2424
F. W. HINZNER A N D D. A. STEVEK-SON
Acknowledgments.-We wish to express our deepest appreciation to Professor Leo Brewer for fruitful discussion aiid encouragement of this work. The authors
Vol. 67
are grateful to the United States Army Research Office (Durham) and the Research Corporation for financial support.
KINETICS OF SOLUTION I N LIQUID METALS. SOLUTION RATE OF ZINC, SILVER, AND TIN INTO LIQUID MERCURY BY F. W. HIKZNER AND D. A. STEVENSON Department of Jfaterials Sczence, Stanford University, Sian,foold, Calzjornza Received Afay 7 , 1963 The solution rates of solid polycrystalline zinc, silver, and tin into liquid mercury were studied undcr turbulent flow conditions (rotating solid cylinders with and without simultaneous ultrasonic fields) and a t different temperatures using radioactive tracers t o follow the reaction. In addition, zinc single crystals and a zinc-mercury intermediate phase mere studied to gain further insight into the sulution mechanism. The solution rate constant was constant when ultrasonics were applied but varied with concentration under pure rotation for all the systems studied. The variation could not be explained by the dependence of diffusion coefficients on concentration. The solution rate constants were related to Reynolds and Schmidt numbers using dimensionless correlations. The exponents on the Reynolds and Schmidt groups were found to be consistent with transport control except in the zinc-mercury system, which appeared to be mixed controlled. Experiments performed on single crystals of zinc showed a more rapid rate of solution for planes perpendicular to the basal plane than for the more densely packed basal planes. The solution rate of zinc from the ?-intermediate phase (46 wt. y6 Zn-54 wt, % Hg) was more rapid than from pure zinc, indicating that the formation of the intermediate phase would not act as a solution barrier for zinc. The viscosity compensated activation energy agreed, within experimental error, with the literature values for the rorresponding diffusion activation energies. I t was concluded that the solution rate of zinc into mercury is mixed controlled while the solution rates of tin and silver into mercury are transport controlled for the range of experimental conditions investigated.
Introduction The kinetics of solution of solids in aqueous and organic systems have been studied quite extensi~elyl-~ but only recently have similar studies been made in liquid metal ~ y s t e r n s . ~ -I ~t ~is unclear whether the solutioiz mechanisms are similar and whether the rate equations used in aqueous and organic systems describe the behavior in metallic systems. It is the purpose of this study to evaluate the applicability of existing rate equations and investigate the rate controlling steps in the solution process in simple liquid metal systems. Solution Rate Equations.-The solution of a pure component consists of two steps iii series: (1) surface solvation aiid (2) transport of the solvated species from the interface. The most general solution rate equation, the Berthoud equation, expresses the rate of approach t o saturation as12
(1) C. 1 '. Kine, T i i n s . iY. Y . Acad. Sei., 10, 262 (1948). (2) M . Eisenberi., C. W. Tohias, a n d C. It. Wilke, Chom. Eng. P r o p . Sump. S e r . , 61, 1 (185.5). (3) D. W. van Krevclm and J. T. C . Krekcls, Rec. trar. chim., 67, 512 (IR58). (4) 31. Davion, Ann. chim. (Paris), 8, 259 (19.53). P k y . ~Colloid . Chcm., 63, 1030 (1948). (6) A. R. Cooper, Ju., and W. D. Hingery, J . Phys. Chcm., 66, 665 Ks).13 For cases of transport control, Kin reduces to D/6,. The equation in this form is known as the Xoyes-Nernst equation1* and ha,s been found to apply to a broad range of experimental conditions. The solution rate constant niay be evaluated by plotting In (1 - C/C,) VS. (A/V)t. The linear nature of this plot in aqueous systems indicates a constant value of Knz. I n some metal systems, however, nonlinear plots have been observed a t higher values of C/CS.l0 Inasmuch as the experimental precision of the quantity hi (1 - C/C& decreases as saturation is approached, an evaluation of t'his region requires refined analytical techniques. In order to document the nonlinear region and contribute to its explanation, a radioactive txacer technique was developed for precise determination of solution rates in liquid metal systems. Rate Controlling Steps.-A number of criteria have been employed by previous investigators to dist'inguish (131 For a general discussion of the rate equations, of. (a) E. E. Ruckley, "Crystal Growth," John Wileyand Sons, Inc., New York, N. Y., 1 9 6 1 , ~147; . (b) I?. A. Moelmyn-Hughec, "The Kinetics of Reaptions in Solution," Oxford University Press, London, 1S47, p. 367. (14) (a) A. A. Noyes a n d W.R. Whitney, Z. p h y s i k . Chem. ILeipzig), 23, 689 (1892); (b) W. Nernst, ibid., 47, 52 (1904).
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KINETICS OF SOLUTION IN LIQCIDnIETALS
Nov., 1963
the cases of transport coiitrol, interface control, and mixed control. These criteria are summarized in subsequent sections Influence of Hydrodynamic Conditions.-Interface controlled solution rate should be independent of hydrodynamic conditions, whereas a transport or mixed control solution process should depend on hydrodynamic conditions. The relative motion of the fluid and solid influences the thickness of 6, and this influeiice has been analyzed for simple geometries.16 In complex processes such as fluid flow, hestt transfer, and mass transfer, useful phenomeiiological relations have been developed using dimensionless groups of variables. The relevant relation for the present investigation is given as
A
H
I
\$here a , a, aud b are dimensionless numbers, dg is the relevant dimension, ]Re and Sc are the Reynolds and Schmidt numbers, and the other symbols ha\-e their usual meaning.' lo For transport coiitrolled processes, the exponent on the Reynolds number varies between 0.6 and 0.8, and the exponent on the Schmidt number between -0.6 and -O.8.l-l1 The exponent on the Reynolds iiurnber would be 0.00 for interface control, and some value between 0.0 and 0.6 for mixed control. Activation Energy. -The activation energy for the solution rate constant is determined from an Arrhenius plot. The activation energy is then compared with the activation energy for diffusion and with interfacial energies.8 Etching Effect.-The appearance of a uniformly bright surface after solution has occurred is interpreted as evidence for transport control, whereas an etched surface is considered evidence for interface or mixed control. l7 General Aims and Approach.-The clearest interpretation may be made for solution kinetic studies which are done with high precision over wide ranges of hydrodynamic conditions and temperatures. It is also essential to obtain adequate concentration-time resolution so that a variation in Km may be observed. Many previous investigations have determined average Kin values o n l ~ ~ . The ~ ' most common means of varying the hydrodynamic conditions for such studies has involved rotating solid cylinders in a static fluid and restricting the solution process either to the base of the cylinder or to the periphery. Although the hydrodynamics of flow about the base of the cylinder is understood better for the transition region from laminar to turbulent flow, the flow about the periphery has the advantage that the relative velocity of solid and liquid is constant over the entire peripheral area and the niomentuni boundary layer at high Reynolds numbers may be considered constant over this area. Equipnieiit was designed ab described in the Experimental section to permit variation of temperature and hydrodynamic conditions over wide limits and to be compatible with tracer analysis. The solution rate of solid metals into (15) L L. B ~ r c u m s h a na n d A. C Rlddlford, Quart Ber (London), 6, 157 (19 52). ( 1 6 ) A S Foust, L 4. Wen/rl, C W Clump L LTaup, and L B 4ndersen, "Prinriples of Unit Operatisns, John Wilej and Sons, Inc Neu York, N. I ' , 1960, Chapter 13 (17) J. AI Lommel and W. Chalmers, Tions. A I M E , 215, 499 (1959)
TO CHUCK OF DRILL PRESS
R FOR CUUM-TIGHT BEARING
I/
\~---I~L-RUBBER
STOPPER
STEEL SHAFT COUNTING TUBE
RADIOACTIVE SPECIMEN MOLYBDENUM CAP MERCURY
Fig. 1.-Solut8ion cell for experiments using rotation only.
liquid mercury mas studied over a wide range of hydrodynamic conditions by the rotation of solid samples with and without concurrent ultrasonic fields. The results were then compared with existing rate equations and the rate controlling steps implied from the experimental information. Systems Studied.-The systems zinc-mercury, silvermercury, and tin-mercury were selected for study for the followiiig reasons: (1) limited solubility of the solid in the liquid and consequently small changes in dimension during solution; (2) the absence of intermediate phases iiivolving solute and solvent; and ( 3 ) a suitable isotope for the solute-a 7-emitter with energy greater than 0.5 Slev. and a half-life greater than one month. I n addition to these systems, zinc single crystals and a zinc-mercury intermediate phase were studied for reasons given in the discussion. Experimental Apparatus and Procedure.--The solution cell used in these studies is shown in Fig. 1. Determination of the solution rate was carried out by immersing the rotating specimen into mercury for an appropriate period of time, removing the specimen, and transferring the liquid to the counting tube to measure the activity of the liquid metal. The transfer of the niercury to the counting tube was carried out iii argon by detaching the cell from the drill press and tilting the entire cell. Since the amount of solute dissolved in mercury wa8 low, the fluidity of mercury was excellent and no dificulty was encountered in effecting complete transfer t o and from the counting tube. The process was repeated until saturation was achieved. The entire experiment was performed in an argon atmosphere and the temperature controlled by immersing the solution cell in a thermostatic bath regulated to &0.5". The rotational velocity of the specimen was varied and the rotational speed, once set, fluctuated less than 1%. A solution cell was designed l o permit experiments involving simultaneous rotation and ultrasonic vibration and was essentially the same as that discussed above with added provisions for acoustical contact between the liquid and the vibrating plate of the ultrasonic transducer. The temperature during the latter
F. W. HIKZKER ASD D. A. STEVENSON
2426
A
(I -2) 0%;
4,000 R PM 4,000 R PM
I
004
of mercury was counted. Approximately 3000 counts were taken in each measurement corresponding to a standard deviation of 1.8%. I n order to avoid high background interference, the base line was set a t the minimum prior to the 1.12-Mev. peak of zinc-65. For silver and tin, the base line was set at 0.58 and 0.34 Mev., respectively Materials.-Zinc (99.98%) and tin (99.967,) were acquired from the Baker Chemical Co., and si1 r (99.99%) from Western Gold and Platinum Co., Belmont, Cal The radioactive metals were prepared by irradiation of ti small specimen in the Stanford reactor and diluting and homogenizing the resulting high activity material by melting under hydrogen in a graphite crucible. Half-inch diameter cylindrical rods chill cast and half-inch long specimens prepared. The sp n was mounted on the rotating shaft of identical diameter aqd the bottom face covered with a molybdenum cap. I n this way only the sides of the cylinder were exposed to the mercury. The specimen was pre-amalgamated by cleaning it in hydrochloric acid, dipping it in mercury, and then washing it with distilled water in order to prevent a delay in the solution process due to a surface barrier film. Zinc single crystals were grown by a modified Bridgeman technique to study the influence of crystallographic orientation on the solution rate. Sections of radioactive zinc single crystals were mounted on the periphery of cylindrical epoxy resin holders in such a wayithat either faces perpendicular or parallel to the basal plane ofthe zinc single crystal were exposed to the mercury.
1,300 00RP RM PM
~~
200 R PM
I
45QC
0
a 3oc.
I
I
I
200
Fig. 2.-Plot
of In ( I
I
I
300
IAN)t
- C/C,) as a function of (9/ V ) t ,Zn-Hg
sys-
tem. I
r
\
/
I
,4,OOOR PM 00 R P M 300RPM P 2 0 0 PM
i
11 /1 I
O ' r
0 4 c
A 45oc. 0 30"C.
IkYl
, 0
Fig. 3.-Plot
200
400
Vol. 67
(AIVlt.
of In ( 1 - C/C,) as a function of ( A / V ) t ,Zn-Hg system under ultrasonic vibrations.
experiments was regulated t o &lo, A 20-kc. magnetostrictive transducer driven by a 400-w. GU 400 General Ultrasonic Co. signal generator (10 kc. to 1.2 Mc.) provided the ultrasonic field. Analytical.-The andytical system was an integral part of the solution cell. The solute concentration was determined by measuring the activity of the mercury solution. All the isotopes selected (Zne5, Ag110, Sn113)were adequate 7-emitters to permit use of a Baird Atomics Co. Model 8100 ?-ray spectrometer. The counting tube of the solution cell containing mercury with radioactive solute was inserted in the well of the counting crystals. This method gave a counting efficiency of approximately 60%, and the geometry was reproducible since the entire volume
Experimental Results Zinc-Mercury System.-The zinc-mercury system mas the most extensively studied. Solution kinetic data for this system were obtained for four different temperatures: 3, 30, 45, and 59". At each temperature, data were obtained for three different rotations: 200, 1300, and 4000 r.p.m., a total of 12 combinations of temperatures and rotational speeds. These rotational speeds correspond to surface velocities of 12, 85, and 250 cm. /sec., respectively. The zinc-mercury system was also studied under conditions of simultaneous rotation and ultrasonic fields (20 kc./sec.) in five conibinations of temperature and hydrodynamic conditions. For all the systems studied, a minimum of two sets of data were obtained for each experimental condition in order to establish the reliability of the data. The tracer techniques developed allowed K m values to be obtained with a precision of 2% up to C/C, values of 0.90. Systematic errors are not expected to exceed 27, and a niaximuln error of 491, is predicted for this range of relative concentration. When ultrasonic fields were applied, it was found necessary to establish conditions so that cavitation occurred ; otherwise no appreciable effect was noted. I n all the studies involving ultrasonics, cavitation was oocurring. Figure 2 shows a graph of In (1 - C/C,) us. ( A / V ) t , according to eq. 2, for a few typical experiments for pure rotation, while Fig. 3 gives the same curve for ultrasonic vibrations and concurrent rotation. In evaluating the quantity ( A / V ) t ,any changes in A or V due to solution were considered. X similar graph is given in Fig. 4 for the solution of zinc single crystals into mercury a t 30" wit4 rotational speeds of 800 and 4000 r.p.m. Figure 4 also shows the solution kinetics of an intermediate phase (46y0Zri-54% Hg by weight) into mercury a t a temperature of 30" with rotational speeds of 1300 and 4000 r.p.ni. Silver-Mercury and Tin-Mercury Systems.-Sohtion kinetic data for these systems have been obtained for two different temperatures and four different hydrodynamic conditions. Because of the low solubility of (18) The complete experimental d a h in tabular form w1ll be turnished on request.
KISETICSOF SOLUTION IN LIQUID METALS
Xov., 1963
2427
silver in mercury, the experiments nere performed a t the temperatures of 59 and 85". The rotational speeds were 800 and 4000 r.p.m. both with and without simultaneous ultrasonic fields. These speeds correspond to surface velocities of 50 and 250 cm./sec. 4 plot of In (1 - C/C,) us. ( A / V ) tfor the silver-mercury system for rotation oiily is shown in Fig. 5, and the same plot a t 59 O under simultaneous rotation aiid ultrasonic vibrations is shown ~nFig. 6. In the tin-mercury system, the experiments were done at 3 and 30" a t the same velocities as in the last system and studies with Simultaneous rotation and ultrasonic vibration were made a t 30". Figure 7 gives a plot of In (1 - C/C,) us. ( A l V ) tfor the tin-mercury system for rotation only, and Fig. 6 gives the same plot at 30 O under concurrent rotatioii and ultrasonic vibrations.
Discussion The dependence of solution rate on relocity is demoiistrated by curves of In (1 - C/C,) us. ( A / V ) t ,as in Fig. 2-6, the slope of these curves being equal to Km according to the Berthoud equation. These curves shorn that K m is not constant for cases of pure rotation, in agreement with earlier studies in the lead-nickel and leadcopper systems.10 ]For rotatioii with concurrent ultrasonic fields, however, the linear relation between ln (1 C/Cs) and ( A / V ) t I S obeyed as seen in Fig. 3 and 6. The former behavior could be explained by a variation in D or 6, with time (or concentration) or lack of interface equilibrium. Reference to literature D-values for zinc and tin in liquid mercury shows that D decreases less than 5 aiid 10% for the respective systems over the concentration ranges in q u e ~ t i o n . ' ~ - ~This l decrease is insufficieiit to explain the decrease in Km with increasing conceiitration. The possibility of 6, changing with concentration was considered. 2 2 A mathematical analysis of the solution process was made assuming: that 6, increases with Lime; a constant value of D over the concentration range in question; and Couette flow (linear variation of fluid velocity with distance from the solid interface) within the range of the concentration boundary layer. For systems in which the momentum boundary layer 6, is significantly greater than 6,, such as the present systems with Sc >> 1, this flow condition is reasonable. The solution of the resulting equation led to the relationz2 111
(1 - (7/C,)
=
--K'(A/V)t2'8
(4)
Although this rate law provides significant improvement in the empirical representation of the experimental information (as seen, ftor example, in Fig. 8)) subsequent estimates of the moineiitum boundary layer and concentration boundary layer were madez3and the time required for development of the concentration boundary layer calculated to be less than 1 see. This time is such a small portion of the entire dissolution experiment that the transient behavior may be neglected ; consequently, eq. 4 must be considered empirical. The main portion of the nonlinearity is probably due to lack of interface (19) H M' Schadler and R E Grace, 77anq .41.1.11' 816,659 (1959 ) ( 2 0 ) hI 1%.Narhtneb and J Pet t , J Chem. Phyq , 24, 746 (1956) (21) UT. C. Cbovrr and N. H F u r m a n , J 4m Chem Soc 74, 6183 (1952). (22) F. Hinmer, Ph.D. T h e m Stanford Univerqitj, 1962 (23) R J. Donnelly and N J. Slmon, J Fluid M e ~ h 7 , 401 (1000)
Fig. 4.-Plot of In (1 - C/C,) as a function of ( A / V ) t ,Zn single crystals-Hg and ?-intermediate phase-Hg systems (30').
C
800RPM
04
06
n
85OC
0
53°C
08 I
0
Fig. 5.-Plot
200
of In (1
-
4 00
C/C,) aa a function of ( A / V ) t ,Ag-Hg system.
equilibrium. A detailed explanation of this behavior is not yet apparent and may result from a number of causes. The zinc-mercury phase diagram shows a peritectjc reaction a t 43' resulting in the formation of an intermediate Below 43' this phase is stable in the concentration range from 44 to 48 wt. % of zinc. This intermediate phase could form a t temperatures below 43' by reaction of the mercury with the zinc sample. If this phase goes into solution into mercury more slowly than zinc, it could act as a barrier aiid decrease the solution rate. Since the flux of mass away from the interface is smaller toward the end of the solution process, the formation of the intermediate (24) hl. Hansen and K Anderko, "Constitution of Binary Alloys," AleGrau-Hi11 Book Co,, In,., Ne\% York, K. Y . , 1958.
F. W. HINZNER AND D. A. STEVENSON
2428
A A
Ag,5S0C C. Sn, 0°C Sn, 33OOC.
400
(A/V)t.
0
200
Vol. 67
I
Fig. 6.-plot of In (1 - C/C,) as a function of (AIVjt,Ag-Hgand Sn-Hg systems under ultrasonic vibrations. Fig. 8.-Plot
Fig. 7.-Plot
of In ( I - C/C,) as a function of ( A / V ) t ,Sn-Hg system.
phase by back diffusion of mercury Lvould more likely occur in the latter stages of solution. If this were the case, the decrease in the solution rate constant toward the end of the experiment might be explained by the formation of the intermediate phase. In order to experimentally prove or disprove this possibility, the intermediate phase was prepared and its solution rate into mercury was studied. The reaction mas followed by observing the increase in activity of ZnG5in the liquid amalgam. The results are shown as a graph of In (1 - C/Cs) us. ( A / V ) tin Fig. 4. Comparing this result with pure zinc under the same conditions it is seen that the solution
of In (1 - C/C,) as a function of ( A / V ) t 2 / 3Zn-Hg , system.
rate of the intermediate phase into mercury is faster than the solution rate of zinc into mercury. Additional studies on the diffusion of mercury into zinc indicate that the time required for conducting the entire experiment is shorter than the time necessary for the formation of the intermediate phase; hence it would go iiito solution as soon as it f0rms.~5 It is concluded that the formation of the intermediate phase does not act as a barrier for the solution reaction and does not lower the solution rate. It is of interest to evaluate the criteria for predicting the rate controlliiig step as discussed in the introduction. One of the criteria is the comparison of the dimensionless exponents in eq. 3. The relevant dimension in evaluating Re for the present geometry has been shown to be the gap distance between the sample cylinder and container walls.2 The values of K were taken from the initial slope of the In (1 - C/Cs) us. ( A / V ) t curves. The exponents a and b - 1 in eq. 3 were obtained by appropriate log-log plots of dPKm/v us. Reynolds number and Schmidt numbers, respectively. The following values for exponents are obtained : for zincmercury, a = 0.20, 1 - b = -0.60; for tin-mercury and silver-mercury, a = 0.62, 1 - b = -0.60. These may be compared with the average of 22 previous investigations for systems believed to be transport controlled, namely, a = 0.60; 1 - b = -0.58.22 The exponent of 0.20 Re in the zinc-mercury system shows that K in that system is less dependent on Re than for a normal transport controlled reaction, indicating that the reaction is mixed controlled. The exponents for the tin-mercury and silver-mercury systems comply with transport controlled kinetics. The orientation dependence of the solution rate is seen by referring to Fig. 4, which shows a significantly slower rate for the basal planes in contrast to the faces ( 2 5 ) The diffusion of mercury into zinc was studied a t 298’K. using radioactive mercury, a n d some preliminary results show D t o be of the order of 10-16 cm.2/sec.
Nov., 1963
KINETICS OF SOLUTION IN LIQUID METALS
perpendicular to the basal plane. This is qualitatively explained by tlie fact that nine nearest neighbor bonds in the solid are broken in the former case in coiitrast to seven or eight bonds in the latter case. This correlates with the higher heat of sublimation for the atoms in the former case.26 The demolistrated dependence of the rate of solution both. on orientation and hydrodynamic conditions further substantiates the postulated mixed control. Arrhenius plots for the solution rate group (at a constant Reynolds number of 100,000) were made and corresponding activation energies for the systems evaluated. Previous authors have compared these activation energies directly with the activation energy for diffusion or to the interfacial energy to imply the rate controlling step." In the former case the interpretation is not straightforward since the iiifhence of temperature on both the diffusion coefficient and boundary layer thickness must be considered. The boundary layer thickness depends on viscosity which is temperature dependent. The analysis of the activation energy must therefore consider tlie simultaneous change of D and v with temperature. The solution rate data may be represented by an equation similar to eq. 3 aiid may be rearranged as
K
(l/d,)a(Re)aubD1-b
=
(3
Both mass and momentum transport are thermally activated aiid D and u may be expressed as
D
=
Do exp [ - (AEl/RT) ]
u = voexp(AE2F2,/RT)
(6)
(7)
Combiiiiiig the above equations
or for a constant Reynolds number and d,
I
