Research Article A Theoretical Model for Metal Corrosion Degradation

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1 Internatonal Corroson Volume, Artcle ID 794, 7 pages do:.//794 Research Artcle A heoretcal Model for Metal Corroson Degradaton Davd V. Svntradze, and Ramana M. Pdapart Department of Mechancal Engneerng, Vrgna Commonwealth Unversty, 4 W Man St., P.O. Box 843, Rchmond, VA 384-3, USA OCMB Phlps Insttute, Vrgna Commonwealth Unversty, Rchmond, VA 384, USA Correspondence should be addressed to Ramana M. Pdapart, rmpdapart@vcu.edu Receved 6 February 9; Revsed 7 July 9; Accepted October 9 Academc Edtor: Omar S. Es-Sad Copyrght D. V. Svntradze and R. M. Pdapart. hs s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Many alumnum and stanless steel alloys contan thn oxde layers on the metal surface whch greatly reduce the corroson rate. Pttng corroson, a result of localzed breakdown of such flms, results n accelerated dssoluton of the underlyng metal through pts. Many researchers have studed pttng corroson for several decades and the exact governng equaton for corroson pt degradaton has not been obtaned. In ths study, the governng equaton for corroson degradaton due to pttng corroson behavor was derved from sold-state physcs and some solutons and smulatons are presented and dscussed.. Introducton Pttng corroson s known to be one of the major damage mechansms affectng the ntegrty of many materals and structures n cvl, nuclear, and aerospace engneerng. Corroson pts generally ntate due to some chemcal or physcal heterogenety at the surface, such as nclusons, second phase partcles, flaws, mechancal damage, or dslocatons. Pttng corroson, a result of localzed breakdown of such flms, results n accelerated dssoluton of the underlyng metal. he corroson mechansms depend on the materal composton, electrolyte and other envronmental condtons 3]. he most corroson models n lterature are focused on the electrochemcal reacton durng the corroson process 4 6]. Mathematcal models are utlzed to get a better understandng of the pttng corroson process 3]. Models are ncluded n a set of governng equatons. Numercal modelng s especally mportant wth the pttng corroson because most governng equatons do not have close form solutons to descrbe the process. Pttng corroson s a very complex process and may nvolve many mechansms. In general, the corroson degradaton modelng should nvolve not only physco-chemcal and envronmental factors. he pttng corroson mechansms may ntate at multple levels startng from nano to mcro and macrolevels. Zavadl et al. 7] nvestgated the contrbuton of vods and ther shapes at nanoscale at the Al/oxde nterface to pt ntaton and propagaton n passve metals. Interestngly, Renner et al. 8] observed the ntal corroson at the atomc scale n sngle crystal alloy n sulphurc acd soluton. Martn et al. 9] conducted n stu AFM detecton of pt onset locaton on stanless steel and concluded that pts were randomly dstrbuted at the nanoscale and revealed that 7% of the pts ntated at stran hardened areas resultng from mechancal polshng. At mcroscale, the pt ntaton and propagaton mechansm may depend on the metal mcrostructure along wth chemcal electrochemcal reactons, physcal flm composton, surface texture and mechancal surface stresses. Expermental and theoretcal studes have largely clarfed the mechansm for the ntaton of these mcroscopc pts as beng caused by localzed electro dssoluton of metal at surface defects and nclusons ]. here are no studes dealng wth exact soluton of tme profle of pt growth and evoluton of corroded degradaton. In the paper, the general equatons of pttng corroson degradaton mechansms based on sold state physcs are derved. hese derved equatons are exact solutons of tme profle of corroded volume, whch mght be very useful for estmatng the determnstc characterstcs of metal corroson.

2 Internatonal Corroson Electrolyte Metal Metal flm Metal after corroson Metal before corroson Fgure : Schematc drawng of corroson metal degradaton under an electrolyte soluton. Atoms n metal flms are shown n yellow and metal atoms n red. If we assume that ε f = ε M, then the energy to move atom n metal flm s to be same as energy needng to move atom n metal. he lateral force to move atom for Cu was measured F = 7 ± 3 pn 7] andx = 3 pm, then approxmate energy per atom was measured E = F x =. J; recalculatng t per mol of atom we wll have ε f = ε M 3 J mol. ε f = ε M. Ja per atom; and ε f = ε M 3 J mol bpermolnumberofatoms.weuse as an estmaton as we are calculatng everythng on number of atoms or on per atom.. heoretcal Model Let us dscuss solated metal system and consder that the metal surface havng two phases. On the metal surface, the metal flm oxde layer s n equlbrum wth the metal. he essental condton/parameter, s defned from the equlbrum condton as μ M, P = μ f, P, where μ M, P andμ f, P are the chemcal potentals of the components present n metal and flm, respectvely, at a gven, temperature,andp, pressure. Based on the statstcal physcs, and Debye nterpolaton formula, Helmholtz free energy of the solated body 6] metal can be wrtten as F = Nε + Nν 3ln e θ/ θ, where Dx = 3/x 3 x z 3 /e z dz s Debye functon,ν = VωM/6π 3 u 3 N, θ = ω M s Debye characterstc temperature of the body, ω M s vbraton frequences of atoms, N s number of atoms, and u s average velocty of sound n the metal. Snce, we have two phases that are n equlbrum on metal surfaces, we can determne the chemcal potentals as FM μ M P, =, N V, 3 F f μ f P, = N. V, Usng and 3, and startng wth the equlbrum condtons, we can determne the connecton between free energes per atom at the border of metal-flm surface, as ε f = ε M, 4 where ε f,m are free energy per atom n flm and metal, respectvely. It should be mentoned that ε f,m s fxed energy, and for metal t can be calculated accordng to recently done atomc force mcroscopy data 3]Fgure. Now, let us consder that the body the metal s n contact wth the electrolyte soluton Fgure and the system, metal wth the electrolyte soluton, s solated, then the flm of the body starts to dssolute n soluton 8, ]. he task s to fnd and to predct tme evoluton of the corroson degradaton, n other words to derve the equaton descrbng pt rad growth n tme. As result of dssoluton due to chemcal reactons, we have changes n free energy. Snce the process goes by tself, and we are dscussng solated system, the varaton of the free energy of the system must be zero. herefore, we have, he Equaton canberewrttenas δf M + δf f + δf S =. F M n M δn M + F f n f δn f + F S n S δn S =, 6 where F M, f,s ndcates free energes of metal, flm, and soluton, respectvely throughout the manuscrpt we are usng desgnaton wth multple ndexes M, f, S whch smply ndcate meanng of the parameter for the dfferent system, for example, F M, f,s = F M, F f, F S ndcates free energes of metal, flm, and soluton, resp.. It s clear that varatons n partcles n the metal, the flm, and the soluton cause varatons of the corroded volume. herefore, the, V, and P parameters n are no longer constants. By takng nto account that F M, f,s n M, f,s = μ M, f,s, P, 7 and chemcal potental of the soluton 6]s μ S = μ P, + and usng, 6 8, we get, N ln n ε M + ν M 3ln e θ/ θ + n ψ P,, 8 ] δn M dv + P M M dn M ε f + ν f 3ln e θ/ θ dv f + P f dn f + μ P, + N ln n ] δn f n ψ P, δn S =. 9

3 Internatonal Corroson 3 Also, the conservaton of number of partcles requres that δn M + δn f + δn S =. hs s gven as ε M + ν M 3ln e θ/ ] θ dv + P M M δn M dn M + ε f + ν f 3ln e θ/ ] θ dv f + P f δn f dn f = μ P, + δnf + δn M ]. N ln n n ψ P, It was mentoned earler that the degradaton process s always accompaned by chemcal reactons that change the volume of corroded metal. Let us consder that chemcal reactons go n a manner that A + B AB] product then number of the elements A, B can be calculated as dn A = dn B = kn A N B, where k s gven by Collns-Kmball expresson 8, 9]as k = k rk D k r + k D ; 3 k D = 4πrD s rate constant; k r s knetc rate of the reacton; r s dstance of the A and B closest approach equal to the sum of ther rad; D s ther relatve dffuson coeffcent. he dffuson coeffcent can be determned accordng to Ensten equaton D = Rσ Z F C, 4 see ] where C s concentraton of the reactng atoms n our case, t wll be metal atoms and some dssolved ons, σ s electrcal conductvty, Z s the valence of the on, F the Faraday constant, R the gas constant, and the temperature. From 3and4, t follows that k = k rk D 4πrRσk r = k r + k D Z F Ck r +4πrRσ. akng nto account the theory of knetc rate calculatons, 8] we can calculate the volume of corroded metal. After some algebrac manpulatons, we wll have dv M, f dn M, f where = dv M, f dn M, f D = = k M, f N M, f N, f,m dv M, f, 6 Rσ f,m Z, f,m F C, f,m, 7 and C, f,m concentratons are concentratons of ons reactng wth flm molecules and metal atoms, respectvely, V M, f V M, V f s volume of the metal and the flm, respectvely. akng all of these nto account n, we wll have ε M f + ε + ν M + ν f 3ln e θ/ ] θ P M k M N M N,M dv M +P f k f N f N, f dv f = μ P, + N ln n en δn M δn M + δn f δn f δn M + δn f + n ψ P,. 8 Now, let us dscuss corroson degradaton step by step. Step s breakdown of the flm, meanng δn f δn M and δn f + δn M δn M leads to the followng smplfcaton: dv f P f k f N f N, f = ε M + ε f + ν M + ν f 3ln e θ/ ] θ μ P, + N ln n n ψ P,. 9 By smlarty, Step leads to δn M + δn f δn f and δn M δn f Usng 8, we have, dv P M k M N M N M,M = ε M f + ε + ν M + ν f 3ln e θ/ ] θ μ P, + N ln n n ψ P,. Equatons 9 and completely predcts the behavor of the corroded metal volume n tme and can be appled to any case, n general. However, t should be mentoned that the soluton to 9and s not possble because functons P f = P M Pt, f = M t, N M, f N M, f t are unknown, n general. In order to fnd the soluton, let us consder some specfc models and functons that take for parameters descrbed n 9,, and. Before the dscusson of models, we can further smplfy general equatons f we determne

4 4 Internatonal Corroson N M, f N M, f t functons. As dscussed above, we can determne N f,m = N f,m, e kn, f,mt. herefore, fnally from 9 wemusthave: P f k f N f, e kn, f t dv f N, f = ε M + ε f + ν M + ν f 3ln e θ/ ] θ μ P, + N ln n n ψ P, 3 P M k M N M, e kn,mt dv f N,M = ε M + ε f + ν M + ν f 3ln e θ/ ] θ μ P, + N ln n n ψ P,. 4 It should be mentoned that n the frst step on the left sde of the 3, we have chemcal potental of pure solvent whle n 4, we have mpurty rch soluton wth dfferent chemcal potental. he last two equatons can be smplfed f make some notatons as, f t Mt ε M + ε M + ν M + ν f 3ln e θ/ θ/ ] P f k f N f, N, f μ P, + /N lnn /en + n ψ P, ], P f k f N f, N, f ε M + ε M + ν M + ν f 3ln e θ/ θ/ ] P M k M N M, N,M μ P, + /N lnn /en + n ψ P, ]. P M k M N M, N,M hen we wll have: dv f = f te k f N, f t, 6 dv M = Mte kmn,mt. 7 Equatons 7 and8 wll completely descrbe the corroded volume f we wll be able to determne f t andmt functons. After n tmes partal ntegratons of 6and7, we can fnd t dv f = f te k f N, f t V n + 8 = f te k k = f N f N, f t,, f t dv M = Mte kn,mt V n + 9 = M te kmn,mt. k = M N,M akng nto account that k M, f = 4πr M, f Rσ M, f k r,m, f Z,M, f F C,M, f k r,m, f +4πr M, f Rσ M, f, 3 r M, f s radus of metal atom and oxde layer molecule resp. depends on dffuson coeffcent, t s clear that the process can be further accelerated by varables from 3. Equatons 8 3 are the exact solutons of tme profle of corroded volume whch mght be very useful for estmatng the determnstc characterstcs of metal corroson. he model system always has fnte volume; so corroson process s endng after a fnte tme. We menton that f corroson process has at least quasequlbrum manner, then δn f = δn M so n 8wewllhave that ε M = ε f, = const and s not changng durng the process and P M = P f = Pt, then 8 can be smplfed as t dv M = Mte kmn,mt V n = k M N,M = + M te kmn,mt, where Mt functon s determned accordng to. 3. Results and Dscusson 3 In order to see the trends n corroson degradaton process, specfcally how the pt profle rad changes wth tme, computer smulatons based on 3 usng Matlab software were carred out wth preposton that the pt has semsphercal shape wth rad R pt. he followng assumptons were used n the analytcal smulaton. Frst, we propose that corroson process takes place at least n a quas-equlbrum manner. hs assumpton allows to further smplfy 8 3to3, where ε M = ε f, = const and are not changng durng the corroson process, and P M = P f = Pt. However, 3 stll has parameters k M N M whch are drectly connected to chemcal reactons. Second, the assumptons are related to the estmatons for Mt functon varaton and the k M N M parameters. Functon Mt s related to the specfc metal characterstcs

5 Internatonal Corroson t s..6.4 ncton M t fu.8 t s 6 a.6.4. cton fun M t b t s..4 M t cton fun c 7 t s.6.4. cton fun M t d Fgure : Smulatons of tme profle of pt rad for varous parameters/functons. he range of Mt varous functon: a Mt,, km Nm 8, b Mt, 4, km Nm 8, c Mt,, km Nm 7, d Mt,, km Nm 9. Preposton of vew Mt > because chemcal potental of the components present n metal are always more than the chemcal potental of any soluton the lne to near of Mt are nonrealstc plots so these dctate us to estmate borders of Mt functon more carefully. = temperature, ωm = vbraton frequences of metal atoms, and u = average velocty of sound whch s related to the chemcal potental of the electrolyte. All the above parameters are drectly or ndrectly responsble for the degradaton propertes of the metal. We estmate that the Mt functon usng and expermental data 7]. Specfcally, we have used the expermental data 7] of dffuson coefficents to estmate the range of on concentratons for n the term f /km NM,O N,M,O and AFM data 7] to estmate ε = εm. J value Fgure dependng on the varaton of dfferent chemcal ons and km NM,O N,M,O values whch are number of oxdzed metal atoms and ons reactng wth metal atoms and wth oxygen. We have taken several dfferent ranges for Mt functon and km NM parameters, whch are gven as: a Mt, and km Nm 8 ; b Mt, 4, km Nm 8 ; c Mt,, km Nm 7 ; and d Mt,, km Nm 9 see Fgure. Also, for some fxed ranges of Mt, parameters km NM takes values 7, 8, 9 Fgure 3. he reason for takng many ranges for the parameters s to demonstrate that even such rough estmaton for the rate of chemcal reactons durng the corroson degradaton, the Mt functon, can be consdered as a characterstc of the metal, gvng reasonable results for pttng corroson degradaton. he range of Mt functon was determned based on f and expermental data 7] for a value of ε = M ε, chemcal potentals for solutons as gven by standard tables, and parameter km was determned accordng to 3. he tme profle of pt rad s shown n Fgures and 3 for varous parameters and functons. he present predctons are compared qualtatvely to the expermental data ] by estmatng Mt and km NM parameters through a range but not strctly quanttatvely. A good agreement s found between the present predctons and the expermental data for the trends n corroson degradaton. However, the quanttatve predctons and comparson to the expermental data need more knowledge about the exact value of forces that are necessary to move atom from metal surface that defntely depends on the knd of metal, exact knowledge of on content of envronment where the metal s placed and must be taken nto account tme varable nature of km NM parameter. Smulatons of tme profle of pt rad for varous parameters/functons showed that f Mt Fgure then the metal s very stable aganst the corroson

6 6 Internatonal Corroson t s..4.6 Mtfuncton t s..4.6 Mtfuncton.8 4 a b t s c..4.6 Mt functon t s d..4.6 Mtfuncton.8 6 Fgure 3: he range of Mt functon was estmated based on the desgnaton of and expermental data 7] and usng ε f = ε M. J value. Smulatons were run for dfferent range of Mt functon and for dfferent values of k M N m parameter. Fgures a & b ndcate pt rad tme profle for range of Mt 7, 6 ]andmt, ], respectvely. Estmaton ntervals of Mt functons were taken by calculatons based on gven by standard tables of chemcal potentals for solutons and for free energy per atoms, k M N M 8 whch was estmated based on 3. Fgures c & d ndcate pt rad tme profle for fxed range of Mt functons Mt 7, 6 ]butfordfferent k M N M parameter, k M N M 7 and k M N M 9,respectvely. degradaton and pt rad s almost wth very slow changes f any, but we should menton that Mt > alwayschemcal potental of the components present n metal are always more than the chemcal potental of any soluton so plots near to zero are nonrealstc plots. Estmatons of the Mt functons based on the desgnaton of and expermental data 7], usng ε f = ε M. J value, gve us followng range Mt 7, 6 ]andmt, ]for dfferent values of k M N m parameter Fgure 3. hough even ths rough estmatons gve us reasonable results. All the above parameters complcate fndng an analytcal soluton of 3. More exact quanttatve descrptons by the proposed theory of corroson pt rad for specfc metals under specfc envronments need further studes. hs aspect of the research s beng pursued as future work. As we dscussed above, to estmate range of varaton for Mt functon, the value for ε f = ε M whch s free energy peratomnmetalneedstobeestmated.hsfreeenergy per atom s estmated based on the AFM expermental data where the mnmal force necessary to move the atom on the surface of the metal was measured. he lateral force to move atom for Cu was measured to be F = 7 ± 3pN7]andx = 3 pm then approxmate energy per atom E = F x =. J. After recalculatng ths value per mol of atom, ths wll be ε f = ε M 3 J mol. ε f = ε M. J peratom. In addton, the range of k M N m parameter also needs to be estmated n whch the value of k M s related to rate of chemcal reacton n the metal durng the corroson degradaton and the parameter N,M s related to the concentraton of the ons reactng wth metal atoms. akng nto account that k M, f = 4πr M, f Rσ M, f k r,m, f Z,M, f F C,M, f k r,m, f +4πr M, f Rσ M, f 3 r M, f s radus of metal atom and oxde layer molecule, resp. depends on dffuson coeffcent, whch s gven by formula D = Rσ f,m Z, f,m F C, f,m 33

7 Internatonal Corroson 7 C, f,m are concentratons of ons reactng wth flm molecules and metal atoms, resp.. hs value s estmated from the concentratons or number of reactng ons n the dffuson coeffcent. he range for k M N m parameter s estmated based on dffuson usng the expermental data ].heseestmatonsarenotverystrct,sowehaveused the range for the parameters, nstead of one fxed value n the smulatons. 4. Conclusons In ths study, the governng equaton for corroson degradaton due to pttng corroson behavor was derved. After several assumptons, some solutons and smulatons are presented and dscussed. he smulaton results obtaned ndcate that the results are reasonable and further valdaton of the theory requres more expermental work. he theoretcal model developed n ths study has the followng lmtatons. Frst, 8 3 completely descrbe corroson degradaton n tme whch cover many unknown parameters that are strctly mathematcal, and determnaton of these parameters may be dffcult. Second, we formulated the corroson degradaton problem as an solated system rather than open system. hrd, n our smulatons, we consdered an deal stuaton, related to pttng degradaton whch has at least quas-equlbrum manner whch s good estmaton as the corroson process s very slow. It should be mentoned that the developed corroson degradaton model s very general and wthout makng any assumptons t wll be very dffcult task n order to get a soluton for pttng corroson degradaton. Acknowledgment hs work s supported by the Natonal Scence Foundaton through a grant DMR-39. References ] P. Marcus and J. Oudar, Eds., Corroson Mechansms n heory and Practce, Marcel Dekker, New York, NY, USA, 99. ] L. L. Shree, R. A. Jarman, and C.. Bursten, Eds., Corroson Metal/Envronment Reactons, Butterworth & Henemann, Oxford, UK, 3rd edton, ] H. H. Strehblow, Mechansms of pttng corroson, n Corroson Mechansms n heory and Practce, p., Marcel Dekker, New York, NY, USA, 99. 4] R. P. We, C.-M. Lao, and M. Gao, A transmsson electron mcroscopy study of consttuent-partcle-nduced corroson n 77-6 and 4-3 alumnum alloys, Metallurgcal and Materals ransactons A, vol. 9, no. 4, pp. 3 6, 998. ] M. J. Palakal, R. M. V. Pdapart, S. Rebbapragada, and C. R. Jones, Intellgent computatonal methods for corroson damage assessment, AIAA Journal, vol. 39, no., pp ,. 6] B. Malk and B. Baroux, Computer smulaton of the corroson pt growth, Corroson Scence, vol. 47, no., pp. 7 8,. 7] K. R. Zavadl, J. A. Ohlhausen, and P. G. Kotula, Nanoscale vod nucleaton and growth n the passve oxde on alumnum as a prepttng process, the Electrochemcal Socety, vol. 3, no. 8, pp. B96 B33, 6. 8] F. U. Renner, A. Sterle, H. Dosch, D. M. Kolb,.-L. Lee, and J. Zegenhagen, Intal corroson observed on the atomc scale, Nature, vol. 439, no. 777, pp. 77 7, 6. 9] F. A. Martn, C. Batallon, and J. Cousty, In stu AFM detecton of pt onset locaton on a 34L stanless steel, Corroson Scence, vol., no., pp. 84 9, 8. ] C. Punckt, M. Bölscher, H. H. Rotermund, et al., Sudden onset of pttng corroson on stanless steel as a crtcal phenomenon, Scence, vol. 3, no. 687, pp , 4. ] D. E. Wllams, R. C. Newman, Q. Song, and R. G. Kelly, Passvty breakdown and pttng corroson of bnary alloys, Nature, vol. 3, no. 63, pp. 6 9, 99. ] R. Ke and R. Alkre, Surface analyss of corroson pts ntated at MnS nclusons n 34 stanless steel, the Electrochemcal Socety, vol. 39, no. 6, pp. 73 8, 99. 3]M.P.Ryan,N.J.Laycock,R.C.Newman,andH.S.Isaacs, he pttng behavor of ron-chromum thn flm alloys n hydrochlorc acd, the Electrochemcal Socety, vol. 4, no., pp. 66 7, ] G. S. Frankel, Pttng corroson of metals: a revew of the crtcal factors, the Electrochemcal Socety, vol. 4, no. 6, pp , 998. ] M. P. Ryan, D. E. Wllams, R. J. Chater, B. M. Hutton, and D. S. McPhal, Why stanless steel corrodes, Nature, vol. 4, no. 6873, pp ,. 6] L. D. Landau and E. M. Lfshtz, Course of heoretcal Physcs, Statstcal Physcs, vol., Pergamon Press, Oxford, UK, 97. 7] M. ernes, C. P. Lutz, C. F. Hrjbehedn, F. J. Gessbl, and A. J. Henrch, he force needed to move an atom on a surface, Scence, vol. 39, no. 866, pp , 8. 8] K. L. Ivanov and N. N. Lukzen, A novel method for calculatng rate constants of dffuson-nfluenced reactons, Chemcal Physcs, vol., no., pp. 9 4, 4. 9] F.C.CollnsandG.E.Kmball, Dffuson-controlled reacton rates, Collod Scence, vol. 4, no. 4, pp , 949. ] R. H. Doremus, Dffuson n alumna, JournalofAppled Physcs, vol., no., Artcle ID 3, 7 pages, 6. ] G. Engelhar and D. D. Macdonald, Unfcaton of the determnstc and statstcal approaches for predctng localzed corroson damage. I. heoretcal foundaton, Corroson Scence, vol. 46, no., pp. 7 78, 4.

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