Mie scattering from submicron-sized CO<sub>2</sub> clusters formed in a supersonic expansion of a gas mixture

S. Jinno; Y. Fukuda; H. Sakaki; A. Yogo; M. Kanasaki; K. Kondo; A.Ya. Faenov; I.Yu. Skobelev; T.A. Pikuz; A.S. Boldarev; V.A. Gasilov

doi:10.1364/OE.21.020656

1. Introduction

At the present time, cluster-gas targets are widely used as one of the promising media for the femtosecond laser-matter interaction [1], including nuclear fusion [2], x-ray emission [3, 4], betatron x-ray radiation [5], laser-driven ion acceleration [6,7], and so on. Because the cluster-gas targets are produced using typically a gas jet nozzle, the design of the gas jet nozzle is the key to such experiments success.

Therefore, we have designed the gas jet nozzle based on a mathematical model of the clusters formation in the gas flows in the nozzle [8]. It is considered to be able to produce the clusters of about 1 μm size for pure rare gases (Ar, Kr, and Xe) according to our computations. The cluster-gas targets have been demonstrated to be useful as a clean x-ray source for x-ray diffraction experiments [4]. However, the x-ray emissions are limited due to the absorption by a significant quantity of residual gas around the laser interaction region.

To avoid this problem, we have used the cluster-gas target produced by the gas jet of the mixture gas of 90% helium and 10% CO₂ instead of the pure CO₂ gas through the nozzle [9]. Since the helium gas helps clusterization of molecular gases, a small quantity of CO₂ gas is enough to produce large CO₂ clusters, while the helium gas does not clusterize and remains as a residual gas. In this case, the absorption of soft x-ray emission in the residual gas is significantly reduced due to the transparency of helium to the soft x-rays. This results that the number of photons for the CO₂/He mixture gas is four to seven times larger than that produced with the pure CO₂ gas.

Furthermore, we have observed a long plasma channel such as promising laser-driven particle accelerations using the cluster-gas target produced from the CO₂/He mixture gas. In fact, efficient generation of high energy ions up to 50 MeV per nucleon was achieved [6, 7] beyond traditional methods. Thus the targets consisting of clusters embedded in a background gas produced by the appropriate mixture gases have a potential as a variety of radiation sources.

In this manner, some experiments using the binary gas mixture have been already performed, whereas the mathematical model of the clusters formation for the binary gas mixture has not been reported in the past. Therefore, we introduce the mathematical model of the clusters formation in the supersonic expansion of the binary gas mixture with one inert component in vacuum and evaluate experimentally the model in this study. According to the numerical modeling, the existence of submicron-sized clusters in the target is predicted. Because the conventional Rayleigh scattering method cannot apply to this size range, we have developed an apparatus capable of measuring micron-sized particles using Mie scattering of the second-harmonics of Nd:YAG laser pulses [10]. Because the detailed knowledge about the initial target parameters is the key for complete understanding of laser-cluster interactions, furthermore, we have measured total gas densities and cluster mass fractions besides the cluster size in this study. There was no accepted method for measuring the cluster mass fraction with the exception of frequency domain holography (FDH) recently reported [11]. In contrast, we have introduced another approach extracting the cluster mass fraction taking advantage of the Mie scattering theory which is applicable to large clusters, i.e., it was derived in combination with the Mie scattering method, an attenuation measurement of the laser beam intensity, and an interferometry.

In this paper, we consider the mathematical model in Sec. 2. This model is the generalization of the clusterization theory [12] on the case of the mixture of several gases and give calculated values of the different cluster parameters for various initial gas pressures and the gas compositions. In Sec. 3, we describe measurement and analytical methods using Mie scattering in order to investigate the spatial structure of the cluster-gas targets. In addition, the interferometer and the laser attenuation are employed to spatially measure the total gas densities and cluster densities. We describe the comparison of the numerical modeling and the experimental results in Sec. 4 and discuss in Sec 5. Such comparison shows that considered clusterization theory can be used successfully to model submicron-sized cluster production. Finally, we conclude in Sec. 6.

2. Mathematical modeling of the clusters origination in nozzle flows of the binary gas mixture

The mathematical model of the clusters formation in a single gas flow is described in details in [8] and [12]. Here we present its generalization for the case of the binary mixture with one inert component. In fact, this new model is based on the same physical suppositions, but takes into account the presence of the inert component.

We consider a gas mixture with two components: inert (which we denote by index i) and clusterizing, or condensing one (denoted by index c). The condensing component, in turn, is subdivided into gas phase (index g) and liquid (or condensed) one, denoted by index l. So, we have, in fact, three components of our medium: inert component i, condensing gas cg and clusters cl.

Since we assume that the clusters are comparatively small and do not move with respect to the surrounding gas, we may consider the composition of our mixture (namely, mass or molar relation between i and c components) to be constant and known. So, we introduce the value ϕ as the mass fraction of the condensing component. So, we can write for the densities of the components

ρ_{c} = ϕ ρ, ρ_{i} = (1 - ϕ) ρ .

We introduce also the so-called dryness degree β which is the mass fraction of the cg subcomponent in the whole c component. So, we can also write

ρ_{c g} = β ρ_{c} = β ϕ ρ, ρ_{c l} = (1 - β) ρ_{c} = (1 - β) ϕ ρ, ρ_{i} = (1 - ϕ) ρ .

In these expressions ρ_cg, ρ_cl, and ρ_i are densities of the components of the mixture, i.e., mass of a component per single volume of the whole mixture. Since the volume of the clusters is negligibly small, for the gas components these are also the densities of these components themselves, which appear in all thermodynamic relations for this component. For the clusters, of course, ρ_cl is much smaller than the density of the liquid phase itself, which we denote as $ρ_{l}^{0}$ .

The process of clusters formation is described in kinetics terms and, of course, this process is essentially nonequilibrium. But we consider two gas components i and cg to be in thermodynamic equilibrium with each other. So, we suppose that the temperature of two components of the gas phase is the same T_i = T_cg = T (and may differ from the temperature of the clusters T_cl). Another supposition is that the pressure of the mixture, in accordance with Dalton law, is the sum of the partial pressures of the gaseous components, P = P_i + P_cg. The liquid phase does not render the pressure, since it is not continuous.

These suppositions allow us to consider the density and internal energy of the whole mixture, not of the individual components. So, the governing system of equations practically does not differ from the single-gas case:

\frac{\partial ρ}{\partial t} + div (ρ v) = 0,

\frac{\partial ρ v}{\partial t} + div (ρ v \otimes v) = - grad P,

\frac{\partial}{\partial t} (ρ ε + ρ \frac{v^{2}}{2}) + div (ρ v (ε + \frac{v^{2}}{2})) = - div P v,

\frac{\partial ρ Ω_{n}}{\partial t} + div (ρ v Ω_{n}) = I r_{*}^{n} + \dot{r} n ρ Ω_{n - 1}, n = 0, 1, 2,

\frac{\partial ρ β}{\partial t} + div (ρ v β) = - \frac{4}{3} \frac{π ρ_{l}^{0} I r_{*}^{3}}{ϕ} - \frac{4 π ρ ρ_{l}^{0} \dot{r} Ω_{2}}{ϕ} .

Here Ω_n are the moments of the distribution function of the clusters with respect to their radii:

ρ Ω_{n} = \int_{0}^{\infty} r^{n} f (r, x, t) d r .

So, we can write for the third moment

1 - β = \frac{4}{3} \frac{ρ_{l}^{0} π Ω_{3}}{ϕ} .

We use the same model of kinetics of the phase transitions as in the single-gas case, but with taking into account the presence of the inert component which changes the gasdynamics parameters but does not take part in the clusters origination process itself. To close the system Eqs. (3)–(7), we use the following relationships. The critical radius

r_{*} = \frac{2 σ (T)}{ρ_{l}^{0} R T ln (P_{c g} / P_{s} (T))} .

The nucleation rate

I = \frac{1}{ρ_{l}^{0}} \sqrt{\frac{2 σ (T) μ}{π N_{A}}} {(\frac{P_{c g}}{k T})}^{2} exp (- \frac{4 π σ (T) r_{*}^{2}}{3 k T}) .

The rate of clusters growth

\dot{r} = \frac{P_{c g}}{ρ_{l}^{0} \sqrt{2 π R T}} (1 - \sqrt{\frac{T}{T_{s} (P_{c g})}}) .

The model includes also some thermodynamic relationships which express the properties of the working gas and close the system of equations. To close the system, we need expressions P(ρ, ε, β), T(ρ, ε, β) and the expressions for some values at the saturation line. In the single-gas model we used expressions

P = P (ρ, ε, β) = ρ \frac{ε + (1 - β) L_{s}}{\frac{γ}{β (γ - 1)} - 1} . T = \frac{P}{β Z ρ R},

which are the consequences of the simple equation of state of the ideal gas and the relationships between the heat of phase transition and the enthalpies of the phases,

L_{s} = H_{g} - H_{i} .

In the gas mixture model, we apply Eq. (13) to each of the components i and c. As a result, instead of direct expressions Eq. (13) we obtain a system of algebraic equations, which includes Eq. (13) for both components, equality of the temperatures of the components and the relationship between the specific internal energies of the components:

ε = (1 - ϕ) ε_{i} + ϕ ε_{c} .

In Eq. (13), we assume that the gas can be described by the equation of state of the ideal gas, and introduce the compressibility coefficient Z to take into account the properties of the actual working gas. When the gas is in the condition close to the saturation line, this supposition becomes very inaccurate. So, the use of some realistic equation of state is desirable. In fact, the equations of state of various gases are known, for example, they are available in NIST database [13].

To close the gas dynamics system of equations (or its modifications such as Eqs. (3)–(7)), the equation of state should be resolved in the form P = P(ρ, ε) (or, for our case, P = P(ρ, ε, β)). But, usually, the equations of state of the real gases are given in the form of functions of some other pair of parameters, for example, (ρ, T), since for this pair of parameters there is a thermodynamic potential — Helmholtz energy. So, in order to use such an equation of state, we must solve a system of algebraic equations with respect to the unknown variables. In addition, since we consider the multiphase medium (even in the single-gas case), some additional relationships should be added to this system to obtain the equation of state of the whole medium. In the case of the single working gas which obeys the equation of state of the ideal gas, such a system may be resolved analytically (resulting in Eq. (13)), If the gas obeys some complicated equation of state, the system cannot be resolved analytically, and some numerical procedure is needed to resolve the system each time when the access to the equation of state takes place in the computation.

In the case of two gases, the numerical solution of an algebraic system is needed even if we consider both gases to be ideal. So, the transition to the real equation of state is not so complicated as in the case of the single gas.

In the computations, usually, we know the values ρ, ε, and β characterizing the whole mixture. To define other thermodynamic parameters, we solve the system

T_{c l} = T_{s} (P_{c g}),

P_{i} = P_{i} ((1 - φ) ρ, T),

P_{c g} = P_{c g} (φ β ρ, T),

P_{i} + P_{c g} = P_{c l} (ρ_{l}^{0}, T_{c l}),

ε = (1 - φ) ε_{i} ((1 - φ) ρ, T) + φ β ε_{c g} (φ β ρ, T) + φ (1 - β) ε_{c l} (ρ_{l}^{0}, T_{c l})

with respect to the values T, T_cl, P_i, P_cg,

ρ_{l}^{0}

.

With use the model Eqs. (3)–(13), we explored the clusters formation in the mixtures He+CO₂, and H₂+CO₂. A series of one-dimensional computations has been performed with different initial pressures and concentrations of the condensing component. We took the three-staged nozzle [8]. In Tables 1, 2, 3 the results of these computations are presented. Table 1 contains the values of dryness at the nozzle’s outlet section, Table 2 presents the densities of clusters, and Table 3 gives the mean value of clusters’ radius. Comparison of the results shows that the presence of the inert component significantly changes the parameters of the clusters. While for the pure CO₂ the characteristic values of dryness is about 74–90%, in the mixture the mass fraction of clusterized CO₂ reaches 65–75%, i.e., the main part of the clusterizing component of the mixture can be turned to clusters.

Table 1. Dryness for various molar concentrations of the condensing component and initial pressures.

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Table 2. Clusters densities, 10⁹ cm⁻³, for various molar concentrations of the condensing component and initial pressures.

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Table 3. Cluster radius, μm, for various molar concentrations of the condensing component and initial pressures.

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Since the gas flow in the nozzle is practically adiabatic, one usually considers that this process is characterized by constant entropy. It is the case for the single-gas flow as well as for the binary gas mixture. But, in the mixture, only the total entropy of the mixture is constant, while there is sufficient entropy exchange between the inert and condensing components. That is why the cluster formation process progresses completely different in the single-gas case and in the binary mixture, even if the initial partial pressure of the clusterizing component is the same as in the single-gas case. In other terms, the condensation process results in some heat release. In the single-gas case, all this heat of phase transition is absorbed by the condensing gas, its temperature increases (the condensation shock), and this results in deceleration of the condensation. Meanwhile, in the binary mixture, some part of the heat of phase transition is absorbed by the inert component, so, the heating of the condensing component is less than in the single-gas case. Consequently, the presence of the inert component facilitates the clusterization process.

In order to investigate the spatial structure of the cluster target, some of these variants were also computed in the 2D formulation (see below Sec. 2.1).

2.1. Spatial structure of cluster targets formed by gas jets

Although the processes of clusters origination and growth take place mostly inside the nozzle, some gasdynamical processes take place in the free jet exiting the nozzle, for instance, the lateral expansion of the jet into the surrounding vacuum. That, in turn, may result in some additional phase transitions and, as a result of these processes, the cluster target is not completely spatially homogeneous but has some spatial structure.

Usually, the nozzles with low expansion angles are used, and the flow in the outlet section is practically homogeneous. Nevertheless, since outside the nozzle the vacuum is maintained, downstream the nozzle outlet section the expansion of the jet is observed (Fig. 1). At the periphery of the jet, the gasdynamic rarefaction wave is formed. Thus, the gas density profiles at different distances from the nozzle’s outlet have different width: more far from the nozzle, the higher is the width of the jet. Besides, there is essential difference between the near and far sections. If we consider the sections more close to the nozzle than the point C (intersection of the leading edge of the rarefaction wave with the axis), there is a domain of unperturbed jet near the axis. So, the density profiles for different distances from the nozzle have a plateau near the r = 0 point. When we increase the distance from the nozzle, the plateau becomes narrower, but the maximum value of density (e.g., the height of this plateau) practically remains the same, since the unperturbed jet exiting the nozzle is practically homogeneous. On the contrary, for the sections more far from the nozzle than the point C we observe the decrease of the maximum density value with the moving off from the nozzle. At Fig. 2 we present the results of numerical modeling which confirm these positions.

Fig. 1 Scheme of spatial structure of a supersonic jet expanding to vacuum and density profiles at different distances from the nozzle.

		10 bar	20 bar	30 bar	40 bar	50 bar	60 bar
He+CO₂	10%	1	1	1	0.992	0.760	0.341
	20%	1	0.989	0.545	0.408	0.326	0.269
	30%	1	0.678	0.535	0.456	0.400	0.371
	40%	0.999	0.678	0.583	0.525	0.489	0.442
	50%	0.979	0.704	0.633	0.595	0.547	0.505
	100%	0.905	0.836	0.795	0.766	0.737

H₂+CO₂	10%	1	1	1	1	1	1
	20%	1	1	1	1	0.987	0.701
	30%	1	1	0.998	0.722	0.613	0.579
	40%	1	1	0.756	0.669	0.630	0.581
	50%	1	0.916	0.734	0.690	0.641	0.596
	100%	0.905	0.836	0.795	0.766	0.737

		10 bar	20 bar	30 bar	40 bar	50 bar	60 bar
He+CO₂	10%	0	0	0	0.0325	0.577	3.89
	20%	0	0.0180	1.27	4.97	9.69	15.6
	30%	0	0.506	2.61	5.38	9.51	15.0
	40%	0.00252	0.837	2.37	4.85	9.03	19.9
	50%	0.0173	0.786	2.00	4.29	11.4	27.4
	100%	0.0222	0.253	1.73	17.7	5420

H₂+CO₂	10%	0	0	0	0	0	0
	20%	0	0	0	0.00139	0.103	0.427
	30%	0	0	0.0199	0.282	1.19	1.70
	40%	0	0.00280	0.224	0.854	1.32	2.13
	50%	0	0.0449	0.494	0.879	1.63	3.29
	100%	0.0222	0.253	1.73	17.7	5420

		10 bar	20 bar	30 bar	40 bar	50 bar	60 bar
He+CO₂	10%	0	0	0.111	0.178	0.227	0.180
	20%	0	0.228	0.230	0.177	0.161	0.150
	30%	0.134	0.279	0.212	0.194	0.180	0.168
	40%	0.207	0.262	0.234	0.213	0.194	0.164
	50%	0.298	0.281	0.257	0.231	0.188	0.156
	100%	0.632	0.447	0.302	0.168	0.0280

H₂+CO₂	10%	0	0	0	0	0	0
	20%	0	0	0	0.0738	0.157	0.349
	30%	0	0	0.145	0.401	0.299	0.294
	40%	0	0.119	0.417	0.327	0.322	0.307
	50%	0	0.454	0.358	0.350	0.327	0.289
	100%	0.632	0.446	0.301	0.168	0.0280

Abstract

1. Introduction

2. Mathematical modeling of the clusters origination in nozzle flows of the binary gas mixture

2.1. Spatial structure of cluster targets formed by gas jets

3. Measurement and analytical methods

3.1. Derivation of cluster size distribution

3.2. Spatial distribution of clusters

3.3. Interferometry

4. Results and comparison with numerical modeling

4.1. Characterization of 30% CO2 + 70% H2 mixture

4.2. Cluster density

4.3. Total gas density profile

4.4. Dryness

4.5. Characterization of 10% CO2 + 90% He mixture

5. Discussion

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (15)

Tables (3)

Equations (32)

Optics Express

4.1. Characterization of 30% CO₂ + 70% H₂ mixture

4.5. Characterization of 10% CO₂ + 90% He mixture