UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
ELECTROCHEMICAL MEASUREMENTS FOR THE DETERMINATION OF DYNAMIC STATES IN THE BRIDGMAN CRYSTAL GROWTH CONFIGURATION BY BRIAN R. SEARS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1990 ACKNOWLEDGEMENTS I would like to acknowledge the support of family, friends, and coworkers whose help and encouragement have contributed to the timely completion of this work. First, I would like to thank my faculty advisors, Dr. Tim Anderson and Dr. Ranga Narayanan, for their guidance and encouragement in my research endeavors. At the same time, I thank Dr. Archie Fripp, also for his guidance and encouragement, but additionally for his willingness and enthusiasm to act as research advisor during my two years of residence at the NASA Langley Research Center. I am grateful to the National Aeronautics and Space Administration for monetary support through a university grant. I would like to thank my coworkers at NASA for the technical and intellectual support which helped me to get past many problems. In particular, I thank Glenn Woodell and Bill Debnam for their invaluable technical support. Also, I thank Ivan Clark, Dave Knuteson, Jim Hurst, and Wayne Gerdes for their support as well as for many enlightening discussions. These people, and others, have freely offered their professional guidance as well as their friendship during my stay at NASA. I am grateful to my parents for their support in my education. I am also deeply grateful to my wife, Paula, for her continued support and especially for her sacrifices during the final stages of my dissertation preparation. TABLE OF CONTENTS Rae ACKNOWLEDGMENTS ................................. ii LIST OF SYMBOLS .................................... v ABSTRACT .......................................... ix CHAPTERS 1 GENERAL BACKGROUND .................... 1 Introduction ................................ 1 Vertical Bridgman Crystal Growth ............... 2 Growth of Compound and Doped Semiconductors ...... 4 Buoyancydriven Convection ..................... 8 Literature Survey ............................ 13 Experimental Approach ........................ 24 2 SOLIDSTATE ELECTROCHEMICAL MEASUREMENTS 28 Introduction ................................ 28 Yttriastabilized Zirconia ....................... 29 Oxygen Concentration Cell ...................... 36 Summary ............... ....... ........... 40 3 OXYGEN DIFFUSIVITY IN LIQUID TIN ........... 42 Introduction ................................ 42 Experimental ............................... 44 Results .................................... 58 Discussion ................................. 77 Summary ...... ....................... ... 81 4 FLOW VISUALIZATION ....................... 83 Introduction ........... .... ..... .... ........ 83 Experimental ............ .... .. ............ 85 Numerical Simulations ....................... 94 Results .................... ...... .... .... 99 Discussion ................................. 118 5 MULTIPLE DETECTOR FLOW VISUALIZATION ..... 121 Introduction ................................ 121 Experimental .............................. 122 Results ............... 126 Discussion ................................. 135 6 SUMMARY AND CONCLUSIONS ................ 138 APPENDICES A NUMERICAL OUTPUT ........................... 142 B EXPERIMENTAL ELECTROMOTIVE FORCE DATA ... 161 REFERENCES ..................... ......... ............... 259 BIOGRAPHICAL SKETCH ............................... 264 LIST OF SYMBOLS ao thermodynamic activity of oxygen A crosssectional area of the experimental cell C concentration C dimensionless concentration, (C C2)/(C1 C2) Co initial concentration C, concentration at the lower surface of the fluid cell C, concentration at the upper surface of the fluid cell CL concentration of the liquid phase in equilibrium with a solid phase C, concentration of the solid phase in equilibrium with a liquid phase C degree Celsius cm centimeter Do molecular diffusivity of oxygen Do' effective diffusivity of oxygen E potential or electromotive force E0 initial potential E, activation energy EMF electromotive force F Faraday constant F dimensionless body force, g/g g magnitude of gravitational acceleration g gravity vector G Gibb's free energy H height of the fluid cell I electric current ID inside diameter J joule k Boltzmann constant K degree Kelvin kg kilogram m meter m index mm millimeter mV millivolt n number of charge equivalents NO oxygen flux OD outside diameter P pressure VP defined as (VP pog)H2/v2 Pr Prandtl number r radial coordinate R radius of the fluid cell R gas constant Ra Rayleigh number Ras solutal Rayleigh number RaT thermal Rayleigh number Ra,, first critical Rayleigh number Ra, second critical Rayleigh number Sc Schmidt number sec second t time t dimensionless time, tv/IH T temperature T dimensionless temperature, (T Ts)/(Ti Ts) T1 temperature at the lower surface of the fluid cell T, temperature at the upper surface of the fluid cell ti ionic transference number AT temperature difference (between top and bottom of fluid cell) V volt v velocity vector V dimensionless velocity vector, vH/v W watt X mole fraction z axial coordinate a Seebeck coefficient P aspect ratio, H/R Ps solutal expansion coefficient pT thermal expansion coefficient y activity coefficient K thermal diffusivity JA dynamic viscosity juo chemical potential of oxygen v cinematic viscosity, W/p p density o, electron conductivity Oh hole conductivity ao ionic conductivity Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements of the Degree of Doctor of Philosophy ELECTROCHEMICAL MEASUREMENTS FOR THE DETERMINATION OF DYNAMIC STATES IN THE BRIDGMAN CRYSTAL GROWTH CONFIGURATION By Brian R. Sears December, 1990 Chairman: Dr. Timothy J. Anderson Major Department: Chemical Engineering An electrochemical flow visualization technique for characterizing natural convection in liquid metals and semiconductors in the vertical Bridgman meltgrowth configuration was developed and tested. The ceramic electrolyte yttriastabilized zirconia was incorporated into the boundaries of the fluid container to act as a window through which the dilute oxygen tracer could be injected, extracted, or measured at the surfaces of the fluid volume. An experimental cell was designed and used to measure the effective diffusivity of oxygen across tin melts in geometries characteristic of Bridgman cells. The technique was able to discern transcritical points in the dynamic state of the melt as a function of imposed temperature gradient. The electrochemical technique was modified and shown to be capable of describing the orientation of flow in Bridgman simulations. An improved method for measuring the binary diffusion coefficient of oxygen in the absence of thermallydriven convection in liquid metals was designed. The oxygen diffusivity in liquid tin was then studied experimentally as a function of temperature, and the results were compared to less wellcontrolled experimental studies. CHAPTER 1 GENERAL BACKGROUND Introduction Vertical Bridgman meltgrowth is a proven method for production of bulk compound semiconductors. However, inherent compositional inhomogeneities and extended defects have limited the electronic and optoelectronic applications of these materials. The electronics industry would greatly benefit from minimization of these imperfections, but a better understanding of their origin must be obtained. Since convection in the melt during solidification is known to be responsible for segregation of the component elements as well as for crystallographic defects, an increased under standing of convection is needed. Previously, the nature of convection was inferred from postgrowth analysis of crystals and also through temperature measurement on growth ampoule surfaces during growth. The opacity of liquid metals and semiconductors preclude optical visualization techniques and high growth tempera tures limit the applicability of other visualization methods. This dissertation presents a novel flow visualization technique capable of characterizing convection in Bridgman growth simulations. The present technique involves the application of a solid state electrochemical cell to introduce, extract, and monitor trace quantities of oxygen across surfaces of the growth ampoule. Methods by which this flow visualization technique can be applied to actual crystal growth experiments are discussed. 2 Vertical Bridgman Crystal Growth Vertical Bridgman crystal growth is one of several methods used to produce bulk semiconducting materials. It is a preferred method for production of semiconduc tors containing volatile or toxic elements since the materials can be easily sealed from the environment. A Bridgman cell is generally composed of a vertical cylindrical ampoule containing the semiconductor melt and is housed within a series of heaters which maintain a thermal gradient along the axis of the cylinder. Directional solidification of the melt is achieved by one of two commonly used procedures: mechanical translation of the ampoule relative to an established thermal gradient (Bridgman method) or translation of the thermal gradient relative to the ampoule by continuously varying the power input to the heaters (gradient freeze method). The rate at which the solidliquid interface moves along the length of the sample is controlled by, but not necessarily the same as, the rate at which the sample moves relative to the furnace. A single crystalline sample can be produced by seeding the end at which solidification will begin. Single crystalline semiconductor materials can be grown in the Bridgman configuration with either top or bottom seeding, although most commonly with bottom seeding. Figure 11 is a schematic of the basic Bridgman growth configuration. Bridgman, Czochralski, and float zone growth of semiconductors are the most commonly used methods ofbulk production of semiconductor materials from the melt. Czochralski growth, in which the crystal is pulled from a molten pool, is the preferred method of growth for the group IV elements, silicon and germanium, although the float zone technique is also used. The group IIIV compound semiconductors are commonly grown by a liquid encapsulated Czochralski technique in order to minimize 3 Hot Hot Zone Zone Insulation // Insulation Zone \ Zone DSolid 10z" Cold / Cold Zone Zone Figure 11. Schematic of the vertical Bridgman crystalgrowth configuration with a partially solidified sample. 4 evaporative losses of high vapor pressure components. Likewise, group IIVI compound semiconductors often contain one or more elements which have relatively high vapor pressures at growth temperatures. This poses two problems, depletion of one component from the melt as it condenses on cool spots of enclosing walls and possible leakage of highly toxic elements into the environment. Bridgman growth offers improved containment of volatile components and is easily modified for growth of crystals over a wide range of pressures. Also, the Bridgman method appears to be the best suited technique for growth of solid solution semiconductor materials since minimization of convection is desired. Forced convection is inherent in Czochralski growth as the boule and/or crucible are rotated. In Bridgman growth, however, natural convection is predominant and can be controlled by application of a magnetic field or reduction of gravity during growth. More detailed discussions of the Bridgman crystal growth technique are available in the literature [1,2]. Growth of Compound and Doped Semiconductors The compositional homogeneity of doped crystals and of compound semicon ductors is of great importance in applications of the materials. In general, com positional homogeneity is desired in semiconductor materials. This is especially true for alloys of various compound semiconductors when the composition controls electronic properties. Compositional homogeneity, however, is not easily obtained in directional solidification of a multicomponent melt. This difficulty results from the nature of the phase equilibrium established between the liquid and the solid solution at the corresponding melting temperature. Figure 12 shows the solidliquid phase diagram [3,4,5] of a pseudobinary mixture, PbxSn.xTe, which forms a completely 5 920 900 BBO C CL. 1880 820 800 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SnTe Mole fraction PbTe PbTe Figure 12. Phase diagram of the pseudobinary mixture, PbxSnl.xTe [3,4,5]. C, and CL identify the equilibrium solid and liquid compositions at a given temperature. 6 miscible solid solution. The symbol Cs indicates the composition of the solid solution in equilibrium with the liquid of composition CL. As solidification of the melt proceeds, solute (SnTe) is rejected at the interface and a diffusion boundary layer is developed in the melt adjacent to the solidliquid interface. The resulting crystal will show a compositional trend like that of curve A in Figure 13 if the only method of mass transport is diffusion, i.e. diffusionlimited growth. A quantitative analysis of solute redistribution for cases in which the distribution coefficient, Ca/C, is constant (e.g., dilute solutes) is given by Smith, Tiller, and Rutter [6]. As shown in the diffusioncontrolled growth curve in Figure 13, after the initial transient, a period of steadystate growth is achieved during which com positional homogeneity is predicted. Such a crystal would be considered to be very high yield. Conditions which eliminate convective mass transfer and thus allow diffusioncontrolled growth are very difficult to realize, however. Application of magnetic fields to stabilize the melt has been successful in reducing convection in some materials, but other materials with low magnetic susceptibilities resist stabilization. The microgravity environment of space, then, offers the only viable location to conduct convectionfree experiments for these materials. Witt et al., in two separate space experiments, were able to grow doped InSb [7] and Ge [8] crystals. Analysis of the crystals indicated that ideal, exclusively diffusioncontrolled growth occurred. On earth, the gravitational field introduces a body force which acts upon variations in liquid density to cause natural convection in fluids. The convection disrupts the diffusion boundary layer and enhances the distribution of solute throughout the melt. Solute rejected at the growth interface is swept away by 0.1 0.2 0.3 0.4 0.5 Fraction of length 0.8 0.7 0.8 0.9 along crystal Figure 13. Characteristic axial compositions of binary solid solutions grown under diffusioncontrolled and fullymixed conditions. The distribution coefficient is 0.7, and the initial melt composition is Co. 2.0 1.5 ) 1.0 U 0.5 0.0 0.0 Diffusioncontrolled  . Fullymixed  I A  I I Curve A ,'  Cure B  Curve B mm, alm a. i.m nl mum a1m.. nnnu 111n n.1111.n11111111 ^^ 8 currents and mixed with the bulk melt. Crystals grown under the influence of steady convection in the melt show a monotonically increasing composition approaching that shown for complete mixing in Figure 13 (curve B) [6]. Under certain conditions, convection in the melt can be oscillatory or turbulent in nature. This has been shown to cause transient backmelting of the crystal which results in compositional striations throughout the final crystal [7,9,10]. The compositional striations resulting from oscillatory flow are much more damaging to the properties of a semiconductor than are the monotonic variations in composition resulting from purely steady flow. Convective transport in meltgrowth configurations has become an area of intense research because the performance of most electronic devices depends largely upon spatial variations in the composition of the material [11,12]. Buoyancydriven Convection Buoyancydriven convection, or natural convection, in fluids results from the effect of the gravitational field on density variations within the fluid. These density gradients can be attributed to thermal or compositional variations. The stability and corresponding dynamic state of a fluid with respect to the density variation depend on the orientation of the density gradient with respect to the lines of gravity, the magnitude of gravitational acceleration, the magnitude of the density gradient, the fluid properties, and the properties of the boundaries of the fluid volume. This study is restricted to physical geometries associated with the vertical Bridgman configuration, i.e. right circular cylinders of fluid bounded by rigid walls on the sides and bottom. The top surface may be either free or bounded by a rigid wall. Flow due 9 to gradients in surface tension, termed Marangoni convection, can be eliminated in the case of a rigid boundary. Some of the parameters which affect the dynamic state of the melt during directional solidification in the vertical Bridgman configuration are defined in the non dimensional forms of the equations of change. The equations of continuity of mass, momentum, energy, and species are, respectively, as follows: V = 0 (11) + (+V)= VP + CF + TF + 2 (12) at Sc Pr Pr + (VV) = V2T (13) t( aI Sc + ( = 92C (14) (t ) In this mathematical model, the OberbeckBoussinesq [13,14] approximation has been applied. This states that the density of the fluid can be assumed constant in all terms except the body force terms in the equations of motion. Because the density, temperature, solutal, and velocity fields are interdependent, retention of a variable density in all terms creates an analytically intractable differential problem. The equations have been cast into simple, nondimensional form by assuming a linear dependence of density on temperature and concentration (for a two component mixture) in the body force terms only. For fluids in which these dependencies are 10 small, such as most fluids used in the Bridgman system, these approximations are valid [15,16,17]. So the equations of change appear in the simple forms shown above where dimensionless parameters scale the various terms. The Prandtl number, Pr, is defined as the ratio of momentum diffusivity to thermal diffusivity of the fluid. This parameter is a measure of the rate of diffusion of vorticity relative to the rate of diffusion of heat in the fluid. The Schmidt number, Sc, appears in the presence of solutal gradients and is defined as the ratio of momentum diffusivity to mass diffusivity. Similar to the Prandtl number, the Schmidt number is a measure of the rate of diffusion of vorticity relative to the rate of diffusion of component species in the fluid. The solutal and thermal Rayleigh numbers, Ras and RaT, are parameters which, in general, will change throughout the course of a crystal growth. These parameters are defined as = (C,C2)gH3 (15) vDy S= PT(r, T)gH' (16) VK where P, and PT are the coefficients of solutal and thermal expansion, respectively; C, are the concentrations at the lower (1) and upper (2) surfaces of the fluid cell; T, are the temperatures at the lower (1) and upper (2) surfaces of the fluid cell; g is gravitational acceleration; H is the fluid cell height; and v, K, and Dg are the momentum, thermal, and molecular diffusivities, respectively. 11 The Rayleigh number can be thought of as a parameter which characterizes the driving force for convection resulting from exclusively vertically oriented density gradients. Considerable attention has been given to the study of hydrodynamic stability and the dependence of dynamic state on Rayleigh number for fluid layers having vertically oriented density gradients. A short review of this work as it relates to vertical Bridgman meltgrowth is given in the following section; however, a simple description of the Rayleigh number as presented by Busse [18] offers an intuitive description of its relevance. In Busse's description, a fluid layer which is heated on the bottom and cooled on top will exist in different dynamic states depending on the magnitude of the imposed temperature gradient. Consider an initially motionless fluid layer subjected to a thermal gradient as stated; the fluid layer will have a certain potential energy associated with it as a result of gravity acting on layers of dense fluid existing on top of layers of lessdense fluid. The fluid also has a viscous nature to it which serves to dissipate any potential energy which is released. The Rayleigh number is a parameter proportional to the ratio of the rate of release of potential energy to the rate of dissipation of energy by viscous forces. The onset of convection in the stagnant fluid requires that the Rayleigh number exceed a certain threshold value at which the potential energy is released more rapidly than it can be dissipated by static viscous forces. Convection is simply the mechanism by which excess potential energy is released. Further changes in the dynamic state will occur as the Rayleigh number exceeds additional threshold values, e.g., oscillatory convection and turbulence. One might expect that the geometry of the fluid cell would also influence the dynamic state in a Bridgman cell. Indeed, another dimensionless parameter emerges 12 when the equations of motion are written in component form for cylindrical geometries. This parameter is the aspect ratio, P, and is defined as the ratio of the height of the fluid cell to the radius. To complete this general discussion of buoyancydriven convection, attention must be given to the effects of horizontally oriented density gradients in the fluid. Unlike the conditional stability of fluids having vertical density gradients, fluids having horizontal density gradients are unconditionally subject to natural convection [19,20]. Muller et al. [21] presented a twoRayleighnumber model of buoyancy driven convection to account for flow in meltgrowth configurations resulting from two dimensional density gradients. In actual crystal growth, horizontal thermal gradients are inherent due to mismatching of material thermal conductivities and also to furnace imperfections. These horizontal gradients provide a driving force for convection in addition to any driving forces due to vertical gradients. The resulting dynamic state is then a nonlinear interaction of the independent flows driven by horizontal and vertical gradients. In summary, the parameters which determine the dynamic state in ideal, vertical, Bridgman meltgrowth are the dimensionless quantities Pr, Sc, Ras, RaT, and P. Nonidealities are introduced, however, through the boundaries of the fluid cell and also by latent heat effects during solidification. Since growth materials must be contained in an ampoule during growth, lateral heat fluxes are inevitable. These may be reduced, though, by using better insulating ampoule materials or by better matching ampoule and growth material thermal conductivities. Nonuniform heating by heater elements also introduces imperfections in thermal boundary conditions, but these may be reduced by improved furnace designs. 13 For the sake of completeness, the other modes of convection in Bridgman growth will be mentioned. Marangoni convection results from gradients in surface tension, generally due to thermal gradients, on a free surface. This type of convection can be completely eliminated by replacing any free surfaces with a rigid boundary to impose a noslip condition at that boundary. Additionally, forced convection can result during solidification when volumetric expansion or contraction occurs during the phase transformation at the growth interface. Accounting for these flows is important when buoyancydriven convection is not important, such as in lowgravity melt growth. In space, the gravitational field can be decreased to 10"3 10"6 g, decreasing the resulting buoyancy forces accordingly. Under conditions of meltgrowth on earth, however, buoyancydriven convection is by far the most dominant source of fluid flow. Literature Survey This section encompasses a review of studies on hydrodynamic stability and natural convection phenomena in fluid layers as they relate to the Bridgman cell. This discussion will begin with an overview of the classic RayleighBenard problem, which serves as the foundation for subsequent studies on the hydrodynamic stability of fluids in various geometries. Extensions of this work which include solutal effects and geometries characteristic of Bridgman meltgrowth will then be discussed. The earliest work in this area focused on the stability of horizontal fluid layers heated from below and cooled from above. In this orientation, strata of increasing density exist on top of one another. One might intuitively conclude that this arrangement would be statically unstable and would break down into a convective motion. It has been shown, however, through experiment and mathematical 14 treatments that the fluid layer heated from below will remain stable for temperature gradients up to some threshold value. Beyond this critical value the fluid layer will break down into a convective motion. It should be noted that fluid layers having decreasing density with increasing height (e.g. horizontal fluid layers heated from above) are statically stable for all magnitudes of density gradient. The earliest reported experiments which spawned an interest in hydrodynamic stability were performed by Thomson [22] and Benard [23]. Benard gave a detailed account of experimental observations of the flow patterns developed as a result of heating from below a fluid layer with a free upper surface. The cellular flow patterns observed in shallow layers of fluid are consequently referred to as Benard cells. Lord Rayleigh [24] was the first to give an analytical description of the Benard flow. In his analysis, Rayleigh determined the conditions under which a fluid layer heated from below would break down into a convective roll pattern. Rayleigh's approach was a linearized perturbation analysis of the momentum and energy equations. Although Rayleigh did not consider surface tension driven convection at the free surface [25,26] (which is quite substantial in the Benard experiments) in his treatment, his work inspired more general analyses over a broader range of boundary conditions. Jeffreys [27,28], Low [29], and Pellew and Southwell [30], in particular, extended the linear analysis to include both free and rigid boundaries at the upper and lower surfaces. They assumed that these boundaries had infinite thermal conductivity and heat capacity (i.e. fixedtemperature boundary conditions), which is a very idealistic approach. Two books which give detailed 15 descriptions of the linear convection problem are by Chandrasekhar [311 and Gershuni and Zhukovitskii [32]. The effects of imposing thermal boundary conditions other than fixed temperatures at the surfaces were introduced by Sparrow, Goldstein, and Jonsson [331. These results are important to the experimentalist, who is constrained in the use of boundary materials of finite thermal conductivity and heat capacity. Sparrow et al. extended the linearized analysis to include a constant heat flux condition at a rigid boundary, as well as a Newton's law of cooling condition at both free and rigid upper boundaries. The critical Rayleigh number for the onset of convection, hereafter referred to as the first critical Rayleigh number, for a constant heat flux condition was found to be significantly lower than for the constant temperature condition. This result is intuitively correct as an isothermal boundary would damp thermal perturbations, and thus, stabilize the stagnant fluid. For both free and rigid surfaces, the first critical Rayleigh number was found to increase monotonically with increasing heat transfer coefficient at the boundary and asymptotically approached the values predicted for the constant temperature boundary conditions. Again, this is intuitively correct as an increased heat transfer coefficient at the surface would tend to damp thermal perturbations more efficiently. Sparrow et al. also noted that the critical Rayleigh numbers were higher for an upper rigid surface than for an upper free surface by approximately 600 for all values of heat transfer coefficient. For reference, the critical Rayleigh number for a laterally unbounded fluid layer between upper and lower rigid walls at constant temperatures is 1707.8. For a free upper surface at constant temperature, the critical Rayleigh number is 1100.7. 16 The preceding results are particularly applicable to the Bridgman meltgrowth system. Typically, Bridgman crystals are grown with a free upper surface. By introducing a rigid surface at the top of the melt, by floating a thin fused silica disk on the surface for example, the fluid would not only be more resistant to transitions to higher dynamic states (e.g., oscillatory and turbulent regimes) but would also be subject to additional viscous slowing near that surface. The effect would not be as dramatic as in the case of infinite horizontal fluid layers, however, since the upper surface in the Bridgman cell makes up a rather small portion of the total surface area. A plethora of additional work has been completed in characterizing flows in infinite horizontal fluid layers. Nonlinear perturbation analyses have been used to test the stability of various flow patterns as well as to test for the existence of other critical and subcritical dynamic transitions. These studies are quite relevant in atmospheric sciences, but I shall diverge here and examine some of the subsequent work relating to thermally driven flows in laterally bounded fluid layers (primarily right circular cylinders). Solutal effects will then be discussed in relation to Bridgman meltgrowth. Early experimentalists soon began to find that approximating infinite horizontal fluid layers in finite structures presented interesting problems. The cellular structure expected was often dominated by flow patterns characteristic of the shape of the bounding side walls. Koschmieder [34], in approximating an infinite fluid layer in both rectangular and circular dishes, observed rolls of rectangular and circular shape, respectively. Stork and Miller [35,36], in wellcontrolled experiments, made similar observations. The results of theoretical analyses on the effects of side walls on the preferred flow plan agree well with these experimental 17 observations. Davis [37] performed a linear stability analysis taking into account the side walls in a rectangular geometry and found that the patterns observed by Koschmieder were indeed stable near the first critical Rayleigh number. Davis' analysis was not wellposed, however, in that slip was assumed at two of the four side walls to facilitate the solution of the problem. This assumption was proven by Davies Jones [38] in later work to be valid only for certain rectangular aspect ratios, those which were studied by Davis. DaviesJones numerical results agreed very well with the reported results of Davis. Segel [391 used a modified perturbation analysis, based on nonlinear predictions, for a rectangular field to support the predictions of Davis. Charlson and Sani [40] later performed a thorough analysis of flows in shallow fluid layers heated from below in cylindrical containers. The flow patterns in these cylindrical geometries, as observed by Koschmieder [34] and Stork and Miiller [36], consist of concentric toroidal rings, the number of which depends on the aspect ratio of the fluid domain. Charlson and Sani analyzed the perturbation equations by recasting them in a variational formulation and applied the RayleighRitz method to approximate the solution. They were able to calculate upper and lower bounds to the first critical Rayleigh number as well as determine the number of toroidal rolls for aspect ratios, 0 = H/R where R is the radius of the cylindrical container, ranging from 0.1 to 2. Pellew and Southwell [30] and Zierep [41] had previously attempted linear analyses of the cylindrical case, but assumed "slip" walls in order that the method of separation of variables could be used to solve the differential equations. The solutions consequently violated the continuity equation for realistic cases in which noslip boundaries are present. Ostrach and Pneuli [42] solved the corresponding linear 18 differential equation for the vertical component of velocity to obtain an upper bound to the first critical Rayleigh number, but an incorrectly specified boundary condition instead lead to a predicted lower bound. Sherman and Ostrach [43] subsequently published an analysis intended to establish a lower bound to the first critical Rayleigh number. The method of Charlson and Sani, however, has given the most reliable predictions of the first critical Rayleigh number thus far. They performed the calculations assuming fixedtemperature upper and lower boundaries for two cases, insulating side walls (dT/dr = 0) and conducting side walls (T(at r=R) = Tw). The predicted critical Rayleigh numbers for a given aspect ratio are, in general, lower in the case of insulating side walls than conducting side walls. This, again, is due to the damping effect of a highly conducting surface on the thermal perturbation field in the fluid. Also, for decreasing aspect ratio, the critical Rayleigh number can be seen to approach the well established value of 1707.8 for a laterally unbounded fluid layer. In an extension of their original work, Charlson and Sani [44] investigated the conditions for the onset of convection in cylindrical geometries for aspect ratios greater than unity. This linear stability analysis accounted for the possibility of three dimensional, nonaxisymmetric flow states. The estimates for the lower bounds to the first critical Rayleigh number are improved over previous calculations by several authors [45,46,47,48]. Additionally, a transition is predicted in the initial dynamic state from an axisymmetric to a nonaxisymmetric flow as the aspect ratio is increased above P = 1.23 for insulating side walls and p = 1.64 for conducting side walls. Fixedtemperature upper and lower boundaries were assumed. The existence of the nonaxisymmetric flow state for larger aspect ratios has been established experimentally in several studies. Ostroumov [49], Slavnov [50], and Slavnova 19 [51] observed the nonaxisymmetric state in transparent fluids of high Prandtl number. Miller, Neumann, and Weber [52] reported observing the axisymmetric flow state at P = 1.0 and a nonaxisymmetric pattern at P a 2.0 for both water (Pr = 6.7) and liquid gallium (Pr = 0.02). Miller, Neumann, and Weber's observations are welldocumented in the case of the transparent fluid, water, as they were able to inject visual tracers in the flow. Thermocouples attached to the outer surface of the ampoule were employed to infer the flow pattern in liquid gallium, however, and extremely wellcontrolled thermal conditions must have been maintained in order to extract accurate thermal data. In hightemperature crystal growth, this method has not been used effectively to determine flow patterns under growth conditions. Miller, Neumann, and Weber were, however, able to infer the level of the dynamic state (i.e., steady, oscillatory, or turbulent) from thermal measurements in the vertical Bridgman meltgrowth with topseeding of GaSb. Steady temperature measurements indicated steady flow, while periodic and nonperiodic temperature fluctuations indicated periodicoscillatory and turbulent flows, respectively. The general dynamic states which occur in a typical vertical Bridgman melt growth are steady, periodicoscillatory, and turbulent flows. The stagnant state is not included since it has never been realized in groundbased growth. As previously mentioned, stagnation, and consequently diffusioncontrolled growth, is presently feasible only in a zerogravity environment. Theoretically, the first critical Rayleigh number delineates the boundary between stagnant and steady flow for vertically oriented density gradients. In application, however, steady flow is realized at sub critical Rayleigh numbers due to horizontal density gradients. This has been observed 20 even in wellcontrolled experiments [36,42]. In this case, the existence of the first critical Rayleigh number is no longer valid since it defines the onset of convection from a stagnant fluid. A change in dynamic state is still expected in the vicinity of the first critical Rayleigh number, however, but is defined as a transcritical change. It can be shown through the NavierStokes equations that conditions of fluid motionlessness require that density gradients be oriented only in the vertical direction [19]. For zerovelocity, the combined NavierStokes equations given in Equation (12) become VP + R CF + TF=r = 0 (17) Sc Pr By taking the curl of Equation (17), the pressure field drops out and Vx R CFS + Vx RaTF O. 8) Sc Pr Assuming a conservative body force, Equation (18) becomes Ra  Ra, .. s(VC x F) + (VT x F)= 0 (19) Sc Pr and since there is no fixed relation between the concentration and temperature gradients, Equation (19) can, in general, only be valid for concentration and thermal gradients oriented in the vertical direction. So, the trivial solution of zerovelocity will not satisfy the NavierStokes equations in the presence of horizontal density gradients. 21 The second critical Rayleigh number is generally defined as that value of the Rayleigh number at which the flow changes from steady to periodicoscillatory. This point of transition cannot be predicted through linear stability analyses and as yet has not been predicted through nonlinear analyses. Experimental observations, however, have shown a dependence of the second critical Rayleigh number, Rad, on both the Prandtl number and aspect ratio for fluids heated from below. Krishnamurti [53] performed experiments with fluids having Prandtl numbers ranging from 0.71 (air) to 8500 (silicone oil) in a layer of large lateral extent bounded by rigid surfaces on top and bottom. Ra. is shown to increase with Pr up to Pr = 50, above which Ra. = 55,000 and is Pr independent. A similar result was reported by Silveston [54]. At low Prandtl numbers, Rad approaches RaE1, and Ra.L was shown experimentally by Krishnamurti to be Prandtl number independent. Krishnamurti also observed higherorder dynamic transitions with increased Rayleigh number. As Ra was increased above Rad, a critical point was reached at which perioddoubling occurred, that is, the frequency of thermal oscillations was doubled. As Ra was increased further, a regime of nonperiodic turbulent convection was encountered. These last two transitions show the same trend with increasing Prandtl number as the second critical Rayleigh number. The experimental results of Miller, Neumann, and Weber [52] with liquid water (Pr = 6.7) and liquid gallium (Pr = 0.02) show the dependence of Ra1, Rae, and the onset of turbulence on the aspect ratio in vertical cylinders. As was predicted by the linear theory of Charlson and Sani [44], Rae1 is observed to increase with increasing aspect ratio. Ra, and the onset of turbulence show the same trend. In agreement with the results of Krishnamurti, Rad occurs at much higher values in the case of water (Pr = 6.7) than in the case of liquid 22 gallium (Pr = 0.02) in vertical cylindrical geometries. For example, at an aspect ratio of 4, Ra8 in gallium was 5x104 and in water was 107. Knuteson [55] also performed experiments to measure the onset of oscillatory flow with surface temperature measurements for vertical cylinders of liquid tin (Pr = 0.01). The values of Ra8 reported for several aspect ratios are slightly higher than those reported by Miller, Neumann, and Weber [521 for liquid gallium (Pr = 0.02) at aspect ratios ranging from 3.3 7.0, although uncertainty in the measurement of the vertical temperature gradient could account for this offset. One would expect Ra, for tin to be slightly lower than for gallium based on the trend of Rae with Pr from Krishnamurti's work. Knuteson also studied the frequency of oscillations as a function of aspect ratio and Rayleigh number. Frequencydoubling and turbulence were also noted in these cylindrical geometries. An interesting phenomenon observed for the first time in 1983 by Miller [56] is the appearance of steady flow at Rayleigh numbers above Rad. In his experiments, Miller grew a Tedoped InSb crystal in a centrifuge (in the thermally unstable orientation) to allow for variation in the magnitude of the body force. The Rayleigh number was varied by changing the angular acceleration of the ampoule during growth. The results showed that steady convection was present initially as the angular acceleration was increased. A critical centrifugal acceleration was eventually reached, however, at which oscillatory flow ensued. This was evident both in temperature measurements and in compositional striations in that region of the final crystal. As the centrifugal acceleration was increased further, the thermal oscillations ceased, and steady flow was again realized. One might be inclined to believe that this relaminarization, as Miller terms it, is a centrifuge effect, but Muller also reports a 23 similar observation in liquid water at normal earth gravity. These are presently the only observations of this phenomenon reported in the literature. Solutal convection in Bridgman crystal growth does not manifest itself in the same manner as thermal convection. Axially directed thermal gradients are generally linear, and the resulting density gradients are linear as well. The driving force for convection is then identical at all points in the fluid. On the other hand, the solutal gradient appears as a result of solute rejection at the growth interface and decreases exponentially with distance from the interface due to diffusion into the bulk melt. The driving force for convection is consequently nonuniform. When the two effects are combined, as must be the case in multicomponent crystal growth, the conditions for and the nature of convection become quite complex. This type of convection is termed doublediffusive convection owing to the difference in diffusion velocities of heat and solute. The simpler case of fluid layers having both linear thermal gradients and linear solutal gradients was examined for conditions of stability by Stern [57]. This case was modelled after an oceanographic phenomenon in which both the temperature and saline concentration of the ocean's water increase with height. The water is considered thermally stable but solutally unstable. Stern recognized that the condition for the onset of convection was not determined by the density gradient itself, but rather by the combined Rayleigh number, Rag RaT. That is, convection can result with a net stabilizing density gradient. This type of motion has been termed doublediffusive convection because it is enabled by the disparity in the diffusivities of heat and salt. A detailed linear analysis of this problem is given by Veronis [58,59]. 24 In the case of doublediffusive convection in directional solidification of single phase binary mixtures, the effects of convective onset are discussed in several papers [60,61,62,63,64]. Since the destabilizing solutal gradients are developed near the solidliquid interface during solidification, the onset of convection is observed to strongly affect the morphological stability of the interface. Nonuniformities in composition and crystallographic defects resulting from thermosolutal (double diffusive) convection are, consequently, of primary concern to crystal growers. Experimental Approach A significant amount of flow visualization in Bridgman geometries has been reported, although primarily for transparent fluids such as water and silicone oils. These fluids have different physical properties than the liquid metals and semicon ductors of interest here, typically having Prandtl numbers at least two orders of magnitude larger than the metals. It is not known whether the flow states observed in highPr fluids model the flow states in lowPr fluids, and it has already been shown that the critical transitions in dynamic state occur within different ranges of Rayleigh number depending on the Prandtl number of the fluid. It is the objective of this research to develop a flow visualization technique which can be used to examine both critical transitions in dynamic state and corresponding flow patterns for lowPr fluids in the vertical Bridgman meltgrowth configuration. The Microgravity Sciences Group at the NASA Langley Research Center, under whose auspices this work was completed, is currently interested in vertical Bridgman meltgrowth of PbxSnl.xTe, a material which has applications in optical detection of wavelengths in the infrared spectrum as well as in making tunable diode lasers. 25 Directional solidification of this pseudobinary material results in compositionally inhomogeneous crystals because of rejection of SnTe at the growth interface. Consequently, growth of this material with bottomseeding results in a thermally stable, but solutally unstable melt due to buildup of the lighter component, SnTe, near the growth interface. Conversely, topseeding results in a solutally stable, but thermally unstable melt. PbxSnl.xTe is, therefore, a good model fluid for the study of natural convection phenomena in the vertical Bridgman system. A flow visualization technique which can be applied to the PbxSnxTe system would, therefore, allow for a better understanding of the effects of various convective motions on crystal properties. An electrochemical technique for introducing, extracting, and monitoring dilute concentrations of oxygen in liquid metals and semiconductors is presented as a viable technique for visualizing flow in Bridgman crystal growth. In this technique, the oxygen anionconducting properties of a dense, inert ceramic, yttriastabilizedzirconia (YSZ), are applied in the construction of multiple electrochemical cells in the growth sample. The melt serves as one electrode for each of the cells in the structure, while independent reference electrodes are maintained at the outside surfaces of the electrolyte. By fashioning sections of the ampoule walls out of the YSZ, oxygen can be introduced or extracted from surfaces of the fluid volume by imposing an electric potential of required polarity between the melt and the reference electrode. Alternatively, the concentration of oxygen at surfaces of the fluid volume can be monitored by measuring opencircuit EMF's between the melt and reference electrodes. Chapter 2 is devoted to a discussion of both the properties of YSZ and the theory describing the operation of a solidstate electrochemical cell. 26 The oxygen serves as a dilute tracer element with limited solubility in liquid metals. Because of its limited solubility, the oxygen was originally thought to be completely unobtrusive to the flow. This does appear to be true under certain circumstances, and a full discussion of its effect on the flow will be given in Chapters 3 and 4. Since this flow visualization technique is in the formative stages, the simplest possible flow scenario was chosen for this research. Pure tin was employed as a model fluid so that purely thermallydriven flow could be established. Lead and tellurium were not used because of safety considerations in heating the materials to temperatures well above 500C. The application of the techniques outlined in this work can be generalized, however, to multicomponent melts with only minor modifications for safety reasons. To use oxygen as a tracer in the Bridgman simulations, its molecular diffusivity in tin must be known. An electrochemical cell specifically designed to measure the molecular diffusivity of dilute oxygen in liquid tin is presented in Chapter 3. This design is superior to cell designs used in similar studies in the past because it minimizes thermal gradients in the melt which cause natural convection. A general discussion of difficulties and misconceptions in experimental diffusivity measurements will also be given in Chapter 3. An approach similar to that used in the oxygen diffusivity measurements is used to study the dynamics of flow in simplified Bridgman simulations. The method involves measuring the effective rate of mass transfer of the tracer across the Bridgman cell for various applied temperature gradients. Here, the trends in mass transfer rate with changing Rayleigh number are indicative of the dynamic evolution 27 of the melt. This technique is discussed in Chapter 4. An extended approach to flow visualization involves the design of a containing ampoule which will enable determination of actual flow orientations within the Bridgman cell. This can be accomplished by using multiple electrochemical detectors on the surface of the ampoule to observe the migration of oxygen pulses introduced at a known location to the surface of the melt. The feasibility of this technique is tested and discussed in Chapter 5. CHAPTER 2 SOLIDSTATE ELECTROCHEMICAL MEASUREMENTS Introduction The flow visualization technique described in this work is based on the electrolytic properties of stabilized zirconium dioxide. The solidoxide electrolyte actually serves a dual purpose in these experiments, structural and electrochemical. The material's rigidity, chemical inertness, and impermeability to atmospheric gases make it ideal for containment of high temperature liquid metals and semiconductors. Stabilized zirconia can be cast into a variety of shapes for virtually any application. The electrolytic properties of stabilized zirconia are manifested in the high conductivity for divalent oxygen anions under certain conditions of temperature and oxygen partial pressure. This makes it an ideal medium through which the oxygen tracer may be added or removed from the metallic or semimetallic melt. This chapter is devoted to a discussion of the stabilized zirconia electrolyte, its applications and limitations. The thermodynamic theory describing the relationship between chemical and electrical processes in these electrochemical cells is also presented. Additionally, attention is given to sources of error in electrochemical measurements using the solidoxide electrolytes. The purpose of these discussions is simply to provide a foundation for the subsequent electrochemical studies in Chapters 3 and 4. 29 Yttriastabilized Zirconia Zirconia (ZrO,) can be doped with yttria (Y20) to form a stable solid solution which ranges from approximately 8 to 50 mole percent yttria [65,66]. This solid solution (YSZ) is arranged in the cubic (fluorite) structure [67]. The conductivity in zirconia can be attributed to electrons, holes, and oxygen vacancy defects. Heavily doping zirconia with rare earth metal oxides such as CaO, Y20,, or MgO is known to increase the concentration of oxygen vacancies. In the case of yttria, these vacancies are necessary to maintain charge neutrality due to the valency difference between Zr4' and Y*. The ionic conductivity of YSZ is consequently increased to a level which significantly dominates any electronic conductivity, at least over a broad range of temperature and pressure. The electrolytic domain of solidoxide electrolytes is generally defined as the realm in which the ionic transference number, t., is greater than 0.99. The transference number is given by S= (21) 0m + 0k + im where ao is the ionic conductivity, ao is the electronic conductivity, and ah is the hole conductivity. The sum of all transference numbers is unity. For a given dopant concentration, the electrolytic domain is determined by the temperature and oxygen partial pressure. The effects of each of these variables will now be discussed in relation to the electrolytic nature of the material. The ionic conductivity of YSZ initially increases with increased levels of doping. A maximum is reached, however, at concentrations of yttria between 8 and 10 mole 30 percent, corresponding to the monocliniccubic solid solution phase boundary [681. Increasing the yttria composition beyond 10 mole percent results in a decrease of ionic conductivity. The corresponding activation energy for ionic conduction is observed to reach a sharp minimum at the phase boundary as well. The optimal doping level for YSZ appears to be the minimum concentration of yttria necessary to produce the fluorite structure, 8 mole percent. Oxygen partial pressure has a strong influence on the electron and hole carrier concentrations. Under conditions of low oxygen partial pressure, oxygen is removed from the lattice according to the equilibrium equation Oo Vo + 2 e + (Po) (22) where Oo is an oxygen atom in its designated lattice site, Vo** is an oxygen vacancy site, e is a mobile electron, and O, is gaseous oxygen. Two electrons must be liberated to accommodate each vacant oxygen site, and consequently, the electronic conductivity due to free electrons becomes important at low oxygen pressures. At the other extreme, high oxygen partial pressures will force oxygen into the lattice according to the equilibrium equation 102(P) + 0 + 2n* (23) where n* is a mobile hole, having an equal but opposite charge of an electron. Here, the filling of each previously vacant oxygen site must be accompanied by the liberation of two holes. Thus, at high oxygen pressures, the electronic conductivity due to holes becomes important. The electrolytic domain of solidoxide electrolytes must, then, exist for intermediate oxygen partial pressures. The width of the active pressure 31 window will, in general, be a function of temperature since the electron and hole carrier concentrations are temperature dependent. The electrolytic domain of YSZ is defined as the range of temperatures and oxygen partial pressures over which the ionic transference number is greater than 0.99. That is, the electronic conductivity within the electrolyte is less than 1% in this range. If the ionic and electronic conductivities are known as functions of temperature and pressure, then the limits of the electrolytic domain can be determined. At high oxygen partial pressures, conduction by electrons is insignificant and the upper pressure boundary of the electrolytic domain is defined by t = 0.99 = o (24) Oi + oim Similarly, at low oxygen partial pressures, conduction by holes is insignificant and the lower pressure boundary of the electrolytic domain is defined by S= 0.99 = o (25) O, + o. The electronic conductivities of YSZ were investigated by Kleitz et al. [69]. The particular samples were 9 mole % yttriadoped zirconia, and the temperature range investigated was 1170 1550C. These results will be extrapolated to lower tempera tures in order to estimate the electrolytic domain in the temperature range used in the present work (550 800C). The equations given by Kleitz et al. for the electron and hole conductivities are, respectively, I a, =5.5xlOP10 P 4e) (26) I .= L4P4eaxp( (27) where o, and oh are in (Qcm)'1, the Boltzmann constant, k, is in eV/K, and Po, is in atm. The ionic conductivity of YSZ electrolytes was studied by Strickler and Carlson [70] and Schouler et al. [71], among others. The two references mentioned give results in reasonable agreement, and the data of Strickler and Carlson, given by the following equation, shall be used here: o. = 115exp 78 (28) The temperature dependence of the upper and lower pressure limits (i.e., the pressures at which Equations (24) and (25), respectively, are satisfied) of the electro lytic domain can be calculated from Equations (24) (28). The functions describing these limits are given by the following equations: InP = 2.88 9.956 (29) kT n Po = 52.27 11.76 (210) kT 33 where Po0 is the upper pressure limit at which hole conduction becomes important, and Po" is the lower pressure limit at which electron conduction becomes important. A graphical representation of these limits is shown in Figure 21. A lower tempera ture limit also exists for the YSZ electrolyte. It can be noted from Equation (28) that the ionic conductivity decreases with temperature. This is due to a decrease in the mobility of oxygen ions as the thermal lattice energy is lowered, that is, the distribution of ions with sufficient energy to overcome the electrostatic binding forces becomes less. A critical lower temperature will eventually be reached at which the ionic carrier density will be insufficient to maintain an appropriate ionic conductivity. This lower temperature limit is generally observed to be in the range of 550 600*C. Operating outside of this electrolytic window in experiments involving oxygen concentration cells will result in uncertainties due to nonequilibrium conditions. Consequently, it is important to choose oxygen atmospheres which fall within the electrolytic domain for a given operating temperature. The oxygen concentration cells used in these studies maintained oxygen atmospheres which fall well within the electrolytic domain at the chosen operating temperatures (550 800C). Figure 22 shows the Gibb's energies of formation [72] of the various oxides used in the construction of the experimental cells. The equilibrium oxygen partial pressures may be read from the dashed oxygen isobars. A copper/copper(I) oxide reference system was used in some of the experiments, while a platinum/air reference system was used in others. Each of these can be seen to provide oxygen atmospheres which ensure proper electrolytic behavior of the electrolyte. T(K) 0.8 0.9 1.0 Temperature" x 1000 Electrolytic domain (t. > 0.99) of 9 mole % yttriastabilized zirconia. Figure 21. i I I I I I I I I I I I I i i 1 1 11i i I 1 i ( tm ) ^ PQ, (otm) S25 105 . 0 c CuO(s) 010 050 0   1025 1 100 150 o f n '(ll 0 1  SiO (s) 175 =  . 10 5 200 101 225 1040 1050 45 250 1 1 1 I I I f t.. I 1 1 500 600 700 800 900 1000 Temperature (OC) Figure 22. Gibb's energies of formation of the most stable oxides of materials used in the electrochemical cells [72]. The dashed lines are oxygen isobars. 36 Oxygen Concentration Cell The oxygen concentration cell is a thermodynamic system consisting of a solid oxide electrolyte separating two electrode compartments having independent oxygen chemical potentials. Each compartment must have a metallic electrode contacting the electrolyte interface to physically couple the chemical and electrical processes in the cell. In operation, a reversible opencircuit electric potential, E, is developed across the electrolyte which is related to the variation in oxygen chemical potential, go, across the electrolyte. Wagner [73] derived this relationship which is stated as Po E = 1 F tdpo (211) nF, where n is the valence of oxygen in the electrolyte (n = 2), F is the Faraday constant, and IO' and Eo" are the oxygen chemical potentials at each electrode (Oo' < Ao"). The integration can be carried out when the variation of ti with J is known. As shown earlier, ti is a function of temperature and oxygen pressure, and the variation of t. with these independent variables must be determined experimentally. In fact, a fair amount of experimental work has been completed in an attempt to characterize the conductive properties of solidoxide electrolytes (Kleitz et al. summarize many of these studies [69]). The results of the various investigations are often in extreme disagreement, however, for a number of reasons. First, the stabilized oxides used in experiments are of varied compositions and have been synthesized by a number of different processes. The singlecrystal grain sizes, in particular, can vary substantially from one process to another. Ionic and electronic transport mechanisms 37 within and across the grain boundaries will then have a significant effect on the overall charged carrier conductivities. Second, even for a given composition and grain structure, the carrier conductivities may not remain constant over time. Extended use of stabilized oxide electrolytes at high temperatures results in changes in the material such as (for YSZ in particular) segregation of yttriarich layers at the grain boundaries, formation of tetragonal ZrO, [74], and impurity segregation at the grain boundaries [75]. These temporal changes due to annealing rule out the effective use of Equation (211) in its general form, and the need for mathematical simplification is noted. The most obvious simplification of Equation (211) is to take advantage of the invariance of t.. within the electrolytic domain. As long as the electrolyte is maintained within the previously described range of temperature and oxygen partial pressure, ti. can be assumed to be constant and equal to unity. For the case t, > 0.99, Equation (24) reduces to AG = ~o Lo" = 2FE (212) In terms of oxygen activity, ao, Equation (212) is given by RTIn a0 2FE (213) ao" where R is the universal gas constant, and T is the absolute temperature. The oxygen concentration cell generally uses a reference electrode and a working electrode. The oxygen activity in the reference electrode is fixed by using a metal/metal oxide chemical system or a gas mixture of known oxygen composition. 38 In the case of the diffusion measurements and flow visualization cells used in this work, the working electrode is the tin melt. Having chosen an appropriate reference electrode, the activity of oxygen in the tin can be calculated directly from Equation (2 13) with experimental electromotive force (EMF) measurements. The EMF measurements in oxygen concentration cells are subject to experi mental errors, however. A general discussion of the sources of these errors will be given here as they relate to the experiments in this study. The first source of measurement error results from nonisothermal operation of the electrochemical cell. The previous equations relating the EMF to the oxygen chemical potential difference were derived under the assumption of isothermal and isobaric conditions. Additional terms must be included, however, in the case of nonisothermal cells. Goto and Pluschkell [76] presented the following simplified equation relating the EMF to oxygen chemical potentials at electrodes of different temperatures: E = 4'F [o(TP.' ) P po(Tlto + a[ T"]. (214) Goto and Pluschkell define a as a constant parameter which is related to the partial molar entropy and heat of transfer of oxygen ions in the electrolyte and the partial molar entropy and heat of transfer of electrons in the electrodes. It is essentially the overall Seebeck coefficient of the electrochemical cell. This Seebeck coefficient is not constant, however, but is a function of oxygen partial pressure in the cell. Fischer [77] measured a(Po,) for 9 mole percent yttriastabilized zirconia with dual platinum electrodes in the temperature range 687 1037C and reported the following empirical equation, a(P = 0.492 0.02201n5 (215) where the units of a are (mV/C) and the oxygen pressure is in mmHg. In a separate experiment, Fridman et al. [78] measured the Seebeck coefficient of 10 percent yttriastabilized zirconia at 1175*C in air and obtained a value of 0.47 mV/*C, agreeing very well with the value of0.492 mV/C from Fischer's results. The order of magnitude of a from these experiments is characteristic of most stabilized zirconia electrolyte materials [79]. The contribution of the Seebeck coefficient of the electrode materials to the overall Seebeck coefficient is very small since the partial molar entropy and heat of transfer of electrons in metals is small in comparison to the identical properties of oxygen ions in the electrolyte. Further errors in EMF measurements may result when currents are passed through the electrochemical cell by applied voltages different from the equilibrium opencircuit EMF. Passing currents through the cell can result in electrode and electrolyte polarization. For large currents, IR drop in the electrode and extension wires leading between the cell and current source can be important. For this reason, electrode materials of high electrical conductivity are generally preferred. Polarization may also occur at the electrodeelectrolyte interface due to the buildup of oxygen resulting from kinetic limitations of the halfcell reactions. Electrode polarization may also occur in an electrode depleted of oxygen, e.g. when diffusion of oxygen to the interface is the limiting kinetic factor. IR drop through the electrolyte due to the resistance to flow of ions in the electrolyte lattice may also become important for large current densities. In general, large currents in an electrochemical cell lead to the 40 production of an irreversible EMF, and the equations relating the reversible EMF as given previously will not be valid. The flow visualization experiments which will be described in Chapter 4 are especially subject to the EMF measurement errors described for nonisothermal cells. In these experiments, the effective diffusivity of oxygen across a tin melt is calculated for various Rayleigh numbers. The temperature gradients which are developed across the electrolytes range up to 120C, for which the corresponding Seebeck voltage is approximately 6 mV. However, since the diffusivities are calculated from the slope of the measured EMF versus time curve, the temporallyconstant Seebeck voltages do not enter into the calculation. Also, EMF measurement errors due to electrode polarization in the diffusivity experiments do not present a problem because of the small currents developed in the potentiostatic removal of oxygen. Summary Electrochemical measurements involving solidoxide electrolytes provide an effective means of determining thermodynamic as well as kinetic properties of oxygen in liquid metals. The accuracy of these measurements is, however, dependent upon maintenance of proper experimental conditions. The solidoxide electrolytes are known to function properly only under certain conditions of temperature and oxygen partial pressure. Additionally, errors resulting from nonisothermal operation of the cells and polarization under high current loads must be accounted for in the analysis of EMF data. A mathematical relationship has been presented which relates the reversible EMF developed across the oxygen concentration cell to the chemical potential 41 difference of oxygen. This relationship will be used in the following chapters to gain insight into the kinetics of oxygen transport through liquid tin. Chapter 3 focuses on measurement of the binary diffusion coefficient of dilute oxygen in liquid tin. Chapter 4 then extends the methods used in Chapter 3 to measure the combined diffusive and convective mass transport of oxygen in Bridgman crystal growth simulations using liquid tin as a model fluid. CHAPTER 3 OXYGEN DIFFUSIVITY IN LIQUID TIN Introduction The binary diffusion coefficient of dilute oxygen in liquid tin is investigated by using an oxygen concentration cell. The flow visualization experiments discussed in Chapter 4 are based upon the measurement of effective diffusivities of oxygen across a tin melt which is subjected to thermal gradients. The resulting effective diffusivity is a measure of the total rate of mass transfer of oxygen due to diffusion and convection. Since it is the overall effect of convection on mass transfer within the melt that is of interest, the diffusion effects must be subtracted out. This can be done since the concentration field depends linearly on the velocity field. Hence, the first goal of this study is to experimentally determine the molecular diffusivity of oxygen in liquid tin in the absence of convection. In order to remove convective effects in these diffusivity measurements, thermal gradients which induce natural convection must be eliminated or at least substantially minimized. Previous investigations of oxygen diffusivities in liquid tin [80,81,82,83], as well as in other liquid metals, have generally not taken suffi cient care to eliminate thermal gradients. On occasion, researchers have even imposed small thermal gradients (hot on top) in order to "stabilize" the melt; however, this technique has adverse effects in the creation of small horizontal thermal gradients due to mismatching of thermal properties of the melt and its container. 43 Otsuka and Kozuka [84], in particular, presented oxygen diffusivities in liquid lead under these conditions. They imposed a 1.5C/cm vertical temperature gradient across the electrochemical cell used in the measurements. Hurst [83] modelled this case numerically and observed lowlevel convection which was significant enough to cast doubt on the reported results. A detailed discussion of the effects of thermal gradients in diffusion measurements is given by Hurst [83]. In addition to thermallydriven convection, solutal effects must also be considered in diffusion experiments. In particular, the orientation of the diffusion cell must be assessed in terms of the direction of the resulting concentration gradient with respect to the direction of gravity. Ramanarayanan and Rapp [80] and Hurst [83] each employed radial diffusion schemes, where the resulting oxygen concentration gradient is perpendicular to the direction of gravity. As discussed in Chapter 1, this orientation causes unconditional hydrostatic instability, resulting in natural convection. The other orientation, used previously by Otsuka, Kozuka, and Chang [82] and in the present work, is axial diffusion. In this case, the resulting oxygen concentration gradient is aligned with the direction of gravity. The presence of a driving force for convection is then conditional upon the solutal Rayleigh number and aspect ratio for a given experiment. Consequently, a need was recognized for improved design of oxygen concentra tion cells for diffusion measurements. An improved cell design is presented here for making oxygen diffusivity measurements under isothermal conditions. The mass diffusivity of oxygen in liquid tin is then given as a function of temperature and compared to the results of the previous investigators. 44 Experimental A schematic of the experimental cell is given in Figure 31. As shown, the cell is actually composed of two oxygen concentration cells which share a common working electrode in the tin melt. Copper/copper(I) oxide reference electrodes are used to establish a known oxygen potential at the reference side of the electrolyte disks. Copper was chosen for several reasons: availability and low cost, machinability, and its physical properties. Copper's high electrical conductivity makes it an ideal electrode material with high resistance to polarization. The high thermal conductivity of copper helps to minimize thermal gradients which will induce convection in the melt. The cylindrical cell was designed for axial diffusion measurements. The purpose of this is to align the concentration gradients, which are developed over the course of an experiment, with the gravity vector. Since the side walls of the diffusion cell are impermeable to oxygen, horizontal concentration gradients are not a consideration. As described in Chapter 1, vertical density gradients developed as a result of solutal gradients may or may not be of sufficient magnitude to cause an onset of convection. The criterion for the onset of convection is the magnitude of the solutal Rayleigh number. Certainly, conducting the experiment so that the fluid will be less dense on top (corresponding to removal of oxygen at the bottom) will avoid develop ment of solutallydriven convection. However, removing oxygen from the top creates an unstable density distribution, and convection may ensue depending on the solutal Rayleigh number. It is not possible to calculate Ras apriori since no data on the coefficient of solutal expansion for the tinoxygen system is available in the literature. Verhoeven [19] reported that the low oxygen concentrations of oxygen in liquid metals Rheniun Extension Wire Alunina Overflow Tube Copper/Copper Oxide Reference  Copper Lead Wires Fused Silica Spacer Copper Electrodes Fused Silica  Tin Melt Figure 31. Schematic diagram of the experimental cell used for oxygen diffus ivity measurements. 46 will not produce density gradients large enough to surpass the first critical Rayleigh number. However, this statement can be shown to be incorrect. The solutal Rayleigh number is given by H ap 1 & p, (31) RaS = & Do. where ap/az is the density gradient and p. is the mean density of the fluid. The required density gradient for the onset of convection can be estimated by inserting typical parameters: Ra.1 = 2500 (for p = 1), H = 0.5 cm, g = 980 cm/sec2, Do = 6x10" cm2/sec, and v = 1.65x103 cm2/sec. The value of the minimum density gradient for instability thus obtained is ( ap 1 4xlO %Icm (32) az p.) which is not an unreasonable variation in a liquid metal system [19]. The diffusivities are calculated in this study for removal of oxygen from both the top and bottom faces of the fluid cell to test the effect of the direction of diffusion on the experimental diffusivities. Cell Construction The tin sample was contained within a fused silica cylinder (General Electric Co., Quartz Products Division, Cleveland, OH) sandwiched between two yttriastabi lized zirconia disks (ZIRCOA Products, Solon, OH). The quartz cylinders used in this 47 series of experiments were varied in height from 0.389 cm to 0.750 cm and measured 0.729 cm ID and 0.945 cm OD. The YSZ disks were 0.953 cm diameter by 0.158 cm thick with a reported composition of 8 wt.% yttria (4.5 mole %). The disks were cemented to the fused silica cylinder with Aremco 571 magnesiabased ceramic adhesive (Aremco Products, Inc., Ossining, NY) after placing the tin sample inside. The tin sample itself was cast from zonerefined bars of 99.9999% purity (Cominco American, Inc., Spokane, WA), which, after machining, was cleaned and etched with a 5 volume % bromine in hydrobromic acid solution. Since the axial diffusion experiment requires that the tin sample be completely confined and make full contact to the upper and lower electrolyte surfaces, a modification had to be made to allow for thermal expansion of the liquid sample upon heating. Consequently, a 1mm diameter hole was ground through the upper electrolyte disk at its center to allow for escape of excess tin. A 0.318 cm OD alumina overflow tube (McDanel Refractory Co., Beaver Falls, PA) was cemented, again with Aremco 571 ceramic adhesive, to the upper surface of the disk to contain the overflow and keep it electrically isolated from the copper electrode. The copper electrode components were machined from 2.54 cm diameter bars of copper (Defense Industrial Supply Center, Philadelphia, PA) of unknown purity. Electrical contacts were made to the copper electrodes with 0.5 mm diameter copper wire of 99.9% purity (Johnson Matthey Inc., Seabrook, NH) by drilling a small hole in the electrode, inserting the end of the wire, and mechanically pressing the junction to form a pressfit around the wire. The reference electrode system consisted of a 1:1 mole ratio of copper and copper(I) oxide powders (Alfa Products, Danvers, MA). The copper and copper(I) oxide powders were each 99.95% pure. A 4:1 mole ratio of copper 48 to copper(I) oxide was used initially, but the cell lifetime was quite short due to reduction of the oxide in the inert environment of the ambient argon stream. The 1:1 mole ratio system exhibited equally good electrical conductivity, however. The powder mixture was packed loosely into recesses machined into the copper electrode pieces, and then compressed by inserting the reference side of the electrolyte disks into the recesses on top of the powder. The upper electrode consisted of a single piece of copper having a 0.320 cm hole drilled through it axially to hold the alumina overflow tube. The bottom electrode consisted of two pieces, an outer cylindrical sheath, 2.54 cm OD, and a smaller cylinder which recessed the reference system. A fused silica spacer was inserted between the upper copper electrode and the outer copper sheath to electrically insulate the two electrodes from one another. The primary purpose of the outer sheath was to maintain isothermal conditions within the melt. Electrical contact was made to the tin sample by extending a 0.25 mm diameter rhenium wire (Johnson Matthey) through the top of the overflow tube. The rhenium wire was electrically contacted to a copper extension wire by twisting the two together over a length of approximately 3 cm. The copper extension wire was drawn into a narrow fused silica capillary tube (General Electric Co.), and the copperrhenium contact was forced into the end of the capillary to maintain pressure on the twisted junction. Resistance measurements of the combined copperrhenium extension wire before and after experiments showed no change. The copperrhenium junction extended no more than 4 cm above the top of the cell, well within the isothermal region of the furnace. 49 The temperature of the cell was measured with a type R (platinum/platinum 13% rhodium) thermocouple which was inserted approximately 2 cm into the top of the outer copper sheath. The accuracy of type R thermocouples is reported to be 0.25% [85]. The cell was contained within a 2.8 cm ID fused silica tube (General Electric Co.) which was capped at the top by a brass cell head. The brass cell head had four ports, three of which sealed 0.318 cm diameter feedthroughs with oring fittings. The fourth served as a connection to the vacuum system and also the outlet for the purified argon stream. Alumina feedthroughs were used to isolate the electrical connections passing through the cell head. The core of each electrical feedthrough was then plugged with RTV sealant (Dow Corning Corp., Midland, MI). The purified argon was introduced through one of the alumina feedthroughs. The argon (Air Products and Chemicals, Inc., Allentown, PA) was purified in two steps. First, the gas stream was passed over a catalyst to react hydrogen with oxygen to form water, which was then removed as the gas was passed through a canister containing sodium aluminosilicate desiccant (Matheson Gas Products, East Rutherford, NJ). Second, the gas stream was passed through a bed of 800C titanium sponge (Alpha Products). The final purified argon stream had a measured residual oxygen partial pressure of 1.6 x 102 mmHg (see Chapter 5). The furnace components were Kanthal wound resistance heater blocks controlled by Eurotherm (Eurotherm Corp., Reston, VA) temperature controllers. An isothermal liner (Dynatherm Corp., Cockeysville, MD) 30.5 cm in height was inserted into the furnace core to establish an isothermal region in which the diffusion cell could be placed. The liner was 3.4 cm ID by 6 cm OD. The cell, 5.6 cm in height, was 50 positioned in the middle of the isothermal liner. The temperature along the liner was measured by extending a narrow type R thermocouple into the furnace, between the liner and fused silica tube containing the cell. The temperature did not vary by more than 0.27*C along the length of the liner. Additionally, another isothermal liner (10.2 cm in height) and heater block were placed on top of the other components to increase the height above the cell which was heated. It is believed that this may reduce any conductive heat losses through the connections leading between the cell and the cell head at the top of the apparatus, as well as limit any radiative heat losses from the top of the cell. These precautions, as well as the copper sleeve encasing the diffusion cell, are believed to significantly reduce any thermallydriven convection in the melt to levels which will not noticeably affect the diffusivity measurements. ProcedureTransient Diffusion Experiments The oxygen diffusivity was determined experimentally by a combined potentiostatic and EMF method. Initially, a uniform oxygen concentration was established within the melt by pumping oxygen in or out of the tin through an applied voltage across either or both oxygen concentration cells. Recall, the experimental cell is a combination of two oxygen concentration cells which share a common tin working electrode and are represented as follows: Cu,Cu2O (1) II YSZ II Q in Sn I Re I Cu (I) Cu I Re I 0 in Sn II YSZ I Cu,CuaO (2) 51 The apparatus was then left open circuit until the measured EMF's across each cell were steady and equal. Then, at the start of the diffusion experiment, a zero oxygen concentration boundary condition was established at one of the tin surfaces (upper or lower) by applying a large voltage, 1.2 2.0 V, between the two electrodes of the corresponding oxygen concentration cell. The opencircuit EMF at the other oxygen concentration cell was then measured over time to yield an oxygen depletion curve for the surface of the tin sample contacting that electrolyte. Figure 32 shows typical EMF versus time data for three different sample heights. The diffusivity can be shown to be related to the slope of the EMF versus time curve at long times by solving the corresponding onedimensional boundary value problem. The diffusion equation is given by aC C12 C a =Do2C (33) at a z where C is the oxygen concentration in tin, t is time, z is axial position, and Do is the binary diffusion coefficient of oxygen in liquid tin. The initial condition and boundary conditions used in the solution of this problem are the following: C=Co at t=O, OszsH (34) C=0 at t>0, z= (35) ac 0 at t 0, z=H (36) az 0 ' I * Figure 32. 1 30 40 Time, t (min) Experimental EMF data for axial diffusion of oxygen through liquid tin. Representative data for three sample heights are given. 53 The general solution to this boundaryvalue problem is given by expf Dolmw2 WC()t) H2\ sin M (37) Co _o x+ I 1\W H 2 22 Since the experimental EMF measurements are related to the concentration of oxygen at the axial position z = H, Equation (37) will be expressed as C(H,t) 2(1) e D(38) Co m +1 [ H2 2 t At large times, however, only the first term is meaningful and Equation (38) reduces to C(H,t) A [4 [(o! i2 (39) C, H2 4 The relationship between the oxygen concentration at z = H and the measured EMF is obtained from an examination of the halfcell reactions in the electrochemical cell. For the transfer of oxygen from the copper/copper(I) oxide reference electrode to the tin melt, the halfcell reactions are as follows: Cu2O + 2e (in Sn) * 2Cu + 02" (in YSZ) (III) O2 (inYSZ) 0 + 2e (inSn) (IV) where 0 is oxygen dissolved in liquid tin. The overall cell reaction can then be expressed in terms of chemical and electrical processes. Cu2O 2Cu + (V) 54 2e" (in Cu) 2e (in Sn) (VI) At electrochemical equilibrium, the cumulative Gibb's energy change for the overall cell reaction must be equal to zero. AG, + AGG, = 0 (310) The Gibb's energy change for the chemical process given in Equation (V) is AGV,= AGO + RTln[ aOacc.2 (311) where AG* is the standard state Gibb's energy change of Equation (V), and the activities of the pure phases, ac, and aco, are equal to unity. The activity of oxygen in tin can be expressed as a = C (312) where y is a function only of temperature, assuming Henry's law holds for dilute oxygen in tin. For the electrical process, AG, = 2FE (313) From Equations (310) (313), the resulting relationship between E and C is 2FE = AG + RTIn(yC) . (314) 55 Since Equation (314) can be written for any arbitrary time in the diffusion experi ment, the following relationship is obtained for the ratio C(H,t)/Co: 2F(E(t) Eo) = RTIn C(Ht)) (315) where E(t) is the measured EMF at any arbitrary time, t, and Eo is the initial EMF. Combining Equations (39) and (315), this relationship is obtained: 2F(E(t) E) = 2D 4 (316) RT 4 H2) Thus, at large times, the rate of change of the measured EMF becomes constant and is proportional to the oxygen diffusivity. The experimentalist must be careful to wait sufficiently long to be certain that the approximation given in Equation (39) is valid. Generally, Equation (39) is considered valid when the second term in Equation (38) is less than 1% of the first term. This implies that only data taken after a minimum length of time, t., should be used to calculate the diffusivity. The value of tm is given by H2 t = 0.178 (317) Do From this, it can be seen that samples of small height are preferred. The time required for the experiment decreases as the square of the sample height. Addi tionally, experiments requiring long times will be more prone to errors resulting from 56 lowlevel convection in the melt. First, the cumulative amount of oxygen carried by convection is greater for long duration experiments than short duration experiments. Second, larger sample sizes are more prone to convection simply due to their size. The viscous damping of convection afforded by the side walls becomes less effective as the bulk is removed further from the walls. Maintaining isothermal conditions in the sample also becomes more difficult as the size is increased, especially in high temperature furnaces which often have only small, truly isothermal zones. A least squares analysis of the EMF data according to Equation (316) should result in an intercept at E(t) Eo = (RT/2F) ln(4/I). This has been used as a measure of the reliability of the data [82]. An intercept deviating from this predicted value would indicate nonideal experimental conditions, i.e. convection or oxygen leakage. Diffusion experiments were carried out for sample heights ranging from 0.389 to 0.750 cm. Ideally, the oxygen diffusivities calculated from each cell height should be identical at a given temperature. Should the measured diffusivities show a variation related to cell height, then the data would be suspect. This, again, would indicate the presence of convection or oxygen leakage into or out of the cell. ProcedureSteadystate Diffusion Experiments The diffusivity of oxygen in tin can be determined from a steadystate experiment as well as the transient experiment. The same apparatus is used in both. In this case, however, a steady current was passed through the entire cell, from one reference copper electrode to the other. Since the current is transferred in the form of oxygen ions through the electrolytes, a linear oxygen gradient is established axially across the tin melt in the absence of convection. Since the current in equals the current out, the total amount of oxygen in the tin sample does not change over time. 57 In order to calculate the diffusivity in the steadystate experiment, the concentration gradient must be known. The flux of oxygen, No, is proportional to the concentration gradient of oxygen, and the proportionality factor is the diffusivity: No = DOC (318) Assuming ionic conduction only, the flux of oxygen is calculated from the cell current, No (319) 2A where I is the cell current, A is the crosssectional area of the tin sample, and the 2 originates from the divalency of oxygen anions in YSZ. So, the diffusivity is given by the equation I Do o *dC (320) EMF measurements across each of the oxygen concentration cells will give the relative concentrations of oxygen at the opposing surfaces of the tin melt. The unfortunate aspect of this method, though, is that the concentration gradient cannot be determined exclusively from the EMF measurements. Knowledge of the absolute concentration of oxygen at some reference EMF must be obtained. One reference state for which experimental data is available in the literature is the saturation point for oxygen in tin [80,86,87]. The results of the three references are in reasonable agreement, and the results of Ramanarayanan and Rapp [80] are plotted in Figure 33 58 for the saturation mole fraction of oxygen as a function of temperature. The corresponding saturation potential developed between the copper/copper(I) oxide reference and the saturated tin in the present cell was measured to be 491(4) mV at 700"C, the temperature at which the steadystate diffusion experiments were carried out. With this information, the oxygen diffusivity in steadystate galvanic cells can be calculated from the measured EMF's between the tin and the individual copper reference electrodes. The steadystate experiments were carriedout for both positive and negative gradients of oxygen. The case where the oxygen concentration increases with height is expected to be hydrostatically stable since the density of the fluid decreases with height. Conversely, a decreasing oxygen concentration with height will create a hydrodynamic stability problem completely analogous to the RayleighBenard problem. The diffusing component in this case, however, is the oxygen solute rather than thermal energy. Should the solutal Rayleigh number exceed the critical value for onset of convection, then the measured diffusivity will reflect the additional mass transport afforded by the convective flow. This information is critical in assessing the merit of a flow visualization technique which uses dilute oxygen as a tracer element in liquid metal samples. Results Transient Diffusion Experiments The experimental results for the transient diffusion experiments will be presented in two groups: 1.) oxygen depletion from the bottom of the tin sample and 2.) oxygen depletion from the top of the tin sample. A marked difference is observed 800 Temperature (C) Figure 33. Oxygen solubility in liquid tin as a function of temperature [80]. 0.001 0.0001 0.00001 l 600 60 between these two cases in the general trends of the measured EMF data. Figure 34 shows the two general trends of the measured EMFs which were observed in the experiments. The first group of experiments (bottomdepletion) is characterized entirely by normal diffusion curves, that is, EMF trends like that shown in the lower curve of Figure 34. This is the expected trend assuming that no convection or oxygen leakage is present in the liquid sample. The second group (topdepletion), however, is characterized by both trends in EMF, and the type of trend is seen to be correlated with the initial oxygen concentration and cell height. The transient diffusion results for depletion of oxygen from the bottom of the sample will be presented first. Table 31 summarizes the experimental conditions for each of the isothermal diffusion experiments in which oxygen was removed from the bottom face of the tin sample by applying a large voltage between the tin and the lower copper reference electrode. The temperature range investigated was 547C to 827C. The initial oxygen concentration in the samples was varied from approximately 8.5 x 108 to 5.7 x 10" mole fraction (based on the oxygen saturation data of Ramanarayanan and Rapp [80] and saturation EMF measurements in the present work). The diffusivity of oxygen was calculated from linear EMF data like that shown in Figure 32. The linear region of the data was fit by a leastsquares method and the corresponding lines were drawn through the data. A summary of the results from each run is given in Table 32. The diffusivities were calculated from the slopes of the linear data, and the corresponding ordinate intercepts may be compared to the predicted intercepts from Equation (316). It can be noted that the experimental intercepts are less negative than the predicted values in each case. This would indicate that either lowlevel convection is present in the sample despite precautions Time, t (min) Figure 34. Representative EMF versus time curves for transient diffusion experiments for removal of oxygen from both the top (upper and lower curves) and bottom (lower curve only) surfaces of the tin sample. Experimental conditions of the bottomdepletion diffusivity studies including sample height, temperature, initial oxygen mole fraction, and applied voltage. 32 37 38 39 315 317 320 322 326 327 328 333 334 335 338 339 341 342 343 344 345 346 347 348 363 364 365 366 367 368 369 370 371 372 373 380 381 Height(cm) 0.389 0.389 0.389 0.389 0.389 0.389 0.389 0.389 0.389 0.389 0.389 0.750 0.750 0.750 0.750 0.750 0.742 0.742 0.742 0.742 0.742 0.742 0.742 0.742 0.467 0.467 0.467 0.467 0.467 0.467 0.467 0.467 0.467 0.467 0.467 0.691 0.691 Initial Mole Fraction. X0 Applied Volt.(V) Temp.(*C) 590 547 547 550 641 723 721 722 720 719 718 730 727 726 729 730 688 690 689 770 771 771 827 827 633 633 633 677 677 677 727 725 724 779 782 705 705 1.5x10' 6.3x10' 1.O0x10 8.5x10' 5.9x105 2.6x10' 2.3x10' 2.5x104 1.6x10' 1.8x10' 2.1x10' 1.8x10' 2.0x10' 1.4x10' 1.8x10' 2.0x10' 9.1x10s 1.1x10' 7.8x10s 2.8x10' 2.8x104 2.8x10' 5.7x10' 5.6x10' 4.8x10s 4.7x10s 5.0x105 8.6x10s 8.4x10' 9.2x105 2.0x10' 1.6x10' 2. x10' 3.5x10' 4.3x10' 1.0x10' 1.3x10 Table 31. 1.2 1.2 1.2 2.0 1.5 1.5 1.5 1.5 1.5 2.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.2 1.8 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 Calculated oxygen diffusivities and corresponding intercepts from a statistical fit of the linear data for each experimental run in which oxygen was removed from the bottom of the tin sample. Exy. # 32 37 38 39 315 317 320 322 326 327 328 333 334 335 338 339 341 342 343 344 345 346 347 348 363 364 365 366 367 368 369 370 371 372 373 380 381 Intercept (mV) Do (cm'/sec) 5.llxlO 2.80x10lO 4.00x105 4.30x105 5.80x105 7.54x105 7.57x10'" 8.76x105 8.12x105 8.40x105 8.32x105 6.37x105 6.88x105 6.30x105 7.67x105 7.72x10" 5.86x105 6.23x105 5.72x105 6.91x105 6.96x105 7.28x105 1.14x104 1.38x10 4.80x105 4.94x10' 5.55x105 5.95x105 6.06x105 6.13x105 7.63x105 7.30x105 7.62x105 8.44x105 1.00x10' 6.91x105 7.16x10"5 Predicted Intercept (mV) 4.69 4.27 5.34 4.98 6.41 7.96 7.46 6.54 7.97 8.40 8.54 6.70 7.52 6.29 7.68 7.94 6.37 7.85 6.10 6.68 7.08 7.07 9.50 9.78 5.87 6.08 7.28 6.67 6.95 7.10 8.21 7.78 7.96 8.24 9.12 7.32 8.20 Table 32. 8.98 8.53 8.53 8.57 9.51 10.37 10.34 10.36 10.33 10.32 10.31 10.44 10.41 10.40 10.43 10.44 10.00 10.02 10.01 10.85 10.87 10.87 11.45 11.45 9.43 9.43 9.43 9.89 9.89 9.89 10.41 10.39 10.38 10.95 10.98 10.18 10.18 64 to eliminate thermal gradients or oxygen leakage into the cell is causing a slow drift in the measured EMF's. The diffusivities from Table 32 are plotted as a function of reciprocal temperature in Figure 35. An Arrhenius dependence on temperature is assumed, and the line drawn through the data corresponds to a linear leastsquares analysis. The following equation is obtained: D = 1.65x103 exp 6150 (321) where R is 1.987 cal/gmole K Two interesting trends are observed, however, when the data from the experiments for each individual sample height are analyzed separately. Figure 36 shows a leastsquares analysis for each of the three sample heights which were studied over reasonable temperature ranges. The first trend is observed in the activation energies calculated from the Arrhenius relationship. These are summarized in Table 33 for the various data sets. E, is lowest in the case of the H = 0.389 cm data set and highest in the case of the H = 0.742 cm data set. The point, however, is not that the activation energy should change with sample dimensions. On the contrary, the activation energy is dependent only on the "activity" of oxygen in the solvent and not on the geometry of the sample. The point is that the temperature ranges studied for each sample height (also shown in Table 33) are different. The activation energy shows an increase with temperature, at least as calculated from this limited data. The second trend observed in Figure 36 indicates an overall lowering of the calculated diffusivities as the experimental sample height is increased. This trend leads to the conclusion that convection is not a likely cause of experimental error, but 0.9 1.0 1.1 Temperature1 x 1000 (K1) Figure 35. Experimental oxygen diffusivities from transient diffusion experi ments using five different cell heights. The solid line is a least squares fit of the data. 0.001 0.0001 0.00001 0.1 III1 1 1 I IIIIIII I I I I I I IIiI  I I 000o0 H = 0.389 cm suoms H = 0.467 cm ooooo H = 0.691 cm AAAAA H = 0.742 cm AAAA H = 0.750 cm A A U ,I 50 1.3 """""""""""""""""""" 1 3 1.3 0.9 1.0 Temperature 1 1.1 x 1000 (K) Figure 36. Linear leastsquares analyses of the experimental diffusivities given for individual sample heights. Analyses were not made for H = 0.691 cm and H = 0.750 cm since the temperature was not varied for these sample heights. 0.001 0.0001 0.00001 0.8 II I I I I I III I I II II 11111  11111 a i !I I II oDoP H = 0.389 cm enmi. H = 0.467 cm ooooo H = 0.691 cm AA.AAA H = 0.742 cm AAAAA H = 0.750 cm A ''a'a i l a a a a a  * 67 Table 33. Experimental activation energies for the diffusion of oxygen in liquid tin. Values are given for the individual data from sample heights of 0.389 cm, 0.467 cm, and 0.742 cm as well as for all data combined. E, (cal/gatom) Calculated from Data: Temperature Range 7340 H 0.389 547 7230C 7690 H = 0.467 633 7820C 9930 H 0.742 688 8270C 6150 All data 547 827C that oxygen leakage is most likely the primary factor in experimental inconsistencies. If convection were present in the sample, then the experimental diffusivities would be expected to increase with sample height. Small sample sizes provide a greater resistance to bulk convection due to the closeness of the bounding walls, while these viscous effects will be less apparent to the bulk in larger samples. So, the most probable explanation for these variations is a source of oxygen in the experimental cell. The larger sample heights require longer times to complete a single diffusion experiment, and consequently, the oxygen leakage will have a more pronounced effect on the calculated diffusivities than in the case of smaller sample heights. For this reason, the diffusivities calculated from the smaller cells are likely the most accurate of all the reported results. The most probable source of oxygen in the diffusion cell is the overflow tunnel through the upper YSZ disk. Although the diameter of the hole is small (1 mm), the oxygen in the small volume of tin which exists there acts as a virtual oxygen leak. Since the oxygen concentration cell defined by this upper YSZ disk is used to monitor 68 the oxygen depletion from the upper surface of the tin sample, the virtual leak directly affects the EMF measurements. Further evidence supporting this theory can be gathered from the data plots in Figure 32. The EMF curves begin to noticeably sag at long times, indicating an inclusion of oxygen somewhere in the tin sample. Certainly, however, more than one oxygen source could exist in the cell. Other possibilities include outgasing from the silica container or YSZ due to chemical reduction as well as leakage of ambient oxygen through micropores in the cemented junctions of the container. Outgasing from the ceramics is not likely, however, considering the stability of those particular oxides in the oxygen atmosphere within the cell. Also, no visual evidence of degradation on the surfaces of the materials is apparent. Leakage of oxygen through micropores, on the other hand, is a possibility since the ceramic cement used to fix the pieces together is porous upon curing. Thus, two possible sources of oxygen leakage have been identified (overflow channel leakage and micropore diffusion), but the magnitude of the error introduced into the calculated diffusivities is uncertain. The second group of transient diffusion experiments were performed by applying the voltage across the upper electrolyte to remove oxygen from the top of the tin sample. The EMF was then measured across the bottom electrolyte to allow for calculation of the oxygen diffusivity. The two general shapes of the EMF versus time curve observed in the transient diffusion experiments were shown in Figure 34, and both of these trends were actually observed in the case of oxygen depletion from the top of the sample. The lower curve is the normal diffusion curve, resulting from purely diffusive mass transport in the melt. The upper curve, however, exhibits an interesting phenomenon which is uncharacteristic of a wellcontrolled diffusion 69 experiment. A sharp rise in the EMF a short time into the experiment indicates that oxygen is being removed quite rapidly from the tin which is adjacent to the lower electrolyte. Based on the results from the diffusion experiments under stabilizing density gradients, this depletion cannot be a result of purely diffusive mass transport. The apparent cause of this rapid disappearance of oxygen from the bottom surface must, then, result from convective transport within the melt. The sudden rise in EMF at short times into the experiment apparently corresponds to the initial onset of convection in the fluid. As the fluid begins to flow, oxygenpoor tin from the upper portion of the melt is swept to the bottom, resulting in a corresponding rise in the equilibrium EMF. However, the slope of the EMF is then observed to decrease after the initial sharp rise. This can be attributed to the sudden decrease in the driving force for convection. The initial onset of convection causes the oxygen to be mixed throughout the melt, and the solutal gradients are thereby reduced. The convection will then subside and settle into a less energetic flow. One would expect the flow to slowly decrease as the oxygen concentrations are reduced from the continued removal of oxygen at the upper surface. Several of the experiments in this group resulted in normal diffusion curves, and the diffusivities were calculated in the usual manner. The remaining experi ments, however, had to be analyzed differently. An effective diffusivity can be defined which is a measure of the combined rate of mass transport of oxygen due to both diffusion and convection. This effective diffusivity is calculated in much the same manner as the binary diffusion coefficient is calculated, from the slope of the EMF versus time curve. For these experiments, the effective diffusivity is calculated from the slope at the point where the rapid rise in EMF is observed. The experimental 70 parameters of each run are detailed in Table 34. The resulting diffusivities from this group of experiments are tabulated along with the experimental and predicted ordinate intercepts from the linear fit in Table 35. The runs which exhibited an apparent onset of convection are marked as such. The diffusivities from Table 35 are plotted in Figure 37 for comparison with results from the bottomdepletion experiments. Each of the data points which exhibited a normal diffusion curve lies very close to the linearized data from the bottomdepletion experiments. The cluster of points which lie well above the linearized data, on the other hand, all exhibited the sharp rise in EMF due to the apparent onset of convection. The onset was not observed in the experiments with Table 34. Experimental conditions of the topdepletion diffusivity studies including sample height, temperature, initial oxygen mole fraction, and applied voltage. Initial Mole Ex. # Height(cm) Temp.(OC) Fraction. X, Applied Volt.(V) 31 0.389 589 2.0x105 1.2 33 0.389 590 1.5x105 1.2 310 0.389 550 8.7x10' 1.5 311 0.389 551 9.2x10' 1.5 312 0.389 639 5.0x10O5 1.5 314 0.389 640 4.4x10 1.5 316* 0.389 724 1.6x10' 1.5 319* 0.389 721 2.3x104 1.5 321* 0.389 721 2.1x104 1.5 324* 0.389 721 2.4x104 1.5 336 0.750 727 2.1x104 1.5 354* 0.742 867 8.3x10"4 1.5 indicates observation of the sharp change in slope in the EMF versus time curve 71 Table 35. Experimental oxygen diffusivities from topdepletion experiments. The ordinate intercepts from a leastsquares analysis of the data are also listed for comparison with the predicted intercepts. Predicted Exp. # D (cm'/sec) Intercept (mV) Intercept (mV) 31 4.32x10s 5.0 8.97 33 4.73x10s 6.2 8.98 310 4.06x10 6.3 8.57 311 4.33x10" 5.7 8.58 312 6.66x10 9.1 9.48 314 6.23x10' 7.8 9.50 316* 2.12x104 25.9 10.38 319* 2.70x104 26.6 10.34 321* 2.69x10 28.6 10.34 324* 2.26x10 15.6 10.34 336 8.78x10 10.5 10.41 354* 3.90x10' 5.2 11.86 indicates observation of the sharp change in slope in the EMF versus time curve bottomdepletion because the resulting density gradient in the melt was oriented such that the more dense fluid was underneath the less dense fluid. In the case of the top depletion experiments, however, the density gradient was oriented in the opposite direction, and the hydrostatic stability of the fluid was then conditional upon the magnitude of the density gradient and the geometry of the fluid cell. The stability problem is analogous to the RayleighBenard problem for vertical thermal gradients, except that the resulting density gradient in this case is not linear, but decreases exponentially away from the depletion surface. Since the hydrostatic stability of the melt is dependent upon the magnitude of the solutally induced density gradient, one might expect a correlation of the observed convective onset with the initial oxygen concentration in the melt. This is indeed the case. In considering only the experiments listed in Table 34 which used a sample 0.8 0.9 1.0 1.1 1.2 Temperature' x 1000 (K1) Figure 37. Effective diffusivities calculated from transient diffusion experiments with topdepletion of oxygen. The solid line corresponds to the least squares analysis of data from experiments with bottomdepletion of oxygen. 0.001 0.0001 0.00001 0 0.' I i i I i I I I I I l I I T i I Ii i I I I I  4  2 data points Bottomdepletion experiments Q0000 H = 0.389, topdepletion experiments ooooo H = 0.750, topdepletion experiments Doo0D H = 0.742, topdepletion experiments I I I I ""i II "p 7 73 height of 0.389 cm, the onset of convection was observed in only those having an initial oxygen concentration greater than 10 mole fraction. This is strong evidence of the ability of very small, solutallyinduced density gradients to drive natural convection in these liquid metal systems. A final observation in these results involves the influence of the aspect ratio on the condition for hydrostatic instability. Experiment number 336 in Table 34 showed no evidence of convection despite an initial oxygen mole fraction of 2.1x104. This mole fraction is shown to be sufficient to cause convection in the 0.389 cm tall samples, but appears to be insufficient to initiate convection in this 0.750 cm tall sample. The diffusivity calculated from experiment number 336 agrees well with measurements from bottomdepletion experiments. Experiment number 354, on the other hand, evidenced a strong onset to convection for an initial oxygen mole fraction of 8.3x104. The sample dimensions were almost identical to those in experiment number 336. Thus, the initial oxygen concentration and cell dimensions both influence the hydrostatic stability of the tin sample during a transient diffusion experiment. This is not surprising since the aspect ratio and Rayleigh number are the two parameters which were found to define the stability criterion in the Rayleigh Benard problem. Steadystate Experiments The results of the steadystate experiments carriedout at 700*C do not agree well with the diffusivities calculated from the transient diffusion experiments; but, further evidence is found to substantiate the occurrence of convection in the presence of destabilizing oxygen gradients. Recall, the steadystate experiments are carriedout by passing a current from one reference electrode to the other, thus creating a linear 74 oxygen gradient across the tin sample. Measurement of the equilibrium EMFs across each of the electrolytes then allows calculation of the oxygen concentrations from solubility data in the literature. The calculated diffusivities are tabulated for both positive (concentrated on top) and negative (concentrated on bottom) oxygen gradients in Table 36 along with the applied cell currents. The height of the sample in each case is 0.691 cm. The diffusivities are also plotted in Figure 38 as a function of the applied cell current. The diffusivities calculated for negative oxygen gradients are approximately a factor of three larger than those calculated for positive oxygen gradients. This discrepancy leads to the conclusion that convective mass transport must be playing a role in the negative oxygen gradient runs. Indeed, this orientation does lead to an increasing fluid density with height, and the hydrostatic stability of the fluid is again dependent upon the magnitude of the density gradient and aspect ratio. In these experiments, the aspect ratio is maintained constant and the density gradient varied by changing the oxygen concentration gradient across the sample. The melt is statically unstable to all of the applied gradients, however, indicating that the first critical Rayleigh number is exceeded in each case. The diffusivities calculated from the positive oxygen gradient runs are substantially lower than the corresponding values determined from the transient diffusion study. The results from the transient experiments are more accurate, however, owing to the minimal experimental error involved in data measurements. The steadystate results are subject to error both internally and externally. The internal errors result from uncertainties in the EMF measurements due primarily to polarization effects as indicated by an increasing effective diffusivity with current. 75 Oxygen diffusivities calculated from steadystate experiments. The cell current and corresponding oxygen gradient are also listed for each experiment. (positive gradients indicate increasing concentra tion with increasing height) Cell Current (uA) 5 10 15 20 10 15 20 25 30 35 40 45 50 55 60 65 65 70 75 80 85 Oxygen Gradient. dX,/dz (cm'1) 12.4x10s 13.4x10s 18.8x10' 21.7x10s 3.65x10s 5.46x10s 7.38x10s 9.25x10s 11.0x10 12.7x105 14.2x105 15.9x10 18.0x10s 19.7x10 20.8x105 22.4x10s 20.2x10s 22.2x105 24.3x10 26.6x10s 28.9x10s Do (cm/lsec) 0.9x10' 1.6x105 1.7x101 2.0x105 6.0x105 6.Ox105 5.9x105 5.9x105 6.0x10 6.1x105 6.2x105 6.2x101 6.1x10 6.1x105 6.3x105 6.4x105 7.1x105 6.9x105 6.8x105 6.6x105 6.4x105 The measurements are made across oxygen concentration cells which are subjected to continuous electrical currents. By the time a steadystate is reached, a significant layer of oxide can be builtup at the surface of the electrolyte at the most negative reference electrode. This leads to measurement errors of varied magnitude depending on the extent of the oxide layer. For example, the saturation potential measured for oxygen saturated tin before one experiment was 491(2) mV, but after applying a 40 pA current for 3 to 4 hours, the measured saturation potential was 510(4) mV. The saturation potential eventually drifted back to its original value, but only after being Table 36. 1.OE004 .OE005 0 6.0E005 S24.0E005 S2.0E005 0.OE+000 I I I I I I i I I I I I  ooooo Decreasing concentration with height 00000 Increasing concentration with height 0 0 O O o oo o 0 00 00 0 O 0 0  0 0 0 0 S, 0 10 20 30 40 Applied 50 60 current 70 (uA) 80 90 100 Figure 38. Steadystate diffusivities plotted versus applied current at 700*C. The cell height was 0.691 cm. I 77 held opencircuit for several hours. A 1 mV error in the measured EMF results in a 2.5% error in the calculated oxygen mole fraction at the electrolytetin interface. Additional errors in the diffusivity calculations from steadystate data are external to the experiments. As mentioned previously, oxygen solubility data from literature sources must be used in the calculation of the diffusivity. The available sources report oxygen solubilities which are in disagreement by approximately 20%, although this is considered to be reasonable agreement for this type of study. A 1% error in the solubility results in a corresponding 1% error in the calculated oxygen diffusivity. Discussion The diffusivity of oxygen in liquid tin has been studied by other investigators previously, and their results are compared to the present results in Figure 39. The studies of Ramanarayanan and Rapp [80] and Hurst [83] used radial diffusion measurements in cylindrical geometries, while Otsuka and Kosuka [81] and Otsuka, Kozuka, and Chang [82] used axial diffusion measurements similar to those reported in this work. The radial diffusion results are in significant disagreement with the axial diffusion results. A probable cause is the presence of buoyancydriven convection in the radial geometries due to the development of horizontal density gradients as the oxygen is depleted from the lateral boundaries of the fluid volume. At first, this argument may seem to contradict the physical evidence since the reported diffusivities in the radial geometries are lower than those reported for axial geometries. This cannot be easily judged, however, since the two types of experiments are carriedout differently. 0.8 0.9 1.0 1.1 Temperature1 x 1000 (K1) 1.2 1.3 Figure 39. Comparison of experimental diffusivities from the present work, Ramanarayanan and Rapp [80], Otsuka and Kozuka [81], Otsuka, Kozuka, and Chang [82], and Hurst [83]. 0.001 0.0001 0.00001 o 0.' i I I I I I l I I I I I I i I i i I 1 1 1 1 I  eP9 Present work S Raomanaryanan and Rapp Otsuka, Kozuka, and Chang  Otsuka and Kozuka Hurst Ia o "" "" " I I0 7 79 The presence of convection in the present studies is shown to yield an effective diffusivity larger than the molecular diffusivity. This can be justified intuitively by considering the effect of convection on the surface concentration of oxygen at the surface opposite to the depletion surface. In the absence of convection, the oxygen diffuses away from this surface into the bulk relatively slowly as the concentration gradient in the bulk evolves. The corresponding change in the equilibrium EMF across the oxygen concentration cell will increase at a corresponding rate. In the presence of convection, however, the oxygen is removed more rapidly from this surface, and the corresponding change in the equilibrium EMF will reflect this. Since the diffusivity is proportional to the rate of change of the EMF with time (at long times), the apparent diffusivity must increase in the presence of convection. However, the diffusivities are calculated differently in the case of radial diffusion experiments. Here, a potential is applied to deplete oxygen from the lateral boundaries of the cylindrical cell, and the ionic current (corresponding to the flux of oxygen out of the tin) is measured as a function of time. The diffusivity is then found to be proportional to the negative logarithmic slope of the ionic current over time (at long times). In the absence of convection, the ionic current decreases relatively quickly as the oxygen is depleted from the melt adjacent to the lateral wall. The presence of convection, however, will replenish oxygen at the lateral wall with oxygen rich fluid from the bulk, altering the evolution of the ionic current with time. The resulting effective diffusivity will then be in error, but whether it should decrease or increase with convection is not intuitively obvious. Diffusion studies of oxygen in liquid metals have been carriedout in the past under the assumption that the oxygen was present at such small concentrations that 80 any resulting density gradients would be insufficient to drive natural convection. This is simply not true, and care must be taken in constructing oxygen diffusion cells such that any density gradients are aligned with the gravity vector. The cell design presented in this study appears to be superior to designs used in the past primarily due to the minimization of thermal gradients in the melt The solutoconvective driving forces can be minimized simply by choosing an orientation which aligns the solutal gradient with gravity, but elimination of thermal gradients is not as simple owing to the high temperatures at which these oxygen diffusion studies are carried out. The highly conductive copper sheath which encloses the diffusion cell is a simple modification which greatly increases the isothermal character of the diffusion cell. The only drawback of the present design is the oxygen leakage which occurs through the overflow tunnel in the upper YSZ disk. However, this problem is averted when the cell is run in the topdepletion mode because the EMF measurements are then taken across the bottom oxygen concentration cell. Care must be taken, however, to reduce oxygen gradients such that the critical instability for the onset of convection is not surpassed. This implies that the initial oxygen concentration must be maintained as low as possible while still remaining within the electrolytic domain of the electrolyte. A point of interest not yet mentioned concerns the applied voltage for creation of the zeroconcentration boundary condition in the transient diffusion experiments. The applied voltages are listed in the tables of experimental parameters for each of the case studies. Overall, they were varied from 1.2 to 2.0 volts with no noticeable effect on the calculated diffusivities. In reality, the equilibrium oxygen concentration at the electrolytetin interface is not zero, but several orders of magnitude lower than 81 the bulk oxygen concentration depending on the value of the applied voltage. At the temperatures studied here, an increase in the applied voltage of 100 mV reduces the equilibrium oxygen concentration by approximately one order of magnitude. So, at 700C, an applied potential of 1.2 volts establishes an oxygen concentration of 1011 mole fraction at the interface. An interesting observation, however, is that applied voltages a 1.5 volts caused a rather large infusion of copper into the YSZ disks. Apparently, the copper migrated into the grain boundaries of the zirconia and was even observed to diffuse across the entire thickness of the disk in some cases. In all likelihood, the presence of copper in the grain boundaries will adversely affect the conductive properties of the electrolyte and should be avoided. Summary An improved experimental cell design for measuring the binary diffusion coefficient of dilute oxygen in liquid metals is presented. The vertically oriented cell is applied to the tinoxygen system for determination of the oxygen diffusivity in the temperature range 547 827C. The results compare favorably with previous investigations which used similar experimental procedures, although the present results are considered to be more reliable due to better controlled thermal conditions. Results from investigations employing radial diffusion techniques show significant differences from the axial diffusion cases. These differences are explained in terms of natural convection phenomena resulting from density variations in the radial diffusion orientation. Convection resulting from vertical gradients in oxygen concentration is also shown to be significant when the resulting density gradient is positive and exceeds a 82 threshold value. The threshold value is found to be dependent on the magnitude of the density gradient and the geometry of the fluid sample. The fluid is found to be hydrostatically stable for all negative vertical density gradients. CHAPTER 4 FLOW VISUALIZATION Introduction The basic experimental approach used to measure the diffusivity of oxygen in liquid tin (Chapter 3) is extended to study certain aspects of the dynamic states in a simplified Bridgman configuration. The vertical diffusion cell is modified and used to measure the transport of oxygen in tin melts which are subjected to axial thermal gradients. The measured transport rates reflect the overall ability of the fluid to transfer oxygen across the fluid cell. This rate is then indicative of the tendency of the fluid to disperse solute which is introduced at one boundary (as in solute rejection during meltgrowth of multicomponent semiconductors which form solid solutions) throughout the bulk. By defining an overall mass transfer coefficient (the effective diffusivity), the relative level of convection in a fluid cell can be studied as a function of certain external parameters. For example, in the RayleighBenard problem (which concerns the stability of dynamic states in horizontal fluid layers heated from below), the dynamic state of a fluid layer is observed to change with the imposed vertical temperature gradient. Similarly, the dynamic state of a confined fluid volume, such as in Bridgman meltgrowth configurations, changes with imposed vertical temperature gradient. These flow transitions have been observed in many fluids, but have only been characterized in detail for high Prandtl number, Pr, fluids since these fluids can be studied visually. The flow characteristics of low Pr fluids, which are of 84 interest in Bridgman meltgrowth, have not been studied in much detail (as outlined in Chapter 1) owing to the absence of a comprehensive flow visualization technique. It is for this reason that the applicability of solidstate electrochemical techniques in flow visualization is investigated. Hurst [83] initially proposed an experimental technique similar to that used in this study. The particular design which he adopted was, however, not capable of maintaining wellcontrolled thermal conditions across the melt. The conditions must be controlled to the extent that measurements are reproducible as the thermal gradient is cycled. Hurst's data show significant scatter due to the inability to accurately characterize the thermal conditions within the fluid cell. The experimental design proposed in this work is much simpler and affords more accurate temperature measurements. The results obtained from the diffusion studies in Chapter 3 indicate that the basic diffusion cell design and technique applied in the transient studies can be applied as a flow visualization tool in low Pr fluid systems. The purely diffusive mass transfer of oxygen in liquid tin was shown to be substantially lower than the combined mass transfer in the presence of convection. This low diffusivity is fortuitous in that oxygen can then be used effectively as a tracer which can be sensed at the boundaries of the fluid volume. Consequently, the goal of this study is to test the viability of using electrochemical sensors to trace dilute oxygen in dynamic fluid systems associated with the Bridgman configuration. The physical problem chosen for these initial tests is that of a vertically oriented cylinder of fluid heated from below. This configuration is primarily of academic interest since Bridgman meltgrowth is generally carried out by heating 85 from above and directionally solidifying from the bottom up. The appearance of dynamic transitions when heating from below, however, provides an ideal medium for testing the electrochemical technique. Two fluid aspect ratios (p = H/R) are studied in these initial experiments. A cell having p = 1 is studied first since it can be modelled numerically as a twodimensional flow [441, and the numerical results are compared with experimental observations. The oxygen transport rates, in particular, are compared for experimental and numerical results for various applied vertical temperature gradients. The experimental technique is then applied to a fluid cell having p = 5.3, again, to study the effects of varying the imposed vertical temperature gradient across the melt. The details of the experimental and numerical approaches are also presented. Experimental The experimental approach is analogous to that used in the oxygen diffusivity studies. The experimental cell design is modified, however, to allow for the application of a thermal gradient along the axis of the cell. The cell design is shown schematically in Figure 41. The outer copper sheath used in the diffusivity studies is removed and the upper and lower copper reference electrodes are extended to protrude into the upper and lower zones of the Bridgman furnace. In this manner, a thermal gradient may be imposed simply by maintaining the upper and lower zones of the furnace at different temperatures. Liquid tin is used as a model fluid for these studies primarily due to its low vapor pressure and low toxicity, although the experimental method may be extended to most any metallic or semimetallic fluids. Overflow Tube Thermocouples Liquid Tin Copper Electrode Copper Extension Wires Fused Silica  YSZ Copper Copper + Oxide Figure 41. Experimental cell design for measuring the effective diffusivity of oxygen across liquid tin. Cell Design The tin melt was held within a fused silica cylinder sandwiched between two YSZ disks. The ID of the cylinder was 2.2 cm in each case, and the YSZ disks were 2.54 cm in diameter by 1.58 mm thick. The heights of the fluid cells used were 1.1 cm and 5.79 cm to yield aspect ratios of 1 and 5.3, respectively. A 1 mm diameter hole was ground through the upper YSZ disk to allow excess tin to flow up the 4.75 mm OD alumina overflow tube which was cemented to the upper surface of the disk with Aremco 571 ceramic cement. The reference electrodes were constructed from 2.55 cm diameter copper bars (Defense Industrial Supply Center). The ends facing the YSZ were hollowed out to create a recess for the 1:1 mole ratio copper/copper(I) oxide powder reference system. The upper copper electrode had a 4.8 mm diameter hole drilled along its axis to receive the overflow tube. Type R thermocouples were inserted into small holes which were drilled into the sides of the two copper electrodes at the ends nearest the fluid cell. The tips were coated with ceramic cement to maintain electrical insulation from the electrodes. The thermocouples were made from the same wire stock, and calibration relative to one another showed no measurable difference at cell temperatures. Electrical connection to each of the copper electrodes was made by inserting a copper wire into small holes drilled into the electrodes and pressing the walls of the hole around the wire. Electrical connection to the tin was made by twisting a copper extension wire to a short length of rhenium wire which could then be fed down the overflow tube into the tin melt. 88 A cast tin ingot was initially placed within the quartz container, and the cell was constructed around it. The cell was then placed inside of a fused silica tube which was capped at the top by a brass cell head. The entire apparatus was then connected to a vacuum source as well as a purified argon source. A schematic of the entire experimental setup is shown in Figure 42, including the cell, furnace, vacuum, argon, and electronic instrumentation. A detailed schematic of the furnace itself is shown in Figure 43. The furnace is a three zone Bridgman furnace which allows heating of the two copper electrodes to desired temperatures by the upper and lower zones. The central zone is required to eliminate any lateral heat losses from the sample. The brass heat pipe located in this zone is designed to create a smooth temperature gradient between the upper and lower zones, and thus reduce any heat transfer to or from the sides of the sample. The cell tube shown in Figure 43 is the fused silica container used to isolate the experimental cell from the oxygenrich atmosphere. Procedure The experimental procedure is completely analogous to that used in the transient diffusion studies in Chapter 3. The boundary value problem, however, is not so easily solved in the case of a fluid in motion. The convective term must be included in the species balance for oxygen in order that the problem be fully posed. The oxygen balance is then given by the following equation: a + ( .V)C = Do2C (41) W II 01 I 0 =Ea . 6a Cell Tube Isotherma I Liner 'Insulation Furnace Brass Heat Blocks Pipe Insulation Isothermal I Liner Figure 43. Schematic of the furnace used in the effective diffusivity studies. 91 The velocity field is not analytically tractable, however, due to the coupling between the energy and momentum equations as well as the complex thermal character of the experimental cell. The problem can be solved numerically, though, and is done so by using the FLUENT computational fluid dynamics code (creare.x Inc., Hanover, NH) to model the experimental cells. The experimental approach used here is based on a onedimensional approximation to mass transfer of oxygen across the dynamic fluid cell. The one dimensional diffusion equation is used to model the transfer of oxygen in a dynamic diffusion experiment across a convecting melt. C Do a2 (42) t az2 The effective diffusivity, Do", then accounts for both diffusive and convective mass transport. Since the experiments are carried out at constant mean temperatures, the variation of the effective diffusivity is expected to be due solely to changes in the dynamics of flow in the melt (i.e. the temperature dependence of the binary diffusion coefficient is less than 0.5% per *C in the temperature range studied here and is not expected to have a significant effect on the effective diffusivity as the thermal gradient is changed). The boundary value problem to be solved for calculating the experimental effective diffusivity is then identical to the problem solved in Chapter 3 for calculation of the binary diffusion coefficient from transient diffusion experiments. The effective diffusivity is related to the measured cell EMF by the following equation at large times: 