毕业论文(英文版)Interface free energy or surface tension_ definition and basic properties.pdf
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1、 Interface Free Energy or Surface Tension:definition and basic propertiesC.-E.PfisterEPF-L,Institut d5analyse et calcul scientifiqueBatiment MA,Station 8CH-1015 Lausanne,Switzerland e-mail:char les.pfister epfl.chNovember 2009AbstractInterface free energy is the con tribution to the free en erg y of
2、 a sy stem due to the presen ce of an in terface separatin g tw o coexistin g phases at eq uilibrium.It is also called in terfacial free en erg y or surface ten sion.The con ten t of the paper is 1)the defin ition of the in terface free en erg y from first prin ciples of statistical m echan ics;2)a
3、detailed exposition of its basic properties.We con sider lattice m odels w ith short ran g e in teraction s,like the Isin g m odel.A n ice feature of lattice m odels is that the in terface free en erg y is an isotropic so that son ic results are pertin en t to the case of a cry stal in eq uilibrium
4、w ith its v apor.The results of section:2:hold in full g en erality.1 Interface free energy in Statistical mechanical1.1 Definition of the interface free energyCon sider a phy sical sy stem at equilibrium in a v essel V,at a first order phase tran sition poin t,w here tw o bulk phases,say A an d B,c
5、oexist(for exam ple,an Isin g m odel at zero m ag n etic field an d low tem perature).If,w hen w e brin g in to con tact the phases A an d B,the state of the sy stem is in hom og en eous an d there is spatial separation of the tw o phases,then at the com m on boun dary of the tw o phases em erg es a
6、 spatially localized structure,called the in terface.In spite of the low dim en sion ality of these in terfaces an d their n eg lig ible con tribution tow ards the g lobal ov erall properties of the phy sical sy stem their presen ce is essen tial for a w ealth of im portan t processes in phy sics,ch
7、em istry an d biolog y.Here w e con sider on ly sy stem s at eq uilibrium,w hich is a rather sev ere restriction.The in terface free en erg y is a therm ody n am ical q uan tity an d it is best explain ed w hen w e con sider the m acroscopic scale,in w hich the len g th of the v essel con tain in g
8、12the sy stem is the referen ce len g th.Un der this scale the in terface is w ell-defin ed an d localized.In the case of a flat in terface perpen dicular to a un it v ector n it is described m athem atically by a plan e perpen dicular to n.w hich separates the bulk phases,an d the state of the sy s
9、tem is specified abov e this plan e by g iv in g the v alue of the orderparam eter of on e of the bulk phases,say A,an d below the in terface that of the other bulk phase.The in terface free en erg y or surface ten sion(per un it area)r(n)describes the t her m o dy n am ical properties of the in ter
10、face at eq uilibrium.How does on e obtain r(n)on ce the in teratom ic in teraction s of the sy stem are g iv en?We can an sw er this q uestion so that w e g et in terestin g in form ation about r(n)on ly for few m odels.How ev er,the w ay of defin in g r(n)is q uite g en eral an d can be applied in
11、prin ciple to m ost sy stem s,an d its orig in can be traced back to the m on um en tal w ork of J.W.Gibbs,On the E quilibrium of Heterogeneous Substances(1875-1878).In statistical m echan ics the therm ody n am ical fun ction s are obtain ed by com putin g the partition fun ction.The sy stem is en
12、closed in a v essel V;takin g in to accoun t the in teration s of the sy stem w ith the w alls of V w e can w rite an expression for the ov erall free en erg y of the sy stem.The basic postulate is that w e can separate the v arious con tribution s to the ov erall free en erg y F(V)at in v erse tem
13、perature(3 in to tw o parts,up to a sm all correction term;on e part is proportion al to the v olum e of V,w hich is the bulk free en erg y of the sy stem,an d an other on e is proportion al to the area of the surface of V,w hich is in terpreted as the w all free en erg y.Thus,at a poin t of first o
14、rder phase tran sition,w hen on ly phase A is presen t,FaV=-iln Zx(V)=/buik(A)IH+fwA)dV+odV),(1.1)w here Za(V)den otes the partition fun ction of the sy stem for phase A,|V|the v olum e of V an d dV the area of the boun dary dV of the v essel.A sim ilar expression holds for phase B.The bulk term s/b
15、ulk(4)an d 九3“3)are the sam e because the sy stem is at a first order phase tran sition poin t,but the surface term s fWSL(A)an d m ay be differen t.Un der specific con dition s on the w alls,w e can obtain(m acroscopic)in hom og en eous states w ith plan ar in terfaces separatin g the tw o coexisti
16、n g bulk phases.In such cases there is an addition al con tribution to the ov erall free en erg y an d w e postulate that the free en erg y can be w ritten asFAb(V)=-1 InZAB(V)=/buik(lB)|V|+/w all(AB)|5V|+r(n)|/(n)|+o(|aV|),(1.2)*w ith/w all(AB)a/w all(4)+(1 a)/w all(3).The term|Z(n)|=OdV)is the are
17、a of the in terface perpen dicular to the un it v ector n,an d a is the proportion of the w alls in con tact w ith phase A.S in ce the sy stem is at a first order tran sition poin t/bulk(4B)=/bulk(4)=/buik(B).Extractin g w all free en erg ies is n ot easy,but it is n ot n ecessary to do this if our
18、postulate is correct,because w e can elim in ate the term s in v olv in g/w an(AB)an d/w an(AB)by con siderin g the ratio of partition fun ction s,小马舞产+(1.3)Notice that QI.3。is alw ay s a term of order O(dV).3An obv ious difficulty in g ettin g r(n)is that w e m ust kn ow the v alues of therm odyn a
19、m ical param eters of the sy stem for w hich there is phase coexisten ce.In deed,for other v alues of these param eters the sy stem has on ly on e bulk phase an d there is n o in terface.Hen ce the surface ten sion is n on-zero on ly for a specific ran g e of v alues of the therm odyn am ical param
20、eters of the sy stem.This is w hy in m an y situation s on e proceeds differen tly in Phy sics.O n e m odels directly the in terface in order to by pass these problem s an d then the in terface free en erg y is sim ply iden tified w ith the free en erg y of the m odel for w hich on e has stan dard m
21、 ethods for ev aluatin g it.This is often an adeq uate w ay to proceed,but it can n ot be applied alw ay s.For exam ple w hen on e is study in g how the coexistin g phases are spatially distributed in side the v essel V,w e can n ot av oid con siderin g the free en erg y of in terfaces between coexi
22、sting phases.1.2 A paradigm,the Ising modelWe iin plein en t the ideas of section 11.11 for the Isin g m odel,for w hich the m athem atical results are the m ost com plete.We con sider the three-dim en sion al Isin g m odel.The tw o-dim en sion al case is also of in terest.Let Z3:=t=(力i,力2,力3):&G Z
23、an dAlm:=t e Z3:m ax(|,t2)L,t3 0,a fixed v alue.This in teraction fav ors the alig n m en t of spin s,sin ce the en erg y is in in iin al w hen(r(t)=(r(力).To m odel the in teraction of the sy stem w ith the w alls,w e in troduce an in hom og en eous m ag n etic field actin g on ly on the spin s loca
24、ted at the boun dary of the box 八”河,畋.3)=-为。,or max(任Y2I)=Lw ith J 0 an d r)(t)=1 or r)(t)=1,but fixed.Differen t kin d of w alls are m odeled by specify in g differen t v alues for 77(力)(see below)an d choosin g differen t v alues for J.The ov erall en erg y of the sy stem is Hlm+Accordin g to stat
25、istical m echan ics,the state of the sy stem at eq uilibrium an d at in v erse tem perature(3 is the Gibbs m easure,so thate+w2M)Prob(a)=w here Z3/=e一处71此+四标2)(T,4The n orm alization con stan t ZM is the partition fun ction an d the ov erall free en erg y of the sy stem in the box Mm is理m(瓦/):=-iln
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