Modeling Polycrystals with Regular Polyhedra

Transcription

1 Mateials Reseach, Vol. 9, No., -6, Modeling Polycystals with Regula Polyheda Paulo Rangel Rios a *, Matin Eden Glicksman b * a Univesidade Fedeal Fluminense, Escola de Engenhaia Industial Metalúgica de Volta Redonda, Av. dos Tabalhadoes, 40, Volta Redonda - RJ, Bazil b Rensselae Polytechnic Institute, Mateials Science & Engineeing Depatment Toy, NY USA Received: Octobe 9, 005; Revised: Febuay 6, 006 Polycystalline stuctue is of paamount impotance to mateials science and engineeing. It povides an impotant example of a space-filling iegula netwok stuctue that also occus in foams as well as in cetain biological tissues. Theefoe, seeking an accuate desciption of the chaacteistics of polycystals is of fundamental impotance. Recently, one of the authos (MEG) published a pape in which a method was devised of epesentation of iegula netwoks by egula polyheda with cuved faces. In Glicksman s method a whole class of iegula polyheda with a given numbe of faces, N, is epesented by a single symmetical polyhedon with N cuved faces. This pape biefly descibes the topological and metic popeties of these special polyheda. They ae then applied to two impotant poblems of iegula netwoks: the dimensionless enegy cost of iegula netwoks, and the deivation of a D analogue of the von Neumann-Mullins equation fo the gowth ate of gains in a polycystal. Keywods: micostuctue, gain gowth, analytical methods, foams, polycystals. Intoduction The polycystalline stuctue is of paamount impotance to mateials science and engineeing. It is an example of a space-filling iegula netwok stuctue that also occus in foams as well as in cetain biological tissues. The technological impotance of polycystals deives fom the fact that the majoity of industial cystalline mateials ae used as polycystals. Theefoe, accuate desciption of the chaacteistics of polycystals is of fundamental impotance. A complete desciption of such a netwok involves the knowledge of the geometic chaacteistics of the individual cystals, of thei cystallogaphic oientation, i.e., thei cystallogaphic textue, of the natue of the intefaces between individual cystals, among othes. Not only is it impotant to know these chaacteistics at one instance in time, but also to pedict thei dynamic behavio when such stuctues change as a function of time because of gain gowth. Nevetheless, a puely geometic chaacteization of an iegula netwok of gains is not easy. One needs to know the volume and shape of the individual gains. This equies tedious and time-consuming expeimental techniques such as seial sectioning, o even applying moe demanding methods such as disintegating the polycystal, fo example, by adding gallium to an aluminium polycystal and inspecting the individual gains. In pactice, one seldom does this, and, instead, often a single measuement is used: one detemines just the gain size using eithe the ASTM numbe o the mean intecept length. Even if detailed measuements wee available, making sense of a vaiety of iegula polyheda would still be a daunting task. Recently, one of the authos (MEG) has published a pape in which a method of epesentation of iegula netwoks by egula polyheda with cuved faces was devised. In Glicksman s method a whole class of iegula polyheda with a given numbe of faces, N, is epesented by a single symmetical polyhedon with N faces. These polyheda ae egula polyheda with cuved faces, constucted in such a way that they satisfy the aveage topological constaints * pios@metal.eeimv.uff.b, glickm@pi.edu imposed by a space filling netwok. Glicksman called these special egula polyheda with cuved faces aveage N-heda o ANHs. In this wok they will be called ANHs, fo bevity. This appoach significantly simplifies the mathematical teatment of iegula netwoks. The geometic popeties of the ANHs can be calculated exactly, as closed-fom expessions ae available fo all thei impotant geometic and kinetic popeties. The ANHs act as poxies fo analyzing iegula netwok gains, allowing igoous teatment of seveal longstanding poblems -5 petaining to these netwoks. This pape biefly descibes the topological and metic popeties of the ANHs. The ANH concept is then applied to two impotant poblems of iegula netwoks: the dimensionless enegy cost of iegula netwoks (in essence thei stoed fee enegy) and the deivation of a D analogue of the von Neumann-Mullins equation, to pedict the gowth ates of polycystals. This pape will focus on the popeties of ANHs athe than on the detailed mathematical deivations that can be found elsewhee -5.. Constuction of ANHs A polycystalline stuctue is not just an aggegate of contiguous polyheda. The gain bounday enegies o suface tensions impose cetain topological constaints on the netwok polyheda. These netwok conditions must be satisfied by the ANHs as well, if they epesent the aveage behavio of space-filling gains. Any polyhedon must satisfy Eule s fomula elating numbe of edges, E, vetices, V and faces, N: N E + V = () In a gain netwok, thee faces always meet along a common edge, and fou faces always meet at a common vetex. Fo an isolated polyhedon this is equivalent to equiing that two polyhedal faces meet along a common edge, and that thee edges meet at a vetex.

2 Rios & Glicksman Mateials Reseach Smith 6 found an expession fo the numbe of edges pe face, p, fo a polyhedon having tihedal vetices: p = 6 - N () Moeove, equilibium conditions equie that the thee faces meet with dihedal angles equal to π/ = 0, and the edges meet at the common vetex with the tetahedal angle equal to acos( /) These conditions ae known as Plateau ules, which ae ovely estictive, inasmuch as the polycystal must be in full equilibium at evey edge and vetex. A much moe geneal, and fa less estictive, condition is to equie that the thee faces meet with an aveage dihedal angle equal to π/ = 0 and edges meet with the aveage tetahedal angle equal to acos(- /) These aveage netwok conditions ae obeyed by ANHs. The constuction of an ANH may be exemplified by constucting a 4-hedon. The stating point fo the constuction of a symmetical cuved face 4-hedon is a egula polyhedon with fou flat faces: i.e., the egula tetahedon. In ode to make the tetahedon satisfy the aveage netwok constaints the flat faces must be cuved into spheical caps. When this is done the edges must also be cuved. Figue shows a schematic dawing of this cuved-face tetahedon. The angles between the nomals at the cente of the faces, α, emain the same afte they have been cuved, but the angle between nomals adjacent to the edges ae now smalle than α. If the faces ae sufficiently cuved, then this angle can be made to be equal to 60, coesponding to an inteio dihedal angle equal to 0. It is also necessay to cuve the edges so that they meet at the common vetex with the coect tetahedal angle, This pocess can be epeated fo all egula flat faced polyheda. Unfotunately, not all egula polyheda can be constucted. The eason fo this estiction can be undestood using Smith s fomula above. The numbe of edges pe face, p, is detemined by the numbe of faces. Fo a polyhedon to be constuctible, p must evidently be an intege numbe. This only happens fo N =,4,6 and, giving p =,,4 and 5, espectively. All the constuctible ANHs ae illustated in Figue. In Figue the volumes of these ANHs ae scaled so that the distance between neaest vetices, called the gauge, λ, is equal is all cases. This example shows why the ANHs must have cuved faces. The cuvatue of these faces is an essential popety demanded by the polycystalline netwok.. Popeties of ANHs Even though ANHs cannot be constucted fo evey value of N it is nonetheless possible to find all thei topological and metic popeties: cuvatue, volume, aea, edge length. These will be summaized hee. An ANH with N cuved faces has (N - ) edges and (N - ) vetices. The adius of cuvatue of a face of an ANH, R, is: R = csc b a - l c cot ~ sin a - csc ~ sin a - cot 6 p m m whee λ is the metic gauge, o scale facto, chosen hee as the distance between neaest neighbos vetices. The paametes ae defined as: α is the angle between nomals at the cente of the faces, it is also equal to the angle between face nomals of the coesponding flat faced polyhedon; p is the aveage numbe of edges pe face; and ω is the angle by which an edge must be cuved in ode that thee edges meet with the tetahedal angle of () N / Figue. Geomety of the aveage 4-hedon. Depiction of the intefacial angle, α 09.5, is shown between adjacent face-centeed nomals, N and N. Owing to the cuved face the face nomals otate and meet at the edge with an angle equal to 60 pescibed by equilibium condition at the tiple lines. Notice that the angle between face nomals, 60, is half the dihedal angle, 0. Figue. The fou constuctible ANHs of unit vetex to vetex distance. Fom left to ight: N =, 4, 6, and, with thei volumes appoximately popotional to 0.50, 0.764,., and 9.5, espectively. Each unitay ANH in the inteval N consists of a fixed volume enclosed by N identical convex cuved faces intesecting at identical cuved edges of unit chodal length between vetices. Edges meet thee at a time at identical tihedal vetices. Vetices ae symmetically disposed about the volume centoid. All ANHs fo N 4 have concave faces, and none ae constuctible. N: α, p and ω ae given below as a function of the numbe of faces, N N (4a) p = 6 - N (4b) (4c) The fomula fo the numbe of edges pe face, p, Equation, is epeated hee fo convenience. The adius of cuvatue of a face allows two impotant quantities to be calculated: the mean cuvatue, H, and the Gaussian o total cuvatue, K. The mean cuvatue of an inteface is the aveage of its two pincipal cuvatues, k and k. Fo a spheical suface of adius R the pincipal cuvatues ae equal, so that: H = ^k k + h = b + l R R = R (5) The mean cuvatue is impotant because it is elated to the inteface velocity, v, in cases of cuvatue-diven inteface movement by means of the well-known expession: v = γmh (6)

3 Vol. 9, No, 006 Modeling Polycystals with Regula Polyheda The minus sign applies because H is positive fo convex sufaces and the inteface always migates towads the cuvatue cente on the concave side. The Gaussian cuvatue of an inteface is the poduct of its two pincipal cuvatues. Full discussion of the fundamental significance of the Gaussian cuvatue is beyond the scope of this pape. Fo a spheical suface of adius R the Gaussian cuvatue is: K = (7) R The mean cuvatue of the face of an ANH can be bette visualized in Figue whee it is plotted against the numbe of faces. Fo compaative puposes the scaling facto, λ, was taken to have unit value fo all ANHs. The mean cuvatue is positive fo N < and negative fo N > 4. The exact point at which H changes sign is N = N C. N C is given by: NC = +, actan tan (8) The ANH possessing N C faces is called the citical ANH o N C hedon. All faces of the N C -hedon possess zeo mean cuvatue; in othe wods, they ae flat. If it wee possible to constuct a netwok just using N C -heda thee would be no pessue diffeence acoss the flat intefaces o, equivalently, the chemical potential would emain steady thoughout the polycystal. Inteface motions would cease, as the pessue diffeence acoss flat intefaces is zeo. Such a netwok would theefoe have the minimum fee enegy, and it would be in state of metastable equilibium. Although the N C -hedon is not constuctible, one can calculate its popeties exactly as will be shown below. Mean cuvatue, H = (R C ) Numbe of faces, N Figue. Mean cuvatue, H = /(Rλ) vs. numbe of faces on ANHs. The values of N cove the pactical ange encounteed in netwok stuctues such as polycystals and foams. The mean cuvatue divides the population of ANHs: fo N the mean cuvatues ae positive, implying shinkage, wheeas fo N 4 the mean cuvatues ae negative, implying gowth. It is inteesting to note that none of the ANH s have exactly zeo mean cuvatue, so that in thee dimensions netwok cells ae, on aveage, eithe shinking o gowing; i.e., none ae conditionally stable polyheda. Fo N < N C the ANHs have convex faces, see Figue, and thei intefaces tend to migate towads the centoid of the ANHs, so that, the ANHs tends to shink and disappea. The opposite is tue fo N > N C. The intefaces ae concave and they tend to move away fom the cente of the ANHs and thus ANHs with N > N C tend to gow. Volume, aea, and edge length ae othe impotant popeties of the ANHs. The aea, A, is given by: A = g(n)4πr whee Ω = π - accos(-/) The volume, V, is: (9a) (9b) V = f ( N) 4R (0a) f ( N) N N accos 6 = e - + (0b) The total edge length is: L = ( N - ) ~ R () These expessions, howeve, cannot be used to find the aea and volume of the citical ANH, because R itself is infinite. Each of the metical quantities listed above may be calculated by diect geometical methods 5 : J tan accos N A C m K b lo = K O, m K actan tan O L P V = ma C C cot b accos l 6 tan, 6 m # # Dimensionless Enegy Cost of Polycystals () () (4) Following Cox and Fotes 7, we define a dimensionless enegy cost, G(N), to constuct a polyhedon with N faces, (5) The dimensionless enegy cost, Equation 5, may be intepeted as (one-half) the suface fee enegy equied to ceate an isolated N-faced polyhedon of abitay size scale, given that G(N) is independent of the polyhedon s gauge o linea metic, λ. By summing G(N) fo all the polyhedal cells compising a space-filling polycystal, one aives at an estimate of the dimensionless excess fee enegy of the gain bounday netwok. The sum of G(N) fo all polyhedal gains, divided by the numbe of gains, yields the aveage dimensionless suface enegy pe gain to constuct the polycystal, < G >, namely, i = n < G > = n! Gi ( N) (6) i =

4 4 Rios & Glicksman Mateials Reseach Figue 4 displays a composite plot of many dimensionless enegy costs, G, vs. the numbe of faces, N, calculated fo a vaiety of constuctible iegula polyheda. Aeas and volumes fo these (constuctible) polyheda wee obtained by high-accuacy simulations pefomed by Cox* using Bakke s Evolve pogam 8, and then conveted to dimensionless enegy costs via Equation 5. The values calculated fo all intege unitay ANHs in the ange 4 N 4 ae also shown. The G-values fo the ANHs wee calculated on the basis of thei theoetically deived aeas and volumes shown in section. Note the nea-pefect ageement achieved between the simulated and theoetical G-values fo the thee constuctible ANHs that appea on this plot, whee N = 4, 6, and. Two impotant obsevations can be made fom Figue 4. The fist is that the ange of values assumed by the dimensionless enegy costs is naow amounting to only.6% of the aveage value. The second obsevation is that the G-values fo the set of ANHs shown fom a shap lowe bound to the lowest values calculated fom Cox s Suface Evolve data fo a selection of constuctible polyheda in each topological class. The most efficient constuctible intege polyheda fall as close as within appoximately 0.% of the dimensionless enegy cost of thei coesponding ANHs. In not a single instance, howeve, does any constuctible polyhedon have a lowe dimensionless enegy cost than its ANH countepat. Whee the ANHs happen to be constuctible, the simulation data and the analytical value agee within the numeical toleance of the Evolve pogam. As a esult, one can conclude that ANHs povide both a lowe bound to the enegy cost and an excellent appoximation to constuctible iegula polyheda. In othe wods, these figues ae consistent Dimensionless enegy cost, G(N) G (ANH) Cox (evolve) Cox () Cox () Cox (0.5) Cox (5) Cox (ANH) G (Kelvin) G (Weaie-Phelan) Gc with the idea that ANHs povide an accuate epesentation o close poxies of iegula polyheda. Of special significance is the dimensionless enegy cost fo the citical ANH, (7) If one imagines a netwok constucted entiely of citical ANHs its dimensionless enegy cost would be < G > = G c. The citical ANH value,.678, is lowe than the dimensionless enegy cost of tiling space with Kelvin s classical tetakaidecahedon 9, G =.6577, and with Weaie-Phelan s moe ecent duplex tiling 0, G = The Weaie-Phelan tiling consists of two, iegula pentagonal dodecaheda and six tetakaidecaheda. When combined into a unit cell, these eight polyheda pack to fom a duplex unit cell with the aveage value of < N > =.5 faces. Kelvin s and Weaie-Phelan s tiling ae illustated in Figue 5. The dimensionless enegy cost of the citical ANH is significantly smalle than the othes. Glicksman and Rios 5 have ecently shown that the citical ANH has the lowest possible dimensionless enegy cost among othe possibilities of tiling space. 5. D Analogue of von Neumann-Mullins Equation Polycystalline netwoks ae often subjected to some kind of pocessing involving heating to tempeatues sufficiently high to assist the movement of the cuved intefaces. As a esult, gain gowth takes place. Duing gain gowth gains with convex faces, N < N c, shink and gains with N > N c gow. The von Neumann-Mullins expession,, valid in two dimensions, fo the gowth/shinkage of an individual gain, is well known fo many yeas, da = M c( n - 6) (8) whee a is the aea and n is the numbe of sides of a two-dimensional gain; M and γ - ae the gain bounday mobility and gain bounday fee enegy pe unit of length. Gains with less than six sides shink, wheeas gains with moe than six sides gow. An expession analogous to von Neumann-Mullins, but valid in thee spatial dimensions, has been sought fo ove fifty yeas,-6. Such an expession may be obtained based on the assumption that a polycystalline netwok can be epesented, albeit abstactly, by Numbe of faces, N Figue 4. Dimensionless enegy cost, G, vs. numbe of faces on a polyhedal cell, N. Data ae shown fo a naow (.6%) ange of G-values fo ANH s (black diamonds); fo a selection of constuctible iegula polyheda and ANHs (cosses and open squaes, espectively), as measued by Cox* using compute simulation; fo Kelvin s othic tetakaidecahedon at N = 4; fo the Weaie-Phelan duplex tiling at N =.5; and fo the citical ANH at N =.97. The G-values fo highly symmetic ANHs fom a lowe bound fo evey class of polyheda tested in vaious topological face combinations. The ange of G values exhibited by vaious polyheda with a fixed numbe of faces inceases with N, as many moe topological combinations become possible. * Cox SJ. Tinity College, Dublin, Ieland, Pesonal Communication; 004. (a) Figue 5. a) Kelvin s elaxed tetakaidecaheda. Illustation suggests how the flatfaced vesion of these polyheda would stack by tanslation to tessellate -d space and fom a body-centeed cubic (BCC) lattice. BCC lattice points would be located at the centoids of each Kelvin cell; and b) The Weaie- Phelan duplex tiling. This tiling consists of eight stacked polyheda: two iegula pentagonal dodecaheda (shown as open fames)each isolated by six medial tetakaidecaheda (shown as solid polyheda). Figues adapted fom and demon.co.uk/polyheda/wp/wp.htm. (b)

5 Vol. 9, No, 006 Modeling Polycystals with Regula Polyheda 5 ANHs. This expession can be obtained by noticing that the inteface velocity is given as: v = A (9) Fom Equations 6 and 9: =- McHA (0) Both H and A ae known as a function of numbe of faces fo the ANHs, Equations 5 and 9. Equation 0 can be put in a moe convenient fom, in which the ate of volume change depends only on the numbe of faces. This can be done noticing that: so that: = V () () In Equation the scale facto, λ, cancels out and the ight hand side is a function of the numbe of faces, only: () whee g(n) and f(n) ae defined above, Equations 9b and 0b. Equation is the ANH-based -dimensional vesion of the von Neumann-Mullins equation. A plot of Equations is shown in Figue 6 along with data fom seveal compute simulations of evolving isotopic -dimensional netwoks. The ageement between the analytic theoy and data fom the simulations of Cox and Fotes 7 is excellent, especially whee the data ae fo constuctible ANHs. It is inteesting that even the ate data shown in Figue 6, simulated fom iegula isotopic netwoks, ae in ageement with the pesent theoy. The close coespondence between the gowth ates pedicted fo ANHs and those found in simulations of iegula -dimensional netwoks is doubtless due to the obust chaacte of the topological aveages incopoated in the set of ANHs. In othe wods, ANHs act as accuate poxies fo all the iegula polyheda in a polycystal exhibiting N faces. Thus, they povide good estimates within each topological class fo the aveage face cuvatue, suface aea, and volume, all of which ae elevant to detemining the gowth ate within each class of gains. The dependency of the gowth ate on the numbe of faces in D is a athe moe complicated function than the elegant (n - 6) facto in D, deived in 95 by von Neumann 6. Simple, lineaized foms can be obtained fo D by ecognizing that / / itself is nealy linea in the vaiable N /. Thus, one may wite in lieu of the exact expession, Equation, the linea fom: Mc = A + B N (4) It is impotant to insue that the lineaized appoximation fo / /, Equation 4, vanishes at pecisely the same value of N whee the exact expession, Equations vanishes. Specifically, one can show that the oot of Equation occus at the citical value, N c. This constaint on Equation 4 may be applied by choosing the constant A in the fom A =- B N c (5) As a consequence of inseting Equations 5 into Equation 4, one is left with a linea kinetic equation fo the gowth of ANHs containing only a single paamete, B, viz., Mc = B _ N - Nci (6) Fitting the one-paamete expession, Equation 6, to discete kinetic data fom the exact expession, Equation, ove the inteval fom N 49 yields the esult that B.709 9/4, with an associated egession coefficient R = Figue 7 shows the discete function / / plotted against N / togethe with the staight line epesenting Equation 6.Thus, one finds that the linea kinetic law in D fo the scale-independent ate of volume change fo ANHs is descibed as, 9 Mc _ N - Nci 4 (7) Equations 6 o 7 ae the D analogues to Equation 8 in D. The ate of aea change vanishes in D fo a polygon with six sides, Volume change ate, (/)d(v / ) Theoy Hilgenfel et al. (poly.) Hilgenfel et al. (mono.) Cox and Fotes Numbe of faces, N Figue 6. Compaison of the theoetical scale-fee ates of volume change fo vaious ANHs with Suface Evolve simulation data. The data fom Cox and Fotes 7 ae Suface Evolve simulations of isolated N-heda, wheeas the data fom Hilgenfel et al. 4 ae based on Suface Evolve simulations of iegula isotopic netwok stuctues containing ove 500 polyhedal cells. / / 0 - y = m* x M0 Value Eo m Chisq R NA NA N / / - N c Figue 7. Linea egession of (/Mγ)/ / / vs. (N / -N c / ). Thee is good ageement between the exact values, Equation, and the staight line, Equation 6.

6 6 Rios & Glicksman Mateials Reseach wheeas in D the ate of change of aea and volume vanishes fo the citical ANH with.9 faces. Acknowledgments The authos ae deeply indebted to D. Simon Cox, Physics Depatment, Tinity College, Dublin, Ieland, fo shaing his Evolve simulation data with us. The authos expess thei appeciation fo the financial suppot of this study deived fom the John Tod Hoton Distinguished Pofessoship in Mateials Science and Engineeing, at Rensselae Polytechnic Institute. One of the authos (PRR) is gateful to the Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, and to the Fundação de Ampao à Pesquisa do Estado do Rio de Janeio, FAPERJ, fo his financial suppot. Refeences. Glicksman ME. Analysis of -D netwok stuctues. Philosophical Magazine. 005; 85():-.. Glicksman ME. Enegetics of polycystals. Jounal of Mateials Science. 005; 40(9-0): Glicksman ME, Rios PR. Micostuctual chaacteistics of -D netwoks. Zeitschift fü Metallkunde. 005; 96(0): Rios PR, Glicksman ME. Self-simila evolution of netwok stuctues. Acta Mateialia. 006; 54(4): Glicksman ME, Rios PR. Minimal netwok patition using aveage ANH. Unpublished eseach, Smith CS. Gain shapes and othe metallugical applications of topology. Metal Intefaces. Cleveland, OH: Ameican Society fo Metals; 95. p Cox SJ, Fotes MA. Popeties of thee-dimensional bubbles of constant mean Cuvatua. Philosophical Magazine Lettes. 00; 8(4): Bakke K. The motion of a suface by its mean cuvatue. Pinceton: Pinceton Univesity Pess; Thomson W. (Lod Kelvin) On the division of space with minimum patitional aea. Philosophical Magazine. 887; 4(5): Weaie D, Phelan R. A counteexample to Kelvin s conjectue on minimal sufaces. Philosophical Magazine Lettes. 994; 69(): Von Neumann J. Witten discussion of gain shapes and othe metallugical applications of topology. Metal Intefaces. Cleveland, OH: Ameican Society fo Metals; 95. p Mullins WW. -dimensional motion of idealized gain boundaies. Jounal of Applied Physics. 956; 7(8): Mullins WW. The statistical self-similaity hypothesis in gain-gowth and paticle coasening. Jounal of Applied Physics. 986; 59(4): Hilgenfel S, Kaynik AM, Koehle SA, Stone HA. An accuate von Neumann s law fo thee-dimensional foams. Physical Review Lettes. 00; 86(): Weaie D, Glazie JA. Relation between volume, numbe of faces and thee-dimensional gowth laws in coasening cellula pattens. Philosophical Magazine Lettes. 99; 68(6): Hilgenfel S, Kaynik AM, Reinelt DA, Sullivan JM. The stuctue of foam cells: isotopic Plateau polyheda. Euophysics Lettes. 004; 67():