In recent years, a major breakthrough in inertial confinement fusion (ICF)1,2 has been the achievement of ignition, as robustly and repeatably demonstrated by the generation of more energy from nuclear fusion than the laser energy input.3,4 This success can be attributed to optimization of laser, hohlraum, and capsule parameters,5 including the mixing between the ablator and fuel.6 A layered ICF capsule is typically composed of very different materials, and this leads to hydrodynamic instabilities [e.g., Rayleigh–Taylor (RT)7,8 and Richtmyer–Meshkov (RM)9,10 instabilities] near the material interfaces during implosion compression.11 This growth of instabilities can create spatially localized mixing regions, where, for example, high-Z ablator material is mixed with low-Z gaseous fuel, which can greatly affect radiative transfer,12 degrading the performance of the implosion.13,14 Accordingly, in the ICF context, understanding and modeling radiative transfer coupled to materials in binary stochastic mixtures is an essential task.

Miller et al.15 presented one-dimensional (1D) benchmark results for radiative transfer in participating binary stochastic mixtures, i.e., with coupling of radiation to material. For various mean chord lengths,16 they compared the accuracy of three forms of approximate closure equations.17–19 In two and three dimensions, Olson20–23 studied a substantial number of cases over a wide range of parameters, thereby improving our understanding of stochastic radiative transfer in more realistic physical scenarios. However, the reported results are far from benchmark ones, because the computational model used was based on P_1/3 equations (i.e., a low-order angular approximation to the radiative transfer equations) coupled to material, which is likely to notably overestimate the transmission flux at the outgoing surface.24 In three dimensions, Brantley and co-workers25–28 have used the implicit Monte Carlo (IMC) method and approximate algorithms to perform numerous simulations for nonparticipating25,26,28 and participating27 binary stochastic mixtures. To the best of our knowledge, however, benchmark results for radiative transfer in participating binary stochastic mixtures are still absent in the case of multiple dimensions.

Regarding the influence of stochastic mixtures on the radiative transfer in multi-dimensions, there appears to be an inconsistency between the results of Olson20,23 and Brantley and Martos.25 Specifically, Olson20,23 obtained similar radiation fluxes and energy densities even when using very different particle sizes, whereas Brantley and Martos25 found that the particle size noticeably affected the transmitted and reflected radiation fluxes. It should be noted, however, that these simulations were carried on the basis of different codes and algorithms and using different physical parameters. To quantitatively resolve this discrepancy, multidimensional benchmark simulations of radiative transfer in participating binary stochastic mixtures are required for the two problems,20,25 for which a physical interpretation may further improve understanding. This is one of the main goals of the present study.

Moreover, a few experiments29,30 have been carried out on the OMEGA laser facility to examine the influence of particle size on radiative transfer. By using intense lasers incident on a hohlraum, a strong radiation source with a peak temperature of about 210 eV was produced, which irradiated a cold binary mixture consisting of a fraction of gold particles randomly distributed in foam material. When the size of the gold particles was increased to 5 μm diameter, an inhomogeneous mix of gold particles and foam was found to have been obtained, since the temperature front position of the heat wave visibly deviated from that in the atomic mix. In this case, an inhomogeneous radiation transport model must be applied in simulations to capture the essential physics. Therefore, the question naturally arises as to how large the high-Z particle need to be to create an inhomogeneous mixture. In the present work, we shall make an attempt at a theoretical interpretation of this problem.

In the present study, we present two-dimensional (2D) benchmark results for radiative transfer in participating binary stochastic mixtures (see Fig. 1 for an illustrative physical configuration), which have not yet been reported elsewhere. For this purpose, we have developed an accurate massively parallel code from scratch for simulating radiative transfer in binary stochastic mixtures in two dimensions (RAREBIT2D), which has been thoroughly validated.24 On the basis of two different sets of physical parameters available in the literature,20,25 we perform systematic 2D simulations to obtain benchmark results. To understand the parameter-dependent influence of radiation fluxes, we analyze the relationship between the mean free path of photons l_p and the system size L. We find that the discrepancy between the results of Olson20 and those of Brantley and Martos25 is actually due to the distinct relationships between l_p and L in these studies, resulting in optically thin and thick mixtures, respectively. Additional 2D simulations further confirm this interpretation. Moreover, nonlinear effects are analyzed in detail. As a final remark, we employ a dimensionless parameter given by the ratio of l_p to L to interpret the effect of particle size on the inhomogeneous mix, and a critical particle size is analytically derived.

The formal theoretical framework is presented in Sec. II. Benchmark results are tabulated and analyzed in Sec. III, and this is followed by an analysis of nonlinear effects in Sec. IV. Section V focuses on the particle size effect on the inhomogeneous mix. Finally, conclusions are drawn in Sec. VI.

We assume that radiative transfer occurs in a 2D Cartesian (xy) domain, which is restricted to 0 ≤ x ≤ L and 0 ≤ y ≤ L, where L is the system size. By taking into account a variety of radiative transfer processes31 of (a) absorption, (b) scattering, (c) emission, and (d) transmission, as depicted in Fig. 1, radiative transfer coupled to material in the gray approximation can be written as1c∂∂t+η∂∂x+ξ∂∂y+σt(t,x,y)ψ(t,x,y,Ω)=cσa(t,x,y)4πaT4(t,x,y)+σs(t,x,y)4π∫ψdΩ′,(1a)ρ(t,x,y)Cv(t,x,y,T)∂∂tT(t,x,y)=−cσa(t,x,y)aT4(t,x,y)+σa(t,x,y)∫ψdΩ,(1b)where t represents the temporal variable and Ω is the direction of flight. The angular variables η and ξ denote the x and y components of Ω, respectively. The radiation specific intensity is denoted by ψ, which is a function of temporal, angular, and spatial variables. For simplicity, the energy variable is omitted. The material temperature is denoted by T. The mass density and heat capacity are denoted by ρ and C_v, respectively. σ_t, σ_a, and σ_s are the total, absorption, and scattering opacities, respectively, which measure the strengths of interaction between radiation and material. a is the radiation constant and c is the speed of light in vacuum.

Owing to the stochasticity of the mixtures considered here, it is difficult to directly solve Eq. (1) for a binary stochastic mixture, since the material-dependent coefficients cannot be uniquely determined at any position.18 An alternative approach15,17,20,32,33 is to sample a number of physical configurations for the binary stochastic mixture and then solve the radiation–material coupled equations for each configuration and perform an averaging of the transport outputs over the ensemble. Figure 1 schematically shows one of the physical configurations for the binary stochastic mixture, which consists of a fraction of high-Z particles (i.e., material 1) randomly distributed in the low-Z background medium (i.e., material 2). It should be emphasized that the absorption opacity of material 1 is higher than that of material 2 by at least one order of magnitude (see Table I). For any position in the domain, the occurrence probability of material 1 in the domain is denoted by p₁, which is calculated as the fraction of the area occupied by 2D particles. In this study, p₁ is also referred to as the “mixing probability.” Therefore, the probability of any point inside the background medium is p₂ = 1 − p₁. To fully characterize the binary stochastic mixture, the particle size distribution and average particle size should also be considered.

Initially, the binary stochastic mixture is cold, with a uniform temperature T0m, and the material is equilibrated with the radiation at any position, and thus we can set the initial conditions asψ(t=0,x,y,η,ξ)=σSBπ(T0m)4,(2a)T(t=0,x,y)=aT0m4.(2b)

Furthermore, the mixture is continually irradiated by an isotropic radiation source installed near the left surface (x = 0) with an initial temperature T0r, while nothing is implemented near the right surface (x = L). The top (y = L) and bottom (y = 0) surfaces are purely reflective boundaries. Accordingly, the boundary conditions are specified asψ(t,x=0,y,η>0,ξ)=σSBπ(T0r)4,(3a)ψ(t,x=L,y,η<0,ξ)=0,(3b)where σ_SB is the Stefan–Boltzmann constant.

Since the binary stochastic mixture is only known statistically, a certain number N of physical configurations are needed to represent it for a given set of physical parameters (i.e., mixing probability, average particle size, and particle size distribution). In this study, the physical observables concerned are the ensemble-averaged radiation transmission flux ⟨J_trans⟩, the reflection flux ⟨J_reflec⟩, the radiation energy density ⟨E_rad⟩, and the material energy density ⟨E_mat⟩, which are computed by〈Jtrans(t,x,y)〉=1N∑n=1N∫η>0[ηψn(t,x,y,Ω)]dΩ,(4a)〈Jreflec(t,x,y)〉=1N∑n=1N∫η<0[|η|ψn(t,x,y,Ω)]dΩ,(4b)〈Erad(t,x,y)〉=1N∑n=1N∫04πψn(t,x,y,Ω)dΩ,(4c)〈Emat(t,x,y)〉=1N∑n=1NaTn4(t,x,y).(4d)

To facilitate the analysis, the radiation fluxes and energy densities along the direction of the initial radiation source are extracted by〈q(t,x)〉=∫0L〈q(t,x,y)〉dyL,(5)where ⟨q(t, x, y)⟩ denotes the physical quantities described above. In particular, we are concerned with the fluxes at the outgoing and entrance surfaces of the binary stochastic mixture, i.e., ⟨J_trans(t, L)⟩, ⟨J_reflec(t, 0)⟩, ⟨E_rad(t, L)⟩, and ⟨E_mat(t, L)⟩, as shown in Tables II–IV. By examining the convergence of the number of physical configurations, we find that ten are more than enough to provide converged results.

In addition, the standard deviation Δ about the ensemble-averaged value is evaluated asΔ=〈q2〉−〈q〉2,(6)which represents a measure of the uncertainty of the ensemble-averaged mean value. It should be mentioned that the uncertainty originates from the variability of physical configurations.

Within the present framework, the simulation code RAREBIT2D has been developed.24 A detailed introduction to this code is beyond the scope of the present study. Instead, we give a brief explanation in the Appendix.

In this subsection, we first employ two different sets of physical parameters to simulate the problems described by Olson20 and Brantley and Martos.25 Relevant parameters are summarized in Table I, and informative results are shown in Figs. 2 and 3. We also present data obtained using the RAREBIT2D code in Tables II and III, specifically for the steady-state time. For clarity, the case notations are the same as in the previous studies.20,25 These data should be useful for validating other codes, theoretical models, or approximate approaches developed by other authors.

For the problem considered by Olson,20 we employ the set 1 parameters in Table I. In accordance with the previous study, the radiation constant is set to a = 1 and the heat capacity is in the form C_v = 4T³. Figure 2 presents the ensemble-averaged radiation transmission flux at the exiting surface as a function of time. Uncertainties are not indicated on top of the mean values, since they are lower than 1% in most cases. Figure 2(a) presents results for three average particle sizes considered, i.e., cases A, B, and C in Table II. By varying the average particle size, the number of particles is changed, provided that the mixing probability and particle size distribution are the same. As can be seen, the transmitted fluxes are nearly identical before 50 ps, after which a divergence is visible, especially at the steady-state times (see Table II). Clearly, the transmission is greatest for the particles with the largest average size (case A in Table II). When the average particle size is decreased by a factor of 3 (case B) and 10 (case C), the ensemble-averaged radiation transmission flux is reduced by about 10% and 15%, respectively. The dependence of the mean radiation energy on the average particle size is similar, but the material temperature is modified only within ∼4%. This observation is entirely different from that in Ref. 20, where the ensemble-averaged fluxes and energy densities are nearly independent of particle size.

On varying the particle size distribution (see the Appendix for analytical expressions) in Fig. 2(b), the results are hardly affected,34 which agrees well with the observation by Olson.20 By varying the mixing probability [see Fig. 2(c)], the smallest mean radiation transmission flux is obtained for p₁ = 0.2 (case F), which is lower than that for p₁ = 0.02 by about 16%. However, Olson concluded that it was insensitive to the mixing probability.

It should be noted that a particularly interesting case was explored in Ref. 20, namely, case G in Table II. Compared with case A, the particle opacity in case G is increased by more than one order of magnitude. Olson20 found that the mean radiation transmission flux at the steady-state time was decreased by ∼1.9% and therefore concluded that making the disks more opaque had little effect. By contrast, the present results show that it is reduced by about 18.5%, which is a more appreciable reduction than in the previous study. The difference may be attributed to the use of a P_1/3 equation in Ref. 20, which is the lowest-order approximation in angle to the radiative transfer equations.35

For the problem considered by Brantley and Martos,25 the set 2 parameters in Table I are used to perform 2D simulations. Systematic results are tabulated in Table III. For comparison, time-varying ensemble-averaged radiation transmission fluxes at the outgoing surface are displayed in Fig. 3 for selected cases. For p₁ = 0.1 and a constant size distribution, Fig. 3(a) shows that increasing the particle size results in more radiation energy being transmitted, which generally agrees with Fig. 2(a). It seems that the effect of particle size on the radiation flux is obvious from very early times. When the particle size (case 3 in Table III) is decreased by a factor of 2 (case 2) and 4 (case 1), the mean transmitted flux at the steady-state time (i.e., 20 000 ps) is decreased by ∼20% and ∼37%, respectively, both of which reductions are much greater than that in Fig. 2(a). If the average particle size is varied within an order of magnitude, which is exactly what Olson20 did, it can be speculated that the mean transmitted flux may be decreased more. For three kinds of particle size distributions, a variation of less than 5% is obtained in Fig. 3(b), which is almost undetectable. It should be noted that the particle size distribution has little effect in either Fig. 2(b) or Fig. 3(b). In Fig. 3(c), the influence of the mixing probability is very significant when p₁ is increased from 0.05 to 0.3, since the mean transmitted fluxes at the steady-state time are decreased by a factor of ∼4.3. In addition, the arrival time of the radiation at the outgoing surface is apparently delayed when p₁ is increased. Similar results can also be found for other cases in Table III.

In summary, the present 2D results show that varying the particle size distribution has very little effect for either the Olson or the Brantley and Martos parameters, provided that the mean chord length16 is identical among the distributions considered. By varying the average particle size and mixing probability, a weak dependence of radiation flux is found for the Olson parameters, while for those of Brantley and Martos, a remarkable dependence is observed. Basically, our results agree qualitatively with the findings in Refs. 20 and 25, respectively. Our results should be more consistent, since they have been obtained by using the same simulation tool in an independent study.

Indeed, our calculated results demonstrate that physical parameters can strongly affect the influence of stochastic mixtures on radiative transfer, which has not been properly understood in previous studies. Basically, radiative transfer is closely related to the photon mean free path37l_p, i.e., the average distance to a collision. If the number of l_p’s through the domain (i.e., the optical depth) is less than 1, then the stochastic mixture is optically thin, the probability of the photon interacting with the mixture is small, and thus the influence of the binary stochastic mixture on radiative transfer is weak. On the other hand, if the number of l_p’s through the domain is much larger than 1, then the mixture is optically thick, and the photon most probably interacts with the mixture multiple times before it finally traverses the domain, and consequently the radiation fluxes and energy densities are remarkably affected by the stochastic mixture. Keeping this in mind, a number of calculations have been carried out by varying the system size L for the two sets of parameters in Table I. Taking into account negligible standard deviations by adding more physical configurations, only one physical configuration is generated for each case. For clarity, the enhancement factor is defined as the ratio of the highest radiation transmission flux to the lowest value when changing the physical parameters.

Figure 4 presents calculated 2D results at the steady-state time at the outgoing surface. For the cases in Table II, l_p is roughly equal to 1.71 cm in case A, which is regarded as the reference. By tuning the particle size distribution, the average particle size, and the mixing probability, l_p is changed to 1.86 cm (case D), 1.1 cm (case C), and 1.0 cm (case F), respectively. Obviously, l_p is always larger than or equal to the system size L = 1 cm used by Olson,20 i.e., l_p ≥ L, and the mixture is then close to optically thin, which explains the insensitivity of the radiation to the physical parameters observed in Ref. 20. In addition, it is expected that decreasing L will result in l_p ≫ L (e.g., L = 0.1 cm), for which the radiation flux is totally independent of the details of the stochastic mixture, which is exactly what is seen in Fig. 4(a). Physically speaking, when the mixture is made optically thick by increasing the system size L, a pronounced influence of the mixing probability and the average particle size will show up, which agrees well with the 2D simulation results in Fig. 4(a).

Of the cases in Table III, we take case 1 as an example. For p₁ = 0.05 and a constant particle size distribution, l_p is calculated as ∼3.64 cm, and it is modified to roughly 3.89, 6.47, and 0.65 cm for an exponential size distribution, λ₁ = 11/10 cm, and p₁ = 0.3, respectively. Compared with the system size L = 10.0 cm given in Ref. 25, the relationship of l_p ≪ L is always satisfied, corresponding to an optically thick mixture. For this reason, a remarkable dependence on the mixing probability and size is completely understandable, since l_p is significantly varied in this case. Also, it can be speculated that the influence becomes stronger when L is increased and is suppressed when it is decreased [see Fig. 4(b)].

Our 2D results and analysis have confirmed that the observed discrepancy between Olson20 and Brantley and Martos25 arises from distinct relationships between l_p and L owing to the differences in the physical parameters, resulting in optically thin and thick mixtures, respectively. In this sense, previous multidimensional results are basically consistent. Moreover, we have demonstrated in 2D that the dependence of the radiation fluxes on the stochastic mixture can be made (in)significant by varying the system size and/or opacities.33

Two types of nonlinear effects are of interest in this study, namely, the scattering effect and the radiation–material coupling effect, neither of which have been analyzed in detail in previous studies. The scattering effect couples the photons at all angles. To quantify its strength, the scatter albedo is defined as ω = σ_s/σ_t. Since we consider only isotropic scattering, the transmitted radiation flux should be unaffected, but the equilibrium between the radiation and material will be changed. To verify this, it is not practical to recalculate all cases presented in the above tables with the scattering term included, and instead some typical cases are investigated. For case F in Table II, Fig. 5 presents temperatures at the outgoing surface.

We indeed find that varying ω(<1) hardly changes the amount of transmitted radiation flux (not shown), but it makes a great difference for temperature equilibration (see Fig. 5). On increasing ω, the growth of material temperature T_m is considerably slowed down, with, for example, the maximum temperature being achieved at about 1000 and 10 000 ps for ω = 0 and 0.9, respectively. However, the final radiation temperature is almost unaffected. In the extreme case of ω = 1 (not shown), the material is decoupled from radiation, and pure transport by scattering is dominant. In this case, more radiation is likely to be transmitted, owing to the absence of the material’s energy channels.

On the other hand, the radiation–material coupling effect is analyzed using the temperature-dependent opacity. The physical parameters are taken from Table 4 in Ref. 20, and six sets of parameters have been simulated. Results are listed in Table IV, in which the opacity values σ_a are those of the coefficient in the opacity function σ(T) = σ_a/T², where T is the material temperature in units of keV. For comparison, results obtained for constant opacity are included in the parentheses in Table IV. Generally, the radiation flux and energy density, and the material temperature, are smaller for temperature-dependent than for constant opacity; for example, the transmitted radiation flux is underestimated by factors of ∼2.2, 2.8, 3.6, 4.0, 4.5, and 4.0 for cases Ht1–3 and Ft1–3 in Table IV, respectively. It seems that increasing p₁ will enhance the influence of temperature-dependent opacity. For instance, compared with case Ht1, the mixing probability p₁ for case Ft1 is increased by a factor of 4. In this case, we see that the radiation flux for constant opacity is lower by a factor of ∼1.7, while for temperature-dependent opacity, it is reduced by ∼3.1.

Figure 6 presents the transmitted radiation flux and material temperature at the outgoing surface as functions of time for case Ft1. It is found that the radiation flux for constant opacity initially grows rapidly, but then levels off within about 300 ps, while for temperature-dependent opacity it increases gradually and reaches an asymptotic value within 1500 ps. The material temperature behaves similarly before ∼300 ps, after which it gradually increases until 5000 ps, reaching T = 197.8 eV for constant opacity. By contrast, it rapidly attains a leveled-off value of T = 138.9 eV for temperature-dependent opacity. Accordingly, it is speculated that the respective evolutions of the temperature distribution for constant and temperature-dependent opacities may also differ remarkably.

The material temperature distribution is shown in Fig. 7 for four snapshots: ct = 1.0, 5.0, 12.5, and 40.0 cm. For constant opacity (upper panels), at early times, the particles are hotter than the background material [see Figs. 7(a) and 7(b)], which may be attributed to incident radiation transferring more energy to the particles. As the particles are heated up to appreciable temperatures [see Figs. 7(c) and 7(d)], radiation emission from particles plays a role in warming up the nearby background material. No definite temperature interface can be identified. For temperature-dependent opacity (lower panels), the incident radiation rapidly heats up the particles near the left surface (x < 0.2 cm) [see Fig. 7(e)], while particles in other regions remain relatively cold (black colors), which results in a pronounced temperature gradient. Compared with Fig. 7(a), the radiation is evidently delayed, because particles are more opaque. As time evolves, the temperature interface is shifted toward the right surface [Fig. 7(f)] and is even nearly smeared out in Fig. 7(h).

A binary stochastic mixture has two mixing limits.38 (1) If the particle size is at the atomic scale (e.g., several nanometers), it is considered to be a microscopically homogeneous mixture, also known as an “atomic mix.”18 In this situation, an effective opacity σ_eff can be expressed as σeffatomic=p1σ1+p2σ2,15,21 where p_i and σ_i are the occurrence probability and opacity of the material i (i = 1, 2), respectively. (2) I the particle size is at the macroscopic scale (e.g., from submicrometers to micrometers), the mixture is considered to be macroscopically homogeneous, a “chunk mix.” The effective opacity can be written as σeffchunk=1/(p1/σ1+p2/σ2).21 In this context, it can be inferred that there may exist a critical point at which a microscopically inhomogeneous mix can be initiated by varying the particle size from that corresponding to an atomic to that corresponding to a chunk mix. Therefore, the question naturally arises as to how to determine this desired particle size—put differently, what is the particle size that results in a distinct mixing different from an atomic mix?

Actually, the particle size effect on radiative transfer has previously been investigated for gold particles in experiments.29,30 Specifically, the positions of the temperature front were measured to determine whether the mixture was inhomogeneous or not. For instance, the front positions after 2 ns for a particle size of 5 μm were larger than those for an atomic mix by ∼100 μm, which is experimentally regarded as an inhomogeneous mix. This certainly is a valid approach. However, if other high-Z elements are considered, the critical particle size for the inhomogeneous mix is most probably different. For this reason, we attempt here to seek another means from a theoretical perspective.

We employ a dimensionless parameter β, which is defined as the ratio of the mixture’s mean free path l_p to the size of system L, i.e.,β=lpL=1+(p1σ2+p2σ1)qp2λ1(p1σ1+p2σ2+σ1σ2qp2λ1)L,=1+σ1σ2lchunkqp2λ1(1/latomic+σ1σ2qp2λ1)L,=1+σ1σ2qp2λ1lchunk1+σ1σ2qp2λ1latomiclatomicL,(7)where we have used the expression for l_p proposed by Levermore et al.39l_atomic and l_chunk are the mean free paths of atomic and chunk mixes, respectively, which are defined as the inverses of the corresponding effective opacities described above. λ₁ denotes the average chord length of the particle,16,39 which is related to the mean particle size ⟨r⟩ by λ₁ = π⟨r⟩/2 for the constant size distribution.36q represents a correction factor due to the departure of the chord length distribution from a Markov distribution.39 For the given material parameters, it should be noted that the range of β is easily determined, i.e., l_atomic/L ≤ β ≤ l_chunk/L. If there exists a critical dimensionless parameter β_c satisfying the relationship β ≥ β_c for the inhomogeneous mix, then a critical particle size can be derived asλ1≥Lβc−latomicσ1σ2qp2latomic(lchunk−Lβc).(8)

At this stage, the remaining task is to determine β_c. For this purpose, a series of 2D simulations are carried out using the physical parameters from our recent work.33 In practice, a squared domain with a side length of L = 0.15 cm contains 10% spherical high-Z particles (material 1) and 90% low-Z background medium (material 2). The absorption opacities are fixed to σ_a,1 = 1000 cm⁻¹ and σ_a,2 = 5 cm⁻¹. For simplicity, the scattering term is neglected, a single physical configuration is generated, and the constant size distribution is applied. With these physical parameters, β ranges from 0.064 to 1.201. Figure 8 shows the simulated transmitted flux at the outgoing surface vs β. Note that the fluxes have been normalized by those of the atomic mix.

Immediately, two distinct regimes can be seen in Fig. 8. (a) For β < 0.13, the radiation flux is linearly dependent on the photon mean free path of the mixture. In this regime, it seems that the radiation flux for small particles can be linearly extrapolated from that for the atomic mix, provided that the photon mean free path of the stochastic mixture is attained. (b) With increasing particle size, the flux satisfies a different linear dependence (red line) on a logarithmic scale. Our understanding is that this regime is characterized by a microscopically inhomogeneous mix. As indicated in Fig. 8, a crossing point is found at β = 0.093. As a rough estimate, this can be taken as β_c, corresponding to a critical particle size R_c = 6.4 × 10⁻⁴ cm = 6.4 μm. It should be mentioned that this value cannot be directly compared with that obtained in experiments,30 since initial conditions and material parameters are quantitatively different. According to Eq. (8), the criterion for determining the inhomogeneous mix is obtained as λ₁ ≥ 0.001 cm, which coincides with the mean free path of photons through the particles, i.e., the inverse of the particles’ opacity (1.0/σ_a,1).18 This finding is supported by additional simulations with other parameters.

Furthermore, Fig. 9 shows the 2D material temperature distribution for particle sizes varying from 0.0005 to 0.01 cm at ct = 0.15 cm. For comparison, the atomic and chunk mix cases are also considered. It should be noted that the particle profiles in Figs. 9(b)–9(g) can be clearly seen, because the particle temperature is higher than that of the surrounding medium. Comparison with the atomic mix with β = 0.064 in Fig. 9(a) shows that the material temperature distribution for β = 0.088 [i.e., R = 0.0005 cm in Fig. 9(b)] is similar, except for the temperature fluctuation along the y direction. If β is increased to 0.113, the temperature front is visibly shifted owing to a deeper penetration of radiation [see Fig. 9(c)], which is consistent with the criterion β > β_c = 0.093 for an inhomogeneous mix. A close inspection reveals the existence of particle shielding of radiation, which is more obvious for larger β in Figs. 9(d)–9(f). Owing to the random sampling of particle locations, some particles near the left boundary may shield the downstream particles, resulting in shielded particles being colder (see the particles in the shadow region). On the other hand, particle self-shielding of radiation can also be observed, especially for β = 0.505 in Fig. 9(g), where a colder region behind the particles is seen. When the particle size is sufficiently large, the temperature distribution tends to that of the chunk mix.

In this study, we have theoretically investigated radiative transfer in participating binary stochastic mixtures in two dimensions. Our 2D benchmark results have been systematically presented for two different sets of physical parameters used in previous studies. To interpret parameter-dependent results, we have evaluated the relationship between the photon mean free path l_p and the system size L. We have found that it actually describes optically thin stochastic mixtures using the parameters for the problem considered by Olson,20 and thus an insensitivity of ensemble-averaged transport outputs to physical parameters is observed, whereas the parameters for the problem considered by Brantley and Martos25 correspond to optically thick mixtures, resulting in a remarkable dependence on the mixture properties. In both cases, we have confirmed that the particle size distribution makes no difference (<5%), provided that the same mean chord length is considered, which is attributed to a negligible variation of l_p.

The present 2D results and analysis support the basic consistency of the multidimensional results obtained in Refs. 20 and 25, with the deviations being due to the distinct relationships between l_p and L resulting from the use of different physical parameters. To further confirm this interpretation, extra simulations have been performed by varying the system size, and we have found that the dependence of radiation on the physical parameters can be made (in)significant by varying the system size and/or opacity.

With regard to nonlinear effects, it has been found that the scattering effect has little influence on the transmitted radiation flux, but the temperature equilibration time between radiation and material is enhanced by almost one order of magnitude for the case considered. On the other hand, the nonlinear effect due to radiation–material coupling dramatically reduces the radiation flux, energy density, and material temperature. A more pronounced difference in the evolution of the temperature distribution has been found between the cases of constant and temperature-dependent opacities.

In contrast to our analysis of temperature fronts in experiments, to understand the relationship between particle size and the inhomogeneous mix, we have employed a dimensionless parameter β given by the ratio of l_p to L. A critical particle size has been derived, which is material-dependent. Combining the results of 2D simulations has provided a criterion to determining the inhomogeneous mix: specifically, the mixing can be considered inhomogeneous for l_p/L > 0.1, below which a homogeneous mix emerges. This finding has been further confirmed by additional simulations with other parameters. Moreover, a particle size effect is clearly seen in the 2D material temperature distributions, which is consistent with the criterion for an inhomogeneous mix. It also reveals that particle shielding of radiation occurs, i.e., particle self-shielding and particles close to the radiation source may shield downstream particles, and this effect can be enhanced by increasing l_p/L.

In the present work, we have used the gray approximation to describe radiative transfer. In realistic problems, the radiation flow is contributed by various photon frequencies, corresponding to very different photon mean free paths, which will result in a significant dependence of the mixture’s optical depth on photon frequency. As radiation is transmitted and blocked in the optically thin and thick cases, respectively, this effect may spread out the profile of the transmitted radiation flux and energy density compared to those in the gray approximation. Although the gray approximation has its own limitations, it has been widely used to understand the radiation flow in stochastic mixtures,15,17,20,26,32,33,38,40–44 and we expect that the present results will be helpful in verifying effective opacity models or other approximate approaches developed for 2D stochastic radiative transfer within this approximation. In the next step of our work, which is currently under way, an extension to multigroup transport is necessary.

Acknowledgment. The authors thank the anonymous referees for their insightful comments. The first author (C.-Z.G.) acknowledges financial support by the National Natural Science Foundation of China (Grant No. 12374259). Z.-F.F. was funded by the National Natural Science Foundation of China under Grant No. 12375235.