)V1531_} RETURNING MATERIALS: Place in book drop to LJBRARJES remove this checkout from “ your record. ‘FINES will be charged if book is returned after the date stamped below. THE DIFFUSIVITIES OF GASES IN PURE LIQUID METALS By Leo Bernardino Tentoni A THESIS Submitted to Michigan State University in partia] fulfiIIment of the requirements for the degree of MASTER OF SCIENCE Department of ChemicaI Engineering 1984 ABSTRACT The Diffusivities 0f Gases In Pure Liquid Metals By Leo Bernardino Tentoni The diffusion of gases in liquid metals is important for industrial and theoretical reasons. This research investigates the applicability of four selected theories for predicting diffusion coefficients of hydrogen. nitrogen. and oxygen in pure liquid metals. Experimentally- determined diffusion coefficients are compared with values predicted by four theories which were chosen for their predictive capacity and their theoretical value: hydrodynamical theory. absolute reaction- rate theory, vibrational displacement theory. and modified Enskog theory. No experimental datum is more accurate than 122, no theoretical prediction is more accurate than ilOZ; the most difficult cases are those with hydrogen, those with silver, and those with gallium. For the oxygen—silver system. experimental diffusivity values are larger than those expected from any theory. The modified hydrodynamical theory and the modified Enskog theory appear the most useful for predicting coefficients for nitrogen in liquid iron and for oxygen in liquid metals. A proposed extension of these theories is necessary for accurate prediction of the coefficient of hydrogen. TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES CHAPTER 1: 1.1. 1.2. Chapter 2: 2.1. 2.2. Chapter 3: 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. OVERVIEW AND REVIEW OF THE FOUR SELECTED THEORIES Importance of Diffusion of Gases in Liquid Metals Four Theories of Liquid-State Diffusion 1.2.1. The Hydrodynamical Theory 1.2.2. The Absolute Reaction-Rate Theory 1.2.3. The Vibrational Displacement Theory 1.2.4. The Modified Enskog Theory EXPERIMENTAL METHODS IN GAS-LIQUID METAL DIFFUSION STUDIES Capillary-Cell Methods Electrochemical Methods RESULTS AND DISCUSSION Experimentally-Determined Diffusion Coefficients of Gases in Pure liquid Metals Comparison of Experimental Data With Predictions From Hydrodynamical Theory Comparison of Experimental Data With Predictions From Absolute Reaction-Rate Theory Comparison of Experimental Data With Predictions From Vibrational Displacement Theory Comparison of Experimental Data With Predictions From The Modified Enskog Theory Conclusions Regarding The Overall Applicability Of The Four Selected Theories. LIST OF REFERENCES 0mm 13 15 20 21 28 28 36 40 4O 49 Table Table Table Table Table LIST OF TABLES Diffusion data for gases in pure liquid metals Comparison of experimental data with the hydrodynamical theory Comparison of experimental data with the absolute reaction-rate theory Comparison of experimental data with the vibrational displacement theory Comparison of experimental data with the modified Enskog theory 11 22 29 32 37 41 Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES Schematic of the unsteady-state capillary-cell (after El-Tayeb and Parlee, 1967). \ Schematic of the steady-state capillary-cell (after Velho et al., 1969). Schematic of electrochemical cell for the galvanostatic method (after Sano et al., 1970). Schematic of electrochemical cell for the potentiostatic method (after Oberg et al., 1973). Hydrogen diameter versus differences in Pauling electronegativity values. The H-Fe system. The O-Ag system. 14 16 17 18 45 47 48 Chapter 1: OVERVIEW AND REVIEW OF THE FOUR SELECTED THEORIES l.l. Importance of Diffusion of Gases in Liquid Metals Diffusion of gases in liquid metals is important for several reasons. First, in liquid—metal cooled fission and fusion reactors. it appears that many important corrosion mechanisms are limited by diffusion of contaminant gas in the liquid metal (Addison. 1972: Johnson, 1977; Klueh, l984; Steinmeyer et al., l98l; Stewart and Sze. 1977). Second. the chemical, electrical. mechanical. and thermal properties of solid structural metals depend on the rate of gas absorp- tion or desorption before and during solidification which is determined by diffusion of gas in the liquid metal (Birnbaum and Wert. 1972; Fromm and H6rz, 1980: Guy and Hren, 1974). Third. gas diffusion data and theories are necessary to provide information about the structure of liquids in general and liquid metals in particular (Ashcroft and Lekner. 1966: Reddy. 1984; Swamy and Reddy. 1984; Weber and Stillinger. 1984). There is no consistent general theory or model of the liquid metallic state which at the present predicts the physical. thermodynamic. and transport properties (Coffey et al., 1984; Hafner, 1977). l.2. Four Theories of Liquid-State Diffusion In historical order, the four selected theories of liquid-state diffusion are: hydrodynamical theory (Einstein. 1905), absolute reaction-rate theory (Eyring, 1936). vibrational displacement theory (Nachtrieb. 1967). and modified Enskog theory (Protopapas and Parlee. l976). The first three theories yield simple mathematical expressions containing physical constants and measured properties of the liquid metal and provide a reasonable fit for self-diffusion data in liquid 2 metals. The mathematical expressions of the modified Enskog theory. while somewhat more complicated than those of the other authors. have only one free parameter (covalent diameter of the gas) and are relatively simple to use. Other excluded theories of diffusion include cluster theory (Egelstaff, 1962), free-volume theory (Cohen and Turnbull. 1959). corresponding—states theory (Pasternak and Olander, l967), fluidity theory (Hildebrand, l97l; Sridhar and Potter, 1977), fluctuation theory (Swalin. 1959). and mobility theory (Walls and Upthegrove. l964). These theories generally yield complicated mathematical expressions containing a number of physical properties of the gas—liquid pair. some of which are difficult to determine. A brief overview of the four selected theories follows. l.2.l. The Hydrodynamical Theory The hydrodynamical theory assumes solute particles are enough larger than solvent particles so that the solvent behaves as a continu- um. The solute particle is often assumed to be spherical with a size independent of solute-solvent interaction; recently however, corrections have been proposed to account for these effects (Brenner, l979. l98l; Emi. l972). For dilute solution with constant activity coefficients. the Nernst-Einstein equation gives (Levine, Chapter 16, 1978) vA D where DAB is the diffusion coefficient of A in B, k is Boltzmann's constant. T is absolute temperature. VA is the average velocity of A. and FA is the average frictional force acting on A. For low Reynolds number flow. this force is (Sutherland, l905: Lamb, 1945) 211 + rA A = 6nuB VA rA 3“ F B 8AB (2) B T "A'BAB where “B is the solvent viscosity. rA is the solute radius. and BAB is a coefficient of sliding friction. Two limiting cases of Equation (2) are the no—slip condition. BAB = m. and the no—stick condition. BAB = O. For the no-slip condition. Equation (2) reduces to Stokes's law: FA = 6nuB VA rA (3) For the no-stick condition. Equation (2) becomes FA = 4% VA rA (4) Substituting Equations (3) and (4) into Equation (1) gives two limiting expressions for the diffusion coefficient: _ kT DAB “'EEUE‘FX' (5) -_-___'SI_.__ and DAB 4WUB rA (5) For diffusion of gases in liquid metals, where rA is assumed to be the solute van der Waals radius. Equation (6) gives better results than Equation (5). tHowever, according to the theoretical analysis of Lamm (1938) and of Lamm and Sjostedt (1938), if the diffusing parti- cles are smaller than the solvent particles. the numerical factor 4 in Equation (6) is still too large. Evans et al. (l98l) showed experimentally that the hydrodynamical theory is not applicable for small solutes (e.g.. argon and methane) diffusing in polyatomic organic liquids because this theory assumes solutes are much larger than solvents. They state that. based on a corresponding states-type analysis. past agreement between the hydro- dynamical theory and experimental diffusivity data was purely accidental due to the narrow atomic density ranges over which experiments were performed. Their explanation cannot be tested for gas-liquid metal systems because there are not enough experimental data on a variety of gas-liquid metal systems over a broad atomic density range. Equation (6) can be modified to , _ kT DAB - fiEfiE’FZ' (7) where r;. the solute radius. is now a function of temperature and is given by (Protopapas and Parlee. l976) r’ - r (1 — o 112 (T/T )32‘)1o‘8 (in cm) (8) A — c ' m where Tm is the melting point of the liquid metal and rC is the covalent radius of the solute. Protopapas and Parlee give no reason why the covalent radius is used in Equation (8). Examination of the data suggest a better fit is available if the radius were related to the ionic character of the solution. Thus. rg must be considered a free parameter which, by nature. is always close to the covalent radius. The results of the modified hydrodynamical theory. Equation (7). are seen to be consistently smaller than experimental results. but they are better than the results of the original hydrodynamical theory. Equation (6). 1.2.2. The Absolute Reaction-Rate Theory In its original form. the absolute reaction-rate theory assumes particles are all of the same kind. Diffusion occOrs through the action of holes (i.e.. vacant sites). Accordingly. the diffusing particle jumps from its equilibrium position over a potential energy barrier into a nearby hole in the surrounding liquid (Eyring. 1936: Glasstone et al.. Chapter 9. 1941). The sum of the energy to form a hole and the energy needed to surmount the barrier is called the activation energy for diffusion. Once a particle jumps into a hole. it must remain there long enough to dissipate the energy it acquired. For viscosity and self-diffusion. respectively. it can be shown that u - U k] 1 (9) - 2 B 12 A3 Av kv _ 2 where A3 is the interparticle distance in the direction of viscous flow. A1. and A2 are interparticle distances perpendicular to A3. and Av and kv are the jump-distance and absolute rate constant. respective- ly. for viscous flow. Similarly. AD and k0 are the jump-distance and absolute rate constant for diffusive flow. Consistency requires that the viscosity and self-diffusion coefficient of a liquid be related because the models are identical and because the jump-distance and absolute rate constants of the two processes are identical: sz kv = 102 kD (11) Substituting Equation (11) into Equation (9) and then Equation (9) into Equation (10) gives kT l D = -- --- (12) SELF “B A2 A3 For spherical particles in a quasi-crystalline lattice. this equation simplifies further because the interparticle distances equal the jump distance. which equals twice the solvent radius: A ‘= A — A - 2rB (13) where rB is the solvent radius: thus kT D = -——-—— (14) SELF 2 “3 r3 In 1958. Eyring modified this theory by assuming gas-like proper- ties for particles which jump into holes. and assuming solid-like properties for all neighboring particle-hole pairs. The idea is that a liquid should have thermodynamic and transport properties intermediate between those of the gas and solid states. For the self-diffision coefficient of a liquid this (significant structures theory) gives (Eyring et al.. 1958. 1960) kT kT DSELF = g b5'AB" 1E‘EE’FE' (‘5) where E is the number of nearest neighbors in the same plane. For quasi—hexagonal close-packed liquids. E = 6 (Ree et al.. 1958). Alter- natively (Eyring and Ree. 1961: Ree et al.. 1964). kT = 51_ NAv 1/3 DSELF = 5 A2 A3 6 V572 (16) A1 “3 where NAv is Avogadro's number and VS is the molar volume of the solid metal at its melting point. For self-diffusion. Equations (14) — (16) are similar to expres— sions derived from hydrodynamical theory (e.g.. Equations (5) and (6)): however. for Eyring's self-diffusion correlation to apply to impurity diffusion. all properties of the liquid metal solution (density. mean frequency of vibration. viscosity. jump distance. and absolute rate constant) must be identical to those in self-diffusion. These assump- tions imply that the diffusion coefficient is independent of the diffusing gas species - a result which is not supported by experiment. Absolute reaction-rate theory remains valuable. as Equations (14) and (15) can be rewritten as kT D = m (17) , _ kT and D - TE—EE-FE- (13) respectively; where r6 . the solvent radius. is now a function of temperature. Choosing r6 as that given by Protopapas and Parlee (1974). AW ”3 1/2 . 1 8 = E1.288 — 1 - 0.112(T/Tm) ( 10' in cm)(19) where AW is the liquid metal atomic weight. pm is the liquid metal density at its melting point. and Tm is the melting temperature. The results from this modification of absolute reaction-rate theory are seen to bracket all the experimental results (Chapter 3. Section 3). 1.2.3. The Vibrational Displacement Theory Nachtrieb (1967) proposed his vibrational displacement theory because of the observation that the self-diffusion coefficient of liquid metals is linearly proportional to temperature. Also. Nachtrieb was able to show that experimental data is just as well represented by a linear relation (D vs. T) as by an Arrhenius—type. activated state. relation (ln D vs. 1/T). This implys that diffusion in liquid metals is not a simple. thermally activated process. Vibrational displacement theory proposes that every particle is enclosed in a cage composed of its nearest neighbors. From equipartition of energy (which is valid at high temperatures). every particle is assumed to vibrate with energy E'= 3kT = %K77' (20) where E is the average energy. K is the force constant of the metal (Pauling. Chapter 7. 1960). and F7 is the mean-squared thermal ampli- tude. From Equation (20). Fri-E— (21) At this point. F7 is inserted into the Einstein diffusion equation for random walks: D = 'FT'I (22) OS)-' where D is the diffusion coefficient and T is the characteristic time of diffusion. This result provides an accurate prediction for self- diffusivity at the melting point if the vibrations are all in phase. and if T is approximately equal to the Debye frequency (VD) of the solid metal. The Debye frequency is usually given in terms of a Debye temperature: e — D = [LE-E (23) where OD is the Debye temperature and h is Planck's constant. Combining Equations (22) and (23) yields an equation which predicts identical diffusion coefficients for all gases in a given liquid metal: k2 on T D = “11'?— (24) The results from this theory are seen to be consistently much smaller than experimental data (Chapter 3. Section 4). 1.2.4. The Modified EnskoggTheoery Thorne. in an unpublished work (Hirchfelder et al.. Chapter 9. 1964: Chapman and Cowling. Chapter 16. 1970). extended the Enskog theory of pure liquids (Enskog and Svenskii. 1921: Enskog. 1922) to binary liquid mixtures. According to Thorne. the binary diffusion coefficient is given by D (25) 12 = D12/812 where 012 is the binary diffusion coefficient in dilute solution and g12 is the pair—correlation (or radial distribution) function. For hard spheres in dilute solution. kT (m + m ) l/2 012 = 8n 3 3* an 1m 2 (25) 12 l 2 10 where m1 and m2 are atomic masses of the two components. n is the total particle density. and 012 is the center-to—center distance between two dissimilar hard spheres in contact: 0 + o l 2 C12 ‘ 2 (27) Here (LI and 02 are the hard—sphere diameters of the gas and liquid metal. respectively. Lebowitz (1964) derived expressions for the pair-correlation function at contact between similar and dissimilar hard spheres. In an early paper. Protopapas and Parlee (1975) corrected Lebowitz's expression for dissimilar hard spheres: (0 - O )(n - n) _ 2 - n. g, _ 1 2 1 2 g12 “(2 .+ nT(l - n1“ “ 2 l 3 (a. + 021m. + no. (28) where n1 and n2 are the packing fractions of the two components given ”1 =‘<%:>"1013 (29) and n = n1 + n2 (30) by Taking a different tack. Alder. et al. (1974) used molecular dynamics to determine correction factors for the Enskog theory based on mass ratio. diameter ratio. and packing fraction of the hard-sphere parti- cles. Thus. Equation (25) becomes D12 = (CF) D12/812 (31) where CF is the correction factor. ll Protopapas and Parlee used the data of Alder et al.. and an idea first suggested by Dymond and Alder. to obtain temperature-dependent expressions for hard-sphere diameters which are then incorporated into the correlation (Dymond and Alder. 1966. 1968. 1970; Protopapas and Parlee. 1974. 1976). The idea is to use hard-sphere diameters which shrink with increasing temperature. thus giving. effectively. soft spheres. This idea combines the mathematical simplicity of hard spheres with the reality of soft spheres. Expressions obtained by Protopapas and Parlee for the hard-sphere diameters are given by Equations (8) and (19) (multiply these equations by 2 to get diameters). Generally. the results from this modified Enskog theory are larger than experiment- al results (Chapter 3. Section 5). 12 Chapter 2: EXPERIMENTAL METHODS IN GAS-LIQUID METAL DIFFUSION STUDIES 13 The main experimental techniques for measuring diffusion coefficients of gases in liquid metals are the capillary-cell methods and electrochemical methods; the latter are used for oxygen diffusion only. A brief overview of these methods follows. 2.1. Capillary—Cell Methods Capillary-cell measurements are either steady-state or unsteady- state. In either case. gas is dissolved in a stagnant sample of liquid metal and the volume of gas absorbed is measured as a function of time. The rate of gas absorption is an accurate measure of diffusivity because convection and surface effects appear to have little effect (El-Tayeb and Parlee. 1967: Birnbaum and Wert. 1972). In the former method. the system is allowed to reach steady state before gas volumes are recorded (Sacris and Parlee. 1970). The unsteady-state method (Figure 1) typically assumes semi- infinite boundary conditions (e.g.. Mizikar et al.. 1962): d2 p c (60w Dt)]/2 v _ m 5 (32) 200pg where V is the STP (standard temperature and pressure) volume of gas absorbed. d is the capillary cell diameter. pm is the liquid metal density at temperature T. pg is the STP density of absorbed gas. CS is the gas saturation concentration in the liquid metal. D is the diffu- sion coefficient. and T is time. A plot of V vs. t% should be a straight line whose slope can be used to calculate D. l4 GAS ALUMINA -A ABSORPTION ____... TUBE : ALUMINA : _;= COMBUSTION : Pd '-1 - TUBE - )- 2 / I :2 y: : // ‘ LIQUID METAL : ¢ t a; : . / : SUPPORT .': .. . TUBE - ~ ~ : : E— THERMOCOUPLE -1 1 '- Figure 1. Schematic of the unsteady-state capillary-cell (after El-Tayeb and Parlee. 1967). 15 For the steady—state method (Figure 2). the solution is (Velho et al.. 1969) V = D-—- t (33) pg A1 L2 + A2 L1 7100 where A1 is the internal cross-sectional area of the capillary tube. A2 is the annular cross—sectional area between the capillary tube and the crucible wall. L1 is the depth Of immersion Of the capillary tube. L2 is the depth of stagnant liquid metal. C.' is the saturation concen- tration of gas at the liquid metal surface inside the capillary tube. and Co is the saturation concentration of gas at the liquid metal surface outside the capillary tube. A plot of V vs. t should be a straight line whose slope can be used to calculate D. 2.2. Electrochemical Methods The electrochemical methods are either galvanostatic (Figure 3). or potentiostatic (Figure 4): both use a solid electrolye. either calcia—stabilized zirconia (ZrO2 - CaO) or yttria-doped thoria (Th02 - Y203). As before. gas is dissolved in a sample of stagnant liquid metal. and the rate Of gas absorption is measured as the change in electromotive force (emf) between the two electrodes for the galvano— static method (Sano et al.. 1970: Honma et al.. 1971) or the change in cell current between the two electrodes for the potentiostatic method (Ramanarayanan and Rapp. 1972: Szwarc et al.. 1972: Oberg et al.. 1973: Klinedinst and Stevenson. 1973). 16 GAS ~1— QUARTZ OR ALUMINA TUBE PLANE 1 —_I""““ ...... ”-1 ....... LIQUID METAL SURFACE U ’ P'LANE‘z’ " ' F"-'--"' ..... _ Figure 2. Schematic Of the steady-state capillary-cell (after Velho et al.. 1969). POTETIO- METER l7 Figure 3. STAINLESS STEEL WIRE QUARTZ CAPILLARY TUBE ALUMINA CRUCIBLE LIQUID METAL Pt ELECTRODE (REFERENCE) Schematic of electrochemcial cell for the galvanostatic method (after Sano etal.. 1970). Ni - Cr ARGON GAS INLET ELECTRODE / +—_Zr 02 - CaO TUBE l WWW/M2 Pt LEADWIRE ——> "*— Pt ELECTRODE LIQUID METAL CONTAINING DISSOLVED OXYGEN Figure 4. Schematic of electrochemcial cell for the potentiostatic method (after Oberg et al.. 1973). 19 For both the capillary-cell methods and the electrochemical methods. the diffusion coefficient is assumed to be independent of gas cOncentration; this is equivalent to assuming that the activity coefficient is independent of gas concentration (Hahn and Stevenson. 1977). 20 Chapter 3: RESULTS AND DISCUSSION 21 3.1. Experimentally—Determined Diffusion Coefficients of Gases in Pure Liquid Metals Table 1 lists (alphabetically. by system) all the experimental diffusion data known to the author for hydrogen. nitrogen. and oxygen in pure liquid metals. Where available. Do and Q are listed for a fit of the data to an Arrhenius relation: D = Oo exp (-Q/RT) (34) Equation (34) is used only for the convenience it gives. it is no less accurate than the data it correlates. 00 may be related to the struc- ture and packing of particles in the liquid metal. but little is known about the independent character of DO at this time (Wilson. 1965: Hahn and Stevenson. 1977). In the table. data marked with an asterisk is considered questionable. Hydrogen-Liquid Metal Systems--For H—Al. the data Of Byalik et al. are reasonable. but the temperature range is narrow. The data of Vashchenko et al. do not compare with those Of other researchers. ' For H-Cu. the data of Sigrist et al. do not compare with those Of other researchers. For H—Fe. the data of Arkharov et al. and Nyquist do not compare with those of other researchers. For H-Li. the data of Alire below 800°C are suspect because of an erroneous correlation used in determining experimental diffusivity values (Buxbaum and Johnson. 1982). Nitrogen—Liquid Metal Systems--For N—Fe. the data Of Chesnokov et al. and Lee and Parlee do not compare with those of other researchers. TABLE 1 Diffusion data for gases in pure liquid metals. An asterisk indicates questionable data. Number in parentheses indicates reference. 00 0 Temp. Range System (cm2/sec) (cal/mOl) (°C) Researcher(s) H-Ag 4.54 x 10-2 1359 985-1208 Sacris & Parlee (106) *H-Al 7.47 x 10-1 7900 900-1000 Byalik et al. (18) H-Al 3.8 x 10‘2 4600 780-1000 Eichenauer & Markopoulos (33) *H-Al 2.34 x 10-3 15000 670-985 Vashchenko et al. (124) H-CO --- --- 1600 Depuydt (26) H-Cu 1.46 x 10-2 4500 1100-1250 Chernega & Vashchenko H-Cu 10.91 x 10-3 2148 1103-1433 Sacris & Parlee (106) *H-Cu 5.12 x 10-3 5880 1103-1361 Sigrist et al. (111) H-Cu --- --- 1101,1201 Wright & Hocking (130) H-Cu --- --- 1090-1205 Yang & Lee (131) *H-Fe 5.21 x 10-2 10000 1560-1650 Arkharov et al. (6) H-Fe --- --- 1600 Bester & Lange (12) H-Fe 4.37 x 10-3 4134 1550-1680 Depuydt & Parlee (27) H-Fe 3.2 x 10-3 3300 1547-1726 El-Tayeb & Parlee (37) H-Fe --- --- 1550-1750 Ershov & Kasatkin (42) H-Fe --- --- 1550-1650 Linchevskii & Shal'kevich (76) *H-Fe 1.86 x 10-3 9370 1550-1650 Nyquist (83) H-Fe 2.57 x 10-3 4100 1550-1720 Solar & Guthrie (112) H-Li 13.0 25000 625-900 Alire (3) H-Li --- --- 800-900 Buxbaum & Johnson (17) 23 TABLE 1 (Continued) 00 0 Temp. Range _ System (cmZ/sec) (cal/mol) (’C) Researcher(s) H-Ni --- --- 1500-1750 Ershov & Kasatkin (42) H-Ni --- --- 1450-1550 Linchevskii & Shal'kevich (76) H-Ni 7.47 x 10-3 8550 1478-1600 Sacris & Parlee (106) *H-Ni 5 x 10-2 9736 1468-1550 Wright & Hocking (130) H-Sn --- --- 1081-1105 Ebro (31) H-Sn ~—- ~-- 1000-1300 Sacris & Parlee (106) N-Co --- --- 1600 Benner (10) N-Fe 1.07 x 10"3 11000 1600-1700 Arkharov et al. (5) N-Fe --- --— 1600 Atarashiya (8) N-Fe 2.86 x 10‘3 14600 1550-1700 Bogdanov et al. (14) *N-Fe 0.4 15200 1550-1650 Chesnokov et al. (23) N-Fe --- --- 1550-1750 Ershov & Kasatkin (42) N-Fe 1.07 x 10-3 11000 1560-1700 Ershov & Kovalenko (43) N-Fe --- --- 1550-1680 Inouye et al. (61) N-Fe --- --- 1550-1700 Kunze (68, 69) *N-Fe 1.83 x 10-2 23120 1560-1680 Lee & Parlee (74) N-Fe --- --- 1600 Schwerdtfeger (108) N-Fe --- --- 1600 Svjazin & El-Gammal (120) N-Ni --- --- 1600 Benner (10) N-Ni 6.602 x 10--2 6580 1500-1700 Chesnokov & Linchevskii (22) 24 TABLE 1 (Continued) Do 0 Temp. Range System (cmzlsec) (cal/mol) ('C) Researcher(s) N-Ni --- --- 1550-1750 Ershov & Kasatkin (42) N-Zn --- 10100 466-574 Pechenyakov et al. (97) O-Ag --- --- 950-1150 Besson et al. (11) O-Ag 5.15 x 10'3 9900 970-1200 Masson & Whiteway (77) O-Ag 1.47 x 10-3 7100 1000-1200 Mizikar et al. (79) O-Ag 2.8 x 10-3 8300 1000-1350 Oberg et al. (85) O-Ag 1.85 x 10-3 7500 980-1130 Otsuka et al. (88) O-Ag --- --- 1000-1150 Otsuka & Kozuka (89) O-Ag 1.47 x 10-3 7100 950-1200 Parlee & Zeibel (94) *O-Ag 2.63 x 10-3 7300 990-1220 Rickert & El-Milligy (105) O-Ag 3.0 x 10-3 8700 1000-1200 Sano et al. (107) O-Ag 2.2 x 10-3 7900 1000-1200 Shah & Parlee (109) O-Ag --- --- 1100 Suito et al. (117) *O-Cu 7.25 x 10--3 15722 1100-1300 El-Naggar & Parlee (36) *O-Cu 2.51 x 10-2 16700 1100-1400 Gerlach et 61. (52) O-Cu --- --- 1200 Kramss et al. (67) O-Cu 6.9 x 10-3 12900 1000-1350 Oberg et al. (85) *O-Cu 2.63 x 10-3 9370 1100-1350 Ogterwaid & Schwarzlose O-Cu 5.7 x 10-3 11900 1128-1322 Otsuka & Kozuka (91) *O-Cu 1.22 x 10-2 14400 1100-1250 Rickert & El-Miligy (105) TABLE 1 (Continued) 25 Do 0 Temp. Range System (cm2/sec) (cal/mol) (‘0) Researcher(s) *O-Cu 1.55 x 10-4 8330 1100-1300 Shurygin & Kryuk (110) *O-Fe 6.23 x 10-3 10800 1550-1650 Ershov & Bychev (41) O-Fe --- --- 1550-1750 Ershov & Kasatkin (42) O-Fe --- --- 1550 Kawakami & GotO (64) O-Fe --- --- 1560,1660 McCarron & Belton (78) *O-Fe 3 34 x 10'3 12000 1550-1680 Novokhatskii & Ershov (82) O-Fe --- --- 1560 Otsuka & Kozuke (93) O-Fe --- --- 1610 Schwerdtfeger (108) O-Fe --- --- 1610 Shurygin & Kryuk (110) *O-Fe 5 59 x 10-3 19550 1560-1660 Suzuki & Mori (119) O-Ga 3 68 x 10-3 8370 750-950 Klinedinst & Stevenson (65) O-In 8 22 x 10-4 12600 750-950 Klinedinst & Stevenson (65) O-Ni --- --- 1550-1750 Ershov & Kasatkin (42) O-Ni --- --- 1500 Otsuka & Kozuka (93) *O-Pb 6 32 x 10-5 3580 700-900 Arcella (4) O-Pb --- --- 750 Bandyopadhyay & Ray (9) *O-Pb 9 65 x 10-5 4800 BOO-1100 Honma et al. (60) O-Pb --- --- 800 Kawakami * Goto (63) O-Pb 1 90 x 10-3 5000 800-1100 Osterwald & Charle (86) O-Pb 1 48 x 10-3 4660 800-1135 Otsuka & Kozuka (90) 26 TABLE 1 (Continued) DO 0 Temp. Range System (cmZ/sec) (cal/mol) (°C) Researcher(s) O-Pb 1.44 x 10-3 6200 740-1080 Szwarc et al. (123) O-Sb 3.07 x 10-4 938 697-857 Fitzner (50) O-Sn 6.4 x 10'4 4600 700-930 Otsuka & Kozuka (92) O-Sn 9.9 x 10'4 6300 784-1000 Ramanaryanan & Rapp (101) 27 Oxygen-Liquid Metal Systems——For O-Ag. the data are fairly uniform: moreover. D values calculated from Equation (34) are larger than those expected from any theory. Oberg et al. point out possible problems with the experiments of Rickert and El-Miligy. For O—Cu. the El-Naggar and Parlee and Gerlach et al. Q values are larger than the others. Oberg et al. point out possible problems with the experiments of Osterwald and Schwarzlose and Rickert and El-Miligy. The data Of Shurygin and Kryuk do not compare with those of other researchers. For O-Fe. the data of Suzuki and Mori do not compare with those of other researchers. The data Of Ershov and Bychev and Novokhatskii and Ershov are reasonable. but Ershov and Bychev used a narrow temperature range and Novokhatskii and Ershov took data at only three temperatures. For O-Ga and O-In. identical temperatures and temperature ranges were used. but the diffusion coefficient is much larger in gallium. Hahn and Stevenson noted that one possible explanation for this difference is the anomolous structure (from X—ray diffraction studies) of gallium. For most simple liquid metals with normal (i.e.. random) structure (e.g.. alkali metals. copper. gold. indium. and silver). the X-ray diffraction pattern gives a sharp symmetrical first peak. For some liquid metals (e.g.. antimony. bismuth. gallium. germanium. lead. mercury. and zinc). the first peak is not symmetrical. but has a shoulder on the high-angle side (Wilson. 1965). This indicates some form of bonding or other interaction between particles. which lends credence to a cluster model for diffusion. such as that proposed by Egelstaff. For O-Pb. Otsuka and Kozuka point out possible problems with the experiments of Arcella and Honma et al. 28 3.2. Comparison of Experimental Data With Predictions From Hydro- dynamical Theory Table 2 compares experimental data with predictions from hydro- dynamical theory. In this table. DEX is the experimental value (from Equation (34)). DH is given by Equation (6). and Dfi is given by Equation (7). Solvent viscosity values were taken from literature (Elliott and Gleiser. 1960) or calculated from a corresponding-states correlation (Chapman. 1966). Hydrogen-Liquid Metal Systems--Except for H—Li. DEX values are very much larger than DH and Dfi values. Nitrogen—Liquid Metal Systems--For N-Fe. DEX values compare very favorably with D; values. Oxygen-Liquid Metal Systems-—As also seen in Figure 7 for O-Ag. DEx values are about twice as large as DH values and about four times as large as Di values. For 0-Sn. DEX values are smaller than DH and DH values. 3.3. Comparison Of Experimental Data With Predictions From Absolute Reaction-Rate Theory Table 3 compares experimental data with predictions from absolute reaction-rate theory. In this table. DEX is the experimental value DMR' and DSS (which will be referred (from Equation (34)): DR' 0’ D R' MR' to collectively as DART) are given by Equations (14). (17). (15). (18). and (16). respectively. The Goldschmidt radius is used for rB because it is nearly identical with the covalent radius and is readily available in the literature (Chapman. 1966). Molar volumes were taken from the literature (Elliott and Gleiser. 1960: Protopapas and Parlee. 1974. 1976). 29 TABLE 2 Comparison Of experimental data with the hydrodynamical theory. An asterisk indicates quesitonable data. Number in parentheses indicates reference. DEX x 10-5 OH x 10-5 Ba x 10--5 System t(’C) (cm2/sec) (cmZ/sec) (cmZ/sec) Researcher(s) H-Cu 1200 314 4.21 15.5 Chernega & Vashchenko (21) 524 Sacris & Parlee (106) * 68.7 Sigrist et al. (111) H-Cu 1300 549 4.98 18.4 Sacris & Parlee (106) * 78.0 Sigrist et al. (111) *H-Fe 1600 355 2.81 10.3 Arkharov et al. (6) 144 Depuydt & Parlee (27) 132 El-Tayeb & Parlee (37) * 15.0 Nyquist (83) 85.4 Solar & Guthrie (112) H-Fe 1700 138 3.22 11.8 El-Tayeb & Parlee (37) 90.3 Solar & Guthrie (112) *H-Li 700 3.15 38.7 150 Alire (3) H-Li 800 10.5 46.8 183 Alire (3) N-Fe 1600 5.57 2.25 5.14 Arkarov et al. (5) 5.66 Bogdanov et al. (14) * 673 Chesnokov et al. (23) 5.57 Ershov & Kovalenko (43) * 3.67 Lee & Parlee (74) N-Fe 1700 6.47 2.58 5.93 Arkarov et al. (5) 6.90 Bogdanov et al. (14) 6.47 Ershov & Kovalenko (43) N-Ni 1500 1020 2.44 5.58 Chesnokov & Linchevskii (22) N-Ni 1600 1127 2.99 6.87 Chesnokov & Linchevskii (22) 30 TABLE 2 (Continued) DEX x 10'5 OH x 10'5 Dfi x 10'5 System t(°C) (cmZ/sec) (cmZ/sec) (cm2/sec) Researcher(s) O-Ag 1000 10.3 2.57 5.49 Masson & Whiteway (77) 8.88 Mizikar et al. (79) 10.5 Oberg et al. (85) 9.54 Otsuka et al. (88) 8.88 Parlee & Zeibel (94) * 14.7 Rickert & El-Miligy (105) 9.62 SanO et al. (107) 9.68 Shah & Parlee (109) O-Ag 1100 13.8 3.18 6.84 Masson & Whiteway (77) 10.9 Mizikar et al. (79) 13.7 Oberg et al. (85) 11.8 Otsuka et al. (88) 10.9 Parlee & Zeibel (94) * 18.1 Rickert & El-Miligy (105) 12.4 Sano et al. (107) 12.2 Shah & Parlee (109) O-Ag 1200 17.5 3.86 8.34 Masson & Whiteway (77) 13.0 Mizikar et al. (79) 16.4 Oberg et al. (85) 13.0 Parlee & Zeibel (94) * 21.7 Rickert & El-Miligy (105) 15.4 Sano et al. (107) 14.8 Shah & Parlee (109) *O-Cu 1100 2.28 2.98 6.35 El-Naggar & Parlee (36) * 5.73 Gerlach et al. (52) 6.10 Oberg et al. (85) *O-Cu 8.48 Osterwald & Schwarzlose 87 * 6.22 Rickert & El—Miligy (105) * 0.73 Shurygin & Kruyuk (110) 31 TABLE 2 (Continued) DEX x 10-5 OH x 10-5 0g x 10-5 System t(°C) (cmZ/sec) (cm2/sec) (cmzlsec) Researcher(s) *O-Cu 1200 3.37 3.61 7.72 El-Naggar & Parlee (36) * 8.68 Gerlach et al. (52) 8.41 Oberg et al. (85) * 10.7 Osterwald & Schwarzlose (87) 9.78 Otsuka & Kozuka (91) * 8.91 Rickert & El-Miligy (105) * 0.90 Shurygin & Kryuk (110) *O-Cu 1300 4.74 4.27 9.18 El-Naggar & Parlee (36) * 12.5 Gerlach et al. (52) 11.1 Oberg et al. (85) * 12.2 Osterwald & Schwarzlose (87) 12.7 Otsuka & Kozuka (91) * 1.08 Shurygin & Kryuk (110) *O-Fe 1600 34.2 2.41 5.14 Ershov & Bychev (41) * 13.3 Novokhatskii & Ershov (82) * 2.96 Suzuki & Mori (119) *O-Pb 900 1.36 8.22 18.4 Arcella (4) * 1.23 Honma et al. (60) 22.2 Osterwald & Charle (86) 20.0 Otsuka & Kozuka (90) 10.1 Szwarc et al. (123) *O-Pb 1000 1.45 9.60 21.7 Honma et al. (60) 26.3 Osterwald & Charle (86) 23.5 Otsuka & Kozuka (90) 12.4 Szwarc et al. 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