Surface segregation in dilute solid solutions

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<ul><li><p>SURFACE SEGREGATION IN DILUTE SOLID SOLUTIONS </p><p>H. P. Sl~WE and 1. J,\GER </p><p>Erich Schmid lnstitut fur Festkiirperphksik. Leoben. Austria, </p><p>Abstract-There is an interaction energ? between solute atoms and a free crystal surface causing surface segregation. Size and shape of the interaction potential is estimated and the kinetics of the segregation process are discussed. </p><p>R&amp;m&amp;-Lne cnergie dinteraction entre les atomes du sol&amp; et la surface librc dun cristai produit une segregation superficielle. On estime la port&amp;e et la forme du potentiel dinttraction et on discute la cinetique de la segregation. </p><p>Zusammenfassung-Zwischen gel&amp;ten Fremdatomen und einer freien KristalloberRache gibt es eine Wechselwirkungsenergie. die eme Obertlachenseigerung bewirkt. CiroRe und Gestalt des Wechselwir- kungspotentiales werden abgeschatzt und die Kinetik des Entmischungsvorgangcs wird diskutiert. </p><p>Several new experimental techniques such as SIMS and Auger-spectroscopy have renewed interest in the chemical composition of metallic surfaces. Frequently such experiments reveal the presence of Tme.mcted elements. This is usually explained by esretrral con- tamination (dirt): sometimes several atomic layers are removed before studying the properties of the real surface. Impurities, however, may also come from the interior of the crystal to contaminate its surface iayers if there is an attractive interaction potential between the solute and the free surface. The present paper discusses the kinetics of this sort of surface segregation. </p><p>POTENTIAL. OF A FOREIGX ATONI NEAR A FREE SURFACE </p><p>Solute atoms in a crystal have an excess energy which is lowered in the vicinity of a free surface. To estimate its magnitude we consider the model shown in Fig. 1. The crystal is represented by a continuum in a semispace containing a hole of radius rt at the distance .Y from the surface. </p><p>LO- </p><p>0.9 - </p><p>P 0.8 - </p><p>5: 3 </p><p>0.7 - </p><p>0.6 - </p><p>0.5 - </p><p>-- -_l-_--- + -+-T--- </p><p>A </p><p>i 1- A </p><p>I f </p><p>P I I / t I 2 3 k ; 6- </p><p>x R </p><p>Fig. I!. Excess energy of a solute atom as a function of its position: solid curve: continuum model. 0. +. A: </p><p>atom&amp;tic model for f.c.c.. b.c.c.. cub. pr. respectively. </p><p>A sphere with equal elastic constants G (shear modulus) and 11 (Poissons ratio) but of different radius r2 is squeezed into the hole. Theory of linear elasticity then gives [l] a strain energy </p><p>where the excess energy in the interior of the crystal is </p><p>and the *-misfit parameter E is </p><p>Fig. L. Model of solute atom in the vicinity of a free surface. </p><p>605 </p></li><li><p>606 S-l-iWE AND JAGER: SURFACE SEGREGATION IN DILLTE SOLID SOLtiTIONS </p><p>Table I </p><p>r=zoc t- -10 YC :quation( I ,Yl </p><p>Sol\cnr b, - I h ,cnl,mol:, Formula co h,lYl </p><p>Au </p><p>Pd FC </p><p>M&amp;T Si </p><p>Q % A Fe Zn </p><p>5.3 x IO-' 6.4 x lO-6 </p><p>0. I 2 0.21 0.90 0.265 </p><p>6.Y x IO- 0.019 I.3 0.30 0.02 I .A I5 </p><p>3.73 x lo- </p><p>1.J 0.4 </p><p>29scQ </p><p>20100 3?200 35600 30550 15300 22400 :72OO 5l5iM 44100 20100 </p><p>&gt; &gt; 6 5 5 5 5 </p><p>&gt; 7 8 9 9 </p><p>- 1698 - 2434 </p><p>7x5 1186 </p><p>4887 1916 I330 1330 </p><p>-913 460 </p><p>6553 3984 7640 6070 2092 1031 </p><p>ill x IO-: O.Oli </p><p>3.85 1.5x </p><p>4unl </p><p>x.9 9.84 9.84 0.2 I </p><p>I?1 x.9 x lo- </p><p>3??00 a9 </p><p>0 __ 35 0.1 I 107.7 ~I.05 I36 0.009 - I?.0 </p><p>0.1' 0.02 10.4 45 </p><p>61.5 34s 3. I 7 2.71 0.54 1.3s </p><p>471 7.7 </p><p>M.? 349 </p><p>3.1 I.70 </p><p>2.0 I 190 2.62 136.7 2.4 I Z?3 3.37 93.9 </p><p>2.33 x IO ?x IO </p><p>IO1 ?x IO 5x IO 4% 102 </p><p>I370 I.5 x IO 1.8 x lo </p><p>97 4.9 x IO. I.1 x IO? </p><p>5i.3 3.36 3.0 I 2.64 0.55 I.37 </p><p>421 7.29 </p><p>36.1 </p><p>3% 243 175.0 523 5.27 307.5 335 5.56 I 17.4 308 2.68 </p><p>4w 2.30 113 3.87 x2.1 502 0.45 301.5 456 1.54 261 273 4.7x10 II7 737 11.5 177 </p><p>775 I?6 504 </p><p>477 3.06 ?.?I </p><p>7.8 x lOA 439 ?.?I 263 1.77 </p><p>Cr Yn </p><p>59cm 55800 </p><p>940 710 2.57 741 3.04 245 I.88 658 2.85 710 3.0 </p><p>IO 9 </p><p>3.7 Y IO 3.4 Y IO .v </p><p>Ni Si </p><p>lx400 </p><p>587043 61500 </p><p>306 161 352 101 3.0 68i 4.4 439 9 </p><p>9 36.4 </p><p>5.89 1.67 7.u 3.0 473 </p><p>If we use for r, and r2 the atomic radii of solvent and solute of a substitutional solid solution, then equs. (I). (2) and (3) give an estimate for the excess energy and its variation with distance from the sur- face. (Equation 1 is valid only for x 2 rl ; V(x) will drop to zero as the atom leaves the crystal.) As can be seen from equation (1) the interaction between the atom and the free surface is noticeable over a distance of only a few atomic diameters. It might therefore appear doubtful whether V(X) should be estimated using a continuum model. However, an amistic model using interatomic potentials yielded rather similar results [2] as can be seen in fig. 2. If r, : rz, equation (2) gives unreasonably low values. For inter- stitial solutes. equation (2) gives unreasonably high values if the radius of the octahedral gap is used for r,. In both cases V(X) should be estimated in a differ- ent way. since strain energy is not the only com- ponent contributing to the excess energy (there are chemical contributions as well). </p><p>For instance V(X) can be estimated from the distri- bution coefficient K near the melting temperature T,,, </p><p>c31: </p><p>In either case. however. equation (1) was assumed to describe the change of V(X) with distance from the surface. </p><p>EQUILIBRIUIM CONCENTRATION IN THE CRYSTAL </p><p>Imagine the crystal to be built up of layers of thick- ness a = Zr, and numbered 1, 2. 3 . ..i...rr starting from the surface. Then, using p a l/3. discrete values of 6 can be ascribed to each layer, namely </p><p>[ </p><p>I y=v, I </p><p>-3(2i. 1 In thermal equilibrium. the local concentration of dilute* solutes of average concentrations c,, will then be given by: </p><p>where </p><p>Ci(t -+ -X:) 3 Ci,l = C* exp f/i. (6) </p><p>v,. - v VX --= Vi- RT 3RT(2i - 1)3 </p><p>(7) </p><p>= RT,In K, (4) TP.7 </p><p>and c* can be obtained from n </p><p>nco = x Ciq. I </p><p>(8) where c, and c, are the concentrations of solidus and liquidus. respectively. Contrary to equation (2), equa- tion (4) may yield negative values of V(z) (for K &lt; 1) in which case the impurities would tend to migrate nr~a!~fiorn the free surface. Table 1 indicates whether C( x ) has been taken from equations (2) or (4). </p><p>l At higher concentration equation (9) must be modified so as to prohibit concentrations larger than 1. The deriva- tion of equation (6) neglects the condition that one atomic site may be occupied by one atom only. Consequently, high values of c* and vi seem to lead to concentrations &gt; 1. A more rigorous treatment [3] leads to </p><p>I c,,q = 1+(1- c*/c*) exp( -vi) (6a) </p><p>Since. however. for high solute concentrations q, depends on concentration. the problem becomes more complex and will be treated in a later paper. </p><p>Fig. 3. Schematical plot of the total potential energy of an atom in the crystal. </p></li><li><p>STUUE AND JAGER: SURFI-\CE SEGREGATION IN DILUTE SOLID SOLUTIONS 60: </p><p>-$ -cl., , / / </p><p>/ I /- </p><p>/ / i </p><p>Fig. 1. Plot of cl/c0 vs r/r for different values of q,. </p><p>Since n usually is a very large number it will in-most The system of equations (12) and (13) was solved cases be sufficiently accurate to write numerically for the initial condition </p><p>Ciq Z Cg exp vi (9) Ci(C = 0) = cg (14) </p><p>i.e. the concentration in the interior of the crystal is hardly influenced by surface segregation so that and the boundary conditions equations (10) and (13). </p><p>ci+cg for i-+-z. (10) The result is seen in Fig. 4 showing the concen- </p><p>tration in the first atomic layer as a function of time for different values of qI. Extrapolation of the linear part of these curves permits the definition of a time </p><p>DIFFUSION OF SOLUTE ATOMS r0 necessary to approach equilibrium concentration IS THE POTENTIAL </p><p>Figure 3 shows schematically the total potential of the foreign atoms in the crystal. The subsequent layers are separated by barriers of height Q. In the interior of the crystal the jump frequency of the atoms is </p><p>lO6- </p><p>1 IOS- ,, = _ = v0 e-Q:RT. (11) -2 </p><p>s 01 c lO4- The balance of particle flow in layer i then gives </p><p>ci_L exp'y + cicl exp'! </p><p>'Ii - vi-1 exp ___ li - Vi+1 - ci 2 </p><p>+ exp? - &gt;I </p><p>At. (12) </p><p>Since we assume that particles will neither leave nor o-1 enter the surface. the first of these equations is 7, </p><p>c,expy -c,expv Fig. 5. Plot of r,,,r vs I,: f : computed from equations </p><p>A!. (13) (IO). (II). (13) and (17): dashed curve: estimated using equation (I 5a). </p></li><li><p>608 STi;[WE AND JAGER: SURFACE SEGREGATION IN DILUTE SOLID SOLUTIONS </p><p>in the first layer. In Fig. 5 various computed values of r0 are plotted vs I,. </p><p>Qualitatively the strong dependence oft,, on I, can be understood as follows: Assume that in equilibrium the concentration in the first layer is increased n-fold. Then the atoms to fill that layer must come at least from the first n layers. If they move by random walk this will take about the time </p><p>t, t n2r = re- . ql (154 </p><p>This estimate is entered into Fig. 5 as a dashed line. The computed points seem to lie on a curve </p><p>approaching equation (15a) for high values of I,. In the region of metallurgical interest. however, they can be better approximated by the straight line entered in Fig. 5. the equation of which is </p><p>to = 11.7rexp(l.7lq,). (15) </p><p>For Aq/2 . 6. </p><p>7. 8. </p><p>9. </p><p>*Also. equation (17) is a stiff equation posing certain difficulties for numerical treatment! </p><p>10. </p><p>1. If the excess energy is negative in certain alloy systems, the surface should be depleted of foreign atoms rather than enriched. The time necessary for this will be only a few c because the impurities do not have to travel very deep into the crystal to make the surface effect noticeable. (This is quite different from surface enrichment which usually involves a large number of atomic layers!) </p><p>5. If an alloy is cooled from high temperatures with a cooling rate T then the surface layers will assume the equilibrium concentration given by equation (6) as long as diffusion is fast enough. Below a critical temperature T,, diffusion will become too slow and the surface will retain the concentration clrr corre- sponding to T,,. </p><p>An estimate for T,, is given by: </p><p>(19) </p><p>and, using equations (7). (I 5) and (18) </p><p>D,,R T;, </p><p>- = 11.70(0.57C:, + Q) exp - O.51h;Tf Q. (20) </p><p>~ </p><p>Table I gives numerical data for a number of alloy systems. It should be noted that surface enrichment can become quite noticeable, even though the inter- action energy is estimated very conservatively by applying equation (5) to the first layer. .Actually. q, might become considerably bigger (of the order of V, IRT) if the atoms occupy half-crystal sites at the surface. </p><p>Acknowlrrlgmlents-The authors are indebted to Prof. Dr. W. Imrich for his patient discussion of their numerical problems and to the Rechenzentrum Graz as well as to VGEST-Alpine for use of their computer facilities. </p><p>LITERATCRE </p><p>Eshelby J. D.. Progress in Solid Mecharrics (edited by Sneddon and Hill), Vol. 2. pp. 86140. Sorth Holland, Amsterdam (1961). Jlger I., Phys. Srnrus Solidi (a) 33. 167 (1976). Liicke K. and Stliwe H. P.. in Rrcocrry anti Recrysm/li- xzrion of Metals. Wiley. NY (1963). Chandrasekhar D.. Rec. Mod. Phvs. 15. I (1943). Seith W.. D@rsion in Meralh Springer. NY (19%). Sorensen K. and Trumpy G., Php. Ret. (B) 7. 1719 (1973). Bernardini J. and Cabane J.. ,-tcru Mrt. 21. 1361 (1973). Kucera J.. iMillion B. and Peskova J. PhJs. Status SoLdi. Il. 361 (1972). Eckstein, Warmebehandlung von Stahl, VEB Deutscher Verlag f. Grundsroffindustrie. Leipzig (1971). Nohara K. and Hirano K. Proc. Im. Corrf: Sci. Tecknol. Iron Steel, Tokyo 6. 1267 (1970). </p></li></ul>

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