Thermal oscillations in melts

Rag. Crystal Growrh Charact. 1981, Vol. 4. pp. 195 - 219. @ Peqamon Press Ltd. Printed in Great Britain.

0146 - 3B35/81/0701

- OlSWB.C0/0

THERMAL OSCILLATIONSIN MELTS J. A. Milsom and 6. R. Pamplin School of Physics, Univemity of Bath Bath BM 7AY, England ISubmiWdS&h

1.

December 1SRl

INTRODUCTION

Free or natural convection is within those physical phenomenonwhich have been studied since the time of Archimedes. This form of convection is generated when some strong inhomogeneityis present in a medium or an interactionoccurs between transport processes and external parameters. The foundationof the understanding of such processes lie within the encompassmentof hydrodynamicalequations. However, in spite of their intractablenature these equations do ascribe some simple flow of patterns which are termed stationary solutions,and are only realised for a particular range of functionswhich classify them. The introductionof a small perturbationin some instances drives the classifyingfunctions out of their range of validity and then the fluid becomes unstable. There is already a large body of literaturewhich has particular reference to the stability of flows such as Chandrasekhar (1961), Ray (1958), Monin and Yaghom (1971). Crystal growers have only recently found it necessary to consider the hydrodynamicsof melts from which the crystals are grown, see in particular excellent articles such as Hurle (1973), Donaghey (1980) and Laudise (1973). Steady free convection in a gravitationalfield is characterisedby five parameters thermal diffusivity,kinematic viscosity, temperaturedifference,characteristic length, and the product of the gravitationalaccelerationand the volume coefficien of expansion. From these parameters it is possible to form two non-dimensional quantities which are the Prandtl and the Rayleigh numbers. The Prandtl number is a measure of the relative iqortance of the heat conduction and the viscosity of the fluid and is defined as p _ kinematic viscosity thermal diffusivity Now this ratio, P, is an index of the capacity of the fluid to diffuse momentum as compared with its capacity to diffuse heat energy. Also this ratio between the two most significantrelaxation times in a real fluid is very important. In low Prandt number fluids the heat diffuses significantlyfaster than the vorticity which is a typical situation in liquid metals in which the effective transport of heat energy is electronic in nature. The Rayleigh number is defined as: 195

196

J. A. Milsom and B. R. Pamplin

Ra = ag TX d4 vx

and represents the balance between the properties governing the natural convection; alternatively, it can also be viewed as the ratio of energy liberated by buoyancy to the energy dissipated by heat conduction and viscous drag. From these two non-dimensional parameters it is possible to form another called the Grashof number, and the three are related as follows: Grashof number = Rayleigh x Prandtl Low values of this number imply that the transport energy is almost entirely by conduction - a molecular process. Conversely, high values correspond to convective regions and the larger this number the greater is the convective current. Two flows are similar if the Prandtl number and the Grashof number are the same. However, convective heat transfer, created by gravity forces is also characterised by another number the Nusselt number, and it is a function of the Prandtl and Grashof numbers. There is naturally no Reynolds number for free convection, due to the fact that there is no characteristic velocity parameter, and the onset of turbulence is determined by the magnitude of the Grashof number which becomes very large, as turbulent region is approached. Consider a horizontal fluid layer with gravity acting vertically downwards. The fluid layer is heated from below and a fixed temperature difference is maintained across the layer. The buoyancy forces generated by the heating cause convective motions in the fluid. These motions can, under certain conditions, appear regular, time independent and cellular when viewed from above or from the side. The simplest sort of cellular motion is that of pairs of counter-rotating rolls, which experimentally appear to be two-dimensional. That is, all fluid motions and changes in temperature and pressure occur in a plane perpendicular to the axes of the rolls. Under certain restrictions, to be described later, these rolls can be analysed theoretically, and their stability to arbitrary infinitesimal disturbances investigated. Other forms of time independent convection cells have been observed. Those which are rectangular when viewed from above are known as bi-modal. Hexagonal cells have also been reported. A certain critical temperature difference is required to initiate convective motions. Generally, when this temperature difference is first exceeded rolls are observed. If the temperature difference is further increased the rolls may change to bi-modal convection. Higher temperature differences will cause time dependent motions. The rolls may oscillate and the bi-modal cells may evolve into more complex convection patterns. Finally the fluid flow becomes turbulent. The ability of fluids to exhibit these types of motions varies a great deal from fluid to fluid. Some fluids appear to become turbulent very easily - i.e. a small increase in the temperature gradient above the critical one necessary to initiate motion causes turbulence. Whereas others will sustain rolls and then bi-modal convection for an increase in the temperature gradient many times above the critical value. The fluid parameter that is used in classifying liquids according to these different sorts of behaviour is the Prandtl number. This is a non-dimensional number - the ratio of the fluid's kinematic viscosity to its thermal diffusivity.

The physical system described above is known as Bdnard convection. For practical considerations the fluid layer is enclosed between rigid boundaries which are good conductors of heat, and the fluid depth is kept constant. In order to simplify

Thermal Oscillations in Melts

197

experimentalinvestigationsof convection cells and their instabilitiesthe depth of the fluid layer is chosen such that the temperaturedifference across it is small. This, therefore, implies that variations in the fluid properties due to changes in temperatureand pressure can be ignored excluding,of course, the change in density necessary to cause the buoyancy forces. Under these constraints the Rayleigh number - a dimensionlessparameter - is a measure purely of the temperature gradient across the layer. The two parameters - Rayleigh number and Prandtl number - together with the wave length of the convection cell, serve to describe the types of Bdnard convection with which this study is concerned. A theoreticalstudy of Lord Rayleigh (1916) - but for the stress free boundaries - showed that for the values of the Rayleigh number Ra less than a critical value Racr no motion would occur, conductionalone being sufficient to transport the heat across the layer. This conducting fluid layer becomes unstable to infinitesimalperturbationsfor values of the Rayleigh number greater than Ra,,. Lord Rayleigh also showed that the critical Rayleigh number is the same for all fluids, whatever their Prandtl number. At the critical Rayleigh number there is one perturbationmode with positive growth rate, and this has wave number k Chandrasekhar (1961) gives for the case of rigid boundaries, the value of 3.11$:' A number of experimentalinvestigationshave been made recently of the motions resulting when the Rayleigh number exceeds critical. The nature of these motions is dependent on the Prandtl number of the fluid. For moderate and large Prandtl numbers, i.e. greater than or equal to 6.7 these motions are stable until the Rayleigh number is Racr, which for water is 17,000 and for fluids with larger Prandtl numbers is 22,000. For values of the Rayleigh number greater than Ra,, three-dimensionalsteady motions occur. These are in the form of regular rectangularcells, i.e. bi-modal convection. However, as the Rayleigh number is increased, this form of convectionbecomes unstable - for water at a Rayleigh number of 31,000 and for oil of Prandtl number 100 at a Rayleigh number of 56,000. Time dependent three-dimensionalflows t_henresult which in turn at still higher Rayleigh numbers the fluid motion becomes turbulent. For small Prandtl number fluids the situation is markedly different. Time dependent flows occur in mercury at Rayleigh number = 2300, and in air at Rayleigh number = 5600. These flows appear to be two-dimensionaland oscillatory,taking the form of waves along the axes of the rolls. Rosby (1969) found with his experimentson mercury that no steady flows occurred once the critical Rayleigh number was exceeded. In experiments to find the wave number of two-dimensionalconvection cells, Willis, Deardoff and Sormzrville(1972) also corroborateKrishnamurti's(1970) findings that the wave number is always less than kc,. Their experimentswere on air, water and a silicone oil with Prandtl number = 450. With water they found that the wave number, for a Rayleigh number of 10,000 is about 2, and that increasing the Rayleigh number decreases the wave number. From these experimentalstudies it appears that fluids with Prandtl number greater than or equal to that of water (i.e. 6.7) exhibit much the same behaviour: rolls with well defined wave number for Rayleigh numbers up to about thirteen times critical, then bi-modal convection for Rayleigh numbers up to about 30 times critical and time dependent three-dimensionalflows beyond this. This implies that the behaviour of water is like a fluid with infinite Prandtl number. The other fluids with small Prandtl number (i.e. mercury) will only sustain steady convection rolls for a small increase in Rayleigh number above critical. Higher Rayleigh numbers lead to two-dimensionaltime dependant flows. Thus it follows tha, the behaviour of convecting fluids with Prandtl numbers near to 1.0 is strongly dependant on their Prandtl number.

198

J. A. Milsom and B. R. Pamplin 2. LANDAU THEORY AND HYDRO-DYNAMICINSTABILITY

The next section of our review will be centred on Landau's theory. Linear stability theory gives a critical Reynolds number Re,, where there occurs a marked transition in the flow pattern. Following the historical approach of Landau we will consider only the Reynolds number, but in some cases initiationof instabilityor change of flow state can be viewed from another non-dimensionalparameter of the same type, for example the Rayleigh is of particular interest in our discussion. The general foundations,on the behaviour of a finite disturbancewith Re in the region of Recr which are in fact independenteven of the actual form of the equation of fluid dynamics were formulatedby Landau (1944). Two assumptionsare now made, first that Re > REcr and secondly the differenceRe - Re,, is small. Now when Re = Re,, there will appear a disturbancewhich has a 'complex frequency'which has a zero imaginary part. There will also exist an infinitesimaldisturbancewith a velocity field having the form V(x,t) = A(t) F(o)

1.1

where A(t) = eBiwt = e-gt-iwlt and F(x) is the eigen function of the associated eigen value problem. Now A(t) will satisfy the equation

i!lg

= 2ylA[*

1.2

As A(t) increases with time (1.2) may be viewed as a truncated expression and should be replaced by a series expansion in powers of A and A* which represent the terms in the dynamical equations that are non-linear disturbances. The motion will be accompaniedby rapid periodic oscillationswhen compared to the characteristictime 1 -. However, these periodic terms are not relevant and they are excluded from 1! dA2 considerationby averaging the expression-. Now since the third order terms dt A and A* will contain a periodic factor, they will vanish during the averaging procedure. For the fourth order terms, after averaging, there will be only a contributionproportional to lA14. The resulting differentialequation will be: 1.3 The coefficient 6 can be positive or negative and in special circumstanceswill be zero. The general solution of equation (1.3) can be written as

IA(t)l*

=

ce,‘y” 2yt

1.4

1+2yce

where c is a constant of integration. When 6 > 0 and t = 0 then

PI WI2 = 1 + ;,2y

1.5

199

Thermal Oscillations in Melts Now if (1.5) is sufficientlysmall, the amplitudeA(t) will initially increase exponentiallywith time and as t approaches finite values the rate of increase decreases. As t approaches infinity the amplitude approaches the value

A(m) = {a,' which is independentof A(0). 6 When Y is a function of the Reynolds number, (it is zero when Re = Re,,) it is possible to expand it in terms of a power series constructedfrom term of the form Hence it follow that Y is now of the order of (Re - Re,,)! , and also 2-T zt?imax - (Re - Recr)j. Turning now to the condition that 6 < 0, the final amplitudeA(m) will be nearly finite and constant and can only be evaluated when higher powers of A are included; implying that equation (1.3) is not now valid for Re > ReCr. Alternatively,for Re < Re,, any small disturbancesgenerated of the form (1.1) will be damped to zero . amplitude. Returning again to Re > Re,, with 6 > 0 the increase of disturbances representedby (1.1) just above Re > Re,, may be viewed as the soft-excitationof an elementary oscillator, leading to the establishmentof steady periodic oscillation with a small but nevertheless finite amplitude proportionalto (Re - Re,,)! . Equation (1.3) only specifies the amplitude of the oscillations. The phase is not defined by the external constraints and in fact the phase may be arbitrary in nature. Thus the final form of the steady oscillationsof such an oscillator is characterizedby a single degree of freedom. Further increasingof Re, this final periodic motion may itself become unstable to small disturbancesof This disturbancewill take the form V - eq(-iwt)f(z,t), where the form U(z,t). 2n A new frequencyL)= w2 now f(x,t) is the periodic function of t with period -. Wl

appear when Re = Re2cr, the positive imaginary part of the 'complex frequency'. Therefore as t approachesvery large values quasi-periodicoscillationsoccur with two periods a

Wl

and 2

now having two degrees of freedom. Hence with a further

increase in Re a series of new oscillatorswill appear; the intervalsbetween the correspondingcritical Rayleigh numbers will decrease continuouslyand the oscillationswhich appear will be of a higher frequency and smaller scale. Then fol sufficientlylarge enough Re the motion will possess very many degrees of freedom and be very complex and disordered.

3.

STABILITY OF A IIEATEDFLUID LAYER

Turning now back to the problem of convection in a fluid layer heated from below results which are also equally applicable to a fluid layer heated from the side. The loss of the condition of rest in a fluid is characterizedwhen some critical temperatureis exceeded which then introduces steady convection. Themain contributionof heat transfer is changed from conduction to convection. These features for a fluid layer heated from below have been very clearly described Rayleigh (1916) who analysed the problem of convection in a fluid layer enclosed between two free boundaries. The problem reduced essentially to an eigen problem for the differentialequation: (&_k2)($-@+i~)[$-1(

+iw.~]~~+kRa.e-O

1.h

1.5b

200

J. A. Milsom and B. R. Pamplin

To estimate the eigen value spectrum , the value of the critical Rayleigh number and the critical wave length is to let W = 0 in (1.5a). Then (1.5a) reduces to the equation:

( $ - k213 e + k’Rae

= 8

Kernel solutions of this equation which satisfy (1.5b) for 5 = 0 and 5 = 1 are all of the form 6 = sin xng, with n = 1, 2, 3 ... etc. Therefore is derived a series of neutral disturbances where w = 0 and the wave numbers satisfy:

(a2n2 + k2)3 = @

1.5d

The minimum, for every given value of (rr2+ k2)3 n = 1, hence giving Ra,, = 2, 2Wk minimum value of k: Ra,, = -where 4 It is evident that the values of Ra,,

k, will correspond to a disturbance with and by suitable analysis gives for the 42s k cr=T* and kc, depend upon the boundary surfaces.

To obtain meaningful results for convection between rigid temperature boundaries similar calculations are required, as has just been indicated, however numerical methods are now employed. These methods have been carried out by Low (1929); Pellew and Southwell (1940), Lin (1955), Reid and Harris (1958) and Chandrasekhar (1961). The parameters characterising the marginal states are tabulated below: Nature of boundary surface Both free One rigid, and one free Both rigid

Racr 657.51 1100.65 1707.76

kcr 2.22 2.68 3.12

Now the value of k yields only the periodicity of the flow in the (z,y) plane and not information on its amplitude characteristics. Naturally, for fluid layers which are subjected to heating the non-dimensional parameter for consideration is the Rayleigh number. When Ra > Ra,, stationary solutions describing states of rest are not unique, in addition to these, further stationary cellular solutio;; ,rii-e of which the amplitude is proportioned to (Ra - Ra,,)i but for small

Ra cr.

This feature agrees well with the concept

that for Ra > Ra,, in a heated fluid layer there occurs softly excited oscillations, which are spatial in character, corresponding to Landau's theory with Re > Re,,. Further cellular solutions of non-linear Boussinesq equations were investigated by Sorokin (1954), Gor'kov (1957), Malkus and Veronis (1958), Kuo (1967), Bisshopp (1962). An important feature, which is observed experimentally, is the existence of a single preferred disturbance mode. The question of such a single mode is closely linked with stability theory for cellular convective motions which is at present far from being complete. The Landau theory proposes that for Ra > Ra,, there will exist an infinite set of unstable infinitesimal disturbances corresponding to a particular spectrum of values of wave numbers, k, surrounding the specific value k = kc, at which the onset of instability appears. However, the most unstable perturbations will correspond to one specific k value, but there will also be an infinite set of such perturbations which can be described by an arbitrary function. Experments have illustrated that under each specific set of boundary conditions there will always arise a single perturbation having a strictly defined form corresponding in a particular instance to a division of the horizontal plane into a

Thermal Oscillations in Melts

201

set of regular hexagonal cells, having a particular amplitude. The Landau theory only yields information that the steady amplitude may be found. However, the theory affords no explanation of why perturbations having several values of k may never arise in the fluid, and why among all the perturbations with value k only those with one special form are actually observed. Segel (1962) explained by calculations, the fact that in a number of specific cases, non linear interactions of perturbations differing in wave number may lead to the dominant growth of perturbations on one wave number, with the resulting suppression of all the rest. In this work a simple "pair interaction" of two rolls independent of the y coordinate in layer bounded both above and below by the unusual plane-free boundary conditions was considered. Segel investigated the evolution of a perturbation for which the velocity component w(x,t) = w&,n,c,t) where 5 = X/D, II = Y/D, 5 = ZfD takes the form of: W (z,t)

= Al(t)

cos

Wl(c.)

+ AZ(~)

cos

lsf2(cJ

1.6

+ other small contributions. Then applying the techniques of Stuart (1960) and Watson (1960) Segel derived as a first non-linear approximation the following "amplitude equations" for the functions Al and A2 having the following form:

dt

= Y,

A,

-

(&Al2

1.8

+ 62A22)A2

Now when A2 = 0 or A1 = 0 the equations will yield a differential equation which is equivalent to the Landau equation, for the amplitude of a single disturbance. The equations (1.7) and (1.8) have the following steady state solutions: Al

= A2 = 0

I

Al = 0, A, = (Y2B2+

II

Al = (Y1/61+, A, = 0

III

Al =

O’,b,

-

Y2Bl)

A,

0’261

-

Y182)

=

1 1

(6162

-

8182)

(6162

-

8182)

-1 -1

)I”

The stability of these solutions may be confirmed with the ordinary methods of stability theory of differential equations. The important case for consideration is when Yl > 0, Y2 > 0, 62 > 0. Now when either solution II or solution III are stable it is implied that solution IV cannot be stable. Thus there is a large class of solutions in which the final state will contain only one roll cell, having a definite wave number, never a mixture of both. A further analysis of the equations when Ra is just greater than Racr shows that if the linear theory growth rate yl of the first roll is more than at least twice the growth rate y2 of the second roll then only III is the stable steady solution. Likewise if y2 > y1 only solution II is the stable one. This implies if one of two competing primary perturbations has a great advantage, in the frame-work of linear stability theory, only the advantaged perturbation will appear while the other one will decay. Now when y2 > y1 > 2~2 both solutions II and III are locally stable; hence in the final state either II or III appear; however which one appears is a function of the initial conditions. In the case with yl > 0 and y2 > 0 and in a forced convection flow system it may lead to an increase of the unstable perturbation corresponding to a solution of type IV.

202

J. A. Milsom and B. R. Pamplin

For thermal convection problems when Ra > Ra,, with instability of both primary rolls one of the rolls will decay. This explains the fact that the interaction between the various perturbations when Ra - Ra,, is small, out of a spectrum of unstable perturbations only a single value of wave number k is always observed. A more general approach to the problem of the way of selection of a single preferred wave number from a whole spectrum of unstable wave numbers for Ra > Ra,, is given by Pomarenko (1968). Az has been previously stated out of an infinite set of different perturbation modes perturbations only arise with a wave number k. This is confirmed experimentally that only a single mode is observed under specified external conditions. This implies that the investigation of the stability of the different steady solutions of the non-linear Boussinesq equations for Ra > Ra,, is most important. Schluter, Lortz and Busse (1965), these investigators considered all these steady solutions which are generated from infinitesimal perturbations arising from (1.4) at a slightly super-critical value of Rayleigh number and studied the stability of the finite amplitude cellular motions derived. They draw an unusual conclusion namely: that all the cellular solutions with the exception of the simplest two-dimensional rolls are always unstable. For the exceptional case of the rolls with a particular given horizontal wave number k, Schluter et al. showed that they are stable to all infinitesimal perturbations with the same wave number k if only this wave number is contained in a band of the unstable wave numbers. Furthermore the authors investigated the stability of two-dimensional rolls of finite amplitude to infinitesimal perturbations of horizontal wave numbers. They found that when Ra - Ra,_, is small enough the rolls wave number k < kcr, where k,, is the wave number of the infinitesimal disturbance which is neutrally stable at Ra cr, cannot be stable to disturbances with arbitrary wave numbers. Figure 1 shows the stability range of rolls at Rayleigh numbers close to critical. Schluter, Lortz and Busse obtained their results by an asymptotic expansion procedu in powers of a small parameter E and valid only for Rayleigh numbers in the region of the critical value. The general stability analysis for solutions of the Boussin equations for higher Rayleigh numbers becomes an intractable problem. However, in the case of a fluid having an infinite Prandtl number the Boussinesq equations are considerably simplified and the stability problem becomes within the encompassment of mathematical analysis.

I

i I

Fig. 1.

Thermal Oscillationsin Melts

203

Employing numerical methods Busse (1967) computed steady solution equationswith Pr = m in the form of rolls for a wide range of Ra numbers and investigatedthe stability of the solutionswith the aid of linear stability theory. He found that the stable rolls correspondedto a narrow elongated region within the (Ra, k) see Pig. 2. The range of stable wave numbers for all Rayleigh nutiers below 22,600 is confined to a small region surroundingk,,.

Fig. 2. At I& - 22,600 all two dimensional solutions of the Boussinesq equations become unstable. This value of Ra at which a second discrete transitionoccurs has been observed experimentally. The results in Figures 1 and 2 are in qualitative agreement with the fact that the characteristicscale of the horizontal cells does not change considerablywhen the Rayleigh number increases. Two dimensional convective cells have been observed by several investigators:Silveston (1958, 1963), Koschmider (1966). Chen and Whitehead (1968), Dubois and Berge (1978) and indirectlyby Bolt (1975) and Milsom (1978). The study of natural convectionof an enclosed fluid, subject to a horizontal temperaturegradient has focussed considerableattention. The majority of the work has been centred on geometry with large aspect ratios. Batchelor (1954) investigatedtheoreticallythe conduction dominated and transitionareas. Further work by Elder (1965), experimentallyinvestigatedthe laminar boundary layer region and reported the existence of secondary and tertiary flows before the corrmencement of turbulence in boundary layers. The stability of the flow has been investigated by Gill (1966); Gill and Davey (1969) and more especially the heat transfer characteristicsby Newel and Schmidt (1970). The work has shed considerablelight on the features of internal natural convectionwith improved temperaturegradients for large aspect ratios. The annular problem has been investigatedby Thorsas(1970) He employed numerical models and obtained the following flow characteristicsfor small aspect ratio when the Reyleigh number is less than 400 the heat transport process is by conduction,while if the Rayleigh number is greater than 3,000 heat transport processes are confined to the boundary layer region. Elder has confirmed experimentallyfor an aspect ratio in the region of unity and with a Rayleigh number of the order of 1,000 the vertical temperaturegradient is zero. However in the interval of the Rayleigh number greater than 1000 but less than 100,000 there are larger temperature gradients in the proximity of the vertical walls and the flow away from the horizontal boundaries is concentrated mainly in the vertical layers.

204

J. A. Milsom and B. R. Pamplin

A detailed experimental study of the velocity field in Rayleigh-Bernard convection has been carried out by Dubois and Berg& (1978). Local velocity measurements were carried out in a convecting layer of fluid and showed that the velocity field can be described by a dominant fundamental velocity mode with also a small proportion of superimposed second and third harmonics. It is well known Davies (1968), Stork and Muller (1972) that in a rectangular box the convective structure just above Ru,, consists of straight rolls parallel to the shorter sides. When Ru - Ra,_ is small the rolls are set up preferentially with a critical wavelength which is of the order of twice the depth of the fluid. An important feature is that the structure of the rolls is essentially two dimension

Ra In the interval 0 < E < 2 where E =

- Racr

Ra

the second and third harmonics are

very small compared with that of the fundamental. Thus the domain is called the "linear domain". On comparison between the amplitudes of the horizontal and vertical components there is a conservation of mass flow in a roll. Busse (1967) was the first to give the local variables in a convective fluid. Norman et ail.(1977) have calculated the amplitudes and spatial dependences of the velocity field. These calculations are based on the first three harmonics. The series is founded on an expansion in terms of, E, namely: R = Rc + E Rczl + a??2Ru2 + E3 Ra3 + . ..

1.19

NOW for the case of two dimensional rolls, we have Ral = Rug - 0 and for value of E

1.20 The amplitude of the vertical fundamental mode is given by V = 0.96

where k = $!

’ Ed

x 2

d giving a dimensionless wave number.

1.21

At the critical wavelength

k = kc = 3.117, an important feature emerges here, that for a fixed value of E the velocity amplitude is only a function of the thermal diffusivity of the convective However for large d the fluid breaks up into fluid and depth of the fluid layer d. a series of rows of convective cells and the theory is no longer valid.

4.

APPLICATION TO CRYSTAL GROWTH

Workers in the field of crystal growth have brought to light a stability problem, which occurs in the melt from which crystals are grown. The problem is centred upon internal natural convection, due to an imposed horizontal temperature difference; however, having aspect ratios which are less than unity. Turning now to the production of crystals the method which will be of chief interest is the zone melting technique. Employing this technique, the material is contained within a rectangular box with typical dimensions 100 mm length 30 mm or 20 mm in width and height. The material crystallises out preferentially at one end of the melt due to an imposed temperature gradient. If the crystals are grown from other than pure elements, striations are observed in the resulting crystals at right angle to the growth axis. Deda (1961) reported the existence of resistivity striations ir

Thermal Oscillationsin Melts

205

germaniumwhich he concludedwas due to the super-coolingof the crystal. He also demonstratedthat the striation repeat distance is a function of the temperature gradient. Further workers Cole and Winegard (1964, 1965) observed thermal oscillationsin liquid tin when the boat waa heated from below in the vertical plane and heated from the side in the horizontal plane. They proposed that the thermal oscillationsmarked the coamrenceumnt of thermal turbulence,in addition demonstratedthe convectivemixing had an important influence on the separationof the solute in the growing crystal. Komarov and Saga1 (1963) showed that when a temperaturegradient in excess of 6' C/ma, was applied to a partially melted bismuth ingot the solid-liquidseparation boundary oscillated steadily. The oscillationswere uniform and could be maintained for several hours; furthermore,the addition of a small amount of tin, as an impurity, did not influence the nature of the oscillations. It was also possible to repeat the observationswith an ingot of lead. The consensus of the opinion was that phenomenon was linked closely with the coammncementof turbulence,at the solid-liquidboundary. Wilcox and Pullmer (1965) worked with calcium fluoride, in a Czochralskicrystal growth system. The authors observed visually the convectivemotions, and recorded the temperatureoscillationswhich reached a maxim value of 30' C, having a frequency of the order of 4 Hz. Several methods were carried out to overcome the effect of the oscillations:rotating the crucible, and the provision of after heaters were effective methods of reducing the oscillations,while baffles appeared to have little influence. Morizane et czz.(1966) used the same experimentalsystem as the previous experimenters;however, the working fluid they employed was an indiunrantimony alloy. With their apparatus the isolated melt did not exhibit temperature oscillations,but when a crystal seed was introducedoscillationscormaenced.The nature of the oscillationswere irregular and present under static and growth conditions. However, large oscillationswere induced when the seed-holderwas water cooled. Muller and Wihelm (1964) continued the work of the previous authors with the horizontal solidificationof an indium-antimonyingot. They observed, when directional solidificationoccurred in ingots of indium-antimonya loy, in the molten region oscillationsof the order of 10' C and frequency of L Hz were obtained. They concluded with a series of further experiments,that the oscillation could not be attributed to periodic freeeing and melting of the ingot. With a further series of experiments the authors observed oscillationsin mercury and liquid indium (low Prandtl nu&er fluids) but not in water and ethylene glycol (relativelyhigh Prandtl number fluids). Cockayne and Gater (1967) and Cockayne et ai?. (1969) have investigatedtemperature oscillationsin oxide melts during crystal growth by Cxochralskimethods; now in melts of calcium compounds regular oscillationswere found to occur. As with previous experimenters,the addition of other elements such as neodymium to the melt did not effect the oscillations. In opaque melts it was possible to observe visually the stream lines within the liquid; the liquid flowed down the centre of the vessel and described approximatelya circular pattern of approximately2-3 mn with a frequency having a period of 10-15 seconds. This convective loop is a prerequisite for the initiation of temperatureoscillations. Other larger convective flow patterns occurred which produced temperatureoscillationsof greater magnitude. The main component of the flow also had a radial spoke pattern and the sideways oscillationof the spokes produced a component which was added vectorially to the main oscillations. Temperatureoscillationsin mercury were also observed by Bradshaw (1966) and reported by Pamplin (1967). In addition, Hurle (1966) has demonstrated,under suitable constraints,the temperaturefluctuationstake the fors

206

J. A. Milsom and B. R. Pamplin

of sinusoidal oscillations which persist for long periods of time. Hurle has suggested that the oscillations are an example of overstability which is discussed fully by Chandraskhar (1961). Hurle (1966) and Utech and Fleming (1966) show that the oscillations can be damped by a magnetic field and this implies that the oscillations are hydrodynamic in character rather than the early observations which ascribed the process as the periodic release of latent heat of fusion during solidification. It is clear that steady convection will not produce strations but that an oscillating convective cell must be present causing the growth rate to fluctuate in time, thus causing more or less impurity to be incorporated into the growing crystal. Further work has been carried out to gain a fuller insight into the understanding of the nature of the oscillations. Hurle (1966, 1967); Hurle, Gillman and Harp (1966) and Jakeman and Hurle (1972) reported some initial experiments employing a horizontal cell containing molten gallium which was subjected to axial heat flow. These authors all observed the onset of regular sinusoidal oscillations of temperature in the melt at a critical axial temperature gradient. A comprehensive investigation was carried out by Hurle, Jakeman and Johnson (1974). An experimental study of the occurrence of temperature oscillations in molten gallium contained in a rectangular boat and heated from the side was carried out. The following observations were found. For a fixed depth of working fluid and width of boat the frequency of oscillation was found to vary linearly with boat length (see Fig. 3 below) 127

IO-

6-

.’ x’

6-

4-

z-

0

.’

/!--2

Boot length,

3

4

cm

Fig. 3. There appeared no correlation between the frequency of the oscillation and the uelt depth for a fixed boat width 13 mm and a fixed length of 30 mm (see Fig. 4).

Thermal Oscillationsin Melts

:e

207

1.4 Melt

depth,

cm

Fig. 4. The dependence on depth was also investigatedas a function on the critical Rayleigb number, for depths less than 5 mm than Rcz,, _ 300, while for depths greater than 5 mn Ra,, is proportional to 8, where n = 3.4 k 0.5. The effect is illustrated in Fig. 5.

Meltdspth,

cm

Fig. 5. The critical Rayleigh number was a function of the square of the Hartmann number; the constant of proportionalityappeared to increase with decreasing length of the melt over the range investigated (see Fig. 6). The following conclusionswere drawn from the experiments. The oscillationswere a perturbationon the basic flow rather than the onset of an entirely different mode of flow. This idea was supportedby the fact that at the onset of oscillationsthere was no significantincrease in the Nusselt nunber. Johnson (1967 using liquid gallium as well, found that the frequency of the oscillationswas proportionalto one quarter power of the depth. Also the critical temperature,by suitable experimentalinvestigations,increased as the square of the Hartmann number on the applicationof a transversemagnetic field. Further work was also conducted on the mean temperature field; however, any correlationwith the oscillationswas not recorded.

208

J. A. Milsom and B. R. Pamplin

MZ Fig. 6. Bradshaw (1966) was able to show that the thermo-soluatalinstabilitydid not influence the frequency of the oscillations,by varying the purity of mercury. He also found by severely reducing the boat length the oscillationscould be eliminated This feature has been confirmed theoreticallyby Gill (1974). Bolt (1975) also carried out experiments in a box geometry configurationwith mercury using small aspect ratios 0.05-0.08. He concluded for depths of mercury in the range 5-8 mm the frequency of the oscillationswas in direct proportion to the depth. However, he could detect no clear-cutminimum horizontal temperaturegradient for the onset of oscillations. He also suggested the existence of a set of rolls having a cell length of the order of IO nm. Experiments frequentlyexhibited a regular amplitude modulation. This feature had been noted by another workder Caldwell (1974); he had found a regular amplitude modulation, possibly the result of the movement of convectioncells relative to his temperaturemeasuring thermistor. Bolt (1975) alsc found during a series of experimentsemploying a differentialthermometer,the results showed a rhythmic rise and fall of amplitude inferring a wavelength. Skafel (1972) conducted the majority of his experimentswithin an annulus subjected to a radial heat flow. He measured the mean temperatureprofiles and the power spectra of the oscillationsjust above their onset. Spatial correlationmeasurement suggested the existence of a progressivewave in the azimuthal direction;he concluded that the frequency of the oscillationswas directly proportionalto the fluid depth. He also described a so called 'resonancecondition'when for particule aspect ratios relatively large temperatureoscillationswere obtained and he attributed their formation to standing waves created in the azimuthal direction. Milsom (1978) used an apparatus similar to that of Skafel. It was an annulus with a temperaturedifferencemaintained between the inner and outer cylindrical surfaces, both mercury and gallium were used as working fluids. The lower surface was insulated and the upper surface exposed to air or covered with a thin layer of silicone oil. The majority of the experimentalwork had been conducted in rectangular boats of relatively large length of the order of 100 mm, with aspect ratios of the order of lo-&. The effective range of boat length employed in Milsom's experiments were 15-30 nunwhile the range of aspect ratios varied from 0.2-0.6. The follc results were found by the author: the existence of a critical temperaturegradient for the onset of oscillationsis confirmed by the results illustratedin Fig. 7.

Thermal Oscillationsin Melts

%?

L ; lz

209

7(01 e

6-

=I.6

(b)j$ = I.66 (c) R = 2.68

5-

4-

RO _ (d) Fir-=

3-

(e) 5 RI

=4

2l-

0

0.1

02

0.3

0.4

0.5

0.6

I 07

Asp& ratio Pig. 7. It was found that the magnitude of the oscillationswere not limited by the temperaturegradient, however, they are seen to vary dramaticallywith aspect ratio. This variation is shown very clearly in figure 8. Increasing the cell lengths another oscillationpeak was observed at an aspect ratio of 0.2 and a diagram similar to figure 7 showing this feature will be found in Milsom (1978).

The influence of the theoreticalwork will now be discussed. Gill (1974) using the work of Hart (1972) as a foundation,derived a model for the explanationof thermal oscillationscreated spontaneouslyin a crystal growth melt. The mechanism of the basic state could be described as follows: a horizontal buoyancy gradient generated a vorticity leading to a shear strain rate. The advection created by the associated velocity will generate a vertical temperaturegradient. Following linear perturbationanalysis, the Landau scheme, velocity and temperatureperturbations are introduced into the basic flow state. Three hydrodynamicequations are formed: x component of momentum, vorticity equation and temperatureequation. Then an eighth order partial differentialequation is derived in terms of a two dimensional stream function. The partial differentialequation is now made non-dimensional. Tl resulting algebraic equation is truncated to the s-11 Prandtl number limit. The following expressionswere obtained in the small Prandtl number limit Ra2cr =

where

2k0

t2 ‘4 - “B k2)

C2 - a2d2 and c2 + n2 - k2

1.22

1.23

210

J. A. Milsom and B. R. Pamplin

20-

15-

IO-

05-

0

I

I

I

01

02

03

I

I

0.4

0.5

I 0.6

Aspect ratio Fig. 8. Non-dimensionalfrequency 5

dT =

Ra2

2k6 + .2

T

=

Ez

,y

1.24

c2 Ra2cr?f'z k2

1.25

Gill (1974) concluded that the oscillationswere primarily longitudinalin nature and could be explained in terms of a diffusion dominated inviscid model. A simple roll convection cell is envisaged, then the plane z will lie between fluid layers with opposing directions of movement along the x axis. Molecular diffusion will occur across the z plane for liquids of low Prandtl number it will be almost instantaneous. Now the z momentum of the particles will be conserved and a perturbationvelocity is superimposedin opposition to the main convective flow and this advection induces a correspondingtemperatureperturbation. The particles are at their minimum temperatureat maximum elevation hence the restoring force is a maxirmm. The fluid is effectively inviscid and simple harmonic motion is the result Figure 9 shows successivepositions, thick line, of the particles undergoing oscillations. The oscillationswill have a frequencyf given by: 1.26

211

Thermal Oscillations in Melts

(ii)

*

Jt Y

Fig. 9. However, in a box geometry, which has finite dimensions the following constraints apply: (i)

when the ratio of the width to depth 5 is less than 0.58, both Ra,, and the frequency

(ii)

f become

infinite and oscillations are not possible.

when the ratio of 5 is less than 2.0 the disturbance wavelength is less than half the optimum value and the wavelength is selected by the geometry so that then

1.27 (iii) when 3 is greater than the critical value of 2.0 then k is fixed at the value for the fastest growing wave and n a i then

fad

1.28

Hurle et al. (1974) predict an oscillation period and Rayleigh number corresponding

1.29

Ra - 4n 2 (l+Y21x( Y where

)1

Yp3?2 mW

These equations are considered in detail later.

1.30

1.31

212

J. A. Milsom and B. R. Pamplin

Milsom (1978) transformed the Gill model into partial cylindrical polars as the geometry of his system was an annulus with a temperature difference maintained between the inner and outer curved surfaces. The critical Rayleigh number is given by: r&8

Ra2cr

4k8p

+

+

4k”

32 pi

+ 2k8p2

+ 4k6

32 P312

+ 234k’+p

=

1.32 32 (Es - Fzk2p

- rzk2

- F,k2pi)32

The non-dimensional frequency is: Pka + 32 Ra2 cr uz + 32 Ra2cr 2= OF

P ? k2 + 32 k6 p3/2 z

1.33

k4 + 2k4P

and 2k6 P; + k6 $12 uT

+ 32 Ra2cr

pi

TZ + 2P 32 k4

2=

1.34

k2p;

In the small Prandtl number limit, the expressions now simplify to: Ra2cr

2k*

=

32 (uz 32 Ra2

cr k’+

2= OT

1.35

- Fz k2) v

a

2k6 + 32 Ra2cr 2

1.36 Fz

=

OT

1.37

k2

These equations (1.35-1.37) are identical to equations ( 1.23 1.24 1.25 ). The non dimensional shear strain rate and temperature gradients are different in two cases. The Prandtl number has a significant effect whether oscillations occur. The Gill model predicts for a Prandtl number greater than 3 that the stabilising effect of tl vertical temperature gradient becomes too great for instability to occur. In addition, for a high Prandtl number fluid in the transition region between conduction and convective heat transfer the magnitude of the velocity is usually not great enough to provide a suitable large shear strain rate. Hurle et al. (1974: quotes the following formulae for the Rayleigh number and the oscillation period: Ra

- 4rr4 (1 + Y2) Y

1.38

where

y=zLG

1.39

mW

and p and m are positive integers. actual experimental range of Ra,,

Now @cr is

is equal to 1030 for Y2 = 0.29.

from 410 to 8600.

The ratio{$j2>

with p = 1 there would be one wavelength contained with the boat.

The

0.29; thus

However, Gill's

Thermal Oscillations in Melts

213

theory cannot predict the effects of the finite length of the containing boat but it affords a good explanation of the phenomenon and is consistent with many observations. From the experimental work of Hurle et al. (1974) the basic flow is i siIPpleconvective loop with the liquid metal rising at the hot thermode, flowing along the boat and descending at the cold thermode; observations which are consistex with the observations of Cormack, Lea1 and Imberger (1974), for high Prandtl number fluids.

Two conditions are required for the occurrence of reasonably stable temperature oscillations. First the fluid must have a low Prandtl number; secondly a critical Rayleigh number usually of the order of a 1000 should be exceeded so that the dominant heat transfer process is convective rather than conductive. Furthermore, for high Prandtl number fluids such as water, when heated below, the simplest sort of cellular motion is a pair of counter-rotating rolls. In these fluids, which are transparent, the cellular motion has been observed by two parallel and coherent beams from a He - Ne laser, by Dubois and Berg; (1978). Under some specific conditions similar convective patterns occur in liquid metals. Milsom (19; found that the amplitude of the temperature oscillations reached a peak of lo C at certain aspect ratios, analogues to resonance. It is believed that these peaks are caused by the presence of two or four counter rotating rolls in the liquid. Busse (1972) investigated the instability of convection rolls in a heated fluid layer for small Prandtl number. The two dimensional rolls which are initially set up become unstable to oscillatory three dimensional disturbances when the amplitude of convection exceeds a particular critical value. It is proposed that the instability corresponds to the generation of vertical vorticity mechanism which will operate in the case of a variety of roll like motions. The instability can manifest itself as a wave travelling in either direction along the axis of the rolls or as a standing wave. The phenomenological picture of this type of instability compares well with the time-dependent perturbations observed by investigators. Also with the work of MOIR (1976) on finite amplitude Biemard convection. It appears then, that the temperature oscillations when arising in a crystal growers apparatus are subject to a third condition a geometrical constraint. It is well known Schulter, Lortz and Busse (1965); Davies (1968), and Stork and Muller (1972) that in a rectangular box the convective structure just above the onset of convection comprises of straight rolls parallel to the shorter side of the rectangular frame. For Ra - Ra,, > 0, but with a small difference the rolls were set up preferentially with a wavelength of the order of 2d. The 'structured' states of Milsom (1978), which are similar to the resonance conditions, of Skafel (1972) which correspond to aspect ratios of 0.4 and 0.2 could be due, it is conjectured to respectively, two and four convective loops. These loops it is suggested are a necessary condition, in some cases, for large temperature oscillations. The suppression of temperature oscillations has been carried out by workers in the field of crystal growth, a typical example is now given. A baffle system has been constructed by Whiffen and Brice (1971) for the suppression of temperature oscillations and used with a crucible for barium strontium niobate crystal growth. The baffle and crucible were constructed from platinum and the support wire from platinunrrhodium alloy. The baffle was mounted horizontally and reduced the effective length of the liquid and possibly suppressed the convection cells. With the baffle absent, oscillations had an amplitude of lo C peak to peak. Then with the baffle in the optimum position , which was found to be one auarter the depth of the crucible from the upper surface, the oscillations were reduced to less than 0.1' C peak to peak.

214

J. A. Milsom and B. R. Pamplin

The implication of the above observations suggest the crystal growers, especially from the melt, should pay special attention to the geometry of their vessels. The crucible which they employ are heated such as to create in some instances large temperature oscillations which could be induced from the setting up of convective loops.

5.

CONCLUSION

In conclusion, the creation of impurity striations in melt grown crystals has been definitely attributed to unwanted convective temperature oscillations in the melt. There has been some light thrown on the essential physics of the problem and the parameters upon which these temperature oscillations depend. At present there appears to be the following possible methods for their suppression in the melt: 1.

A reduction in temperature gradients.

2.

The application of an optimum magnetic field in horizontal crystal growth methods , just sufficient to damp out the oscillations; however the frequencies present in puller melts are lower than the horizontal melts with approximately the same volumes.

3.

The geometry of the vessel so constructed that convective cells are avoided.

4.

The introduction of baffles into the melt which could inhibit the formation of square rolls. Typical stable convective loops are shown in Fig. 10 for the box geometry, and in Fig. 11 for unstable loops the annular geometry and stable loops for the box geometry.

r(

---______

----

--_--

C-;-

(01 Aspect mtiozO.4

Fig. 10.

Thermal Oscillations in Melts

215

Fig. 11. The implication of the above observations for crystal growers is that special attention should be paid to the geometry of the vessel in which the melt is contained. When there is a temperature gradient in the crucible temperature, oscillations, large in comparison with the oscillations coming from the temperature controlling system, can be induced. The problem is certainly a complex one and theoretical understanding of the phenomena is limited. This is probably due to the fact that liquids, with relatively small Prandtl number, the thermal conduction proceeds rapidly and also the internal velocity develops very quickly. These facts together with the non-linearity of the intertial terms generates an almost intractable differential problem. Nevertheless, the agreement between theoretical predicti and experimental results is remarkable considering the severe approximations which are introduced into the theoretical models.

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J. A. Milsom and B. R. Pamplin

218 Glossary $7

Acceleration due to gravity

CE

Thermal expansion coefficient

P

Prandtl number

Ra

Rayleigh number

Racr

Critical Rayleigh number

Recr

Reynolds number

Recr

Critical Reynolds number

G

Grashof number

u

Velocity field where U(x,t) = {tr(x,ti>, V(y,t), W(Z,r}

U

Velocity in the ;7 direction

V

Velocity in the y direction

W

Velocity in the z direction

d

Depth of the working fluid

R

Vertical wave number

%

?a *T V

Non-dimensional shear Non-dimensional temperature gradient Non-dimensional theoretical frequency Thermal diffusivity

X

Kinematic viscosity

w

Angular frequency

Tz 8

Temperature gradient

k

Wave number

k cr

Critical wave number

Temperature perturbation

Other symbols are defined in the paper for convenience

Thermal Oscillations in Melts

219

THE AUTHORS

J. A. Milsom

Tony Milsom was awarded a BSc (Mathematics) from London University in 1961. He worked as a research scientist in the aircraft industry and completed by part-time research a MSc (Theoretical Physics) in Quantum statistical mechanics in 1966. He completed a PhD at Bath University in January , 1979. He is a Senior Lecturer at a Technical College in Wiltshire and a Fellow of the Institute of Mathematics and Its Applications. Dr. Brian Pamplin is Senior Lecturer in Physics at Bath University. From Bristol Grammar School he won an open major scholarship to Trinity College Cambridge in 1951 and subsequently served as a radar officer in Korea. He completed his Ph.D. at Nottingham in 1960 after a period in the semiconductor industry. He has led a small group of researchers in the crystal growth and evaluation of new adamantine semiconductors at Bath for the past eighteen years and published over 100 papers. He has lectured in the U.S.A., Canada, Venezuela, South Africa, Hungary, Japan, India, Germany, the U.S.S.R., France, Italy and England. Brian Pamplin is editor in chief of this journal and of the Solid State Science series of books in which the second edition of his "Crystal Growth", to be published shortly, is volume 18. Besides his interest in semiconductors, Brian Pamplin has had an active political career as a Liberal Councillor in Bath and candidate for Parliament. He is also interested in problems relating to science and religion and has published in this area too.