Part 2

INTRODUCTION

2.1 CRYSTAL GROWTH

The industrial crystal growth of a range of electronic materials is a core technology of the electronic industry. For example, growth from the melt of bulk single crystals of semiconductors provides the wafers for integrated circuits; oxide crystals are used for laser host lattices. The most utilized crystal growth techniques from the melt are:

1. The directional solidification (Bridgeman technique);

2. The crystal pulling technique named after Czochralski;

3. The zone melting method, the floating zone method is one of a variants of its vertical configuration.

Each method has its advantages and disadvantages. Before going to the main subject of the research, let us to briefly introduce and compare to each other the Czochralski and the floating zone technologies.

Refining of silicone by the Czochralski technology is the most common method of pro- ducing monocrystalline silicone. Large diameter monocrystalline silicone can be grown with the Czochralski technique. As in the case of floating zone technology growth of Czochralski monocrystalline silicone starts from a small diameter monocrystalline seed crystal. Czochralski silicone is pulled from a quartz crucible in which the silicone is kept in the molten phase by heating the rotating crucible either by induction or resistive heating. Segregation of impurities between the liquid phase and the solid phase is as good as is the case for float zone silicone, but care must be taken during Czochralski growth in order not to contaminate the molten silicone from the surrounding atmosphere. Because of the open furnace principle it is difficult to grow Czochralski silicone with resistivities above 1.000 Ohm

cm. Czochralski silicone is mainly grown with resistivities below 100 Ohm

cm which is sufficient for the making of most ICs, including low- to medium power ICs.

In comparison with float zone silicone, Czochralski silicone can be grown to very low resistiv-

ities in the order of 1−2 mOhm−cm. This is important for low and medium power applications,

where the Czochralski silicone wafer serves as a carrier substrate for the actual components built

on epitaxial material deposited on top of the Czochralski silicone wafer. When growing to these

very low resistivities it becomes important to control the oxygen content in the silicone. Big

efforts are being invested into oxygen control issues. Also, at these low resistivities issues resis-

tivity control becomes important. The Czochralski technique is the one that gives the largest,

roundest rods, and that is important. The larger the substrate the more chips can be made

in a single step. The industry is tooled up to use certain standard diameters. The most com-

mon standard now is 200mm, but the state-of-the-art is 300mm. The Czochralski technique

accounts for approximately 90% of the world market of silicone crystals, mainly used for memory chips and other integrated circuits, the remaining 10% are float zone crystals used for power applications and sensors.

The floating zone method, one of the most fascinating purification techniques, is of technical importance for the growth of silicone. The main technological advantage of the method is containerless which aids to prevent contamination. Zone refining of crystals in order to increase their purity has been known for more than 50 years (was first patented in 1952 by Theuerer and used for the first time by SIEMENS AG). The purification method takes advantage of the concentration change by segregation for most impurities during the liquid – solid transition.

The floating zone is suspended between the melting and the freezing interfaces. The floating zone technology has been perfected for the growth of silicone monocrystals. Being freely floating with no contact to crucibles and other possible contaminant sources make it possible to grow monocrystalline silicone with the highest purity which is important for a number of electronic and optoelectronic applications. Floating zone monocrystalline growth starts from a high purity, small diameter seed crystal. The seed crystal is prepared in the right crystalline direction in order to grow pure silicone with no crystalline defects.

Floating zone technology for growing silicone monocrystals is by far the most pure method and results in silicone with unique properties as opposed to any other growth technology. Float zone silicone can be grown with resistivities exceeding 100.000 Ohm

cm because of the intrin- sic process purity. Czochralski silicone does not have this high level of purity. Floating zone technique has a superior performance with respect to crystal defects like vacancies or interstitial agglomerates. In addition, material grown by this method contains basically no oxygen (concen- tration of oxygen is lower by three orders of magnitude in float zone silicone in comparison with Czochralski silicone where crucibles needed for suspending the liquid silicone). Perhaps the most important feature of the floating zone technology is the ability to exactly control the resistivity of the crystal. This is particularly important for applications using the bulk of the silicone wafers for manufacturing devices. There are two practical ways of obtaining the very good resistivity control. One is by doping the crystals when they are pulled by introducing controllable amounts of gaseous dopants into the growth chamber. Most common dopants are phosphorus and boron for n

and p

type, respectively. This technique is called

or

The other technique is by doping the crystal after it has been pulled. This technique is called

and it is done in neutron irradiating reactors. The starting material for ex-situ doping is high resistivity silicone that after being irradiated with a controllable dose of neutrons changes its resistivity by transforming silicone atoms to dopant atoms. Crystals produced by this method in a terrestrial environment are not large in size due to the weight of the melt which tends to destroy the liquid zone held by surface tension. Silicone crystals are at present industrially grown at diameters up to 150mm, weighing more than 35kg.

Briefly, the floating zone method could be described as a long semiconductor (e.g. silicone) or metallic (e.g. GaAs alloy) rod which is at first locally melted and then re-crystallized. The floating zone method uses a small heater to locally melt a small portion of the bar (Fig. 2.1).

The bar is slowly pulled through the heater and thus the melted zone will be re-solidified.

Since the ring heater creates temperature gradients along the free surface of the melt, con-

vection of different types occurs in the melt and it influences the quality of the crystal made

by this method [36]. Among the known crystal defects, which were grown by the industrially-

used methods, are voids, inclusions, distortion, strain, dislocations, inhomogeneities, striations

[45]. Among the mechanisms causing the imperfections there are gravity–dependent and those

that take place both on ground and under microgravity conditions. The gravity–dependent are

strain fields dislocations caused by weight of crystal; grain boundaries and voids being a result

Figure 2.1: Floating zone. On the left: a sketch of the method; on the right: a photo of the real technological process of a silicone crystal production by Topsil. The bar is melted by the ring heater and pulled down for the melted silicone to re-crystallize.

of constitutional supercooling; macroscopic inhomogeneities and striations (result of steady and unsteady buoyancy convection respectively); too small crystal pressure caused by hydrostatic pressure. Strain fields dislocations are also a result of spatial and temporal temperature varia- tions being caused by thermocapillary convection only, as well as the striations which are created by both steady and unsteady Marangoni convection. The processes caused by the interface ki- netics (microscopic mechanism), independent of gravity, lead to forming micro- and macroscopic inhomogeneities in crystal.

Imperfections induced by gravity.

Because of the difference between the compositions of crystal and co–existing melt even in state of equilibrium, macroscopic inhomogeneities with respect to the distribution of the dopants occur in crystal as a result of segregation phenomenon. Strong convective mixing makes dopant’s concentration profile being strongly growing with solidified fraction. Camel and Favier [9] have made a systematic analysis of the dependence of longitudinal segregation profiles in the presence of buoyancy. They found ranges of different segregation behavior in directional solidification, and presented the result in terms of dimensionless convective heat transport Gr

Sc (Sc is the Schmidt number) versus dimensionless growth rate.

Microscopic inhomogeneity with respect to the distribution of dopants is very common in mixed crystals. It is gravity–driven unsteady convection and as a consequence unsteady heat transport that cause temperature fluctuations on the solid-liquid interface and creates striations via the growth rate dependence of the segregation.

These imperfections along with the other gravity–caused ones, such as dislocations, strain fields, voids and grain boundaries, can be eliminated by reduction of gravity.

Gravity–independent imperfections.

Later researches, [31, 149], have reported that buoyancy is not the only mechanism of the

oscillatory instability and thermocapillary convection also may be a reason of the oscillations.

Eyer

[31] grew a silicone crystal of 8mm. in diameter on a sounding rocket using the floating zone technique. Investigations of the crystal revealed striations, even though the process took place under microgravity. Even if the microgravity environment on board the rocket is always far from absolute 0

g, buoyancy can not overshadow the thermocapillarity, that made thinking of an important role of Marangoni convection in crystal growth.

Strain fields, dislocations, voids and striations can also occur in absence of gravity. Each in- homogeneity has its own mechanisms but it is always the Marangoni convection that stays behind all them. It does not matter if the thermocapillary convective flow is steady or time–dependent, it is always undesirable. For example, microscopic striations are caused by time–dependent Marangoni convection while steady convection causes the increase in macrosegregation.

The oscillation phenomenon in floating-zone melt was first reported in 1979 by Chun and Wuest [19] and by Schwabe and Scharmann [110]. Later, several demonstrations of oscillatory thermocapillary flow in model floating zones have been given. Schwabe et al. [94] showed that stationary convection exists in the form of an axisymmetric roll bounded by the free surface.

Having taken a zone of length less than 3.5 mm they observed a transition to the oscillatory regime in the form of azimuthally traveling wave for M a = Re

P r

7

10

. In [20], Chun made a study of coupling of the oscillatory regime to the rotation of the crystal rods, observing transition to turbulence.

The final quality of the crystal is degraded resulted from the oscillatory convective motion leading to fluctuations in both heat- and mass–transfers at the liquid-solid interface where the melt re–solidifies and thus causing dopant striations in the crystal. Aiming at eliminating these defects and improving the homogeneity, investigations of the convection in melting zone, its stability and dependence upon different factors started. Originally, it is natural convection that was thought to cause the oscillatory regime. That is why the microgravity environment was chosen for industrial production of crystals. Beginning with the early seventies, a lot of microgravity experiments were performed. Since Apollo mission up to year 1995 at least 77 experiments were successfully done.

While in floating zone method there are a free surface and a longitudinal temperature gra- dient, one can not suppress completely the thermocapillary convection, but it is possible at least to somehow weaken the oscillatory regime. Most of the works aiming at suppressing the oscillations use methods of altering the steady state and thus decreasing the effective Marangoni number to attenuate the fluctuations. Among the methods the most popular are to put the floating-zone formed by electromagnetically active melt under magnetic field (Dold et al.[25], and Cr¨ oll et al. [24]); to create the counter flow of the ambient gas (Dressler and Sivakumaran [29]); to impose vibrations of the end-walls (Anilkumar et al. [3]) or rotate the whole system;

or simply to use surrounding gas of certain temperature and pressure that decreases the surface tension (Azumi et al. [5]). More sophisticated methods consist for example in local heating the free surface by actuators using feedback control algorithm defined by the oscillations in the liquid bridge themselves. The input of the algorithm is signals from the local sensors (Petrov et al. [91]).

2.2 LIQUID BRIDGE (LB)

The study of convection in liquid bridges is related to the floating zone crystal–growth process.

What are the model used for theoretical description of the real phenomenon and its origin?

The heater is situated in the middle of the melting zone held by rigid rods at the top and the

bottom (Fig. 2.1), where the temperature profile has its minimal values. Thus, the temperature

reaches is maximum at the mid-height of the liquid–gas interface. Spreading from the hot to the cold areas along the free surface, two sets of convective rolls (above and below the heater) are developing. Hence, to simplify the theoretical consideration of the problem, only a half–zone can be focused on (Fig. 2.2). A model used for analyzing the processes taking place in melt during the floating zone procedure of crystal growth is called liquid bridge.

Figure 2.2: Liquid bridge (half zone) model

In the half–zone model (liquid bridge) a small volume of liquid is held between two coaxial circular disks, which are kept at different temperatures yielding a temperature difference ∆T = T

T

. As the applied temperature gradient is parallel to the interface, motion from the hot to the cold region appears for any non-zero value of ∆T. When the temperature difference between the disks exceeds the critical value, ∆T > ∆T

, unique for a given set of parameters, the flow is three-dimensional and/or unsteady. Thus, the definition of the critical temperature difference for the liquid bridge problem is the minimal ∆T when the thermocapillary flow either steady and gets a three-dimensional structure (small P r) or becomes oscillatory (large P r). The transition from steady to oscillatory flow, as illustrated in Fig. 2.3, is an important feature of Marangoni convection in the liquid bridge configuration. Generally, two hydrothermal waves propagating in opposite directions bifurcate from two-dimensional state at the critical point.

They result in standing (SW) or traveling (TW) wave depending on the ratio of their amplitudes [34, 152, 122, 76]. For high P r liquid bridges a mechanism of hydrothermal instability was proposed by Wanschura et al. [152] is identical with that suggested by Smith and Davis [132] and Smith [133] (velocity – temperature coupling in case of an infinite layer subjected to horizontal temperature gradient). It is also

coupling that could be responsible for oscillatory instability. With high P r fluids, as the flow becomes faster, the temperature field gets more distorted resulting in big gradients on the cold corner.

Thus, more hot fluid is transported by the surface flow and the return flow becomes cooler.

In return, this temperature change by the flow changes the thermocapillary driving force on

the interface, and the flow rate is changing. Kamotani and Ostrach [47] suggested that the

free surface dynamical deformations could play an important role in the development of the instability. Their argument is that one could not neglect the interface deformations since the thermal boundary layer is very thin, and thus its thickness is altered. This is not the case with small P r liquids (P r

O(10

)) which have a feature of establishing almost linear liquid temperature gradients. In this case, the type of instability was identified by Levenstam and Amberg [64] as a purely hydrodynamic instability similar to the one of a vortex ring.

Type of the wave which will onset in the system is an open question. There is no concensus reached on this point. It is clear that the stable type of the hydrothermal wave depends upon the parameters of the system (Prandtl number, temperature difference, gravity level, liquid volume, ambient conditions etc.). Different experimental and numerical works, performed even for the same parameters, report different results. Having mentioned that their results being in agreement with microgravity experiment [75], Savino and Monti [103] showed that the instability appears as SW. They performed calculations for P r = 30, Γ = 0.5, 1.For P r = 4, 7 and aspect ratio Γ

(0.5, 1.3), Leypoldt et al. [68] obtained the TW the only stable solution. In annular configuration, for ν = 2cSt and a wide range of aspect ratios, Kamotani et al. [48] reported that at the onset of oscillations TW is stable. This result was confirmed by calculations of Sim and Zebib [130] made for P r = 17. But calculations by Lavalley et al. [63] gave SW as a stable solution. Carrying out laboratory experiments with silicone oil liquid bridge, ν = 1

5cSt, P r

18

90 (as they used temperature of the cold rod of

C), and varying aspect ratio in a wide range Γ

(0.3, 2.0), Ueno et al. [148] observed only SW at the onset of oscillatory regime.

The calculations performed by Melnikov et al. [74] for ν = 1cSt, Γ = 1 case confirmed the experimental results.

The hydrothermal waves are characterized by (azimuthal) wave number m which reflects the spatial symmetry of the flow. In oscillatory regime, one may see in temperature distribution a set of m hot and m cold patterns that rotate (TW) or pulsate (SW). Wave number depends upon the Prandtl number, the aspect ratio, gravity level and may be upon the temperature difference

∆T . The azimuthal wave number at the threshold of instability is called critical mode. The flow structure in liquid bridge in the supercritical parameters’ region is very similar to that observed for a flow in infinite liquid column (Γ =

) by Xu and Davis = [156]. However, the system’s spatial limitation changes the critical mode. For Γ = 1, Gr = 0, it is not m = 1 but m = 2.

The first empirical correlation for the determination of the azimuthal wave number, m

2.2/Γ, has been suggested by Preisser

[94] by analyzing the experimental data for a fluid of P r = 8.9. Here Γ = d/R is the aspect ratio (see Fig. 2.2). The slightly different correlation, m

2.0/Γ, has been obtained numerically for P r < 7 assuming pure Marangoni convection by Wanschura et al. [152], Leypoldt et al. [68].

By analogy to low P r instability and that of a thin vortex ring [64], it follows that the azimuthal wave number may satisfy the relation m = 2.5/Γ. In reality the coefficient of the proportionality for liquid bridges was found smaller than ’2.5’. The numerical calculations by Wanschura

[152] showed that the relation m

2.0/Γ remains also valid for small Pr number, although the mechanism of the instability has different origin for relatively high, 0.5

P r

7, and low Prandtl liquids.

The discussed above empirical relation m

2.0/Γ does not hold for high Prandtl number

liquid bridges P r

30. Indeed, it has been theoretically obtained by Xu and Davis [156] for high

Prandtl numbers fluids that the critical azimuthal wave number m

is equal to one for infinitely

long cylindrical jet. Experiments [12], [125], [77] with different silicone oils demonstrate that

the critical azimuthal mode corresponds to m

= 1 for P r

30 for the unit aspect ratio and

aspect ratio close to one. The 3D numerical calculations by Shevtsova

[122] have also

confirmed that this empirical relation is not valid for P r = 35.

Understanding the evolution of the thermocapillary flows is valuable for material processing in space. The first experimental and numerical studies analyzed the dependence of ∆T

on the properties of the liquids, e.g. Prandtl number, on the aspect ratio and on the presence of the gravity vector. Lately the attention was aimed at the investigation of the role of a liquid bridge volume and conditions in the gas phase, surrounding the liquid bridge. Azumi

[5]

investigated the effect of the oxygen partial pressure in the ambient atmosphere. Shevtsova

[127] focused their attention on the role of thermal conditions around the liquid bridge.

To sum up the results of experimental approach to the liquid bridge problem and show its importance, below is given a brief summary of microgravity experiments on liquid bridges formed by different substances.

1981. Principal investigators: Schwabe, D.; Scharmann, A. Experimental goal: critical Marangoni number – temperature oscillations in floating zones. Sample materials: N aN O

. Results: The experiment chamber housed a single, 6.0-mm diameter, 3.4-mm long N aN O

sample. Temperature oscillations with the same higher frequency under 1g

and 10

g

and with a more complex spectrum occur. The average temperature under 10

g

(343

344

C) is smaller than under 1g

(345

C).

1981. Principal investigators: Chun, Ch.-H.; Wuest, W. Experimental goal: observation of Marangoni convection in floating zones. Sample materials: 10 cSt silicone oil with T i and T iO

tracer particles. Results: A 10 mm long, 10 mm diameter, silicone oil liquid bridge was formed between two co-axial copper discs, subjecting the column to a constant two temperature difference of 7.82 K and 18.9 K. An axisymmetric steady Marangoni flow pattern was observed.

1982. Principal investigators: Schwabe, D.; Scharmann, A. Experimental goal: critical Marangoni number – temperature oscillations in floating zones. Sample materials: N aN O

. Results: The objective of this experiment was to determine critical Marangoni number. Analysis of the thermal data resulting from both the rocket and terrestrial experiments indicated that the onset of temperature oscillations occurred at a temperature difference of 23 K. The cor- responding critical Marangoni number for the flight was 9.6

10

, which was identical to the critical Marangoni number in the 1g

reference experiment.

1983. Principal investigators: Chun, Ch.-H.; Wuest, W. Experimental goal: observation of oscillatory Marangoni convection in floating zones. Sample materials: Methanol with titanium metal powder. Results: The objectives of this experiment were to examine liquid bridges whose corresponding Marangoni numbers were far beyond the critical values, to observe the oscillatory convection and determine if the oscillatory state progresses to a turbulent state, and to study the role of the liquid column aspect ratio. An oscillating flow was demonstrated by the up-and downwards oscillating Marangoni vortex near the free surface.

1983. Principal investigators: Napolitano, L. G.; Monti, R. Experimental goal: Observation of free convection in low gravity. Sample materials: 100 and 5 cSt silicone oils. Results:

The objectives of SL-1 experiments were to establish a stable float zones of 10 cm height, to

create Marangoni flow in the zone, and to investigate these Marangoni flows under a number of

parameters to determine the local and global properties of the system. Liquid bridges of 7 to 8

cm in length with aspect ratios on the order of one were established. Most importantly, it was

reported that the established Marangoni flows were of the boundary layer type.

1984. Principal investigators: Monti, R. Experimental goal: study of thermal Marangoni convection in a floating zone. Sample materials: 5 cSt silicone oil. Results: The specific objectives of the investigation were to study the unsteady thermal conditions in the liquid bridge and characterize the different thermal and flow regimes at relatively high Marangoni numbers. The upper disk was heated to 90

C at a rate of 1K/s. During the low-gravity phase of the experiment, three different flow regimes were established and examined: laminar (Stokes and boundary layer), oscillatory, and non-periodic.

1985. Principal investigators: Napolitano, L. G. Experimental goal: Marangoni flows – a study of surface-driven convection phenomena in very low gravity. Sample materials: 5 cSt silicone oil. Results: The objectives were quantitatively examine thermal Marangoni flows in a single-liquid system, examine the thermal and/or solutal Marangoni flows in a two-liquid system, and analyze the effects of certain parameters.

1986. Principal investigators: Monti, R. Experimental goal: study of thermal Marangoni convection in a floating zone. Sample materials: 5 cSt silicone oil. Results: The specific objective of the experiment was to investigate the onset of Marangoni oscillations under different thermal conditions. The experiment was fully controlled from Italy, using Telescience approach.

1989. Principal investigators: Monti, R. Experimental goal: critical Marangoni flow.

Sample materials: 2 cSt silicone oil. Results: The onset of oscillation regime depends on several parameters. Four oscillation onsets were obtained at different temperature ramps. The experimental results are discussed and compared with numerical simulation

1991. Principal investigators: Azuma, H. Experimental goal: observation of Marangoni convection. Sample materials: 10 cSt silicone oil. Results: This experiment observed Marangoni convection in steady state and the surface tension waves. The experimental results are in roughly agreement with the result of analytical data.

1992. Principal investigators: Enya, S. Experimental goal: study of Marangoni induced convection in materials processing under microgravity. Sample materials: Paraffin. Results:

The liquid bridge was formed in microgravity, but any flow could not be observed.

1992. Principal investigators: Hirata, A. Experimental goal: generation and control of Marangoni convection. Sample materials: 6 cSt silicone oil. Results: This experiment inves- tigated micro- mechanism of Marangoni convection. The transition process from laminar flow to oscillatory flow behavior was observed by in-situ observation. The amplitude of Marangoni oscillating flow was very small at initial and increased with time.

1992. Principal investigators: Monti, R. Experimental goal: determination of the influence of the temperature ramp and of the aspect ratio on the onset of Marangoni oscillatory regime.

Sample materials: 2 cSt silicone oil. Results: Critical value of ∆T and Marangoni number

are respectively 7

C and about 60.000. A non–uniform oscillatory regime was observed. It

implies that the damping of oscillatory flow, obtained by forcing the temperature difference to

a value of about 5

C, and by observing the transition from an oscillatory to an axial-symmetric

pattern is not sufficient to ensure a proper resetting to the initial conditions. Oscillatory regime

is effectively periodic, having a frequency of about 0.08 Hz.

1993. Principal investigators: Chun, Ch. -H. Experimental goal: higher modes and their instabilities of oscillating Marangoni convection in a large cylindrical liquid column. Sample materials: 5 cSt silicone oil. Results: This experiment investigated transitions from steady thermal Marangoni convection to non-periodic states and of higher oscillating modes in liquid columns as a function of column characteristics.

1993. Principal investigators: Monti, R. Experimental goal: onset of oscillatory Marangoni flows. Sample materials: silicone oil. Results: This experiment investigated the transition from steady to oscillatory thermal Marangoni convection in liquid columns as a function of column aspect ratio, Prandtl number, and thermal profile.

1993. Principal investigators: Kuwahara, K. Experimental goal: experiment on generation of Marangoni convection flow and controlling method. Sample materials: 6 cSt silicone oil.

Results: The experiment verified that electric convection is generated in Silicone oil when DC voltage is applied. Laminar Maranogni convection flow is generated at the temperature difference of 10K between disks. The flow could be accelerated or restricted controlling the flow caused by electricity.

1994. Principal investigators: Kawamura, H. Experimental goal: To experimentally ob- serve,the three-dimensional structure of oscillatory Marangoni convection under microgravity, and to compare the observations with the numerical simulations to clarify the mechanism by which it occurs. Sample materials: silicone oil. Results: 1) Images were obtained at the end and cross section of flow patterns Technology developed to measure velocity profile in three dimensions using image processing techniques. 2) Problems remain in terms of a quantitative discussion due to the unexpected deformation of the liquid bridge. 3) The temperature at which the transition from axial flow (laminar flow) to non-axial flow (vibratory flow) occurred was considerably higher than expected. It is probably due to the deformation of the liquid bridge and to the rather high increasing rate of the temperature.

1995. Principal investigators: Kawamura, H. Experimental goal: 3D velocity measurement of Marangoni convection in liquid column. Sample materials: 2 cSt silicone oil. Results: The measurement of the three-dimensional velocity profile of the Marangoni convection in a liquid bridge was done by the use of four CCD cameras. A liquid bridge was formed between two 50mm diameter coaxial disks with separation of 33mm. In the first stage with the temperature difference of 10 K , the axisymmetric Marangoni convection was observed. As the temperature difference become 50 K, the convection had enhanced and a non-axisymmetric flow profile, oscillatory Marangoni convection, was observed.

1997. Principal investigators: Nishino, K. Experimental goal: simultaneous observation of

3D fluid flow and liquid bridge surface temperature of unsteady Marangoni convection. Sample

materials: silicone oil. Results: All experimental procedures were successfully performed as

scheduled and every measurement apparatus functioned as designed. Unexpected leakage of

some working fluid during the formation of the liquid bridge occurred and it prevented the

ultrasonic technique and the thermocouple rake insertion method from acquiring data worthy

of detailed analysis. The unsteady complex flow field in the bridge was measured with the 3D

particle tracking technique, and the presence of vortical recirculation regions were revealed and a

maximum flow rate inside the bridge was found to be about 8 mm/sec. The IR imaging technique

showed the presence of periodic surface temperature variations with a period of approximately

10 seconds. In contrast, surface oscillations measured from image analysis revealed a period of 3-9 seconds in dynamic surface deformations. The measurement of surface velocity using the photochromic technique showed the existence of strong tangential velocity component of approximately 10 mm/sec.The experiment provided, for the first time, an understanding of the correlation between 3D fluid flow, surface temperature, surface velocity and surface dynamic oscillation of a temperature-driven unsteady Marangoni convection generated in microgravity.

It makes not much sense to list the later experiments. Starting from mid-nineties, the nu- merical results have overtaken the experimental ones. With the spectacular rise in its industrial importance, the initially largely empirical development of the subject received a theoretical con- tinuation. Many studies have been carried out aiming at the process enhancement and control.

To shed some light on the instability mechanism, theoretical studies of thermocapillary convec- tion began. The transition from the steady to oscillatory flow is well comprehended due to the numerical modeling (see for example Wanschura

[152]; Leypoldt

[68]; Lappa

[61]). These results are supported by the experimental studies, see recent review by Schatz and Neitzel [105].

Figure 2.3: Transition Phenomena of Thermocapillary Flow in Liquid Bridge (after D.

Schwabe et al.). M a

means critical Marangoni number.

Since oscillations have significant implications to crystal growth (see explanations above), it

is important to understand how and when the transition occurs. Despite the fact that much

research has been conducted on the oscillation phenomenon in the past twenty years, the cause

of oscillations is not yet fully understood. For example, recent experiments under microgravity conditions showed that the Marangoni number corresponding to the onset of oscillations is dependent on the size of the liquid bridge, suggesting that the Marangoni number alone cannot determine the conditions of the transition. The subject is obviously very complex and requires extensive experimental, numerical, and theoretical efforts to solve it.

Investigations of some important features of oscillatory flows are conducted through the experiments or analysis of partial differential equations; see reviews by Schatz and Neitzel [105], and by Kuhlmann [55]. For the high Pr liquids the 3D flow is oscillatory from the very beginning of its onset. In the low Pr range (Pr less than about 0.07), the flow is known to become at first three-dimensional steady due to a hydrodynamic instability. Levenstam and Amberg [64]

showed that with further increase in ∆T beyond the transition this flow undergoes a second bifurcation, from a stationary to an oscillatory (see Fig. 2.3). Ohtaka et al. [83] reported experimental results obtained for liquid tin liquid bridge with P r = 0.01 and aspect ratio equal 2. According to their experiment, the second transition occurs at Re of about 5000 (about five times larger than Re when transition to 3D steady regime takes place [152]). It is worth mentioning that some discrepancies between the experiment and numerical simulations regarding the critical conditions exist, especially when Γ

1 [83].

Considering spatio-temporal convective flows in liquid bridges Frank and Schwabe [34] have presented results supplementary to the well-known transitions to chaos. Except for quasi- periodic and period-doubled flow states, they have observed the onset of the time-dependent flow via chaotic intermediate flow states, the splitting of sub harmonics in the Fourier spectrum, etc. The stabilization of thermocapillary flows for remote unstable states by the application of a nonlinear control algorithm was considered in liquid bridge configuration by Petrov [90].

Due to non-linearity of the system only a few analytical results are available.

2.3 MATHEMATICAL MODELS

Fluids, considered as continuous media in classical mechanics, obey the general laws of conser- vation of mass, linear momentum and energy. In mathematical representation of equations the form they could be written depends upon a representation for the state of fluid. Two different approaches could be distinguished. One is the Lagrangian representation, where the state of fluid ”particle” is considered as a function of time and its initial position. The other approach (adopted throughout the work) is Eulerian representation, where at each time t and spatial position

= (X, Y, Z ) the state, e.g. velocity

Written at first in dimensional form, the governing equations used further for the investiga- tion of the problem will be given in non-dimensionless form. Liquid is incompressible, and the liquid density ρ is varied only with temperature. With all the assumptions made, the equations for transport of momentum and energy are the two basic partial differential equations of sec- ond order to be solved together with continuity equation for any momentum and heat transfer coupled problem:

ρ ∂

∂t + ρ(

)

=

P +

σ

+

(2.1) ρC

∂T

∂t + ρC

(

)T = λ

(

T), (2.2)

∇ ·V

= 0, (2.3)

where

is velocity, T is the temperature, P - pressure,

σ is the stress tensor.

is vector of

volume forces acting on the system. In the energy equation, coefficients λ and C

are thermal conductivity and specific heat. Operator

is nabla operator.

Velocity and temperature fields are coupled as the material properties (density ρ and surface tension σ) of liquid depend on temperature.

ρ = ρ(T

)(1

β(T

T

)), σ = σ(T

)

σ

(T

T

)),

β and σ

are coefficient of thermal expansion and σ

=

∂σ/∂T . T

is reference temperature, chosen for this work to be equal to minimal temperature in the system.

Further transformations of the conservation of momentum equation eq.( 2.1) necessitate additional physical assumptions. Assuming the fluid is Newtonian (it is incompressible as it was assumed above)

σ = η(∂V

/∂X

+ ∂V

/∂X

) (i, k = (X, Y, Z), X

, X

= (X, Y, Z), η is dynamic viscosity. Hereinafter we consider that there is only one volume force, weight due to the gravity

approximation [58]:

ρ ∂

∂t + ρ(

=

+ η(∇ · ∇)

ρβ

T

), (2.4) where ρ is taken at T

but for simplicity it is not written in the equation.

Figure 2.4: Schematic illustration of flow in liquid bridge.

The validity of the Boussinesq approximation is discussed in [58], where it is derived a necessary condition gH/c

β∆T which must be valid. Here c is velocity of acoustic waves in the media, H is the size of the system and ∆T is characteristic temperature difference.

Temperature–dependence of the surface tension comes in play via the surface forces acting

on the liquid–gas interface and thus it influences the flow via the boundary conditions. Let us

assume that the liquid is bounded by a passive gas of negligible density and viscosity and the pressure of the surrounding gas is constant. The stress balance between the viscous fluid and inviscid gas on the free surface is given by (see Shevtsova

[115, 117])

[P

P

+ σ(∇ ·

= σ

+ ∂σ

∂ τ , (2.5)

where P

is the pressure of the ambient gas, the term σ(

τ and

are unit tangential and outward unit normal vectors (Fig. 2.4).

Projections of eq.(2.5) on normal and tangential take the form P

P

=

σ

σ 1

R

+ 1

R

, (2.6)

τ

σ

+ τ ∂σ

∂ τ = 0, (2.7)

where R

, R

are main radii of curvature of the free surface. We will rewrite the second term in left side of eq.(2.7) using the linear dependence upon temperature of the surface tension σ in the form

σ

.

The tangential projection eq.(2.7) defines the thermocapillary force. The interface location is determined by the normal projection eq.(2.6) and it depends upon hydrodynamic, hydrostatic and Laplace pressure.

To complete mathematical description of the problem one should define the rest of boundary conditions. On the liquid–solid interface velocity is zero,

= 0, and the temperature has constant values, T (z = 0) = T

= T

, T (z = d) = T

= T

+ ∆T . On the liquid–gas interface normal to the free surface velocity is zero

= 0, and heat flux is given,

λ

= Q.

In our case we assume that Q = 0.

It is always convenient when doing numerical work to non–dimensionalize the governing equations, i. e., to find a set of units where all quantities become dimensionless. For this reason, one should use a scaling for all the variables. Let us take the velocity scale V

, for the length L

, the time and pressure scales t

, P

, and the temperature variation ∆T for scaling the temperature as (T

T

)/∆T . Assuming

=

V

,

=

L

, t

= t t

p = P

P

, Θ

= (T

T

)

∆T ,

=

L

, the governing non–dimensional equations read:

∂

∂t

+ V

t

L

(

)

=

P

t

ρL

V

p + ν t

L

(

)

β

V

Θ, (2.8)

∂Θ

∂t

+ V

t

L

(

)Θ = k t

L

(∇

Θ), (2.9)

∇^∗·

= 0, (2.10)

where k =

p

is thermal diffusivity of liquid.

Boundary conditions eq.(2.7) will take the following dimensionless form:

τ

σ

= σ

∆T ηV

τ

∂Θ

∂ τ

, (2.11)

Or, in form of dimensionless numbers (dropping the super index

):

∂

∂t + 1

St (

)

=

Eu

St

p + 1

Re

St (

)

Gr Re

St

g Θ, (2.12)

∂Θ

∂t + 1

St (

)Θ = 1

Re

P rSt

(

Θ), (2.13)

∇ ·

= 0, (2.14)

τ

σ

= Re

Re

τ ∂Θ

∂ τ , (2.15)

where St =

cht_ch

is Strouhal number, Eu =

ch

is Euler number, Gr =

- Grashof num- ber, Re

=

- ”classical” Reynolds number, Re

=

- thermocapillary Reynolds number.

Everywhere below

Re

= Re), and by choosing an appropriate scaling V

=

ch we put

Re

= 1.

2.4 SCALING ANALYSIS

In many cases, appropriate assumptions and approximations can be made to reduce the com- plexity of the governing equations and arrive at quick and simple solutions. This section begins with a basic discussion of scaling, an analysis technique that reveals terms that can be appropri- ately neglected in the governing equations. To start with, let us take t

=

and P

= ρV

. By that choice, we put St = 1 and Eu = 1.

Consider two–dimensional stationary plane flow (

= (v

, v

)) in rectangular corners (α

= α

= π/2) of cavities or liquid bridges when buoyancy effects are negligible and free surface is

τ = (0,

1),

= (1, 0)). Also, the pressure term will be disregarded since it could be eliminating by taking the curl of eq.(2.12).

(

)

= (

)

(2.16) (

)Θ = 1

P r

(

Θ), (2.17)

∇ ·

= 0, (2.18)

∂v

∂x =

∂Θ

∂z . (2.19)

For the boundary layer scaling, presented below, we introduce Marangoni number M a = ReP r. All the notations used below are shown in Fig. 2.5.

Small Prandtl number liquids,

M a

0.

Free surface shear-layer.

From eq.(2.17), in the limit P r

0, it follows that thermal boundary layers are absent since

(∇Θ) = 0 and

1. Thus, with low P r fluids (P r

O(10

), e.g., liquid metals), the temperature in liquid changes almost linearly from the hot to cold walls. Let the shear–layer developing on the free surface has thickness δ

(

means shear-layer). From the boundary condition eq.(2.19) it follows:

∂v

∂x + Re ∂Θ

∂z

v

δ

+ Re 1

1 = 0

v

Reδ

. (2.20)

Figure 2.5: Boundary layers in liquid bridge.

Substituting approximation for v

from eq.(2.20) into continuity equation eq.(2.18) gives

∂v

∂x + ∂v

∂z

v

δ

+ Reδ

1 = 0

v

Reδ

. (2.21) The primary momentum ballance parallel to the liquid-gas interface (y-direction):

v

∂v

∂x + v

∂v

∂z = ∂

v

∂x

+ ∂

v

∂z

, (2.22)

that together with eqs.(2.20,2.21) gives Reδ

Reδ

δ

+ Reδ

Reδ

1

Reδ

δ

+ Reδ

1

δ

Re

, v

Re

. (2.23) This scaling law was obtained by Ostrach [85] and Napolitano [78].

Rigid walls layers.

On the liquid-solid interfaces conventional viscous boundary layers are expected. The free stream velocity v

Re

. Denoted the thickness of the viscous boundary layer as δ, from the continuity equation eq.(2.18) it follows that

∂v

∂x + ∂v

∂z

Re

1 + v

δ = 0

v

Re

δ. (2.24) From the momentum ballance in x-direction:

Re

Re

1 + Re

Re

δ

δ

Re

1 + Re

δ

δ

Re

, v

Re

. (2.25)

Layers near the cold corner.

In [55] a reference given to numerical evidence that in the vicinity of the cold corner thermal gradient is characterized by a single length scale and the temperature variation over this length is of the order O(1). Let us use δ

(

means cold corner) to denote the boundary layer thickness near the cold corner. For an insulating free surface the temperature weakly varies in x-direction, perpendicular to the liquid-gas interface, and thus

= O(1). Canright [10] obtained several scalings for different limits of Re, M a.

From the boundary condition eq.(2.19) it follows:

∂v

∂x + Re ∂Θ

∂z

v

δ

+ Re 1 δ

= 0

v

Re δ

δ

. (2.26)

Substituting approximation for v

from eq.(2.26) into continuity equation eq.(2.18) we get

∂v

∂x + ∂v

∂z

v

δ

+ Re δ

δ

= 0

v

Re δ

δ

₂

. (2.27)

From the momentum and energy conservation equations using eqs.(2.26,2.27) it follows:

Re δ

δ

1 δ

+ 1

δ

, (2.28)

M a δ

δ

(1 + δ

)

1 + 1

δ

, (2.29)

In [10] the following limits were considered:

Re

0, M a

0. Then

δ

1, δ

1, v

Re.

Re

,

1. Then

δ

Re

, δ

1, v

Re

.

Re

M a

1, M a

Then

δ

M a

, δ

M a

, v

Re.

Re

1. Then

δ

M a

, δ

M a

P r

, v

ReP r

.

The last two cases, the so-called convective–viscous and convective–inertial flows, have a

peculiarity. Namely, that the boundary layer near the cold corner becomes very thin with the

length scale δ

being very small. The latter means that the cold corner implies some difficulties

when the flow in liquid bridge with the parameters belonging to the limits

1, M a

or Re

,

1 is resolved by means of direct numerical simulations. For having a

good spatial resolution of the cold corner the mesh’s intervals must be very small and therefore

calculations will drastically slow down. It was demonstrated numerically for finite-size systems,

like high Marangoni number thermocapillary convection in containers [157] and high Prandtl

number liquid bridge [127], the temperature gradient on the free surface varies considerably

within a small distance from the cold and hot walls. Over the rest of the free surface, temperature

profile may be considered to be almost constant. Due to the steep local temperature gradients,

profile of the velocity v

will have in the vicinity of the cold corner extremely high values that are difficult to resolve numerically. To overcome this difficulty, a special technique of

”regularization” exists. Its main idea is to introduce a so-called regularization function f (y) to smooth the temperature gradients in the corners. This approach was proved to work well in case of high Prandtl number liquid bridges.

High Prandtl number liquids,

M a

The analysis for the high P r, while M a

, liquid bridge problem is more complicated since the convective flow distorts the isotherms. The scaling of the viscous flow in the hot corner region under assumption that Re

O(1) was performed by Cowley and Davis [23]. The hot wall was kept at a constant high temperature within a finite distance from the free surface. The length and thermal scales were taken to be equal to the distance from the free surface to the hot wall and to the temperature difference between the hot part of the wall and the rest of the wall that could be taken as the cold wall temperature. In frames of their model the following scaling was derived:

δ

M a

, δ

M a

, where δ

is the thickness of the boundary layer in the hot corner.

For the velocity in the core far away from the boundaries, the following scaling is valid:

v

ReM a