E46 · Environmental electron microscopy

Environmental-cell TEM: Bubbles and Pressure

How do pressure and beam-driven bubbles change an environmental TEM image? Gas scatters electrons out of the selected image channel and can dim the image. A beam-produced bubble replaces liquid, shortening the liquid path and making that region brighter. Brightness therefore depends on what the electrons pass through. A clearer movie alone cannot establish that a reaction is less disturbed; dose and flow controls are needed.

See a beam-driven bubble replace liquid, then switch to gas and measure pressure from transmitted counts. Compare both with the retained liquid-reaction movie.

Count-derived radius, pressure or rate $\text{—}$

Difference from known input $\text{—}$

Mean counts per sampled pixel $\text{—}$

Model selected-channel transmission $\text{—}$

More measurements and target

Known input reference $\text{—}$

Beam to intrinsic hazard ratio $\text{—}$

Selected exposure midpoint $\text{—}$

Exposure duration $\text{—}$

Total incident areal dose $\text{—}$

Usable sampled frames $\text{—}$

Assumed radiolysis species · dimensionless $\text{—}$

Retained mean gas inventory · molecules $\text{—}$

Gas inventory removed by liquid renewal $\text{—}$

Numerical molecule-balance residual $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Medium: 0 reaction · 1 bubble · 2 gas

c

1

Applied gas pressure

P

80\,\mathrm{kPa}

Gas path between windows

L_g

1000\,\mathrm{nm}

Liquid path length

L

300\,\mathrm{nm}

Thickness of each window

w

50\,\mathrm{nm}

Effective liquid renewal rate

\nu

0.001\,\mathrm{s^{-1}}

Incident electron flux

J

500\,\mathrm{\frac{e^{-}}{nm^2\,s}}

Blank beam between exposures: 1 on

b

0

Advanced settings

Assigned gas yield per incident electron

y_g

0.002

Counting noise: 1 on

n_P

1

Exposure cadence

f

0.5\,\mathrm{s^{-1}}

Exposure fraction of each interval

d

0.5

Horizontal specimen drift

v_d

0.3\,\mathrm{\frac{nm}{s}}

Follow known drift: 1 on

a

1

Exposure-integrated fit: 1 on

m

1

Selected stored frame · zero-based

n

12

Learning goal and model limits

Choose the medium, record, then infer

How do pressure and beam-driven bubbles change an environmental TEM image?

Drag the gold handle to change beam flux or gas pressure. Bubble target: resolve at least four exposures, keep all 24 complete frames and recover the selected radius within 15 percent or 0.5 nm, whichever is larger. Pressure target: error at most 10 percent or 2 kPa and conditional counting uncertainty at most 2 kPa. Both require at least 8 counts per pixel; the reaction branch keeps its separate rate target below.

Model assumptions

The bubble branch is a pinned spherical hydrogen bubble in an 80 nm square, flat-window cell at 298.15 K and 100 kPa ambient pressure. Both illuminated area and liquid volume use this same square field. The assigned effective yield is adjustable from zero to 0.01 gas molecules per incident electron. It is not a radiochemical yield for 200 keV electrons in water. Dissolved molecules, bubble molecules, production and removal by liquid renewal are tracked separately. Surface tension is rounded to 0.072 newtons per metre; the dilute Henry coefficient is 0.0000078 mole per cubic metre per pascal, converted to molecules. A predetermined 8 nm heterogeneous nucleus is introduced only when the dissolved inventory can supply its gas and the corresponding equilibrium dissolved inventory. Gas is transferred from solution, not created at nucleation. The bubble uses ideal gas and Laplace pressure, and an assigned interface velocity of 0.0000001 metre per second. Ideal-gas and linear Henry relations are extrapolated to the high internal pressure of a small bubble; nonideality, water vapor, diffusion, coalescence and detailed water chemistry are omitted. These are continuum mean inventories, not individual molecular events. A sphere touching a window or reaching 35 nm radius stops the model before the invalid step; earlier full exposures and any shortened final exposure are retained with their actual durations. Bubble intensity follows the remaining liquid chord, with the same assigned attenuation lengths as the reaction branch. A constant exposure-equivalent radius is fitted from counted spatial profiles using known center, window and liquid thickness; non-detection remains distinct from zero dissolved gas. The 0.25 nm fit grid and deviance cuts are teaching diagnostics, not confidence coverage. The gas branch uses a known uniform pressure from zero to 100 kPa, ideal number density and an assigned selected-channel cross-section of one times ten to the minus twenty square metres. Pressure is inferred from acquired mean counts conditional on known gas path, cross-section and both windows. Negative apparent estimates from noise are retained. Uncertainty is conditional on the count calibration; thickness or cross-section uncertainty is omitted. Each normal run stores 24 finite exposures. The default image is already acquired; presentation motion does not advance physical time. The gas curve is a separate known calibration, with one measured transmission at the applied pressure. Apparatus axial distances are compressed separately from lateral nanometres: the displayed gap is 40 plus 40 times the square root of liquid thickness divided by 1500, or gas thickness divided by 15000. Bubble height uses the same axial factor as its cell, so the displayed ellipsoid represents a sphere under this mapping. Window display thickness is its physical thickness divided by ten. Representative molecules and electron trajectories are schematic; images and fits stay outside the apparatus viewport. The retained reaction-only branch uses the following separate model. The beam passes through two windows and the liquid between them, and the specimen sits on the inner face of the exit window. The effective selected-channel lengths are assumed: 200 nm per window and 350 nm for the liquid. They are not published inelastic mean free paths. Without the beam, the intrinsic conversion rate is 0.1 per second. The beam creates one lumped, dimensionless radiolysis species. It starts at zero, is produced at 0.002 times the electron flux, and is removed at 0.5 per second plus the flow-renewal setting; it raises the conversion rate by 0.12 times its concentration. These are teaching coefficients, not water chemistry or a calibration for a real material. Flow is an effective renewal rate, not a pump's volume flow or a fluid simulation. In this reaction-only branch, gas transport and bubbles are not coupled to the converting patch. Window bowing, pressure mechanics, spatial diffusion, radical networks and quantitative multiple scattering are omitted. Electrons that leave the selected channel are not all absorbed. The beam is fixed at 200 keV and travels along positive z, with x to the right and y up. The 80 nm square field has 64 by 64 pixels at 1.25 nm pitch, and each run stores 24 exposures. The selected-channel intensity is an assumed linear mix of two unresolved states with 0.45 contrast; it is not atomic phase contrast and does not identify a material. The converting patch has a uniform 14 nm radius and a 4 nm cosine edge. Each image sums 16 midpoint samples across the exposure, including the horizontal drift during that time. The detector records 80 percent of electrons, using fixed, independent random seeds: counts follow Poisson statistics below 40 expected counts and a normal approximation above. The display uses a fixed linear scale from zero to one of the incident level; it clips colors only, never counts. Conversion fractions come from a 6 nm core region, using known flat-field transmission and contrast calibrations. Following the known drift moves the sampling region with the reference motion; it does not estimate registration from the images. A frame whose field misses part of the region is marked unavailable. The rate fit sees only the counted fractions and the exposure timing, never the known specimen or the intrinsic rate. Noise can push fractions outside zero to one, and they stay in the fit. The fit finds one constant apparent rate between zero and five per second, so it cannot tell what caused the conversion. Changing the selected frame, the sampling alignment or the fit method reuses the stored counts. On-screen motion runs separately from acquisition time. Lens paths, thickness and magnification are schematic. There is no wave optics, diffraction, absorption spectroscopy, temperature gradient or damage calculation.

Physics tutorial

Environmental-cell TEM: Bubbles and Pressure

BackgroundTwo thin windows keep liquid or gas inside a TEM cell. Electrons can scatter out of the selected image channel in that medium. In liquid, the beam can also produce gas: a bubble replaces part of the liquid path and makes that part of the image brighter. The cell and the recorded counts must describe the same path.

Why it mattersIn the bubble example, infer the selected radius within 15 percent or 0.5 nm, whichever is larger, from at least four resolved exposures. Keep all 24 exposures complete and collect at least 8 mean counts per pixel. In gas, infer pressure within 10 percent or 2 kPa, with conditional counting uncertainty at most 2 kPa and at least 8 counts per pixel. The retained reaction preset has its separate rate target: 10 percent error, beam hazard at most 8 percent of intrinsic hazard, 20 usable frames, 35 percent intrinsic conversion and an exposure-integrated fit; intrinsic rate times frame interval must be at most 0.2.

Start with the essentials

Focus question: How do pressure and beam-driven bubbles change an environmental TEM image?
One-sentence intuition: A bright region can appear because a bubble has replaced liquid. It need not mean that the specimen became easier to observe without disturbing it.

Core mathematical model

Count the gas before creating a nucleus

\begin{aligned}\dot N_g&=y_gJ_{\rm on}A\\\dot N_d&=\dot N_g-\nu N_d-\dot N_b\end{aligned}

The dissolved inventory supplies the bubble and loses gas through liquid renewal. The predetermined nucleus appears only when that inventory can supply both the seed and its equilibrium dissolved gas. This rule is assigned; it does not calculate a nucleation barrier.

Bubble inventory and internal pressure

\begin{aligned}P_b&=P_{\rm ext}+2\gamma/R\\N_b&=\frac{P_b}{k_BT}\frac{4\pi R^3}{3}\end{aligned}

The bubble is a pinned sphere. Its gas follows an ideal equation of state; exchange uses a linear Henry relation. Both are teaching extrapolations at the large internal pressure of a small bubble. The model stops before the sphere touches a window or reaches the lateral bound.

A bubble shortens the liquid path

\begin{aligned}c(r)&=\begin{cases}2\sqrt{R^2-r^2}&r<R\\0&r\ge R\end{cases}\\\ell_l(r)&=L-c(r)\\\mathcal T(r)&=e^{-2w/\Lambda_w-\ell_l(r)/\Lambda_l}\end{aligned}

Only points inside the projected circle lose a liquid chord. Each frame averages these transmissions over its actual exposure. A spatial count fit estimates one exposure-equivalent radius from known center and thickness calibrations. An unresolved profile is not proof of zero dissolved gas.

Pressure from calibrated transmission

\begin{aligned}\widehat\tau_g&=-\ln(\overline N/N_{\rm win})\\\widehat P&=\frac{k_BT}{\sigma L_g}\widehat\tau_g\\N_{\rm win}&=\eta Jp^2\Delta t\,e^{-2w/\Lambda_w}\end{aligned}

The gas number density follows the ideal-gas law. Pressure requires an independently known gas path, cross-section and window transmission. The plotted pressure curve is that calibration; the measured point comes from counts at the applied pressure. Thickness uncertainty and nonideal gas effects are omitted.

Radiolysis products: production and renewal

\begin{aligned}\dot c&=\alpha J_{\rm on}-(\beta+\nu)c\\\dot H&=k_0+\gamma c\end{aligned}

The radiolysis products are lumped into one dimensionless species. The Lab does not solve water chemistry.

Conversion measured from counts

\begin{aligned}\widehat u&=\frac{1-\overline N/N_{\rm flat}}{C}\\N_{\rm flat}&=\eta Jp^2\Delta t\,\mathcal T\end{aligned}

The flat-field level and the contrast are known calibrations, and the fraction is computed from the recorded counts themselves.

Fitting a rate to finite exposures

\begin{aligned}\overline u(k)&=1-e^{-kt_0}\frac{1-e^{-k\Delta t}}{k\Delta t}\\u(t)&=1-e^{-H(t)}\end{aligned}

Each frame averages the conversion over its exposure, and the fit matches one constant apparent rate to those averages. A good fit does not prove a thermal cause.

Electrons lost from the selected channel

\mathcal T=\exp\!\left(-\frac{L}{\Lambda_l}-\frac{2w}{\Lambda_w}\right)

The assumed effective lengths describe electrons leaving the selected image channel, which is not the same as being absorbed.

Common difficulties

Brighter is not always less perturbed

Typical misconceptionA bright patch means that the original liquid specimen is easier to measure.

Better mental modelA bubble can brighten the image because it replaces liquid along the electron path. Gas formation has changed the cell itself. The known-input radius curve is separate from the count-derived fit; neither establishes a rate for an unperturbed reaction. Use independent dose, flow and thickness calibration.

Run the experiment

01
Track a bubble made by the beam
Compare The beam makes a bubble with Renewal suppresses the bubble. Move through stored exposures in the advanced controls.
What to observe: Renewal removes dissolved gas and can prevent a visible bubble while leaving the windows unchanged. The image fit can resolve a radius only after a spatial bubble signal emerges.
02
Reach the window and stop
Choose The bubble reaches a window, then try a long exposure and a larger gas yield.
What to observe: The model stops before invalid sphere geometry. A shortened final exposure keeps its true duration and count budget; the 24-complete-exposure target fails.
03
Measure gas pressure
Choose Gas pressure removes counts and drag the gold pressure handle. Compare with Pressure hidden by lost counts.
What to observe: Higher pressure removes electrons from the selected channel. Known path and window calibrations let the remaining counts estimate pressure; too few counts leave it unresolved.
04
Keep reaction kinetics separate
Choose Fit the liquid reaction rate. In this retained branch, adjust flux, cadence, renewal and drift processing.
What to observe: This branch measures conversion in a separate prescribed reaction model. The beam can bias its apparent rate even when the movie looks cleaner; its cause plot and rate target remain separate from bubble-radius and pressure inference.