Quasi‐Newtonian Environmental Scanning Electron Microscopy (QN‐ESEM) for Monitoring Material Dynamics in High‐Pressure Gaseous Environments

Abstract Environmental scanning electron microscopy (ESEM) is a powerful technique that enables imaging of diverse specimens (e.g., biomaterials, chemical materials, nanomaterials) in a hydrated or native state while simultaneously maintaining micro‐to‐nanoscale resolution. However, it is difficult to achieve high signal‐to‐noise and artifact‐free secondary electron images in a high‐pressure gaseous environment due to the intensive electron‐gas collisions. In addition, nanotextured substrates can mask the signal from a weakly scattering sample. These drawbacks limit the study of material dynamics under extreme conditions and correspondingly our understanding in many fields. In this work, an imaging framework called Quasi‐Newtonian ESEM is proposed, which introduces the concepts of quasi‐force and quasi‐work by referencing the scattering force in light–matter interactions, to break these barriers without any hardware changes. It is shown that quasi‐force is a more fundamental quantity that has a more significant connection with the sample morphology than intensity in the strongly scattering regime. Experimental and theoretical studies on the dynamics of droplet condensation in a high‐pressure environment (up to 2500 Pa) successfully demonstrate the effectiveness and robustness of the framework and that the overwhelmed signal of interest in ESEM images can be reconstructed through information stored in the time domain, i.e., frames captured at different moments.


The transverse mechanical force in a 2D imaging system
The field equation of a paraxial beam propagating along the z-direction satisfies where   u r and    r are the spatial-variant transverse amplitude and phase of the field   E r .   where n b , μ 0 , and ν denote the background index, the permeability of vacuum, and the speed of wave in vacuum. Here n b and μ 0 are constants that are meaningful in optics; however, we show in the following that these constants can be eliminated to provide an analogy to the electron wave function. For a non-magnetic small object positioned in the paraxial beam, the momentum of the wave will result in the movement of the object, indicating the existence of mechanical force and work (provided the object is freely moveable). In optics, the mechanical force primarily consists of the scattering force because the trapping force is negligible due to the adoption of paraxial illumination. Therefore, the transverse component of the mechanical 3 force exerteded on the object arises from the transverse momentum flux of wave.
Mathematically, the transverse force has a concise form, i.e., where σ is the extinction cross-section of the small object. Note that for a given illumination, σ is a constant; thus, F only depends spatially on E and  .

Measuring the mechanical force and work in a 2D imaging system
The field equation of a paraxial beam propagating along the z-direction satisfies the paraxial  Figure S1. Schematic showing the dynamic changing of best focal plane z 0 as a condensed droplet grows from second t 0 -Δt to t 0 +Δt. The distance between the tip of the focused e-beam and the substrate, known as the position of the preset focal plane, is a constant during the horizontal scanning, leading to an automatic depth scanning along the optical axis to collect both the in-focus and out-of-focus information. [1] 5 Applying Equation S7 to an SEM is non-trivial because the electron-matter-interaction induced by a high-energy electron beam is nonlinear and polychromatic (i.e., the backscattered and secondary electrons correspond to longer de Broglie wavelengths compared to the excitation source due to energy loss). However, we can still apply Equation S7 to reconstruct   w r and   fr with the knowledge that no ideally monochromatic source exists in practice. Therefore, the reconstructed   w r and   fr are "polychromatic quantities."

Features of quasi-force and quasi-work reconstruction in scanning electron microscopy
For a dynamic object, the corresponding irradiance for the condensed droplet, the center (best focal plane) of a droplet rises due to the dynamic growth during condensation, leading to a "sample-induced" variation in the irradiance along the longitudinal direction. Therefore, where S = Q * TQ is diagonal, K = Q * WQ, and H = Q * GQ. Note that T, Q, and G are known matrices. The elements of K can be computed directly by after which the unknown normalized quasi-work matrix W is obtained via QKQ * . The  Figure S2. The a) measured intensity map, b) normalized quasi-work map, and c) phase map.

Operator properties in QN-SEM and the comparison with phase reconstruction
There is a discontinuous line in the intensity map (marked by a pair of arrows in a)) due to an instrument error. The condensing experiment was implemented at 1500 Pa on a silicon substrate.
The gradient and Laplacian operators in Equation S7 are local ones. By local, we mean that applying the operator does not enable a local feature to have significant impact on the other areas of the image. This indicates that if there are "abnormal pixels" in an SEM image such as those caused by system instability or measurement error (for example, a zero-intensity pixel), the local abnormality will not propagate to the other areas in the image when reconstructing normalized quasi-work and quasi-force. However, in the matrix-based solver (Equations S8 and S9), the introduction of the global operator matrices such as Q and Q * may alter the local properties and result in a spread of a local abnormality. The application of regularization techniques suppresses this phenomenon and maintains the high-contrast in the normalized quasi-work and quasi-force maps. However, in the phase-based SEM imaging framework, 1 the abnormality of the problem is further deteriorated because of dividing   w  r by the raw irradiance   2 E r (see more details in ref. [4]). To alleviate this abnormality, we can, for instance, add a nonzero value to the "abnormal pixels" or to all the pixels. But because the levels of systematic errors are difficult to know, determining a reasonable correction value requires a trial-and-error effort in practice (see Figure 2g). This issue in phase reconstruction in turn reflects the fact that quasi-work and quasi-force reconstruction is more robust to perturbations in the SEM imaging process than phase reconstruction.
To further demonstrate the advantage of the proposed QN-SEM over phase-based ESEM, we consider another case of "abnormal pixels" where a discontinuous line exists on several frames of a video when recording the growth dynamics of condensed droplets in a conventional ESEM. This phenomenon is commonly seen in ESEM due to the mismatch between the recording rate of the detector and the scanning frequency of the electron beams.
As shown in Figure S2a, there is a blurry horizontal line (indicated by two arrows) in the intensity map captured at 1500 Pa. The reconstructed normalized quasi-work map, as shown in Figure S2b, not only successfully recovered the inclined illumination-induced shadow of each droplet, but also maintained the binocular vision with a high fidelity. However, using phase reconstruction, the images of droplets were severely distorted-there were no more shadows and the droplets are no longer spherical. See two representative droplets marked by "1" and "2" in the phase map and compare them with those in the quasi-work map. This result demonstrates again that the proposed framework outperforms the phase reconstruction method in resisting system errors. To further demonstrate the advantage of the proposed framework, we reconstructed quasiwork and quasi-force images from raw intensity images captured at 1000 Pa, 1500 Pa, 2000 Pa, and 2500 Pa, respectively. As depicted in Figure S3e-h and i-l, quasi-work and quasi-force imaging precisely recovered the size and morphology of each droplet, where the image contrast was significantly improved, and the background noise was successfully suppressed.
We next study the intensity and reconstructed quasi-work distribution (the white solid curves) along a line (see the white dotted lines) crossing the center of a droplet in Figure S3a-h.
Apparently, the intensity signal is comparable to the intensity of background noise, making it difficult to distinguish the droplets using the intensity distribution. In contrast, one can easily find a peak that corresponds to the droplet from the quasi-work distribution. To quantitatively analyze the enhancement, we define the image contrast as the ratio of the strength of the peak signal (I p ) from a droplet to the peak value [max(I b )] of background noise, i.e., According to Eq. (S10), a smaller Contrast indicates a lower image quality and implies that it is more difficult to find droplets. Figure S3m shows the contrast of the intensity and quasiwork as a function of pressure. The contrast of intensity always stayed near 1, indicating that the peak signal of the droplet was comparable to that of the background noise. Using the quasi-work reconstruction, the contrast was improved by a factor of five at 1000 Pa and decayed with pressure because of increased random electron scattering at higher pressure.
However, an enhancement of approximately two in contrast at 2500 Pa was still achieved, which is enough for us to retrieve the droplets from the quasi-work images. Note the contrast analysis was only performed on the raw intensity and quasi-work images. The quasi-force exhibits peaks at the boundaries of the droplet, which makes it difficult to define the contrast 10 in a way that can be directly compared with the contrast of the intensity and quasi-work images. Figure S4. Reconstructed quasi-work and quasi-force images under different dwell times. a-c) Quasi-work images and d-f) quasi-force images. The first, second, and third rows correspond to the reconstructed quasi-quantities with the dwell time set as 1 s, 2 s, and 3 s, respectively.

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The quasi-work and quasi-force distribution along a line crossing the center of a representative droplet is presented in g) and h), respectively.
To demonstrate the repeatability, we reconstruct the quasi-work and quasi-force for different dwell times. As presented in Figures S4a-c, the reconstructed quasi-work maps are visually indistinguishable. A quasi-work distribution along a line crossing the center of a representative droplet (see the dotted lines in Figure S4a-c) shows that the morphologies have only minor difference due to the different noise distribution and levels in the frames captured at different time. Through the quasi-force images, we can more clearly observe the difference induced by different dwell time, but the morphologies of droplets still agree well. A quasiforce distribution along the same line crossing the center of a representative droplet clearly validates that different dwell times do not change the morphologies of droplets, which means the quasi-force is repeatable as well. (see Figure S4h). The stronger fluctuations in the curves demonstrates that quasi-force is more sensitive to the environments than quasi-work. Here, we should mention that the dwell time should not be arbitrarily large; this is because the three adjacent frames (used to reconstruct the quasi-work and quasi-force) captured at a very large dwell time may contain entirely different droplets due to the significant growth and coalescence of droplets. Figure S5. Reconstructed quasi-work image for backscattering electron image. a) Raw ESEM image of condensing droplet using the backscattered electron detector, where the elastically backscattered electrons are dominant during the imaging process. The raw ESEM image has very low contrast due to the electron-gas scattering at high pressure (1000 Pa). b) Retrieved quasi work image using the developed approach where the image quality was significantly improved, and a few micron size droplets were well resolved.

Effectiveness of the proposed framework on backscattering electron images
Our imaging framework is relatively weakly dependent on the elastic and inelastic scattering conditions. To demonstrate this point, we further imaged the droplet condensation using the backscattering electron detector in ESEM ( Figure S5), where the elastic scattering is more significant because the backscattered electron has larger energy than the secondary electron [3,4]. Similar to the results observed using the secondary electron detector, the raw ESEM image ( Figure S5a) has very low contrast due to the intense electron-gas molecule scattering at high pressure (1000 Pa). However, the image quality can be significantly improved using the "quasi work" treatment as shown in Figure S5b, where a number of droplets that are only a few microns in size can be well resolved. Considering similar enhancement can be achieved using both the backscattered electron and the secondary scattered electron imaging, we believe the imaging technique developed in this work is useful and robust.

Captions for Supporting Information Movies 1 and 2
Figure caption of Movie 1. Comparison among intensity, quasi-work, and quasi-force maps.
The condensing experiment was implemented at 1300 Pa on a silicon substrate.

Figure caption of Movie 2.
Comparison among intensity, quasi-work, and quasi-force maps.
The condensing experiment was implemented at 2500 Pa on a silicon substrate.