized by self-assembly typically are highly oriented but have planar disorder. As expected, the spot patterns produced from these thin films are highly dependent upon the orientation of the sample. Figure 7 shows reciprocal space simulations of an Im3hm symmetry structure with three different orientations, (100), (110), and (111). The striking differences between the patterns simulated for the three orientations is expected and can be used to identify orientation. However, the patterns observed from different space groups (in the same crystal class) but with the same orientation can be quite similar. The diffraction pattern produced by an Im3hm nanostructure is very similar to those patterns produced

by nanostructures with Fm3hm and Pm3hm when oriented similarly. Consider the reciprocal space patterns of all three nanostructures oriented such that the (111) is parallel to the substrate where each structure has the same “nearest neighbor” spacing that would result from assembly with the same surfactant. In Figure 8, these are shown for a distance of 150 Å. One notices the strong similarity in the patterns for the lowest-order peaks with only the relative distances between peaks different. However, even from these lowest-order peaks, one can identify the different Bravais lattices by the relative positions of peaks in reciprocal space. The Im3hm structure looks very similar to the Fm3hm but is just stretched out along sz. Also note the (hkl) indices are different for each Bravais lattice (see Figure 8). Other typical patterns that are observed in nanostructure films are shown in Figure 9. The patterns simulated for each of the space groups in Figures 7, 8, and 9 employ only the “general” reflection conditions as reported in the International Tables for Crystallography40 and do not include any special reflection conditions or suppressions that may further reduce the number of experimentally observed diffraction spots. However, significant peak suppression is required to explain the observed scattering patterns for the Pm3hn7 and P63/mmc12 structures. A recent, detailed investigated of the Pm3hn structure by Anderson and co-workers41 revealed that changes in the void fraction greatly changed the intensity of the {110} and the {200} peaks. The suppressed peaks are identified as hollow spots in Figure 9e,f for the Pm3hn and P63/mmc structures, respectively.

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Figure 6. SAXS patterns of Fmmm mesoporous SnO2 films collected at angles of incidence from perpendicular to low angle. The circles are the overlays of the simulated pattern.

Figure 7. Reciprocal space calculated by NANOCELL for different orientations of same space group Im3hm with many domain orientations about the substrate normal where: (a) [100], (b) [110], and (c) [111] are perpendicular to the substrate. Note that the substrate normal is horizontal (same substrate orientation as shown in Figures 1c and 3b).

4. Conclusions We have reported a method to calculate the location of all Bragg diffraction peaks from nanostructured thin films for arbitrary angles of incidence from just above the critical angle to transmission perpendicular to the film. At grazing angles, the positions are handled accurately using a code developed within the framework of the DWBA. By neglecting the intensity information, one can quickly identify the space group, domain orientation, type of disorder, and lattice constants. Further, by including the ability to simulate patterns at all angles of incidence from grazing angle to 90° transmission, the full 3D structure of reciprocal space may be simulated. There are two important limitations for the peak position predictions in GISAXS geometry that are more or less restrictive depending on the sample. Simply, the incident and exit angles must be greater than the critical angle. These limitations become

restrictive when the critical angle becomes large (very electrondense films), or the unit cell of the sample is very large, which decreases the exit angle for each hkl. For cases of surfactant self-assembled films, the limitations are not restrictive at all because, typically, the critical angle is around 0.2° but the largest d spacings occur around 0.8° 2θ. In the end, one finds that the DWBA changes the predicted spot positions only slightly and only for peaks close the critical angle. Experimentally, it becomes most advantageous to collect GISAXS scattering patterns at an incident angle just above the critical angle so as to probe as low of sz values as possible. We also note that the structures reported for self-assembled nanomaterials are typically described as very high-symmetry structures. Some common space groups with space group (SG) numbers are: Ia3hd (SG 230, the highest symmetry 3D periodic structure possible), Im3hm (SG 229), Fm3hm (SG 225), Pm3hn

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Figure 8. GISAXS simulations of different space groups but with the same orientation for: (a) Im3hm, (b) Fm3hm, and (c) Pm3hm. Each structure is oriented with the [111] perpendicular to the substrate and has a nearest-neighbor distance of 150 Å. Additionally, the labeled peaks are highlighted red for clarity, and the film orientation is congruent with that of Figure 1c. Note that the substrate normal is horizontal (same substrate orientation as shown in Figures 1c and 3b).

Figure 9. Characteristic reciprocal space patterns for self-assembled nanostructured films. Simulations of different space groups where (a) p6mm with the c-axis parallel to the substrate, (b) [110] oriented Im3hm, (c) [111] oriented Fm3m (d) [211] oriented Ia3hd, (e) [211] oriented Pm3hn, and (f) a [001] oriented P63/mmc. The structures in (b-f) are oriented such that the [uVw] direction reported is perpendicular to the plane of the substrate. Also, note that the substrate lies in the sz ) 0 plane, and thus, the substrate normal is horizontal (same substrate orientation as shown in Figures 1c and 3b). These patterns are characteristic of those observed on the detector in GISAXS or low-angle transmission SAXS data. Note, the hollow circles in (e) and (f) are allowed diffraction peaks but have been shown to be experimentally suppressed in many cases.

(SG 223), and P63/mmc (194, the highest symmetry 3D hexagonal structure). This high level of symmetry is somewhat deceiving because we are only observing scattering from the coarse-grained electron-density distribution. Microscopically, or at the atomic level, no such symmetry exists. In most cases for films, these high-symmetry structures are broken due to contraction of the film. Upon contraction, the patterns become stretched along the sz direction but typically do not contain additional peaks. This is most likely due to the preservation of coarsegrain symmetry in the electron density. In addition to this, each space group has a given orientation that is almost always observed (i.e., (10) orientation for c2 mm, (110) orientation for Im3hm, etc.). The three facts: (1) small number of high symmetry space groups, (2) typically no extra peaks upon film contraction, and (3) common film orientations mean that the total number

of expected qualitative scattering patterns is quite small. In fact, most of them are shown in Figure 9. Using this fact, the phases may be identified quickly, almost by fingerprint comparison with the prototypical patterns shown in Figure 9. Also, lattice constants may be calculated accurately from the DWBA corrected peak positions. Further, hypotheses about the presence of mixed phases or orientations may be settled quickly. In addition, artifacts produced by reflection events can be easily identified. Diffraction spots originating from terms 1 and 2 are typically shifted from their BA position only slightly. However, additional peaks appear on the detector from terms 3 and 4 that may be misinterpreted as real features of reciprocal space. These additional peaks can be quickly identified by changing the angle of incidence slightly (0.2° f 0.3°). These peaks will shift dramatically (even for Rf > 2Rc) because the

9892 J. Phys. Chem. B, Vol. 110, No. 20, 2006 main beam is reflected before diffraction. As a result, patterns that were previously difficult to interpret are now relatively straightforward. Though not discussed above, the results may be used to interpret neutron scattering patterns as well (SANS and GISANS). Finally, the NANOCELL code is available (from the author by email) and is free for academic use. The source code is written in Mathematica and may be modified by others. Several example files are posted on the website under the menu item “Research: NANOCELL” at http://www.ecn.purdue.edu/∼hugh. Acknowledgment. We acknowledge financial support from National Science Foundation under the CAREER Award (0134255-CTS) and the use of the NSF-funded facility for Insitu X-ray Scattering from Nanomaterials and Catalysts (MRI program award 0321118-CTS) to collect GISAXS data. Additionally, we 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. W-31-109-ENG-38. We thank Jin Wang and Xuefa Li at APS for their help and C. Jeffery Brinker for sharing beam time. 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. 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. Science 1993, 261, 1299-1303. (3) 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. Science 1995, 267, 1138-1143. (4) 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. Nature 1997, 389, 364-368. (5) Zhao, D.; Yang, P.; Melosh, N.; Feng, J.; Chmelka, B. F.; Stucky, G. D. AdV. Mater. 1998, 10, 1380-1385. (6) Brinker, C. J.; Lu, Y. F.; Sellinger, A.; Fan, H. Y. AdV. Mater. 1999, 11, 579-585. (7) Grosso, D.; Cagnol, F.; Soler-Illia, G.; Crepaldi, E. L.; Amenitsch, H.; Brunet-Bruneau, A.; Bourgeois, A.; Sanchez, C. AdV. Funct. Mater. 2004, 14, 309-322. (8) Zhao, D. Y.; Feng, J. L.; Huo, Q. S.; Melosh, N.; Fredrickson, G. H.; Chmelka, B. F.; Stucky, G. D. Science 1998, 279, 548-552. (9) Zhao, D. Y.; Huo, Q. S.; Feng, J. L.; Chmelka, B. F.; Stucky, G. D. J. Am. Chem. Soc. 1998, 120, 6024-6036. (10) Zhao, D. Y.; Yang, P. D.; Margolese, D. I.; Chmelka, B. F.; Stucky, G. D. Chem. Commun. 1998, 2499-2500. (11) Besson, S.; Ricolleau, C.; Gacoin, T.; Jacquiod, C.; Boilot, J. P. J. Phys. Chem. B 2000, 104, 12095-12097. (12) Grosso, D.; Balkenende, A. R.; Albouy, P. A.; Lavergne, M.; Mazerolles, L.; Babonneau, F. J. Mater. Chem. 2000, 10, 2085-2089.

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