毕业论文（英文版）Interface free energy or surface tension_ definition and basic properties.pdf

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 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 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 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,coexist(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 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,chem 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 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 stem 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 terface 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 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 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 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 order 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/bulk(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 coexistin 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 area 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 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 am 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 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 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 coexisting 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 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 located 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 statistical 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 ZM.pAt the therm ody n am ical lim it(in fin ite-v olum e lim it),the bulk free en erg y per spin is in depen den t on the choice of J 0 an d of 77,/bulk(a h)=lim也(反 h,J).L00The m odel exhibits a first order phase tran sition w ith the m ag n etization as order-param eter if the extern al m ag n etic field h=0 an d the in v erse tem perature 0 0c(d),w here/3C(/)is the in v erse critical tem perature of the(/-dim en sion al Isin g m odel(0 仇(3)(3cd)the spin-flip sy m m etry of Hlm is broken.There is a phase w ith positiv e m ag n etization m*(/3)an d an other w ith n eg ativ e m ag n etization m*;the bulk free en erg y/buik(/5,h)is n ot differen tiable at 九=0,d d0 0 the in teraction s w ith the w alls fav or the bulk phase w ith positiv e,respectiv ely n eg ativ e,m ag n etization.We are also in terested in the case of m ixed boun dary con dition s,w hich is related to the em erg en ce of a plan ar in terface perpen dicular to n.Let n=(%，,%).We set/、+1 if tin1+力2n2+13n3 0,77：=.1 if+12n2+13TI3 0c(),d 2,3.We choose 0 a 0 the probability m easure on the den sity profiles becom es con cen trated on the un iq ue m ag n etization profile p(x)=m*(3)；this con stan t profile describes the m acroscopic state of the+-phase of the m odel.The m acroscopic lim it correspon ds to the reg im e of the law of larg e n um bers in probability theory.To obtain r(n)w e use(11.3b an d the fact that by sy m m etry of the m odel ZL=so that w e on ly n eed to com pare ZL an d ZL.We setr(n)=例7r(n)n V l-8 Ldr(1.4)The term|7r(n)Q V is the area of the in tersection of the plan e 7r(n)w ith V.O n e can prov e:the limit Q1.4。is independent on J J,and for 0 0c(d)the function r(n)verifies properties a),b)and c)of d2.7h:in the macroscopic limit the measure on the density profiles is concentrated on the unique magnetization profilePn（N）：=if x is abov e 7r(n)m*if x is below 7r(n).This profile describes a m acroscopic state w ith a plan ar in terface perpen dicular to n.Therefore r(n)can be in terpreted as the free en erg y of that in terface perpen dicular to n.The con dition J J is im portan t,because for som e v alues of J J has a sim ple phy sical in terpretation;it en sures that the w alls of the box V are in the com plete w ettin g reg im e,so that the in terface can n ot be pin n ed to the w alls.In the literature the stan dard choice for ferrom ag n etic m odels is J=J,so that(jl.4l)g iv es the correct defin ition of r(n).These results illustrate the fact that it is im portan t to choose correctly the in teraction s of the sy stem w ith the w alls in order to use defin ition(11.3h.O n e m ust av oid the possibility of pin n in g of the in terface to the w alls.An y w all in teraction s w hich in duce a m acroscopic state w ith an in terface perpen dicular to n an d such that otherw ise(Q is in depen den t of the chosen in teraction s are adm issible for defin in g the in terface free en erg y.S ev eral other defin ition s for r(n)hav e been proposed for the Isin g or sim ilar m odels.Most of them in v olv e a ratio of partition fun ction s an d are based on the sam e pattern leadin g to Q)so that w e shall n ot rev iew them here(see section for referen ces).A possibility of av oidin g the abov e problem w ith the w alls is to suppress(partially)the w alls of the sy stem by takin g(partial)periodic boun dary con dition s.Then on e im poses a con dition im ply in g the existen ce of a sin g le plan ar in terface perpen dicular to n.There are also v arian ts of(I1.4F)w here on e con siders a box in stead of All an d take first the lim it A/oo before takin g L oo.When J J,then on e can take the lim its in an y order,first L oo an d then Af oo or v ice-v ersa,or sim ultan eously oo an d M oo.The reason is that the w alls are in the com plete w ettin g reg im e an d the in terface is n ot pin n ed to the w alls.In g en eral it is n ot easy to show that reason able defin ition s g iv e the sam e v alue for r(n).The surface ten sion for the tw o-dim en sion al Isin g m odel can be com puted exactly.O n sag er com puted the in terface free en erg y for n=(0,1),/5t(0,1)=2(K K*),(3 仇 an d t(0,1)=0,otherw ise,w here K*is defin ed by exp(2K*)=tan h K an d K=(3J.O n sag er did n ot use the defin ition(H3D;the com putation of t(0,1)defin ed by Q1.4j)is due to Abraham an d Martin-Lof.The full in terface free en erg y has been com puted by McCoy an d Wu.1.2.2 Inequalities for t in the Ising modelWe set in this subsectionn():=(0,sin 0,cos 0)an d r():=r(n().There are tw o in eq ualities w hich relate the in terface free en erg y an d the order-param eter,t(0)0 iff there is a phase tran sition.S in ce t(0)is a m in im al v alue,w e hav e r(n)0 iff 九=0 an d(3 Be.7We n ow in troduce the step free energy 7tep-This q uan tity is in terestin g on ly for d=3.Let be defin ed as*/、+1 if 力30 or if 力3=0 an d t2 0,n(t)c c1 if 力3 0 or if 力3=0 an d t2|sin|Tstep(1.7)g iv es in form ation about the n on-differen tiability of r at 6*=0.Positiv ity of 7tep im plies that 7(。)is n ot differen tiable at 9=0 sin ce t has a m in iin um at 9=0 an ddlilll T(6)Tsten-010 de j pThe phy sical con seq uen ce of the n on-differen tiability of t is explain ed at the en d of section 12.2.31 n on-differen tiability of t im plies the existen ce of a facet for the eq uilibrium shape WT defin ed in(12.11).The secon d in eq uality relates the step free en erg y of the three-dim en sion al m odel to the tw o-dim en sion al in terface free en erg y,w hich w e w rite here,step 7-2(0).(1.8)Therefore,if/3 a(2)then 7tep 0-2 Basic properties of the interface free energyWe exam in e in this section w hat are the basic properties of t an d w e discuss the therm ody n am ical stability of in terfaces.2.1 Convexity of the interface free energyWe assum e that r(n)0 for each un it v ector n.We con sider the three-dim en sion al case.Elem en ts of the Euclidean space E3 are w ritten x=(%i,%2,?3)；w e den ote the Euclidean scalar product by(x|y):=71g l+x2y2+&券,an d the Euclidean n orm by|x|.By con v en tion r(n),w ith|n|=1.is the phy sical v alue of the in terface free en erg y of an in terface perpen dicular to n.It is con v en ien t to exten d the defin ition of t to E3,as a positiv ely hom og en eous fun ction,by settin g7(x)：=|x|r(x/|x|).8We in troduce the half-spaceH(n):=x:(x|n)r(n),w hose boun dary is the plan e dH(n)=x:(x|n)=r(n).Notice that=H(tn)for all 力0.Besides w e defin e the equilibrium shape WT as the con v ex set,w hich is the in tersection of the half-spaces that isWT:=x:(x|n)r(n),V n .(2.1)The n ext arg um en t is due to Herrin g|H(1951)1;it is reproduced alm ost in its orig in al form.Let T(Aq,A,A2,A3)be the tetrahedron w ith v ertices Ao,Ai,A2,A3.The face opposite to the v ertex is den oted by Ai;it con tain s all v ertices Aj,j*i,an d its area is I 小.Let rii be the outw ard un it n orm al to the face We hav eIo Ino+|i lni+仆21112+|Ab In；3=0.S etAln=一.=国小+We com pare the free en erg ies|Ao|(n)an d|Ai|r(n i)+人河山)+|A3|r(n 3),or w hich is the sam e,r(n)an d目Nm)+4(n2)+搭心)|Ao|4o|Let mi,m2,m3 be reciprocal v ectors to ni,n2,n3,defin ed by the relation s mJ%)=0if E#j an d(m jn i)=1 otherw ise.We hav e(m jn)=|A.J/|Ao|,so that 昌 7(m)=7(112)01,111)=(y|n).The v ector y=r(n?；)m7：v erifies the iden tities(y|n?；)=r(n?；),i 1,2,3,i.e.y isthe in tersection poin t of the three plan es 5Z/(n i),5Z/(n2)an d 57/(n3).We hav e three cases.(1).If(y|n)r(n),then the in tersection of the plan e dHn)w ith WT is em pty,sin ce WT is a subset of 7/(n i)n/(n2)ri7/(n 3).A(hy