CHAPTER 4 : Beam-Sample Interactions
4.8 Finite Beam Irradiation with Diffusion
In most Liquid Cell Electron Microscopy experiments, only a fraction of the liquid in the cell is irradiated. Radiolysis products generated within the irradiated region will diffuse out and continue to react outside the irradiated volume. To examine the effects of diffusion, a model system, similar to the one used in the temperature rise analysis is employed. The system is comprised of a cylindrical electron beam of radius a located in the center of a one-dimensional radial chamber of radius W. For the calculations presented in this section, a = 1 µm and W = 50 µm. It is important to note that W may be as large as a few mm in practice. We focus here on circumstances relevant to TEM, where the electron beam is maintained at a fixed position for a prescribed time interval.
In the irradiated domain 0< r < a, the equations include the relevant source terms
Ri. In the region outside the beama < r < W,ψ = 0 (and thereforeRi = 0). Figure 38 (left column) depicts H2, eh–, O2, and H+ concentrations as functions of the radial position r at various times. Figure 38 (right column) depicts the concentrations of the species at the beam’s center (r= 0), the beam’s edge (r=a), and at the device’s outer, impermeable surface (r=W) as functions of time.
All species reach steady state concentrations within seconds - much longer than the equilibration time in the homogeneous case. This longer equilibration time results from the slow diffusion process that takes place over time scales on the order of the diffusion time,tD,i∼W2/Di. Typical diffusion coefficients of solutes in water are on the order of 10−9m2/s, and so it is easy to estimate the order of magnitude of the time to steady state by using the diffusion time scale. Due to the chemical reactions between species, the actual steady state concentration is a function of position within
Figure 38: Heterogeneous model predictions for the spatial and temporal evolutions of H2 (a and b), eh– (c and d), O2 (e and f), and H3O+ (H+) (g and h). The left column depicts the concentrations of the selected radiolysis products of neat water as functions of the radial distance from the center of the irradiated region at various times. The right column depicts the concentrations of the same products at the center and edge of the irradiated region and at the perimeter of the liquid cell. The beam and liquid cell radii are, respectively, 1 µm and 50 µm. The dose rate is 7.5×107 Gy/s.
the liquid cell.
It is very important to make two notes about the time it takes to establish a steady state. First, although it may take minutes or hours to reach the long time steady state in a large device, the species concentrations within the irradiated volume will be on the same order of magnitude as their long time values extremely quickly (milliseconds as described in the homogenous case). This means that changes in the local chemistry are unavoidable even for a large liquid cell. Consequently, a second caveat is that flowing the solution through the cell is also unlikely to help mitigate beam-induced phenomena, as the flow rate required to sweep away primary products would need to be large. We can estimate when flow will be important via the Damk¨ohler number, the ratio of the production rate to the advective mass transfer rate. When
Da= Ril
Civ
(4.32)
is less than unity, then flow will have a significant role in the solution chemistry. Here,
l is the length scale (irradiated volume radius ∼ 1 µm) and v is the flow velocity. SettingDa= 1, we solve for velocity under the conditions ofC ∼10−6 (M),l ∼10−6 (m), ψ ∼ 107 (Gy/s), and G ∼ 1 (molecule/100eV) to get v ∼ 1 (m/s). Meaning the flow rate will have to be sufficiently high to make the fluid velocity greater than 1 (m/s) in the irradiated volume for advection to be dominant. This large flow velocity requirement means the radiolysis products will likely play the dominant role in solution chemistry.
Figure 38 illustrates the spatial and temporal behavior of some species. The most well-behaved (least reactive) species, H2, is continuously produced within the irradiated region and diffuses away (Figure 38). At early times (t < 10−4s), the
production exceeds the diffusion and the H2 concentration increases rapidly at the beam’s center. At later times, the production is balanced by diffusion, which is reflected by the plateau in CH2(0, t) (Figure 38b). As the H2 concentration outside
the beam builds up and the diffusive flux decreases, the H2 concentration in the irradiated region resumes its growth to eventually approach its maximum, equilibrium value.
The more reactive radiolysis products, such as eh–, H , OH , H3O+, and O2 exhibit more complex behaviors than H2. The highly reactive eh–, H, and OH persist mainly in beam region and their concentrations drop quickly outside thereof. As they diffuse away from the irradiated region they are consumed through chemical reactions and drop rapidly to zero (Figure 38c). The somewhat complex behavior of eh– (Figure
38c and d) can be understood by considering its interactions with O2 (Figure 38e and f), the dominant scavenger of eh– in neat water (via eh– + O2 ⇒ O2–). Oxygen production is delayed somewhat, and initially the eh– concentration in the irradiated region increases rapidly. Once the production of O2 ramps up (t > 10−5s), the O
2
scavenges eh– and reduces its concentration. As a result, the concentration of eh– as a function of time exhibits a peaks at t ∼ 5µs. As time increases further, the e–
h concentration declines to its equilibrium value. The peak in the spatial distribution of eh– next to the edge of the irradiated region at intermediate times (i.e.,t∼10−4s) is attributable to the mass transfer of O2 by diffusion away from the irradiated region. The temporal and spatial distributions of the oxygen concentration, causally, exhibit opposing trends to that of eh–.
H3O+(or H+, Figure 38g andh) behaves similarly to the other radical products, but with somewhat higher concentration outside the irradiated region than eh–. After the onset of irradiation, the H3O+ concentration in the irradiated region quickly
peaks and then drops to a steady state. Outside the irradiated region, the H3O+ concentration grows slowly by diffusion. Eventually the entire liquid cell will have an increased H3O+concentration and reduced pH. For a neat water initially at pH 7 exposed to a 1µm radius beam of 1nAat 300kV (a dose rate of 7.5×107 Gy/s), the steady state the pH will drop to∼4.9 within the irradiated region and ∼ 6.1 outside the irradiated region.
As in the homogeneous case, we can fit a power law similar to the one inEquation (4.30) to specify the concentrations of radiolysis products at the beam’s center as functions of the dose rate. The calculations were carried out for a liquid cell with radius W = 50 µm, but the results should be approximately applicable for any sufficiently large W (i.e., W > 10 µm). The pre-factor αi and the exponent βi (0.3< βi <1.5) are listed in Table 3.
4.8.1. Diffusion Coefficients
When solving the heterogeneous problem, we need to account for mass transfer by diffusion. For the reader’s convenience, we reproduce the diffusion coefficients of radiolysis products in water at room temperature in Table 4.