2.1 Fuel layering stage

Figure 3.

The FST layering method provides rapid symmetrization and freezing of solid ultrafine fuel layers. (a) Schematic of the FST layering module. (b) Target before layering (‘liquid $+$ vapor’ fuel state). (c) Target after FST layering (uniform solid layer). (d) Single-spiral LC (1) in the working assembly. (e) Single-spiral LC (1) shown with magnification. (f) Double-spiral LC.

The FST layering method is based on rapid fuel cooling via the contact heat conductivity^{[Reference Aleksandrova and Koresheva1–Reference Aleksandrova, Akunets, Koresheva, Koshelev and Timasheva5]}. A batch mode is applied, and high cooling rates ( $q=1{-}50~\text{K/s}$ ) are maintained to form isotropic ultrafine solid layers in moving free-standing targets (Figure 3). During FST layering, two processes are mostly responsible for maintaining uniform layer formation.

– First, as the target rolls along the spiral layering channel, the forced target rotation results in liquid layer symmetrization.

– Second, heat transport outside the target via conduction through a small contact area between the shell wall and the wall of the layering channel (evacuated metal hollow tube, helium-cooled outside) results in liquid layer freezing.

At the same time, high cooling rates ( $q=1{-}50~\text{K}/\text{s}$ ) combined with high-melting-point additives to the fuel content in the range $\unicode[STIX]{x1D702}=0.5\%$ –25% (neon, argon, tritium) result in the creation of stable, ultimately disordered structures with a high defect density or an isotropic medium. The effect of additives is as follows.

– They initiate growth of dislocations that prevent the formation of a coarse-grained crystalline phase and enhance the mechanical strength of the layers.

– They decelerate diffusion transport processes and raise the diffusion activation energy.

– They work as stabilizing agents, keeping the grain size stable, and, as a consequence, maintaining the thermal and mechanical stability of the ultrafine cryogenic layers.

The total layering time for targets of ${\sim}$ 2 mm in diameter is typically less than 15 s, which is a necessary condition for tritium inventory minimization. An important parameter is the target lifetime within a temperature interval, $\unicode[STIX]{x0394}T_{\text{ex}}$ , in which a stable ultrafine fuel structure can exist. Our experiments showed that this interval for FST layering, $\unicode[STIX]{x0394}T_{\text{ex}}=\unicode[STIX]{x0394}T_{\text{FST}}$ , has the largest possible range, from 4.2 K right up to the temperature of solid fuel melting at the triple point. The ultrafine fuel structure presents the challenge of making smooth fuel layers and keeping them smooth under temperature variations (free of local defects). The FST layering method is promising for the formation of a stable ultrafine layer from D–T with the molecular composition: 25% $\text{D}_{2}$ , 50% D–T molecules, and 25% $\text{T}_{2}$ . In this case, $\text{T}_{2}$ in D–T is considered as a high-melting-point additive with respect to $\text{D}_{2}$ and D–T. Thus, the ultrafine layers obtained by FST are layers which can be referred to as layers with inherent survival features because they have enhanced mechanical strength and thermal stability. This is a significant factor for the retention of layer quality during target delivery^{[Reference Aleksandrova, Koresheva, Ospov, Timasheva, Tolokonnikov, Panina, Belolipetskiy and Yaguzinskiy4, Reference Aleksandrova, Akunets, Koresheva, Koshelev and Timasheva5]}.

Note that a conventional approach to solid layering (known as beta-layering method^{[Reference Kozioziemski, Mapoles, Sater, Chernov, Moody, Lugten and Johnson.14, Reference Mapoles15]}) involves crystallization from a single seed crystal in the fixed target under extremely slow cooling ( $q\sim 3\times 10^{-5}~\text{K}/\text{s}$ ) and precise cryogenic temperature control ( ${<}100~\unicode[STIX]{x03BC}\text{K}$ ) to obtain single-crystal-like layers. In a uniform thermal environment, the beta-layering method can form a spherical fuel layer, but it is not efficient in preventing local defects. The target lifetime (layer roughness is less than 1- $\unicode[STIX]{x03BC}\text{m}$ rms) is of a few seconds after reaching the desired temperature^{[Reference Mapoles15]}. This is a consequence of the D–T layer formed by beta-layering being obtained as a result of an almost equilibrium process of crystal growth, such that all the features of the equilibrium crystalline state are inherent in such a layer, including the temperature-dependent behavior of the local defects on the inner surface of the D–T layer. Besides, the total layering time is more than 24 h. Thus, the beta-layering method is not efficient for mass target fabrication for IFE usage. It is a one-of-a-kind technique, and very expensive^{[Reference Aleksandrova and Koresheva1, Reference Kozioziemski, Mapoles, Sater, Chernov, Moody, Lugten and Johnson.14, Reference Mapoles15]}.

In contrast, the FST layering method works with nonfixed moving targets at high cooling rates according to the ‘layering $+$ delivery’ scheme. The layering channel is a major element in the practical implementation of this scheme. Figure 3 schematically shows the operational scenario, which is as follows (see details in Refs. [Reference Aleksandrova and Koresheva1–Reference Aleksandrova, Belolipetskiy, Koresheva and Tolokonnikov3]).

– The FST layering module (LM) works with a target batch at one time.

– The transport process is target injection between the basic units of the LM: shell container (SC), layering channel (LC) and test chamber (TC).

– The LC is a special insert into the cryostat, a certain part of which must be at cryogenic temperatures. The medium immediately surrounding the target inside the LC is vacuum ( ${<}10^{-4}$  Torr, 1 Torr  $=$  133.3 Pa).

– Targets move downwards from the top in the LC in rapid succession, one after another, that allows high repetition rates of target injection to the TC.

– The TC is used for cryogenic target quality control: precise tomography and threshold characterization.

– The TC is a prototypical interface unit between the layering module & target injector (LM&TI).

For IFE usage, such a scenario allows economical fabrication of large quantities of targets for injection at the laser focus. When injecting, the targets will be tracked in flight by on-line holographic methods^{[Reference Aleksandrova, Koresheva, Koshelev and Ospov2]} to provide the data needed for precise target irradiation to achieve ignition and high energy gain.

For a better understanding of the time-integral performance (TIP) criterion for the FST layering, we emphasize that the LC is a special insert into the LM cryostat, a certain part of which must be at cryogenic temperatures. The medium immediately surrounding the target in the LC is a vacuum, and the layering process does not require that the target surface be almost isothermal. For successful FST layering, the LC must have a well-defined geometry in order to satisfy the TIP criterion:

(2) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\text{Form}}<\unicode[STIX]{x1D70F}_{\text{Res}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{\text{Form}}$ is the FST layering time and $\unicode[STIX]{x1D70F}_{\text{Res}}$ is the target residence time in the LC.

We start our analysis with a target cooling procedure, which is a preparatory stage before the FST layering. After shell filling at 300 K, the SC with the filled shells is transported at the same temperature from the fill system to the LM to carry out the FST layering experiments. Then the SC is cooled down to a certain depressurization temperature $T_{\text{d}}$ , which is significantly lower than room temperature. This is required for the SC depressurization stage (i.e., for gas removal from the dead volume of the SC). A level of decrease in temperature should be estimated.

The possibility of performing the SC depressurization procedure under conditions excluding both shell damage due to internal pressure and fuel leakage from the shells due to reverse diffusion exists only under a temperature decrease, when the gas pressure drops (Figures 2 and 4), the gas permeability decreases, and the shell material strength rises^{[Reference Merkuliev, Akunets, Borisenko, Bushuev, Gromov, Dorogotovtsev, Isakov, Koresheva, Nikitenko, Tolokonnikov and Isakov16, Reference Aleksandrova, Koresheva, Koshelev and Nikitenko17]}. As the gas pressure in the shell does not depend on the shell material, the possibility of performing one or another variant is determined only by the tensile properties of the shell and its configuration. Two cases are possible for the BODNER-Target upon cooling (Figure 2): at some values of $T_{\text{d}}$ , the tensile strength, $\unicode[STIX]{x1D70E}_{\text{st}}$ , becomes sufficient for fuel removal from the dead volume of the SC when the fuel is still gaseous, whereas in some cases the necessary pressure drop can be achieved only at $T_{\text{d}}<T_{\text{CP}}$ (see Table 2), when the fuel in the SC becomes liquid.

Figure 4.

The gas pressure in the shell versus the fuel density near the critical point for (a)  $\text{D}_{2}$ and (b) D–T.

Below we discuss the safe depressurization of the SC, or the BODNER-Target stability during excess gas removal from the SC with the fuel-filled shells at $P_{\text{f}}=1100$  atm at 300 K ( $\text{D}_{2}$ /D–T). The calculations were made according to the following formula:

(3) $$\begin{eqnarray}P_{\text{st}}=2\unicode[STIX]{x1D70E}_{\text{st}}\unicode[STIX]{x1D6FF},\end{eqnarray}$$

where $P_{\text{st}}$ is the difference in pressure inside and outside the shell just before shell damage occurs, and $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x0394}r/r_{0}$ is the inverse aspect ratio of the shell ( $\unicode[STIX]{x0394}r=r_{0}-r_{1}$ is the shell thickness, $r_{0}$ and $r_{1}$ are the outer and the inner shell radii, respectively). In addition, we use the van der Waals equation to impose the condition for depressurization:

(4) $$\begin{eqnarray}P=\frac{R_{\text{G}}T}{\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}-b}-\frac{a\unicode[STIX]{x1D70C}^{2}}{\unicode[STIX]{x1D707}^{2}},\end{eqnarray}$$

where $R_{\text{G}}$ is the gas constant, which is dependent on temperature and density, $\unicode[STIX]{x1D707}$ is the molecular weight, $T$ is the absolute temperature, and $a$ and $b$ are temperature-dependent parameters. The condition for the SC depressurization can be written in the form of

(5) $$\begin{eqnarray}P(\unicode[STIX]{x1D70C}_{\text{f}},T_{\text{d}})<P_{\text{st}},\end{eqnarray}$$

where $P$ is the pressure inside the shell at $T=T_{\text{d}}$ . Using Equations (3)–(5), we determine the reference values, which are the limits for the BODNER-Target to depressurize the SC when the fuel is gaseous (see Figure 4 and Table 4). A generalized value for gas fuel removal at $T_{\text{d}}>T_{\text{CP}}$ makes $\unicode[STIX]{x1D70E}>2500$  MPa, which is almost ten times more than the value given in Ref. [18] (spherical polyimide shells have the following reference data at 300 K: elastic modulus ${\sim}$ 15 GPa, ultimate tensile strength of 300 MPa, i.e., $P_{\text{st}}=3$  atm). In other words, SC depressurization is possible if the fuel is in the state of ‘liquid $+$ vapor’ (Figure 2).

Figure 5.

Depressurization temperature in the case of the BODNER-Target for $\text{D}_{2}$ , $\text{T}_{2}$ and D–T.

Table 4.

Required tensile strength near the critical point temperature.

The calculation results are presented in Figure 5: the values of $T_{\text{d}}$ are 27.5 K ( $P_{\text{d}}=2.66$  atm) for $\text{D}_{2}$ and 28 K ( $P_{\text{d}}=2.56$  atm) for D–T. They could be improved if we had the necessary, precise information related to the cryogenic tensile properties for the BODNER-Target design. As shown in Ref. [Reference Aleksandrova, Koresheva, Koshelev and Nikitenko17], temperature decreasing may enhance the mechanical behavior of the polymer shells, and the values of $T_{\text{d}}$ could be much higher.

As the BODNER-Target is cooled, there is a point where the fuel gas in the shell is liquefied (Figure 2). Since solid layer formation must pass through the liquid phase, the time for which the liquid phase exists, $\unicode[STIX]{x1D70F}_{\text{Liquid}}$ , is a key parameter and must be sufficient for symmetrization of the cryogenic layer. The initial amount of the liquid fuel before symmetrization depends on $T_{\text{in}}$ . Shell rotation causes spreading of liquid fuel on the interior of spherical shells (Figure 6(a)), and under certain conditions it results in uniform layer formation (Figure 6(b), see details in Section 3). This important effect (which is referred to as dynamical self-symmetrization) makes it pertinent to study the dynamical spread of the liquid fuel inside the moving target and develop numerical models of the process. The obtained results (theoretical and experimental) can be found in Refs. [Reference Aleksandrova, Bazdenkov and Chtcherbakov19, Reference Aleksandrova, Bazdenkov, Chtcherbakov, Gromov, Koresheva, Koshelev, Osipov and Yaguzinskiy20]. If $T_{\text{in}}$ is close to $T_{\text{TP}}$ (see Table 3), then layer freezing has begun before the achievement of layer uniformity (Figure 6(c)). Special measures are required to prevent this effect.

Figure 6.

Dynamical layer symmetrization during FST layering: (a) schematic of the target rolling along the LC; (b) $T_{\text{in}}=21$  K and (c) $T_{\text{in}}=15$  K show the influence of $T_{\text{in}}$ on the layer uniformity. Both targets have the same parameters. But in case (c) during target rolling the liquid $\text{H}_{2}$ begins to spread onto the inner shell surface, and as $T_{\text{in}}=15$  K is close to $T_{\text{TP}}=13.96$  K for $\text{H}_{2}$ , then quick freezing has begun before the achievement of layer uniformity.

Technologically, implementation of the dynamical layer symmetrization is provided by precise control of the following parameters of the FST layering method: the initial target temperature, $T_{\text{in}}$ , before the FST layering starts, the temperature profile along the LC and the LC geometry for the given target design. In the optimal case, the layer symmetrization time $\unicode[STIX]{x1D70F}_{\text{Sym}}$ is defined by the ratio:

(6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}_{\text{Sym}}\sim \unicode[STIX]{x1D70F}_{\text{Liquid}}. & & \displaystyle\end{eqnarray}$$

From physics, the maximum temperature $T_{\text{in}}$ corresponds to the obvious condition $T_{\text{in}}\sim T_{\text{S}}$ ( $T_{\text{S }}$ is the temperature of fuel separation into the liquid and vapor phases). However, in practice the choice of $T_{\text{in}}$ should be given for each particular formation isochore, taking into account the value of $T_{\text{d}}$ , i.e., the maximum working temperature $T_{\text{in}}=T_{\text{d}}$ , or slightly below $T_{\text{d}}$ (Figure 7).

Figure 7.

The relative radius of a vapor bubble ( $\unicode[STIX]{x1D6FC}$ ) under the BODNER-Target cooling (filled with $\text{D}_{2}$ up to 1100 atm at room temperature); $\unicode[STIX]{x0394}T_{\text{max}}$ and $\unicode[STIX]{x0394}T_{\text{work}}$ are the maximum and working temperature ranges for uniform layering ( $T_{\text{S}}=36.5$  K,  $T_{\text{d}}=27.5$  K).

Consider a BODNER-Target history at the formation isochore $\unicode[STIX]{x1D703}=1.54$ in accordance with Figure 2.

– SC depressurization at $T_{\text{d}}>T_{\text{CP}}$ (when the fuel is gaseous) is impossible due to the low tensile strength of the shell (Figure 4 and Table 4). This means that the required pressure reduction can be obtained only at fuel liquefaction in the SC, and the temperature $T_{\text{in}}$ will be less than the critical temperature.

– SC depressurization at $T_{\text{CP}}>T_{\text{d}}>T_{\text{S}}$ when the fuel is a compressed liquid (Figure 2) has no practical interest as from the view point of layer symmetrization, no vapor bubble exists inside the shell, while from the view point of fuel layering time, there is a compressed liquid fuel, and it requires an additional time of cooling, which, in principle is a ‘dead time’.

– SC depressurization at $T_{\text{d}}<T_{\text{S}}$ (when the fuel is ‘Liquid $+$ Vapor’) is the only practical route for the BODNER-Target design, and the following temperatures

(7) $$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\text{D}_{2}:~18.73~\text{K}=T_{\text{TP}}<T_{\text{in}}<T_{\text{d}}=27.5~\text{K},\\ \text{and}\\ \text{D}{-}\text{T}:~19.79~\text{K}=T_{\text{TP}}<T_{\text{in}}<T_{\text{d}}=28.0~\text{K},\end{array} & & \displaystyle\end{eqnarray}$$

are the working temperature intervals for developing the FST layering method. Note also that the temperature $T_{\text{in}}$ plays an important role in the vapor bubble dynamics upon a temperature drop (Figure 7).

Thus, in the case of the BODNER-Target there are three options for development of the FST-LM.

– The tensile strength of the shell is sufficient to depressurize the SC at temperature $T_{\text{d}}$ , and fuel layer symmetrization can be reached at $T_{\text{in}}=T_{\text{d}}$ .

– In the opposite case, it is essential to use temperature profiling along the LC with the purpose of increasing the time $\unicode[STIX]{x1D70F}_{\text{Liquid}}$ for fuel symmetrization. Of course, at sufficiently high temperatures $T_{\text{in}}$ (in comparison with $T_{\text{TP}}$ ) the temperature profiling can be moderate.

– In the case when $T_{\text{d}}$ is close to $T_{\text{TP}}$ , the FST layering approach remains feasible because it works with a gravitationally sagged fuel in the shell immediately before the fuel layering (Figure 3(b)), and only after layer symmetrization does freezing begin. In other words, if the temperature $T_{\text{in}}$ is slightly above $T_{\text{TP}}$ , then special temperature profiling along the LC becomes necessary to develop the FST-LM for BODNER-Scaled-Targets (FST-LM-BTs) for high-gain direct-drive target production. Note also that, in this case, injection filling of the polymer shells with a cryogenic liquid fuel (if it is workable for direct-drive target) can be used in the stage of ‘fuel filling’^{[Reference Aleksandrova, Koresheva and Koshelev10]}.

Thus, the temperature $T_{\text{in}}$ is one of the major parameters: (1) fuel just before the FST layering can have different phase states that depend on $T_{\text{in}}$ (Figure 2); (2) the maximum temperature range for uniform layering depends on $T_{\text{in}}$ (Figure 7). To illustrate this issue, we computed the ratio $\unicode[STIX]{x1D6FC}=r_{\text{v}}/r_{\text{i}}$ ( $r_{\text{i}}$ is the inner shell radius, and $r_{\text{v}}$ is the vapor bubble radius) for two values of the parameter $\unicode[STIX]{x1D703}$ , which limit, from above and below, the $\text{D}_{2}$ -formation isochore:

(8) $$\begin{eqnarray}\text{D}_{2}:~1.47<\unicode[STIX]{x1D703}_{\text{D2}}=1.54<1.61\quad (\unicode[STIX]{x0394}T=35{-}20~\text{K}).\end{eqnarray}$$

From Figure 7 it is clearly seen that the vapor bubble dynamics is almost linear over a large range of $\unicode[STIX]{x1D703}$ for the entire temperature range $\unicode[STIX]{x0394}T$ in Equation (8). However, under $T_{\text{in}}\sim T_{\text{d}}$ the balance (Equation (6)) can be violated, and the liquid phase existence time will be insufficient for layer symmetrization. In this case, temperature profiling along the LC becomes necessary. Note also that the working temperature range for uniform layering can be effectively increased by using an LC in the form of a double spiral (Figure 3(f)) that maintains the gain in the uniform layering time. We also plan to study FST layering within LCs like a three-leaved figure (trefoil) in cross-section in the next experimental campaign on testing and benchmarking the operational conditions of the key elements of the FST transmission line.

2.2 BODNER-Target layering time

In this section we consider the issue of modeling the FST layering time for the BODNER-Target. From physics, fuel ice formation by FST layering is based on a number of effects, including the surface properties and stress states under point-contact conduction-cooling of the moving targets, elastic shell deformation, thermal contact area expansion over the shell surface, phase transitions and dynamic symmetrization of the liquid fuel layers, fuel ice formation inside a shell having a high-Z coating, the influence of cooling rate on the layering results, etc.

During FST layering, heat transport outside the target occurs via conduction through a small contact area (CA) between the shell wall and the LC wall. The geometrical CA (GCA)^{[Reference Aleksandrova, Bazdenkov and Chtcherbakov19, Reference Aleksandrova, Bazdenkov, Chtcherbakov, Gromov, Koresheva, Koshelev, Osipov and Yaguzinskiy20]} is defined by the elastic shell deformation during the process of the target rolling along the spiral LC:

(9) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{\text{g}}=S_{\text{ca}}/S_{\text{sh}}=\sqrt{\frac{3G}{4\unicode[STIX]{x1D70B}r_{0}^{2}\unicode[STIX]{x1D6FF}E}},\end{eqnarray}$$

where $S_{\text{ca}}$ and $S_{\text{sh}}$ are the surface areas of the contact and the shell, respectively, $E$ is the Young’s modulus and $G$ is the normal reaction of support (LC wall, see Figure 6(a)). The GCA expansion caused by a metal tube (LC) radius $R_{\text{tube}}$ can be taken into account by the factor $\unicode[STIX]{x1D6FE}$ :

(10) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\frac{1}{(1-r_{0}/R_{\text{tube}})^{1/3}}.\end{eqnarray}$$

The greatest effect on GCA expansion is caused by heat transfer along the shell surface under heat exchange with the fuel (thermal contact area (ThCA)^{[Reference Aleksandrova, Belolipetskiy, Chtcherbakov, Kalabuhov, Koresheva, Koshelev, Kutergin, Nikitenko, Osipov, Panina, Safronov, Timasheva, Timofeev, Usachev, Tolley, Edwards and Spindloe21]}). Heat transfer in the tangential directions results in considerable enlargement of the GCA and formation of the so-called ‘effective CA’ ( $\unicode[STIX]{x1D712}_{\text{eff}}$   $=$ ‘geometrical $+$ thermal’). The value of $\unicode[STIX]{x1D712}_{\text{eff}}$ depends on the target material and composition, on the temperature $T_{\text{in}}$ , and on the course of the target cooling. All these factors make it possible to optimize the FST layering process.

When using FST layering, three stages of target formation can be distinguished.

– Stage 1: Liquid fuel cooling from $T_{\text{in}}$ down to $T_{\text{TP}}$ ( $\unicode[STIX]{x1D70F}_{\text{Liquid}}$ ).

– Stage 2: Liquid-to-solid transition at $T_{\text{TP}}$ ( $\unicode[STIX]{x1D70F}_{\text{Solid}}$ ).

– Stage 3: Solid fuel cooling from $T_{\text{TP}}$ down to $T_{\text{cool}}$ ( $\unicode[STIX]{x1D70F}_{\text{Cool}}$ ).

Then, the total layering time $\unicode[STIX]{x1D70F}_{\text{Form}}$ includes the characteristic time for all the stages:

(11) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\text{Form}}=\unicode[STIX]{x1D70F}_{\text{Liquid}}+\unicode[STIX]{x1D70F}_{\text{Solid}}+\unicode[STIX]{x1D70F}_{\text{Cool}}.\end{eqnarray}$$

The temperature $T_{\text{cool}}$ is a requirement from the target specifications for the amount of D–T in the vapor phase inside the shell (see Figure 1). The BODNER-Target before the shot must be at $T=18.3$  K to decrease the D–T vapor pressure to $0.3~\text{mg}/\text{cm}^{3}$ so as to avoid Rayleigh–Taylor instabilities during the implosion process (i.e., after fuel freezing at $T_{\text{TP}}$ the target must be cooled down to $T_{\text{TP}}$  – 1.5 K). No data are available about the required vapor pressure for $\text{D}_{2}$ fuel.

Figure 8.

Cooling time of several thin metal overcoats for different target designs ( $\varnothing$  – diameter,  $W$  – cryogenic layer thickness).

The rate of heat removal from the target and, hence, the characteristic time of the FST layering process, are mainly controlled by the CA size. In the case of the BODNER-Target (having a thin gold overcoat) this value will be determined by the ThCA, because the corresponding cooling time of such a gold overcoat is tenths of a millisecond (Figure 8). Nevertheless, we consider both cases to have a reference point for determining the layer symmetrization time.

– The minimal CA (GCA: BODNER-Target without a thin gold overcoat) is $\unicode[STIX]{x1D712}_{\text{g}}=1.2\times 10^{-3}$ for D–T and $\unicode[STIX]{x1D712}_{\text{g}}=1.08\times 10^{-3}$ for $\text{D}_{2}$ . For simplicity, we take the same value $\unicode[STIX]{x1D712}_{\text{g}}=10^{-3}$ for $\text{D}_{2}$ and D–T to estimate the time $\unicode[STIX]{x1D70F}_{\text{Form}}$ (Equation (11)).

– The maximal CA (ThCA: BODNER-Target with a thin gold overcoat) is equal to $\unicode[STIX]{x1D712}_{\text{eff}}\sim 6.5\times 10^{-2}$ ( $\text{D}_{2}$ /D–T) and shows a significant drop in the fuel freezing time during FST layering (Table 5).

Table 5.

The BODNER-Target layering time.

Analyzing the filling stage for the BODNER-Target^{[Reference Ito, Nagai, Norimatsu, Nikitenko, Tolokonnikov, Koresheva, Fujimura, Azechi and Mima8]} we considered three different materials for the shell: polyimide, polystyrene and glow-discharged polymer. However, when analyzing the layering stage, the necessary set of shell material parameters is available only for polystyrene (PS)^[22]. For this reason, we computed the FST layering time for two options: ‘PS –  $\text{D}_{2}$ ’ and ‘PS – D–T’. In the case of a porous layer, the effective conductivity could be obtained from the Clausius–Mossotti equation^{[Reference Aleksandrova, Belolipetskiy, Chtcherbakov, Kalabuhov, Koresheva, Koshelev, Kutergin, Nikitenko, Osipov, Panina, Safronov, Timasheva, Timofeev, Usachev, Tolley, Edwards and Spindloe21]}. As for the temperature of target entry to the LC, $T_{\text{in}}$ , we chose two values:

1. (a) Maximum value $T_{\text{in}}$  – close to the temperature $T_{\text{S}}$ for fuel separation into the liquid and vapor (assume using shells with a high tensile strength).

2. (b) Working value $T_{\text{in}}$  – close to the temperature $T_{\text{d}}$ under the SC depressurization before layering (assume using shells with a weak tensile strength).

In the course of modeling, we used a simulation code for rapid fuel layering inside moving free-standing IFE targets^{[Reference Aleksandrova, Bazdenkov and Chtcherbakov19–Reference Aleksandrova, Belolipetskiy, Chtcherbakov, Kalabuhov, Koresheva, Koshelev, Kutergin, Nikitenko, Osipov, Panina, Safronov, Timasheva, Timofeev, Usachev, Tolley, Edwards and Spindloe21]} and optimized it for the BODNER-Target computations within the FST layering method. It is based on solving the Stephen’s problem for moving boundaries between the fuel phases (gas, liquid and solid) and for a nonlinear boundary condition on the outer shell surface. Heat transport outside the target is via conduction through a small contact area. The computational tools allow one to model the layering time as a function of the target and LC parameters, as well as other experimental conditions. The obtained results are summarized in Table 5. The main conclusion is as follows – the FST layering time does not exceed 23 s for $\text{D}_{2}$ fuel and 30 s for D–T fuel.

The next step is the computation of a set of optimization parameters related to the LC geometry to maintain the process of BODNER-Target fabrication by the FST layering method. Our study has shown that in the case of GCA (which is the reference point for ThCA under temperature profiling along the LC) the target can be fabricated by the FST layering method using a double-spiral LC manufactured during the experimental modeling (Table 6).

Table 6.

Double-spiral LC (mockup testing results).

a Note: PS shell is used in this study as a surrogate target.

The following results were obtained for two temperatures $T_{\text{in}}$ of target entry into the LC (compare Tables 5 and 6).

For $T_{\text{in}}\sim T_{\text{d}}$ , for both $\text{D}_{2}$ and D–T, the double-spiral LC specifications are those at which the TIP criterion $\unicode[STIX]{x1D70F}_{\text{Form}}<\unicode[STIX]{x1D70F}_{\text{rol}}$ is valid for the BODNER-Target.

For $T_{\text{in}}\sim T_{\text{S}}$ and $\text{D}_{2}$ fuel, the TIP criterion is valid.

For $T_{\text{in}}\sim T_{\text{S}}$ and D–T fuel, the TIP criterion is not valid ( $\unicode[STIX]{x1D70F}_{\text{Form}}=28.5$  s and $\unicode[STIX]{x1D70F}_{\text{rol}}=23.5$  s).

Nevertheless, the double-spiral LC works in this case as well because if the spiral angle varies slightly along the spiral length then one can scale down or scale up the target speed and, correspondingly, the rolling time $\unicode[STIX]{x1D70F}_{\text{rol}}$ . In addition, the length of Spiral 2 can be extended to ${\sim}1.7$  m to meet the TIP criterion.

Another option for BODNER-Targets is a three-fold-spiral LC manufactured during the experimental modeling (Table 7).

Table 7.

Three-fold-spiral LC (mockup testing results).

a Note: PS shell is used in this study as a surrogate target.

The computation has shown that within ${\sim}$ 5 s after the start (which corresponds to $S=0.7$  m along the spiral path), target motion proceeds with a constant velocity $V_{\text{max}}=0.3~\text{m}/\text{s}$ . As the total length of the spiral (Specifications #1) is 9.187 m, then we have $(9.187-0.7)/0.3=28.29$  s. Thus, the total rolling time is $\unicode[STIX]{x1D70F}_{\text{rol}}=28.29~\text{s}+5~\text{s}$ (elapsed time before achievement of $V_{\text{max}}$ ) $=$ 33.29 s, which is in good agreement with the experiment (Table 7). Thus, we can realize the rolling conditions for BODNER-Target fabrication and satisfy the TIP criterion in the case of a three-fold-spiral LC even with a certain time margin. Nevertheless, further development of computational models of the targets response during their movement in the LC is required to optimize the FST layering time (especially $\unicode[STIX]{x1D70F}_{\text{Sym}}$ ) and to plan experiments with different IFE targets or under special experimental conditions.

For example, the proposed three-fold-spiral LC (Specifications #1, Table 7) can have a short extra spiral (Specifications #2, Table 8) so that two spirals ‘Spiral 3 $+$ Spiral 4’ make a combined layering channel (CLC)^{[Reference Aleksandrova and Koresheva1]}, which, as a matter of fact, consists of these spirals assembled one after another: acceleration Spiral 3 $+$ deceleration Spiral 4. This allows one to reduce the target velocity at the CLC output (if needed). It takes no more than 1.5 s at $\unicode[STIX]{x1D6FC}=3^{\circ }$ .

Table 8.

Combined three-fold-spiral LC.

Thus, BODNER-Targets with $\text{D}_{2}$ /D–T fuel can be fabricated by the FST layering method using $n$ -fold-spiral LCs at $n=2,3$ .

A few comments should be made regarding the ThCA. The GCA expansion caused by heat transfer in the case of a 0.03- $\unicode[STIX]{x03BC}\text{m}$ gold layer in the BODNER-Target design gives $\unicode[STIX]{x1D712}_{\text{eff}}=6.5\times 10^{-2}$ . For both $\text{D}_{2}$ and D–T fuel, $\unicode[STIX]{x1D70F}_{\text{Form}}<0.5$  s (Table 5), and LC-temperature profiling becomes necessary to increase $\unicode[STIX]{x1D70F}_{\text{Liquid}}$ and obtain a uniform layer. In this instance, $T_{\text{in}}$ can be ${\sim}$ 21 K, as the hydrogen isotope vapor pressures near the triple point determine the minimum operating pressures (Table 3) to consider an injection filling procedure. In Ref. [Reference Ito, Nagai, Norimatsu, Nikitenko, Tolokonnikov, Koresheva, Fujimura, Azechi and Mima8], it is noted that filling of the polymer shells with a cryogenic liquid fuel is suitable for the FST method because the fusion fuel in the shell directly before the FST layering has a two-phase state ‘liquid $+$ vapor’ (Figures 2 and 3). As stated in Ref. [Reference Wittman23], future direct-drive inertial fusion cryogenic targets will be filled with D–T mixtures through fill tubes of a few tens of micrometers in diameter. A testing facility has been constructed at the Laboratory for Laser Energetics (University of Rochester, USA) to determine the temperature and pressure requirements.

2.3 BODNER-Target self-heating from the beta-decay of tritium

The target self-heating is a natural process driven by bulk fuel heating from the beta-decay of tritium contained in the D–T mixture. The radioactive transformation parameters have the following values: the half-life period is $T_{1/2}=12.33$ years and the average decay energy is 5.54 keV^{[Reference Souers13]}. The law of radioactive transformation is quite simple: $N=N_{0}\exp (-\unicode[STIX]{x1D706}t)$ , where $N_{0}$ is the initial number of radioactive nuclides at $t=0$ and $\unicode[STIX]{x1D706}$ is the decay constant. If in the obtained equation we substitute the half-life period $T_{1/2}$ instead of time $t$ , then we can find the relationship between the decay constant $\unicode[STIX]{x1D706}$ and the period $T_{1/2}$ . Indeed, since the following ratios are valid:

$$\begin{eqnarray}\displaystyle & & \displaystyle N(T_{1/2})=N_{0}/2,\quad N_{0}\exp (-\unicode[STIX]{x1D706}T_{1/2})=N_{0}/2\quad \text{and}\quad \nonumber\\ \displaystyle & & \displaystyle \exp (-\unicode[STIX]{x1D706}T_{1/2})=1/2,\nonumber\end{eqnarray}$$

then the decay constant is equal to $\unicode[STIX]{x1D706}=\ln 2/T_{1/2}$ . Subject to the obtained equality, we can rewrite the radioactive transformation law in a form suitable for estimation:

$$\begin{eqnarray}N=N_{0}\exp (-t\ln 2/T_{1/2}).\end{eqnarray}$$

Since the half-life period (12.33 years) is much longer than the FST layering time (tens of seconds), $N\sim N_{0}$ and the decay rate can be estimated as $\text{d}N/\text{d}t=N_{0}\ln 2/T_{1/2}$ . Then the average output capacity from the beta-decay $P_{\unicode[STIX]{x1D6FD}}$ is

$$\begin{eqnarray}P_{\unicode[STIX]{x1D6FD}}=\frac{N_{0}\ln 2}{T_{1/2}}\langle E_{\text{decay}}\rangle ,\end{eqnarray}$$

where $E_{\text{decay}}$ is the decay energy. The average value of this energy is 5.5 keV. To estimate self-heating of the BODNER-Target, the average beta-decay capacity $(P_{\unicode[STIX]{x1D6FD}}=790~\unicode[STIX]{x03BC}\text{W})$ should be compared to the minimum heat removal from the target (target without gold overcoat) due to contact conductivity between the shell wall and the LC wall ( $Q_{\text{out}}>\text{63,000}~\unicode[STIX]{x03BC}\text{W}$ ). The desired ratio $\unicode[STIX]{x1D6EF}$ will be equal to

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}=P_{\unicode[STIX]{x1D6FD}}/Q_{\text{out}}=0.0125.\end{eqnarray}$$

When assessing the influence of the beta-decay of tritium, we proceeded from the fact that all the generated heat assimilates. Therefore, the obtained value $\unicode[STIX]{x1D6EF}<1.3\%$ means that this additional heat source is negligible in the process of BODNER-Target fabrication by the FST layering method.