On Resolution in Electron Tomography of Beam Sensitive Materials

Laboratory of Materials and Interface Chemistry and CryoTEM Research Unit, Department of Chemical Engineering and Chemistry, Eindhoven University of ...
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On Resolution in Electron Tomography of Beam Sensitive Materials Delei Chen,†,‡ Heiner Friedrich,*,† and Gijsbertus de With† †

Laboratory of Materials and Interface Chemistry and CryoTEM Research Unit, Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, Den Dolech 2, 5612 AZ, Eindhoven, The Netherlands ‡ Dutch Polymer Institute (DPI), P.O. Box 902, 5600 AX, Eindhoven, The Netherlands S Supporting Information *

ABSTRACT: Electron tomography is a powerful tool that is widely used in materials science to unravel the 3D morphology of functional nanostructures such as organic solar cells. However, analysis of the 3D information is limited by the data quality which is dominated by the beam sensitivity of specimens. Therefore, it is necessary to develop advanced acquisition schemes to obtain high-quality 3D morphological information. In this work we simulate the electron tomography workflow to study the influence of the electrons dose, tilt range, and tilt scheme on the reconstruction quality. We employ a slab model of rodlike filler particles in a polymer matrix that is directly compared with the simulated 3D reconstruction using statistical means and by characterizing edge intensity profiles. We find that using counterintuitive acquisition schemes, i.e., large tilt increments at low electron doses, the reconstruction quality is preserved or even improved. Furthermore, inaccuracies in determining the position of interfaces between materials are quantified in dependence of the acquisition scheme with implications for volume and connectivity determination. Finally, it is shown that for the chosen specimen geometry at a tilt range of ±75° the resolution becomes isotropic regardless of the total electron dose and the tilt increment, which is a clear advantage for subsequent quantitative analysis.

1. INTRODUCTION Polymer-based materials, like block copolymers, polymer blends, or polymer nanocomposites, have a wide range of properties that are dependent on their structure at the nanoscale.1 Insight into the structure−property relations is vital for designing new materials with desired functions and properties. Electron tomography (ET) is frequently used for understanding these relationships,2 since it can provide intuitive real-space representations of three-dimensional (3D) sample morphologies.3,4 In electron tomography, the 3D morphology of a specimen is reconstructed from a series of projections which are acquired in a (scanning) transmission electron microscope ((S)TEM) by tilting the specimen over a large angular range at small tilt increments.5,6 Even though electron tomography has been developed for decades, it is still a challenge to obtain high-quality reconstructions for quantitative measurements at the nanoscale.3 This is on account of several limitations which will be discussed below. In the ideal case, the tilt series should cover an angular range of ±90° at tilt increments as small as experimentally possible. However, this is usually not feasible due to the limitation caused by the sample holder, the sample stage,7 and more importantly the beam sensitivity of the sample.8 Shadowing of the area of interest by the edges of the sample holder, mechanical limits of the sample stage, or the common slab geometry prevent tilting to high angles. Thus the limited tilt range creates a missing angular range of information referred to as a “missing wedge”. The missing wedge © 2013 American Chemical Society

significantly influences the quality of reconstruction as it causes elongation of structural features along the beam direction thus degrading resolution. One possibility to limit the effects of the missing wedge is by acquiring a second tilt series perpendicular to the first (“dual-axis tomography”).9,10 In dual-axis tomography the missing wedge becomes a “missing pyramid”, and therefore the reconstruction quality is improved.11 However, even in dual-axis tomography, the problem of missing information still persists. Additionally, due to the slab geometry of most polymer samples, tilting to high angles is impossible as the effective thickness of the sample, i.e., electron beam path length, increases with the inverse of the cosine of the tilt angle, e.g., by a factor of 3 at 70°. This problem can be alleviated using a needle-shaped specimen which can nowadays be prepared by focused ion beam.12,13 Such samples also need to be mounted to a special specimen holder that allows tilting over ±90°, thus completely eliminating the missing wedge. However, this approach is very time-consuming and, henceforth, rarely applied. Moreover, it is only possible for materials that are mechanically strong enough for making needle-shaped specimens, a requirement which many polymeric systems do not fulfill. Furthermore, the ideal reconstruction requires that all projections are aligned with respect to a common coordinate Received: September 13, 2013 Revised: December 12, 2013 Published: December 19, 2013 1248

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Figure 1. Schematic diagram of the simulation approach. A model system is defined first. The subsequent image formation process has three stages: calculating ideal projections from electron−specimen interaction, blurring these projections by the optical system used as described by the point spread function (PSF), and adding noise to the blurred projections as introduced by electron source (shot noise) and by CCD read-out noise. Thereafter, reconstruction is done using the projections obtained with various acquisition schemes. Finally, based on the model system, a comparison between model and reconstructed system is made.

system.4 In the experiment, local shifts from one projection to another are hardly avoidable.14 Therefore, accurate alignment of the tilt series should be carried out before reconstruction. The most widely used alignment methods are based on pattern matching by cross-correlation or on tracking fiducial markers.4,14,15 The cross-correlation method aligns the tilt series by calculating the relative shift between the neighboring projections.16 The advantage of the cross-correlation method is that it can provide moderate accuracy for thin samples with a fast and straightforward implementation.4,16 In the fiducial marker tracking method, the tilt series is aligned by determining the relative shifts and rotations between images according to the measured coordinates of fiducial markers through the entire tilt series.17 The advantage of fiducial marker tracking method is that it not only corrects for the translational shifts and tilt axis direction, but also for image rotations and magnification changes. Nevertheless, small misalignments on the order of 1− 2 pixels (residual error mean for fiducial marker method) are inevitable, further degrading the resolution in experiments.15,17 The probably most important challenge influencing the quality of reconstructions of polymeric systems arises from low contrast, i.e., low signal-to-noise ratio (SNR) in the projections, and the beam sensitivity of specimens.8 This is because polymer materials mainly consist of light elements with only small differences in density or composition. Moreover, polymers are extremely sensitive to electron irradiation and can be quickly degraded by the cumulated electron dose as compared to inorganic materials.18−20 Often it is observed that polymers will shrink during acquisition of the tilt series.21 The shrinkage

effect will lead to changes in the sample geometry which can be partially modeled and corrected for, but still adversely affect the reconstruction quality. In general, radiation damage, and thus, shrinkage, can be significantly reduced by applying electron microscopy at low temperatures and using a low electron dose, which is our preferred strategy. Alternatively, biological thin sections are often preradiated until shrinkage ceases and then imaged at higher electron doses to optimize contrast and SNR. As radiation damage limits the total dose that can be utilized for electron tomography, it is fair to say that the challenge in ET of polymers and nanocomposites is primarily dose related. Hence, the main question arises of how to distribute the available dose most efficiently for recording a tilt series to obtain the best possible reconstruction. While we focus in this article on the effects of dose and tilt scheme, extensions to include sample thickness, defocus, or aperture sizes can be envisioned. Considering insufficient tilt range, finite tilt increments, and limited total dose, a further understanding of the influence of the acquisition scheme on the reconstruction quality is imperative. There are several studies that have tried to address this problem experimentally.12,22 However, such studies lack the ground truth of the samples at hand and are therefore limited. An alternative way to study the interplay between acquisition scheme, dose, and imaging parameters, as proposed in this work, is via the use of TEM image simulation. We simulate the entire bright-field electron tomography workflow of a model system from projections of a tilt series to reconstruction, ending with a qualitative and quantitative comparison between the initial model and the reconstruction. 1249

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Series of projections based on different acquisition schemes and different total number of impinging electrons are generated, including all relevant noise contributions, and subsequently reconstructed via the conventional reconstruction technique. The fidelity of the different reconstructions on dependence of the acquisition parameters is then assessed by image quality, SNR, and edge intensity profiles. More specifically, edge intensity profiles which characterize the transition between different materials are analyzed locally and globally by fitting an edge spread function to provide a direct link to the attainable resolution.

Table 1. Parameters Used To Calculate Projections symbol

value

unit

E D Cs f Δz σ

300 70 3 2.89 −5 0.17

keV μm mm mm μm e/pixel

transferred image is a convolution of the intensity of the ideal projection with, in first approximation, a Gaussian PSF which is given by h(x, y) = (2πσb2)−1 exp(−(x2 + y2)/2σb2). The standard deviation σb is the blurring factor and defines the maximum resolving power (resolution). To determine σb, we take two factors into account. The first factor is the diffractionlimited resolution σr. Given the focal length f, the wavelength λ, and the diameter of the objective aperture D, σr can be calculated as σr = 1.22 fλ/D, according to Rayleigh’s criterion.30 The spherical aberration of the objective lens and the applied defocus introduce the dominant blurring factor σc. This factor is determined as the inverse of the spatial frequency at which the first zero-crossing of the contrast transfer function (CTF) occurs.31 This is because the information beyond this spatial frequency cannot be directly interpreted. As a result, the total blurring factor σb can be treated approximately as σb2 = σr2 + σc2. Additional details on the CTF can be found in the Supporting Information and Figure S2. The effects of chromatic blur are commonly minimized experimentally by zero-loss energy filtering and are, therefore, not taken into account. Finally, the blurred projection is recorded by the electron detector, i.e., a charge-coupled device (CCD) camera. This recording process introduces two types of noises in the final image, namely, shot noise and CCD noise.32−34 Shot noise is generated in the conversion process from electrons into photons in the scintillator and is following Poisson statistics with the number of the incoming electrons as the variance. CCD noise is introduced during the readout process and mainly consists of noise from the dark current and readout noise. This noise can be described by a Gaussian distribution with zero mean. We determined the standard deviation of this Gaussian distribution from a dark image (acquired without illuminating the CCD detector) of a Gatan MSC 794 camera as installed at the TU Eindhoven CryoTitan. In the final step of image simulations shot noise and CCD noise are added to the blurred image. 2.3. Acquisition Schemes. All the TEM parameters for calculating the bright-field projections have been taken from typical experimental conditions as used with the TU Eindhoven CryoTitan (Table 1). As mentioned above, polymer specimens are highly sensitive to electron irradiation and can be easily damaged by cumulative electron exposure.18,20 Considering the beam sensitivity, we employ three different total electron doses Itotal, which roughly correspond to electron dose conventionally applied for cryo-tomography of vitrified samples in their native liquid environment (highly beam sensitive, Itotal = 102 e Å−2), polymers and composites thereof (beam sensitive, Itotal = 104 e Å−2), and inorganic materials (least beam sensitive, Itotal = 106 e Å−2). The total electron dose is evenly divided into N portions according to the number of projections (N) in the tilt series. Thus, the incident electron dose for an individual projection is determined by I0 = Itotal/N. To include the most common experimental maximum tilt ranges, we cover in our simulations

2. THEORETICAL METHODS AND SIMULATION The simulation approach, shown in Figure 1, consists of four steps: model definition, image formation and recording, and reconstruction followed by comparison. In the following each step of the ET workflow will be discussed in more detail. 2.1. Model Definition. The chosen model system tries to resemble actual examples of functional polymer nanocomposites. For instance, phase separation, crystallinity, and connectivity of P3HT nanorods in a P3HT/PCBM bulk heterojunction influence the power conversion efficiency of the corresponding polymer solar cells.23,24 Analogously, distribution and connectivity of carbon nanotubes in nonconducting polymer matrices impacts the electrical properties (i.e., percolation) of the system.25,26 Henceforth, our model consists of a polymer matrix (background) and rodlike nanofillers (objects), as shown in Figure 1 and Supporting Information Figure S1. The rodlike filler particles such as P3HT nanowire or CNTs are approximated by cylinders with spherical end-caps, having a 200 pixels length and 20 pixels diameter with a sampling of 1 nm/pixel. The thickness of the matrix is 200 pixels. The applied sampling of 1 nm/pixel in the model bridges the realms of medium resolution cryo-electron tomography to high-resolution 3D imaging and is used in much of our experimental work. 2.2. Image Formation. Image formation in bright-field electron microscopy can be divided into three steps with (1) electron−specimen interaction, (2) the optical system, and (3) the detector.27,28 As an incoming electron beam I0 passes through a specimen with thickness t, interactions between them occur. Thereafter, an image is formed from the transmitted electrons It. Since these images of the 3D specimen have no blurring and noise, they are referred to as ideal projections. We modeled the electron−specimen interaction by mass−thickness contrast.28,29 The intensity of the transmitted electron beam It is therefore calculated by the equation It = I0 exp( −t /Λt )

parameter acceleration energy objective aperture diameter spherical aberration focal length under defocus CCD noise

(1)

where Λt is the mean-free path length. In our case of a 200 nm polymer thin film, the effective beam path even at high tilts of 75° (∼770 nm) is still significantly smaller than the mean-free path, e.g., 1087 nm in P3HT imaged at 300 kV;29 thus, one operates in the single scattering regime. Subsequently, the two-dimensional (2D) ideal projection is transferred by the optical system of the TEM into the detector plane, which blurs the projection depending on the specifications of the instrument and the applied defocus (see Table 1 for details). For simulation purposes, the electron optical system is commonly approximated by a thin objective lens including an aperture in its backfocal plane which can be modeled as a convolution operator with the point spread function (PSF) h(x, y) of the optical system.27 In other words, the intensity of the 20,28,29

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a maximum tilt range of ±50° to ±80° at 5° intervals. Furthermore, we evaluate sampling schemes of Δθ0, 1°, 2°, and 5° for constant tilt increments, and two Saxton schemes,35 with 3° and 5° as starting increments at zero degree tilt angle. For clarity, the next tilt angle θi is calculated by θi = θi−1 + Δθ0 in constant increment tilt schemes and by θi = θi−1 + Δθ0 cos θi−1 in Saxton schemes. The entire acquisition parameter space is shown in Table 2. By combining with the total electron dose, the tilt range, and the tilt increments, we finally obtain a total of 105 simulated tilt series. Table 2. Parameters for 105 Acquisition Schemes (Tilt Series) parameter

value −2

total electron dose (e Å ) tilt range tilt increment

2

10 , 104, 106 ±50°, ±55°, ±60°, ±65°, ±70°, ±75°, ±80° constant increment: 1°,2°, 5° Saxton scheme: 3°, 5°

2.4. Reconstruction. Reconstruction algorithms, like weighted back-projection (WBP) or iterative techniques, assume a linear relationship between the object and the projections.36,37 Therefore, the projections are “log-corrected” first according to eq 1. After this “linearization” of the projections, reconstruction is implemented by the WBP method. To reduce high-frequency noise, in the WBP a lowpass filter with cutoff frequency set to 0.3 cycles/pixel and a Gaussian roll-off with sigma set to 0.05 is used. It should be pointed out that the value of the cutoff frequency affects the spatial resolution in reconstruction. The raw reconstruction data need to be normalized with a scaling factor for intensities before comparisons. This scaling is intended to preserve initial intensities and to reduce artifacts introduced by incomplete sampling and influence of the slablike sample geometry. To find the scaling factor, a homogeneous object (density one) of which the dimensions are identical to the model system was used. By employing the same acquisition scheme, reconstructions of the homogeneous object are calculated. The scaling factor is calculated as the ratio of the mean intensity in the reconstruction and the mean intensity in the model. The final reconstruction intensities are calculated by multiplying the raw intensities with the scaling factor. In this way, the mean intensities of the reconstructions calculated from different acquisition schemes, but at the same total electron dose, are kept constant.

Figure 2. Effects of the total electron dose on the interpretability of the reconstructions using a tilt range of ±65° and an increment of 1°. (a) XY slice from the ideal model, where the Y direction is the tilt axis. (b) Corresponding slice from reconstruction at electron dose 102 e Å−2. One can find the reconstruction quality is greatly improved at higher electron doses (c) 104 e Å−2 and (d) 106 e Å−2. Scale bars are 100 nm. For visualization the images are shown in the same dynamic range.

3. RESULTS 3.1. Reconstruction Results. The fidelity of the reconstructions is first compared on the basis of arbitrary cross sections through the volume as seen in Figures 2 and 3. The effects of the three electron doses are illustrated in Figure 2. As expected, increasing the total electron dose while keeping the same acquisition scheme for the tilt series can greatly improve the interpretability of the reconstruction. Comparing this cross section with the original XY slice from the model system (Figure 2a), we found it is almost impossible to distinguish any features in the corresponding slice reconstructed using a 102 e Å−2 dose (Figure 2b). The interpretability of the reconstruction is greatly improved by increasing the electron dose to 104 e Å−2 and to 106 e Å−2, as displayed in Figure 2c and Figure 2d, respectively.

Figure 3. Effects of tilt increments on the quality of reconstruction shown on XZ slices using a tilt range of ±65° and a 104 e Å−2 electron dose for three different increments. (a) XZ slice from the ideal model, where the Z direction is the beam axis. The corresponding slice from reconstruction using the tilt increments of constant 1° with 131 projections (b), Saxton 3° with 57 projections (c), and constant 5° with 27 projections (d). Scale bars are 100 nm. For visualization the images are shown in the same dynamic range.

Figure 3 shows the influence of the tilt increments on the reconstruction. It is surprising to see that there is only a minor difference between the reconstructed systems using the five different tilt increments. From Figure 3, we can clearly observe elongation effects along the beam direction Z, which may result 1251

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with increasing maximum tilt range. Moreover, Figure 4 shows that for ideal tilt series the tilt increments of S5° and 2° or S3° and 1° produce similar SNR, illustrating the benefit of the Saxton schemes for slab specimens.35 In case of noisy projections even in the case of 106 e Å−2 the same trend, however, at much reduced magnitude, is observed. The influence of the tilt increments on the SNRs in the reconstruction is massively reduced as the electron dose decreases to 104 and 102 e Å−2, where only minor variations between SNRs are observed. Notably, the SNRs for all reconstructions, even for the ideal reconstructions, does not meet the generally accepted Rose criterion,38−40 i.e., SNR ≥ 5, which means that distinguishing object features at the (1−3 × 10−7) significance level is not directly possible due to noise arising from the sampling scheme and WBP reconstruction algorithm. 3.3. Global and Local Edge Intensity Profiles. Next, we quantify the resolution of the reconstructed system by comparing with the model system. Several approaches have been explored to describe the resolution in electron tomograms, but no straightforward method exists.41−45 For a single-tilt axis scheme with an ideal tilt range ±90°, Crowther et al.41 proposed that the resolution dx in the X direction, perpendicular to the tilt axis (Y direction) and the optical axis (Z direction), is dependent on the number of projections N and the diameter of the object D and can be written as dx = πD/N. For a limited tilt range, an elongation is introduced in the reconstructed system along the Z direction due to the “missing wedge”. The elongation factor exz has been estimated by Radermacher et al. from the maximum tilt angle α and the resolution dz in the Z direction and is given by42

in artificial connectivity between objects. A comparison of reconstruction slices obtained from different maximum tilt ranges is provided in the Supporting Information, Figure S3. It should be pointed out that long rods oriented perpendicular to the beam direction almost disappeared after reconstruction, even with a higher tilt range. As a result, it can lead to misinterpretation or erroneous quantification. To minimize this artifact, one should choose the tilt axis in the direction in which the specimen has as few as possible objects perpendicular to the tilt axis. 3.2. Signal-to-Noise Ratio (SNR). While above qualitative conclusions give a good impression of the overall tendencies, a quantitative comparison of model system and reconstructed system is mandatory. First, we quantified the overall contrast in the reconstruction, based on the signal-to-noise ratio (SNR), which is determined by μ − μ0 SNR = 1 σ0 (2) where μ1 and μ0 are the mean density of the objects and the background, respectively, and σ0 is the standard deviation of the background. We plot the SNRs as a function of the tilt range and the tilt increment, as shown in Figure 4. For comparison, the SNRs of

dz = dxexz = dx

α + sin α cos α α − sin α cos α

(3)

However, above definitions for resolution are based on sampling considerations alone and do not take noise and blurring into account. On the basis of the simulations, we are able to evaluate the resolution in a much more direct way by actually comparing the initial model with the reconstruction. The most direct assessment of imaging, recording, and processing effects is encoded in the transition from one phase to another, i.e., from polymer matrix to the nanofiller. We take the edge intensity profile across the background-object edge with an equal number of pixels on each side, which is 20 pixels long and 1 pixel wide (Figure 5). In the initial model, the edge intensity profile is binary and therefore a step function, while the corresponding edge profile from the reconstruction is degraded leading to a sigmoidal profile. Characterizing such edge intensity profiles will provide information on the intensity range a, the background intensity b, the edge position x0, and the edge steepness k. Therefore using an edge spread function (ESF) containing these four parameters is a quantitative means for comparing structures present in the model and in the reconstruction. We employ one of the simplest continuous sigmoidal functions,46 namely, the hyperbolic tangent function, for the ESF: 1 S(x) = a[1 + tanh(k(x − x0))] + b (4) 2 To determine the four unknown parameters, we fitted the function to the edge profile. The resolution is defined by the full-width-at-half-maximum (fwhm) of the first derivative of the edge intensity profile, which is given by fwhm = 1.7626/k. In

Figure 4. SNR showing the overall quality of reconstruction with different acquisition schemes at three different total electron doses, low 102 e Å−2 (blue area), medium 104 e Å−2 (red area), and high 106 e Å−2 (gray area). For comparison, the SNRs of the reconstruction from ideal projections (without blurring and noise) are also calculated (green). S3° and S5° represent the Saxton increments 3° and 5°, respectively. Using the same tilt range, the number of projections is generally increasing from increments 5° to 1°. Using the tilt increments of Saxton 3° generates a larger number of projections than using the constant tilt increments of 2° at the tilt range ±75° (77 by S3° and 76 by 2°) and at the tilt range ±80° (91 by S3° and 81 by 2°).

the reconstruction from the ideal tilt series acquired with the same acquisition schemes but without blurring and noise are also calculated, as shown in Figure 4. The number of projections N is generally increasing from 5°, S5°, S3° to 2° and 1°. At the tilt range ±65°, for example, N is 27, 33, 57, 66, and 131, respectively. We found that the SNRs of reconstructed system at 102 e Å−2 are around 0.13, while the values reach ∼3.5 at electron dose 106 e Å−2. Expectedly, the SNR increases 1252

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tilt range of ±75°, regardless of electron dose and tilt increment. In other words, an isotropic resolution in the reconstruction is achieved by using a tilt range of ±75°, which has advantages for subsequent quantification. Surprisingly, the results show that the steepness and the edge position of the global edge profiles are hardly dependent on the tilt increments, i.e., number of projections, which is counterintuitive to the sampling considerations as expressed by the Crowther criterion.41 It is also worth to point out that the original dynamic range a = 1 and background b = 0 cannot be reproduced (see also Supporting Information Figure S4), although a careful normalization procedure was used. 3.4. Effects on Local Edge Profiles. The global edge profiles are virtually noise-free and therefore can only reflect the overall quality of the reconstruction. In contrast, local edge profiles will also contain dose-related properties. In this section, we focus on the analysis of local edge profiles based on the edge spread function. For the total electron doses of 104 and 106 e Å−2, the individual local edge profiles are 1 pixel wide. However, for the total electron dose of 102 e Å−2, the reconstruction is too noisy and the 1 pixel wide edge profiles are too chaotic. Therefore each local edge profile is calculated by averaging over a 5 × 5 pixels plane perpendicular to the direction of the edge profiles. Nevertheless, the intensity difference between background and object is in some of the local edge profiles imperceptible. As a result, many profiles cannot be fitted properly and should be disregarded. The classification procedure that was used to distinguish between suitable and nonsuitable profiles for ESF fitting is schematically represented in Figure 7. In the first step we employ the t test at a 0.05 significance level to examine all N profiles whether a significant difference between mean density of background μ0 and object μ1 in an individual edge profile exists, i.e., whether μ0 < μ1. If the difference is significant, the profile will be fitted to the edge spread function to find the four parameters; otherwise, it will be considered as chaotic and will be disregarded. The fitting procedure for the remaining N1 profiles employs the Levenberg−Marquardt algorithm which is an iterative technique.47−49 In most cases proper values for four parameters are obtained after 20−30 iterations. We therefore set the maximum number of iterations to 100 and considered all solutions that would be found after 100 iterations as unstable fittings and removed them, leaving over N2 profiles. One should bear in mind that the number of accepted edge profiles is determined by the choice of the significance level and the number of the maximum iterations. Finally, the fitting results are screened with respect to edge position and steepness. Due to a cutoff frequency at 0.3 cycles/ pixel used in the WBP algorithm, the maximum resolution of reconstruction will be approximately 3 pixels. Therefore, to make sure the edge profile always includes background and object, the edge position should be in the range (3, 18), with respect to the 20 pixels length of the edge profile. Any solution of the edge position not in this range will be also disregarded, leaving over N3 profiles Furthermore, the fwhm of the first derivative of the edge profile should be below 20 pixels because of the 20 pixels length of the profile. As a result, the minimum steepness, as defined by k = 1.7627/fwhm, is k1 = 1.7627/20 ≅ 0.09. On the other hand, because the maximum resolution is 3 pixels, the maximum steepness is determined via same equation as k1 = 1.7627/3 ≅ 0.6. Consequently, we divide the fitted results into three groups

Figure 5. Illustration of the edge intensity profile from the model system (top), which is taken across the background-object interface from the slice showing a plot of the edge profile from the model (blue), a sample of a local edge profile from the reconstructed system (dark triangles), and its fitted curve (dark dash). Also shown in this graph is the globally averaged edge profile (red).

other words, the steepness k is the inverse of the resolution. It should be mentioned that the original edge position is at x0 = 10.5 pixel position in this work. To quantify the quality of the reconstruction, i.e., the resolution, we investigated the edge intensity profiles globally and locally. Due to the missing wedge, it is clear that the edge profiles in the reconstruction are inevitably anisotropic, which means that the profiles should be characterized separately for the X, Y, and Z directions. First, we focus on the global edge profile (GEP). For each reconstruction, the GEPs in the X, Y, and Z directions are obtained by averaging all the local edge intensity profiles, the total number of which is about 3 × 106, for each direction and fitted these GEPs to the ESF given by eq 4. The results of the steepness k and the edge position x0 are shown in Figure 6. It can be seen that the influence of the total electron dose on the GEPs is negligible. The reason is that, although the total electron dose dominates the SNR in the tilt series and therefore in the reconstruction, the procedure of obtaining GEPs by averaging over approximately 3-million edge profiles results in virtually noise-free GEPs. As expected, the tilt range is the main factor influencing the resolution. The steepness k is increasing in the Z direction with increasing tilt range, slightly decreasing in the X direction, and remains almost constant in the Y direction. The edge position approaches the actual position x0 = 10.5 with increasing tilt range. From Figure 6, we can see that edge position is shifted by 2 pixels in the Z direction at the tilt range of ±50° and reaches its actual position at the tilt range of ±80°. This is a particularly important effect as it will influence volume quantification significantly. For instance, with a 1 nm edge shift along the direction Z for reconstruction using a tilt range of ±60° and 1° increment, the volume of a spherical particle with a 20 nm diameter will increase in size by 10%. Moreover, we observe that the steepness becomes the same for the X, Y, and Z directions at a 1253

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Figure 6. Steepness k (a, c, e) and edge position x0 (b, d, f) of global edge intensity profiles along X (blue), Y (gray), and Z (red) directions for various total electron doses and the acquisition schemes. Changing the total electron dose from 102 e Å−2 (a, b) via 104 e Å−2 (c, d) to 106 e Å−2 (e, f) does not alter the steepness and the edge position globally. As expected, the steepness (resolution) of the GEP improves with increasing tilt range. Surprisingly, there is only minor difference between the tilt increments used.

with respect to steepness. The first group is the group of the edge profiles for which the steepness is less than k1. In this group, the line profiles have no distinguishable transition between the background and the object, and therefore they will not be discussed further. The second group is the group of the edge profiles for which the steepness belongs to the range [k1, k2]. For values in this range, different values for the steepness k have a physical meaning. The rest of the edge profiles, for which k > k2 and hence different values for k cannot be distinguished, are put into group III. The remaining three parameters are still interpretable and can be analyzed. Finally, we calculate the probability densities distribution for each parameter in groups II and III.

The probability density distributions of the steepness and the edge shift along the beam direction Z in group II are shown in Figure 8 for a tilt range of ±75°, and along the direction X and the tilt axis Y shown in Supporting Information Figure S5. Instead of using the edge position x0, in Figure 8b, d, f, we plot the probability density distribution as a function of the edge shift, i.e., Δx0 = x0 − 10.5, to indicate the number and direction of pixels shifted. A negative edge shift means that an object along the edge profile is elongated to the background side which results in a volume expansion, while a positive shift means that the length of the object is decreased leading to volume shrinkage. The influence of the total electron dose can 1254

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be clearly observed from Figure 8, an effect which is clearly not present in the global edge profiles. At a dose of 106 e Å−2 (Figure 8e), the distribution of steepness values is a quasi-Gaussian distribution whose mean is close to the steepness of the corresponding global edge profile kg. As the electron dose decreases, the distribution of k widens and skews toward lower values. The concomitant degradation of resolution is only slightly reflected in the curve mean; e.g., for ±75° tilt range and 1° increments, the mean of k changes from 0.22 (106 e Å−2) to 0.22 (104 e Å−2) to 0.20 (102 e Å−2). This dose effect becomes clearer when looking at the resolution in the Z direction calculated from the mean of k ± standard deviation of k (see paragraph 3.3 and eq 4), which broadens from 8.2 ± 3.0 nm (106 e Å−2) to 8.1 ± 4.5 nm (104 e Å−2) to 8.7 ± 4.6 nm (102 e Å−2). In comparison, at a tilt range of ±75° and 5° increments, the resolution in Z improves to 7.5 ± 3.2 nm (106 e Å−2) to 7.7 ± 4.5 nm (104 e Å−2) to 8.2 ± 4.6 nm (102 e Å−2). More importantly, the number of fitted line profiles in group II (Figure 7) decreases from ∼80% (106 e Å−2) to ∼20% (104 e Å−2) to only 8% (102 e Å−2) independent of tilt increments which renders resolution estimations under cryo-conditions (102 e Å−2) particularly unreliable. Moreover,

Figure 7. Procedure for analyzing the individual edge profiles, N in total. Only the N1 profiles fulfilling the t test are fitted. Of these, only N2 profiles remain which can be fitted properly with less than 100 iterations. The fitted edge profiles then are screened according the constraints of the edge position x0 ∈ (3, 18), resulting in N3 remaining profiles. In the end, with respect to the steepness k, we divide the fitted results into three groups. An example result is shown on the right.

Figure 8. Probability density distribution of the steepness k (a, c, e) and the edge shift Δx0 = x0 − 10.5 (b, d, f) at a total electron dose 102 e Å−2 (a, b), 104 e Å−2 (c, d), and 106 e Å−2 (e, f), along the beam direction Z with the tilt range ±75°, and five tilt increments: 1° (blue), 2°(red), 5°(magenta), and Saxton 3° (green) and 5° (black). 1255

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the reconstruction quality is the tilt range. The edge steepness k along the beam direction Z roughly doubles from ±50° to ±75° tilt range, thus doubling resolution to approximately 8 nm. For the studied slab geometry and instrumental conditions, isotropic resolution is realized at a tilt range of ±75°, regardless of the total electron dose or the tilt increments, a fact which can be profitably used for subsequent quantification. Also for quantification, the positions of edges are found to be quite ambiguous at doses of 104 and 102 e Å−2 with the initial position being achieved on average at a tilt range of ±75°. Most importantly, we find that tilt increments have only a minor influence, i.e., large tilt increments at low electron doses preserve or even improve the reconstruction quality, leading to counterintuitive acquisition schemes. Hence the number of projections can be significantly reduced without sacrificing reconstruction quality, which will be extremely beneficial for the analysis of beam sensitive materials. Our findings on doselimited imaging, contradict Crowther’s sampling-based resolution estimate41 and Radermacher’s elongation factor (resolution anisotropy between Z axis and X axis), which are commonly cited in the literature.42 The presented approach illustrates that the problem of resolution in electron tomography of beam sensitive specimen can and needs to be revisited to substantiate interpretation and quantification of all kinds of 3D nanostructured materials such as organic solar cells or polymer composites.

only small effects of the tilt range on k and, thus, the resolution are observed, counterintuitive to eq 3.42 Finally, also the elongation factor given by the ratio of the resolution in different directions, e.g., exz = 1.06 (106 e Å−2) or exz = 0.96 (102 e Å−2) for ±65° tilt range and 1° increments, is smaller than predicted from eq 3, i.e., exz = 1.42. On the other hand, the distribution of edge shift values at a dose of 106 e Å−2 (Figure 8f) is also a quasi-Gaussian distribution of which the mean is close to the corresponding global edge shift. As the electron dose decreases, the distribution of the edge shift Δx0 is spreading wider around the mean, which indicates that the edge position becomes more uncertain, leading to an extra challenge for connectivity studies. In addition, we note that the probability distribution of the edge shift, especially along the beam direction Z, becomes narrower with increasing tilt range (106 e Å−2). For example, at 1° tilt increments, the standard deviation of the edge shift is 2.45 nm for a tilt range of ±65° (Supporting Information Figure S6) and decreases to 2.05 nm for a tilt range of ±75° (Figure 8f). If the total electron dose decreases to 104 and 102 e Å−2, the distribution of the edge shift Δx0 widens significantly and slightly skews to the background side. The increased uncertainty in edge position leads not only to challenges for connectivity studies but also strongly effects volume quantification. For instance, at a tilt range of ±65° and 1° tilt increments (102 e Å−2), the edge position (Supporting Information Figure S6) is negatively shifted to 0.4 nm with a standard deviation of 4.0 nm, which results in quantification errors for a sphere of 20 nm in diameter by −36% to 43% in volume. A study on connectivity via gap intensity profiles is provided in the Supporting Information. While gaps of 6 nm and above are resolvable in the X direction, in the Z direction parallel to the electron objects separated by less than 9 nm will appear connected. Furthermore, the influence of the tilt increments, i.e., the number of projections, on steepness and edge position is mainly noticed along the Z direction at the electron dose 106 e Å−2, and at a high tilt range, e.g., ± 75°. As shown in Figure 8f, the distribution of the edge position is slightly wider using 5° tilt increment than using others. But the tilt increments generally have limited influence on the reconstruction quality. This suggests that, instead of using small tilt increments, one can preserve the reconstruction quality with large tilt steps, thus significantly reducing the number of projections, e.g., number of projections reduced by a factor 5 from 1° to 5°, an effect obviously beneficial for the study of beam sensitive materials. For group III edge profiles, we observed similar results for the probability distributions of edge shifts as for group II, and the corresponding graphs are displayed in Supporting Information Figures S8 and S9.

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ASSOCIATED CONTENT

S Supporting Information *

Full model system with a slab geometry; full description of point spread function; additional results of the effects of the tilt ranges, dynamic range a, and background b of global edge profiles along the X, Y, and Z directions; probability density distributions of the steepness and the edge shift from group II, probability density distribution of the edge shift from group III. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +31-40-247-3041. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank prof. M. Dijkstra and Dr. R. Ni (Utrecht University, The Netherlands) for providing the random packing code and Maarten Wirix for characterization of CCD noise. This research forms part of the research program of the Dutch Polymer Institute (DPI), projects #615.

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4. CONCLUSIONS We have simulated the entire electron tomography process to investigate the influence of beam sensitivity and acquisition scheme on the quality of reconstructions for polymer specimens with a slab geometry. Our results show significant influence of the electron dose with the SNR in reconstructions being below the Rose criterion, i.e., SNR < 5, even if obtained at the maximum dose of 106 e Å−2 which significantly exceeds the allowed dose for polymer specimens. The influence of dose, maximum tilt range, and tilt increment were quantified from edge intensity profiles, i.e., a global and local edge spread function. At constant electron dose, the main factor influencing

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