Estimation of the FST-Layering Time for Shock Ignition ICF Targets

Abstract

The challenge in the field of inertial confinement fusion (ICF) research is related to the study of alternative schemes for fuel ignition on laser systems of medium and megajoule scales. At the moment, it is considered promising to use the method of shock ignition of fuel in a pre-compressed cryogenic target using a focused shock wave (shock- or self-ignition (SI) mode). To confirm the applicability of this scheme to ICF, it is necessary to develop technologies for mass-fabrication of the corresponding targets with a spherically symmetric cryogenic layer (hereinafter referred to as SI-targets). These targets have a low initial aspect ratio A_cl (A_cl = 3 and A_cl = 5) because they are expected to be more hydrodynamically stable during implosion. The paper discusses the preparation of SI-targets for laser experiments using the free-standing target (FST) layering method developed at the Lebedev Physical Institute (LPI). It is shown that, based on FST, it is possible to build a prototype layering module for in-line production of free-standing SI-targets, and the layering time, τ_form, does not exceed 30 s both for deuterium and deuterium-tritium fuel. Very short values of τ_form make it possible to obtain layers with a stable isotropic fuel structure to meet the requirements of implosion physics.

1. Introduction

The objective of this article is to expand the FST-layering method developed at the LPI on SI-target fabrication. A schematic representation of the FST-layering method is shown in Figure 1a,c, where the following designations are used: layering module (LM), shell container (SC), layering channel (LC), test chamber (TC). The SI-targets include a polymer shell and a cryogenic fuel layer (Figure 1b).

The SI-target specifications are given in

Table 1. The SI-targets’ specifications.

Target Design Options	Polymer Shell				Fuel Layer
	R (μm)	R0 (μm)	ΔR (μm)	Ash	Rvapor (μm)	W (μm)	Acl
SI-1	1080	1049	31	34,8	888	161	5,5
SI-2	902	880	22	41	733	147	5
SI-3	815	791	24	34	593	198	3

, where the following symbols are accepted: R and R₀ are the outer and inner radii of the shell, ΔR is its thickness, R_vapor is the radius of the gas cavity (contains saturated fuel vapor), W = R₀—R_vapor is the thickness of the cryogenic layer, A_cl is the initial aspect ratio of the cryogenic layer (A_cl = R_vapor/W). For low-aspect SI-targets, A_cl is in the range of 3–5 [1,2].

The FST-layering method works with free-standing (or unmounted) and line-moving targets and allows one to fabricate large quantities of such targets and continuously inject them at the laser focus [3]. The FST-layering method is highly compatible with a new approach to the target delivery system based on noncontact target transport with levitation [4] which is a necessary condition for high-repetition-rate laser facilities. Below, we evaluate the prospects of the FST-layering method for in-line production of the SI-targets.

The algorithm for conducting the FST-layering experiments is implemented as follows:

(1)

The process starts with the ramp filling of a batch of unmounted shells located in the shell container with fuel gas at room temperature (300 K) and transporting them at the same temperature from the filling system to the layering module. In order to obtain a thick cryogenic layer (Table 1, W = 198 − 147 μm), the shells are filled to a high internal pressures P_f with deuterium (D₂) and deuterium-tritium mixture (DT). The filling stage for SI-targets was studied in detail in [5] where we presented the results of modeling the D₂/DT-fill time and rates for SI-targets. It was found that the fill pressures were P_f = 678–1250 atm for D₂ and P_f = 690–1230 atm for DT. When implementing the optimal filling procedure with a constant pressure gradient, the calculations showed that, on average, the SI-targets can be filled to the required pressures in a time from 1.65 h (polyimide) to 7 h (polystyrene) with Young’s modulus E = 3 GPa and a safety factor δ = 0.55 (i.e., with half the pressure drop across the shell wall relative to the maximum allowable value). As the value of δ increases, the filling time is even shorter.

(2)

The next stage includes mounting the shell container in the layering module, followed by its cooling to the depressurization temperature T_d, which is significantly lower than room temperature. The depressurization procedure is necessary to drop the pressure in the shell container and remove the fuel outside the shells. Taking into account that the fill pressure P_f is very high (as mentioned above), it is necessary to determine the conditions that exclude both the shell damage by internal pressure and the fuel leakage from the shells due to back diffusion. The fulfilment of these conditions is possible only at the temperature decrease, when the gas pressure drops down, the gas permeability decreases, and the strength of the shell material increases. We have found that for polystyrene shells with a tensile strength σ < 50 MPa, the required pressure reduction for a safe depressurization of the shell container (considering both D₂ and DT) can be achieved only by liquefying the fuel inside it, i.e., for T_d < T_cp (T_cp is the critical point temperature). A gravitationally sagged liquid remains in the shells, and this is an initial fuel state before the FST-layering (Figure 1c, at the top). For stronger shells (σ ~ 110 MPa), the depressurization temperature can reach the values T_d = 45 K > T_cp, i.e., the value of σ is sufficient to depressurize the shell container when the fuel is gaseous (it can be, for example, for polyimide and glow discharge polymer).

(3)

The final step is freezing the spherically symmetric layer in the shells during their rolling inside the spiral layering channel, i.e., the fabrication of the cryogenic target itself (Figure 1c, at the bottom). It is to this stage that this work will be devoted.

2. Mathematical Modeling of the SI-Target Fabrication by the FST-Layering Method

Fabrication of a solid fuel layer in the batch of moving shells by the FST-layering method is carried out during the shells’ transport by injection between the main functional elements of the FST-layering module: SC—LC—TC (Figure 1a). The layering channel is made from a metal hollow tube cooled outside by liquid helium. The optical test chamber is used to characterize the finished cryogenic targets, and also serves as an intermediate unit between the FST-layering module and the target injector.

During the FST-layering, two processes are mostly responsible for maintaining a spherically symmetric layer formation (Figure 2):

―

A random target rotation when it is rolling down along the layering channel (single, double, or triple spirals) results in a liquid layer symmetrization (Figure 2a).

―

A heat transport outside the target via conduction through a small contact area between the shell wall and the layering channel wall results in a liquid layer freezing (Figure 2a,b).

Below we evaluate the prospects of the FST-layering method for in-line SI-target production. The main parameter to be determined is the layering time (τ_form) during which a cryogenic layer is formed inside the shell as it moves in the spiral layering channel (Figure 2c). Therefore, simulation of the FST-layering process is necessary for FST-layering module construction to produce spherical cryogenic targets for their shock ignition in laser thermonuclear fusion.

In [6], a model of rapid fuel freezing inside moving free-standing shells was proposed and further adapted for different classes of targets. This model is based on the solution of a Stefan problem [7] related to phase transitions in matter, in which the boundary between the phases can move with time. It is assumed that all three fuel phases can coexist inside the shell, whereas the real number of phases is determined by the thermal target history during the FST-layering process. The thermal conductivity and heat capacity of the fuel are assumed to be known and determined by interpolation of existing experimental data [8,9,10]. The densities of the liquid and solid phases as a function of temperature, ρ_liquid and ρ_solid, are also known from [8,9,10]. The boundary condition on the outer shell surface describes the heat removal from the target. In our case, the cooling rates are realized when the target is cooled by heat conduction through a small contact area between the shell and the layering channel wall (Figure 2a,b), i.e., the heat is removed when only a part of the shell is in thermal contact with the layering channel.

The contact area, S_ca, occurs due to the shell deformation during its motion in the layering channel. An estimate of S_ca depending on the target characteristics can be found in [6], where the geometrical contact area is determined by the following relationship:

χ_g = S_ca/S_sh = (½)(3N/(πR²_targetδ_shE))^1/2, δ_sh = A_sh⁻¹,

where χ_g is the dimensionless parameter, S_sh is the shell surface, E is the Young’s modulus of the shell, N is the normal reaction of support of the layering channel wall (Figure 2a). The expansion of S_ca due to the LC curvature can be taken into account using the coefficient γ [11]:

γ = 1/(1 − R_target/R_tube)^1/3,

where R_tube is the radius of the hollow metal tube from which the layering channel is made (Figure 2b).

Note that a major influence on the S_ca expansion is conditioned by the heat transfer along the shell surface under the heat exchange with fuel. This is related to the thermal conductivity, λ, of the hydrogen isotopes, which is much greater than that of the shell (in our case, polystyrene). Indeed, the value of λ of polystyrene varies from 0.029 W/mK at 4.2 K to 0.074 at 20 K [12]; for D₂ λ = 0.46 W/mK at 4.2 K and 0.27 W/mK at the triple point temperature T_tp = 18.7 K; for DT λ = 0.54 W/mK at 6 K and 0.24 W/mK at T_tp = 19.7 K [8,9,10]. These data indicate that the tangential heat flows cannot be neglected. Modeling and taking into account the tangential and radial heat flows made it possible to obtain an accurate estimate of the S_ca expansion: it is almost an order of magnitude greater than the geometrical contact area [3]. This leads to a significant increase in the real values of S_ca and, as a consequence, to the formation of the so-called effective contact area characterized by the parameter χ_eff.

In a general case, the value of χ_eff depends on the material and composition of the target, as well as on the course of target cooling. Summarizing the above, we will have:

χ_eff = (ξγ/2) · (3G/(πR²_targetδ_shE))^1/2,

where ξ is the thermal factor of the S_ca expansion.

The process of the target cooling is an isochoric process shown in Figure 3a, in which the following nomenclature is accepted: T_cp is the critical point (CP) temperature, T_tp is the triple point (TP) temperature, and T_s is the temperature of starting the separation process into liquid and gaseous phases. The entire process can be conventionally divided into four stages:

• Target cooling from T = 300 K to the temperature T_d. This stage is important for the shell container depressurization.
• Target cooling from T_d to the temperature T_in, which corresponds to the target entry into the layering channel (T_in is an initial target temperature before the FST-layering). The value of T_in can be between T_s and T_tp, i.e., there is already a certain amount of liquid fuel inside the shell (Figure 1c, at the top).
• Formation and cooling of the liquid phase in the temperature range of T_in − T_tp. The value of T_in determines one of the key parameters—the time of liquid phase existence, τ_liquid, which must be sufficient (~35–40% of τ_form) to symmetrize the liquid layer when the shells are rolling along the layering channel [3]. Note that as they cool down, the role of fuel vapor becomes negligible (Figure 3b). In the triple point its pressure is ~0.2 atm for all hydrogen isotopes [8,9,10].
• Liquid phase freezing at the triple point temperature T_tp.
• If necessary, cooling the target from T_tp to a certain operating temperature T_form.

For computation of the FST-layering time, it is necessary to know the SI-target parameters in the two-phase state of “Ice + Vapor” (Figure 1b) at a temperature T_form. According to [1,2] they have the following values:

―

DT fuel: T_form = 18.3 K, vapor density ρ_vapor = 0.3 mg/cm³, ice density ρ_solid = 250 mg/cm³;

―

D₂ fuel: T_form = T_tp = 18.7 K, vapor density ρ_vapor = 0.448 mg/cm³, ice density ρ_solid = 196.687 mg/cm³.

This allows one to calculate the fuel mass parameters, i.e., those initial data that will determine the course of the FST-layering process, namely: m_solid—solid fuel mass, m_νapor—fuel vapor mass; ρ_fill—gaseous fuel density in the shell at 300 K, M_fuel—total mass of fuel. The obtained results are shown in

Table 2. Initial data for calculating the layering time of the SI-targets.

Fuel Mass Parameters	SI-1		SI-2		SI-3
	D2	DT	D2	DT	D2	DT
ρfill (mg/cm3)	77.6	98.5	83.3	105.7	114.0	144.8
msolid (μg)	374.1	475.5	237.0	301.2	236.0	299.9
mνapor (μg)	1.31	0.88	0.73	0.49	0.39	0.26
Mfuel (μg)	375.4	476.4	237.7	301.7	236.4	300.2

.

Another key factor is related to the fact that the quantitative ratio between the liquid and gaseous components of fuel changes with a temperature drop, which plays an important role in determining the dynamics of the relative radius of the gas cavity. In the two-phase region (T < T_s), the masses of gas (this is saturated vapor) and liquid are equal, respectively:

m_vapor = (4/3) π (R₀ − W)³ · ρ_vapor(T),

(1)

m_liquid = (4/3) π · (R₀³ − (R₀ − W)³) · ρ_liquid(T),

(2)

where the thickness of the liquid layer is found from the law of conservation of mass:

M_fuel = m_vapour + m_liquid,

(3)

M_fuel = (4/3) π · R₀³ · ρ_fill,

(4)

or

(1 − W/R₀)³ · ρ_vapor(T) + (1 − (1 − W/R₀)³) · ρ_liquid(T) = ρ_fill.

Here, T is the target temperature, and the phase densities ρ_vapor and ρ_liquid are known from the phase diagram [8,9,10].

Assuming R₀ − W = R_vapor is the radius of the gas cavity (Figure 1b) we have:

(1 − W/R₀)³ = R³_vapor(T)/R₀³ = (ρ_liquid(T) − ρ_fill)/(ρ_liquid(T) − ρ_vapor(T)).

Let us introduce the following parameter

α (ρ_fill,T) = (ρ_liquid(T) − ρ_fill)/(ρ_liquid(T) − ρ_vapor(T))

(5)

and write the value of ρ_fill as

Θ = ρ_fill/ρ_cp,

(6)

where ρ_cp is the critical density (ρ_cp = 69.8 mg/cm³ for D₂, and ρ_cp = 87.06 mg/cm³ for DT). Then the parameter α can be written in the form:

α (ρ_fill,T) = (ρ_liquid(T) − Θρ_cp)/(ρ_liquid(T) − ρ_vapor(T)).

(7)

From equalities (1)–(4), taking into account (7), it is easy to obtain a number of useful relations:

(V_vapor(T)/V₀) = α, R_vapor(T)/R₀ = α^1/3, (V_liquid(T)/V₀) = 1 − α

(8)

m_liquid(T)/m_fuel = (ρ_liquid(T)/ρ_fill)(1 − α), m_vapor(T)/m_fuel = (ρ_vapor(T)/ρ_fill) · α

(9)

A_cl = 1/(1 − α^1/3).

(10)

Note that if the target temperature is below the triple point temperature, then there is already a solid cryogenic layer inside the shell. Relationships (1–10) remain valid in this case as well, with the replacement of the index “liquid” by “solid”.

To select the temperature T_in we use relation (8) for α^1/3, which determines the dynamics of the relative radius R_vapor/R₀ = α^1/3 during the target cooling. Figure 4 shows the calculated data for R_vapor/R₀ = α^1/3 in the case of deuterium. For SI-targets the parameter ϴ (see formula (6)) is equal to ϴ₁ = 1.1, ϴ₂ = 1.19, ϴ₃ = 1.63 so that the range Δϴ = 1.0–1.63 covers all three SI-targets (see Figure 4).

From Figure 4 it is clearly seen that starting from T = 30 K the function R_vapor/R₀ = α^1/3 is almost linear, which is extremely important for the process of fuel layer symmetrization (see Figure 2).

Let us make a few remarks regarding the choice of the shell material. Analyzing the filling stage for the SI-targets [5] we considered three different shell materials: polyimide, polystyrene and glow discharged polymer (GDP). However, analyzing the FST-layering stage, a necessary set of shell parameters (tabulated data on heat capacity and thermal conductivity at cryogenic temperatures, see

Table 3. Heat capacity and thermal conductivity of the polystyrene.

T (K)	λ (W/mK)	T (K)	λ (W/mK)	T (K)	C (J/kgK)	T (K)	C (J/kgK)
1	0.011	70	0.1111	5	10.05	80	381.50
4.2	0.029	80	0.1150	10	32.16	90	420.49
10	0.0541	90	0.1184	20	102.11	100	460.24
20	0.0744	100	0.1231	30	170.45	120	523.00
30	0.0863	150	0.1326	40	226.73	140	594.13
40	0.0947	200	0.1407	50	270.55	160	661.07
50	0.1012	250	0.1472	60	311.95	180	728.02
60	0.1066	300	0.1539	70	346.52	200	799.14

[12]) is available only for polystyrene (PS). For this reason, computation of the layering time was made for two options: “PS ― D₂” and “PS ― DT”. The results of calculations for two values of T_in are presented in

Table 4. FST-layering times for SI targets.

Target Design Options	Tin = 30 K		Tin = 26 K
	D2	DT	D2	DT
SI-1	29.6 s	28.2 s	21.9 s	20.5 c
SI-2	19.9 s	18.5 s	14.9 s	13.4 c
SI-3	27.0 s	25.1 s	20.1 s	18.2 c

.

The main conclusion is as follows: the FST-layering time for all three designs of SI-targets does not exceed 30 s for both D₂ and DT fuel. This is of great importance for reducing the time and space scales of all production steps in the target fabrication facilities. In addition, the lifetime of the liquid phase at T_in = 30 K is ~60% for D₂ and ~63% for DT, and at T_in = 26 K it is ~41% and ~46%, respectively, which is a sufficient condition for the symmetrization of the fuel layer.

3. Discussion of the Obtained Results

In this section, we will discuss the obtained results in terms of constructing a prototype of the FST-layering module for the layering times shown in Table 4. Indeed, the question of the layering channel geometry in which the SI-targets can be fabricated remains open. A key issue is the spiral type (n-fold spirals, n = 1, 2, 3) and its parameters (inclination angle, diameter, height and width of the spiral). These values directly control the target residence time in the layering channel, τ_res, which must be longer than the layering time, namely: τ_form ≤ τ_res. By varying the above parameters, one can optimize the FST-layering method for any target class, including SI-targets.

Generally, one can view the target motion in the following rolling conditions:

―

Target slides on the layering channel surface (no rotation: sliding and only sliding or pure S&S-mode);

―

Target combines rolling with sliding (rolling with sliding or mixed R&S-mode);

―

Target rolls on the layering channel surface without sliding (rolling and only rolling or pure R&R-mode).

During target fabrication it is necessary to realize only the R&R-mode to avoid the outer shell roughening and to achieve fuel layer uniformity. Therefore, the time-integral performance criterion can be written in the following type (τ_rol is the time of pure target rolling):

τ_form ≤ τ_res = τ_rol

Thus, determination of the rolling conditions is one of the main problems, which influences the choice of the layering module operation including simplifying the physics design and modifying the specifications. First of all, we should to determine the spiral angles β for realizing the pure target rolling (R&R-mode).

In [13], calculations were performed for two versions of the layering channels (double and triple spirals), which were fabricated and tested just in R&R-mode for two spiral angles β = 11.5° and β = 16.7°.

For SI-targets, we start with a layering channel in the form of a double spiral. Its parameters are: spiral angle β = 11.5°, radius of each spiral R_cyl = 21 mm, height of each spiral H_cyl = 450 mm, τ_rol = 23.5 s for PS-shell with a diameter of ~2 mm, which corresponds to the SI-target dimensions (see Table 1). In other words, this 2S-LC can be used to carry out the FST-layering experiments at T_in = 26 K, since the main condition for SI-targets is met: τ_form < 22 s < τ_rol = 23.5 s (see Table 4).

Consider now the layering channel in the form of a triple spiral. Its parameters are: β = 16.7°, R_cyl = 21 mm, H_cyl = 880 mm, τ_rol ~ 35 s. In this case, the FST-layering experiments can be carried out both at T_in = 26 K and T_in = 30 K (see Table 4). The latter is extremely important, since the value of τ_liquid increases with an increase in the temperature T_in from 26 K to 30 K.

Thus, the modeling results have shown the benefits of the FST-layering method to develop and validate rapid layering technologies that are applicable to mass SI-target fabrication. This is due to the fact that:

―

The FST-layering method works with free-standing and line-moving targets. A specific future here is the possibility to build a prototype of the FST-layering module, which must be integrated in an FST-production line operating in high-repetition-rate conditions [3,4]. In [13], the key elements of the FST-production line and their functional description are given in detail. Additionally, the development strategy of such line creation seeking to develop commercial power production based on laser IFE has been discussed.

―

A short layering time (τ_form < 30 s for D₂ and DT) is required to induce the formation of multiple crystals of different orientations for obtaining ultra-fine fuel layers with a stable isotropic fine-grained or nanocrystalline structure and avoiding instabilities caused by grain-affected shock velocity variations. A Fourier-spectrum of the bright band of the cryogenic layer is given in [14]. It has shown that surface imperfections of the cryogenic layer formed by the FST-layering are less than 0.15 microns for modes N_f = 20–30. Note also that under granularity growth (in which the grain size decreases) the material strength increases. This means that the ultra-fine fuel layers have an adequate thermal and mechanical stability which supports the fuel layer survivability under target injection and transport through the reaction chamber. Additionally, such short layering times are also promising in terms of tritium inventory reduction in the target fabrication facilities [15].

Recall that a conventional approach such as beta-layering uses with a single target and requires more than 17 h [16] for its fabrication with an aniisotropic layer such as a single crystal. A long-run beta-layering process in very strict isothermal conditions (target temperature must be controlled down to 1 mK precision) leads to the roughening of the layer surface and the provoking of implosion instabilities in the case of deviation from the specified conditions. Another important issue is related to the fact that each target must be mounted on a special suspension which excludes the target positioning at the laser focus by injection. Note that target injection is a necessary condition for achieving a high symmetry of irradiation by a laser, as well as plasma generation with an intensive thermonuclear reaction in high-repetition-rate laser facilities.

4. Conclusions

Shock ignition is a recently proposed ICF scheme, in which the stages of compression and hot spot formation are partly separated. An SI-target is composed of D₂/DT gas surrounded by a cryogenic D₂/DT solid layer as a fusion fuel. A distinctive feature of the design of these targets is a low initial aspect ratio (inner fuel layer radius/fuel layer thickness, A_cl = 3 and A_cl = 5-design) to provide greater implosion stability. The paper discusses the issues of SI-target fabrication using the FST-layering method. It has been shown that on this basis it is possible to build a prototype of the FST-layering module, operating with a batch of moving free-standing SI-targets under high-repetition-rate conditions. The layering time does not exceed 30 s for both D₂ and DT, which offers the potential for obtaining an isotropic fuel that is very important for the progress towards ignition.