Thermal diffusion in liquids and gases has been the focus of a number of studies and is considered, at present, a reasonably wellknown topic. In solids though, the phenomena is far less understood namely in what regards the effects of thermal gradients upon the integrity and characteristics of
materials. Furthermore it is not clear whether or not thermal diffusion can be used as a purificating method for solids. Yet, many current techniques such as the laser, used in metal working and other
solids, induce strong thermal gradients being, therefore, potential cause for the occurrence of thermal diffusion. In this work, a survey and discussion of theoretical and experimental studies of thermal diffusion in solids is undertaken

Morgado, M. Fátima

Pinheiro, J. D. R. S.

Universidade do Minho: RepositoriUM

t of er-tal 35/1 THERMAL DIFFUSION IN SOLIDS - AN OVERVIEW M. Fatima Morgado 1 · 2 Joao de Deus Pinheiro 1. C.Q.P.A. (INIC), Universidade do Minho, Largo do Pa9o, 4700 BRAGA 2. Institute Superior de Engenharia, Rua S .To~. 4200 PORTO ABSTRACT :.;.:;..--Thermal diffusion in liquids and gases has been the focus of a number of studies and is considered, at present, a reasonably wellknown topic. In solids though, the phenomena is far less understood name-ly in what regards the effects of thermal gra-dients upon the integrity and characteristics of ~terials. Furthermore it is not clear whether or not thermal diffusion can be used as a purificatim method for solids . Yet, many current techniques such as the laser, used in metalworking and other solids, induce strong thermal gradients being, therefore, potential cause for the occurrence of thermal diffusion . In this work, a survey and dis-cussion of theoretical and experimental studies of thermal diffusion in solids is undertaken. INTROCAJCTI ON Under a temperature gradient a demixing of solid solutions occurs: in the initially homogeneous multicomponent solid material a flow of atoms ap-pears and a concentration gradient is created . This is known by thermal ~iffusion, thermal transport, thermomigration effect or SQret effect. The fi'BJ1itude and sign of this effect can be .ex-Pressed in terms of the heat of transport (Q*)of t~ components, since the thermal gradient pro-~ces a difference in the heats of transport of solute and solvent that is possible to mesure . ~e questions can be asked about the Soret ef-fect in solids: ·are the direct and indirect effects of migration induced by the temperature gradient perceptible enough in a solid body ? - how can the temperature gradients in solid bodies be used to increase the purity of materials such those used in some technologies, like elec-tronics, nuclear technology, etc.? THERMAL GRADIENTS IN MATERIALS One of the first purifications of materials under the influence of thermal gradients was studied by BUDA (1) in monocrystalline silicone samples pre-viously saturated by prolonged diffusional doping with radioactive ·Ni 63 by using electrical meas-urements of resistivity layer, radioisotope ana-lysis and analysis on a mass spectrometer. BUOA observed that: a) the temperature gradient causes intensive migr~ tion of the nickel in a direction conformable to the direction of the temperature gradient, and a ' concentration gradient of the nickel appears; b) the concentration gradient grows with time; c) the concentration gradient also grows with the temperature gradient; d) the nickel always concentrates in the high temperature side independently of the temperature gradient value. OOA, SON and MIYAKE (2) studied the thermomigra-tion of carbon in niobium by a steady-state meth-od using radioactive tracer. They observed that under a thermal gradient the interstitial carbon atoms in niobium are subjected to a driving force towards the hotter region. Nuclear power reactor components and cementedcar-bide cutting tools are two examples of materials in which the problem of thermodiffusion is encoun-tered. Niobium metal and its alloys are al ternathe structural materials for fast reactors and they can probably be used as first wall for fusion re-actors, where large temperature gradients exist (3). This is the reason why studies of thermal diffusion with niobium are very important, partie-ularly in the cases where tritium and carbon mo-bilities in it are concerned . In Japan, SUGISAK et al. (4) studied the motion of tritium in niobium prodJced by the driving for-ce due to the thermal gradient for temperatures 35/2 ditions, us ing radioactive tracer te h c ni~s CARLSON et al . (5). The results of th . etr exn.. ...:tt ments show rather conclusively that carbon is thermotransported to the hotter region f 0 rtiObt~ in both one-phase and two-phase cond't· l lens . The heat of transport has the rather large val•~ "" Of -13 : 1 K cal mol- l for either condition . ~ first paper with quantitive results in the~!. fusion on glasses appeared in 1980 (6). Two lithium silicate glasses were prepared, subj~~ to a constant temperature gradient and lithi~ concentration was measured. It was observed that under a temperature gradient the lithium ion mi. grates to the higher temperature region thr~ the rigid silicate network. Heats of transportin glasses were measured and compared with ther~-transport experiments in metals. It was noticed that glasses'heats of transport are of the s~ order of magnitude of metal's. It means that thermodiffusion occurs not only in crystalline solids but also in amorphous ones and so that thermotransport is not due to the existence of free electrons in the solid. One can concl~ between 100-300°C. They concluded that the ratio that: between the partial pressure of tritium in the higher temperature side and that in the lower temperature side, is modified by one order of magnitude by the effect of thermal diffusion. That means that thermal diffusion has an appreci-able influence on tritium permeation in the niobium and that physico-chemical behaviour of tritium in a first wall is considerably more complex under a thermal gradient than in isothermal conditions. The thermodiffusion behaviour of carbon in niobium was investigated in one-phase and two-phase con--using temperature gradient to force and to guide the stream of one or several elements is effective and can be utilized as a procedure to remove impurities in solid state; - factors limiting migration of impurities (puri-fication ) were not found; - the sublimation of impurities from the surface of samples under a perpendicular temperature gr~ dient is more intensive than under isothermal conditions. :Pert. is ictJi e or he that 1 mi-lrt in . ced same 1St hat of ude to ls to CAL INTERPRETATIONS ANO•MATHEMATICAL MODELS ~·mplest physical interpretation of the phe-rhe sl ~na is the interstitial diffusion of impurity l·n a heavy host lattice atoms under the in-at()TlS nee of a temperature gradient. As the mobility flue of the solute atoms is sufficiently higher than that of host lattice atoms, one can safely assume that only the migration of interstitial solute at()Tls is significant for the redistribution and that, as the host lattice atoms can be considered a rigid one, it provides a convenient frame of reference for the flux. This fact permits a much simpler analysis of the phenomena than if the roobi lilies of both components should be considered and no one could be taken as a reference to the move-ment of the other . ~ture of driving forces in thermaldiffusion has been discussed by several authors ( 7 , 8, 9, 10) . The so-called intrinsic contribution and the scat-terlng effects of heat, carriers, such as phonons and electrons, are, in general, considered as the ~in contributions to the driving forces in ther-~ldiffusion and is has generally been concluded that electron scattering is dominant ( 10, 9, 2) . ~e electronic contribution is the inelastic scat-tering of electrons by solute atoms, resulting fr001 the electric field present in metallic con-Q.jctors under a temperature gradient. This is the well-known thermoelectric effect. KIRDi£IM and FRC!t+! ( 11) derived an expression re-lat· lng the heat of transport to the analogous ele.£ trotransport parameter z * Q = u z* ( 1) e the QUantity Ue being defined as the energy of 35/3 transport of electrons. If the electron scatter-ing contribution is dominant and as electronsmove from a higher energy level to a lower energy leve l * * Ue will always be negative, i.e . Q and Z should have opposite signs. This sign correlation has been noted by other authors and the experimental study of different metal interstitial solute systems shows a very little number of exceptions . According to the model proposed by BHAT and SWALIN ( 12) for the particular case of liquid metals, Q * can be decomposed in 2 parcels: one resulting from the interaction of the conduction electrons with solute ions and the other resulting from the in-teraction of the difusing ion with the other one~ The latter can be calculated by assuming that the liquid metal is a dense gas and applying the ther-motransport theory in binary gas mixtures . The sign of this part is determined primarly by the difference of the masses of the two ions , the heavier impurity migrating to the colder end. According to SORBELLO'S (10) analysis,if phonon scattering is the dominant factor and if the in-terstitial solute atoms a~e lighter than the ma-trix atoms, the solute atoms migrate against the phonon wind , i.e. , towards the cold region. The flux of solute atoms (J) is calculated from irreversible thermodynamics (1 3, 14 ) and is gen-eraly written as J = -0 [grad N + (Q* N!RT2) grad T (2) The steady-state J=O and equ. 1 can be wri t ten as * RT2 grad N _ -RT 'iJ (Ln N) Q - -N- · grad T - 'iJ (Ln T) since +'iJ(Ln T) - \1( 1 = r Q* :R V' Ln N) '\} _1 ) T For the one dimensional case * Q R d Ln N d (..!. ) T (3) * From equ. (3 ) it is obvious Q can be obtained from the slope of a plot Ln N versus T for the steady-state condition. For the particular case of the thermodiffusion of 1 carbon in niobium the plot of Ln N versus T is shown in f igure 1. 1200"C 11 oo ·c , 10 • IH •1820 "C/Cm • llT •1 400"C/cm 1 OS .. 1,00 7.0 7.5 10jTIKI FIGURE 1 - Ln specific carbon activity vs. reciprocal tempera t ure Relative carbon concentr.ation along the axial di-rec t ion of the specimen is plotted as a function of the reciprocal t emperature, assuming the temper ature dis t ribut ion along the axial direction of t he specimen is l i near . The graph shows that carbon in niobium becomes richer in the higher t emperat ure region and that the heat ?f transport has a negative value . 35/4 Another example can be the thermodiffusion of lithium in silicate glasses. The logarithm of the lithium concentration rer err~ to the initial concentration is plotted vers us the reciprocal temperature. 0 2 - ..... 0 , d 1 5 , 6 1 7 FIGURE 2 - Steady-state concentration of lithium vs. reciprocal temperature Once again the graph shows the lithium ion mi.qrates to higher Lemperatures. The so-called Soret coefficient is defined as sT = f ( 4 ) and it can be shown that it relates with the tern-* perature (T) and the heat of transport ( Q ) by the following expression Q* RT2 ( 5 ) Usually ST is very small and therefore thermaldi f -fusion may be neglected as compared to ordinarY diffusion . From (2) and (5) the impuri ty flux i s 

Thermal diffusion in solids : an overview

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