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By Gerald Edmunds

The liquidus line on the phase diagram for temperature versus composition of a binary alloy system, representing the boundary between the homogeneous melt and the heterogeneous melt plus solid, besides showing the minimum casting temperature, is useful in determining eutectic compositions and the presence of phase changes such as peritectic transformations and intermetallic compound formation. The liquidus often is determined by the cooling curves of thermal analysis, but supercooling frequently occurs, so that the temperature values are low. An alternative method, the one employed here, consists of determining the composition of samples of the melts from heterogeneous alloys that are at equilibrium. With meticulous attention to fundamentals, this method is capable of yielding values limited in precision only by the temperature control and measurement and analyses. Apparatus By starting, on the one hand, with a supersaturated solution and allowing the excess solute to precipitate, and on the other hand with an undersaturated solution and allowing more of the solute to be dissolved, both processes being carried out at the same temperature, equilibrium may be said to have been approached from both directions; that is, from above and from below. If the analyses show the solutions to contain the same fraction of solute when equilibrium has been approached from above and from below, one is assured that equilibrium has been reached in both determinations. In order to carry out such determinations, it is necessary to have equipment for obtaining and maintaining the proper temperatures; for holding the melt, measuring its temperature, and stirring, sampling and analyzing it. The present work required temperatures from 420° to 900°C. The equipment consisted of: A furnace, which was a vertical alundum tube, 13/4-in. diameter inside and 18 in. long, wound with Nichrome resistance wire, surrounded by a 3-in. i.d. alundum tube wound for 5 in. of length near each end, for compensation of end losses. This muffle was contained in an iron shell with Transite end plates, the intervening space being packed with Sil-O-Cel. The outside dimensions of the furnace were 12 in. in diameter by 181/2 in. in height. Heating current was supplied by three independent variable-voltage transformers (Variac), one connected to each end compensator winding and manually controlled, and one connected to the main winding through a series resistance of 5 ohms. This resistance was short-circuited for

Jan 1, 1944

By Roman Šejnoha, Karel Mazanec

A pronounced tendency for delayed fracture zoas observed in the martensite structure of low-alloy steels in the as-quenched condition. Cracks of predominantly intercrystalline nature nucleated and propagated at stresses substantially lower than the short-titne tensile strength of- the steels, and were observed even in hydrogen-free speciunens. The occurrence of delayed fractures in the as-quenched condition is interpreted as a supevposition of the effect of free dislocations generated during the martensite transforlnation, and of a weakening of the prior austenite grain boundaries caused by dynamic shocks exerted on the boundaries by growing martensite plates. AN important metallurgical problem, connected with the production and use of high-strength steels, is delayed fracture in martensitic structures. Such fractures occur in quenched steels as a result of prolonged loading with external stresses far lower than the short-time tensile strength of the steel. The same effect can be produced by sufficiently high internal stresses: it is well-known that quenched structures will in some cases crack intergranularly at room temperature if not tempered quickly enough even in the absence of any externally applied stress. In Troiano's papers1,2 on delayed fractures in high-strength steels, the fundamental role of hydrogen migrating under the influence of multiaxial stress towards the nuclei of cracks was found. Fig. 1 shows the typical relationship between stress and the time to fracture found by Troiano in hydrogen-charged high-strength steels. On the other hand, Shurakov 3,4 explains the realization of delayed fractures in martensite as due to a "quasi-viscous" behavior of the austenite grain boundaries immediately after the martensitic transformation. 1) MATERIALS AND EXPERIMENTAL METHODS Two low-alloy steels of commercial purity were used for the experiments reported in this paper. Their chemical composition is shown in Table I. The heats were melted in a laboratory furnace, cast to ingots, forged to bars of 3/4 in. diameter, and annealed. Machined test pieces were heat-treated as follows: Ni-Cr-Mo steel held at 900°C for 30 min and quenched in oil; Mn-Si-Cr steel held at 900°C for 30 min and quenched in water. For the delayed-fracture tests, notched cylindrical test pieces, Fig. 2, stressed in torsion, and unnotched cylindrical tensile-test pieces, 1/4 in. diameter, were used. The specimens were normally loaded with a constant static load 45 min after quenching for the torsional tests and 10 min

Jan 1, 1965

By John W. Cahn

The spinodal has long been regarded as a limit beyond which a homogeneous phase could no longer be metastable. But only recently has it become apparent that a phase beyond the spinodal would decompose by simple diffusional clustering mechanism quite different from the nucleation and growth mechanisms encountered for metastable phases. The theory of spinodal decomposition, which is based on the diffusion equation modified by thermodynamic requirements , is phenomenological, and each parameter can be measured by independent thermodynamic or diffusion experiments. The details of the theory have been carefully verified in a number of simple systems. It appears that the spinodal mechanism is commonly encountered in clustering reactions in solids and glasses. HISTORICAL INTRODUCTION The understanding of the earliest stages of precipitation in alloys has been the subject of three prior Institute of Metals Lectures. In 1932 Paul ~erica' reviewed the puzzling evidence that hardening of duralumin is completed before precipitation of CuAlz takes place and suggested that hardening occurs when the copper atoms "assemble in random groups on lattice positions of the duralumin, not of the CuAlz lattice". He called these random groups knots. This modification of his dispersion hardening theory seemed to fit all the facts, but Merica worried that the required diffusion "is just the reverse of what we deem normal; although quite spontaneous, it proceeds uphill, if you please". The reality of Merica's knots was quickly confirmed by Guinier's' and preston's3 X-ray work and these knots are today known as G.P. zones in their honor. Theoretical justification for uphill diffusion was also quickly provided by Becker and Dehlinger~ who noted that the diffusion coefficient could be negative in certain regions of phase diagrams. The boundaries of such regions could be described by a 90-year-old thermodynamic concept first described by ~ibbs' in 1877 as a limit of metastability for fluid phases, which by 1890 had received the name spinodal (see Appendix A). Spinodal in fluids is as old as nucleation theory. Both appear for the first time in the same paper by Gibbs who called the spinodal the limit of metastability. For the first 50 years, nucleation theory was forgotten and the spinodal was extensively treated in every good thermodynamics textbook.81Q The idea was simple and logical and it was accepted as part of a rigorous thermodynamic development of metastable phases, and for a while it was extensively investigated theoretically.

Jan 1, 1969

By A. A. Rauth

In this paper, the author derives a series of formulas from basic principles and illustrates the application of these formulas to practical grinding charge problems. The paper establishes the nearly proportional relationship of mill power to charge level and shows how this relationship can be used to advantage in the design, manufacture and operation of grinding mills. The procedures outlined offer a rapid means of anticipating the power requirements for a given charge or determining makeup charges for a given power reading. The theories which have been developed concerning the dynamics of grinding have been motivated largely by the desire to derive workable formulas for the accurate determination of grinding mill power. This accurate determination has always been handicapped by the large number of variables to be considered, some of which are mill diameter, length, speed, liner design, grinding charge and material density, volumetric mill loading, internal friction, and charge size and gradation. In efforts to determine mill power, attempts have been made to utilize torque concepts. The formulas derived consider the grinding mass as a rigid body and utilize factors that are based on empirical data so that the movement of the charge within the mill is compensated for. These formulas are of principal interest to the mill designer since he is charged with the responsibility of analyzing all of the variables and from them predicting a mill size and a motor size which are compatible both with each other and with the grinding job for which they were intended. The mill operator, on the other hand, has an entirely different problem. The mill dimensions are already established and so is the driving motor. The operator's main concern is to obtain maximum performance from the mill he already has. For him, the many variables which perplex the mill designer are fixed values. Mill diameter, length and speed are set and, to a large extent, so are the conditions inside the mill from which the torque formula advocates derive their empirical data. It is conceivable that if the major mill dimensions which affect power demand are held constant, and if the variables influencing empirical factors are held fairly constant, as they are in an operating mill, then the only major variable remaining which affects power is the level of the charge itself. Under this condition.

Jan 1, 1970

By S. J. Hruska, G. M. Pound

The critical supersaturations for appreciable nucleation rate of cadmium crystals on copper and glass substrates at 186°Kwere measured as a junction of thermal-beam energy over a range of source temperatures between 616" and 839OK. There was no effect of source temperature on critical super saturation. This implies that the adsmbate prior to nucleation is in thermal equilibrium with the substrate. THE deposition of metallic vapors on solid substrates has been found to exhibit a critical behavior in that the rate of formation of deposit increases markedly when the intensity of the vapor source is increased slightly.* Evidence of the ability of sub- where Je is the flux which would arrive if the substrate were equilibrated with bulk condensate at the adsorbate temperature. The following work was undertaken to test this hypothesis. A) Energy Exchange Between Metallic Vapor and Substrates. When a freely translating atom collides with a stationary solid body its energy and momentum undergo changes which determine whether the atom is reflected or adsorbed. An atom is considered to be reflected if it returns to a freely translating state after remaining under the influence of the body for only a time comparable to the period of oscillation of atoms on the surface. An atom which remains for a longer time is said to be in an adsorbed state. The parameter which describes the energy transfer occurring at the interface is the thermal-accommodation coefficient which may be defined in terms of energy as Ei = average energy of impingent atoms, Eeq = average energy of an emergent atom which has equilibrated with the substrate, and Eem = average energy of an emergent atom. If a temperature is assigned to each of these energies, we have at as defined by Knudsen.2 Since this allows no distinction between reflected and evaporated atoms, a more specific definition is introduced for the present purposes, namely 71 = probability that an impingent atom is transferred to an adsorbed state, aj = thermal-accommodation coefficient for reflected atoms, and aTc = thermal-accommodation coefficient for adsorbed atoms. During a time t,, the mean stay time, prior to evaporation the adsorbed atom (adatom) will interact with the substrate and this will have the net effect of transferring the adatom towards a state of

Jan 1, 1964

By A. Chakravarti, C. C. Harris

First-order kinetics adequately describe the recovery of a single floatable species. The conditions under which cell behazior can be simulated by a single-phase model are discussed; deviations from the multiphase model are accountable. The rate coefficient depends upon operating conditions besides particle size and mineralogy and the physicochemical properties of the pulp constituents. The model for a real flotation system combines the first -order, single-phase model with a distribution of noninteracting species. Methods for determining the distributions are discussed. Applications of the procedure to flotation studies are outlined. The mathematical description of flotation as a rate process can be traced to Garcia1 who reported cases in which the fraction of solid remaining in a cell in semi-batch operation, F (for Notation used throughout, see Table l), decays exponentially with experimental time T. F = e-k>r. (1 The differential equation leading to Eq. 1 is dF ----= -k,F (2) dT which is recognized as the algebraic analog of the chemical kinetics first-order reaction equation.' This has led to suggested mechanisms for flotation and mathematical relationships other than first order: with very little foundation other than analogy. The flotation rate coefficient kl can be accounted for as the product of several probabilities concerning bub-ble-particle attachment and detachment, aggregate entry from pulp into froth and return from froth to pulp, and removal from froth as concentrate. These factors clearly contribute to the magnitude of the overall flotation rate, and there are others also, but describing flotation merely as the overall probability of consecutive gross events hardly appears a significant advance in understanding. More recently, it has been described in terms of a stochastic process, which promises to provide a more satisfactory approach? The present paper will be more concerned with a phenomeno-logical description of the flotation process than with an account of the internal mechanics of the process. This is not a statement of priorities, or a judgement of the relative importance of these options but rather an assessment of what is presently possible. Previous investigators'" have developed methods for interpreting semi-batch flotation data in terms of species distribution. These methods have been used in an attempt to explain flotation behavior in response to changes in operating conditions." Inspection of examples given in the literature" reveals discrepancies between theory and experiment, and data obtained in the Colum-

Jan 1, 1971

By Merton C. Flemings, Harold D. Brody

Analyses that include diffusion of solute in the solid phase are formulated to describe solute redistribution in dendritic solidification of metallic alloys. The analyses are based on conditions that include negligible undercooling before nucleation of solid phases, negligible increase of solute in advance of the tips of growing dendrites, complete diffusion within the liquid over distances the order of dendrite spacings, and a plate-like dendrite morphology. The classical nonequilibrium freezing equation, Cs = kCo(I - fS)k-l accurately describes solute redistribution between dendrite arms for the solidification processes considered, provided diffusion in the solid is negligible. To account for the effect of diffusion in the solid, an analytic expression is given which is similar inform to the classical expression. 172 addition, a numerical-analysis procedure is etnployed to examine in more detail the effect of diffusion both during and after solidification. The analyses are intended for application to solidification of castings and ingots to describe a) final solute distribution after solidification and cooling to room temperature (microsegregation) and b) local fraction solid as a function of ternperatcre within the solidifying casting or ingol. SOLUTE redistribution in cellular and dendritic solidification has been discussed quantitatively in recent papers'-3 and in a series of earlier work.4-6 However, none of the analyses developed have permitted close correlation of theory with final solute distribution (microsegregation) observed in castings and ingots. In this paper, discussion is given of approximations which may reasonably be made in mathematical treatment of microsegregation, and analyses are presented based on these approximations. Numerical results are given for Al-Cu alloys. In a second paper the analyses will be compared with exeriment.7 CLASSICAL ANALYSIS The classical quantitative treatment of solute redistribution within a closed "volume element" has been derived repeatedly, for example by Gulliver,4 Scheil,' and Pfann.' This "classical nonequilibrium solidification equation" is written for constant partition ratio: where Cs = interface composition of the solid when the weight-fraction solid within the "volume element" is fs (weight fraction, wt pct); k = equilibrium partition ratio; Co = initial alloy composition within the volume element (weight fraction, wt pct). Assumptions of Eq. [11 are: 1) There is negligible undercooling before nuclea-tion, or from curvature or kinetic effects. 2) There is no mass flow in or out of the volume

Jan 1, 1967

By J. Chipman, G. G. Hatch

T. ROSENQVIST*—It is a pleasure to see the excellent way in which the experimental part of this work has been handled. There seems to be little doubt that the distribution data obtained corresponds most closely to thermodynamic equilibrium under the prevailing reducing conditions, namely equilibrium with graphite and one atmosphere CO pressure. The desulphurization curves in Fig 10 show the same general feature as the curves given by Holbrook and Joseph, but the distribution ratios are from 20 to 40 times greater—undoubtedly due to a closer approach to true equilibrium. In the theoretical discussion, the authors calculate a theoretical distribution (S) ration -jg-. which they find to be about 50 times greater than the experimental. The deviation is so great that the basis for their calculation needs a more thorough examination. The authors base their thermodynamic calculation on free energy expressions where diluted solutions of FeS and CaS are used as standard states. (The activity coefficient in diluted solutions is taken to equal unity.) Such a standard state will change when the nature of the solvent is changed. Taking the free energy of the reaction [FeS] ? (FeS), Eq 2, which is derived from the distribution of sulphur between an iron and a FeO-melt, it is very unlikely that the free energy of this reaction will be the same for a distribution between pig iron and a calcium silicate slag. Therefore a more fundamental basis for the thermodyuamic calculations seems needed, where all thermodynamic equations are referred to unambiguously defined standard states. The most natural standard states for CaO and CaS are the pure solid substances at the same temperature. As standard state for sulphur in iron, pure liquid FeS can be used. This rules out Eq 2 [FeS] ;=s (FeS) because ?F° = 0. The standard equation will then be: FeS, + CaO6 + Cgraph ?Fei + CaS8 + CO. vFo1773 = 25,000 cal It would be more universal and also simpler to refer the escaping tendency of sulphur in liquid iron to the corresponding H2S/H2 ratio which can readily be determined experimentally. As standard state a gas mixture H2S/H2 = 1/1 can be used. (This corresponds at the temperature of liquid iron closely to one atmosphere S2 vapor.) Thus the standard equation for the sulphur reaction can be formulated as follows: H2S0 + CaO3 + Cgraph ?H2o + CaS8 + COg The standard free energy of this reaction has been calculated from the best available data to AF°m3 = —35,000 cal. This gives for the equilibrium constant at 1500°C Now, the solubility of CaS in blast furnace slags has been determined by McCafferey and Oesterle* and corresponds at 1500°C to about 10 pet S (varying somewhat with the composition of the slag.) If the activity of CaS is assumed linear between 0-10 pet as curve 1, (see Fig 11), then acaO = 0.1 (S); (S) being wt. pet sulphur in the slag. For a diluted solution of sulphur in an iron melt saturated with carbon, the ratio H2S/H2 is, according to Kitchener, Bockris and Liberman,f about 0.01 [S], [S] being wt. pet sulphur in iron. Substituting these values in the expression for Kp we find The value 2.103 is only 4 times greater than the experimental coefficient found by Hatch and Chipman, but the value is very sensitive to a small error in AF°. A better agreement with the experimental distribution coefficient can be obtained if one assumes the activity of CaS to run like curve 2 (Fig 11). This (S) will give a lower theoretical W, value, a value which varies with (S) exactly as Hatch and Chipman learned. Such a shape of the activity curve, which corresponds to a positive deviation from Raoult's law, is actually to be expected from the fact that liquid silicate and sulphide phases usually show incomplete miscibility. A closer agreement between experimental and theoretical data can not be expected before we have more complete data for the individual activities of CaS and CaO in the slag. The activities acaS and Ocao referred to the solid phases as standard states, are exact defined quantities contrary to the somewhat undefined expression "free lime," and they are independent of any theory for the constitution of liquid slag. J. CHIPMAN (authors' reply)—The authors wish to thank Mr. Rosenqvist for his very interesting and useful thermodynamic addition. Curve 2 of his figure offers the needed basis for explaining the increase in the ratio (S)/[S] with increasing sulphur content. Attention is called to an error in the printed paper: Fig 2 and 3 are reversed. M. TENENBAUM*—In the figures showing the relationship between excess base and sulphur distribution (Fig 6, 7 and 9) the slope of the curve tapers off in the negative basicity range. Somewhat the same thing is observed with open hearth slags. In that case, the fact that some sulphur distribution between slag and metal is obtained with negative basicity is interpreted as indicating some dissociation of the lime silicate compounds whose existence in oxidizing basic slags has been used to explain various observed phenomena with regard to other slag-metal reactions. In the case of the blast furnace slags, the reduced slope of the sulphur distribution curve with decreasing excess base is attributed to the amphoteric effect of alumina. Has the possibility of other explanations been investigated ?

Jan 1, 1950

By S. A. Mersol, C. T. Lynch, F. W. Vahldiek

The room-temperature microhardness of as-grown and annealed MoaC single crystals was measured on the (0001), {2110), and1012) planes using Knoop and Vickevs indenters at loads ranging front 25 to 1000 g. The orientatimz dependence of hardness with respect to crystal axes was also studied. The average random hardness of as-grown crystals was determined to be 1520 kg per sq nm. Annealing to 2000°C decreased. the average hardness by 150 units. An increase in hardness after annealing at 2200 ;C was noted. Optical and electron microscopy revealed slip and twin traces on all planes studied, as produced by mi-cvohavdness indentations. Basal (0001)(2i10) slip was determined to be the primary slip system and was substantiated by electron transmission microscopy. A secondary {1010)(2110) slip was produced by mi-crohardness indentations. The lattev also produced twinning- of the {10i2)[0001] type, as proven by electron diffraction. Electrical resistivity and elastic-modulus anisotropy were found and correlated with hardness anisotropy and Mo2C crystal structure. Elastic-modulus values were obtained by microhard-uess and ullrasonic methods. Bonding mechanism of Mo2C is discussed. ROOM-TEMPERATURE microhardness indentations are useful for studying hardness anisotropy, slip, and twinning in brittle materials. Slip has previously been produced in this manner in and WS~~.~ Recently, the authors5 reported slip of the {10i0)(11~0) type produced by high pressure and microhardness indentations on hexagonal TiBz (c/a = 1.066) single and polycrystals. This slip system was also reported by French and ~homas' and Taka-hashi and ~reise~ for hexagonal WC (c/a = 0.976) crystals. These results suggest that prismatic rather than basal slip is favored in hexagonal nonmetallic materials having a c/a ratio considerably less than the ideal (1.633). Buerger precession and cylindrical X-ray rotation patterns were previously' taken on cleaved sections of the Mo2C single crystals studied in this work. Th~y were found to be hexagonal MoaC with a. = 3.0233A, co = 4.7344A, and C/O = 1.5660. The latter ratio is close to that of the beryllium metal (c/a = 1.57), which slips primarily on the (0001) plane: but also slips on the (1010) planes.'0 ~irconium" (c/u = 1.59) and titanium12 (c/o = 1.59) deform mainly by slip on nonbasal planes which contain a close-packed direction. This is due to the fact that for these two metals the initial resolved shear stress for slip on the (10i0) prismatic planes is lower than that on the (0001) plane. The prominence of basal rather than prismatic slip in metals of high c/o ratios is shown by cadmium (c/a = 1.89), zinc (c/a = 1.86), and magnesium (c/a = 1.62) which deform mainly by basal slip. However, in case of the latter, by stressing magnesium crystals in tension or compression parallel to the basal plane, slip on (10i0) planes can also be produced.13 Several hardness values for polycrystalline Mo2C are reported in the literature: Biickle ' and Samsonov'~ give a value of 1800 and 1479 kg per sq mm, respectively, at a 100-g load; and Kieffer and Benesovsk~'~ report a value of 1950 kg per sq mm at a 50-g load. ~ott'~ reports a Vickers hardness value of 2000 kg per sq mm, with the load unspecified. A Rockwell A hardness value of RA = 88 has also been reported.'' In the present work, for comparison with single-crystal Mo2C hardness values, a Khnloo value of 1600 It 150 kg per sq mm was found on 99.6 pct pure and 99 pct dense hot-pressed Mo2C. This work was undertaken partly to explain the considerable differences in hardness values reported for polycrystalline Mo2C. EXPERIMENTAL The Mo2C single crystals investigated were prepared by a Verneuil-type process using an electric arc by the Linde Co. of the Union Carbide Corp.lg The largest specimens grown were boules 7 mm in diam by 40 mm in length. The crystals had an average density of 9.04 g per cu cm, with a Mo + C content of 99.8 wt pct. The major impurities were: 100 ppm each Na, Zr, and Ca; 85 ppm 0, 55 ppm Fe, and 10 ppm each Cr and Ta. The crystals were found to be carbon-poor, the average carbon content being 5.73 wt pct (stoichiometric value is 5.89 pct). The molybdenum content was found to be 94.08 pct, which is nearly stoichiometric. Electron-microprobe traverses of selected specimens were done with a Phillips-AMR microanalyzer. Thin-film and carbon replicas were used to prepare electron micrographs. This work was done with a JEM-6A electron microscope. Prior to optical and electron-optical studies, specimens were mounted in Lucite and polished on a vibratory polisher using diamond-paste grades ranging from 9-1 p and Linde A powder for up to 48 hr. Dilute nitric acid was used for thin-section polishing and chemical etching for 1-15 min. Electrical-resistivity measurements at room temperature were taken with a Rubicon bridge, using gold contacts. For hardness measurements, a Tukon Microhardness Tester Type FB with Knoop and Vickers indenters was used. Measurements were taken at loads ranging from 25 to 1000 g; however, the 100-g load was chosen as the standard load. All measurements were taken at room temperature. Indentations of cracking classes 1 and 2 only were considered for hardness determinations.20)21 (There are six cracking classes, ranging from "class 1" for a perfect inden-

Jan 1, 1967

By R. G. Weavind, R. S. Young, I. Wolf

ONE of the most important fields of investigation at the Diamond Research Laboratory in Johannesburg is concerned with improvements in metallurgical practices for the diamond mining companies, with particular reference to increased recoveries of the smaller industrial stones. Considerable work has been done on the possible application of flotation processes to the diamond industry. From the viewpoint of recovery processes, diamonds in their natural state may be grouped into two classes, the water-repellent diamonds, which adhere to a grease surface, and the water-avid diamonds, which, because of their wettability, will not adhere to a grease surface and must be recovered by means other than the grease tables used for water-repellent diamonds. Experiments in the flotation of small diamonds from both kimberlite and alluvial deposits indicate the following conclusions: Water-repellent diamonds: Satisfactory flotation of —16 mesh diamonds, which are naturally water-repellent, may be accomplished using either Aero-float 25 together with cresylic acid, or Du Pont B23 as frother together with kerosine as auxiliary oily conditioner or froth stiffener. The quantities of reagents required are of the order of 0.15 lb per ton Aerofloat and 0.18 lb per ton cresylic acid. The use of acid circuits is detrimental, and for best results a pH between 7 and 9 should be maintained. R. G. WEAVIND and I. WOLF are with the Diamond Research Laboratory, Johannesburg, South Africa, and R. S. YOUNG, Member AIME, is Director of Research, Diamond Research Laboratory. TN 76B. Manuscript, March 23, 1951. A flotation period of 5 to 10 min is necessary to insure that all floatable diamonds are removed. Diamonds larger than 16 mesh cannot be recovered satisfactorily by flotation. Xanthates do not act as collectors for diamonds, and this is probably associated with their inability to impart a water-repellent surface to this mineral. Water-repellent diamonds smaller than 28 mesh can be floated quite well by the addition of a frother alone, such as pine oil or Du Pont B23. Water-avid diamonds: Partial flotation of the smaller sizes of wettable diamonds can be achieved by preconditioning with fatty acid soaps, cationic amines, and similar reagents, which will impart a water-repellent film to diamonds. Washing to remove excess reagent, followed by flotation, will result in a fair recovery, but the latter is much lower than for water-repellent diamonds. Crushed boart: Recovery of crushed boart from the waste products of industrial uses is best accomplished by first conditioning the diamonds present in the mixture with fatty acids, cationic amines, etc., removing the conditioning agent by washing, and floating in a cell of the pneumatic type. Flotation reagents found to be most suitable are Du Pont B23, and Aerofloat 25 together with cresylic acid. The optimum quantities are 1.2 g Du Pont B23, and 0.55 g Aerofloat plus 1.4 g cresylic acid, per liter of pulp in the cell. A solids-to-water ratio of 1 to 4 is found to be the most suitable, and alkaline circuits are necessary for best results. Diamonds larger than 28 mesh are not recovered satisfactorily by this procedure, but for finer sizes flotation offers an excellent procedure for the reclamation of this strategic industrial material.

Jan 1, 1952

By H. L. B. Gould, Jr. Wenny D. H.

TO date there has been but limited interest in alloys of 80 to 95 pct Co and Fe with or without other additions. In 1932, S. R. Williams' reported practically zero magnetostriction for the 90 pct Co-10 pct Fe alloy. This same composition was used for pole pieces to control the magnetic field in magnetrons operating at high temperatures.' In 1947, a 90 pct Co, 9.5 pct Fe, and 0.5 pct Mn composition was investigated3 for use in ballast lamps because of its high temperature coefficient of resistance. Recently, as part of a survey of ductile and malleable magnetic alloys capable of being processed to thin gages for possible applications in memory devices, the high Co-Fe alloys have been re-examined to determine their fabricating and magnetic characteristics. Alloys from 80 up to 95 pct Co and Fe with and without additions such as vanadium were prepared. In general these all had excellent ductility and malleability. Wire was drawn to thin gages without difficulty and fine hard-drawn wires flattened to 2) The phenomenon may have potential in estimating the carbon content of molybdenum. A given solution-treating temperature and cooling rate could perhaps be calibrated so that the area fraction of surface covered by carbide could be interpreted in terms of carbon concentration. By segregating to the surface, a big "magnification" of the carbon content is obtained, which should make the method cheap, direct, and sensitive and, therefore, perhaps of interest to the molybdenum industry. This work was conducted at Battelle Memorial Institute. Financial support by the Aeronautical Systems Division, United States Air Force, under Contract No. AF 33(657)-7376, is gratefully acknowledged, and the writer is thankful for the experimental assistance of Miss Marjorie Cantin. ribbon without splitting. Similarly, strip was cold-rolled as much as 99.75 pct reduction to a 0.25 mil thickness without intermediate anneals. Samples of round and flat wire and foil, in the severely cold-worked and annealed states, were checked for magnetic characteristics. The alloys exhibited widely divergent magnetic characteristics depending on the processing and heat-treating procedures used in preparing the samples. Direct-current hysteresis loops showing the different magnetic characteristics of 90 pct Co, 9.5 pct Fe, and 0.5 pct Mn alloys are shown in Figs. 1 to 8 inclusive, while Figs. 9 and 10 show the loops for an alloy of 82 pct Co, 11.75 pct Fe, 5.6 pct V, and 0.65 pct Mn. Figs. 1 and 2 show the high remanence and exceedingly square-loop characteristics of the hard-drawn wire and ribbon roll flattened therefrom. With the change in processing from roll flattening to flattening by drawing through flat diamond dies the resultant change in texture is accompanied by the drastic change in the shape of the loop as shown in Fig. 3. This skewed loop on annealing reverts to a more normal loop as shown in Fig. 8. The most unique loops are obtained with flat rolled strip. These are shown in Figs. 4 and 5. In the case of the hard-rolled material, Fig. 4, there is little change in induction until the magnetizing force reaches 50 oe. From 50 to 60 oe there is a sharp increase in induction to over 15,000 gauss, which for all practical purposes is saturation. The induction remains at a constant value while the magnetizing force decreases to 40 oe. At this point the

Jan 1, 1965

By F. H. Ellinger

TI HE crystal structure of neodymium has been re-A ported by Quill1 and by Klemm and Bommer"s to be hexagonal-close-packed, Strukturbericht-type A3, with a, = 3.650.4, c, = 5.890.4, c/a = 1.614. Klemm and Bommer,% owever, pointed out that the observed intensities of the lines of the neodymium powder pattern agreed only in part with the intensities calculated for the type A3 structure, and that the neodymium powder pattern showed extra lines which could not be traced to impurities. They found that these extra lines could be accounted for by doubling the c-axis, but apparently they did not consider their evidence to be sufficiently conclusive to justify reporting the doubled c-axis. In the present work, the structure of neodymium was found to be double hexagonal-close-packed. The neodymium metal used in the present investigation was obtained through the courtesy of F. H. Spedding of the Ames Laboratory. Its major impurity was stated to be 0.03 pct Ca. The author's spectrographic analysis indicated high purity and carbon was determined to be less than 30 ppm. X-ray powder specimens were prepared from filings stress relieved in vacuum at 200°C for 50 hr. Powder photographs were taken in an 114.6 mm diam camera (Straumanis film mounting) at room temperature and in a 19 cm diam Unicam high temperature camera at elevated temperatures. Copper x-radia-tion was used for all the photographs. The powder pattern of neodymium was found to index to a hexagonal unit cell with dimensions, as determined from the high angle Ka doublets, of: a.= 3.655+0.001.4; c, = 11.796+0.004.4; and c/a = 3.227 at room temperature. The measured density of neodymium shows that this cell contains 4 atoms, the calculated density is accordingly D, = 7.02 g per cu cm. The diffraction data for the first 15 lines of the powder pattern are listed in Table I. In a double hexagonal-close-packed unit cell, the 4 atoms would be in the positions (0,0,0), (0,0,%), (1/3, 2/3, 1/4), and (2/3,1/3,%) of space group P 6,/mmc. On the basis of these positions, the relative intensities of the reflections were calculated using the expression: 1+cos2 20 I oc \Frp 1 ' sin2 0 cos 0 where F is the structure factor and p is the multi- plicity factor. Since absorption was not taken into consideration, comparison of observed and calculated intensities should be made only for neighboring reflections. Each atom has 12 neighbors, 6 in the basal plane at a distance of 3.655.4 and 3 above and 3 below at the closer distance of 3.626.4. These distances correspond to an average radius of the neodymium atom of 1.820.4. The description of this structure in terms of A, B, and C close-packed layers of atoms is ABAC ABAC. ...... The structure repeats every fourth layer and the stacking sequence of the second, third, and fourth layers corresponds to cubic-close-packing of atoms. A series of powder photographs taken at successively increasing temperatures up to 680a10"C revealed that the double hexagonal-close-packed structure is stable up to this temperature. Acknowledgments The authw wishes to thank W. H. Zachariasen and A. S. Coffinberry for their helpful comments on this work and also Marion Gibbs for measuring the diffraction films. This work was sponsored by the United States Atomic Energy Commission.

Jan 1, 1956

By R. C. Hall

The magnetic anisotropy and magnetostriction of single crystals of alloys between 25 and 59 wt pct Co in Fe have been determined in the disordered and ordered states. The magrzetostriction is large and positive for all alloys in both states of order (up to 150 x 10-6 for A,, and 40 x 10-6 for 111). The magnetic anisotropy becomes zero rzear 41 pct Co for the disordered state. Ordering shifts the zero-anisotropy composition to about 50 pct Co. COBALT-iron alloys have for many years been of interest as magnetic materials because they have unusually high magnetic saturation values.' This interest was increased recently by the development of Supermendur.' Supermendur is an alloy of 49 pct Co-49 pct Fe-2 pct V* which was found by suitable purification and heat treatments, including a magnetic anneal, to possess excellent magnetic properties. These included a square hysteresis loop with a low coercive force and a high magnetization, and a low total loss. Two factors which are important in determining the magnetic hysteresis properties are the basic magnetic properties of magnetostriction and anisotropy. Alloys having zero anisotropy and zero magnetostriction show improved hysteresis properties.3 Since very meager data4-' are available in the literature on the single-crystal anisotropy and magnetostriction of the cobalt-iron alloys, the present program was undertaken to determine these two properties in the disordered and ordered states. EXPERIMENTAL TECHNIQUES Ingots of the cobalt-iron alloys containing from 25 pct to 59 pct Co in iron were vacuum melted from electrolytic iron and cobalt of 99.9 pct purity. The ingots were cooled at a rate of about 1°C per hr through the transformation temperature (950" to 980°C dependent on the alloy content) for the growth of large grains. From these grains, single-crystal disks were fabricated with either the (110) or (100) plane parallel to the faces of the disk. The crystals were given two heat treatments to produce either the ordered state (cooling rate of 20°C per hr from 850 "C) or the disordered state (water quench from 850 °C). Since the iron-cobalt alloys order very rapidly, the water quench may not have produced complete disorder in all cases. Due to the method of growth, the crystals were not perfect. Most of the samples had low-angle boundaries and a few had some small included grains. These imperfections could and probably did affect the accuracy of both the magnetic anisotropy and magnetostriction measurements. Much of the scatter in the data may be attributed to the crystal imperfections. The methods of determining the spontaneous saturation magnetostriction and the anisotropy have been previously described.419110 MAGNETOSTRICTION The magnetostriction in the [loo] and [Ill] directions (hO0 and ?111 respectively) is illustrated in Fig. 1 for the disordered state. Both ?100 and ?111i became large positive values as cobalt was added to iron. The state of order did not have a large effect on the magnetostriction. The effect of order did not appear too consistent in these data. Tentative curves are shown in Fig. 1 for the ordered magnetostriction near the FeCo composition; here ?100 appeared to be lowered and ?111 raised as the degree of order was increased. These results agree qualitatively with the

Jan 1, 1961

By S. Feuerstein

THE influence of environment on the mechanical properties of metals has been extensively studied and documented.' Most investigations have dealt with an interaction between a particular environment and a specific metal surface and the concomitant change in bulk mechanical behavior. Typical contributing media are oxide and metal film:;, electrolytes, and other surface-active agents. In the last category fall controlled gaseous environs (i.e., pressure or vacuum). Most vacuum exposure experiments prior to 1963 were conducted at pressures s>10-6 torr and elevated temperatures, or in such a manner as to prolong the exposure (e.g., by stress-rupture, creep, and fatigue). Little attention has been given to vacuum environment effects on the mechanical behavior of metals subjected to simple uniaxial deformation at room temperature. Test times and temperatures were assumed too ineffective as far as bulk changes were concerned. Further, the contribution of surface effects to the bulk mechanical behavior of metals was considered minimal. Under vacuums of the order of 1.0- 8torr and higher, molecular densities and, consequently, the rate of monolayer formation are such that exposures from seconds to days would be required for complete gas coverage. Thus, atomically clean surfaces created under ultra-high vacuum could conceivably maintain their cleanliness for time intervals orders of magnitude greater than at higher pressures. The influence of this unusual type of environmental surface phenomenon on the short-term bulk mechanical behavior of aluminum was first observed by Krame and Podlaseck in 1963.' kin equivalent effect was reported for magnesium by Williams and Nelson.3 It was concluded from these studies2,3 that vacuum and, consequently, the monolayer formation rate of the oxide, were responsible for the described changes in flow behavior. The presence of an oxide layer impedes dis- I) EXPERIMENTAL DATA: MOLYBDENUM The tensile behavior of molybdenum was investigated by the author4-6 at ambient temperature over a range of pressures from 760 to 10-10 torr. These studies revealed that the flow behavior of the bulk specimens (0.200 in. diam by 2 in. gage length) was independent of pressure on sintered (0.062 mm grain size) and arc-cast polycrystals (0.012 to 0.149 mm grain sizes) and single crystals. Since the specimens were relatively large in size, it was assumed that any flow behavior effect would be surface-limited because of the short-term deformation procedure. Strain-rate-change experiments performed to determine the activation volumes for individual specimens at 760 to 10- 10 torr further indicated that the surface flow contribution was either minimal or zero, contrary to the findings of Shen el a1.7 on aluminum. Although the flow behaviors did not change, specimen ductility did change. For the sintered material at a strain rate of 4.2 x 10-4, specimen ductility was -9 pct higher at l0- 10 torr than at atmospheric pressure (dry nitrogen). A more comprehensive investigations on this same material revealed a distinct pressure dependence, Fig. 1. Intermediate and atmospheric pressure tests were obtained by backfilling the system from l0- 10 torr with extra dry, prepurified nitrogen. Other environments (air and purified argon) were also utilized for the 760 torr experiments; the results were essentially the same as those for dry nitrogen atmospheres. Departure from the fracture strain values obtained at 760 torr takes place in the vicinity of l0-6 torr, as indicated by the scatter band; a noticeable ductility increase is apparent at l0-8 and 10-10 torr.

Jan 1, 1970

By W. L. Kerlie, J. H. Richards

The variation in hydrogen content of basic open-hearth steels during refining and the effect of changing from steam to air atomization were studied. The water content of the furnace atmosphere, the slag basicity, and the degree of oxidation of the slag-metal system largely control the hydrogen in the steel. SMALL amounts of hydrogen have an adverse effect on the ductility of steel and a wide range of hydrogen contents has an adverse effect on the internal soundness of both ferrous and nonferrous castings. This has been of major concern to the metal industry, both manufacturers and fabricators, for many years. As one of the results of continuing investigations of the hydrogen problem, much has been learned about methods of reducing the hydrogen content of liquid metals below the relatively high levels that are normally associated with such phenomena as rising ingots and excessive porosity in sand casting. However, it was not until a reliable and simple method for sampling and analyzing liquid metal was developed that much progress was possible in quantitatively evaluating the effect of steelmaking practice on hydrogen contents at levels below 6 to 8 ppm. Throughout all the work described in the following presentation the Geffner pintube technique was used to obtain samples of liquid metal1 * and the tin-fusion method, developed at Massachusetts Institute of Technology,2 was used to analyze the samples. Before attempting a systematic experimental program for minimizing the hydrogen content of open-hearth steels, it was necessary to determine the variation in hydrogen in the open-hearth bath during the normal processing of a heat. The logs of ten heats are illustrated in Figs. 1 through 7. The number at each point on the graphs refers to the number of pintube samples analyzed for hydrogen. An increase in the hydrogen content of the liquid metal during the early period of refining is apparent in all heats. There is evidence of a decrease in hydrogen later in the refining period, see Fig. 2, for example. Therefore, the existence of a maximum hydrogen content at some intermediate time during the refining period is anticipated. To test for the existence of a maximum, the data illustrated in Figs. 1 to 5 were grouped together so that they could be examined on a collectively comparable basis. The grouping was effected, as illustrated in Fig. 8, by superimposing Figs. 1 to 5 so that the apparent hydrogen maxima coincided at a fixed position, indicated by the zero on the time axis. Also, by vertical adjustment until the graphs appeared to the eye to be superimposed, the hydrogen scale was made to show the change in hydrogen content. That is, a standard amount was subtracted from the hydrogen values for each heat. Therefore, Fig. 8 represents a generalized presentation of the data and, for this reason, cannot be used to predict the actual hydrogen content of the bath during refining, but rather to show the probable rate of change in hydrogen during refining. Statistical examination of the grouped data in Fi.g. 8 shows that the apparent. increase and suhsequent decrease in hydrogen are real and confirms the existence of a maximum in general. The average increase In hydrogen during the early period of refining is between 0.8 and 1.3 ppm per hr, and the

Jan 1, 1958

By Samuel Storchheim

IT was noted that variations existed in the magnetic properties, namely, coercive force, He, and rema-nence, Br, of 18/8 stainless steel wires of the same analysis when these wires were given the same percentage of reductiori of area at different rates of reduction. In order to determine the extent of these variations, a program was initiated whereby commercial 302 and 304 stainless steels were drawn with a wide variation of speeds. Ultimate tensile strengths and specific resistances, as well as the Hc-BT measurements, were determined. If variations of considerable magnitude were found, it was thought that the knowledge accrued from the investigation could probably aid in determining the particular drawing speed required to obtain desired properties for a given specification for a particular analysis of wire. Drawing Procedure Two specimens (set I, No. 302 and No. 304), the chemical analyses of which are given in Table I, were drawn at different speeds, namely, 45 to 3900 ft per min, directly from 0.016 in. diam wire to 0.004 in. diam wire, a 93.7 pct reduction of area. To obtain this high percentage of reduction, the wire at 0.016 in. was softened by annealing at 1150°C. It was then strung in a No. 16CF variable speed Vaughn drawing unit and passed through 13 B.&S. gage dies of successively smaller diameter, as shown in the accompanying table. Following this step, several hundred feet of the wire was drawn, at a particular speed, through the dies and wound upon a spool. Once the sample of wire was drawn at the desired speed, the spool with the finished sample was removed, tagged for later identification, and an empty spool was placed in the machine. Another wire sample was drawn at a different speed, after making the necessary adjustments upon the Vaughn unit. At the greater speeds, several thousand feet of sample were drawn to insure that the wire to be used for testing would be uniform. This was done because the desired speed was not reached instantly; the wire taking a short while to attain the required drawing rate. All speed measurements were made by placing a tachometer against one of the machine's pulleys during the actual drawing of the specimens. 84.4 pct reduction of area. The 0.072 in. diameter wire was also annealed at 1150°C. The process of obtaining samples was the same as described above. The only difference between the two was that the rates involved were not as extensive, set II covering a range of 625 to 1150 ft per min. Tensile tests were carried out for the 0.004 in. diam wires on a horizontal Scott Tensile Tester, while tensile tests for the 0.028 in. diam wires were made on a vertical Scott Tensile Tester. Each value listed in the tables represented the average of five samples with a maximum deviation of approximately 3 pct among the samples. Resistance per foot of the wires was determined by use of standard bridges accurate to the nearest hundreth of an ohm for the 0.004 in. wire, and to the nearest thousandth of an ohm for the 0.028 in. wire. The resistances per foot were then converted to resistivities by use of the following equation: P = RxCM where R is the resistance per ft in ohms per ft; CM, the area in circular mils; and P, the resistivity in ohms—CM. The values listed in the tables likewise

Jan 1, 1952

By A. A. Bauer, H. A. Saller, F. A. Rough, J. R. Doig

among uranium alloy systems are the U-Zr, U-Ti,and U-Mo binary systems, in that an intermediate phase, of limited solubility range, exists in each system. These intermediate phase alloys offer a promising field of investigation, both as binary alloys and as ternary alloy combinations. While a limited amount of work has been done on the binary alloys, none has been done on ternary combinations of these alloys. As a possible prelude to more intensive investigation of these systems, a study of the phase relationships between the 6 phases of the U-Zr and U-Ti systems was undertaken. The solid-state portions of the binary constitutional diagrams for the U-Ti,1 Zr-Ti,2 and U-Zr3 systems are shown in Fig. 1. The solid to liquid transformations are not shown, as these transformations were not studied in the ternary system. The phase designations employed are based upon those shown in the uranium constitutional diagrams. The crystal structures of the various phases are given in Table I. Experimental Materials and Methods The ternary section presented is based upon the results of thermal, metallographic, and X-ray analyses of a series of alloys ranging in composition between the phases of the U-Zr and U-Ti systems (U-74 atomic pct Zr to U-35 atomic pct Ti). After preparation of the alloys, thermal data were obtained. On the basis of these data, specimens were heat treated and examined metallographically. Specimens were then selected from those heat treated for X-ray determination of the phases present in what were considered critical phase regions. Alloy and Specimen Preparation—Alloys for this investigation were prepared from selected as-reduced uranium and crystal-bar zirconium and titanium. For nominal compositions, see Table 11. Charges were arc melted from five to seven times under a helium atmosphere, finally being cast into wire-bar form for fabrication. A zirconium getter button was used to clean the furnace atmosphere prior to melting. The alloys were homogenized 20 hr at 900 °C and furnace cooled. A portion of each bar was then removed for thermal analysis; the remainder was saved for fabrication. Alloys 8 through 14 were placed in steel tubes along with zirconium getter chips and, after flushing with argon, the tubes were welded shut and rolled at 1500 °F to a 50 pct reduction in diam. The remaining alloys were annealed 24 hr at ll00°F to insure transformation of the ? phase to the 8-U2Ti phase and were then hot rolled from a salt bath. Attempts to roll alloys 6 and 7 at 1100 °F were unsuccessful. Following this, attempts were made to roll pieces of alloys 1 through 5, which had cracked rather severely during casting, at 1200 °F, but these efforts were also unsuccessful. These alloys were, therefore, further treated in their cast and homogenized condition. All alloys were pickled to remove oxide formed during rolling. Thermal data were obtained on these alloys by conventional differential thermal analysis at a heating rate of 30C per min, and by a high-speed heating method at rates of approximately 10°C per sec. For the latter method of analysis, a 36-gage Chromel-Alumel thermocouple is spot welded between two slivers of alloy, the sandwich being en-

Jan 1, 1958

By T. J. Whalen, S. M. Kaufman

Surface-tension measurements for liquid Fe-C-Co and Fe-C-Ni alloys were performed by the sesszle-drop technique at 1350o and 1425°C. Cobalt was shown to he more effective than nickel in lowering the surface tension of Fe-C alloys A minimum value for surface tension of 1640 dynes per cm was found in the Fe-C-Co system at a concentration of 1.5 pet C and 12pet Co. Analysis of these measurements in terms of adsorption and desorption behavior indicated that cobalt and nickel lower surface tension in Fe-C alloys by replacing carbon at the liquid-vapor interface. Comparison of results to date for Fe-C liquid alloys has shown that most substitutional-type elements having about the same atomic diameter as iron also have relatively the same influence on surface tension when present in dilute amounts. It has been shown previously that surface-adsorption behavior in liquid Fe-C ternary alloys is related to the manner in which the ternary addition affects the activity of carbon in the bulk solution.' Certain ternary additions, such as chromium, tending to decrease carbon activity cause adsorption of carbon and chromium to the liquid-vapor interface. Additions, such as silicon, increase the carbon activity, resulting in a desorption of carbon and an adsorption of silicon. Both cases were shown to cause a decrease in surface tension due to adsorption. It is the object of this paper, as part of the writers' continuing interest in the effect of alloying additions on the surface tension of Fe-C alloys, to present and discuss surface-tension data on solutions in the ternary systems Fe-C-Co and Fe-C-Ni. Available activity measurements for these systems in the solid state indicate a strong increase in carbon activity when either nickel or cobalt is added to Fe-C alloys.2, 3 EXPERIMENTAL PROCEDURE The alloy compositions used are shown in Table I. These alloys were prepared by vacuum melting in stabilized zirconia crucibles and then vacuum casting into cold-steel molds. The nominal analyses for the alloys used in this investigation are given in Table I. Surface tensions and densities were measured by the sessile-drop method4,5 in conjunction with the tables of Bashforth and bdams.' Specimens were prepared by slicing 1/4-in. cubes from the center of the vacuum-cast ingots and grinding the edges of these cubes until their shapes approximated spheres with flat bottoms. Proper handling techniques, including an acetone rinse immediately before use, assured minimum contamination from sources extraneous to the actual melting. For each experimental run a sample was placed on a BeO plaque and positioned and leveled inside a vacuum induction heating apparatus. The sample and plaque arrangement sat inside a molybdenum tube which acted as the susceptor. Diagrams of this apparatus may be found in the literature.= After heating to temperatures just above the liquidus at pressures of the order of 10-4 to 10- 5 mm Hg, photographs of the drop profile were made using its natural thermal radiation as a light source. Some measurements were made with partial pressures (-1 cm Hg) of a mixture of H2 and He gases. The H2-He atmospheres proved to be necessary at higher nickel contents to retard excessive vaporization losses. Chemical analysis for carbon and the ternary addition was performed for each alloy sample after completion of the experimental run.

Jan 1, 1964

By Donald Ofte, L. J. Wittenberg

The absolute viscosity coefficients and activation energies for viscous flow of bismuth, lead, and zinc are reported. The viscosities were measured in an oscillating cup viscosimeter from nearly 1000°C to within 1 deg of the melting point of zinc; 2 deg of the melting point of lead; and in the supercooled condition of bismuth, 9 deg below the melting point. Two methods of relating the change in the viscosity coeflicient with temperature were compared. From the plot of the logarithm of viscosity (77) as a function of reciprocal absolute temperature (1/T), the equations of the lines which fit the data are: AS a part of the Reactor Fuels and Materials Research program at Mound Laboratory, precise measurements of the physical properties of liquid metals are of interest because the desirability for a low vapor pressure liquid coolant in high-temperature nuclear power plants has focused attention on the properties of the liquid metals.' Also, a clearer understanding of the liquid metal state can be achieved when better measurements are available. For these reasons, the viscosity values of certain liquid metals have been determined. The viscosities of the liquid metals, bismuth, lead, and zinc are measured in an oscillating cup viscosi-meter. In this apparatus the liquid is sealed in a right-circular cylindrical cup which is attached to a torsion fiber so that the system forms a torsion pendulum. The damping effect of the molten metal upon the normal oscillations of the torsion pendulum is a function of the viscosity of the liquid. This method is advantageous because of the small amount of sample required (8 to 10 cc), the ease of remote operation of the apparatus in a vacuum system, and the containment of the liquid metal in a hermetically sealed, nonreactive, tantalum crucible. With the liquid metal sealed in the crucible, loss of liquid by vaporization and chemical reactions with atmospheric gases are prevented. Additionally, an absolute viscosity determination is obtained which avoids the sources of possible error in a relative measurement. The difficulties in a relative method include the repetitious cup calibrations, and the uncertainty inherent in the practice of calibration of the apparatus at room temperature and extrapolation to the high temperatures necessary for the investigation of most liquid metals. The oscillating cup viscosimeter was first described by Meyer in 1891.2 The first complete mathematical treatment offering an absolute measurement of viscosity with this type of apparatus was proposed by Andrade and chiong3 in 1936 for an oscillating sphere filled with a liquid. This solution was modified to include a right circular cylinder by Hopkins and Toye4 in 1950, and R. Roscoe5 in 1958. Roscoe's treatment for the calculation of the viscosity of a liquid at a particular temperature from the logarithmic decrement and period of oscillation of the torsion pendulum was used in this investigation. A detailed description of the method of calculating the viscosity from these measurements has been reported earlier.' This calculation has been programmed for the IBM 704 and IBM 1620 computers for expedition of the otherwise laborious calculations. APPARATUS The cutaway view of the apparatus in Fig. 1 shows the suspension system within the high vacuum enclosure. The low residual gas pressure. maintained between 1 x 10-4 and 5 X 10-6 mm Hg during a determination, avoided both air drag on the pendulum and

Jan 1, 1963

By J. S. Aronofsky, J. P. Heller

This paper presents a mathematical analysis of the fluid mixing which occurs during flow through porous media. The analysis is based on the well-known diffusion equation with mass transfer term. It is pointed out that the use of this equation is justified by a single general assumption which does not specify any particular mechanism for the mixing process. Formulas are given for tracer fluid concentrations for two different boundary and initial conditions. Calculated numerical values compare closely with some published results of experiments in which no viscosity or density difference existed between displaced and displacing fluids. INTRODUCTION A phenomenological theory of the mixing and diffusion process is presented for the flow in a porous medium of two miscible liquids of equal viscosity and density. This is based on the classical diffusion equation which has been used to explain such processes as Brownian motion and heat conduction. More recently, this same differential equation has been derived (in a manner analogous to the work in this paper) for the problem of heat transfer during fluid flow in porous media.' The purpose of this paper is to show that the available experimental evidence justifies the use of this simple diffusion equation. The recent data of Koch and Slobod' and of von Rosenberg' are used to test this diffusion model by comparing concentration curve shapes and calculating "effective diffusion coefficients". The same differential equation for miscible displacement, as well as some experimental confirmation, has been published in Japan by K. Yuhara,4 and recently in this country by Day.5 Yuhara arrives at his con-clusions through analogy with turbulent diffusion, de- scribing the microscopic velocities in terms of "eddy motion" — although unlike the ordinary eddies of hydraulics. Day proceeds directly from Scheidegger's6 concept of the dispersion of velocities. After a very clear description of Scheidegger's theory, he applies it to calculate the motion first of a drop and then of a thin layer of brine into a bed of sand. For the latter case, Day derives and solves the same differential equation as Eq. 1. In our presentation no detailed assumptions are made as to the mechanism of displacement. In fact, several microscopic processes may well be involved—the only general assumption made is that their cumulative microscopic effect may be described by the addition of a second-order term to the differentirl equation of continuity. DERIVATION OF EQUATIONS Let us consider the displacement in one spatial dimension of Liquid A from a sample of porous media by Liquid B. A and B are assumed completely miscible and of the same viscosity and density. Both fluids are also assumed to be incompressible and it is specified that the total flow is the same for any cross section. It is evident that the rate at which Liquid B would be carried across such a section would be given by the product of the fluid velocity, u, and the concentration, C, of B in the fluid mixture assuming the constancy of u and C over the cross section. Since the volume of Fluid B is conserved, this leads to the one-way wave equation - u ?C/?x = ? ?C/?l where x increases in the direction of u—the "Darcy velocity", Q/A (or velocity flux per unit area), and 9 is the fractional porosity. Consider the case where fluid B is displacing Fluid A and the concentration distribution is represented by a step funcLon (with a vertical discontinuity). The solution of Eq. 1 predicts no mixing

Jan 1, 1958