THERMODYNAMIC OF URFACE AND INTERFACE 1. Intoduction Eveything has to end somewhee. Fo solids, o liquids that "somewhee" is a suface, o an inteface between phases. Fo liquids, the inteface is between the liquid and a vapo phase o between two immiscible liquids. Fo solids, the inteface can be between the solid and a vapo o liquid phase, between two chemically diffeent solids, o between cystals (gains) of the same chemical composition but diffeing in cystallogaphic oientation. The popeties of these intefaces diffe fom bulk popeties and they influence the popeties of mateials in many ways. An atom at a fee suface of a solid has geate enegy than an atom in the inteio (less tightly bonded). uface atoms ae at highe enegy levels. The sum of all excess enegies (excess with espect to atoms inside) of the suface atoms is the suface enegy. uface enegy () can be defined in tems of U, H, A o G depending on the physical conditions: U H A G, V, P T, V T, P A A A A The most commonly used one is based on G. Then dg = V dp - dt + da Thee ae seveal commonly obseved manifestations of this suface enegy. As a liquid doplet ties to minimize its enegy, it assumes a spheical shape- the shape that minimizes the suface aea. malle doplets tends to agglomeate into lage doplets. In sinteing small paticles educe suface to volume atios.. uface Tension: The Mechanical Analog of uface Enegy A film of liquid pulled by a foce F, theeby ceating new suface (two sufaces top and bottom), the balance of foces and the wok done can be expessed as 1
l F dx l dx = F dx F l The units of suface tension,, ae foce/length. By multiplying both the numeato and denominato of by a length tem, the dimensions become enegy/aea. Thus, can be thought of as suface tension o suface enegy. 3. Appoximate Calculation of olid uface Enegies Assuming binding enegy of an atom to a solid is the esult of discete bonds to its neaest neighbos, then enegy of one bond,, H s 0. 5z N A H s : mola enthalpy of sublimation (beaking all the bonds) z: coodination numbe N A : Avagado's numbe Thee ae 0.5 z N A bonds pe mole. If we cleave an FCC cystal along (111) plane, 3 bonds pe atom will be boken. Because thee ae two sufaces fomed, the wok equied to fom the sufaces will be 3/ pe suface atom. Then the wok pe suface atom is
W 3 H ( fo FCC z 1) 4N A The suface enegy is then H 4N A N A whee (N/A) is the numbe of atoms pe unit aea. Fo FCC, along (111) plane N/A=4/( 3a 0 ). Whee a o is the lattice paamete. Fo a (100) plane thee ae 4 boken bonds pe atom, and the numbe of atoms pe unit aea is / a 0 (a) (b) Atomic packing (a) on the (111) plane, and (b) on the (100) plane of an FCC cystal. 3
Fo coppe H s = 170000 J/mole and a o = 3.615 A. Using (100) plane, = 1440 egs/ Expeimental esults indicate that this value is about 1600 egs/. This shows that computation gives an inaccuate but easonable values. As expected a simple model of this kind is useful in detemining appoximate values and visulizing the physical phonemanon of the of the system. Howeve, fo accuate deteminations othe changes that takes place in the vicinity of the suface has to be taken into account. 4. Effect of uface Cuvatue In physical systems suface between two phases is not infinitely thin, geometical plane. They extend to seval atomic layes. In a system with only one component, C, a geometical suface can be located by assigning no mass to the suface. That is the shaded aea on both sides of the suface ae set equal. A B Phase I [C] Phase II Distance uface uface between two phases: I and II,. If this suface moves between A and B, the volumes of the phases will change: phase I inceases, phase II deceases. chematically A phase I B phase II The total change of Gibbs enegy dg = GI dni + GII dnii + da = Wev 4
dn I and dn II ae changes in the masses as infinitesimal movement of suface takes place. At equilibium W ev is zeo. Fo one component system, G = = Go I dni + II dnii + da = 0 dni = -dnii = dn Then, II - I = da dn If the suface is flat (infinite adius of cuvatue), then da/dn is zeo, because the aea of the inteface does not change as mass moves fom phase I to phase II. Thus, I = II, this is sometimes witten as I = II = chemical potential of a mateial with an infinite adius of cuvatue. Fo a cuved suface, the aea changes as the bounday moves. Intoducing the mola volume V dv dn II - I = da dn = V da dv The atio da/dv is not zeo fo a cuved suface. Fo a spheical suface phase I phase II V = 3/3 and A = whee is the solid angle subtended by the potion of the suface being consideed: dv = d and da = d da/dv = / Theefoe, II - I = V whee V mola volume and is the intefacial (suface) enegy between phases I and II. 5
Compae two conditions fo the same mateial; I-flat suface, II- suface with finite adius of cuvatue. Then, the diffence in chemical potentials of flat suface,, and the suface with finite adius of cuvatue, is given by - = V Thus, chemical potential of a mateial with a finite adius of cuvatue is geate than the chemical potential fo that same mateial with an infinite adius of cuvatue. This diffeence in manifests itself in seveal ways. Fo nonspheical, cuved suface with pincipal adii of cuvatues 1 and, 1, = V ( 1 1 ) 1 Equation that shows the diffeence between the chemical potentials of flat suface, and the suface with finite adius of cuvatue can also be deived using mechanical agument combined with themodynamics. If we think the suface as a membane suounding the condensed phase, and the suface tension as the stess in the membane, we can establish the pessue diffeence accoss the cuved suface though a foce balance. P o P i The foce tending to push the two halves of the sphee apat is the poduct of the coss-sectional aea at the mid point ( ) and the diffeence in pessue between inside and outside of the sphee (P i - P o = P). This foce ( P) is balanced by the suface tension foce ( ). Thus P = Then, P = This inceased pessue on the condensed phase tanslates into a diffeence in chemical potential. d = V dp (at constant T) 6
Integation between two limits yields above elationship - = V Compute the pessue in a one micon diamete wate doplet fomed in supecooled wate vapo at 80 o C. The vapo pessue of wate at 80 o C is 0.5 atm. The suface enegy of wate may be taken as 80 egs/. P = egs 7 80 10 J 4 10 eg m 30000 N 6 ( Pa) 0. 510 m m P = 30000 Pa 3. 16 atm. 101333 atm Pa Pi = 3.16 + Po = 3.16 + 0.5 = 3.68 atm. 4.1. Effect of uface Cuvatue on Vapo Pessue A condensed mateial (solid o liquid) is in equilibium with its vapo. At equilibium, chemical potential of the condensed phase, c, is equal to that of its vapo, v. This is valid fo both flat (a) and cuved (b) sufaces. V V (a) (b) c, = v, c, = v, c, - c, = V Fom chemical potentials c, - c, = v, - v, = R T ln P P o o 7
whee P o is the vapo pessue of a paticle of adius. P o pessue ove a flat suface. Theefoe, ln P o = V P RT o is the equilibium vapo Calculate and plot as function of adius the vapo pessue in a system with liquid zinc doplets suspended in zinc vapo at 900 K.The vapo pessue of zinc ove bulk liquid at this tempeatue is.5x10 - atm; the mola volume of the liquid is 9.5 cc/mol and = 380 egs/. ln P o = V 3 egs 9 5 380 8. = mol 7 9. 6 10 10 J eg P RT 8. 314 900 K ( ) ( ) o J mol K 9. 610 8 ( ) o P ( atm). 510 e = 10-3 ; P o =.500x10 - atm = 10-4 ; P o =.504x10 - atm = 10-5 ; P o =.54x10 - atm = 10-6 ; P o =.75x10 - atm P o x10 (atm).75.7.65.6.55.5.45 0.01 0.1 1 10 x104 () 4.. Phase Bounday hift in Unay ystems 8
Conside a unay two phase (+) system in which the cuvatue of the - bounday plays a ole in the conditions fo equilibium. T = T P = ; P = P + = d = V dp - dt d = V dp - dt = V d(p + ) - dt V dp - dt d = V d(p + ) - dt eaangement yields, ( - ) dt - (V - V) dp - V d = 0 By seting - = t(/) and V - V = Vt(/), above equation can be t(/) dt - Vt(/) dp - V d = 0 Integating between two limits (adius of cuvatue) gives the esult of cuvatue on phase bounday shift due to cuvatue. The fist two tems ae identical with the Clausius-Clapeyon equation in which vaiations due to geometic effects ae not consideed; the thid tem contains the effect of cuvatue on the equilibium conditions. 4.3. olubility of mall Paticles To simplify the poblem, conside an A-B system with a solubility line as shown (assuming vanishingly small solubility fo A in B). T 1 1 T a B 0 X B X B at T 1 the composition of the phase in equilibium with the phase is X B,. Hence a B = 1 at X B = X B,. phase is taken as pue B, fo simplification. 9
If mateial B exists as small spheical paticles, the chemical potential of B as a function of its adius of cuvatue is B, - B, = V The suface enegy, - is the intefacial enegy between pue B and. B, - B, = R T (ln ab, - ln ab,) If B obeys Heny's law in phase, then ab, = B o XB, and ab, = B o XB, Combination of elationships yields, B, - B, = R T ln X X Then, ln X X B, B, = V RT whee X B, is the solubility of B in A () when adius of cuvatue fo B paticles is, X B, is solubility when B has an infinite adius of cuvatue (flat suface). The solubility of B inceases as its paticle size deceases. If B exists as small sphees and lage sphees, the solubility of B will vay with paticle size. In the mateial immediately suounding the small paticles, the concentation of B will be highe. If sample is held at high tempeatue whee B atoms ae mobile, the small paticles will tend to dissolve and the lage paticles will gow. This is coasening. B, B, 4.4. Melting Tempeatue of mall Paticles To examine the effect of cuvatue upon solid-liquid equilibia, compae the solid/liquid equlibia fo (a) infinite and (b) finite adius of cuvatue. (a) (b) When a paticle of solid in equilibium with its liquid. Ts = Tl s = l Pessue diffeence in solid and liquid ae elated by; P = ls Ps = Pl + ls 10
dl = Vl dpl - l dt ds = Vs dps - s dt = Vs d(pl + l s ) - s dt at equilibium, s = l, hence ds = dl Vl dpl - l dt = Vs d(pl + l s ) - s dt eaangement yields, (l - s) dt - (Vl - Vs) dpl - Vs l-s d = 0 Assuming that we ae not concened with oveall pessue changes on the liquid (dpl) and l - s = m m dt = Vs l-s d Integating fom the melting tempeatue of a lage paticle (=, T, adius of cuvatue) to melting point of a paticle with adius (, T ). Assuming m constant, m dt = Vs l-s d m T = - Vs l-s 1 V ls Then, T = m 1 When a liquid is cooled below its bulk melting point the solid phase foms and gows as dendites. The scale of the esulting micostuctue, indicating whethe it is fine gained o a coase stuctue and the ate at which it solidifies, is stongly influenced by the size of tips as they gow. The tip adius is in tun detemined by the tempeatue diffeence between the liquid adjacent to tip and that of the supecooled liquid suounding it. Calculate this tempeatue diffeence fo a silicon dendite with a tip adius of 0.1 micons gowing into suounding liquid that is undecooled 5 o C below the bulk melting point. The suface enegy of the solid-liquid inteface may be taken as 150 egs/. Assuming the liquid and solid at the tip of the dendite ae in local equilibium. The tempeatue of the liquid at the inteface can be calculated fom T = 11. 3 egs V ls 1 mol 1 7 10 J 4 m 46500 mol 0.110 1683K 150 T = -1. K Ti = 1683-1. = 1681.8 K Tl = 1683-5 = 1678 K J eg 5. The Equilibium hape of Cystals 11
If suface enegies of a condensed phase is isotopic, as in liquid, then based on common expience the condensed phase will assume a spheical shape at equilibium. If suface enegy vaies with cystallogaphic oientation, as in a solid, it is expected that the equilibium shape to be detemined by the elative suface enegies of the cystallogaphic planes. The pesence of low suface enegy planes would be pefeed and the equlibium shape would not be a sphee. The equlibium shape of a cystal is detemined by minimizing its suface fee enegy. If a cystal may be found by planes 1,, 3,... whose exposed aea will be A 1, A, A 3,..., the equilibium shape will be the one that minimizes i A i. A gaphical method fo doing this uses a Wulff plot. In this method, the suface enegy of a cystal is ploted in pola coodinates. Planes ae dawn at each point nomal to the adius. The equilibium shape will be defined by the inteio suface of the planes eected pependicula to the pola plot. The development of facetes in two dimensions can be illustated by a simple example. Conside the two-dimensional cubic cystal shown (110) c a (100) (100) a - / c When looking in (100) diection, planes (110) and (100) constitutes the exteio of the suface. a and c dimensions of the cystal will be calculating by minimizing the 1
suface enegy E s at constant cystal aea, A. Cystal is symmetial in the vetical and hoizontal diections; hence only the uppe ight quadant may be isolated fo analysis. Es = a c o + c 1 A = a - c 4 The suface enegy is minimized by setting the diffeential of E s with espect to c equal to zeo at costant A. The esult is c a 1 0 the atio c/a is a measue of the elative amounts of exposed (110) and (100) planes. If the suface enegies of planes ae equal, o = 1, then c = a c and the size of the two facets will be equal. 13