Bailey, CL

Harrison, NM

Mukhopadhyay, S

Searle, BG

Wander, A

Spiral - Imperial College Digital Repository

Ab Initio Surface Thermodynamics inMulti Component Environments C L Bailey, A Wander, S Mukhopadhyay, B G SearleN M Harrison September 2007   Technical Report DL-TR-2007-004©2007 Science and Technology Facilities Council  Enquiries about copyright, reproduction and requests for additional copies of this report should be addressed to:  Chadwick Library Science and Technology Facilities Council Daresbury Laboratory Daresbury Science and Innovation Campus Warrington WA4 4AD  Tel: +44(0)1925 603397 Fax: +44(0)1925 603779 email: library@dl.ac.uk  Science and Technology Facilities Council reports are available online at: http://epubs.cclrc.ac.uk/                         ISSN 1362-0207    Neither the Council nor the Laboratory accept any responsibility for loss or damage arising from the use of information contained in any of their reports or in any communication about their tests or investigations. Ab Initio Surface Thermodynamics in Multi ComponentEnvironmentsC.L. Bailey1, A. Wander1, S. Mukhopadhyay2, B.G. Searle1 and N.M Harrison1,21Computational Science and Engineering Department,Science and Technology Facilities Council,Daresbury Laboratory, Daresbury,Warrington, Cheshire,WA4 4AD, United Kingdom2 Department of Chemistry,Imperial College London,Exhibition Road, London,SW7 2AZ, United KingdomAbstractThe methodology for the calculation of the phase stability of different surfaces in contact witha multi component environment of gases is presented. The evaluation of the surface free energy ofa system is discussed in detail including the use of thermodynamic data tables.1I. METHODOLOGYThe relative stability of surfaces with different stoichiometries is determined by the com-parison of their surface free energy. The methodology used to calculate the surface freeenergy at a finite temperature and pressure has been developed for metal oxide systems1,2,3and extended to multicomponent environments4. Previously we have outlined a methodol-ogy to allow the evaluation of the surface free energy of AlF3(0112) in the presence of HFand H2O5. In the present work we present a full derivation and discussion of this methodol-ogy. Although expressed in terms of AlF3 the method is applicable to any multicomponentgas phase environment.At a given temperature a solid is in equilibrium with its vapour whenµvapour(T, Pvap) = µsolid(T, Ptot) (1)where Pvap is the vapour pressure of the material and Ptot is the total pressure of the system.Ptot enters equation 1 via the pressure dependence of the PV term in the Gibbs free energy ofthe solid. Consequently, the variation depends upon Ptot rather than any individual partialpressure. In the following derivation any terms which have a dependency on Ptot will be inthe solid phase but could be rewritten in terms of the vapour pressure of that phase.As an example we consider an AlF3 slab with adsorbed hydroxyl groups, water and HFexposed to an atmosphere containing gaseous H2O, H2 and HF. The methodology is triviallyextended to other systems. Modelling the AlF3 surface including adsorbed HF, water andhydroxyl groups as a slab of material periodic in two dimensions and of finite thickness inthe third, we define the surface free energy as6γ(T, PF2 , PO2 , PH2 , Ptot) =12A[Gslab(T, Ptot)−NAlµAl(T, Ptot)−NF12µF2(T, PF2)−NO12µO2(T, PO2)−NH12µH2(T, PH2)] (2)where A is the surface area of the unit cell (the factor of 2 accounts for both sides of theslab). Gslab is the Gibbs free energy per unit cell of the slab and NAl, NF , NO and NH arerespectively the total number of aluminium, fluorine, oxygen and hydrogen ions within theslab. PF2 , PO2 and PH2 are the partial pressures of the F2, O2 and H2 molecules respectivelyand Ptot is the total pressure of the system. Note that for aluminium we could express thechemical potential as µAl(PAl) where PAl is the equilibrium vapour pressure of Al atoms2above a solid sample at the given temperature and total pressure. µAl, µF2 , µO2 and µH2 arethe chemical potentials for aluminium, fluorine, oxygen and hydrogen respectively.It is assumed that bulk aluminium fluoride is in equilibrium with aluminium and fluorinein their natural states, consequentlyGbulk(T, Ptot) = mµAl(T, Ptot) + n12µF2(T, PF2) (3)where Gbulk is the Gibbs free energy per formula unit of the bulk crystal having stoichiometryAlmFn. Similarly, HF and H2O are in equilibrium with their constituent atoms, hence12µH2(T, PH2) +12µF2(T, PF2) = µHF (T, PHF ) (4)µH2(T, PH2) +12µO2(T, PO2) = µH2O(T, PH2O) (5)Using equations 3, 4 and 5 we can eliminate µAl, µF2 and µO2 from equation 2.γ(T, PHF , PH2O, PH2) =12A{Gslab(T, Ptot)−NAlmGbulk(T, Ptot)−(NF −nmNAl) [µHF (T, PHF )−12µH2(T, PH2)]−NO [µH2O(T, PH2O)− µH2(T, PH2)]−NH12[µH2(T, PH2)]}(6)Re-arranging this equation we obtainγ(T, PHF , PH2O, PH2) =12A{Gslab(T, Ptot)−NAlmGbulk(T, Ptot)−(NF −nmNAl)µHF (T, PHF )−NOµH2O(T, PH2O)−12(nmNAl −NF − 2NO + NH)µH2(T, PH2)}(7)Alternatively we could express the surface free energy as a function of µO2(T, PO2) by sub-stituting for µH2(T, PH2) from equation 5.γ(T, PHF , PH2O, PO2 , Ptot) =12A{Gslab(T, Ptot)−NAlmGbulk(T, Ptot)−(NF −nmNAl)µHF (T, PHF )−12(nmNAl −NF + NH)µH2O(T, PH2O)−14(NF −nmNAl + 2NO −NH)µO2(T, PO2)}(8)The chemical potential of an ideal gas can be written asµX(T, PX) = µX(T, P◦X) + kT ln(PXP ◦X)(9)3We can therefore calculate the chemical potential at any pressure if we know the value ofµX(T, PX) at a given pressure P◦X .The above derivation is relative to the energy zero of classical thermodynamics wherethe energy of formation of an element in its standard state at standard temperature andpressure is zero. However, we need to convert to the energy zero of the DFT calculations sothat Gbulk and Gslab are simply the DFT total energies of the system. To do this we writeequation 9 asµX(T, PX) = µX(0, P◦X) + [µX(T, P◦X)− µX(0, P ◦X)] + kT ln(PXP ◦X)(10)where the term in square brackets is now the change in the chemical potential in movingfrom T = 0 to T = T at constant pressure P ◦X .µX(T, PX) = µX(0, P◦X) + ∆µX(P◦X) |T=TT=0 +kT ln(PXP ◦X)(11)we can define the chemical potential on the DFT energy scale asµ′X(T, PX) = µX(T, PX)− µX(0, P ◦X) + EDFT (T = 0) (12)Traditionally, phase diagrams in the literature2,3 have been plotted as a function ofµ∗X(T, PX) = ∆µX(P◦X) |T=TT=0 +kT ln(PXP ◦X)(13)which does not include the term µX(0, P◦X) which is the enthalpy of the compound at T=0K.The term ∆µX(P◦X) |T=TT=0 in equations 11 and 13 and implicitly in equation 12 can beobtained from thermodynamical reference tables7, as described in section III. The valuesthat we have used in this paper are displayed in table I. Formally µ is a Gibb’s free energywhile EDFT is an enthalpy, H. These are related viaG(T, P ) = H(T, P )− TS(T, P ) (14)where S is the entropy of the system, but at T = 0, G and H become identical.The Gibbs free energies of the slab and bulk crystal are computed at the athermal limitand their temperature dependence is ignored as it is negligible compared to that of thegas. Correction to finite temperature is possible by either molecular dynamics simulationor the calculation of the lattice dynamics and the use of the quasiharmonic approximation.However, as the current article is concerned with the qualitative behaviour of the surface4TABLE I: Calculated values ∆µ at 1 atm obtained from thermochemical tables7.Temp (K) ∆µH2O |T=TT=0 (eV) ∆µHF |T=TT=0 (eV)300 -0.48 -0.45600 -1.11 -1.03stability rather than a quantitative determination of the absolute surface formation energythese small corrections have not been computed. The small PV term due to the change involume of the surface and bulk phases is also neglected. This approximation is in line withprevious studies1,2,3.Substituting equation 11 into equation 12 allows us to express the variation of µ′X(T, PX)with pressure at fixed T asµ′X(T, PX) = EDFT (T = 0) + ∆µX(P◦X) |T=TT=0 +kT ln(PXP ◦X)(15)Equation 7 can be used along with equation 15 for HF and H2O allows us to evaluate thesurface energies of different clean and hydroxylated AlF3 surfaces as a function of HF andH2O partial pressure at fixed temperature and at a fixed value of µH2 . The lowest energysurfaces can then be plotted as a function of µHF and µH2O.II. LIMITING VALUESIn principle the limiting values of µX(T, PX) can be estimated or calculated. The lowerlimit is the value at which the compound decomposes into its constituent elements. At thispointµXY (T, PXY ) = µX(T, PX) + µY (T, PY ) (16)this limit can be obtained either from thermodynamic tables or via calculation of the energyof the components of the system in their standard states.For example, considering the lower limit of F2, below which AlF3 decomposes into Al(s)and F2, we obtain∆Greaction = GAlF3bulk −GAlbulk −32GF2 (17)5= kT ln(PF2P ◦F2) (18)min[µF2 ] =23[GAlF3bulk −GAlbulk] (19)This value can be obtained via the calculation of the energy of bulk Al and AlF3. However,often it is not possible within the given calculational scheme to obtain accurate resultsfor both a metallic and ionic solid. It may therefore be preferable to obtain values fromthermodynamical tables. The free energy of formation of crystalline AlF3 is -14.8eV atstandard temperature and pressure. This is equivalent to a partial pressure of F2 of 10−170atm at 300K. Thus in this case the lower limit would be determined by the quality of thevacuum in the experiment.Similarly, an upper limit can be defined as the point at which the constituent gasescondense onto the surface of the material.µgaseousX (T, PX) = µCondensed PhaseX (T, Ptot) (20)This value can again be obtained from thermodynamic tables provided data for a condensedphase exists for the given temperature and pressure. If not, it can be approximated as thepoint at whichµ∗X(T, PX) = 0 (21)For example, the vapour pressure of H2O at 300K is 3.6x10−2 atm, this corresponds toµ∗ = -0.57eV. However, in the temperature range that we are typically interested in datafor the condensed phase of F2 is not apparently available. We therefore take the maximumvalue of µF2 to be the total energy of a isolated F2 molecule at T=0K. The vapour pressureof HF at 300K is 1.3 atm, corresponding to µ∗ = -0.44eV. At 600K the condensed phase ofboth HF and H2O are not apparently available hence we set the maximum value of µ∗ tozero.III. OBTAINING ∆µ(P ◦) FROM THERMODYNAMIC TABLESIt is important that the values in equation 15 are evaluated correctly. In particular, careshould be taken in calculating the term ∆µX(P◦X)|T=TT=0 . Values in tables are often given withrespect to T = 298.15 K hence it is important to make a correction to T = 0 K.6As an example, we will consider obtaining ∆µ(P ◦)|T=TT=0 for HF using data from the NISTwebsite8. The quantities C◦p , H◦−H◦298.15 and S◦ are given as a function of temperature. Forinstance, at 600K C◦p = 29.23 J mol−1 K−1, H◦ −H◦298.15 = 8.80 kJ mol−1 and S◦ = 194.2 Jmol−1 K−1.We can write∆µX(P◦X)|T=TT=0 = H◦ −H◦0 − TS◦ (22)Hence we also require the value of H◦298.15 − H◦0 . This has to be obtained separately, forinstance from the CODATA tables9. Values for many molecules are also given on theirwebsite10. For instance, H◦298.15 −H◦0 = 8.599 kJ mol−1 for HF. Hence,∆µHF (P◦HF )|T=600KT=0 = (H◦ −H◦298) + (H◦298 −H◦0 )− TS◦= 8.80 kJ mol−1 + 8.599 kJ mol−1 − 600 K× 0.1942 kJ mol−1 K−1= 99.121 kJ mol−1 = −1.03 eV (23)IV. ENTROPY OF MIXINGIn reality, however, the transition from one phase to another will not be immediate,consequently, mixing of two or more phases will occur. Entropy associated with the mixingof phases will reduce the energy of the surface. Calculating this entropy allows us to estimatethe extent of mixing that will occur at these phase boundaries. Following the methodologydeveloped by Reuter and Scheffler4 we have calculated the configurational entropy, Sconfig,per unit surface area asTSconfigNAsite=kBTAsite[ln(1 +nN)+(nN)ln(1 +Nn)](24)where Asite is the surface area per defect site, N is the number of surface sites and n is thenumber of defect sites in the limit of n/N  1. If we set n/N = 0.1, then,TSconfigNAsite= 0.335kBTAsite(25)This is used to determine regions on our phase diagram where more than 10% mixing betweenadjacent phases is expected.7V. THE ROLE OF µH2The only source of fluorine and of hydroxyl groups in our formulism is HF and H2Orespectively. The addition and subtraction of such groups introduces an explicit dependenceof the surface energy on µH2 . Consequently we need to assign a value to µH2 in orderto plot phase diagrams, although in practice this would be controlled via the gas phaseequilibrium established within the experimental chamber. A first reference point is obtainedby setting the chemical potential of H2 to its saturation limit (µH2=0). This correspondsto an extremely H2 rich environment. At a fixed chemical potential of HF or H2O, loweringthe value of µH2 is equivalent to increasing the values of µF and µOH which stabilises F andOH rich surfaces respectively.In order to assign a realistic value to µH2 we note that the partial pressure of H2 in air at300K and 1 atm is typically 5×10−7 atm11. This can be converted to a chemical potentialusing the data from thermochemical tables7. This results in a classical chemical potentialfor H2 of µH2 = -0.5025 eV.VI. AN EXAMPLE PHASE DIAGRAMFigure 1 shows the phase diagram for the clean and hydroxylated α-AlF3 (0001) (1×2)surface as a function of H2O and HF chemical potentials and partial pressures at 300Kand 600K. Only the stoichiometric surface and surfaces obtained from this surface via thesubstitution of F ions for OH ions are stable. As a consequence the phase diagram isindependent of µH2 . The highlighted areas on the diagram are regions where at least 10%of the neighbouring phase is expected to co-exist at 300K. It can be seen that under normalatmospheric conditions, assuming a partial pressure of HF of 10−10 atm and H2O of 1.8×10−2atm (50% humidity) the 2F 1OH and 1F 2OH are predicted to co-exist. Under reactionconditions (600K) assuming a partial pressure of HF of 1.0 atm and H2O of 10−2 atm the 3Fsurface is predicted to be stable. A more detailed discussion of this diagram will be givenelsewhere12. Within this phase diagram the adsorption of H2O and HF molecules to thesurface has not been considered. However, this is a situation that is likely to occur. We arecurrently in the process of calculating the effect on the surface stability of such adsorbedspecies. Detailed discussion of the resultant phase diagrams will be presented elsewhere12.8FIG. 1: The stability of clean and hydroxylated α-AlF3 (0001) (1×2) surfaces as a function of HFand H2O partial pressures. The pressure scale bars extend across the accessible range of partialpressures and terminate at -0.57eV for H2O and -0.44eV for HF at 300K as discussed in section II.Calculations for the α-AlF3(0112)(1×1) surface result in a very similar phase diagram12.1 I. Batyrev, A. Alavi, and M.W. Finnis. Faraday Discuss., 114:33, 1999.2 X.G. Wang, W. Weiss, S.K. Shaikhutdinov, M. Ritter, M. Petersen, F. Wagner, R. Schlogl, andM. Scheffler. Phys. Rev. Lett., 81:1038, 1998.3 K. Reuter and M. Scheffler. Phys. Rev. B, 65:03506, 2002.4 K. Reuter and M. Scheffler. Phys. Rev. B, 68:04507, 2003.5 S. Mukhopadhyay, C.L.Bailey, A. Wander, B.G. Searle, S.L.M. Schroeder, R. Lindsay, and N.M.Harrison. accepted in Surf. Sci., 2007.6 J.W. Cahn. Interfacial Segregation. American Society for metals, Metals Park, Ohio, US, 1977.7 M.W. Chase, C.A. Davies, J.R. Downey, D.J. Frurip, R.A. McDonald, and A.N. Syverud. J.Phys. Chem. Ref. Data, 14, 1985.98 http://webbook.nist.gov/chemistry/form-ser.html. NIST Chemistry WebBook.9 J.D. Cox, D.D Wagman, and V.A. Mededev. CODATA Key Values for Thermodynamics.Hemisphere Publishing Corp, 1989.10 http://www.codata.org/resources/databases/key1.html.11 D. R. Lide, editor. CRC Handbook of Chemistry and Physics 73rd Edition. CRC Press, BocaRaton, Florida, 1992.12 C.L. Bailey, S. Mukhopadhyay, A. Wander, and N.M. Harrison. In preparation, 2007.10

Ab Initio Surface Thermodynamics in Multi Component Environment

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