A numerical study of shock-induced particle velocity dispersion in solid mixturesK. Yano and Y. Horie Citation: Journal of Applied Physics 84, 1292 (1998); doi: 10.1063/1.368197 View online: http://dx.doi.org/10.1063/1.368197 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/84/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermochemical modeling of temperature controlled shock-induced chemical reactions in multifunctionalenergetic structural materials under shock compression J. Appl. Phys. 111, 123501 (2012); 10.1063/1.4729048 Investigation of ShockInduced Chemical Reactions in NiTi Powder Mixtures Using Instrumented Experiments AIP Conf. Proc. 620, 1123 (2002); 10.1063/1.1483735 New evidence concerning the shock-induced chemical reaction mechanism in a Ni/Al mixture AIP Conf. Proc. 429, 639 (1998); 10.1063/1.55574 Particle velocity dispersion in shock compression of solid mixtures AIP Conf. Proc. 429, 259 (1998); 10.1063/1.55533 Recovery studies of impact-induced metal/polymer reactions in titianium based composites AIP Conf. Proc. 429, 667 (1998); 10.1063/1.55498

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A numerical study of shock-induced particle velocity dispersionin solid mixtures

K. Yanoa) and Y. HorieDepartment of Civil Engineering, North Carolina State University, Box 7908 Raleigh,North Carolina 27695-7908

~Received 9 February 1998; accepted for publication 27 April 1998!

Shock-induced particle velocity dispersion in solid mixtures was numerically investigated using twoapproaches: discrete element simulation and continuum mixture calculation. Results show~i! atrend-wise agreement between the two models,~ii ! nonequilibrium distributions of particle velocitydispersion, and~iii ! particle velocity dispersions of 20–100 m/s for a 10 GPa shock wave in Ni/Almixtures and 5–70 m/s for a 5 GPa shock wave in Ti/Teflon mixtures. Particle velocity dispersionsof this magnitude are thought to be the driving mechanism for initiation of chemical reactions inreactive solid mixtures. ©1998 American Institute of Physics.@S0021-8979~98!02815-1#

I. INTRODUCTION

In the science and technology of shock compression ofsolids, the understanding of micro-scale details, is becomingincreasingly important.1 Much of the current interest in par-ticle velocity dispersion~PVD! at the meso-scale~on theorder of micrometers! is based on observations of unusualchemical reactions in inorganic powder mixtures and theproposal that PVD plays a critical role in the ignition of thesereactions.2,3 A similar suggestion has been made regardingthe ignition of explosives through formation of shear bands.4

In fluid mixtures, PVD is known to cause fine fragmen-tation and high speed mixing and is considered to be theessential mechanism for vapor explosion.5 Thus, it is notunreasonable to expect that even in solid mixtures, PVDplays an essential role in the control of chemical reactionsunder high-pressure shock wave loading. In fact, there isexperimental evidence that even in polycrystalline copper therelative magnitude of PVD is about 18% of the particle ve-locity for plane shock compression at about 3 GPa. Above acertain shock strength, evidence of turbulentlike flow is ob-served in recovered specimens.6

In solid mixtures, however, PVD has never been evalu-ated except by an elementary method where shock imped-ance is used to calculate the difference in relative particlevelocities. In this article, PVD in shock compression of solidmixtures is numerically investigated using two methods: adiscrete element calculation and a continuum mixture theory.Mixtures consisting of Ni/Al and Ti/Teflon are investigatednot only because of observed shock chemistry, but also be-cause of their contrasting mechanical properties.7,8

II. DISCRETE ELEMENT MODEL

Details of the discrete element modeling are described inRef. 9. Therefore, this section summarizes only the essentialfeatures as they relate to motion and deformation. In thistechnique, materials are represented by a collection of meso-scale discrete particles called elements. Figure 1 is an illus-

tration of a discrete binary mixture where constituent ele-ments are distributed randomly. Motion of elements isdescribed by rigid body dynamics. Thus, the governing equa-tions of each element are given by

mid2r i

dt25 (

iÞ j , j 51

N

Fi j , ~1!

Jid2Ui

dt25 (

iÞ j , j 51

N

~qi j 3Fi j !, ~2!

wherei ~or j ! is an element number,N is the total number ofelements,mi is the element mass,Ui is the angular vector,r i

is the position vector,Ji is the inertia tensor,Fi j is the re-sultant force acting on elementi due to elementj , andqi j isthe distance vector from the mass center of elementi to theapplication point ofFi j . In this section, characters with su-perscripti or j represent the properties pertaining to elementi or j , and those with superscripti j pertain to propertiesbetween elementsi and j . Different types of interactionforces are considered depending on the bonding state~linkedand/or contact! that is determined by the distance betweenand deformation history of elements~see Table I for a de-tailed definition!. The types of interaction forces used in thepresent model are illustrated in Fig. 2 and are represented bythe following functions:

~i! Central potential forcef cpi j

f cpi j 5

2a i j mn

r 0i j ~n2m! F S r i j

r 0i j D 2~n11!

2S r i j

r 0i j D 2~m11!G r i j

r i j , ~3!

where a i j is the interaction potential coefficient,r i j is therelative position vector of elementj with respect to elementi , r i j is the inter-element distance, andr 0

i j is the initial inter-element distance.m andn are integer parameters of the in-teraction force potential (m,n).~ii ! Shear resistance forcef sh

i j

f shi j 5Gi j g i j Ai j t i j , ~4!a!Electronic mail: kyano@eos.ncsu.edu

JOURNAL OF APPLIED PHYSICS VOLUME 84, NUMBER 3 1 AUGUST 1998

12920021-8979/98/84(3)/1292/7/$15.00 © 1998 American Institute of Physics

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where Gi j is the effective shear modulus,g i j is the shearstrain, Ai j is the effective linking area, andt i j is the unitvector pointing in the direction of tangential relative velocityof elementj with respect to elementi .~iii ! Central damping forcef cd

i j and tangential viscous fric-tion force f tv

i j

f cdi j 5Cnvn

i j , ~5!

f tvi j 5Ai j m i j

vti j

r i j , ~6!

whereCn is a damping coefficient,vni j is the radial compo-

nent of relative velocity of elementj with respect to elementi , m i j is an effective viscosity coefficient, andvt

i j is the tan-gential component of relative velocity of elementj with re-spect to elementi .~iv! Dry friction force f d

i j

f di j 5Cdf cp

i j t i j , ~7!

where Cd is a Coulomb frictional coefficient, andf cpi j

5uf cpi j u. Dry friction will be present only when a repulsive

central potential force acts between unlinked elements.

Types of interaction forces between elements are chosenbased on perceived material response behavior. For instance,for elements that are linked, central potential force, shearresistance force, and central damping force may be involved,i.e., Fi j 5f cp

i j 1f shi j 1f cd

i j . On the other hand, for elements

that are in contact, but not linked, only the repulsive part ofthe central potential force and the dry friction force may beinvolved, i.e.,Fi j 5f cp

i j 1f di j .

In the model computations of Ni/Al and Ti/Teflon mix-tures, the interaction forces were chosen to compare the re-sults with continuum mixture calculations as follows: a cen-tral potential force@Eq. ~3!# and a central damping force@Eq.~5!# for linked elements and a central potential force and atangential viscous friction force@Eq. ~6!# for contacting ele-ments. The rationale for these selections is that in the con-tinuum mixture calculation, shear resistance is neglected andthe interaction force between and within constituents is ofviscous type. The parameters of the central potential force@a i j , m, andn in Eq. ~3!# were obtained by fitting Eq.~3! toexperimental Hugoniots.10,11 Selection ofCn in Eq. ~5! wasmade by qualitatively matching the shock rise-time in metal-lic components to the measured values.12 A few trial runswere necessary to determine the coefficients. Material con-stants and parameters used in the calculation are listed inTable II.

The initial and boundary conditions for the model com-putations are shown in Fig. 1. Initially, neighboring elementsof the same material were linked. Elements of different ma-terials can be in contact, but no linking was allowed. Theradius of each element was chosen to be 1mm so that ashock wave may be captured with a reasonable spatial reso-lution. Periodic boundary conditions were imposed on thevertical sides of the mixture to avoid edge effects. Threemass fractions of the denser material~0.3, 0.6, and 0.9! wereexamined. For each mass fraction, the impact speedup wasvaried from 150 to 600 m/s, with an increment of 150 m/s.

III. CONTINUUM MIXTURE MODEL

A. Preliminaries

In the continuum mixture model, heterogeneity of a mix-ture is represented by the volume fraction of each constitu-ent. For a mixture ofn constituents,

(i 51

n

a i51, ~8!

where subscripti denotes constituenti and a i its volumefraction. Using this volume fraction, one may introduce par-tial densityr i and partial pressurepi as follows:

r i5a ir io , ~9!

pi5a i pio , ~10!

FIG. 2. Interaction forces in the discrete element model.

TABLE I. Bonding states between elementsi and j .

r i j a<r min

ij b Linked and in contact.

r minij ,rij<r0

ij cLinked and in contact if the elements were linkedpreviously,or in contact otherwise.

r 0i j ,r i j <r max

ij dLinked if the elements were linked previously, or unlinkedand not in contact otherwise.

r maxij ,rij Unlinked and not in contact

ar i j : inter-element distance.b r min

ij : minimum inter-element distance for elements to be linked if theywere previously unlinked.

c r 0i j : inter-element distance at the initial state.

d r maxij : maximum inter-element distance for elements to maintain linking

status.

FIG. 1. Illustration of the discrete mixture model and the boundary condi-tions for propagating a plane shock wave in the mixture.

1293J. Appl. Phys., Vol. 84, No. 3, 1 August 1998 K. Yano and Y. Horie

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wherer io andpi

o are the intrinsic density and the solid pres-sure of constituenti .

B. Momentum conservation equation

Nigmatulin’s formulation of mixtures13 is followed andapplied to the binary system. The major assumptions made inthis analysis are:

~i! one spatial dimension,~ii ! no material strength effect,~iii ! common solid pressure and shock wave speed be-

tween constituents, and~iv! no body force, no chemical reaction, and no phase

transformation.

With these assumptions the momentum conservation equa-tion is given by

]

]t~r iwi !1

]

]x~r iwi

2!5]pi

]x1Pji

~11!~ iÞ j and i , j 51,2!,

wherewi is the particle velocity andPji is the momentumsupply from constituentj to constituenti per unit volume ofthe mixture per unit time. The momentum interaction termPji is expressed as

Pji 5Pda i

dx1D~wj2wi !, ~12!

whereP is the common pressure@P5p1o5p2

o by assumption~iii ! above# and D is the drag coefficient between constitu-entsi and j .

For steady wave propagation, Eq.~11! reduces to thefollowing ordinary differential equation by the transforma-tion: j5x2Ut, whereU is the steady shock wave speed.

d

dj~r iui

21pi !5Pda i

dj1D~uj2ui !, ~13!

where ui is the particle velocity with respect to the shockfront, i.e., ui5wi2U. If the origin of the j coordinate istaken at the point where all the properties~pressure, density,velocity, etc.! have reached equilibrium after shock loading,then the integration of the above equation yields

@r iui21pi #5E

0

dP

da i

djdj1E

0

dD~uj2ui !dj, ~14!

whered is the distance from the origin to the initial state and@ f (j)#5 f (d)2 f (0). The integrals on the right hand siderepresent the result of momentum exchange between con-stituents that are absent in the momentum jump condition forsingle component materials.

C. A model for the drag coefficient

Evaluation of the drag coefficientD in Eq. ~14! is basedon a model for dispersed two-phase flow.14 A similar modelis proposed for reactive granular materials.15 In our model,Dis given by

D53

8CD

adrc

ruvru, ~15!

wheread is the volume fraction of the dispersed phase,rc isthe density of the continuous phase,r is the radius of thedispersed particle,vr is the relative velocity between the twophases, andCD is a nondimensional coefficient. Since theReynolds number~defined below! of the powder materialsinvestigated is on the order of unity or smaller,CD for theStokes regime may be used:

CD524

Re, Re5

2r uvrurc

mc, ~16!

wheremc is the viscosity of the continuous phase. Substitu-tion of Eq. ~16! into ~15! yields

D59

2

mcad

r 2 . ~17!

For binary solid mixtures, the above model is modified toreflect the fact that both constituents can play the role ofcontinuous phase

D59

2

m~a1a2!1/2

r 2 , ~18!

where m is defined bym5a1m11a2m2 ~where m i is theviscosity of constituenti ,i 51,2!.

TABLE II. Material constants and parameters used in the discrete element calculation.

Materials of interacting elements

Ni-Ni Al-Al Ni-Al Ti-Ti Teflon-Teflon Ti-Teflon

a i j ~mJ! 3.311 0.423 1.939 1.926 0.100 0.162m 1 2 1 1 1 1n 2 3 2 2 2 3Cn ~g/s! 117a 36.3a — 37.9a 2.49a —m i j ~Pa s! 20.0b 5.0b 12.5c 20.0b 2.0b 11.0c

Ai j (31028 m2) 2.0 2.0 2.0 2.0 2.0 2.0

aOnly applicable to linked elements. The values are determined to fit shock rise-time in metals.bSame values as those used in the continuum mixture calculation~see Table III!.cAverage of the two materials.

1294 J. Appl. Phys., Vol. 84, No. 3, 1 August 1998 K. Yano and Y. Horie

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D. Derivation of the particle velocity dispersion

Now Eq. ~14! can be solved for the PVD. First, userearrange the equation as follows:

E0

dD~uj2ui !dj5@r iui

21pi #2E0

dP

da i

djdj. ~19!

Since the volume fractiona i is either a monotonically in-creasing or decreasing function over 0,j,d, the integral onthe right hand side of Eq.~19! can be evaluated by using thefirst mean value theorem as follows:

E0

dP

da i

djdj5P~z!E

0

d da i

dj5hPf~a i02a i f !, ~20!

where 0,z,d, h is a positive normalizing factor less thanunity, and subscripts ‘‘0’’ and ‘‘f ’’ indicate the initial andthe final state. Upon substitution of Eq.~18! and Eq.~20!into Eq. ~19!, the average PVD is obtained:

^uj2ui&'@r iui

21pi #2hPf~a i02a i f !

Dd

'2

9

r 2

m~ a1a2!1/2UDt$@r iui

21pi #

2hPf~a i02a i f !%, ~21!

wherea i is the average initial and final values of the volumefraction andd has been replaced byUDt ~shock speed3 risetime!. The right hand side of Eq.~21! can be calculated interms of the three parameters~Dt, r , andh! and flow prop-erties~r i , ui , pi , anda i! for a givenU. The latter are givenby solving a system of algebraic equations: Eqs.~22!–~30!.Equations~22!–~24! represent the jump conditions for mass,momentum, and energy of the mixture, Eqs.~25! and ~26!the mechanical and the thermal equation of state of constitu-ent i ( i 51,2), Eqs.~27! and ~28! the additivity of specificvolume and energy, and Eqs.~29! and~30! the conditions oftemperature and pressure equilibrium at the final state:

wf2U

Vf5

2U

V0, ~22!

Pf5Uwf

V0, ~23!

Ef512Pf~V02Vf !, ~24!

pi fo 5

bTi

b iF S Vi0

o

Vi fo D b i

21G1g i

Vi fo ETi f

, ~25!

Ei f 5bTi

Vi0o

b iH 1

b i21 F S Vi0o

Vi fo D b i 21

21G1Vi f

o

Vi0o 21J 1ETi f

,

~26!

Vf5l1V1 fo 1l2V2 f

o , ~27!

Ef5l1E1 f1l2E2 f , ~28!

ET1

Cv1

5ET2

Cv2

, ~29!

Pf5p1 fo 5p2 f

o . ~30!

Subscripts ‘‘0’’ and ‘‘f ’’ indicate the initial and the finalstates as before. The variables and material properties used inthe above equations are defined as follows:w: particle ve-locity of the mixture.~Note thatwf2U5u1 f5u2 f!; P: pres-sure of the mixture.~Note thatPf5p1 f

o 5p2 fo !; V: specific

volume of the mixture;E: specific energy per unit mass ofthe mixture;bTi

: isothermal bulk modulus of constituenti ;b i : material constant of constituenti ; g i : Gruneisen con-stant of constituenti ; Vi

o : specific volume of constituenti .Vi

o51/r io ; ETi

: Thermal energy per unit mass of constituenti . ETi

5CVi(Ti2Ti0) whereCVi

, is specific heat at constantvolume andTi temperature of constituenti ; Ei : specific en-ergy per unit mass of constituenti ; andl i : mass fraction ofconstituenti .

There are 12 equations for 13 variables:U, wf , Pf , Vf ,Ef , p1 f

o , p2 fo , V1 f

o , V2 fo , E1 f , E2 f , ET1 f

, and ET2 f. Note

that Eqs.~25!, ~26!, and ~30! represent two equations each.Thus, all the variables are uniquely determined once theshock speedU is given. Also, parametric study has shownthat the effect ofh on the evaluation of PVD is small.

Equations~22!–~30! were solved for the mixtures ofNi/Al and Ti/Teflon. The same mass fractions were used asthose in the discrete element simulation. For selection of theparticle radiusr in Eq. ~21!, it is necessary to consider theaggregation effect of elements in the discrete element simu-lation. If mono-sized elements of 1mm in radius were mixedrandomly then the average aggregate radius for the particularcomposition considered was found to be 2mm. Values ofparameters used in the calculation are summarized in TableIII.

IV. RESULTS AND DISCUSSIONS

Since the discrete element mixtures are represented by arandom collection of binary constituents, direct output is notinformative without some kind of a statistical operation.Therefore, a ‘‘sampling window’’ having a width of 4mm~as shown in Fig. 3! is used. The macroscopic flow proper-ties such as pressure and particle velocity were evaluated asthe ensemble averages of elements in the window. The num-ber of elements within the window varied from 300 to 400depending on the window location.

Figure 4~a! shows spatial particle velocity profiles at15.4 ns after impact for the Ni/Al mixture oflNi50.3 andup5600 m/s. The shock speed was 5328 m/s, which is closeto a continuum mixture estimate~5665 m/s! based on Eq.

TABLE III. Parameters used in the continuum mixture calculation.

Mixture m ~Pa s! r ~mm! Dt ~ns!

Ni/Al 20.0a/5.0b 2.0 2.0d

Ti/Teflon 20.0a/2.0c 2.0 2.0d

aAssumed after Ref. 16.bTaken from Ref. 17.cAssumed.dAssumed after Ref. 12.

1295J. Appl. Phys., Vol. 84, No. 3, 1 August 1998 K. Yano and Y. Horie

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~25!. These profiles were obtained by moving the windowfrom the rigid wall to the undisturbed region. It is apparentthat the largest PVD occurs at the wave front. However,there is also a recurrence of low amplitude dispersion in thedownstream. It is thought that the PVD at the wave front isinduced primarily by the inertia difference between the twomaterials, whereas the PVD downstream occurs due to mo-mentum exchange in the equilibration process. The same fea-tures were also observed for the other mass fractions. Theresults for the Ti/Teflon mixture of lTi50.3 andup5600 m/s at 40.7 ns after impact are shown in Fig. 4~b!.The calculated shock speed was 1719 m/s. As in the Ni/Almixture, the largest PVD was observed at the wave front.However, in contrast to the Ni/Al mixture, the wave frontwas wider and the magnitude of PVD was much smaller. Thespreading of the wave front in the Ti/Teflon mixture isthought to be caused by a large difference in wave propaga-tion speeds of the two materials. The reduction in PVD mag-nitude is attributed to the smaller inertia difference betweenthe constituents.

Figure 5~a! shows the probability density of the PVD forthe Ni/Al mixture at four locations that are indicated by A,B, C, and D in Fig. 4~a!. The area under the curve over avelocity interval represents the probability of finding PVDsfor that interval. The distribution within the wave front B hasmultiple peaks and is clearly not that of equilibrium fluctua-tion. However, these peaks disappear downstream, and thedistributions at C and D are close to that of equilibriumdistribution. If location C is used to estimate the relaxationtime constantt by use ofl /U, wherel is the distance to theshock front at the time of sampling,t51.3 ns. The result isevidently different for the Ti/Teflon mixture as shown in Fig.5~b!. Although the distribution has multiple peaks at B, thepeaks are much less prominent. Further downstream, the dis-tribution profiles at C and D were again close to that ofequilibrium. The relaxation time for the Ti/Teflon is 5.2 ns.

At present, the physical significance of the multi-peakdistributions is not well understood. Model calculations areaffected by the fact that there is not yet a definitive procedurefor selecting the damping coefficient,Cn . It is felt that thepresent selection ofCn based on shock rise-time provides anupper bound. But, it suffers from the fact that it becomes afunction of pressure. Since PVD is also influenced by mate-rial properties, the quantitative delineation of model depen-dency of observed phenomena from physical effects is atpresent very difficult, and will be a subject of future study.

Figures 6~a! and 6~b! compare the maximum PVDs cal-culated by the discrete element model~shown by solid sym-bols! with steady PVDs obtained by the continuum mixturecalculation~shown by curves!. There is a trend-wise agree-ment between the two models. The deviation of the con-tinuum calculation from the discrete element calculation islarger in a higher pressure region. The two major contribut-ing factors are:~i! the difference between steady and tran-sient calculations and~ii ! the use of constant shock rise timeDt in Eq. ~21!.

The range of PVD for the Ni/Al mixture varies from 20to 100 m/s for the shock pressure of 10 GPa. For Ti/Teflon,the PVD varies from 5 to 70 m/s for the shock pressure of 5GPa. If one divides the velocity dispersion by the inter-element distance of 2mm, one obtains the maximum strainrate of;107 l/s. This is comparable to the estimated shearstrain rate in a shear band where chemical reactions wereobserved.18

FIG. 4. Spatial profiles of particle velocity for~a! Ni/Al ( lNi50.3) and~b!Ti/Teflon (lTi50.3) mixtures.up5600 m/s. Locations A, B, C, and D areassociated with Figs. 5~a! and 5~b!.

FIG. 3. Illustration of a sampling window considered for statistical averag-ing of flow properties using discrete element calculations.

1296 J. Appl. Phys., Vol. 84, No. 3, 1 August 1998 K. Yano and Y. Horie

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Finally, according to Ref. 19, if the kinetic energy ofsuch a PVD can be transformed to the energy required forcreating new surfaces, then the spatial scale of mixings maybe given by

s536g

r~Dup!2 ls , ~31!

whereg is the surface free energy,r is the density,Dup isPVD, andls is the fraction of mass transformed. Substitut-ing representative values for metals~g51 J/m2, r563103 kg/m3, Dup5100 m/s, andls50.1!, one obtainss560 nm.

If reactive powder materials are mixed down to this spa-tial scale under shock compression, concomitant to a largeincrease in temperature, then massive chemical initiation canoccur at appropriate conditions of pressure and temperature.The reactions in such a region can proceed to completion fortime less than 1ms in metallic powders.20 The above esti-mate gives credence to the idea that sub-grain level mixing

in reactive powder mixtures through the particle velocity dis-persion may be the driving mechanism for ultrafast chemicalreactions at the shock front.

V. CONCLUSION

Shock-induced particle velocity dispersion in Ni/Al andTi/Teflon mixtures was investigated numerically by usingtwo models: discrete element model and continuum mixturemodel. Intense nonequilibrium particle velocity dispersionwas observed at the shock front in the Ni/Al mixture,whereas it was less prominent in the Ti/Teflon mixture. Thevelocity dispersion of 20–100 m/s was obtained in theformer for a shock pressure of 10 GPa. The range was 5–70m/s for the Ti/Teflon mixture at 5 GPa. Localized kineticenergy associated with the velocity dispersion of this magni-tude is found to be sufficiently large to create sub-grain levelmixing on the order of 50 nm. Sub-grain mixing at this levelcan explain the massive initiation of chemical reactions inreactive powders under high-pressure shock compression.

FIG. 5. Probability density distribution of PVD for~a! Ni/Al and ~b! Ti/Teflon mixtures at locations A, B, C, and D indicated in Figs. 4~a! and 4~b!.

FIG. 6. Maximum PVD observed in the discrete element simulation~solidsymbols! and the steady PVD obtained by the continuum mixture calcula-tion ~curves! for ~a! Ni/Al and ~b! Ti/Teflon mixtures.

1297J. Appl. Phys., Vol. 84, No. 3, 1 August 1998 K. Yano and Y. Horie

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ACKNOWLEDGMENT

This work is supported in part by U.S. Army ResearchOffice ~Grant No. DAAH 04-94-G-0033!.

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