L -R184 NO8 STUDY OF BOUNDARY STRUCTURES(U) VASNJNGTON UNIV SEATTLE iniDEPT OF MATERIALS SCIENCE AND ENGINEERING R KIKUCHI

I 31 JUL 87 RRO-19994 3-PH DAG29-85-K-8166p UNCLASSIFIED F/G 11/6 NL

IENE.

1.2 14

I-- 3A

0GoRSTUDY OF BOUNDARY STRUCTURESc-

FINAL REPORT

RYOICHI KIKUCHI

JULY 31, 1987

U,S, ARHY RESEARCH OFFICE

DAAG29-85-K-0166 (U.W.)DAA&29-83-C-O011 (hUGHES)

DEPARTilE,1T OF ;iATERIALS SCIENCE AND ENGINEERINGUNIVERSITY OF WASHINGTONSEATTLE, WASHINGTON 98195

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11 TITLE (Include Security Classification)

Study of boundary structures (Unclassified)

12 PERSONAL AUTHOR(S)

Rvoichi Kikuchi

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Final Report FROM 4/1/83 TO7/31/87 July 31, 1987 116 SUPPLEMENTARY NOTATION The view, opinions and/or findings contained in this report are those

of he authqr($) and shyuld not be constued asan Qfficiail Department of the Army position,

COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identitfy by block number)

FIELD GROUP SUB-GROUP

'9 ABSTRACT (Continue on reverie itf necessary and identitfy by block number)

The project extends a previous study on structures of a 2-D grain boundary (g.b.).

Three major tasks have been done. The previous work in 1980, called A, was for a pure

component system. In the work labeled B, impurities are added to the system and the adsorp-

tion of impurities at the g.b. is calculated. The work labeled C is closely tied to B and

formulates atomic diffusion in a system with the g.b. The work labeled D is for a binary

system of composition close to stoichiometry, which becomes ordered.

The C-7M and PPM. -The basic analytical tool is the cluster variation method (C) and

the path probability method (PPM), both developed by the author. In the CyM, the free energy

is minimized with respect to a large number of variables using a linear iteration method

called the natural iteration method (NIM). These methods work nicely.

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The quadruple square (QS treatment. A square lattice is chosen as the basic lattice

(called the DSC lattice) of the analysis. The first, second, and third nearest neighborpairs are excluded due to a large repulsion. The fourth and fifth neighbors contributeinteraction potentials F, AB and 5 A.. aid so on. In the DSC lattice the CVM is formulatedusing a cluster made o ne lattic6 points in the form of a quadruple square. The QS form-

ulation leads to numerical results that are reliable compared with the MC method.

The pair treatment. 'In low temperatures, atoms hardly deviate from the skeletal stablecrystalline structure, and hence we can ignore DSC lattice points not on the stable crystalwhich is a square lattice. Then we can use the pair cluster as the basic cluster.- The pairmethod is found to give reliable numerical results practically the same as those of QS for

low temperatures including the entire ordered phase in the work D. This result is signifi-cant because when the pair method is used, the two sides of the g.b. do not need to beoriented symmetrically as were done in the present report. Applications of this point of

view for g.b. of different angle combinations and the 3-D cases are now planned.

Hybrid cluster. In the work D for the ordered system g.b., it is found necessary tosupplement the pair cluster by a triangle cluster near the coincident laLtice site. Sucha hybrid choice of clusters was tried for the first time, but it makes sense. We will needmore hybrid choices near the g.b. center in order to take into account the frustration

effect of atoms.

The low temperature "melting" of g.b. In all of A, B, and D cases, we found that the

low T behavior and the high T behavior of g.b. can be clearly distinguished. In low T, itis crystal-like. As T increases, the g.b. opens up as -log (Tm - T), with TM being the

bulk melting temperature. The transition region between the two is about one-half of T .m

Diffusion along the g.b. Diffusion along the g.b. is orders of magnitude larger nearthe g.b. than the bulk diffusion. The diffusion coefficient is proportional to the squareof the local vacancy density. This holds when the impurities are adsorbed to the g.b. andalso when they are repeated.

ion For

'I - I F IF

.T. .SECURITY CLASSIFICATION OF THIS PA.f

,P_ k, X A. A. A? A

I. Introduction and Summary

In any type of crystalline materials, be it metallic, ceramic, dielectnic, or ,emiconductin, grain bound-

aries (g.b.) play the key roll in affecting mechanical properties (strength and tougliness). lio%%ever, the

atomic structure of g.b.has not been known, except the ' = 0 behavior, mainly because of the lack of

techniques which can analyze the structure. 'The present project studies the struclure and properties of

g.b. in a two-dimensional system for finite temperatures using statistical mechanical techniques devcloped

by the author.

In one of the tasks performed under the contract preceding the present one, a 2-I) grain boundary

(g.b.) was studied theor'tically. The free energy I' of tile entire system (including the gb.) was written

in terms of the variables (called the state variables) which represent probabilities of local configurations.

The special feature of this work was that the cluster variation method (('VM) 2 was used in formulating

the entropy in terms of the basic variables. |ly minimi/ing F, the equilibrium state of the g.b. was derived.includinz the density profile and the excess free energy value. Since this work forms tile basis of the tasks

done under the contract being reported here, we attach Ref. I as Appendix A.

One of tile main conclusions of Ref. I was that the g.b. starts, to melt at around one-half of themneltine temperature I',,, and the g.b. broadens near F',,, as - log( l;,, - T). [his conclusion attracted the

interest of workers and several papers have been published on the subject reporting the lonte (Carlo

simulations. The low lemperature g.b. melting was confirmed by the simulations, the latest paper of which

may be the one due to Yip et al.3

%Since the method used in Ref. I is versatile in stud\ing properties of grain boundaries, we extended

it to several cases during the present contract. [he results are in Appendices 13, C', and I), and tIhe are

siutnmariied in subsequent sections.

Appendix B is an extension of Ref. I and includes impurities. Appendix ( is based on 11 ,ndcalculates atomic diffusion along the gb. [his paper is significant in that the technique of irreversible

I Introdction and Stnnmary

statistical mechanics (thle path problalbilitY ine.thod, 1 I11%l Iis used IIin l t h . stiudies It is shown that

(lilbi1sioli Is orders (if, ianitiudc larer near the gbh. kotrpared \ ith the bulk d(loniii. O ne significant

rest ilt is that for cit her case M ien thle imnpurit ics are adsorbed to or rejectcd fron, tie g.. fihe diffusionl

coecientC becme lrig near hle gb. in proportion tO the ;(uare ot' the local vaca'sic% conicentratiOis.

* :ptIentli I) works w%i h i th L. b. he' wee~n tw~o ordered phasecs. I lie s'tnmctrlc bou ndary and asvm-

nlettle boundary are diimiishcd. lhe profile tt the Ionev-i~ineec order across the v, b is calculated.

In) theL bounda1(rN problemns thle numbecr of' ind denmctt vatriables is uisti;dllv large, easily exceeins- 10)00,

and ;I niniinuln1 (it thle [ice energy a;ctuall\ the anid ptential 12) Is to be IhMund wkith respect to flhese%at iecs I I,. linecm iterMitioe tecniu IIaIledI thILI ie N II works ama/iiiel\ well and Q2 aklw decrea-ses

:is th IteII ratIIin proceeds. Ihowe\ cr smll the III decremtTI IMt beC.

I lie Impilanee11 of the 2gr11n boulndaries in) influecwingi- micc~ia propecrties (,f materials comles mv~mslv

from the conicent ratlonl of vacanicies at thle vgb. Statistical meichanics canl Calculate thle amount of vacancies;

;it theC L 1b As %%C eM% haOI Il doeIi tis project Min-rporied inI the appendices. The concentration of vacancies,

hoss ever, is (,in% llth beg-inning of thle 1- 1) studies of pralctical importance. 1t is neceded to proceeti to study

)uo % Ox ~ acn sinflenc kinesic ehiio),o the nueiis

IIIhe diffiIonl stidy i Ill ppend( (' is lIne step aheaCd 0. tile precvailing cffoul of 4tatistical mchlanics.bust till us far shosrt oif the reall> Important ;Isrect oIf the kinetic study, imnely thle plastic def'Ormiwtionl 'Ill(d

fracturec It is p~imne-d that we( %%Ill IIos eOn to theC ;tud% of grain bouiidar\ Contribution to these phentomnta.

I het (AV\I and IT%~I :are iers% ssell suilted in stun)gIT thle stuet~rc of boundaries :miails ticall. .Xt hougli

the ss a> may; still be loiz to aichieve thle goal of unders4tandingv mcclumiical proper-ties of mnaterials fromnrniL-roui opuc belm\ior, wc ssIll continue ouir Mfort tow\ards tile goal by Improving thecse miet hods.

I ITNtIOP 11m)tu mid ;nl mmwi 2

%~~~~.. -I .,.. . %

2. Grain boundaries with impurities in atwo-dimensional lattice gas model (attached as

Appendix B)

The model of Ref. I was extended to include impurities. The work was accepted for publication inPhysical Review B, and is attached as Appendix B in this report. The g.b. is formed on the basic I)SClattice. In the CVM analysis we use a quadruple square (QS) of nine DSC points as the basic cluster.

When the matrix-impurity interaction is repulsive so that the impurities are rejected from the bulkmatrix, thev are adsorbed near the grain boundary. This is Because vacancies are concentrated around theg.b. When the matrix and impurity interaction is attractive, impurities stay awav from the g.b. The

amount of adsorption is calculated as a function of temperature.

As was observed in Ref. I, the low temperature structure and the high temperature structure of theg.b. can be clearly distinguished. In the high temperature region, the g.b. widens as in Ref. I, dependingon T as - log(]',, - T). This behavior is derived based on the calculations of both the entropy and the'adsorption.

,4.

2 (Wirmn bomndaries %.Ih implirilics m a t;I -dmiemmli,um,, lt~ c L Is :1 modcl ( tItaci'cd as Ap-pen'i x .)

3. Theory of diffusion along a 2-D grain boundary(attached as Appendix C)

The papers published so far in this and preceding AR() contracts are all based on the cluster variationmethod (CVM)' 5 of equilibriun statistical mechanics. The basic approach is to minimize the free energy(actually the grand potential) with respect to the basic variables to derive the equilibrium state.

[he CVNI has been extended to non-equilibrium processes by treating the time axis as the fourthspace axis. The extension has been called the path probability method (PPM). 4 It was used successfullyin treating atomic diffusion in cn'stals and in the kinetics of phase transitions.

In the attached Appendix C. we have used the PPIM to study atomic diffusion along a grain boundary.A chemical potential gradient V pi is imposed along the g.b. to induce atomic flux. Qualitatively speaking,the ratio of the flux and V t gives the diffusion coefficient. 'he treatment is based on the equilibriumformulition in Appendix B and works on the impurity g.b. case.

Taking advantage of the finding in the following section 4, the diffusion is formulated in the skeletalcrvst:lline lattice only (ignoring the rest of the I)SC lattice points) using the pair as the basic cluster. Thisis drastically simpler than using a nine point QS as the basic cluster of the diffusion formulation.

The work has a novel t,:chnical contribution In previous I'NI formulations6' 7 of diffusion, only thebulk diffusion was treated, and thus the gradients of state variables are small under the imposcd V1t. In

the present case, the gradients of states variables are not small near the g.b. It is shown in Appendix Cthat even when the space gradients of state variables inI cqluilibriuin arc larte, at( onic flux s can beformulated when small VIL's are imposed.

3 I heory of diffusion along a 2-I) grain bo indary (atlahed as AppCnlix () 4

When a reasonable set of kinetic parameters are used. numerical computations lead to tie results thatthe diffusion is many orders of magnitude faster near the g.b. than inside the bulk. This result holds even

when the impurities are adsorbed to or repelled from the boundary. The diffusion coefficient is roughly

proportional to the square of the local vacancy concentration.

The diffusion coelticient along a line parallel to the g.b. follows the Arrhenius relation nicely, and the

activation energy can be deternined for each line at a certain distance from the g.b. The activation energy

is smaller near the g.b., corresponding to the accumulation of vacancies near the g.b. center.

3. "Ilhcory of diff'usion along a 2-1) grain Imotaii {:1tr am'holid as ,Alpc'nkix C')

- - - - - - - -- - - - -- - - - - - - -

4. 2-D grain boundary between ordered piases

(attached as Appendix D)

l ixtending the work in Appendix 13, a 2-1) g.b. between two ordered phases is stodied. lhe l)S("

lattice of Appendices A and 13 form the basis of the analysis, and the stable crvstallinc state is thv, s(piare

, att e I f t -lisne shape. In this paper, the skeletal squar- lattice is ordered when the sytfl is close to10 tile stoichlormctric AB alloy.

Two approaches were used in the formulation. One is based on the i)SC and uses the quadruplesquire (OS) mentioned in Section 2 as the basic cluster. [he other works only with the skeletal square

lattice and uses a pair of lattice points in this lattice as the basic cluster. A significant findimi is that the

two approaches give identical numerical results for low temperaturcs Including tile entire ordered region

For the cohcren, boundary of the present study, the symmetric boundary and he as rntnc

boundarv are dit in eilhed. Vinder a certain choice of in eraction parameters, the slable boundary is

symmetric in low temperatures but switches to asymmctric in high tempcrat tires.

lhe bulk shape of the skeletal crystalline lattice is the square lattice. I hweve;r. near the ub. the shape

is distorted and thus the pair cluster is insufficient in taking into account the frutration effect. It wasfound that an additional triangle cluster near the coincidence lattice site can remedy the entropy expressionand leads to the correct kIn2 value at the OK limit.

%

4. 2-I) vrai boundary between ordered phases (allached as Appcndix D)

i..4

5. T1ransitionl ayer inI aI attice-gas miodelI of a

solid-melt iunterface (at tached as Appendix L')

V_ V

b %~

% ~

I oI n 1 e 1 1la ((- !1 m I d II I II(I-I c11 7 c

AI~pcndiceS A through E

I. I i: *~~. 1i 1;1 'T i~et !Tntri.Tc,. by JAV. (alin -ind R.

I I I, N.T T ' TI T t ?T1IT 111 t t tIt pr~1-c 1 o111 cflt t

'iI \Ithr(ITL' I

Appendix F

Complete sunnnarj' of the auth or's work on boundary, structure

related to the present contract.

Since the present final report closes thce series of projects onl boundary structure supported by the

U.S. Army Research Office, and a new page of the boundary studies is about to open, it is appropriate

to summrize thle entire work (lone by thle author in the subject of interest.

1. (NI and~ PI

These two methods form the basis of thle boundary studies.

(K51I) R. Kikuchi, "A theory of cooperative phecnomena," Phys. Rev. 91 (1951) 989. ['his is thle firstfull paper onl the CVNM, and is still a good introductionl.

(K1167) R. Kikuchi and S.G. Brush, "Improvement of thle cluster-variation method," .1. C7hem Phlys.

47 (1967) 195. For a 2-1) systmn, 1-1) long clusters arc eniough to obtain good accuracy.

(K 74) R. Kikuchi, "Superposition approximation anol natural iteration calculation in cluster-vaniationl

method," .1. (hem. Phyis. 60 (1974) 1071. 'The NI method is an important computational device to use

iml the (;VM.

(K6) R . K ikuchi, "I' he pat 11 probability methmod," Irog. I heor. Phys., Suppl. No. 35, p. 1 (1966).

Still a good introdluction to (fie PPM.

Appendix 1: 9

2. Early papers

A (K6()) R. Kikuchi, "Statistical dynamnics of boundary mnotion," Annals of Phys. 11 (1969 329. Thiis

is tile first paper by the author onl thle boundary structure. It is interesting that thc wvork was onkiccs

The Ising mnodel boundary between thle +- and - domains is moved perpendicular to itself under anlexternal magnetic field. The velocity of thle motion is derived as, an cigenvalue of a dif'ferential equation.The formulation was done Using thle po-inlt approximation of the PPM1.

(CIK6 ) T.W. ('ahn and R. Kikucii. "ieory Of do01Min Walls in ordered structures 1. P'roperties atabsolute 7cro,"' 1. Phvs. ('hemn. Solids 20) (1Q1 1)4.

(1K('62) R.. 1Kikuichi and J."' (%1h111 Theory oif domain walls in Ordered stru ct ires 1I. Pair approxi -mation for nonzero temperatures,' .v. Ch'lin Solids 23 (1962) 1.37.

((7K66) .W' C ahin and R. K iktchi, -1 hecory of domain walls in ordered st ructutres Ill. Ftlect oifsttstitut ional deviations from \toiclimet ry." .1. l'hys ( 'hem. Solid-, 27 (1966) I . 5. In these three papers.the (A'\f was used. I loiever. since the NJ method In (K 74) below was niot known, only thle pair

appro ximnation was applied for an antiphase boundary ( .\PH) in bcc.

31. 'l1i scalar product (SP) fornuiiation

't'he excess free energy' of the boundaryln IS (alciilatedl ustiv only thle bulk proper-ties of thle twko sides.

(K67) R. Kikuchi, "Cooperative phenomena in thle triangular lattice," .1. Chem,. Phlys. 47 (11Q67) 1664.

TIhis paper shows that the "central hump" of tile free energy curve disappears as, we improve approximationin thle C~vM hierarchy. TIhis conclusion make-, us question thle validity andor accuracy of thle (

treatments of thle boundary structure, for example in the papers in 2 above.

(( 'NN7 I )I.1). L . t layton and( ( i.WN. WVood)) Iry, .1r., .1.( 'ht'mn I Il i. s. 55 ) 7 71 ) V Y)S I his pipecr hiows

how to calculate tile ecess free energyv of aI boiundar im- iic t iilk piopert ics (,ill of thle t% s ites I heir

mnethod is uised in the suhs-cqtient papers.

Appendix 1: In

(K721,11,111) R.. Kikuchi, "Boundary free energy' in the lattice model. 1. General formulation," .1.(Ciem. Phvs. 57 (1972) 777; "If. Applications, of the general formula," .1. Chem. Phys. 57 (1972) 783; "Ill.Solution of thle paradox," .1. Chem. P1hys. 57 (1972) 787. These three papers present detailed discussions

-. of the SP method, and the interpretation of thc (VNI treatment in 2.

(K721V) R. Kikuchi, "Phasc transition within a phase boundary," .J. Chem. Phys. 57 (1972) 4633.U sing thle SP method, it is shown that an order-disorder type phase transition occurs within a phaseboundary, when the boundary itself is 2-I).

(K73) R.. Kikuchi, "Phase transition within a phase boundary Ill," Scripta Mvet 7 (1973) 275. 'Ihiscompares works of several different groups on thle subject.

(K76) P.. Kikuchi, "Natural iteration method and] boundary free energv," .1. Chem. Ilhvs. 65 (2976)4545. Fromn this paper on, the boundary work was supported by tile U.S. Armyv Research O)ffice. Thiepaper uses the NI method to calculate the boundary free energy formulatcd by thle SP~ method.

(K77a? R. Kikuchi, ''The scala'r-produet cxpersslin of boundary free energy for long-range interactingsvstemns." .1. (hem. Phs. 68 ( 1977) 119). 'The S11 formulation is extended to thle case wvhen thle Interactionis not limited to the necarest -nieighbor pairs.

4.hl9utoi fteNN

T11ht.')ntuiral iteration method (N\IN\1) was devised inl ( K74) quoted inl I above. '[he N IN made it*possible to calculate boundar-y stn-icture with basic clusters, larger than thle pair.

(K77b) R. Kikuchi, "Structure of phase boundaries," J. Chem. Phys. 66 (1977) 3352. In addition tofihe NIN11. a spcial technique adlopted from the work of Weeks and Giilmner is used to calculate <~ 110 >boiida ry of thle bcc ks~ig model using a tet raliedron as the basic cluster. Results are compared with t heSP' treatmencrt of thle samne bounda~ry'.

5. Ice Immnmlari' (1111'11 aul AI'll) using ai letralu'dron

\rpcindi I I

(CK79) J.W. ('ain and R. Kikuchi, "(;round state of fcc alloys with inultiatorn interactions," Acta

Met. 27 (1979) 1329.

(KC79) R. Kikuchi and J.W. Cahn, 'Theory of interphase and antiphaes boundaries in fcc alloys,"Acta Met. 27 (1979) 1337. These two papers form a connected pair. ll'13 and APB of CujAu are

calculated using the tetrahedron as the cluster. [he bulk phase is the best analytical formulation calculated

by van 13aal, Golosov, Kikuchi, and de Fontaine using the CVM.

(CKR5) T.W. Caln and R. Kikuchi, 'Transition laver in a lattice-gas model of a solid-melt interface."

Phys. Rev. B31 (1985) 4300. This is the paper in Appendix F of this report.

6. 2-1) grain boundary studies.

This is the last category of the boundary studies done by the author so far. The papers arc in

Appendices A through 1) of this report.

Apd

'

Appendix F 12

\ .%,\*, *A.'%',\.(' :. "- .'4' ' -&. . '..' ... , ,.:*

Bibliography

I1. R. Kikuchi and J.W. Cahin, "Grain -bo undary melting transition in a two-dimensional lattice-gas

model", Phys. Rev. B21 (1980) p. 1893.2. R. Kikuchi, "A theory of cooperative phenomena", Phys. Rev. 81 (1957) p. 988.

3. T. Nguyen, P.S. flo, T. Kwok, C. Nitta, and S. Yip, Phys. Rev. len. 57 (1Q86) p. 1919.

* 4. R. Kikuchi, -'ihe path probability method", ['rog. 'Th. Phys Suppl. No. 35 (Kyoto. 1966) p. 1.

5. R. Kikuchi, "Superposition approximation and natural iteration calculation in cluster-variation

method', .J. (,hem. Phys. 60 (1974) p. 1071.6. R. Kikuchi and 11. Sato, "Correlation factor in substitutional diffusion in binary alloys", .1. (Chem.

l'hvs. 53 (1970) p. 2702.7. R. Kikuchi and 11. Sato, "Diffusion coefficient in an ordered binary alloy", .1. (,'hem Phys. 57

(1972) p. 49)62.8. K. (ischwend, If. Sato, and R. Kikuichi, "Kinetics of order-disorder transformations in alloys.

11", .J. ('hem. Phys. 69 (197R) p. 5006.9. R. Kikuichi and I.WA. ('ahn, 'Theory of interphase and antiphiase boundaries in fke alloys", Ac-ta.

-plet. 27 (1979) p. 1337.

13bilowraphy 13

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