Modeling the polarized X-ray scattering from periodic nanostructures with molecular anisotropy

Abstract

There is a need to characterize nanoscale molecular orientation in soft materials, and polarized scattering is a powerful means to measure this property. However, few approaches have been demonstrated that quantitatively relate orientation to scattering. Here, a modeling framework to relate the molecular orientation of nanostructures to polarized resonant soft X-ray scattering measurements is developed. A variable-angle transmission measurement called critical-dimension X-ray scattering enables the characterization of the three-dimensional shape of periodic nanostructures. When this measurement is conducted at resonant soft X-ray energies with different polarizations to measure soft material nanostructures, the scattering contains convolved information about the nanostructure shape and the preferred molecular orientation as a function of position, which is extracted by fitting using inverse iterative algorithms. A computationally efficient Born approximation simulation of the scattering has been developed, with a full tensor treatment of the electric field that takes into account biaxial molecular orientation, and this approach is validated by comparing it with a rigorous coupled wave simulation. The ability of various sample models to generate unique best fit solutions is then analyzed by generating simulated scattering pattern sets and fitting them with an inverse iterative algorithm. The interaction of the measurement geometry and the change in orientation across a periodic repeat unit leads to distinct asymmetry in the scattering pattern which must be considered for an accurate fit of the scattering.

Graphical abstract

1. Introduction

Nanoscale molecular orientation can govern the behavior of a wide variety of technologically important soft material systems, but characterizing the distribution of molecular orientation within a system remains a significant challenge. For example, the direction and degree of molecular orientation in semiconducting polymers strongly affect their electronic and charge transport properties in transistors and solar cells (Sirringhaus et al., 2000). These polymers typically have highly directional ${\pi }^{\ast }$ transitions in the aromatic rod-like polymer backbones, which tend to have a preferred orientation (DeLongchamp et al., 2009). Another system that can exhibit molecular orientation is that of directed self-assembled block copolymers, which are being used in next-generation lithography to enhance lithographic resolution (Ruiz et al., 2008). As the patterned features become smaller, it is predicted that the polymers will exhibit stronger molecular orientation in the form of highly extended chain conformations (Bates & Fredrickson, 1990). Other systems that can exhibit molecular orientation include aligned liquid crystals (Nickmans et al., 2016) and polymers patterned by nanoimprint lithography (Aryal et al., 2009; Hlaing et al., 2011). In systems such as these with both long-range periodic order and molecular orientation, we can use resonant X-ray scattering to probe the molecular orientation at length scales much smaller than the X-ray spot size.

Resonant soft X-rays are a powerful tool to probe molecular orientation in soft materials. They can excite transition dipole moments such as carbon $1s\to {\pi }^{\ast }$ whose orientation depends on molecular bond orientation (McNeill & Ade, 2013). The absorption and scattering of these X-rays depend on the orientation of the transition dipole moment relative to the electric field vector of the X-rays. The technique known as near-edge X-ray absorption fine structure (NEXAFS) measures the absorbance or electron yield of X-rays near a resonance edge. The peaks observed in the NEXAFS spectrum correspond to individual transition dipole moments. Calculations based on density functional theory allow for detailed interpretation of NEXAFS spectra to gain a quantitative measure of molecular orientation (Patel et al., 2015). By variation of the incidence angle, this technique can determine the average molecular orientation of a film within the beam footprint, but it does not provide information about how the orientation changes with position. Scanning transmission X-ray microscopy (STXM) is essentially the same technique with lateral resolution and has been used to map the molecular orientation of organic thin films (Watts et al., 2011; Collins & Ade, 2012). In traditional STXM, the signal is weak for films that are less than about 100 nm thick and it is not possible to obtain information at length scales less than the probe size. However, this limitation does not apply for ptychographic imaging (Dierolf et al., 2010; Holler et al., 2017), and ptychographic imaging at soft X-ray energies has the potential be a powerful technique for mapping molecular orientation.

Resonant soft X-ray scattering (RSoXS) combines scattering with the use of resonant X-rays and has been used to measure relative molecular orientation and morphology in soft material films (Mitchell et al., 2006; Wang et al., 2010, 2011; Collins et al., 2012; Gann et al., 2016). The scattered intensity is a result of the change in scattering length density as a function of position in the sample. This change in scattering length density has been used to characterize the morphology of block copolymers using resonant X-rays at different energies to isolate the scattering from different polymer blocks (Wang et al., 2011). The ability to control the incident X-ray polarization further allows for the characterization of molecular orientation. Polarized RSoXS has been used to characterize the local orientation of molecules relative to domain interfaces in soft material blends with no long-range net orientation by analyzing how the scattered intensity varies with the scattering direction relative to the incident polarization direction (Collins et al., 2012; Gann et al., 2016). Similar to scattering, resonant soft X-ray reflectivity has been used to characterize polymer interfaces (Wang et al., 2005; Sunday & Kline, 2015) and molecular orientation (Mezger et al., 2011), and calculations based on density functional theory have been used to relate experimental data to quantitative molecular orientation (Pasquali et al., 2014; Capelli et al., 2016).

In this work, we discuss the effect of molecular orientation on a technique known as critical-dimension small-angle X-ray scattering (CDSAXS). CDSAXS is a nondestructive variable-angle transmission measurement that enables the characterization of the three-dimensional shape of periodic nanostructures, including buried morphology. It uses the periodicity of the nanostructure to solve for the local repeating structure, and has been used successfully to fit the shape of inorganic line gratings (Sunday et al., 2016; Sunday, List et al., 2015) and more complex structures such as fin field effect transistors (Lemaillet et al., 2013; Wang et al., 2009). CDSAXS has also been used to characterize the buried interface of directed self-assembled block copolymers (Sunday, Ashley et al., 2015; Sunday et al., 2014, 2013), which form vertical lamellae with long-range order. The morphology of organic thin films is particularly challenging to characterize via CDSAXS owing to the small sample volume and low contrast between components, resulting in low scattered intensity. Therefore, resonant soft X-rays instead of hard X-rays have been used to enhance contrast and obtain higher intensities (Sunday et al., 2014, 2013). This measurement is referred to as res-CDSAXS, although, because of the lower X-ray energies, the scattering angles are much larger than what is typically referred to as SAXS. In previous work, res-CDSAXS has only been used to enhance contrast when probing morphology (i.e. shape) and not to probe molecular orientation. However, the effects of molecular orientation must be included in the scattering model to properly fit the scattering of samples that have local or long-range molecular orientation. By including these effects, we can use res-CDSAXS to probe both morphology and molecular orientation as a function of position simultaneously, averaged over the periods.

CDSAXS requires inverse fitting to obtain information about the sample from the scattering data owing to the phase problem (Pauw, 2013). Because of the large number of parameters that need to be fitted and the rough fitness landscape, evolutionary algorithms have been found to be an effective way of fitting CDSAXS data and finding the global minimum (Hannon et al., 2016). Even with the use of these algorithms, a large number of simulation iterations with different parameters is required to find a scattering pattern that closely matches the experimental data. This is an even greater problem when we want to fit not only morphology information but also molecular orientation. Thus, it is important to optimize both the runtime of a single simulation and the fitting algorithm.

In the present work, we develop a computationally efficient simulation for res-CDSAXS on soft material gratings that takes into account both grating shape and molecular orientation. At resonant energies, the form factor depends on the orientation of the transition dipole moment of an excitation relative to the electric field vector of the incident and scattered X-rays. By changing the orientation of the linearly polarized incident X-rays we can collect scattering data that provide more information about molecular orientation. We first validate the simulation by comparing it with a more detailed but slower rigorous coupled wave simulation. Then we observe how the simulated intensity changes when various parameters such as orientation strength are changed. We use NEXAFS spectra to constrain the candidates of molecular orientation in the res-CDSAXS sample model. Finally, we analyze the ability of various sample models to generate unique best fit solutions by generating simulated scattering patterns from a model system with molecular orientation and fitting the resulting scattering using an inverse, iterative algorithm. We use the simulated data to demonstrate that it would be possible to obtain meaningful and unique solutions from an experimental dataset.

2. Methods

2.1. Born approximation simulation

Because of the transmission geometry, the simulation can be simplified by using the Born approximation, allowing us to account only for single scattering. Therefore, the complex form factor $F$, which is a function of the scattering vector $\mathbf{q}$, is the Fourier transform of the scattering length density $\rho$, which is a function of real-space position $\mathbf{r}$ :

$$F(\mathbf{q})=\int \rho (\mathbf{r})\exp (-i\mathbf{q}\cdot \mathbf{r})\text{d}\mathbf{r}.$$ (1)

In scattering experiments, we observe the scattering intensity, which is proportional to $|F(\mathbf{q}){|}^2$. The phase information of the Fourier transform is lost, so the scattering intensity cannot be simply inverse Fourier transformed to find $\rho$. We use the following expression for scattering intensity $I$, where $S$ is the structure factor, ${\sigma }_{\text{DW}}$ is the Debye–Waller factor which we use to model roughness, $I_0$ is an intensity scaling factor (Pauw, 2013) that combines elements such as incident flux, diffraction efficiency and detector efficiency, and $I_{\text{Bk}}$ is the background intensity:

$$I(\mathbf{q})=I_{0}S(\mathbf{q})|F(\mathbf{q}){|}^2\exp \left(-|\mathbf{q}{|}^2{\sigma }_{\text{DW}}^2\right)+I_{\text{Bk}}.$$ (2)

In CDSAXS experiments, we typically measure structures with one-dimensional periodicity, where the structure factor is

$$S\left(q_x\right)=\sum _n\delta \left(q_x-\frac{2\pi n}{d}\right).$$ (3)

Here $\delta$ is the Kronecker delta function, $d$ is the grating period and $n$ is an integer denoting the order of the diffraction peak. In previous CDSAXS work, we have modeled the scattering length density of multi-phase samples by stacks of trapezoids, each with $\rho$ independent of position, where $\rho$ refers to the difference between the scattering length density of each trapezoid and that of the material or empty space outside of each trapezoid. This allows us to bring $\rho$ outside the Fourier transform and simplify the fitting. The dimensions of each trapezoid (Fig. 1) are fit parameters. For a sample with constant $\rho$ within each trapezoid, the expression for the form factor becomes the following, where $T$ refers to each trapezoid:

$$F\left(q_x,q_z\right)=\sum _T\left[{\rho }_T\underset{z_1}{\overset{z_2}{\int }}\underset{x_a(z)}{\overset{x_b(z)}{\int }}\exp \left(-i{q}_{x}x\right)\exp \left(-i{q}_{z}z\right)\text{d}x\text{d}z\right],$$ (4)

where

$$x_a(z)=x_1+\left(z-z_1\right)\frac{x_2-x_1}{z_2-z_1},$$ (5)

$$x_b(z)=x_4+\left(z-z_1\right)\frac{x_3-x_4}{z_2-z_1},$$ (6)

and $x_1$, $x_2$, $x_3$, $x_4$ , $z_1$ and $z_2$ are shown in Fig. 1.

Figure 1.

CDSAXS measurement geometry (only one trapezoid is shown for simplicity).

The simulation evaluates the closed-form solution [shown in the supporting information, equation (SI1)] of the above expression at each desired $q_x$, $q_z$ point.

We find the following expression [derivation shown in equations (SI3)–(SI9)] for the scattering length density of each trapezoid ${\rho }_T$, which considers the molecular orientation-dependent interaction between the resonant X-rays and the transition dipole moments of the antibonding orbitals that they induce within the sample:

$${\rho }_T\propto \left|{\mathbf{E}}_{\text{scat}}\right|={\widehat{\mathbf{E}}}_{\text{scat}}{\mathbf{\alpha }}_T{\widehat{\mathbf{E}}}_{\text{in}}.$$ (7)

Here ${\mathbf{\alpha }}_T$ is a symmetric second-rank complex polarizability tensor, which contains information about the strength and orientation of the transition dipole moments at a given wavelength. ${\widehat{\mathbf{E}}}_{\text{in}}$ and ${\widehat{\mathbf{E}}}_{\text{scat}}$ are the electric field vectors of the incident and scattered X-rays, respectively. Owing to the periodicity assumed by this simulation, ${\rho }_T$ and ${\mathbf{\alpha }}_T$ refer to the difference in these quantities between each trapezoid and the space outside of the trapezoids. For this work, these quantities are zero in the empty space outside of the trapezoids. Previous CDSAXS work measured isotropic samples at the $S$ polarization, so only a constant ${\rho }_T$ was necessary, while in this work the above expression is used for ${\rho }_T$.

For simplicity, we assume perfect linear polarization parallel $(P)$ or perpendicular $(S)$ to the plane of incidence. For the $S$ polarization, ${\widehat{\mathbf{E}}}_{\text{in}}$ lies along the $y$ axis and does not change with sample angle, while for the $P$ polarization ${\widehat{\mathbf{E}}}_{\text{in}}$ lies in the $zx$ plane and does change with sample angle. Here the incidence angle ${\theta }_{\text{in}}$ is defined with respect to the surface normal:

$${\widehat{\mathbf{E}}}_{\text{in},S}=\left[\begin{array}{lll}0& 1& 0\end{array}\right],$$ (8)

$${\widehat{\mathbf{E}}}_{\text{in},P}=\left[\begin{array}{lll}\cos \left({\theta }_{\text{in}}\right)& 0& -\sin \left({\theta }_{\text{in}}\right)\end{array}\right].$$ (9)

The scattered electric field vectors at each desired $q_x$, $q_z$ point are defined by the following:

$${\widehat{\mathbf{E}}}_{\text{scat},S}=\left[\begin{array}{lll}0& 1& 0\end{array}\right],$$ (10)

$${\widehat{\mathbf{E}}}_{\text{scat},P}=\left[\begin{array}{lll}\cos \left({\theta }_{\text{in}}+{\theta }_{\text{scat}}\right)& 0& -\sin \left({\theta }_{\text{in}}+{\theta }_{\text{scat}}\right)\end{array}\right].$$ (11)

The incident and scattered angles can be expressed in terms of the desired $q_x$, $q_z$ points using geometry:

$${\theta }_{\text{in}}\left(q_x,q_z\right)={\sin }^{-1}\left[\frac{q_z}{{\left(q_x^2+q_z^2\right)}^{1/2}}\right]-{\sin }^{-1}\left[\frac{\lambda {\left(q_x^2+q_z^2\right)}^{1/2}}{4\pi }\right],$$ (12)

$${\theta }_{\text{scat}}\left(q_x,q_z\right)=2{\sin }^{-1}\left[\frac{\lambda {\left(q_x^2+q_z^2\right)}^{1/2}}{4\pi }\right].$$ (13)

We also investigate a more complex case where the orientation strength of ${\mathbf{\alpha }}_T$ varies within a trapezoid. In this case, the orientation strength of ${\mathbf{\alpha }}_T$ exponentially decays with horizontal distance from the interfaces defined by line segments $\overline{\left(x_1,z_1\right)\left(x_2,z_2\right)}$ and $\overline{\left(x_4,z_1\right)\left(x_3,z_2\right)}$ with a persistence length $l_{\text{p}}$. This may apply to systems such as block copolymers where the chains are extended at short distances from the interfaces (Bates & Fredrickson, 1990). Since the change in ${\mathbf{\alpha }}_T$ with orientation strength is approximately linear, we define a constant $\mathbf{\Delta }{a}_{\text{interface}}$ for the difference between ${\mathbf{\alpha }}_T$ at the interface and in the bulk, and scale this with the decay, giving us a form factor that can still be expressed as a closed form solution [shown in equation (SI2)]:

$$F\left(q_x,q_z\right)={\widehat{\mathbf{E}}}_{\text{scat}}(a_{\text{bulk}}\underset{z_1}{\overset{z_2}{\int }}\underset{x_{\alpha }(z)}{\overset{x_b(z)}{\int }}\exp \left(-i{q}_{x}x\right)\exp \left(-i{q}_{z}z\right)\text{d}x\text{d}z$$ (14)

$$+\mathbf{\Delta }{a}_{\text{interface}}\underset{z_1}{\overset{z_2}{\int }}\underset{x_{\text{a}}(z)}{\overset{x_{\text{p}}(z)}{\int }}\left\{\exp \left[-x_1(x,z)/l_{\text{p}}\right]+\exp \left[-x_{\text{r}}(x,z)/l_{\text{p}}\right]\right\}$$

$$\times \exp \left(-i{q}_{x}x\right)\exp \left(-i{q}_{z}z\right)\text{d}x\text{d}z){\widehat{\mathbf{E}}}_{\text{in}}.$$

Here $x_1$ and $x_{\text{r}}$ are the horizontal distances between the point $(x,z)$ and the left and right interfaces, respectively:

$$x_1(x,z)=x-\frac{x_1\left(z_2-z\right)+x_2\left(z-z_1\right)}{z_2-z_1},$$ (15)

$$x_{\text{r}}(x,z)=\frac{x_4\left(z_2-z\right)+x_3\left(z-z_1\right)}{z_2-z_1}-x.$$ (16)

Finally, we use the following expression for the diffraction efficiency prefactor $I_1$, which multiplies $|F(\mathbf{q}){|}^2$ to make up the diffraction efficiency [derivation in equations (SI10)–(SI24)] and which is part of the intensity scaling factor $I_0$ along with incident flux and detector efficiency:

$$I_1=\frac{{\pi }^2\exp \left[-{\mu }_{\text{sub}}{t}_{\text{sub}}/\cos \left({\theta }_{\text{in}}\right)\right]}{{\lambda }^2{d}^2\cos \left({\theta }_{\text{in}}\right)\cos \left({\theta }_{\text{in}}+{\theta }_{\text{scat}}\right)}.$$ (17)

This encompasses a footprint correction $\cos \left({\theta }_{\text{in}}\right)\times \cos \left({\theta }_{\text{in}}+{\theta }_{\text{scat}}\right)$ that takes into account the fact that, at more grazing angles, the beam goes through a larger volume of material that can scatter the beam; a substrate attenuation correction $\exp \left[{\mu }_{\text{sub}}{t}_{\text{sub}}/\cos \left({\theta }_{\text{in}}\right)\right]$ dependent on incident angle, where ${\mu }_{\text{sub}}$ is the substrate attenuation and $t_{\text{sub}}$ is the substrate thickness; and a prefactor based on the grating pitch $d$.

This expression does not include a sample attenuation correction which would take into account attenuation of the incident and scattered beams by the sample. This is technically needed since, in our experiments and in the simulated geometry, the incident beam hits the substrate before the sample. However, this correction was negligible for the cases we simulated, even at resonant energies, owing to the thinness of the polymer gratings.

2.2. Rigorous coupled wave simulation

To check the correctness of the Born approximation simulation, we compare it with a rigorous coupled wave (RCW) simulation for the same shape parameters, optical parameters, energy and ($q_x$;$q_z$) points.

The RCW approach solves Maxwell’s equations for a periodic structure by dividing the structure into layers in $z$, expanding the fields in each layer into a Floquet series in $x$ and $y$, and matching boundary conditions between each layer. The implementation used here is based upon the work of Moharam and co-workers (Moharam, Grann et al., 1995; Moharam, Pommet et al., 1995), as extended by Li (1996), and is available in the public domain SCATMECH (Germer, 2000) library. The RCW simulation is very flexible and has been used to simulate spectroscopic ellipsometry data for optical critical-dimension metrology (Lemaillet et al., 2013; Ding et al., 2007), but it can also simulate X-ray scattering for a variety of measurement geometries. For X-ray scattering applications, the number of Floquet orders in the simulation needs to be at least as high as the number of diffraction orders and the number of layers needs to be sufficiently large to ensure that the structure is adequately approximated. In simulations performed for this study, ±(15–30) Floquet orders were considered (up to twice the number of diffraction orders) and the structures were divided into 50 layers. The simulation returns absolute diffraction efficiencies for all of the diffraction orders and polarizations for any given incident angle and wavelength.

2.3. Sample models

In this work, we analyze the ability of the Born approximation simulation and fitting procedures to find unique and accurate fits of shape and molecular orientation for various sample models. First, a target sample model is used to generate a simulated parameter set including Debye–Waller roughness and constant background noise, meant to simulate readout noise on a CCD detector. The scattering intensity of this target is simulated and an additional Poisson noise, meant to simulate photon shot noise, is applied. The intensity after application of Poisson noise is $I_{\text{f}}(\mathbf{q})=\text{Poisson}\left[I(\mathbf{q})/I_{\text{single}}\right]I_{\text{single}}$, where $I_{\text{single}}$ is the intensity from a single photon hitting the detector and Poisson(x) draws from the Poisson distribution with the average rate of $x$. A fitting run is then initiated using a fit sample model, which may be the same as or different from the target sample model, starting at a random parameter set and attempting to reach the target by fitting the scattered intensity of the target. When we fit simulated data for different sample models, we use $I_{\text{single}}=0.1$ and the target parameters $I_0=1$, ${\sigma }_{\text{DW}}=3\text{nm}$ and $I_{\text{Bk}}=5$. With these parameters, the scattering intensities are of the order of 10⁴ counts for the first order to 5–10 counts for the ninth order.

To generate the shape parameters, we assume that the trapezoids have reflection symmetry about the $yz$ plane, i.e. they are not tilted. Along with the constraint that trapezoids are stacked on top of each other and share adjoining sides, this results in $2T+1$ shape parameters, namely the height and bottom width of each trapezoid and the top width of the top trapezoid.

For these simulations, we assume that the principal axes of $\mathbf{\alpha }$ are $x$, $y$ and $z$, making $\mathbf{\alpha }$ a diagonal tensor:

$$\mathbf{\alpha }(\lambda )=\left[\begin{array}{lll}{\alpha }_x(\lambda )& 0& 0\\ 0& {\alpha }_y(\lambda )& 0\\ 0& 0& {\alpha }_z(\lambda )\end{array}\right].$$ (18)

Without this assumption, $\mathbf{\alpha }$ would require twice the number of optical parameters. We use the Lorentz–Lorenz relation to express the relationship between the ${\alpha }_i$ of each axis and the complex refractive index ${\tilde{n}}_i$ of each axis:

$${\alpha }_i(\lambda )\propto {\tilde{n}}_i{(\lambda )}^2-1,$$ (19)

$${\tilde{n}}_i(\lambda )=1-{\delta }_i(\lambda )+i{\beta }_i(\lambda ).$$ (20)

To guide the generation of potential optical parameter solutions, we obtain the average $\mathbf{\alpha }$ from NEXAFS experiments. The absorptive (imaginary) part of the refractive index, $\beta$, is measured for the $x$ and $y$ axes using normal-incidence NEXAFS at the $P$ and $S$ polarizations, respectively. When using transmission NEXAFS data, the absorption coefficients for each axis ${\mu }_i$ can be directly calculated and ${\beta }_i$ obtained by ${\beta }_i={\mu }_i\lambda /4\pi$. For electron-yield NEXAFS, for a narrow energy range (Henneken et al., 2000) the intensity is proportional to ${\beta }_i/\lambda$. ${\beta }_z$ cannot be directly measured since that would require a NEXAFS measurement at a grazing angle of zero, but it can be extrapolated from variable-angle NEXAFS at the $P$ polarization by finding the fit coefficients $A$ and $B$ in the equation ${\beta }_P\left({\theta }_{\text{in}}\right)=A+B{\cos }^2\left({\theta }_{\text{in}}\right)$ separately at each energy in the spectrum (Stöhr & Samant, 1999; Patel et al., 2015).

In order to calculate ${\delta }_i$, we use the Kramers–Kronig transform, which relates the dispersive and absorptive atomic scattering factors $f_1$ and $f_2$ (proportional to $\delta$ and $\beta$, respectively), with $Z^{\ast }$ as the relativistic correction and $P^{\ast }$ as the Cauchy principal value (Kronig, 1926; Watts, 2014):

$$f_1(E)=Z^{\ast }-\frac{2}{\pi }{P}^{\ast }\underset{0}{\overset{\infty }{\int }}\frac{E^{\prime }{f}_2\left(E^{\prime }\right)}{E^{\prime 2}-E^2}\text{d}{E}^{\prime }.$$ (21)

While an exact calculation of the dispersive atomic scattering length factor requires integration over all energies, we use a calculation of the Kramers–Kronig transform with an open-source implementation called KKcalc (Watts, 2014). This first represents the atomic scattering factors as piecewise Laurent polynomial functions over a wide range of energies, on the basis of a table of elemental atomic scattering factors (Biggs & Lighthill, 1988; Henke et al., 1993), then splices in experimentally measured $\beta$ data at the energies of interest and calculates $\delta$ at those energies.

If we assume that the optical constants in the nanostructure do not change periodically with position, no optical fit parameters are needed as we can just use the measured average $\mathbf{\alpha }$. But as it is likely that optical constants do change with position, we explore methods of generating optical parameters for each trapezoid and energy of interest for the target and candidate parameter sets. The simplest method is to directly use ${\beta }_i$ and ${\delta }_i$ at a single energy for each axis as the optical parameters, letting the fitting algorithm determine them. This can potentially lead to physically unrealistic values (i.e. not Kramers–Kronig consistent) and does not take advantage of the NEXAFS information at a range of energies that we can collect.

The second method, which we focus on, is to generate ${\beta }_i(\lambda )$ candidates on the basis of orientation factors and a combination of experimentally measured ${\beta }_{\text{avg}}(\lambda )$ spectra that are averages over the beam area (Stöhr & Samant, 1999; Gann et al., 2016). For this method, we first need to generate a baseline ${\beta }_0(\lambda )$ spectrum which has close to zero absorption at the resonance peak of interest ${\lambda }^{\ast }$, representing the case where all of the transition dipole moments at ${\lambda }^{\ast }$ within the film are oriented perpendicular to a particular axis. We then linearly interpolate ${\beta }_i(\lambda )$ spectra between the baseline and the case of perfect alignment where all the transition dipole moments at ${\lambda }^{\ast }$ are parallel to that axis. This requires making the simplifying assumption that the relationship between transition dipole moment orientation distributions at each $\lambda$ is the same for each position as in the average film, with principal axes $x$ , $y$ and $z$ . For each candidate we choose fractional orientation factors $f_x$, $f_y$ and $f_z$, with the constraint that $f_x+f_y+f_z=1$. An orientation factor of 1 represents the case where all the transition dipole moments at ${\lambda }^{\ast }$ are oriented parallel to a particular axis. The oriented fraction of the absorption is added to the baseline spectrum to generate ${\beta }_x(\lambda )$, ${\beta }_y(\lambda )$ and ${\beta }_z(\lambda )$ for each trapezoid.

To generate the baseline spectrum, we select two ${\beta }_{\text{avg}}(\lambda )$ spectra, denoted as ${\beta }_A(\lambda )$ and ${\beta }_B(\lambda )$, where ${\beta }_A>{\beta }_B$ at the resonance peak of interest ${\lambda }^{\ast }$. We then find the constant $f_0$ that satisfies ${\beta }_B\left({\lambda }^{\ast }\right)-f_0\left[{\beta }_A\left({\lambda }^{\ast }\right)-{\beta }_B\left({\lambda }^{\ast }\right)\right]=0$. Then the baseline spectrum is

$${\beta }_0(\lambda )={\beta }_B-f_0\left[{\beta }_A\left({\lambda }^{\ast }\right)-{\beta }_B\left({\lambda }^{\ast }\right)\right]$$ (22)

and the candidate spectra are

$${\beta }_x(\lambda )={\beta }_0(\lambda )+f_x\left[{\beta }_{x,\text{avg}}(\lambda )+{\beta }_{y,\text{avg}}(\lambda )+{\beta }_{z,\text{avg}}(\lambda )-3{\beta }_0(\lambda )\right],$$ (23)

$${\beta }_y(\lambda )={\beta }_0(\lambda )+f_y\left[{\beta }_{x,\text{avg}}(\lambda )+{\beta }_{y,\text{avg}}(\lambda )+{\beta }_{z,\text{avg}}(\lambda )-3{\beta }_0(\lambda )\right],$$ (24)

$${\beta }_z(\lambda )={\beta }_0(\lambda )+f_z\left[{\beta }_{x,\text{avg}}(\lambda )+{\beta }_{y,\text{avg}}(\lambda )+{\beta }_{z,\text{avg}}(\lambda )-3{\beta }_0(\lambda )\right].$$ (25)

Note that a set of orientation factors does not provide information about the distribution of molecular orientations within a sample, it just describes the average molecular orientation. Also note that the sum of $\beta (\lambda )$ for any set of three orthogonal axes is equal to the total absorption ${\beta }_{\text{tot}}(\lambda )$ (Stöhr & Samant, 1999).

2.4. Fitting algorithms

The goal of fitting is to find the parameters that minimize a goodness-of-fit function that compares the simulated and target scattering. We use the covariance matrix adaptation evolution strategy (CMAES; Hansen & Ostermeier, 2001) and Monte Carlo Markov chain (MCMC; Sambridge & Mosegaard, 2002) algorithms to do this. These algorithms have been compared for fitting non-resonant CDSAXS data in a recent paper from our group (Hannon et al., 2016). There it was found that CMAES was able to converge to a lower (better) goodness-of-fit value faster than MCMC. MCMC has the advantage of being able to sample the local fitness landscape with a probability distribution proportional to our goodness-of-fit function. Therefore, we first use multiple independent CMAES runs to find the best fit parameter set and to find the parameter correlations and sensitivity of the goodness of fit to each of the parameters in the global fitness landscape. Sixty independent CMAES runs, each with a population of 500, were used for each sample model. A few outlier CMAES runs did not converge to a low goodness of fit, so the runs where the lowest goodness of fit was higher than the third quartile plus 1.5 times the interquartile range of the lowest goodness of fit of each run were dropped. We then use 30–50 MCMC chains per sample model, starting at the best fit, to find parameter correlations in the local fitness landscape. The Spearman correlation coefficient is used as a measure of correlation to be more robust to outliers and so that linear and nonlinear monotonic correlations can be measured. Our fitting code is based on existing open-source implementations of CMAES and MCMC [DEAP (Gagn, 2012) and emcee (Goodman & Weare, 2010), respectively].

To begin a fitting run, we first generate the target scattering $I_{\text{tar}}(\mathbf{q})$. The initial candidate parameters $a_{0i}$, where $i$ is the index of each parameter, are normally distributed about a point that is a random distance away from the target parameter set (sampled from a uniform distribution between 50 and 150% of each target parameter), except for the orientation factors, which are initially set to 1=3. The parameters used in the fitting algorithms $b_i$ are scaled either linearly or logarithmically using a scaling constant for each parameter $s_i$ , selected so that they are initially normally distributed about 0 and have similar sensitivities. To generate the simulation parameters, we use

$$a_i=\{\begin{array}{ll}{10}^{s_i}{b}_i+{\log }_{10}{a}_{0i}& \text{if}\hspace{1em}i=I_0 \text{or} I_{\text{Bk}},\\ s_i{b}_i+a_{0i}& \text{otherwise}.\end{array}$$ (26)

We then generate the simulated scattering $I_{\text{sim}}\left({\mathbf{q}}_i\right)$ and calculate the goodness of fit $\text{Ω}$ using a log error test:

$$\text{Ω}=\frac{1}{N}\sum _{i=1}^N\left|{\log }_{10}{I}_{\text{sim}}\left({\mathbf{q}}_i\right)-{\log }_{10}{I}_{\text{tar}}\left({\mathbf{q}}_i\right)\right|.$$ (27)

This goodness-of-fit function, which we want to minimize, is well suited for data that span several orders of magnitude and is not affected by a constant intensity scaling factor or the number of points. We can also fit datasets at different energies and polarizations by summing their goodness of fits. For the MCMC algorithm, the Metropolis–Hastings criterion is used to determine acceptance of a candidate parameter set. The fitness standard deviation $C$ and the parameter standard deviation are constants that vary the sample acceptance rate and the locality of the fitness landscape to be sampled. We chose these constants to be more local than in the CMAES runs:

$$P(\text{accept})=\{\begin{array}{ll}1& {\text{if Ω}}_{\text{candidate}}\le {\text{Ω}}_{\text{current}},\\ \exp \left[-\frac{{\text{Ω}}_{\text{candidate}}-{\text{Ω}}_{\text{current}}}{C}\right]& \text{otherwise}.\end{array}$$ (28)

The CMAES algorithm is terminated when the goodness of fit stagnates (when the range of best fitnesses during the last ten generations is less than 10⁻⁴, typically less than 200 generations), while MCMC is run for a fixed number of iterations (100 000, resampling every 25 iterations) to adequately sample the local fitness landscape. Each fitting run is done on a single node of a cluster, where multiple threads are used to run multiple chains (MCMC) or individuals in the population (CMAES). Independent CMAES fitting runs with the same target parameters are run on different nodes to collect statistics.

3. Results and discussion

3.1. Characteristics of Born approximation simulation

An easily observable aspect of the scattering intensity in resonant CDSAXS measurements that can indicate preferred molecular orientation is asymmetry in the scattering intensity between $+q_z$ and $-q_z$. This can happen in two cases. The first, which does not require molecular orientation, is when the cross-sectional shape of the trapezoids is tilted, i.e. they do not have reflection symmetry about the $yz$ plane. The second is when the trapezoids have different optical constants, whether the optical constants are isotropic $\left({\tilde{n}}_x={\tilde{n}}_y={\tilde{n}}_z\right)$ or anisotropic. This does not require the optical constants to be asymmetric about the $z$ axis. For the first case, asymmetry would be observed at both resonant and non-resonant energies, and for the second case only at resonant energies. Also, for the first case, if the film is rotated azimuthally by 180°, the direction of the asymmetry flips. We focus on the second case in our simulations because observations of our experimental samples correspond to the second case. The reason for the asymmetry in the second case is that, while the absolute value of the form factor of each trapezoid is symmetric with $q_z$, the complex form factor is asymmetric, so when multiple trapezoids with different optical constants are added in the second case, the absolute value of the sum is asymmetric. For simulations with a single trapezoid, asymmetry is also observed when there is a change in optical anisotropy strength within the trapezoid as a function of position, whether vertical or horizontal. This asymmetry provides information about the molecular orientation of the sample and can lead to more unique fits. For all cases, an additional very weak asymmetry is also observed when the footprint and substrate attenuation corrections are applied, since these corrections are symmetric with ${\theta }_{\text{in}}$ and not with $q_z$. Note that the scattering angles are large, so the relationship between ${\theta }_{\text{in}}$ and $q_z$ is monotonic but offset from the origin by ${\theta }_{\text{scat}}$.

3.2. Comparison of Born approximation and RCW

To validate the Born approximation simulation, we compare it with the RCW simulation for the test case of a grating with pitch $d$ = 140 nm, represented by a stack of three trapezoids with pre-defined optical constants (Fig. 2). In this case, we set the middle trapezoid to have isotropic optical constants and the top and bottom trapezoids to be strongly anisotropic with absorption strongest along the $z$ axis. For $I_0$ we use the diffraction efficiency term $I_1$ from equation (17). No constant background, Debye–Waller roughness or Poisson noise was applied. The trapezoid shape and the data points simulated were chosen to be similar to experimental samples (Sunday, Ashley et al., 2015; Sunday et al., 2017). A significant amount of asymmetry between the scattered intensity at $+q_z$ and $-q_z$ was observed in both simulations, especially for the $P$ polarization. On a log scale the two simulations match quite well, deviating more at intensity minima. When $\delta$ and $\beta$ are smaller, the error between the two simulations decreases, and when $\delta$ and $\beta$ are larger, the error between the two simulations increases. One reason that the RCW is more accurate is that the Born approximation assumes that the incident wave inside the material is the same as that outside the material, and so it does not account for refraction at the interfaces between the substrate, trapezoids and air. The Born approximation also does not account for multiple scattering, which plays more of a role at higher angles and $q_z$ where the path lengths are longer. Despite this, it appears that the Born approximation approach can simulate polarized resonant CDSAXS experimental data reasonably accurately. Although the amount of error appears large near the intensity minima, the scattering intensity in these simulations and in experiments varies by several orders of magnitude, so this amount of error should not be a hindrance to fitting experimental data.

Figure 2.

Comparison of the Born approximation and RCW CDSAXS simulations for a stack of three trapezoids. (a) Trapezoid geometry and (b) optical constants. (c), (d) Scattering intensity for (c) $P$ polarization and (d) $S$ polarization. Each line (vertically offset for display) is for a different diffraction order from 1 to 15 and $-50\le {\theta }_{\text{in}}\le 50°$. Lines are from the Born approximation and points are from the RCW. (e), (f) Difference between the logs of the intensities of the Born approximation and RCW simulations for (e) $P$ polarization and (f) $S$ polarization.

The main advantage of the Born approximation simulation is that it has a much faster runtime than the RCW simulation. A single RCW simulation with ±15 Floquet orders took 83 s to run on one core of a 3.3 GHz processor (Intel¹ Core i5–3550) for the angles shown and for both polarizations, while one Born approximation simulation for a fixed $\delta$ and $\beta$ takes on average 0.0063 s per trapezoid for one polarization and energy, or 0.038 s for the case shown. This is about 20 000 times faster than the RCW simulation. When a Kramers–Kronig transform is required as part of the Born approximation simulation, the simulation takes on average 13 times longer than without the transform, on average 0.080 s per trapezoid for one polarization and energy, or 0.48 s for the case shown. The fitting algorithm and fitness calculation take up a relatively small amount of computation time compared to the simulations themselves for the large number of simulations typically required for inverse fitting.

3.3. Scattering as a function of energy, interface and orientation strength

To investigate how X-ray energy and orientation strength at interfaces can affect the scattering intensity, we simulate and compare various cases of a single trapezoid with isotropic optical constants in the bulk and anisotropic optical constants at the interfaces (Fig. 3c), as given in equation (14). In order to use optical constants that are physically realistic, experimental variable-angle partial-electron-yield NEXAFS of a spin-coated semiconducting polymer film (poly{[N,N]9-bis(2-octyldodecyl)naphthalene-1,4,5,8}, P(NDI2OD-T2)) was measured and extrapolated to the $x=y$ and $z$ axes (Fig. 3a). The film was primarily face on, with the aromatic rings lying parallel to the substrate and the $1s\to {\pi }^{\ast }$ dipole moments perpendicular to the substrate. NEXAFS near the ‘magic’ polar angle of 35.3 from normal (Stöhr, 1992) (54.7° from grazing) was used to generate an isotropic ${\mathbf{\alpha }}_{\text{bulk}}$ that is the same for all the trapezoids, while ${\mathbf{\alpha }}_{\text{interface}}$ for a given set of orientation factors was generated as discussed in equations (23)–(25) (Fig. 3b). An orientation factor of $f_x=f_y=f_z=1/3$ results in approximately the same $\beta$ as the experimental data at the magic angle. No constant background, Debye–Waller roughness or Poisson noise was applied to the scattered intensity for these simulations.

Figure 3.

(a) Experimental and extrapolated NEXAFS of face-on spin-coated P(NDI2OD-T2). The extrapolated 0° spectrum corresponds to the electric field along the $x=y$ axis and the extrapolated 90° spectrum corresponds to the electric field along the $z$ axis. (b) Extrapolated NEXAFS and Kramers–Kronig spectra at the magic angle, at the $x=y$ and $z$ axes of the sample, and at perfect orientation. (c) Illustration of the sample model with isotropic orientation in the bulk and anisotropic orientation at the interface. (d)–(f) Born approximation CDSAXS simulation at $P$ polarization: (d) ${\log }_{10}I\left(f_z=1\right)-{\log }_{10}I\left(f_z=1/3\right)$ for the ninth diffraction order and $l_{\text{p}}$ = 2 nm; (e) range of $l_{\text{p}}$, energy = 284.2 eV, $f_z$= 1 at interface, vertically offset for display; (f) range of $f_z$ at interface, $l_{\text{p}}$= 2 nm, energy = 284.2 eV, vertically offset for display.

Variations of the persistence length, energy and orientation strength all have the effect of changing the optical constants as a function of position and therefore the scattered intensity. We observe that a strong asymmetry between $+q_z$ and $-q_z$ requires that both $\delta$ and $\beta$ are nonzero and that at least one of them is different at the interface compared to the isotropic bulk. The $\text{C}=\text{C}1s\to {\pi }^{\ast }$ absorption peak (Schuettfort et al., 2011) at 284.2 eV was the energy with the strongest asymmetry in the scattering when there was interfacial orientation, and thus the strongest difference between interfacial orientation and no interfacial orientation (Fig. 3d). The difference also increases as the peak order increases. Here only the last peak order is shown. Note that the asymmetry is stronger at the 284.2 eV peak than at the other $\text{C}=\text{C}1s\to {\pi }^{\ast }$ peaks at 285.1 and 285.8 eV because at 284.2 eV not only $\beta$ but also $\delta$ is different at the oriented interface compared to the isotropic bulk (Fig. 3b). The asymmetry was strongest at a persistence length of around 2 nm (Fig. 3e). Very low persistence length means that almost none of the trapezoid is oriented and very high persistence length means that the entire trapezoid is oriented; for both of these cases there is no asymmetry. When $f_z$ is varied (Fig. 3f), an isotropically oriented interface at $f_z$= 1/3 has no asymmetry, while a completely oriented interface at $f_z$= 1 or 0 has strong asymmetry.

3.4. Fitting simulated data for different sample models

We study fit uniqueness and parameter correlations for various sample models dealing with molecular orientation by fitting simulated data with added Poisson noise. For each sample model, we conducted 60 independent CMAES fitting runs of each model starting at random parameters and then 30–50 independent MCMC chains starting at the best fit parameters from CMAES. The sample models (Table 1) were all 140 nm gratings represented by a stack of three trapezoids, unless otherwise stated. Certain variables were kept constant for all models: we use the same target shape parameters with three trapezoids, $I_{\text{single}}$, target $I_0$, ${\sigma }_{\text{DW}}$ and $I_{\text{Bk}}$, CMAES and MCMC settings, number of diffraction orders, and number of $q_x$, $q_z$ points, unless otherwise stated. Target optical spectra were derived from NEXAFS of a spin-coated P(NDI2OD-T2) film as discussed in the previous section. For each of the models, the random seed for the Poisson noise is different for each run, to allow us to study how accurately the parameters can fit experimental data that contain random noise. The target parameters used in each model are shown in Table SI1. For each of the models, $f_z$ is not directly varied but is determined by the constraint that $f_x+f_y+f_z=1$.

Table 1. List of sample models.

Each sample model is composed of a target sample model, which is run once with the target parameters to generate the target scattering, and a fit sample model, which is run many times with iterated parameters to fit the target scattering. Scattering is asymmetric when trapezoids have different bulk orientation factors or when trapezoids have interface orientation factors different from the bulk orientation factors.

Name	Asymmetric	Target model	Fit model	Energies (eV)	Polarizations
A	No	Orientation factors	Same as target	284.2	P
B	Yes	Orientation factors	Same as target	284.2	P
C	Yes	Interface orientation factors	Same as target	284.2	P
D	Yes	Interface orientation factors	Orientation factors	284.2	P
E	No	Orientation factors (8 trapezoids)	Orientation factors (3 trapezoids)	284.2	P
F	No	Orientation factors (8 trapezoids)	Orientation factors (3 trapezoids)	270, 284.2	P, S
G	No	Orientation factors (8 trapezoids)	Orientation factors (3 trapezoids)	284.2 (×4)	P (×4)
H	Yes	Morphology → Orientation factors	Same as target	270, 284.2	P, S
I	Yes	Morphology → Interface orientation factors	Morphology → Orientation factors	270, 284.2	P, S

The first set of sample models, A–D (Fig. 4), involve fitting simulated data at only a single resonant energy (284.2 eV) for the $P$ polarization, which is affected by optical constants along the $x$ and $z$ axes. In model A, each trapezoid has the same orientation parameters, causing the scattering to be symmetric. From the boxplot, it can be seen that for most parameters the variation between runs is quite low, except for the heights of each trapezoid which have larger variation. The heights of adjacent trapezoids are also strongly negatively correlated as shown in Fig. SI3. This is because all of the trapezoids have the same scattering length density, so an increase in the height of one trapezoid and a corresponding decrease in the height of an adjacent trapezoid does not change the scattering much. Model B is similar to model A except that now each trapezoid has different orientation parameters in the target, resulting in asymmetric scattering. In this case the variation in the best fit dimensional parameters (heights and widths) is smaller compared to model A, but the variation in orientation parameters is larger. The fractional variation in $f_y$ is larger than that of $f_x$ because the target value for $f_y$ is larger, except in the case of the topmost (labeled as number 3) trapezoid. The orientation parameters for a given trapezoid tend to be correlated owing to the constraint between them. Also, orientation parameters between different trapezoids are correlated with each other but not correlated with the shape parameters. In model C, the orientation parameters in the bulk of each trapezoid are isotropic and are not fit parameters, and there is a single set of anisotropic orientation parameters at the interfaces with a persistence length of 2 nm. This produces scattering that is slightly asymmetric but less so than model B. (The orientation parameters in the boxplot refer to the interface parameters, not the bulk.) The height parameters again have a relatively large variation and correlation for the same reasons as model A. For each of these three models, ${\text{Ω}}_{\text{fit}}$ (shown in the right-most column of the boxplots as well as in Table SI2) is close to ${\text{Ω}}_{\text{target}}$, about 0.03, indicating that the fitting algorithm was able to find a good fit. In some cases, ${\text{Ω}}_{\text{fit}}$ can even be slightly less than ${\text{Ω}}_{\text{target}}$ when parameters are found that are a better fit to the scattering plus Poisson noise than the target parameters.

Figure 4.

Results from 60 independent CMAES fitting runs each of models A–D. From left to right: illustration of target model (the numbers on the trapezoids refer to $f_x$, $f_y$, $f_z$ for each trapezoid or for the entire shape, and for model D the best fit is also shown in red), the target scattering (vertically offset for display) with Poisson noise (dots) and best fit (lines), and boxplots for percent error between the best fit parameters and target parameters of each run with whiskers showing 1.5 times the interquartile range. (The target parameters that are being compared with are interface orientation parameters for model C, bulk orientation parameters for models A, B and D.)

Finally, in model D, the target parameters are the same as in model C, but the fit parameters are the same as in model B. The best fit parameters are shown in red in Fig. 4. This is meant to simulate a case where the sample model does not perfectly represent the physical sample, which has an unknown model, typically the case for experimental data. In this case the best fit scattering is surprisingly close to the target scattering, with ${\text{Ω}}_{\text{fit}}$ of 0.04, only slightly worse than in the previous models. The variation in best fit parameters between runs is quite high, meaning that the fit is not very unique. For the run (shown bottom left) with the overall best goodness of fit, the overall shape does not match the target very well, but most of the area is taken up by the middle trapezoid which has orientation parameters very close to the isotropic bulk of the target. The small but highly anisotropic top and bottom trapezoids in the best fit have orientation parameters very different from those in the target, but they alter the scattering in a similar way to the anisotropic left and right sides in the target. This shows that when fitting a single resonant energy and polarization, using the wrong model can lead to incorrect solutions that appear to fit well.

Next we assess the impact of changing the amount of Poisson noise applied to the target scattering. Fig. 5 shows three variations of model B with $I_{\text{single}}$ ranging from 0.01 to 1. This has the same effect as varying $I_0$ and $I_{\text{Bk}}$ concurrently. An $I_{\text{single}}$ of 0.1 is the closest to experimental data. Increasing the amount of Poisson noise causes both ${\text{Ω}}_{\text{target}}$ and ${\text{Ω}}_{\text{fit}}$ to increase by the same amount, and causes the variation in best fit parameters between runs to increase, thus causing the fit to be less unique. With a higher amount of noise, it becomes increasingly difficult to fit the scattering with parameters that are close to the target parameters. Again, the correlation coefficients show that orientation parameters tend to be correlated with each other and the shape parameters with each other, with less correlation between these two types. Increasing the amount of Poisson noise causes the parameters to become less correlated.

Figure 5.

Results from 60 independent CMAES fitting runs each of model B with varying Poisson noise, denoted in parentheses. From left to right: the target scattering (vertically offset for display) with Poisson noise (dots) and best fit (lines), boxplots for percent error between best fit parameters and target parameters of each run, and Spearman’s correlation coefficient for best fit parameters of each run.

Next we investigate models E–G, in which a target model with eight trapezoids is fitted with a fit model with only three trapezoids (Fig. 6). In model E, only data from one resonant energy (284.2 eV) at the $P$ polarization were fitted, while in model F one resonant energy and one non-resonant energy (284.2 and 270 eV) at the $P$ and $S$ polarizations were fitted simultaneously, which is four times the quantity of data as in model E. In model G, the same quantity of data was fitted as in model F, but in this case the data consist of four repeated simulated measurements from one resonant energy (284.2 eV) at the $P$ polarization, each with different Poisson noise. In all of the models, in the run with the best fit the three trapezoids in the fit model approximated the eight trapezoids in the target model quite accurately, and the goodness of fit was low. The orientation parameters were not very correlated with each other, while the shape parameters were more correlated with each other.

Figure 6.

Results from 60 independent CMAES fitting runs each of models E–G, with different amounts of data. Column 1: model F target scattering (vertically offset for display) with Poisson noise (dots) and best fit (lines) for $P$ and $S$ , and illustration of target model in black, best fit model in red. Column 2: boxplots for percent error between best fit parameters and target parameters of each run. Column 3: Spearman’s correlation coefficient for best fit parameters of each run.

Interestingly, while model G has a smaller variation in best fit parameters than model E owing to the increased quantity of data, model F has a much larger variation, and the standard deviation of ${\text{Ω}}_{\text{fit}}$ for model F is eight to 18 times that of models E and G (Table SI2). This is hypothesized to be due to the additional information from fitting multiple polarizations and energies in model F, revealing the mismatch between the target and fit models. We also looked at two intermediate cases in between models F and G with the same quantity of data, where two energies and one polarization or one energy and two polarizations were fitted. The standard deviation for both cases was in between that of models F and G. In contrast, we tested similar cases where the target and fit models were the same and found no significant difference between repeated simulated measurements and simulated measurements at multiple polarizations and energies (Fig. SI6).

In the last set of models, H and I (Fig. 7), instead of fitting scattering from multiple energies simultaneously as in the previous set of models, we fit the shape parameters first at the non-resonant energy (270 eV) and a single polarization, and then fit the orientation parameters at a resonant energy (284.2 eV) at the $P$ and $S$ polarizations while holding the shape parameters constant at the best fit of each run. The first step, in which the shape parameters are fitted, is the same for both models. At 270 eV, $\beta$ is practically zero and $\delta$ changes only very slightly with orientation parameter, so the scattering is not affected by molecular orientation and is symmetric. Interestingly, the variation in the shape parameters is significantly lower than that in model A which also has symmetric scattering, suggesting that the presence of the orientation parameters in model A increases the variation in the shape parameters. During the second part of the fit, in model H both the target and fit models use the same bulk orientation parameters. The variation in the orientation parameters here is about the same as in model SI4 (see supporting information), which contains the same scattering data but fitted simultaneously. Thus, it appears that when the target and fit models are the same, the main advantage of fitting the shape and orientation in two steps instead of simultaneously is the removal of all correlation between the shape and orientation parameters, which results in a faster fitting time. The two fitting steps in model H took on average 20 min, while the single fitting step in model SI4 took on average 57 min.

Figure 7.

Results from 60 independent CMAES fitting runs each of models H–I, where non-resonant scattering is used to fit shape and resonant scattering is used to fit orientation. From left to right: the target scattering (vertically offset for display) with Poisson noise (dots) and best fit (lines), boxplots for percent error between best fit parameters and target parameters of each run, and Spearman’s correlation coefficient for best fit parameters of each run. (The target parameters that are being compared with are bulk orientation parameters for model I.)

Model I is similar to model D in that the target model uses interface orientation parameters and an isotropic bulk while the fit model uses bulk orientation factors. Compared to model D which also used different target and fit models, model I has a higher best ${\text{Ω}}_{\text{fit}}$ of 0.05, the fit looks visually worse and the best fit bulk orientation factors are not close to matching the isotropic target. Since the molecular orientation of the sample model does not affect the non-resonant scattering, we can still be relatively confident in the shape parameters found in the first step, even with an incorrect choice of sample model Thus it appears that when the target and fit models are different, fitting the shape and orientation in two steps provides the additional advantage of being less likely to find incorrect solutions.

Finally, for each of the sample models, the best fit parameters out of all the CMAES runs were taken and MCMC chains starting at those parameters were run to sample the local fitness landscape around these minima. The correlation coefficients from these MCMC chains are shown in Fig. SI4. The parameters are much less correlated in the local fitness landscape than in the global fitness landscape as shown by the CMAES correlations. This suggests that the fitness landscape contains multiple correlated minimum valleys surrounded by slopes where the parameters are not as correlated.

4. Conclusions

In this work, we have developed a Born approximation simulation for polarized resonant X-ray scattering of periodic nanostructures that takes into account biaxial molecular orientation. We find that our simulation matches the scattering of a more computationally intensive RCW simulation sufficiently well to fit data with the Born approximation model. Asymmetry in the scattering patterns can result from a change in molecular orientation and optical constants at resonant energies across a periodic repeat unit. We generated simulated biaxial optical spectra based on measured NEXAFS spectra and used them to investigate fit uniqueness and parameter correlations for different simulated sample models based on orientation factors. These models included samples with a vertical orientation change, where each trapezoid has a different preferred orientation, and samples with a horizontal orientation gradient depending on distance from an interface. The method of generating orientation factors from NEXAFS is a good way of generating relatively unique solutions while minimizing the number of orientation parameters.

When scattering from only a single resonant energy and polarization is fitted using a CMAES algorithm, there are large amounts of variation in the best fit orientation factors from different runs. The use of scattering data from more than one polarization adds information because scattering from the $P$ polarization is only affected by optical constants along the $x$ and $z$ axes while scattering from the $S$ polarization is only affected by optical constants along the $y$ axis. Thus, when scattering from multiple energies and polarizations is fitted at the same time and the fit model is correct, the fit uniqueness improves. However, when multiple energies and polarizations are fitted at the same time and the fit model is different from the target model, fit uniqueness actually worsens because of the additional information revealing the mismatch between the models. In contrast, duplicate scattering information with different noise improves fit uniqueness whether or not the sample and fit models match. When it is suspected that the sample model may not accurately represent the physical sample, fitting the shape using non-resonant scattering and then the molecular orientation using resonant scattering in two steps is useful for avoiding incorrect but well fitting solutions that may result from fitting both energies simultaneously. Fit uniqueness is also greatly affected by the amount of noise in the scattering data. Finally, the correlation coefficients from the best fit of each CMAES run show that orientation parameters tend to be correlated with each other and shape parameters with each other, while comparing CMAES and MCMC correlations results in the observation that the global fitness landscape is more correlated than the local one.

This simulation can be used to fit a variety of soft material samples such as directed self-assembled block copolymers and semiconducting polymer gratings, as long as they have both periodicity and preferred molecular orientation. The basic framework of the simulation can even accommodate more complex physics-based models of molecular orientation and density gradients.

5. Related literature

For literature related to the supporting information see Jackson (1999).