Axisymmetric Marangoni convection in microencapsulation

Acta Astronautica 57 (2005) 97 – 103 www.elsevier.com/locate/actaastro

Axisymmetric Marangoni convection in microencapsulation Pravin Subramaniana,∗ , Abdelfattah Zebiba , Barry McQuillanb a Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854, USA b General Atomics, San Diego, CA 85608, USA

Available online 21 April 2005

Abstract Spherical shells used as laser targets in inertial confinement fusion (ICF) experiments are made by microencapsulation. In one phase of manufacturing, the spherical shells contain a solvent (fluorobenzene (FB)) and a solute (polystyrene (PAMS)) in a water–FB environment. Evaporation of the FB results in the desired hardened plastic hollow spherical shells, 1–2 mm in diameter. Perfect sphericity is demanded for efficient fusion ignition and the observed surface roughness maybe driven by Marangoni instabilities due to surface tension dependence on the FB concentration (buoyant forces are negligible in this micro-scale problem). Here we model this drying process and compute nonlinear, time-dependent, axisymmetric, variable viscosity, infinite Schmidt number solutocapillary convection in the shells. Comparison with results from linear theory and available experiments are made. © 2005 Elsevier Ltd. All rights reserved.

1. Introduction Successful ICF experiments require high and spherically uniform laser energies which are absorbed by spherical fuel capsules made of a plasma polymer [1]. The compression of the fuel must be uniform and perturbations on the inner capsule wall would grow during implosion due to Rayleigh–Taylor instabilities. The resulting mixing of the polymer and fuel degrades the fusion ignition. Manufacturing of 1 mm fuel targets [2] has been achieved by microencapsulation (see Fig. 1) to produce hollow PAMS mandrels. These mandrels ∗ Corresponding author.

E-mail addresses: [email protected] (P. Subramanian), [email protected] (A. Zebib), [email protected] (B. McQuillan). 0094-5765/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2005.03.018

are then coated with the plasma polymer. Heat treatment decomposes the PAMS which diffuses through the plasma polymer coating leaving behind the desired target hollow shells. McQuillan [3] has demonstrated that Marangoni instabilities driven by surface tension dependence of the FB concentration are the cause of outer surface deformations and deviation from sphericity observed with 2 mm shells. McQuillan and Takagi [4] were able to prevent the formation of surface ripples in 1–2 mm mandrels by manipulating the prevailing Marangoni numbers in the experiments. These Marangoni instabilities can cause more and serious bumps during the manufacturing of planned larger spherical targets. Hence, it is essential to study the hydrodynamics of the convective patterns which develop in the spherical shells during the drying process.

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P. Subramanian et al. / Acta Astronautica 57 (2005) 97 – 103

PAMS+FB

MICROENCAPSULATION Droplets Generation

2. Mathematical model W1 PAMS+FB W2+FB

W1 centers

Lose FB W1 removed osmotically

PAMS Shell

Fig. 1. Sketch of microencapsulation. Flows through the capillaries of the generator produce droplets of water, surrounded by a mixture of FB and PAMS, and suspended in an aqueous solution. The FB is removed by evaporation over several hours/days leaving a cured solid PAMS shell. The water droplet is then later removed by osmosis into ethanol.

The pattern of motion realized in a convectively unstable system with spherical symmetry can be considered without reference to the physical details of the system. Numerous theories have been developed over the years to study the heat transfer in the mantles of terrestrial planetary interiors which occurs by convection [5]. Busse [6] studied the patterns of nonlinear convection in a homogeneous fluid contained between two concentric spherical boundaries assuming a spherically symmetric gravity force and distribution of heat sources. Nonlinear axisymmetric convective motions of self-gravitating, infinite Prandtl number fluids in spheres and spherical shells for different modes of heating were modeled by Zebib et al. [7]. A theoretical study of the linear and weakly nonlinear variable viscosity convection in spherical shells with an infinite Prandtl number fluid and two modes of heating was also performed by Zebib [8]. Three-dimensional steady thermal convection of an infinite Prandtl number, Boussinesq fluid with temperature-dependent viscosity was examined by Ratcliff et al. [9]. Linear stability analysis of the Marangoni mechanism in spherical shells during microencapsulation was considered by Subramanian et al. [10,11]. In the present paper, we study nonlinear axisymmetric convection in the shells assuming infinite Schmidt number.

We consider a spherical shell of initial thickness Lr = R2∗0 − R1∗ , where R1∗ and R2∗0 are the shell’s inner and initial outer radii, respectively. The aspect ratio of the shell is
= R1∗ /R2∗0 < 1 (all starred quantities are dimensional). The inner boundary is assumed stress free and impermeable, while nonlinear boundary conditions are prescribed at the moving outer boundary. The shell contains a mixture of a solvent and a solute with concentrations C ∗ and (1−C ∗ ), respectively. The ambient is a mixture of the solvent and water into which the solvent is evaporating. Thus, there is a net mass flux across the receding outer surface. The physical quantities are nondimensionalized with respect to r , r , r , Dr , Lr , tr = L2r /Dr , Cr , Dr /Lr , Pr = r /tr for density, dynamic viscosity, kinematic viscosity, mass diffusivity, length, time, concentration, velocity, and pressure, respectively. We assume linear variation of interfacial tension ∗ with concentration C ∗ according to ∗ = r − (C ∗ − Cr ), where subscript r designates a reference state. The Capillary number Ca =
/¯ = (¯ − r )/¯ = Cr /¯ . Here, ¯ = r + Cr and  = −d∗ /dC ∗ . Relevant nondimensional quantities are: the mass transfer Biot number Bi = KLr /r Dr based on an assumed mass transfer coefficient K which is taken as constant, the Reynolds number Re= Cr Lr /r r and the Marangoni number Ma = Re Sc, where the Schmidt number Sc = r /Dr is about 106 and is assumed infinite in our nonlinear model. In the limit of Ca → 0 considered here, it was shown [11] that the O(1) outer surface is a perfect sphere r2 = r2d (t) determined by the diffusive state, while deviations from sphericity are O(Ca). The diffusive state is the solution of the nonlinear dimensionless system:

jC 1 j = 2 jt r jr


r

2 jC

jr

 (1)

with initial condition C(r, 0) = 1. The boundary conditions derived from conservation of the solute and solvent are

jC (r1 , t) = 0 jr

(2)

P. Subramanian et al. / Acta Astronautica 57 (2005) 97 – 103

and at the outer boundary r = r2d (t) given by (4), we have

and

jC (r2 (t), t) = −Bi(HC − C∞ )(1 − CCr ). jr

(3)

The evolution of the outer radius with time is given by dr2 = −BiCr (HC − C∞ ) dt

(4)

with initial condition r2 (0) = 1/(1 −
). Here, C∞ is the concentration of the solvent in the ambient, and H is a partition coefficient (a thermodynamic property) relating the equilibrium concentration of the solvent in the shells to its concentration in the ambient water and is set at 0.0015 in the present model. The O(1) nondimensional equations for axisymmetric, infinite Sc convection in spherical coordinates (r, , ) are: 1 j 1 j 2 (sin U ) = 0, (r Ur ) + 2 r jr r sin  j   jp jU r 1 j 1 0= − + 2 2r 2 + 2 jr r jr jr r sin    
 jU r j U j j + ×  sin   j j j jr r   −2 jU 4Ur 2U cot  + − − r r j r r
 j U +r cot  , jr r   1 jp 1 j j 1 2 jU 0= − + 2 r + 2 r j r jr jr r sin  j   j U j U 2U j − × 2 sin  − 2 + j jr r r jr      jUr  3 jUr 2U cot 2  × + − r j r r j r 2 j + 2 (Ur ), r j 1 j jC 1 j + 2 (r 2 Ur C) + (sin U C) jt r jr r sin  j

 jC 1 j jC 1 j = 2 sin  , r2 + r jr jr r sin  j j with boundary conditions at r = r1 given by
 j U jC Ur = =0 = jr r jr

99

(5)

(6)

(7)

(8)

(9)

· Ur = BiCr (HC − C∞ ) + r2d ,

jC = −Bi(HC − C∞ )(1 − CCr ), jr 
  j U 1 j Ma jC +  r (Ur ) = − . jr r r j r j

(10) (11) (12)

In the full sphere model, we impose the following symmetry boundary conditions at  = 0 and : U =

jUr jC = =0 j j

(13)

which yields convection with even and odd number of cells that correspond to the degree of surface harmonic l (employed in the linear theory). A hemispherical model with the symmetry conditions (13) imposed at  =0 and /2 which results in convection with even number of cells is also investigated. We have assumed constant density and mass diffusivity at their reference values and the viscosity ((C)) is a prescribed function (C) = e (1−C) , where = 25.675Cr for the PAMS–FB system.

3. Numerical scheme A front-fixing coordinate transformation is employed and nonlinear axisymmetric convection is computed by a second-order accurate finite-volume approach. Briefly, the computational domain is divided into cells whose surfaces coincide with the spherical coordinate surfaces. The grid points are located at the geometric centers of these small cells and additional boundary points are included to incorporate the boundary conditions (9)–(13). The concentration and pressure are located at the grid points, while staggered velocity components are adopted to avoid unrealistic pressure fields and associated numerical instabilities. Time marching is accomplished by either a fully implicit first-order forward Euler scheme or by a second-order accurate Crank–Nicholson scheme, and an iterative approach (based on the SIMPLER algorithm [12]) is used. Computations are performed and supercritical patterns are investigated in the relevant parameters space with random initial conditions.

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P. Subramanian et al. / Acta Astronautica 57 (2005) 97 – 103 4.9 4.8

r2e [Experiments]

4.7 Bi = 5

4.5

Recr

r2d(t)

4.6

4.4 4.3 4.2 4.1 4 50

100

150

200

8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

t0 = 50.0

t0 = 40.0

t0 = 0.1 t0 = 1.0

5

250

10 l

15

20

t

Fig. 2. Variation of r2 (t) with time for the 1 mm shells (inner radius = 0.470 mm and the initial outer radius = 0.592 mm). Concentration, time and length scales are respectively, 0.92, 83 s and 0.122 mm and C∞ = 0.0001.

4. Results In [10,11] the diffusive state was computed and linear stability analysis employing normal mode decomposition in surface harmonics was performed for both 1 and 2 mm shells. Frozen-time, quasi-steady-state calculations determined the critical Reynolds number and degree of surface harmonics l (linear convection is independent of the azimuthal wavenumber). The maximum growth rates of perturbations (max ) were also determined by solving the initial boundary value problem subject to random initial conditions which revealed that, modes with increasing l are preferred at later times. In this section, we present results for nonlinear axisymmetric convection in both shells. 4.1. 1 mm shells The evolution of the outer radius r2d (t) for the 1 mm shells as determined from the diffusive state solution with Bi = 5 and C∞ = 0.0001 is shown in Fig. 2. The value of the diffusion coefficient [13] which sets the time scale is not precisely known for our system and is assumed 1.8 × 10−6 cm2 /s. The critical curves for the 1 mm shells in Fig. 3, as predicted by the linear theory, show that at all times during the curing process, Recr always corresponds to mode 1. Macr can be calculated as Ma=Re Sc, where Sc=1.183×105 (with r

Fig. 3. Variation of Recr with wavenumber l for the 1 mm shells at the same parameter values as Fig. 2 and various frozen times t0 .

evaluated at Cr = 0.92). Since the initial state is one of uniform concentration, Recr first decreases with time as a destabilizing gradient develops and then increases with viscosity increase due to loss of solvent. By specifying an operating Reynolds number Reop = 2.0, we compute nonlinear axisymmetric convection in the 1 mm shells, for this set of parameters which mimic the experiments. According to Fig. 3, this value of Reop is supercritical during early times of curing, while is subcritical at later times. Thus, we expect convection to first grow at a supercritical Reop and to gradually weaken at later times when Reop becomes subcritical. Our nonlinear calculations are consistent with this prediction with a one cell motion (l = 1) at t =0.1 as shown in Fig. 4 which develops into the three cell convection in Fig. 5 at t = 40.0. This increase in l is in agreement with the max calculated in the linear theory. At later times the strength of circulation as measured by max diminishes. It should be noted that the computed cellular patterns are in good agreement with the critical curves of Fig. 3 which are almost flat upto modes 3–4 at early times. Fig. 6 gives the C field of the motion in Fig. 5 with the isoconcentration lines pushed in the direction of motion. Hemispherical convection is also possible as shown in Figs. 7 and 8 where a four cell (l =4) motion is displayed at t = 20.0. 4.2. 2 mm shells Fig. 9 shows the evolution of the outer radius r2d (t) as determined from the diffusive state solution with

P. Subramanian et al. / Acta Astronautica 57 (2005) 97 – 103

4

101

4

ψ

C 0 -0.0001 -0.0002 -0.0003 -0.0004 -0.0005 -0.0006 -0.0007 -0.0008 -0.0009

0

0

-2

-2

-4

-4 0

2

4

6 x

8

0.9622 0.9621 0.962 0.9618 0.9616 0.9614 0.9612 0.9611 0.961 0.9609

2

y

y

2

10

0

12

Fig. 4. Full sphere convection stream function () at time t = 0.1 and Reop = 2.0, at parameter values of Figs. 2 and 3. (Negative values of  correspond to anticlockwise circulation also indicated by arrows).

2

4

6 x

8

10

Fig. 6. Concentration (C) contour plot corresponding to Fig. 5.

ψ 4

4

ψ

y

0

4 3 2 1 0 -1 -2 -3 -4

3 y

2 1.68 1.36 1.04 0.72 0.4 0.08 -0.24 -0.56 -0.88 -1.2

2

12

2

1

-2

0 0

-4 0

2

4

6 x

8

10

12

Fig. 5. Full sphere convection stream function () at time t = 40.0 and Reop = 2.0, at parameter values of Figs. 2 and 3.

Bi = 1 and C∞ = 0.0014. The curing takes about 4 days and manipulating the values of Bi and C∞ in the calculations can lead to excellent agreement with the experiments [11]. The critical curves in Fig. 10 show that at early times, Recr corresponds to mode 2, while higher modes are preferred at later times. Macr can be calculated as Ma = Re Sc, where Sc = 1.539 × 106 (with r evaluated at Cr = 0.82).

1

2

3 x

4

5

6

Fig. 7. Hemispherical convection stream function () at time t = 20.0 and Reop = 2.0, at parameter values of Figs. 2 and 3.

For Reop = 2.0 which is supercritical at early times, multiple cell motions develop upto t = 300.0. Fig. 11 shows a nine cell motion (l = 9) at time t = 200.0 from the full sphere calculations. This is consistent with the critical curves (Fig. 10) which are almost flat upto mode 8–10 with multiple modes excited and interacting nonlinearly. Due to upwelling at one pole and downwelling at the other, the C contours are shifted upwards as can be seen from Fig. 12. It is also worthy noting the shifts in the C contours in the remaining parts of the shell due to the multiple cell motions. As expected, at later times when the Reop is subcritical,

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P. Subramanian et al. / Acta Astronautica 57 (2005) 97 – 103 4.5

C 0.9868 0.98675 0.9867 0.98665 0.9866 0.98655 0.9865 0.98645 0.9864 0.98635 0.9863 0.98625 0.9862 0.98615 0.9861

y

3

2

3.5 3 Recr

4

t0 = 1000.0

4

t0 = 300.0 t0 = 0.3 t0 = 100.0 t0 = 1.0

2.5 2 1.5 1

1

0.5 0

5

0 0

1

2

3 x

4

5

10 l

6

Fig. 8. Concentration (C) contour plot corresponding to Fig. 7.

15

20

Fig. 10. Variation of Recr with wavenumber l for the 2 mm shells at the same parameter values of Fig. 9 and various frozen times t0 .

14 r2e [Experiments] 13.95

ψ 10 Bi = 1

5 13.85

y

r2(t)

13.9

1.5 1.22727 0.954545 0.681818 0.409091 0.136364 -0.136364 -0.409091 -0.681818 -0.954545 -1.22727 -1.5

0

13.8

-5 13.75

0

2500

13.20

5000 t

7500

10000

Fig. 9. Variation of r2 (t) with time for the 2 mm shells (inner radius = 1.000 mm and the initial outer radius = 1.077 mm). Concentration, time and length scales are respectively, 0.82, 33 s and 0.077 mm and C∞ = 0.0014.

the convection slowly dies down as two cell motions until there is no further motion in the shells. Finally, Fig. 13 shows the hemispherical solution with a four cell motion (l = 4) at time t = 300.0. 5. Concluding remarks The mathematical model developed here is applicable to the drying phase of microencapsulation of ICF targets. Comparisons between our theoretical predictions and observations indicate reasonable agreement

-10

0

5

10

15

20

25

30

35

x

Fig. 11. Full sphere convection stream function () at time t = 200.0 and Reop = 2.0, at parameter values of Figs. 9 and 10.

given the uncertainty in the values of thermodynamic properties and conditions in the experiments [11]. The axisymmetric calculations performed support the predictions of the linear theory for 1 and 2 mm shells. Hence, we conclude that the Marangoni instabilities could well be the source of the observed surface roughness. Companion O(Ca) surface deformations will be determined and full 3-D nonlinear computations are planned.

P. Subramanian et al. / Acta Astronautica 57 (2005) 97 – 103

References

C 10

0.996212 0.99621 0.996208 0.996206 0.996204 0.996202 0.9962 0.996198 0.996196 0.996194 0.996192 0.99619 0.996188 0.996186 0.996184 0.996182

y

5 0 -5 -10

0

10

20

30

x Fig. 12. Concentration (C) contour plot corresponding to Fig. 11.

14 ψ 12

3 2 1 0 -1 -2 -3 -4

10

y

8 6 4 2 0

0

5

10

103

15

x Fig. 13. Hemispherical convection stream function () at time t = 300.0 and Reop = 2.0, at parameter values of Figs. 9 and 10.

Acknowledgements We gratefully acknowledge support of the National Science Foundation (NSF Grant no. CTS-0211612), the US Department of Energy (Contract DE-AC0301SF22266), and the Rutgers University Engineering Computing Services.

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