Thermochemical Processes Episode 11 pdf

An average value of this function for liquid metals is about 0.47, corresponding to a ratio ofthe distance of closest approach of the atoms in a liquid metal to the atomicradius of about

given by Furth’s equation

Further support for this approach is provided by modern computer studies

of molecular dynamics, which show that much smaller translations than theaverage inter-nuclear distance play an important role in liquid state atom move-ment These observations have confirmed Swalin’s approach to liquid statediffusion as being very similar to the calculation of the Brownian motion ofsuspended particles in a liquid The classical analysis for this phenomenonwas based on the assumption that the resistance to movement of suspendedparticles in a liquid could be calculated by using the viscosity as the frictionalforce in the Stokes equation

where d is the particle diameter, and U is the (constant) velocity of the particlethrough the liquid of viscosity  This, when combined with a diffusion compo-nent obtained from a random walk description yields the Stokes–Einsteinequation for Brownian movement This calculation was then extended to themovement of atoms in liquids, by substituting the diameter of atoms for thediameter of the particles.

3d

The viscosity therefore replaces the restraint on diffusion arising from theinteraction of atoms expressed by the Morse potential in Swalin’s treatment.The introduction of molecular dynamical considerations suggests that thethe use of the atomic diameter in the Stokes–Einstein equation should bereplaced by an expression more accurately reflecting the packing fraction

of atoms in liquids, i.e the volume available to an atom in a close-packedarrangement 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 ofthe distance of closest approach of the atoms in a liquid metal to the atomicradius of about 1.55 Each atom must be considered as moving in a ‘cage’ ofnearest 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 102poise for liquid potassium, and 4.1 ð 102 poise for liquid copper, at theirrespective melting points Empty metals are those in which the ionic radius

is small compared to the metallic radius, and full metals are those in whichthe ionic radius is approximately the same as the metallic radius The processwas described by Andrade as an activated process following an Arrheniusexpression

 D 0exp Qvis/RTpoise

where Qvis 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 energyfor the evaporation of metals Evap is compared with that of viscosity,

Evap/Qvis ð rion/rmetal3D3 to 4

A further empirical expression, due to Andrade, for the viscosity of liquidmetals at their melting points, which agrees well with experimental data is

 D5.1 ð 104MTM0.5Vpoise

where M is the molecular weight in grams, TMis the melting point, and V isthe molar volume in cm3 A further point to note is that the viscosities of liquidmetals are similar to that of water at room temperature, about 102 poise, and

so useful models of the behaviour of high-temperature processes involvingliquid metals can be made easily visible at room temperature using water tosubstitute for metals, and a suitable substitute for other phases, usually liquidsalts or metallurgical slags, which can have up to 10 poise viscosity

In connection with the earlier consideration of diffusion in liquids using theStokes–Einstein equation, it can be concluded that the temperature dependence

of the diffusion coefficient on the temperature should be TexpQvis/RTaccording 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 themaximum pressure required to form a bubble which just detaches from acylinder of radius r, immersed in the liquid to a depth x, is given by

pmaxDxmaxg C2!

rwhere ! is the surface energy, and maxis the maximum radius of the bubblejust before detachment, or the Rayleigh equation for the oscillation frequency

ωin shape of a freely suspended levitated drop of mass m in an electromagneticfield, which is related to the surface energy by

where K is a universal constant, and Vm is the molar volume, which provides

a fair correlation among the data for elements Because of the relatively highdiffusion coefficients in liquids, and the probability of the rapid convectioncurrent distribution of solute elements to their equilibrium sites, the surfaceenergies of liquid metals are found to be very sensitive to the presence

of surface active elements, and to be substantially reduced by non-metallicelements such as sulphur and oxygen, in the surrounding atmosphere Greatcare must therefore be taken in the control of the composition of the gaseousenvironment to assure accurate data for liquid metals Table 10.2 shows somerepresentative results for elements which should be compared with the datafor the corresponding solids (Table 10.2).

Table 10.2 Surface energies of liquid elements

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 solutionyield surface energies approximately one-half of that of the pure metal at aconcentration 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-metallicimpurities 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 electronicstructure Heat is carried through a metal by energetic electrons having

translational energies above the energy distribution of the metal ion cores.The conductivity can therefore be calculated using the Lorenz modification ofthe Wiedemann–Franz ratio

K

%T Da constant

where the constant for liquid metals is about 2.5 W ohm K2 For liquid silvernear the melting point, this value is 2.4, and the corresponding value forthe solid metal is approximately the same The thermal conductivity wouldtherefore be about 3.8 W cm1K1

The heat capacity is largely determined by the vibration of the metal ioncores, and this property is also close to that of the solid at the melting point Ittherefore follows that both the thermal conductivity and the heat capacity willdecrease 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 fromthe liquid state These materials fall into two categories, one containing ametalloid dissolved in a metal at about 15–30 atom per cent, and the othercontaining metals only, with compositions around the 50–50 atom per centcomposition 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 diagramshows 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 systemsform inter-metallic compounds, and others show immiscibility in the solidstate There are also some more complex systems composed of the elementsabove, such as the ternary systems Fe–Pd–P, Ni–Pt–P, and the Ni–Fe–P–Bquaternary

The structures of the first group of glasses are consistent with the suggestionthat the smaller metalloid atoms fill holes in the metal structure, and enable acloser 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 randompacking model of Bernal, in which the co-ordination number of the metalloidcan 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 coolingfrom 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 theglass structure can be obtained for any system by the observation of the rate ofcrystallization 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 theminimum in the rate of cooling required to achieve the formation of a glass.According to homogeneous nucleation theory, the critical Gibbs energy toform a nucleus is given by

expressed in terms of the fusion data thus

exp



 GŁVmRT



where GMis 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 isthe number of atoms at the surface of the nucleus Here, T is the degree ofsupercooling, and Vm is the molar volume of the solid metal

The critical size of the stable nucleus at any degree of under cooling can becalculated with an equation derived similarly to that obtained earlier for theconcentration of defects in a solid The configurational entropy of a mixture

of nuclei containing nŁatoms with n0 atoms of the liquid per unit volume, isgiven by the Boltzmann equation

SmDkln[n0Cn

Ł]!

n0! nŁ!Eliminating the factorials by Stirling’s approximation, which is strictly onlycorrect for large numbers, differentiating the resulting expression for the Gibbsenergy 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 mostprobable value of nŁ, it follows that

!3V2m[SfT]2for homogeneous nucleation For most metals, the entropy of fusion is approxi-mately 10 J K1mol1, the interfacial energy of solids is about 2 ð 105J cm2,and the molar volume is about 8 cc g atom1hence this expression may be simpli-fied to

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 theTTN curve (Figure 10.2)

Using the Stokes–Einstein equation for the viscosity, which is unexpectedlyuseful for a range of liquids as an approximate relation between diffusion andviscosity, explains a resulting empirical expression for the rate of formation

of nuclei of the critical size for metals

where AŁkTŁ is the energy of critical nucleus formation, GŁ, TŁD0.8TM,

Tr is the the ratio of the temperature to the melting point and K is a universalrate constant

Liquid metals in energy conversion

Nuclear and magneto-hydrodynamic electric power generation systems havebeen produced on a scale which could lead to industrial production, but to-datetechnical problems, mainly connected with corrosion of the containing mate-rials, has hampered full-scale development In the case of nuclear power, theproposed fast reactor, which uses fast neutron fission in a small nuclear fuelelement, by comparison with fuel rods in thermal neutron reactors, requires

a more rapid heat removal than is possible by water cooling, and a liquidsodium–potassium alloy has been used in the development of a near-industrialgenerator The fuel container is a vanadium sheath with a niobium outercladding, 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 transportedfrom the fuel to the turbine generating the electric power in stainless steel

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 asFeO–Na2O A number of fuel element types have been tested, including ametal U–Mo alloy and a UO2–PuO2 solid solution The liquid coolant ispumped around the heat exchanger by electromagnetic forces If an electricpotential is applied across a column of liquid metal which is held in a strongmagnetic field, the liquid metal moves through the field in a direction at rightangles 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 electronsfrom 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 stainlesssteel, is equipped inside the magnetic field with electrodes which provide anexternal 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, theliquid passing through the magnetic field and then heat exchanger of a steamturbine which is a second source of electrical power, a so-called co-productionprocess, 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 veryhigh melting points, above 2000 K, and would need a high temperature orplasma furnace to produce any significant consolidation of powders in thesolid 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 sinteringprocess, can act as a transport medium for the dissolution and precipitation ofparticles of carbide The metal additive remains in the fully sintered body as aninter-granular phase which limits the upper temperature of use of the material.This method is used as an alternative to the other successful but technicallymore difficult industrial process which employs hot pressing of the powders

at 2000 K under several atmospheres pressure, using graphite dies, yieldingproducts 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, butthis phase, while leading to consolidation, can also lead to a deterioration inthe high-temperature mechanical properties

In order to produce successful liquid-phase sintering, the liquid phase shouldwet, and to a small extent, dissolve the solid phase to be sintered A majorinitial effect of the wetting of the solid phase is to cause the particles torearrange to a maximum density by surface tension effects during the earlystages of sintering These forces bring about a spreading of the liquid phaseamong the voids in the initial compact, and the process of dissolution proceedsmore 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 thepoint of contact of the particles with the formation of a liquid bridge betweenthem The centre-to-centre distance decreases as material is removed from thisregion by dissolution in the liquid Considering the sintering of two sphericalparticles 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 Pythagorastheorem that

h D X2/2a

The material in the centre of the bridge has a higher chemical potentialthan that remote from the bridge on the free surfaces of the spheres because acompressive force acting between the two spheres accompanies the formation

of the liquid bridge which generates this chemical potential difference Thisforce is proportional to the ratio of the interfacial tension, !, divided by theradius of the bridge free surface, which approximates to half of the centre-to-centre decrease, and can be represented by K!/h, where K is a proportionalityconstant (Figure 10.3)

a

2 2

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 tration difference of the dissolved material in the bridge from that in the bulk

concen-of the liquid can be found from the thermodynamic approximation.

-bridge-bulkDK!/h D RTlnCbridge/Cbulk D RTC/Cbulk

if CbridgeCbulkis small Approximating the circumference of the bridge to

a thin cylinder, the diffusive flux out of the bridge is given by

and integrating the resulting expression

2ah2dh/dt D 4DυK!Cbulk/RTwith h D 0 at t D 0

yields

h3D6DυK!Cbulk/RTt/a

and the centre-to-centre distance decreases as a function of time according to

a t/a1/3law Since the fractional shrinkage of the whole compact, L/L, isproportional to h

L/L D 1/3V/V / t1/3/a4/3 (D.W Kingery, 1959)

Several other sintering mechanisms can become important once the initialphase is over, and the external pressure is no longer a factor in the process.Surface diffusion from the periphery of the particles to the neck (see p 302)and solution-diffusion through the liquid are the two most probable mecha-nisms, since other sintering processes would require the parallel diffusion ofatoms of each species in the carbide in volume or grain boundary migration.There is always the possibility, however, that the rate of the solution-diffusionprocess will be decreased by the kinetics of the transfer of material acrossthe boundary layer separating the carbide and the liquid metal The phasediagram for the nickel–tungsten system, for example, shows the formation

of high-melting solid compounds up to 1525°C, with a wide solubility abovethat temperature, which is therefore the lower limit for the solution-diffusionmechanism.

Bibliography

T.E Faber An Introduction to the Theory of Liquid Metals Cambridge University Press (1972).

R.A Swalin Liquid metal diffusion theory, Acta Met., 7, 736 (1959).

R.M German Liquid Phase Sintering Plenum Press New York (1985) TN695 G469.

The production of crystalline semiconductors

The production of germanium and silicon is carried out by the reduction ofthe oxides, SiO2and GeO2 There is a considerable difference in the stabilities

of the oxides, and GeO2 can be reduced by hydrogen at temperatures around

1100 K, but SiO2 must be reduced by carbon at temperatures around 2300 K.The elements only reach a vapour pressure of 106atmos at 1500 K and 1750 Krespectively, and so the reduction of GeO2 does not lead to any significantvaporization, but this is not so in the production of silicon The gaseouscomposition of a 1:2 mixture of SiO2 and carbon at 2000 K is as follows:pCO D 0.75, pSi2D0.17, pSiO D 0.049, pC3 D0.016 and pSi D 0.015and with a 1:4 mixture of SiO2

pCO D 0.61, pC3 D0.21, pSi2D0.14, pSiO D 0.016, pSi D 0.014atmos as the major components It is probable that this mixture of gases plays

a major role in promoting the SiO2CC reaction

The products of these reduction reactions are crystalline powders Furtherpurification is obtained by conversion to the tetrachloride of germanium, andthe hydrochloride of silicon, SiHCl3 These chlorides boil at 357 K and 298 Krespectively, and hence may be purified by distillation before being reduced tosemiconductor-grade elementary powders with hydrogen Germanium powdercan be melted in a carbon crucible without significant contamination, butsilicon is best obtained in impure form as a bar by condensing the reductionproduct of SiHCl3 by hydrogen on a heated silicon rod similar to the vanArkel refining of zirconium

Zone refining of semiconducting elements

The method of zone refining which was first used in the production of verypure germanium depends for its success on the difference between the thermo-dynamic properties of an impurity, present as a dilute constituent dissolved in

a nearly pure solid, and those of the impurity dissolved in the coexisting liquid.

At solid–liquid equilibrium, the chemical potential of the impurity must bethe same in each phase and since

RTln ! D GxsDH  TSxs

where ! is the activity coefficient If X is the mole fraction of the dilute solutethe distribution of this impurity between the solid and the liquid phases islnXS/XL D H (transfer)  TSxs (transfer) D ln KS – L

where the transfer terms are for the transfer of the impurity from the solid tothe equilibrium liquid phase The solvent, germanium, is present to more than95% concentration, and its activity coefficient may be taken as unity Thusthe partition coefficient of the impurity between the solid and liquid KS – Lisdetermined by the heat of transfer and therefore of the activity coefficient ratio

of the impurity between the solid and the liquid solvent

In zone refining, a bar of the impure material can be heated in a frequency heater to produce a thin layer of liquid which is slowly movedthrough the solid by raising the heater along the bar Since equilibrium is notachieved in one pass of the liquid through the solid, several passes are madebefore a usefully high state of purity is reached Alternatively, the impurebar can be contained in a chemically inert, ceramic tube and the liquid layercan be produced by passing the tube through a short heater The liquid phase

radio-is well stirred by the r.f field, and hence can be assumed to be uniform incomposition except at the boundary layer where atoms of the impurity arere-deposited on one side, and dissolved on the other as the liquid advances.The removal of impurity in one pass can be calculated by making a massbalance for the advancing liquid phase If the original impurity content of thebar is C0, and CL is the impurity content of the liquid, then for an advance

υx of the liquid, the amount dissolved into the advancing liquid minus theamount deposited behind the liquid is equal to the increase in the impuritycontent of the liquid

C0υx  KS – LCLυx D lυCL

where l is the liquid zone length, and in differential form this becomes

ldCLDC0KS – LCLdx

which upon integration, and inserting the boundary conditions CLDC0when

x D0, yields the equation for the resulting content of impurity in the solid as

a function of the distance along the bar, CS,

CSDC0[1  1  KS – LexpKS – Lx/l]

In order to calculate the effect of several passes an iterative calculation isneeded using the initial profile at each pass to represent C0 Clearly for thesecond pass, the concentration profile given in the right-hand side of the aboveequation must be used It is clear that the partition constant of the impuritybetween the solid and liquid is the most significant parameter in the success

of zone refining

The values of the partition coefficients increase with the valency of theimpurity, being about 105 atom per cent for the elements of Group IB (e.g.copper) in germanium, up to about 1 atom per cent for Group IV (tin) Theliquid phases in these systems are approximately Raoultian at the germanium-rich end, showing only small negative deviations in Cu C Ge, and smallpositive deviations in Sn C Ge We may therefore neglect the heats of solu-tion of these impurities in germanium at the zone refining concentrations Itfollows that the highly endothermic heats of solution in the solid phase deter-mine the partition coefficient These are related to the difference in structure

of the solute elements from the Group IV semiconductors For example, it hasbeen shown that the heat of solution of copper in solid germanium is about

180 kJ g atom1

Physical and chemical properties of

glassy and liquid silicates

The structures of the silicate minerals depend on the ratio of silica to theother metal oxides In pure silica, the structure consists of SiO4 tetrahedrasharing oxygen atoms or ions at each corner The co-ordination number ofsilicon is therefore four, and that of oxygen two On the addition of metaloxides, such as Na2O or MgO, the effect, which may be described as ‘bridgebreaking’, involves the separation of two silicon atoms joined by an oxygenion, through the addition of another oxygen ion The accompanying metal ion

is accommodated within the structure thus,

Si–O –Si C MO ! 2Si–O – C M2C

when an alkaline earth metal oxide is added On addition of further amounts ofmetal oxide, the silica structure is increasingly broken down until a sheet struc-ture is formed with the alkali metal oxides of formula M2OÐ2SiO2, corresponding

to an anion of formula Si2O52 At the composition M2OÐSiO2or MOÐSiO2, themeta-silicates, an endless series of chains are formed, in which the silicate ionmay be represented as SiO32, and at the composition 2M2OÐSiO2, or morefrequently 2MOÐSiO2, the ortho-silicates, the structure consists of isolated SiO4units interspersed with metal cations in octahedral co-ordination

The naturally occurring silicates are variable in many more ways than this.For example, the introduction of Al2O3 can lead to the formation of alumino-silicates in which the silica structure can include the aluminium ion providingthat charge compensation is made by the addition of a balancing amount of uni-

or bivalent cations In a similar fashion phosphorus can be accommodated bythe omission of a compensating amount of these cations The structures of all

of the simple and complex silicate structures can be accounted for by the use

of a completely ionic model with Si4C, Al3Cor P5Cions in the main structureassociated with the appropriately balancing number of cations However, there

is certainly a covalent contribution to the bonding in these structures, which

at the covalent extreme would be bonded by sp3 Si–O bonds Bearing this inmind, the ionic model can be accepted as only a working approximation which

is very useful Apart from substituting metal cations on the silicate structure,many examples occur in which fluoride or hydroxyl ions can be substitutedfor oxygen ions when, again, there is the proper charge compensation by thecations