A Uniﬁed Theory for Interphase Transport Phenomena with ... · A Uniﬁed Theory for Interphase Transport Phenomena 205 3.2 Interphase mass transfer between two pla-nar laminarlayerswith

Copyright c© 2007 Tech Science Press FDMP, vol.3, no.3, pp.203-230, 2007

A Unified Theory for Interphase Transport Phenomena with InterfacialVelocity and Surface Tension Gradients: Applications to Single Crystal

Growth and Microgravity Sciences

Akira Hirata1

Abstract: This article is a summary of author’stypical research works (over the last four decades)on interphase transport phenomena in the pres-ence of interfacial fluid motion and surface ten-sion gradients on liquid-fluid interfaces, and re-lated applications to single crystal growth and mi-crogravity sciences. A unified theory for momen-tum, heat and mass transfer on liquid-fluid andsolid-fluid interfaces is proposed, which takes intoaccount interface mobility. It is shown that in-terface contamination and turbulence can be wellexplained, respectively, by suppression and en-hancement of the interfacial velocity induced bysurface tension gradients. Transport phenomenaon solid spheres, liquid drops and gas bubbles arealso treated within the context of the proposedtheory. This theory is then extended to the caseof crystal-melt and melt-fluid interfaces. Resultsprovided by microgravity experiments performedwith drop shafts, parabolic flights, sounding rock-ets and the space shuttle are used as a relevantmeans for further elaboration and validation of theproposed theoretical framework.

Keyword: Transport phenomena, interfacialvelocity, surface tension gradient, Marangoniconvection, single crystal growth, microgravityexperiments, Bridgman method, floating zonemethod, Czochralsky method.

1 Introduction

Interphase transport phenomena are very impor-tant as a fundamental issue for a variety of dif-fusional operations in chemical and biochemi-cal processes, such as gas absorption and liquid-

1 Waseda University, Tokyo, 169-8555 Japan. Professor A.Hirata passed away on April 13, 2007.

liquid extraction. Moreover, these phenomenaplay an important role in single crystal process-ing.

Since there are many variables which influenceinterphase transport phenomena in a complicatedmanner, however, only few theories were pro-posed before the 60’s to explain the underlyingmechanisms (Shirotsuka, Hirata and Murakami,1966).

From this standpoint, the author worked for 50years to shed some light on the mechanisms ofmomentum, heat and mass transfer on solid-fluidand liquid-fluid interfaces, to introduce new theo-ries and extend them to surface-tension-gradient-driven convection phenomena in high quality sin-gle crystal growth and microgravity sciences (Hi-rata, 2005).

The interface mobility, specifically the interfacialfluid motion caused by surface tension gradients,is considered in this study and a unified theory isproposed for the interphase transport phenomenaon solid spheres, liquid drops and gas bubbles.

The interface mobility is the most important vari-able distinguishing interphase transport phenom-ena between a liquid-fluid interface and a solid-fluid interface. Three important non-dimensionalparameters are introduced in such unified the-ory: the non-dimensional interfacial velocity, thenon-dimensional ratio of surface tension gradientdriven force to viscous force (or inertia force),and the non-dimensional contamination degree.With the unified theory, interfacial contaminationphenomena and interfacial turbulence are wellexplained, respectively, by suppression and en-hancement of the interfacial velocity which arecaused by negative and positive surface tensiongradients on the liquid-fluid interface, respec-

204 Copyright c© 2007 Tech Science Press FDMP, vol.3, no.3, pp.203-230, 2007

tively.

In bulk single crystal growth from melt by theBridgman method, the floating zone (FZ) methodand the Czochralsky (CZ) method, the crystalquality is influenced strongly by convection in themelt. Early works only considered forced andnatural convections. Over recent years, the au-thor has provided some insights into the transportphenomena on the crystal-melt interface and themelt-ambient fluid interface, accounting for sur-face tension gradient driven convection. As an ex-ample, single peaked and high efficiency ultravi-olet emission at 345 nm from In0.03Al0.20Ga0.77NLEDs has been achieved with MOVPE under con-tinuous current injection conditions.

Although it was a widespread opinion before1990 that no convection existed in microgravity,the author pointed out that the interfacial-tension-gradient-driven convection, or Marangoni con-vection, could be very important even in micro-gravity conditions. The mechanism of thermocap-illary and solutal convections in two-dimensionalopen boats and liquid bridges, under both gravityand microgravity conditions was elucidated viamicrogravity experiments performed with dropshafts, parabolic flights, sounding rockets and thespace shuttle, as well as on the ground.

Owing to the large amount of articles publishedby the author and related results, the present ar-ticle will be limited to giving some flavor of themost significant results, by showing some exam-ples and by introducing the unified theoreticalframework mentioned before. The reader inter-ested in a more thorough discussion should con-sult the relevant sections of the articles cited here-after.

2 Transport phenomena with interfacial ve-locity on solid-fluid interfaces

For the case of transport phenomena with interfa-cial velocity on solid-fluid interfaces, the readermay consider Hirata & Shirotsuka (1967), Shirot-suka & Hirata (1970), Shirotsuka & Hirata (1967)and Shirotsuka et al. (1969).

In such works, it was made clear quantitativelyfrom both the theoretical and experimental re-

sults that, for the conditions considered therein,the rate of heat transfer with mass transfer de-creases against an increase in the mass transferrate, due to mass transfer cooling, and the effect ofthe Schmidt number on the mass transfer rate de-pends on the interfacial velocity, which promotesmass transfer.

3 Transport phenomena on liquid-fluid inter-face

Interphase transport phenomena on a liquid-fluidinterface are influenced by many variables in amanner more complicated than that on solid-fluidstationary interface. The interfacial mobility isthe most significant variable in distinguishing in-terphase transport phenomena between a liquid-fluid interface and a solid-fluid stationary inter-face. The author has studied interphase transportphenomena on liquid-fluid interface, paying par-ticular attention to this interfacial mobility.

3.1 Transport phenomena in high mass fluxmass transfer

Momentum, heat and mass transfer phenomenain a laminar boundary layer on a liquid-fluid in-terface were theoretically analyzed with tangen-tial and normal interfacial velocities taken into ac-count simultaneously.

Approximate equations governing the effects oftangential and normal interfacial velocities onthese phenomena were proposed by Hirata, Kiraand Suzuki (1979a), Hirata and Suzuki (1980).The following important results were obtained:

1. The interfacial velocity parallel to bulk streamincreases heat and mass transfer rates,

2. The normal velocity to the interface from thebulk stream (such as gas absorption) increasesheat and mass transfer rates,

3. The normal velocity to the bulk stream fromthe interface (such as evaporation) decreasesheat and mass transfer rates.

In all cases, it should be pointed out that theeffects are more important with larger Schmidtnumbers.

A Unified Theory for Interphase Transport Phenomena 205

3.2 Interphase mass transfer between two pla-nar laminar layers with a flowing interface

Both an experimental and an approximate anal-ysis were made about the effect of the interfa-cial velocity on interphase mass transfer with atangentially moving liquid-liquid (immiscible) in-terface between two plane laminar layers (Shirot-suka, Hirata and Niwa, 1972). Preliminary exper-imental results showed that mass transfer coeffi-cients varied unusually with both external flowsin a complicated manner. Once the experimen-tal data were rearranged according to a theoret-ical analysis considering the interfacial velocity,the results were well explained as shown in Fig.1. From these results, three remarkable things canbe noticed:

1. When the interfacial flow has the same direc-tion of the bulk stream, the mass transfer rateincreases with the interfacial velocity.

2. At high dimensionless interfacial velocity (ra-tio of the interafcial velocity to the free streamvelocity), the mass transfer rate is proportionalto the square root of the interfacial velocity.

3. In counter-current flow, mass transfer rate de-creases against an increase in the interfacialvelocity.

3.3 Momentum and mass transfer with an in-terfacial velocity in spherical geometries

A numerical analysis was made for the interphasemomentum and mass transfer through a flowingplane interface (two-dimensional laminar bound-ary layer conditions, Shirotsuka et al, (1969). Ap-proximate equations to fit the numerical solutionswere obtained for both, drag and mass transfer co-efficients as follows (Hirata & Suzuki, 1980):

CD

CDS=∣∣∣∣1− ui

u∞

∣∣∣∣[

1+(2.88−a)ui

u∞

]1/2

(1)

ShShS

=[

1+(

2.88−aSc−b)

Sc1/2(

ui

u∞

)]1/2

(2)

where

a =0.09+(ui/u∞)2

0.10+(ui/u∞)2

b =3(ui/u∞)

2+12(ui/u∞)

CDS and ShS are the values for zero interfacial ve-locity. They can be evaluated by the followingequations for laminar boundary layer:

for a plane interface;

CDS = 0.664Re−1/2 (3)

ShS = 0.332Re1/2Sc1/3 (4)

for a spherical interface;

CDS = (24/Re)(1+0.125Re0.72) (5)

ShS = 2+0.6Re1/2Sc1/3 (6)

Eqs. (1) and (2) with Eqs. (5) and (6) are consis-tently applicable to momentum and mass transferfor solid spheres, gas bubbles and liquid drops asmentioned later (Hirata, 1982, Hirata and Okano,1984, Hirata, Nishizawa and Okano,1992a).

From these results, three remarkable things can benoticed:

1. With increasing the dimensionless interfa-cial velocity, the normalized Sh number in-creases, while the normalized drag coefficientdecreases.

2. The normalized drag coefficient is only a func-tion of dimensionless interfacial velocity, andthe normalized Sh number is a function of di-mensionless interfacial velocity and Sc num-ber. Both are independent of Re number.

3. The effect of Sc number on mass transferchanges with the interfacial velocity. Al-though in the case of a stationary interface,the Sh number is proportional to 1/3 power ofSc number, in the case of Sc1/3(ui/u∞) muchgreater than unity, it is proportional to thesquare root of Sc, Re and dimensionless inter-facial velocity. This means that the Sh num-ber is proportional to the square root of uix/D.The equation is exactly the same as the Boussi-nesque equation.


Figure 1: Mass transfer with a flowing interface separating two-laminar layers

3.4 Momentum heat and mass transfer withsurface tension gradients on a planarliquid-fluid interface

When an interfacial tension gradient caused bya temperature and/or a concentration differenceexists at a liquid-fluid interface, this gradient af-fects the interfacial velocity significantly. Con-sequently it also affects significantly the rates ofmomentum, heat and mass transfer in a compli-cated manner. Simple configurations of the inter-facial motion affected by the interfacial tensiongradient on a co-current flowing plane interfaceare shown in Fig. 2. In Fig. 2, case (ii) corre-sponds to a clean interface without any interfacialtension gradient. Case (i) is with a negative inter-facial tension gradient. Case (iii) is with a positiveinterfacial tension gradient, where the mobility ofthe interface is enhanced.

The two-dimensional laminar boundary layerequations; the continuity equations, momentumequations, energy equations and diffusion equa-tions for two different phases as shown in Fig. 2were numerically solved with the additional inter-

facial boundary condition obtained from the con-tinuity of tangential stress, where the interfacialtension gradient is considered (Hirata, Kawakamiand Okano, 1989). A schematic diagram of typi-cal numerical results showing the effect of the in-terfacial tension gradient on the interfacial veloc-ity, the friction coefficient and the Nu number areshown in Fig. 3.

In Fig. 3, the numerical results are arranged by di-mensionless groups, where the important groupsare newly defined as Eqs. (7), (8) and (9).

modified friction coefficient, which should beused on the flowing interface:

Cf N ≡ τs

ρu∞ |us −u∞| (7)

Ma

Re2 ≡ ∂σ/∂xρu2

∞

≡ Interfacial tension gradient driven forceViscous force

(8)


MaRe

≡ (∂σ/∂x)xμu2

∞

≡ Interfacial tension gradient driven forceInertia force

(9)

Figure 2: Interfacial motion affected by a surfacetension gradient

From Fig. 3, six remarkable things can be noticed:

1. At a particular negative value of the Marangoninumber, the mobility of the interface is sharplysuppressed.Such a value of Marangoni number is newlydefined as the negative critical Marangoninumber (Ma)c−, which is derived here theoret-ically as follows:

(Ma1)c− =

−0.332Re13/2

[1+(

u2∞

u1∞

)3/2(ρ2μ2

ρ1μ1

)1/2]

(10)

2. Around the negative critical Marangoni num-ber, the rates of momentum, heat and mass

transfer sharply decrease to the value at the sta-tionary interface: us = 0, due to the suppres-sion of the interfacial velocity.

3. Beyond a particular positive value of theMarangoni number, the interfacial velocityis gradually enhanced, and consequently therates of momentum, heat and mass trans-fer are gradually increased with increase inMarangoni number. Such a Marangoni num-ber is defined here as the positive criticalMarangoni number (Ma)c+ the absolute valueof which is nearly equal to that of the negativecritical Marangoni number.

4. When the Marangoni number is much greaterthan the positive critical Marangoni number,the interfacial velocity is proportional to the2/3 power of Marangoni number, and the mod-ified friction coefficient and the Nu number areproportional to the 1/3 power of Marangoninumber.

5. In the region between the negative and positivecritical Marangoni numbers, the interfacial ve-locity is independent of the interfacial tensiongradient, consequently, the interfacial tensiongradient does not affect the momentum, heatand mass transfer.

6. The interfacial contamination phenomena andthe interfacial turbulence phenomena (wellknown to be important in liquid-liquid extrac-tion) can be well explained, respectively, bythe suppression and the enhancement of the in-terfacial velocity caused, respectively, by thenegative and the positive interfacial tensiongradient on the liquid-fluid interface.

3.5 Momentum and mass transfer through theinterface of single gas bubbles and liquiddrops in the presence of surface tensiongradients

In order to evaluate the interfacial velocity onsingle gas bubbles and single liquid drops, theNavier-Stokes equations in spherical coordinatesfor two different phases were theoretically ana-lyzed with the surface tension gradient considered


Figure 3: A schematic diagram showing the ef-fect of surface tension gradients on the interfacialvelocity, friction coefficient and Nusselt number.

as the boundary condition at the interface. The in-terfacial velocities were theoretically obtained ina closed form for the cases of both very low andvery large Re number.

us

uT=

38

π−[

38

π −A(1−m)]

exp(−DRe)I0(DRe)

(11)

where,

A =1

8(1+κ)π

B = 2.197

√3π

2+3κ +m1+

√γκ

D =[

38

π −A(1−m)]2/(

2πB2)where, m is defined as the degree of interfacialcontamination as shown below:

m =43

∣∣∣∣MaRe

1sinθ

∣∣∣∣ (12)

When the interface is completely clean and with-out any surface tension gradient, m is zero. In thecase of m > 0, the interfacial tension gradient sup-presses the interfacial velocity. It is the same phe-nomena as the well known interfacial contamina-tion phenomena.

The theoretical results provided by Eqs. (2) and(4) combined with Eq. (11) explain well previousexperimental results for phase mass transfer fromsingle liquid drops (Shirotsuka and Hirata, 1971a)and from single gas bubbles (Shirotsuka and Hi-rata, 1971b). Typical examples for comparison ofthe theoretical results with previous experimentalresults by Thorson et al (1962, 1968) for drag co-efficient and Sh number in single drops are shownin Fig. 4, a) and b). It can be noticed that theexperimental results for both drag coefficient andSh number are well explained by the theoreticalresults for m = 3 (Hirata, Nishizawa and Okano,1992a). The major outcome of Fig. 4 is that aunified (analogy) theory for momentum and masstransfer from single solid sheres, gas bubbles andliquid drops can be established.

The results calculated from the theory on the ef-fects of γ and m on dimensionless interfacial ve-locity, drag coefficient and Sh number are shownin Fig. 5. From Fig. 5, the following remarks canbe deduced:

1. With an increase in the values of κ and m, theinterfacial velocity decreases.

2. The effect of m on the interfacial velocity atlow Re number is much larger than at highRe number. This means that small dropsare more sensitive to interfacial contaminationthan large drops.

3. The effect of m on the drag coefficient can beconsidered the same as the effect of κ on it. Onthe other hand, the effect of m on the interfacialvelocity and Sh number is larger than the effectof κ .

4. The effect of m on the drag coefficient is muchlarger at high Re numbers than at low Re num-bers. On the other hand, the effect of m on theSh number is much lower at high Re numbersthan at low Re numbers.


Figure 4: Comparison of theoretical results with previous experimental results

5. At particular low Re numbers, depending onthe value of m, liquid drops behave like rigidspheres. The related value of m is defined asthe critical contamination degree m∗.

The relationship between Re numbers and m* isshown in Fig. 6.

From Fig. 6, the following can be deduced:

1. When Re<1, the interface become rigid atm∗=1.

2. When Re�1, the following relationships are

obtained:

m∗ ∝ Re1/2 (13)

Mac− ∝ Re3/2 (14)

The negative critical Ma number, Mac− atwhich liquid drops behave like rigid spheresis proportional to 3/2 power of the Reynoldsnumber, the same as in Eq.(10) mentioned ear-lier.

3. With an increase in the value of κ , the negativecritical Ma number increases.


Figure 5: Effects of κ and m on interfacial ve-locity, drag coefficient and Sherwood number(Sc=1000)

Figure 6: Relationship between Reynolds numberand m∗, the critical value of m for which the inter-face behaves as a rigid surface (γ=1)

4 Transport phenomena in single crystalgrowth from melt

Since single crystal growth is a process coupledin a complicated manner with momentum, heatand mass transfer, as well as phase transition andchemical reaction, the quality of single crystals issignificantly influenced by convection.

In bulk single crystal growth from melt by theBridgman method, the floating zone method andCzochralsky (CZ) method, works as late as the1980’s had only considered forced and natural

convection. The author has made clear the inter-phase transport phenomena on crystal-melt inter-face and melt-ambient gas interface with surfacetension gradients driven convection (Marangoniconvection) taken into account. From theseworks, important knowledge for the high qualityproduction of bulk single crystals from melt hasbeen obtained.

4.1 Transport phenomena in Bridgmanmethod

4.1.1 Solidification problem

The author developed a numerical methodfor solving the two-dimensional unsteady one-directional solidification problems by using aboundary-fitted co-ordinate system (Saitou andHirata, 1991a, 1991b, 1993), and proposed theoptimal operational conditions for high qualitycrystals (Saitou and Hirata, 1992c). Important in-formation wa sprovided about the solid-liquid in-terface shape (Saitou and Hirata, 1991c, 1992a),the ratio of liquid to solid thermal conductivity(Saitou and Hirata,1992b) as well as the inter-facial stability, which significantly influence thequality of single crystals.

4.1.2 Thermal and solutal natural andMarangoni convection

Natural and Marangoni convection formed spon-taneously in a melt inside a two-dimensional rect-angular open boat were investigated by meansof an order-of-magnitude evaluation and numer-ical analysis (by the finite difference method). Aquantitative evaluation was made of the Gr num-ber, Ma number, Pr number and melt depth, allof which affect the interfacial velocity and thevelocity distribution of the melt convection. Itwas concluded that Marangoni convection is im-portant when the melt is shallow (Okano, Itohand Hirata, 1989a). Interfacial velocities inducedby solutal Marangoni convection were also mea-sured in In-Ga-Sb systems (initial bulk melt withan additional material contacting the bulk inter-face, Arafune and Hirata, 1999a). Empirical andtheoretical correlation for Re numbers based onmaximum interfacial velocities induced by theMarangoni convection was obtained as follows


over a wide range of Pr numbers (Arafune, Sug-iura and Hirata, 1999b):

Re ≥ 1; Re = 0.28

(Ma

−1/2Pr

)2/3

(15)

Re ≤ 1; Re = 0.049As1/2Ma (16)

4.1.3 Magnetic field effects

The effects of magnetic fields on Marangoniand natural convection were studied numerically.Convective velocities in magnetic fields dependon the Ma number, Gr number, as well as thephysical properties, and the direction and strengthof the applied magnetic field. For suppressionof natural convection, both horizontal and verti-cal magnetic fields are effective. The suppressionof Marangoni convection is more difficult thannatural convection, and a vertical magnetic fieldis more effective than a horizontal one. The in-terfacial velocity induced by natural convectionis inversely proportional to the Hartmann num-ber. That induced by Marangoni convection is in-versely proportional to the square root of the Hanumber (Hirata, Tachibana, Okano and Fukuda,1992b).

4.1.4 Coupled thermal and solutal Marangoniconvection

Experimental and numerical studies were per-formed on thermal and solutal Marangoni con-vection occurring during the growth of InSb frommelt of In1−xSbx.

Figure 7: Schematic concentration distribution(upper) and flow direction (lower): solid-line anddashed-line arrows in lower illustration indicate,respectively, thermal and solutal convection

Fig. 7 shows the schematic diagram of the con-centration distribution of Sb and the interfacialflow direction for three typical cases:

1. Case (a): Acceleration.When 0<x<0.5, the interfacial flow is more ac-celerated than in the normal case (b), becausethermal and solutal flows have the same direc-tion.

2. Case (b): Normal.When x=0.5, it is a normal case, where onlythermal Marangoni convection exists insidethe melt.

3. Case (c): Deceleration.When 0.5<x<0.68, the interfacial flow is decel-eratedwith respect to case (b), because thermaland solutal effects counteract.

Fig. 8 shows that the above considerations explainthe experimental results well. It can be demon-strated from Fig. 8 that the solutal Marangoniconvection is more important than the thermalone in the growth of single crystals of semicon-ductors and chemical compound semiconductors(Arafune and Hirata, 1998, 1999a, Arafune, Ya-mamoto and Hirata, 2001).

4.1.5 Crystal-melt interface shape control

It is worth noting that the solutal Marangoni con-vection has been experimentally considered as anew important control parameter in addition toconvectional control parameters such as the tem-perature gradient and cooling rate in order to ob-tain a planar crystal-melt interface in InSb crystalgrowth. It can be concluded from the experimen-tal results that the utilization of solutal Marangoniconvection can effectively allow the control of thecrystal-melt interface shape without changing thetemperature gradient or the cooling rate (Arafune,Kodera, Kinoshita and Hirata, 2003).

4.2 Transport phenomena in floating zone(FZ) method

Natural and Marangoni flows produced in a meltduring single crystal growth by the floating zonemethod were analyzed by an order-of-magnitude


Figure 8: Relation between crystal length and typ-ical interfacial velocity. Arrows show the flow di-rection near the interface; xinit = (a) 0.5, (b) 0.45,(c) 0.55, (d) 0.6

Figure 9: Comparison of analytical results shownin Table 1 with previous experimental results. 1)McNeil et al.(1984), 2) McNeil et al. (1985), 3)Preisser et al.(1983)

evaluation (Okano, Hatano and Hirata, 1989a,Hirata, Nishizawa, Imaishi, Okano, Saghir andScott, 1991). The analytical results on the effectsof Marangoni and natural convections on Re num-bers based on free interfacial velocity are shownin Table 1. Comparison of the analytical resultswith previous experimental results is shown in

Fig. 9. It can be seen from Fig. 9 that the analyt-ical results are good agreement with previous ex-perimental results for the case of Marangoni con-vection in a vertical capillary liquid bridge (modelliquids and glass melt).

These results have been effectively applied toelucidate the mechanism of Marangoni convec-tion phenomena under microgravity conditions asmentioned later.

4.3 Transport phenomena in Czochralski(CZ)method

The crystal-melt interface shape is one of the im-portant variables in determining the quality ofbulk single crystals.

It is significantly influenced by convection inmelt. In order to obtain high quality uniformsingle crystals, it is necessary that the interfa-cial shape should be kept in a plane (Hirata etal., 1992c). Although previous research had onlyconsidered forced convection due to crystal andcrucible rotations, and natural convection due totemperature distribution in melt, the author con-sidered Marangoni convection in addition to theforced and natural convections, and proposed anevaluation method for determining which convec-tion is dominant in melt, as shown in Fig. 10(Okano, Tachibana, Hatano and Hirata,1989c, Hi-rata et al., 1992a) for the case of boundary layerflow regime as a typical example. Once experi-mental results for the critical rotation rate ω∗, atwhich a plane interface shape is obtained at thecrystal rotation rate, are arranged based on Fig.10, it can be seen that when the value of ordinateis constant, the natural convection is dominant inmelt, and when the value of ordinate is propor-tional to the value of abscissa, the Marangoni con-vection is dominant. The previous experimentalresults have been rearranged based on Fig. 10,and are shown in Fig. 11, where only control-lable variables in the dimensionless groups in Fig.10 are used. Fig. 12 shows an example of com-parison of previous experimental results duringgrowth of Gd3Ga5O12 (GGG) with theoretical re-sults in the case of Marangoni convection domi-nant condition (Fig. 12(a)) and in the case of nat-ural convection dominant condition (Fig. 12(b)).


Table 1: Analytical results on natural and Marangoni convections in a floating zone

Re 1 Re � 1Pr ≤ 1,Pr > 1 Pr < 1 Pr > 1

Natural convectiondominant regime

MaGr < 1 Ma2/3

Gr1/2 < 1 Ma2/3

Gr1/2 Pr1/6 < 1

Re ∝ Gr Re ∝ Gr1/2 Re ∝ Gr1/2

Pr1/2

Marangoni convectiondominant regime

MaGr > 1 Ma2/3

Gr1/2 > 1 Ma2/3

Gr1/2 Pr1/6 > 1

Re ∝ Ma Re ∝ Ma2/3 Re ∝ Ma2/3

Pr1/2

Figure 10: Diagram for determining the domi-nant convection in the melt–The case of boundarylayer flow regime

It can be pointed out from Figs. 11 and 12 that theMarangoni convection is very important and dom-inant in the melt in most of these experiments,and should be considered in order to control thecrystal-melt interface shape. The theoretical andempirical relation between the critical crystal ro-tation and operational conditions is proposed asfollows (Okano et al., 1989c, 1991, Hirata et al.,1993a, 1993b, Tachibana et al., 1990):

ω∗ ∝(

ΔTΔr

)2/3(Δrrc

)1/3

(17)

<1 Pr>1

< 1

G r1

2

M a2

3

G r1

2P r

16 < 1

R e ∝ G r1

2

P r1

2

> 1

M a2

3

M a2

3

G r1

2P r

16 > 1

R e ∝ M a2

3

P r1

2

Figure 11: Evaluation of the dominant convec-tion conditions in terms of dimensionless num-bers (1), (2) Miller et al. (1978); (3), (4) Tak-agi et al. (1976); (5) Miyazawa et al. (1978); (6)Miyazawa et al. (1978); (7) Miyazawa and Kondo(1973); (8) Miyazawa (1980); (9) Lukasiewicz etal. (1992); (10) Berkowski et al. (1987); (11),(12) Berkowski et al. (1991); (13) Brice et al.(1974); (14) Hirata et al. (1993a); (15) Trauth andGrabmaier (1991); (16) Hirata et al.(1993b).

4.4 Chemical compound semiconductor byLaser-MOVPE

4.4.1 Atomic monolayer epitaxy of chemi-cal compound semiconductor by Laser-MOVPE

A new technique for obtaining atomic monolayersof semiconductors via vapor phase epitaxy frommetal organic compounds exposed by laser beam(to prepare three dimensional super lattice LSI)was developed by Meguro et al., (1988, 1990a) for


Figure 12: Comparison of theoretical results with previous experimental results (GGG) (a) dominantMarangoni convection (b) dominant natural convection

GaAs, and Meguro et al., (1990b) for AlAs andGaAlAs. Self-limiting Ga deposition on GaAsduring laser-atomic epitaxicial processing was di-rectly observed and the related mechanism waselucidated by Simko et al. (1992).

4.4.2 Room temperature UV-emission at 345nmfrom InAlGaN-based light-emitting diodes

Optical and electrical properties of 340 nm-band bright UV-light emitting diodes (LEDs)were compared among In0.03Al0.20Ga0.77N andother active regions (Hirayama et al., 2000a,2000b, 2001a, 2001b, Kinoshita et al., 2000a,2000b, 2000c). Single peaked and high ef-ficiency ultraviolet (UV) emission at 345 nmfrom In0.03Al0.20Ga0.77N LEDs were achieved un-der continuous current injection conditions (Ki-noshita et al., 2001a). Significant broadening andpeak shift of the electroluminescence (EL) spec-trum were not observed. Such a good behaviorcan be attributed to the high quality InAlGaN-quaternary active layer (Hirayama et al., 2001b).The EL intensity of the InAlGaN quaternary-based LED showed more than one order of magni-tude higher intensity than that of AlGaN and GaNbased LEDs. Additionally, InAlGaN quaternary-based LEDs show a linear increase in intensitywith increasing injection current from the I-Lcharacteristics, that is also observed for InGaNbased LEDs (Hirayama et al., 2001b). This showsthat the recombination efficiency of the InAlGaN-based LEDs is as high as that of InGaN based

LEDs. From these results InAlGaN quaternary isexpected to be a promising material for UV LEDsand LDs (Hirayama et al., 2002).

5 Transport phenomena under microgravityconditions

As already explained before, when single crys-tals are grown from a melt, the quality of thesingle crystals is seriously affected by convec-tion induced by temperature and concentrationdifferences in the melt. Therefore, thermocap-illary Marangoni convection induced by a tem-perature difference on melt interface, and solutalMarangoni convection induced by a concentrationdifference on melt interface, are very importantfor the quality of single crystals under μg condi-tions. The author has elucidated the mechanismof thermocapilary and solutal Marangoni flowsunder μg conditions via drop shafts, parabolicflights, sounding rocket, space shuttle and recov-erable satellite as well as on the ground.

5.1 Development of equipments and opera-tions for microgravity experiments

Novel equipments and operations are necessaryfor μg experiments significantly differing fromfor ground ones. A facility is required to be com-pact and able to do experiments automatically invery short time, for instance, within 5 seconds incase of a drop shaft experiment. The author de-veloped novel experimental equipments to get astable formation of a liquid bridge even with a se-


rious change of gravity in a drop shaft, parabolicflight or sounding rocket experiment (Azuma etal., 1990, Yoshitomi et al., 1992).

Also a three-dimensional in-situ observation tech-nique to measure velocity and temperature dis-tributions was developed (Yoshitomi et al., 1992;Hirata et al., 1993c, 1994, 1997a).

5.2 Interfacial contamination in a silicone oilliquid bridge

Silicone oils are widely used as model fluids tostudy convection phenomena inside the FloatingZone, because silicone oil is transparent and itsphysical properties are well known. Throughmany experimental studies it has been found formethyl-phenyl silicone oil (KF-56) that interfa-cial contamination is caused by water slightly in-cluded in the silicone oil itself and also by wa-ter vapor in the surrounding atmosphere (Hirataet al., 1993d, 1993e). Since interfacial contam-ination leads to a significant decrease in the in-terfacial velocity and weakens Marangoni convec-tion, it should be pointed out that much attentionshould be paid to complete dehydration of bothsilicone oil and the surrounding gas.

5.3 Laminar thermocapillary convection in aliquid bridge during parabolic flight

Laminar thermocapillary Marangoni convectionin a liquid bridge with silicone oils (KF-96: 5, 10,20, 50 cSt and KF-56: 15 cSt) was observed un-der μg conditions during a parabolic flight (Hirataet al., 1994). Fig. 13 shows that the relationshipbetween Ma numbers and Re numbers based onthe interfacial velocity caused by thermocapillaryMarangoni convection, at the center of the free in-terface of the liquid bridge. It can be seen thatthe experimental results are in good agreementwith the previous theoretical results shown in Ta-ble 1 (Okano et al., 1989b). In the aforementionedworks, Axis-symmetric laminar Marangoni con-vection was studied experimentally and numeri-cally.

The initial temperature conditions inside a liq-uid bridge, heat transfer across the free interface,and temperature dependence on both velocity andtemperature field were also made clear.

Figure 13: Relationship between Marangoninumbers and Reynolds numbers based on the in-terfacial velocity

5.4 Oscillatory thermocapillary convection in aliquid bridge with TR-1A sounding rocket

Laminar and oscillatory three-dimensional ther-mocapillary Marangoni convection in a liquidbridge of a silicone oil: 6 cSt (mixed with 20vol%of KF96L:1cSt and 80vol% of KF56:15cSt) wasobserved during 6 minutes μg condition with TR-1A sounding rocket No. 2 in August, 1992 (Hirataet al., 1993c)

There were two stages for the experimental se-quence: at the first stage, the temperature differ-ence between the ends of the liquid bridge wassmall, 30◦C between 90 and 215 second, and atthe second stage, the temperature difference wasincreased up to 50◦C between 330 and 441 sec-ond.

Laminar thermocapillary Marangoni convectionwas observed at the first stage and oscillatory ther-mocapillary Marangoni convection was observedin the second stage. The experimental resultson the interfacial velocity caused by the laminarMarangoni convection were explained according


to previous results (Hirata et al., 1989) and numer-ical analyses (Hirata et al., 1991a, 1993c). Thetime history of the interfacial velocity at the centerof free interface caused by oscillatory Marangoniconvection is shown in Fig. 14. The Fourier spec-trum for temperature oscillation inside the liquidbridge between 360 and 440 second is shown inFig. 15. The amplitude of both velocity and tem-perature fluctuation increased with time elapsed.The main frequency of the temperature oscilla-tions was 0.097Hz as shown in Fig. 15 which wasthe same as that of the velocity field oscillation.

Figure 14: Interfacial velocity in a liquid bridge(TR-1A sounding rocket experiment)

Figure 15: Fourier spectrum (ν=6cst, D=15mm,L=10mm TR-1A sounding rocket experiment)

5.5 Gravity effects on oscillatory featuresof thermocapillary convection in liquidbridges

Oscillatory thermocapillary Marangoni convec-tion in silicone oil liquid bridges was observedunder both normal gravity (1g) condition (Hirataet al., 1997b) and μg condition with the dropshaft facility of JAMIC (Yao et al., 1997, Hirataet al., 1997c). Examples of the typical oscilla-tory temperature distribution under 1g conditionare shown in Photo 1, where thermal sensitive liq-uid crystals used there change their color fromred (37.5◦C) to green (40◦C) and blue (42.5◦C).In the horizontal view, the cold region, which iscolored red, moved around the center of the liq-uid bridge in a counter clockwise movement. Inthe vertical view, non-axisymmetrical flows wereobserved. As the temperature difference betweendisks increased, both amplitude and frequency oftemperature fluctuations increased.

Under the same temperature differences, as thelength of a liquid bridge increased, the amplitudealso increased, but the frequency decreased.

It was found that the fundamental frequency oftemperature oscillations as well as the criticaltemperature difference (at which laminar convec-tion changes to oscillatory one under μg condi-tions) were different with respect to 1g conditions,and strongly dependent on the ratio of the liquidbridge volume to the volume of a cylinder withthe same radius and height (Hirata et al., 1997c).For instance, the fundamental frequency of tem-perature oscillations under μg conditions wassmaller than under 1g conditions when the non-dimensional liquid bridge volume was V/V0=73.8vol%, (when V/V0=88.1 vol%, however, it waslarger under μg conditions than under 1g condi-tions).

5.6 Transition process from laminar to oscil-latory thermocapillary convection in liquidbridges under gravity (1g) and micrograv-ity (μg) conditions

The critical temperature difference and the tem-perature oscillatory features were carefully ob-served under 1g and μg conditions by drop shaft


with close attention paid to the liquid bridge vol-ume (Hirata et al., 1997d, Sakurai et al., 1999).

Typical examples of the temperature signals andFFT under 1g and μg conditions are shown in Fig.16 where the symbols are related to the tempera-ture signal state as follows:

(S): laminar steady state

(P): periodic oscillatory state

(QP): quasi periodic oscillatory state

(C): chaotic state

(P2): period doubling oscillatory state

(QP+P2): mixture of (QP) and (P2)

As shown in Fig. 16, the temperature at the coolbottom of the liquid bridge under μg conditionswas higher than under 1g, except for the longestliquid bridge (L=3.3mm) for which (P) under 1gconditions was replaced by (S) for μg.

The temperature signals can be clas-sified into 6 patterns when the grav-ity changes from 1g to μg as follows:

for drop shaft experiments: a-i): (S)→(P)a-ii): (P)→(P)a-iii): (QP)→(P)a-iv): (C)→(P)

for ground experiments: b-i) (P2)b-ii) (QP+P2)

Fig. 17 shows the temperature signal states di-agram for V/V0 vs. ΔT , where the plus sign +shows the transition point between (S) and (P) andthe cross sign ×shows the transition between (S)and (P).

From Fig. 17, five remarkable things can be no-ticed:

1. The critical temperature difference is very sen-sitive to the liquid bridge volume.

2. Under 1g conditions, the temperature signalsshow the existence of six states as shown inFig. 17, but under μg conditions, all tempera-ture signals shows only two states: (S) and (P).

3. The shorter the liquid bridge length is, thehigher the critical temperature difference is,under both 1g and μg conditions.

4. For the liquid bridge with length L=1.6 mm,a very high critical temperature difference isobserved between 70 and 80 vol% of liquidvolume under 1g conditions, while under μgconditions, the critical temperature differenceis remarkably lower. It means that there aresome ranges of the liquid bridge volume ratiowhere the regime is (S) under 1g conditions but(P) under μg conditions (cf. Fig. 17 a-i) andb-i)).

5. For the liquid bridge with length L=3.3 mm,there are some ranges of the liquid bridge vol-ume ratio where the convection is (P) under 1gconditions, but (S) under μg conditions.

5.7 Flow structure of time-dependent thermo-capillary convection in liquid bridges

The flow structure of time-dependent three-dimensional thermal Marangoni convection in sil-icone oil liquid bridges was further observed inthe horizontal section by Arafune et al. (2000)and Kinoshita et al. (2001b). It was foundthat there was a polygon-shaped stagnant regionaround the central axis of a liquid bridge, and theperiodic oscillatory state (P) could be further clas-sified into periodically pulsating (Ps) and periodi-cally rotating (PT ) states.

Figure 18 shows schematic motions of the stag-nant region in the horizontal section for (Ps) and(PT ). It can be seen in Fig. 18 that (Ps) exhibits acoherent phase with a different amplitude depend-ing on the azimuthal angle and (PT ) shows a phaseshift for each azimuthal angle with same ampli-tude, while (QP) shows the mixture of these twotypes of temperature fluctuations. With increas-ing the temperature difference, the temperatureand velocity fields exhibit the following states:(S) – (Ps) – (QP) – (PT ). This transition processdepends on the liquid bridge volume and gravitylevel, but occurs for all kinematic viscosities andliquid bridge lengths considered.


Photo 1: Observation of oscillatory Marangoni convection (D=5mm, L=3.0mm, V/V0=67vol%, ν=2.3cSt,ΔT =27.5◦C).


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� !�"Figure 16: Typical temperature sigals and FFT spectra (D=5mm, ν=2cSt)


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Figure 17: Diagram of temperature signal states in the V/V0−ΔT


Figure 18: Schematics of the typical flow structure for pulsating (Ps) and rotating (PT ) phases withwavenumber m = 2 (a) and m = 3 (b).

5.8 Uniform mixing of melt of multi-component chemi-calcompound semicon-ductors by solutal Marangoni convection:Space Shuttle experiment on IML-2Mission

A microgravity experiment on “Uniform mixingof Melt of Muticomponent ChemicalcompoundSemiconductor”, was carried out aboard SpaceShuttle mission in July 1994 (Hirata et al., 1996,Okitsu et al., 1996, 1997). Six samples shownin Fig. 19 which were composed of In-Sb (M-1, D-1) and In-GaSb-Sb (M-1,-2, -3, -3’, D-2)were melted by a uniform temperature furnaceand cooled down rapidly to be solidified. M-samples and D-samples were mixed, respectively,by solutal Marangoni convection due to concen-tration gradient at the melt-air interface, and bymolecular diffusion only. The concentration dis-tributions of In, Ga and Sb in axial direction forD-2, M-2, -3, -3’ samples composed of In-GaSb-Sb are shown in Fig. 20. It can be noticed fromFig. 20 that M-samples are more uniform than D-sample.

It has been also noticed from the comparison

of μg-samples with 1g-samples (Okitsu et al.,1994) that μg-samples show more uniform con-centration distribution than 1g-samples, becausethe gravity convection and segregation can be sup-pressed under μg conditions. Numerical simula-tions also justify the quick mixing by the solutalMarangoni convection (Yasuhiro et al., 1997).

5.9 Dissolution and recrystallization processesof chemicalcompound semiconductorswith the Chinese recoverable satellite andparabolic flights

The effects of gravity and crystal orientation onthe dissolution of GaSb into melt and the recrys-tallization of InGaSb were investigated under 1gand μg conditions with a Chinese recoverablesatellite (Hayakawa et al., 2000) and airplanes(Hayakawa et al., 2001, 2002). Numerical simula-tions were also carried out for the effects of grav-ity on the solid-melt interface and compositionalprofiles (Ozawa et al., 2000, 2001, Hayakawa etal., 2000, 2002, Okano et al., 2002, 2003). TheInSb crystal was melted and GaSb was dissolvedpartially into the InSb melt, and then formed In-


Figure 19: Configuration of samples for the Space Shuttle experiment

Figure 20: Concentration distribution of In, Ga and Sb in axial direction for μg samples D-2, M2, M-3 andM3’

GaSb solution which was solidified during cool-ing process. The experimental and numerical re-sults clearly show that the shape of the solid-melt interface and compositional profiles in thesolution were significantly affected by the gravitylevel (Fig. 21). It can be seen from Fig. 21 thatunder μg conditions, the Ga compositional pro-files are uniform in the radial direction, the inter-faces are almost parallel, and on the contrary, un-

der 1g conditions, as large amounts of Ga movedup to the upper region due to buoyancy, the dis-solved zone broadens towards gravitational direc-tion.

During the cooling process, needle crystals ofInGaSb started appearing and the value of x inInxGa1−xSb crystals increased with a decrease intemperature. The GaSb with the (111)B plane dis-solved into the InSb melt much more than that


Figure 21: Morphology of the cut surface, schematic representation of the dissolved area, and Ga composi-tional profile around the center of the (a) space sample, (b) earth sample. The Ga composition was measuredin the divided areas of 2.0×1.6mm2 by EDS.

with the (111)A plane.

Relative contributions of the solutal naturalconvection, and thermocapillary and solutalMarangoni convection to the melting process of aGaSb-InSb-GaSb system under 1g and μg condi-tions were also investigated numerically (Okanoet al., 1991, 2003). The numerical simulationresults show that thermocapillary convection en-hances the melting near the free interface underboth 1g and μg conditions. Under 1g conditions,the contribution of the solutal Marangoni convec-tion to the flow field is less than that of the thermo-capillary convection since the heavier componentsinks to the lower part of the melt, and the con-centration gradient on the free interface becomesvery gentle. Consequently, the intensity of the so-lutal Marangoni convection is very small.

On the other hand, the interface shape is greatlyaffected by the presence of solutal Marangoniconvection under μg condition, even though thevelocity of the solutal Marangoni convection isrelatively small.

6 Conclusions

A rich summary of author’s typical researchworks over the last 40 years on interphase trans-port phenomena with interfacial fluid motion and

surface tension gradients (and related applicationsto single crystal growth and microgravity sci-ences) has been reported.

The interface mobility (i.e. an interfacial veloc-ity caused by surface tension gradients inducedby temperature or concentration gradients alongthe interface) is very important and affects the in-terphase transport phenomena.

A unified theory for momentum, heat and masstransfer on liquid-fluid and solid-fluid interfaceshas been proposed with interface mobility takeninto account. It has been shown that in such atheoretical framework, interfacial contaminationand interfacial turbulence can be well explained,respectively, by suppression and enhancement ofthe interfacial velocity induced by suface tensiongradients.

Transport phenomena on solid spheres, liquiddrops and gas bubbles have been also treated inthe context of the proposed theory. The theoryhas been also elaborated for crystal-melt and re-lated melt-fluid interfaces.

The mechanisms underlying thermal and solutalMarangoni convection under gravity and micro-gravity conditions have been discussed and elu-cidated with the support of relevant microgravityexperimental results.


For additional relevant and recent results thereader may consider the following works: Am-berg and Shiomi (2005), Lappa (2005), Gelfgat etal. (2005), Lan and Yeh (2005), Tsukada et al.(2005), Li et al. (2006), Matsunaga and Kawa-mura (2006), El-Gamma and Floryan (2006),Okano et al. (2006), Ma and Walker (2006), Kaki-moto and Liu (2006), Sohail and Saghir (2006),Jaber and Saghir (2006), Giessler et al. (2005),Prud’homme and El Ganaoui (2007), Kamotaniet al. (2007), Votyakov and Zienicke (2007).

Acknowledgement: The author would like toexpress sincere appreciation to all of coauthorsand many students for their active works, andalso to Japanese Government which supportedmany of these works, particularly, the Science andTechnology Agency (presently, the Ministry ofEconomy, Trade and Industry) and the NationalSpace Development Agency of Japan (presently,the Japan Aerospace Exploration Agency).

Nomenclature

A,AS aspect ratio, A,AS ≡ L/H [-]Cf N modified friction coefficient when [-]

us −u∞ > 0 Cf N ≡ τs/(ρu∞(us −u∞)/2)CDS drag coefficient of rigid spheres

CDS ≡ τs/(ρu∞2/2) [-]

D diffusivity [m2/s]or disk diameter [m]

d crucible diameter [m]Gr Grashof number, Gr ≡ gβ ΔTx3/ν2 [-]H melt depth [m]h heat transfer coefficient [W/m2K]Io modified Bessele function [-]κ mass transfer coefficient [m/s]L interface length [m]Ma Marangoni number [-]

Ma ≡ (∂σ/∂x)·x2/(ν ·μ)m contamination degree, [-]

defined by Eq. (12)Nu Nusselt number, Nu ≡ hx/λ [-]Pe Peclet number, [-]

Pe ≡ ux/α (for heat transfer)Pe ≡ ux/D (for mass transfer)

Pr Prandtl number [-]Pr ≡ ν/α (for heat transfer)

R radius [m]Re Reynolds number, Re ≡ ρux/μ [-]r radial distance [m]rC crystal radius [m]Δr length of melt interface [m]Sc Schmidt number, Sc ≡ μ/ρD [-]Sh Sherwood number, Sh ≡ kx/D [-]T temperature [K]ΔT temperature difference [K]

between crucible wall and crystal,or between both ends of liquid bridge

t time [s]u velocity parallel to interface [m/s]uT terminal velocity [m/s]V liquid bridge volume [m3]V0 volume of cylindrical liquid bridge [m3]x x∗ coordinate [m]

Greeks

α thermal diffusivity [m2/s]κ viscosity ratio, κ ≡ μd/μc [-]λ thermal conductivity [w/(m·K)]β thermal expansion coefficient [1/K]γ density ratio, γ ≡ ρd/ρc [-]μ viscosity [Pa/s]σ interfacial tension [N/m]ν kinematic viscosity, ν ≡ μ/ρ [m2/s]ρ density [kg/m3]θ angle [radian]τs shear stress at stationary interface, [N/m]

τs ≡ −μ(∂u/∂y)y=0

ω crystal rotation rate [radian/s]ω∗ critical crystal rotation rate [radian/s]

when crystal shape is plane

Subscripts

c continuous phasec+ positive critical pointc− negative critical pointd dispersed phasei interface, or ith phaseinit initialx mass fractions based on interfacial velocity,

or stationary interface, or solidw water phase, which is rate-determining phase


∞ free stream condition0 condition at us = 01 1st phase2 2nd phase

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