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Physical properties and applications of liquid metals 293 given by Furth’s equation pjdj D Cj 2 exp [Ej/kT]dj where C is a constant, and the average distance being j 2 D 1 0 j 4 exp Ej kT dj 1 0 j 2 exp Ej kT dj Swalin then uses a Maclaurin expansion of the Arrhenius term to obtain the energy of formation of a cavity in the liquid which permits a small jump, to obtain the equation j 2 D 3ZN 0 kT 16H v ˛ and hence using Einstein’s equation for the diffusive movement of a particle moving randomly in three dimensions, as in Brownian motion, D D 1 6 j 2 where 1 D kT h Z with obtained from a transition state theory equation for the movement of a particle over an energy barrier with Z neighbours, results in D D 3Z 2 N 0 k 2 T 2 96H v h˛ The Morse function which is given above was obtained from a study of bonding in gaseous systems, and this part of Swalin’s derivation should prob- ably be replaced with a Lennard–Jones potential as a better approximation. The general idea of a variable diffusion step in liquids which is more nearly akin to diffusion in gases than the earlier treatment, which was based on the notion of vacant sites as in solids, remains as a valuable suggestion. Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom move- ment. These observations have confirmed Swalin’s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation F D 3dÁU 294 Thermochemical Processes: Principles and Models where d is the particle diameter, and U is the (constant) velocity of the particle through the liquid of viscosity Á. This, when combined with a diffusion compo- nent obtained from a random walk description yields the Stokes–Einstein equation for Brownian movement. This calculation was then extended to the movement of atoms in liquids, by substituting the diameter of atoms for the diameter of the particles. D D kT 3Ád The viscosity therefore replaces the restraint on diffusion arising from the interaction of atoms expressed by the Morse potential in Swalin’s treatment. The introduction of molecular dynamical considerations suggests that the the use of the atomic diameter in the Stokes–Einstein equation should be replaced by an expression more accurately reflecting the packing fraction of atoms in liquids, i.e. the volume available to an atom in a close-packed arrangement compared to that which is occupied in a liquid. An average value of this function for liquid metals is about 0.47, corresponding to a ratio of the distance of closest approach of the atoms in a liquid metal to the atomic radius of about 1.55. Each atom must be considered as moving in a ‘cage’ of nearest neighbours which is larger than that afforded by close packing, as in a solid. Thermophysical properties of liquid metals Viscosities of liquid metals The viscosities of liquid metals vary by a factor of about 10 between the ‘empty’ metals, and the ‘full’ metals, and typical values are 0.54 ð 10 2 poise for liquid potassium, and 4.1 ð 10 2 poise for liquid copper, at their respective melting points. Empty metals are those in which the ionic radius is small compared to the metallic radius, and full metals are those in which the ionic radius is approximately the same as the metallic radius. The process was described by Andrade as an activated process following an Arrhenius expression Á D Á 0 exp Q vis /RT poise where Q vis has a value of about 5–25 kJ, and Eyring et al. have suggested that the viscosity is determined by the flow of the ion cores and if the energy for the evaporation of metals E vap is compared with that of viscosity, E vap /Q vis ðr ion /r metal 3 D 3to4 Physical properties and applications of liquid metals 295 A further empirical expression, due to Andrade, for the viscosity of liquid metals at their melting points, which agrees well with experimental data is Á D 5.1 ð10 4 MT M 0.5 V poise where M is the molecular weight in grams, T M is the melting point, and V is the molar volume in cm 3 . A further point to note is that the viscosities of liquid metals are similar to that of water at room temperature, about 10 2 poise, and so useful models of the behaviour of high-temperature processes involving liquid metals can be made easily visible at room temperature using water to substitute for metals, and a suitable substitute for other phases, usually liquid salts or metallurgical slags, which can have up to 10 poise viscosity. In connection with the earlier consideration of diffusion in liquids using the Stokes–Einstein equation, it can be concluded that the temperature dependence of the diffusion coefficient on the temperature should be TexpQ vis /RT according to this equation, if the activation energy for viscous flow is included. Surface energies of liquid metals A number of experimental studies have supplied numerical values for these, using either the classical maximum bubble pressure method, in which the maximum pressure required to form a bubble which just detaches from a cylinder of radius r, immersed in the liquid to a depth x,isgivenby p max D x max g C2 r where is the surface energy, and max is the maximum radius of the bubble just before detachment, or the Rayleigh equation for the oscillation frequency ω in shape of a freely suspended levitated drop of mass m in an electromagnetic field, which is related to the surface energy by D 3 8 mω 2 The resulting data for liquid metals indicate a systematic relationship with the bonding energy of the element, which is reflected in the heat of vaporization H ° vap . Skapski suggested an empirical equation D K H ° vap V 2/3 m where K is a universal constant, and V m is the molar volume, which provides a fair correlation among the data for elements. Because of the relatively high diffusion coefficients in liquids, and the probability of the rapid convection current distribution of solute elements to their equilibrium sites, the surface energies of liquid metals are found to be very sensitive to the presence 296 Thermochemical Processes: Principles and Models of surface active elements, and to be substantially reduced by non-metallic elements such as sulphur and oxygen, in the surrounding atmosphere. Great care must therefore be taken in the control of the composition of the gaseous environment to assure accurate data for liquid metals. Table 10.2 shows some representative results for elements which should be compared with the data for the corresponding solids (Table 10.2). Table 10.2 Surface energies of liquid elements Element Surface energy H liquid vap (mJ m 2 )(kJmol 1 ) Sodium 197 104 Antimony 371 244 Bismuth 382 198 Lead 457 190 Indium 556 237 Magnesium 577 138 Germanium 607 325 Gallium 711 267 Silicon 775 400 Zinc 789 123 Aluminium 871 320 Silver 925 274 Copper 1330 326 Uranium 1552 522 Iron 1862 401 Cobalt 1881 408 It will be observed that the surface energy is also approximately proportional to the melting point. Surface energies of liquid iron containing oxygen or sulphur in solution yield surface energies approximately one-half of that of the pure metal at a concentration of only 0.15 atom per cent, thus demonstrating the large change in the surface energy of a metal when a small amount of some non-metallic impurities is adsorbed to the surface of the metal. Thermal conductivity and heat capacity The conduction of heat by liquid metals is directly related to the electronic structure. Heat is carried through a metal by energetic electrons having Physical properties and applications of liquid metals 297 translational energies above the energy distribution of the metal ion cores. The conductivity can therefore be calculated using the Lorenz modification of the Wiedemann–Franz ratio K T D a constant where the constant for liquid metals is about 2.5 W ohm K 2 . For liquid silver near the melting point, this value is 2.4, and the corresponding value for the solid metal is approximately the same. The thermal conductivity would therefore be about 3.8Wcm 1 K 1 . 300 10 35 30 25 20 300 500 700 900 20 30 40 50 60 70 80 90 400 500 600 700 Temperature (K) Temperature (K) 800 900 1000 Thermal conductivity (W m −1 K −1 ) Cp (J g atom −1 ) = Sodium = Zinc = Tin = Lead = Sodium = Zinc = Tin = Lead Figure 10.1 Thermal conductivities and heat capacities of the low-melting elements Na, Zn, Sn and Pb 298 Thermochemical Processes: Principles and Models The heat capacity is largely determined by the vibration of the metal ion cores, and this property is also close to that of the solid at the melting point. It therefore follows that both the thermal conductivity and the heat capacity will decrease with increasing temperature, due to the decreased electrical conduc- tivity and the increased amplitude of vibration of the ion cores (Figure 10.1). The production of metallic glasses A number of metallic alloys form stable glasses when quenched rapidly from the liquid state. These materials fall into two categories, one containing a metalloid dissolved in a metal at about 15–30 atom per cent, and the other containing metals only, with compositions around the 50–50 atom per cent composition. Examples of the first are the systems Ni–P, Au–Si, Pd–Si, Ge–Te and Fe–B. In each of the first group of systems, the phase diagram shows the existence of a low melting eutectic, compared with the melting point of the metallic constituent. In the second group the only similarity appears to be a stronger interaction in the liquid than in the solid state. Some systems form inter-metallic compounds, and others show immiscibility in the solid state. There are also some more complex systems composed of the elements above, such as the ternary systems Fe–Pd–P, Ni–Pt–P, and the Ni–Fe–P–B quaternary. The structures of the first group of glasses are consistent with the suggestion that the smaller metalloid atoms fill holes in the metal structure, and enable a closer approach of the metal atoms and an increased density. The arrangement of the metal atoms can either be in a random network or in the dense random packing model of Bernal, in which the co-ordination number of the metalloid can be 4, 6, 8, 9 and 10, some of the latter three involving 5 co-ordination of the non-metal. The formation of glass requires that the rate of cooling from the melt must be greater than the rate at which nucleation and growth of a crystalline phase can occur. The minimum rate of cooling to attain the glass structure can be obtained for any system by the observation of the rate of crystallization as a function of supercooling below the liquidus. The extremum of the TTN (time, temperature, nucleation rate) curve shows the maximum rate of nucleus formation as a function of undercooling temperature, and hence the minimum in the rate of cooling required to achieve the formation of a glass. According to homogeneous nucleation theory, the critical Gibbs energy to form a nucleus is given by G Ł D 16 3 3 G 2 s In the present case of the nucleation of solid particles from a liquid, the heat capacity change from liquid to solid may be ignored, and hence G s can be Physical properties and applications of liquid metals 299 expressed in terms of the fusion data thus G s D H s TS s D H f T f T T f D S f T (Note: H f DH s ; S f DS s ) The rate of formation of stable nuclei per unit volume of liquid can be described by the general equation Rate/second D nf 0 exp G M RT exp G Ł V m RT where G M is the activation Gibbs energy of diffusion, and describes the rate of arrival of atoms by diffusive jumps at the surface of the nucleus, and n is the number of atoms at the surface of the nucleus. Here, T is the degree of supercooling, and V m is the molar volume of the solid metal. The critical size of the stable nucleus at any degree of under cooling can be calculated with an equation derived similarly to that obtained earlier for the concentration of defects in a solid. The configurational entropy of a mixture of nuclei containing n Ł atoms with n 0 atoms of the liquid per unit volume, is given by the Boltzmann equation S m D k ln [n 0 C n Ł ]! n 0 ! n Ł ! Eliminating the factorials by Stirling’s approximation, which is strictly only correct for large numbers, differentiating the resulting expression for the Gibbs energy of this mixture of nuclei of this size and liquid atoms, with respect to the size of the nucleus, and setting this equal to zero to obtain the most probable value of n Ł , it follows that n Ł n 0 D exp G Ł kT D exp 16 3 3 V 2 m [S f T] 2 for homogeneous nucleation. For most metals, the entropy of fusion is approxi- mately 10 J K 1 mol 1 , the interfacial energy of solids is about 2 ð10 5 Jcm 2 , and the molar volume is about 8 cc g atom 1 hence this expression may be simpli- fied to G Ł ¾ D 10 13 T 2 J with the critical nucleus diameter of 320/T nm. It will be seen that the rate of nucleation depends on one factor which decreases as the temperature decreases, the diffusion rate to the surface of the nucleus, while the other, the critical nucleus size and formation Gibbs energy, also decreases as the 300 Thermochemical Processes: Principles and Models Melting point Temperature Rate of nucleation of solid particles Time Nucleus radius Critical nucleus radius Time to accumulate atoms (decrease in diffusion rate) Figure 10.2 The time–temperature–nucleation curve showing the balance between the rate of nucleation and the critical radius which produces a maximum rate temperature decreases. The two factors have opposing temperature effects, and thus the rate of nucleation goes through a maximum at the ‘nose’ of the TTN curve (Figure 10.2). Using the Stokes–Einstein equation for the viscosity, which is unexpectedly useful for a range of liquids as an approximate relation between diffusion and viscosity, explains a resulting empirical expression for the rate of formation of nuclei of the critical size for metals Rate D K Á exp 0.0205A Ł T 2 r T 3 r where A Ł kT Ł is the energy of critical nucleus formation, G Ł , T Ł D 0.8T M , T r is the the ratio of the temperature to the melting point and K is a universal rate constant. Liquid metals in energy conversion Nuclear and magneto-hydrodynamic electric power generation systems have been produced on a scale which could lead to industrial production, but to-date technical problems, mainly connected with corrosion of the containing mate- rials, has hampered full-scale development. In the case of nuclear power, the proposed fast reactor, which uses fast neutron fission in a small nuclear fuel element, by comparison with fuel rods in thermal neutron reactors, requires a more rapid heat removal than is possible by water cooling, and a liquid sodium–potassium alloy has been used in the development of a near-industrial generator. The fuel container is a vanadium sheath with a niobium outer cladding, since this has a low fast neutron capture cross-section and a low rate of corrosion by the liquid metal coolant. The liquid metal coolant is transported from the fuel to the turbine generating the electric power in stainless steel Physical properties and applications of liquid metals 301 ducts. The source of corrosion in this reactor is oxygen dissolved atomically in sodium, which gives rise to a number of inter-metallic oxides, such as FeO–Na 2 O. A number of fuel element types have been tested, including a metal U–Mo alloy and a UO 2 –PuO 2 solid solution. The liquid coolant is pumped around the heat exchanger by electromagnetic forces. If an electric potential is applied across a column of liquid metal which is held in a strong magnetic field, the liquid metal moves through the field in a direction at right angles to the two applied fields. The converse process is applied in magneto-hydrodynamic power conver- sion, the principle involved being the separation of the conduction electrons from the ion-cores in a liquid metal by pumping a column of the metal through a magnetic field. The container of the liquid metal, usually made of stainless steel, is equipped inside the magnetic field with electrodes which provide an external circuit for the electrons, and thus an electromotive force is gener- ated outside the liquid metal and its container. The propulsive force is appled to the liquid metal which is held in the containing stainless steel loop, the liquid passing through the magnetic field and then heat exchanger of a steam turbine which is a second source of electrical power, a so-called co-production process, which increases the overall efficiency of the plant. The cooler metal is then returned to the magnetic field area. Liquid phase sintering of refractory materials An important industrial procedure involving a liquid–solid sintering reaction is the sintering of refractory carbides of the transition metals which have very high melting points, above 2000 K, and would need a high temperature or plasma furnace to produce any significant consolidation of powders in the solid state at atmospheric pressure. The addition of a small amount of nickel, about 5–10 volume per cent, to tungsten carbide before sintering at about 1800 K leads to a considerably greater degree of sintering than could otherwise be achieved. This is because the nickel, which is liquid during the sintering process, can act as a transport medium for the dissolution and precipitation of particles of carbide. The metal additive remains in the fully sintered body as an inter-granular phase which limits the upper temperature of use of the material. This method is used as an alternative to the other successful but technically more difficult industrial process which employs hot pressing of the powders at 2000 K under several atmospheres pressure, using graphite dies, yielding products which can perform at higher temperatures. In the sintering of such materials as silicon nitride, a silica-rich liquid phase is formed which remains in the sintered body as an intra-granular glass, but this phase, while leading to consolidation, can also lead to a deterioration in the high-temperature mechanical properties. 302 Thermochemical Processes: Principles and Models In order to produce successful liquid-phase sintering, the liquid phase should wet, and to a small extent, dissolve the solid phase to be sintered. A major initial effect of the wetting of the solid phase is to cause the particles to rearrange to a maximum density by surface tension effects during the early stages of sintering. These forces bring about a spreading of the liquid phase among the voids in the initial compact, and the process of dissolution proceeds more rapidly at those parts of the solid grains where there are sharp edges, and hence the highest chemical potential of the solid. This therefore promotes a better packing of the solid particles. The initial sintering process continues by the dissolution of material at the point of contact of the particles with the formation of a liquid bridge between them. The centre-to-centre distance decreases as material is removed from this region by dissolution in the liquid. Considering the sintering of two spherical particles of radius a which have sintered together with a decrease in the centre- to-centre distance of 2h and a bridge of thickness X, it follows by Pythagoras theorem that h D X 2 /2a The material in the centre of the bridge has a higher chemical potential than that remote from the bridge on the free surfaces of the spheres because a compressive force acting between the two spheres accompanies the formation of the liquid bridge which generates this chemical potential difference. This force is proportional to the ratio of the interfacial tension, , divided by the radius of the bridge free surface, which approximates to half of the centre-to- centre decrease, and can be represented by K/h,whereK is a proportionality constant (Figure 10.3). a 3 4 1 2 2 h 1. Surface diffusion 2. Solution transfer 3. Grain boundary diffusion 4. Volume diffusion Central pore being filled by the dissolution precipitation mechanism Figure 10.3 The model for liquid phase sintering of high-melting solids with liquid metals as a sintering aid Using the equation which emerges from these considerations, the concen- tration difference of the dissolved material in the bridge from that in the bulk [...]... dispersed in the solid In modern processing where higher 312 Thermochemical Processes: Principles and Models Repeat pattern in talc Ionic structure of each plane 6 oxygen 4 silicon 4 oxygen + 2 hydroxyl 6 magnesium 4 oxygen + 2 hydroxyl 4 silicon 6 oxygen Repeat pattern in kaolinite 6 hydroxyl 4 aluminum 4 oxygen + 2 hydroxyl 4 silicon 6 oxygen Figure 11. 1 The atomic structures of talc and kaolinite firing... which is the ratio of the Gibbs energy change to the heat change of the cell reaction 2H2 C O2 D 2H2 O(g) where G° D 494 130 C 108T J mol 1 or CH4 C 3O2 D CO2 C 2H2 O(g) G° D 819 110 C 239.3T J mol 1 322 Thermochemical Processes: Principles and Models and so the thermodynamic efficiency, E, for these two fuels is For H2 (g) : E D G° /H° D 1 TS° /H° D 1 0.20 D 80% and for CH4 (g) D1 0.27 D 73% These... cations on the silicate structure, many examples occur in which fluoride or hydroxyl ions can be substituted for oxygen ions when, again, there is the proper charge compensation by the cations 308 Thermochemical Processes: Principles and Models The thermodynamic properties of the solid silicates show the expected entropy change of formation from the constituent oxides of nearly zero, which is typical of... decreasing repulsive interaction between the highly charged Si4C ions as the radius of the metal ion separating them in the silicate structure increases Table 11. 1 shows values of the heats of formation for a number of technically important silicates Table 11. 1 Standard heats of formation of some metal silicates at room temperature (from metal oxides) H° (kJ mol 1 ) 298 Metal ion Composition LiC NaC Li2 SiO3... metal–silicate liquid two-phase systems will influence the boundary layer thickness to a greater extent than in the liquid metals and alloys, mainly because of the higher viscosity of the silicate 310 Thermochemical Processes: Principles and Models Metal solubilities in silicate glasses Some metals are soluble as atomic species in molten silicates, the most quantitative studies having been made with CaO–SiO2... kinetics of the transfer of material across the boundary layer separating the carbide and the liquid metal The phase diagram for the nickel–tungsten system, for example, shows the formation 304 Thermochemical Processes: Principles and Models of high-melting solid compounds up to 1525° C, with a wide solubility above that temperature, which is therefore the lower limit for the solution-diffusion mechanism... been used as nucleating agents for the liquid slag, and on cooling, spinels of composition MgCr2 O4 or magnesiumsubstituted FeCr2 O4 separate as dispersed fine particles in a glassy mixture 314 Thermochemical Processes: Principles and Models denoted as ‘pyroxene’, which can be described by the general formula Ca1 x (Mg, Fe)1Cx (Si, Al)2 O6 The primary nuclei containing the magnesium spinel form at... small, that they may be regarded as covalent compounds The crystal structure of the mineral, orpiment (As2 S3 ) from which the structures of these compounds derive, has 12-member buckled rings 316 Thermochemical Processes: Principles and Models with alternating sulphur and arsenic atoms co-ordinated by the directed bonds of the sulphur atom The crystalline compounds all have congruent melting points, increasing... Richardson, 1965) These results show that there is very little tendency to form branching chains and rings as in the silicates, but the chain lengths increase as the phosphorus/metal ratio increases 318 Thermochemical Processes: Principles and Models The mobilities of ions in molten salts, as reflected in their electrical conductivities, are an order of magnitude larger than those in the corresponding solids... does not increase their electrical conductivities, probably due to the formation of covalently bonded molecular species such as Ca2 Cl2 Ca0 –CaCl2 and similarly Ba2 Cl2 (Figure 12.1) 1200 115 0 Na F Continuous solution 110 0 Na Cl Na I 1000 T° C 1050 Na Br Two phases, metal and halide 950 900 20 40 60 80 Mole % Na metal Figure 12.1 The solubilities of liquid sodium in the liquid sodium halides These facts . calculated by using the viscosity as the frictional force in the Stokes equation F D 3dÁU 294 Thermochemical Processes: Principles and Models where d is the particle diameter, and U is the (constant). the surface energies of liquid metals are found to be very sensitive to the presence 296 Thermochemical Processes: Principles and Models of surface active elements, and to be substantially reduced. Thermal conductivities and heat capacities of the low-melting elements Na, Zn, Sn and Pb 298 Thermochemical Processes: Principles and Models The heat capacity is largely determined by the vibration
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