Simulations of hydrogen outgassing from a carbon fiber electrode Free

High-power devices operating under high voltage conditions are used in numerous applications including vacuum electronics,^1–3 particle acceleration,^4,5 inertial confinement fusion,⁶ and microwave generation.² The use of high electric fields (typically exceeding 100 kV/cm) is aimed at enabling strong electron emission from appropriately fabricated cathode structures.^7,8 However, other accompanying effects such as localized heating at electrodes and explosive emission^9–12 and/or desorption of gases^13–15 also occur. Consequently, gas desorption from materials that can lead to plasma formation and subsequent breakdown or surface flashover can be an important process in such systems. Apart from plasma formation, outgassing has also been known to lead to anode–cathode gap closure with impedance collapse^16,17 and pulse shortening^18,19 or reduce the efficiency and performance of high-power microwave devices.²⁰ Besides potential gap closure and pulse shortening, the pressure rise resulting from outgassing could limit the achievable pulse repetition rate.^21–23 

There can be numerous routes for outgassing. These include gas release from the material surface (vaporization), release of molecules/atoms adsorbed at the surface (desorption), and possible outflow from inside the bulk material (diffusion). Adsorption can be broken into two separate components—physisorption and chemisorption. Of these, physisorption is a process by which molecules are attracted to the surface due to van der Waals forces, and the binding that arises from the induced dipole attraction is relatively temporary, with no change in the electronic or chemical structure. Chemisorption, on the other hand, involves chemical bonds, and a potential barrier usually has to be overcome before the molecules can approach close enough to bind and fall into the chemisorption potential well.²⁴ Assuming that care is taken to clean and/or treat the surface,^25,26 both of these processes can be expected to reduce. This would make the diffusive flow of gas trapped in the bulk become an important process. For this reason then, the issue of diffusive outgassing is examined in this simulation study.

As shown in our recent outgassing calculations based on a coupled molecular dynamics (MD)–density functional theory approach,²⁷ the presence of vacancies can lead to the trapping of gas atoms and decreased diffusive flow within the bulk material. It is conceivable that due to local heating associated with Joule dissipation from electron emissions at the cathode over time, the trapped gaseous entities could acquire sufficient energy to be able to de-trap and escape from the local potential wells. Such releases would then contribute to outgassing through diffusive outflow from the bulk. The resistive Joule heating alluded to above, as has been shown,^14,28 could occur within ultrashort (∼ns) time scales in short emitters. It might even be conjectured that localized heating can lead to an instability at the metal tip, similar to the Rayleigh–Plateau^29,30 process for fluids or solid columns.³¹ Furthermore, the geometric confinement inherent in nanoemitters would likely shorten phonon mean free paths, reduce heat outflow, and lead to decreases in the local thermal conductivity.³² The latter would then hasten the temperature rise and produce stronger and quicker thermal effects. In any case, focusing on outgassing from the bulk would likely remain a germane issue in all cathodes serving as charge suppliers for high-power microwave devices, despite any surface treatments that the samples might have been subjected to.

For large-area emission devices, such as, for example, magnetically insulated transmission line oscillators (MILOs), velvet and carbon fiber (CF) cathodes^33–36 have emerged as promising candidates. Though velvet cathodes have a very uniform emission at relatively low fields, exhibit a low gap closure velocity, and are inexpensive, they do suffer from disadvantages such as short lifetime and high outgassing rates.^9,37–39 In comparison, carbon fiber based cathodes not only have a low threshold field and a low gap closure velocity but also exhibit uniform emission, have lower outgassing, and have demonstrated longer lifetimes.^37,40 A more recent report⁴¹ is indicative of substantial improvement in the operation of a hard-tube MILO upon replacing polymer velvet cathodes with low-outgassing carbon fiber array cathode structures. Their data revealed that the acceptable pressure requirements for MILO operation were hundred times better than those obtained with velvet cathodes.⁴¹ Furthermore, the number of pulses for their hard-tube MILO increased from 7 with the velvet cathodes to about 127 pulses upon the use of carbon fiber array cathodes. Thus, the carbon fiber cathodes appear to hold significant advantages for high-power applications. Given this prognosis, it becomes germane to analyze and study outgassing from such carbon fiber electrodes in greater detail.

In this contribution, we present simulation results for hydrogen transport in carbon fibers based on the molecular dynamics (MD) technique used earlier by our group.^24,38 The first step involves the construction and synthesis of the carbon fibers starting from appropriate molecular building blocks. This initial step involves the interconnection of an array of graphene sheets resembling ladder-like structures that can structurally self-organize. In our study, hydrogen atoms were subsequently placed within the carbon fiber matrix, and their motion followed based on Newtonian dynamics as with the usual MD procedure. Thus, the goals of the present calculations are twofold. First, at the simplest level, the intent is to get quantitative predictions of the diffusion coefficients as a function of temperature for hydrogen within the carbon fibers. Since localized heating is expected to be a natural outcome during the operation of high-power HPM (High Power Microwave) devices, carrying out the calculation at different temperatures provides a more meaningful and relevant exercise for outgassing predictions. For comparison, similar temperature-dependent calculations are also obtained for copper. To the best of our knowledge, comparisons between gas diffusion in carbon fibers and copper have not been reported. Finally, the time-dependent outgassing dynamics are probed and a simple temperature-dependent time constant obtained.

Carbon fibers offer advantages of high stiffness and strength, along with a low density.⁴² Early work in this area focused on carbon fiber manufacture based on cellulose (rayon) based precursors,⁴² though polyacrylonitrile (PAN) is currently the precursor of choice.^43,44 Experimental studies of the microstructure indicate that the fibers consist of long and aligned graphitic sheets, with folded sheets along the transverse cross section.^45,46 There has also been recent computational work to study various mechanical properties or failure mechanisms from an atomistic perspective.^47–49 

Here, the carbon fiber (CF) structure was generated using an approach based on interconnection of an array of graphene sheets resembling ladder-like structures that can structurally self-organize.⁵⁰ In a way, the strategy was similar to that previously used and reported for cross-linked epoxy,^51,52 which included stages of bond creation followed by atomic relaxation. In doing so, a computationally efficient approach that enables generation of atomic configurations of CFs with a range of microstructures similar to those observed in experimental samples is implemented. Graphene sheets of five different lengths were first formed. These ladder-like monomers were used as the structural units of the overall CF structure. The choice of the ladders as the initial structural units is based on previous studies that have reported similar intermediate structures during the manufacturing of real CFs.⁵³ In order to enable the generation of CFs with different microstructures, varying microscopic properties, and inherent heterogeneity, five different lengths of the ladders were chosen as the building blocks. These monomers generated using VMD software⁵⁴ are shown in Fig. 1 based on the OVITO visualization software. Monomers were placed in an orthogonal simulation box with dimensions of 60 × 60 × 160 ų along the x, y, and z directions, respectively. This system packed with monomers was initially relaxed and thermalized at 300 K for up to 15 ns using the many-body MD technique. This entailed letting all the constituent atoms within the structure move dynamically with each subjected to the many-body forces from all the remaining atoms in the ensemble. Standard Newtonian mechanics was applied. The MD simulations were carried out using the LAMMPS software code, which is an open source package.⁵⁵ Such a simulation naturally leads to an evolution into a minimum energy configuration. Next, the well relaxed monomers in the simulation cell were connected with each other to build the CFs. The newly created bonds were then relaxed at their equilibrium bond length using the stepwise bond relaxation method.⁵⁶ Following bond creation, angles and dihedrals were added and the system was again relaxed for a further 20 ns. At the end of this step, the density of the CF ensemble was checked for verity. In our simulations, a density value of 1.86 g/cc was obtained for the CF system at the end of these processes, which falls within the standard range of 1.75–1.93 g/cc⁵⁷ reported in the literature for carbon fibers. Our simulated structure had ∼51 000 particles. Eventually, a stable CF system was obtained in a form that could represent an electrode. The carbon fiber system after creation of bonds, addition of angles and dihedrals, and eventual relaxation is shown in its full form in the two different perspective views shown in Fig. 2. This figure was generated for visualization in OVITO software tool.⁵⁸ 

For purposes of calculating the hydrogen outgassing rates from CF electrodes and related analysis, this system was then filled with 500 hydrogen atoms and relaxed at 300 K for a duration of 10 ns. The goal of this study was to observe outgassing rates of hydrogen from the CF structure at different temperatures. For simulations at the different temperatures, the stable CF system was heated in a stepwise fashion from 300 to 1000 K in steps of 50 K. For every temperature, the system was relaxed for 3 ns using a constant number of particles and fixed pressure and temperature (NPT) conditions. Similar simulations were also conducted for hydrogen outgassing from copper material, the details of which have been reported elsewhere.²⁷ Here, for completeness, results for outgassing from copper and carbon fiber structures of similar dimensions have been compared.

In these MD simulations, the interactions between particles in the carbon fiber system were described by the general AMBER force field.⁵⁹ The parameters found for carbon in the original set of general AMBER force field (GAFF) formulation were used.⁶⁰ A Lennard-Jones (LJ) form has been used to represent van der Waals interactions, with a cutoff distance value of 12 Å⁶¹ was used to model such interactions. The interaction between the hydrogen and carbon atoms has been modeled using Lennard-Jones (LJ) potential.⁶² The LJ potentials used for the hydrogen–hydrogen (H–H) interaction had the following parameters: σ_H–H = 0.265 nm and ε_H–H = 1.50 × 10⁻³ eV,⁶³ while the values for the LJ-based carbon–hydrogen (C–H) interaction were taken to be σ_C–H = 0.298 nm and ε_C–H = 2.70 × 10⁻³ eV. The usual Lorentz–Berthelot combining rules⁶⁴ were used to calculate the heterogeneous parameters σ_C–H and ε_C–H. A time step of 1 fs²⁸ was used for carrying out the simulations. Constant temperature and pressure were maintained through an implementation of the Nosé–Hoover thermostat and barostat, respectively.^65,66 Throughout the simulations, a constant pressure of 1 atm was maintained. Periodic boundary conditions were applied along x, y, and z axes directions during the relaxation process. Following the successful thermalization, the system was put through multiple additional simulations for determination of outgassing rates and thermal diffusivity. For completeness, the force field parameters used are listed in Table I.

It may be mentioned that in theory, the temperature and partial and overall pressures in an actual device [e.g., as in a hard-tube magnetically insulated transmission line oscillator (MILO)] can increase over time with outgassing and device operation. However, this change is not appreciable, though the percentage might not be negligible. In any case, the motion of atoms in an MD system is mainly dictated and controlled by inter-atomic forces, which are orders of magnitudes larger than any external pressure or temperature change that might be operative. Hence, we believe that pressure or temperature changes would not affect the present results and can be neglected.

The diffusion coefficient (D) of hydrogen gas in the carbon fiber and even in the copper lattice was obtained by tracking the mean square displacement (MSD) $⟨|r(t)|2⟩$ of all the diffusing hydrogen atoms over a given period of time at each temperature. The mean square displacement (MSD) is related to time-dependent variations in position for an N-atom system as

where r_i(t) is the position vector of the ith atom at time t. The MSD connects the diffusion coefficient through the relation⁶⁷ 

A quick comment is perhaps in order with regard to our choice of hydrogen atoms rather than the molecular form. Our simulations using both copper and carbon fiber material indicated that under equilibrium conditions, the distance between hydrogen atoms is greater than 2 Å, while the H₂ bond length is of ∼0.75 Å. This implies that H atoms cannot bind together to form a H₂ molecule in the intrinsic bulk. While the compositional variability in the CF materials does inject a degree of uncertainty, here the presence of atomic hydrogen was assumed at both the bulk and interfacial regions for simplicity.

It has long been recognized that time correlation functions show two stages of response: a short time response from particle interaction events and a long time response associated with many particle collective effects.^68,69 This picture remains but refinements have been reported taking account of mode-coupling theory with decay of fluctuations⁷⁰ or improved nonlinear curve-fitting approaches.⁷¹ The early or initial regime represents the ballistic mode with near collision-free particle motion. In the ballistic regime, the MSD can be approximated by a parabola with regard to the time t, i.e., MSD ∼ t². The longer-time regime represents diffusive motion dominated by collisions with gas particles mimicking random walks with roughly MSD ∼ t. Researchers typically rely on plotting the MSD on a log–log scale to ascertain the diffusive mode,⁷² which is commonly achieved in a few nanoseconds. Thus, over longer times (taken to be well over 100 ns in the current simulations), the diffusion coefficient saturates to a constant value at the operating temperature. Results of the MSD over time for hydrogen displacement in both copper and in the carbon fibers are shown in Fig. 3 for different operating temperatures ranging from 400 to 1000 K. To ensure accuracy of results, three sets of simulations were performed for each case. The results were very similar, and the average is reported here. The system consisted of 51 000 host (carbon or copper) atoms and 500 hydrogen atoms at interstitial sites for the diffusion. The hydrogen fraction was thus under 1% and is perhaps reasonable to ensure that the system behavior is controlled by the underlying carbon (or copper) environment. In this context, it may be emphasized that the molecular dynamics method is very computationally intensive and the complexity can typically increase as ∼O(N²), where “N” represents the number of particles. This places a natural limit on the number of atoms used in the simulations to keep the computations within tractable limits in terms of time and memory requirements. Besides, as discussed in the following results, the trends for hydrogen diffusion in copper did yield results comparable to experimental observations, thereby implying reasonable accuracy with the number of atoms chosen.

Since one of the aims was to compare results between copper and carbon fibers and since the melting point of copper is 1358 K, the MSD was studied only up to 1000 K, even though the carbon fibers can tolerate much higher temperatures. The MSD simulations were performed for a time long enough to emerge beyond the transient phase and into the steady state. According to classical mechanics, mean square displacement can show nonlinear behavior for short periods or at the initial stages. However, beyond the initial phase, linearity is observed in mean square displacement with time.⁷³ In the present simulations, linearity was observed in mean square displacement with time for durations beyond ∼50 ns, with the hydrogen atoms entering the diffusive phase. This care to run the simulations for long times was taken at all temperatures, so that the diffusion coefficient D could be extracted from the linear regime of the MSD. The temperature-dependent diffusion coefficients D obtained from the MSD data of Fig. 3 based on Eq. (1b) for copper and the carbon fiber structure are shown in Fig. 4. For copper, some data reported in the literature by Ishikawa and McLellan⁷⁴ and Perkins and Begeal⁷⁵ are also shown for comparison. An Arrhenius-type behavior with D(T) = D₀ exp[−E_a/(k_BT)] naturally emerged from the calculations, with an activation energies of 0.45 and 0.386 eV for copper and carbon fiber, respectively, and the corresponding pre-exponential factors of 2.34 × 10⁻⁶ and 1.251 × 10⁻⁸ m²/s. From Fig. 4, it is evident that the diffusion of hydrogen in carbon fibers is lower than that in copper by about a factor of 15.5 at 400 K and a factor of 86.8 at 1000 K. Thus, the relative benefits of using CFs over copper for reduced outgassing are enhanced at higher temperatures. In the context of high-power devices, where heating is expected, there would be greater utility in CF electrodes over traditional copper material.

Finally, the outgassing dynamics and rate constants for the outflow were computed for hydrogen gas. For these calculations, 500 hydrogen atoms were randomly placed inside the CF simulation volume, and the system was thermalized at different temperatures. To obtain outflow rates, the number of atoms remaining inside the simulation box was tracked as a function of time. Periodic boundary conditions were imposed on the four lateral faces, with reflection at the bottom surface. Atoms reaching the top face were allowed to escape and were removed from the system once their distance from the surface exceeded twice the lattice constant. This allowed for time-dependent calculation of the population N(t) remaining inside the main simulation region. Results obtained from MD simulations are shown in Fig. 5 at three different lattice temperatures. For comparison, results for copper at 1000 K are also included. To ensure statistical robustness, at least five simulations were carried out at each temperature. The associated results at a 95% confidence interval have been shown. Not only are the number of hydrogen atoms expelled from the system predicted to be higher in copper than the carbon fiber, but the slope is also sharper. Approximating the time dependence of the number of atoms [=N(t)] inside the material copper by a first-order process such as N(t) = N(t = 0) exp(−t/τ), with τ being a suitable time constant, the value of this parameter could be obtained. The above time-dependent form can also be obtained from a solution of the one-dimensional diffusion equation: δC(x,t)/δt = D ⋅ δ²C(x,t)/δx² with C(x,t) denoting the hydrogen density, under the assumption of a time-invariant diffusion coefficient D. In any case, the temperature-dependent time constant τ(T) extracted from the simulation data of Fig. 5 for copper was given by²⁷ 

For the carbon fibers, the time constant τ(T) as a function of temperature turned out to have the following form:

From the above, the outgassing rate for CF is seen to be lower.

The predicted trends toward a lower outgassing from CF electrodes compare very well qualitatively with the very recent experimental observations of Li et al.⁷⁶ For example, the output pulse limit of their hard-tube MILO has been increased from 7 pulses with polymer velvet cathode to about 127 pulses with carbon fiber array cathodes. Another advantage with carbon fibers would be their ability to withstand much higher temperatures as compared to copper or velvet cathodes. This would allow for stronger pre-operative bakeout. For example, Li et al.⁷⁶ reported bakeout temperatures of less than 150 °C, a limitation of their velvet cathodes since the velvet and the epoxy would no longer stick at higher temperatures. With carbon fiber array cathodes, on the other hand, bakeout temperature of hundreds of centigrade can be possible.

It must be emphasized that this merely constitutes a preliminary computational study on this emerging area, which is complex. Other interesting aspects not treated here encompass the presence of material defects, variations in the microstructure, possibility of oxide layers, events such as the out-going hydrogen gas reacting chemically with the surface oxide to form water that would subsequently desorb, etc. With the presence of an oxide, one then needs to be concerned with and carefully study oxide diffusion, possible chemical reactions, and determine the limiting step(s) of the overall process. An additional complication could involve more accurate and realistic computations of the surface potentials and changes in the energy landscape, which would differ from that for bulk material. A likely approach could the hinge on a combining the MD with density functional theory, as was done in a previous contribution,²⁷ or a report by Lane et al.⁷⁷ describing a combined grand canonical Monte Carlo (GCMC) for the surface water coverage, and molecular dynamics for desorption. Furthermore, in order to include reactive events and dynamic bond formation, or even dissociation of water at the surface into constituent gases, approaches such as the reactive force field (ReaxFF) interactive potentials⁷⁸ would become necessary.

Outgassing in high-power systems is a known problem. The gas desorption from materials (mainly electrodes) can lead to a host of issues such as plasma formation, subsequent breakdown or surface flashover, anode–cathode gap closure with impedance collapse, and reduction in the efficiency and performance of high-power microwave devices. Hence, attempting to reduce outgassing remains an important consideration in this arena, and in this regard, velvet and carbon fiber cathodes have emerged as promising materials. Between the two, carbon fiber based cathodes not only have a low threshold field and a low gap closure velocity but also exhibit uniform emission, have lower outgassing, and have demonstrated longer lifetime. For example, bimodal CF structures may be even more useful and efficient⁷⁹ and would be discussed elsewhere. Experimental results have already shown the production of electron beams with current densities up to 10 kA/cm² without shorting of the cathode–anode gap by the cathode plasma while sustaining hundreds of pulses without degradation in its emission properties.⁸⁰ Here, a numerical study based on the classical molecular dynamics scheme was used for quantitative assessment of hydrogen outgassing from carbon fibers as a function of temperature. The aim was also to compare the outgassing results for a typical electrode material such as copper.

Our results have shown dramatic reductions of hydrogen diffusion in carbon fibers at all temperatures as compared to copper. An Arrhenius-type behavior was predicted with reductions by factors of 15.5 and 86.8 at 400 and 1000 K, respectively, over copper. The results point to even stronger improvements for operation at higher temperatures (with a diffusion constant ratio of ∼187 in the asymptotic limit), which is encouraging, since such devices are expected to undergo a fair degree of internal heating during active operation. Outgassing simulations were also performed, and the behavior roughly exhibited an exponential decay for the remaining hydrogen population in the material over time. Our results are valid for the short time scales where the diffusion-limited process dominates. With reductions in the sample hydrogen concentration over time, less hydrogen would arrive at the surface per unit time. Eventually, the rate at which hydrogen atoms at the surface combine and desorb as H₂ molecules would become the dominant factor in the outgassing. Similarly, it must be mentioned for completeness that differences in microstructure could alter the results slightly, but the general trends are expected to remain. Such issues are beyond the present scope, as are possible effects of surface contaminants that could alter the potential energy landscape at or close to the surface.

R.P.J. would like to thank S. Desai and Professor A. Strachan (Purdue University) for helpful discussions. This work was supported in part by grants from the Office of Naval Research (No. N00014-18-1-2382) and the Air Force Office of Scientific Research (No. FA9550-19-1-0056).

The data that support the findings of this study are available from the corresponding author upon reasonable request.