Introducton to Interfacal Segregaton Xaozhe Zhang 10/02/2015
Interfacal egregaton Segregaton n materal refer to the enrchment of a materal conttuent at a free urface or an nternal nterface of a materal. In a polycrytallne old, a egregaton te can be a dlocaton, gran boundary, tackng fault, or an nterface wth a precptate or econdary phae wthn the old. There are two recognzed type of egregaton: equlbrum egregaton and non-equlbrum egregaton
Interfacal Thermodynamc Follow Gbb. Interfacal energy denoted by (wth typcal unt of [mj/m 2 ]). It defned a the reverble work needed to create unt area of urface, at contant temperature, volume (or preure), and chemcal potental. Ue lqud-vapor urface a an example,.e., conder a ytem compoed of a lqud and a vapor phae at equlbrum, eparated by an nterface.
Interfacal exce nternal energy, E : E = E - E' E'' and nterfacal exce no. of mole of component : n = n - n ' - n " Excepton: no nterfacal exce volume, V = V' + V'' Internal energe of phae ' and '' are wrtten (a uual): de de TdS PdV TdS PdV dn dn and nterfacal exce nternal energy (no volume) wrtten: de T Interfacal exce properte de TdS da dn ds ds ds P dv dv da dn dn dn de TdS PdV da dn
Interfacal Thermodynamc All other thermodynamc properte can be wrtten from the prevou relaton: e.g. Gbb free energy, G = E + PV - TS dg de PdV VdP TdS SdT SdT VdP da Note! The urface exce quantte are arbtrary, nce they depend on precely where the dvdng urface located wthn the dffue regon aocated wth the nterface. We hall ee later how th ue may be addreed. dn
Gbb adorpton equaton and otherm The expreon for de, wrtten only n term of extenve ndependent varable (ds, da, dn ), whle the ntenve varable (T,, ) are contant n the equlbrated ytem. Th make ntegraton of de traghtforward: Re-dfferentatng yeld: E TS A n de TdS S dt da Ad Comparng wth the expreon for de, we conclude: S dt Ad n d 0 We now defne the followng pecfc nterfacal exce quantte: S / A and n / A where referred to a the adorpton of component dn n d
and obtan: d dt d Gbb adorpton equaton At contant temperature, th mplfe to the Gbb adorpton otherm. For a twocomponent ytem, t can be wrtten: d 1 d 1 2 d 2 Th may be mplfed further by the ue of the Gbb-Duhem equaton: n 1 d 1 n 2 d 2 0 It conventonal to take component 1 and 2 to be the olvent and olute, repectvely, and to elmnate 1 from the adorpton equaton: d 1 d 2 n 2 n 1 2 The l.h.. of the above equaton meaurable, therefore the r.h.. cannot be arbtrary.
Approxmate form of the Gbb adorpton otherm The chemcal potental may be expreed a: 2 = 2 + RT ln a 2, where a, the actvty and the tandard tate chemcal potental. For deal oluton, a 2 = x 2, where x 2 the mole or atom fracton. For dlute oluton, a 2 = k o x 2, where k o Henry' Law contant. In both of thee cae, d 2 = RT d(ln x 2 ). Thu, n both thoe cae the Gbb otherm may be wrtten: 1 RT d 1 d lnx 2 n 2 n 1 2 In partcular, n dlute oluton, n' 2 << n' 1, o that we can wrte: 1 RT d d lnx 2 2 Th the mot commonly ued form of the Gbb adorpton otherm.
Stattcal Thermodynamc Model of The Gbban approach not alway eay to ue Nether, nor t varaton wth compoton, are eay to meaure n old. 2 dffcult to meaure drectly, becaue nterfacal compoton profle can extend ome dtance from the nterface, and one mut determne the compoton profle of both component n the mot general cae. Gbban thermodynamc do not provde a relatonhp between 2 and 2. Wthout uch nformaton, the Gbb adorpton otherm cannot be ntegrated. Thee factor have led to the development of varou model whch can overcome ome of thee problem. Here we hall brefly decrbe one of the mplet of thee model. Segregaton
Monolayer (urface) egregaton model (regular oluton approxmaton) x 1 x x 1 x exp E eg RT x and x are the atom fracton of the olute n the urface monolayer and the bulk, repectvely, and eg the energy of egregaton,.e. the energy change reultng from exchangng a olute atom n the bulk wth a olvent atom at the urface. E eg ( B A ) 2 z (x x ) z v (x 1 ) 2 E el A and B are the urface energe of the pure component (B=olute), the area per mole n the monolayer, = AB ( AA + BB )/2 the regular oluton contant, j are bond energe between -j par of atom, the n-plane coordnaton and z z v half of the out-of plane bond made by an atom, and E el the elatc tran energy of a olute atom.
Typcal reult from th type of model 1 monolayer multlayer 15 x = 0.01; E el = 2727 J/mol A = 1650 mj/m 2 ; B = 918 mj/m 2 0.8 10 layer atom fracton 0.6 0.4 0.2 x (bulk) = 0.1 ln(x /1-x ) 5 0 = 562 J/mol = 0 J/mol = -562 J/mol 0 1 2 3 4 5 6 7-5 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 layer number 1/T(K) Alo, the relatonhp between x and : (x x)/ From th relaton, and the one between x and x, t poble to ntegrate the Gbb adorpton otherm and obtan:
RT where k a urface enrchment factor gven by ln(1 kx) x x kx /(1 kx) (mj/m 2 ) 1500 1400 1300 1200 1100 1000 T=1000 K T=800 K T=600 K T=400 K 900 0.0001 0.001 0.01 0.1 1 Note! The urface energy not proportonal to the adorpton, rather t the lope, d/d, that proportonal to adorpton. x
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