FAD-AIOI 417 CATHOLIC UNIV OF AMERICA WASHINGTON DC VITREOUS STATE LAB F/9 7/4THE COMMINUTION BEHAVIOR OF LIQUIDS. U)JUN 81 T A LITOVITZ, C J NONTROSE DAA029-76-0-0100

UNCLASSIFIED ARCSL-CR-81025 Hi

END

EEEE

E2LEVEL% JAICHEMICAL SYSTEMS LABORATORY CONTRACTOR REPORT

ARCSL-CR-810Z5

THE COMMINUTION BEHAVIOR OF LIQUIDS

Final Report

o?

by

T. A. LitovitzC. J. Montrose

DTICA&ELECTE...%

June 1981 JUL 1 5 1981

B_

CATHOLIC UNIVERSITY OF AMERICAVitreous State LaboratoryWashington, D.C. Z0064

Contract DAAG29-76-DOIO0

w US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDChemical Systems Laboratory

UIAberdeen Proving Ground, Maryland 21010

Approved for public release; distribution unlimited.

7--."

E2LEVEL% JAICHEMICAL SYSTEMS LABORATORY CONTRACTOR REPORT

ARCSL-CR-810Z5

THE COMMINUTION BEHAVIOR OF LIQUIDS

Final Report

o?

by

T. A. LitovitzC. J. Montrose

DTICA&ELECTE...%

June 1981 JUL 1 5 1981

B_

CATHOLIC UNIVERSITY OF AMERICAVitreous State LaboratoryWashington, D.C. Z0064

Contract DAAG29-76-DOIO0

w US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDChemical Systems Laboratory

UIAberdeen Proving Ground, Maryland 21010

Approved for public release; distribution unlimited.

7--."

Disclaimer

The views, opinions, and/or findings contained in this report are those of the authorsand should not be construed as an official Department of the Army position, policy, or decisionunless so designated by other documentation.

Disposition

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Disclaimer

The views, opinions, and/or findings contained in this report are those of the authorsand should not be construed as an official Department of the Army position, policy, or decisionunless so designated by other documentation.

Disposition

Destroy this report when it is no longer needed. Do not return it to the originator.

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11. CONTRACT OR GRANT NUBER(e)

T. AJL1-tovtz DAAG29-76-D-0 100

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Catholic University of America AE I OKUI UBR

Vitreous State Laboratory(

Washington,D.C. 20064

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I6. SUPPLEMENTARY NOTES

Contract Project Officer: Arthur K. Stuempfie (DRDAR-CLB-PO, 301-671-3058)

19. KEY WORDS (Continue on rev'erse side if neceecery mid identify by block numbe r)

F. plosive dissemination Computer simulation') >,Cmiinatiol Molecular dynamics

20, B§STrACT (Cmitwue - inverse ei~ I ne7er an identify by block numtber)

'A phvsical mode-l of the explosive comminution process in liquids is presented. Thekey featui 'I the model involves the development of flaws in the liquid under the actionof large expln)siv,7-v generated stresses. It is at these flaws that the material fails via apro~t-ss tormnf "' liquid fracture." Guided by molecular dynamics (MO) studies, a highlysimp:'fir-d vie-w of the process is proposed. Even this simplified picture is relativelySI1C II.SSf~L1 in yielding semiquantitative agreement with experimental data.

D An 73 14n3 EDITION OF I NOV 65,IS OBSOCLETE UCASFE

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~. EORT 2.GOVT ACCESSION No. 3. RECIPIENT'S CATALOG NUMBER

C ,LrE (mid Su.btti) OF Rd POR ;~ftfOVEREO]9 Final 4 SeP 79 0THE COMMINUTION LQISBEHAVIOR OF LQISd - Maroko ',l

114-eRPOMN1R1W. 'REPORT NUMBER

11. CONTRACT OR GRANT NUBER(e)

T. AJL1-tovtz DAAG29-76-D-0 100

J.iMontrose "- .. 139. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

Catholic University of America AE I OKUI UBR

Vitreous State Laboratory(

Washington,D.C. 20064

II CONTROLLING OFFICE NAME AND ADDRESS I Z.; 4EpT GAT a

Corfander/Wr)iector, Chemical Systems Laboratory Juns8lAITNI: DRDAI-l-CLJ-R 13. NUMBER OF PAGES

Aberueen Proving Ground, MD 21010 42.T NTRING AGENCY NAME A ADORESS(if different fro. Controlling Office) 1S. SECURITY CLASS. (of this report)

Commander/Director, Chemical Sy~tems LaboratoryATTN: DRDAR-CLB-.PO . UCASFEAberdeen Proving Ground, MD 21010r/, U.1 DECLA.SIFICATION DOWNGRADING

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A pproved for public relee distribution unlimited.

____ JL U~A 'A17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, If different from, Report)

I6. SUPPLEMENTARY NOTES

Contract Project Officer: Arthur K. Stuempfie (DRDAR-CLB-PO, 301-671-3058)

19. KEY WORDS (Continue on rev'erse side if neceecery mid identify by block numbe r)

F. plosive dissemination Computer simulation') >,Cmiinatiol Molecular dynamics

20, B§STrACT (Cmitwue - inverse ei~ I ne7er an identify by block numtber)

'A phvsical mode-l of the explosive comminution process in liquids is presented. Thekey featui 'I the model involves the development of flaws in the liquid under the actionof large expln)siv,7-v generated stresses. It is at these flaws that the material fails via apro~t-ss tormnf "' liquid fracture." Guided by molecular dynamics (MO) studies, a highlysimp:'fir-d vie-w of the process is proposed. Even this simplified picture is relativelySI1C II.SSf~L1 in yielding semiquantitative agreement with experimental data.

D An 73 14n3 EDITION OF I NOV 65,IS OBSOCLETE UCASFE

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PREFACE

The research described in this report was authorized by contract IL161102A71A,Research in Defense Systems, Technical Area - Chemical Defense Research. This work wasstarted in September 1979 and completed in March 1980.

Reproduction of this document in whole or in part is prohibited except with permissionof the Commander/Director, Chemical Systems Laboratory, ATTN: DRDAR-CLJ-R, AberdeenProving Ground, MD 21010. However, Defense Technical Information Center and the NationalTechnical Information Service are authorized to reproduce the document for US Governmentpurposes.

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3/1 -

PREFACE

The research described in this report was authorized by contract IL161102A71A,Research in Defense Systems, Technical Area - Chemical Defense Research. This work wasstarted in September 1979 and completed in March 1980.

Reproduction of this document in whole or in part is prohibited except with permissionof the Commander/Director, Chemical Systems Laboratory, ATTN: DRDAR-CLJ-R, AberdeenProving Ground, MD 21010. However, Defense Technical Information Center and the NationalTechnical Information Service are authorized to reproduce the document for US Governmentpurposes.

Di9A

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TABLE OF CONTENTS

Page

1 INTRODUCTION...................................................... 7

z BACKGROUND....................................................... 7

3 PRELIMINARY CONSIDERATIONS....................................... 8

4 A PHYSICAL MODEL FOR COMMINUTION................................. 9

5 COMPARISON WITH EXPERIMENT....................................... 13

6 THE EFFECT OF POLYMERIC ADDITIVES................................ 15

7 SUMMARY AND CONCLUSIONS......................................... 15

APPENDIX............................................................... 17

DISTRIBUTION LIST ........................................................ 39

5/6

TABLE OF CONTENTS

Page

1 INTRODUCTION...................................................... 7

z BACKGROUND....................................................... 7

3 PRELIMINARY CONSIDERATIONS....................................... 8

4 A PHYSICAL MODEL FOR COMMINUTION................................. 9

5 COMPARISON WITH EXPERIMENT....................................... 13

6 THE EFFECT OF POLYMERIC ADDITIVES................................ 15

7 SUMMARY AND CONCLUSIONS......................................... 15

APPENDIX............................................................... 17

DISTRIBUTION LIST ........................................................ 39

5/6

THE COMMINUTION BEHAVIOR OF LIOUIDS

1. INTRODUCTION

The aim of this investigation is to assess the role of nonlinear viscoelasticity in theprocess of liquid comminution under explosive stress. Ultimately the goal of the program is toevaluate the hypothesis that the crucial factor in determining the comminution and dispersalcharacteristics of liquids under these conditions is the nonlinear dynamical response behavior ofthe liquids. In attacking the problem, it is essential first to develop a model of comminutionthat incorporates the mechanism of liquid response and failure when subjected to explosivelygenerated stresses. The central theme of this report is the proposal of such a model, the ex-ploration of its consequences, and, where possible, a comparison of predictions based upon itwith experimental data. This model calculation forms the first part of the report. Followingthis is a brief section which isolates those elements of the proposed mechanism that are con-trolled by the nonlinear viscoelastic characteristics of the liquid. Lastly we address the issue ofutilizing computer-simulation molecular-dynamics (MD) experiments as a tool for probing thesenonlinear characteristics; we conclude that MD experiments offer significant potential forcontributing to the understanding and control of the processes of explosive liquid comminutionand dispersal.

Z. BACKGROUND

The experimental study that provides the principal impetus and guidance for addressingcornminution behavior is that of Gerber and Stuempfle.* In this work a high-explosive projectorwas employed to produce dissemination of a liquid cloud around a preferential vertical axis. Athree-dimensional sampling array was used to measure the mass distribution of the cloud follow-ing its emergence from the projector. The mass distribution along the vertical z-axis wasdes-ribed by a Weibull distribution,

f(z) = (O/z)(z/zo) exp[-(z/z)03

such that f(z)dz gives the mass fraction lost in the height range z to z + dz above theprojector. In the Weibull for', z0 is a characteristic length parameter and 03 is a shape parame-ter. Gerber and Stuempfle present their data in terms of four parameters z 50 - the median ofthe distribution - and the first three moments of the distribution. In terms of the parametersof the distribution

z 5 0 =zo470 2 )l/f,

and the mean or first moment is

<Z> = Zormi/)/.

Values of g3 fall in the range from about 0.6 to 3 with most of the more than 30 liquids studiedbeing described by P 's near Z; <z> values varied from about 65cm to about 335cm. Generallyone finds that larger values of <z> are accompanied by larger values of 9. A major theme inwhat follows is the attempt to understand those liquid properties that are operative in the

* Gerber, B.V. and Stuempfle, A.D. TSD 004.7-I]/Vol. 1. A New Experimental Technique forStudying the Explosive Comminution of Liquids. 1976 Army Science ConferenceProceedings. Vol. I. September 1976. Uncla,,-ified Report.

7

THE COMMINUTION BEHAVIOR OF LIOUIDS

1. INTRODUCTION

The aim of this investigation is to assess the role of nonlinear viscoelasticity in theprocess of liquid comminution under explosive stress. Ultimately the goal of the program is toevaluate the hypothesis that the crucial factor in determining the comminution and dispersalcharacteristics of liquids under these conditions is the nonlinear dynamical response behavior ofthe liquids. In attacking the problem, it is essential first to develop a model of comminutionthat incorporates the mechanism of liquid response and failure when subjected to explosivelygenerated stresses. The central theme of this report is the proposal of such a model, the ex-ploration of its consequences, and, where possible, a comparison of predictions based upon itwith experimental data. This model calculation forms the first part of the report. Followingthis is a brief section which isolates those elements of the proposed mechanism that are con-trolled by the nonlinear viscoelastic characteristics of the liquid. Lastly we address the issue ofutilizing computer-simulation molecular-dynamics (MD) experiments as a tool for probing thesenonlinear characteristics; we conclude that MD experiments offer significant potential forcontributing to the understanding and control of the processes of explosive liquid comminutionand dispersal.

Z. BACKGROUND

The experimental study that provides the principal impetus and guidance for addressingcornminution behavior is that of Gerber and Stuempfle.* In this work a high-explosive projectorwas employed to produce dissemination of a liquid cloud around a preferential vertical axis. Athree-dimensional sampling array was used to measure the mass distribution of the cloud follow-ing its emergence from the projector. The mass distribution along the vertical z-axis wasdes-ribed by a Weibull distribution,

f(z) = (O/z)(z/zo) exp[-(z/z)03

such that f(z)dz gives the mass fraction lost in the height range z to z + dz above theprojector. In the Weibull for', z0 is a characteristic length parameter and 03 is a shape parame-ter. Gerber and Stuempfle present their data in terms of four parameters z 50 - the median ofthe distribution - and the first three moments of the distribution. In terms of the parametersof the distribution

z 5 0 =zo470 2 )l/f,

and the mean or first moment is

<Z> = Zormi/)/.

Values of g3 fall in the range from about 0.6 to 3 with most of the more than 30 liquids studiedbeing described by P 's near Z; <z> values varied from about 65cm to about 335cm. Generallyone finds that larger values of <z> are accompanied by larger values of 9. A major theme inwhat follows is the attempt to understand those liquid properties that are operative in the

* Gerber, B.V. and Stuempfle, A.D. TSD 004.7-I]/Vol. 1. A New Experimental Technique forStudying the Explosive Comminution of Liquids. 1976 Army Science ConferenceProceedings. Vol. I. September 1976. Uncla,,-ified Report.

7

determination of the average height <z>. Exploring the details of the mass distribution patternis beyond the scope of this work.

3. PRELIMINARY CONSIDERATIONS

In this section we give a crude description of the liquid comminution process. Theprincipal result of the section is an equation giving the average height <z> in terms of the sizeof the particles or droplets that compose the cloud.

We consider the effect of the explosion on the liquid to be roughly separable into twomajor components:

(a) The explosive release of energy generates a shock wave that in passing throughthe liquid produces large tensile, compressional, and shear stresses. These cause local internalfissures within the liquid so that it comminutes into many small liquid fragments.

(b) The explosion produces a rapidly expanding gas that intrudes itself into thestress-produced fissures, thereby preventing their healing, and causing the liquid fragments tobe expelled from the projector with a vertical velocity v0 .

The dependence of the initial velocity on liquid parameters can be estimated by the fol-lowing sort of rough argument. We shall assume that relatively little of the energy of explosionis used in the formation of the droplets so that most of the energy E is transferred to thekinetic energy of the droplets. The initial velocity imparted to the droplets then becomes

v0 = (2c/m)l/zx p-1/2 (1)

where m is the mass of a droplet, C is the kinetic energv per droplet, and p is the liquiddensitv. Assuming that E is the same for each explosion, it follows that the initial velocity ofthe droplets is dependent only on the density of the liquid.

The dynamical problem for the projected droplets can now be worked outstraightforwardlv, at least in lowest order approximation. We assume that the droplets arespherical and that the size distribution is characterized by a function f(a) which gives the proba-hility of a droplet's radius falling in the range a to a+da. We take the forces acting on a dropletof radius a to he

(a) gravity: .4 ra 3 pg,and 3

(b) air resistance: -61rnav.

In those Pxprpssions g is the acceleration due to gravity, 71 is the viscosity of the medium (air)into which the droplet cloud is projected, and v is the instantaneous droplet velocity in thevertical direction. Newton's law then takes the form

dv/dt + v/T = -g, (Z)

whore for a droplet of radius a

= Zpaz/9n. (3)

.R

determination of the average height <z>. Exploring the details of the mass distribution patternis beyond the scope of this work.

3. PRELIMINARY CONSIDERATIONS

In this section we give a crude description of the liquid comminution process. Theprincipal result of the section is an equation giving the average height <z> in terms of the sizeof the particles or droplets that compose the cloud.

We consider the effect of the explosion on the liquid to be roughly separable into twomajor components:

(a) The explosive release of energy generates a shock wave that in passing throughthe liquid produces large tensile, compressional, and shear stresses. These cause local internalfissures within the liquid so that it comminutes into many small liquid fragments.

(b) The explosion produces a rapidly expanding gas that intrudes itself into thestress-produced fissures, thereby preventing their healing, and causing the liquid fragments tobe expelled from the projector with a vertical velocity v0 .

The dependence of the initial velocity on liquid parameters can be estimated by the fol-lowing sort of rough argument. We shall assume that relatively little of the energy of explosionis used in the formation of the droplets so that most of the energy E is transferred to thekinetic energy of the droplets. The initial velocity imparted to the droplets then becomes

v0 = (2c/m)l/zx p-1/2 (1)

where m is the mass of a droplet, C is the kinetic energv per droplet, and p is the liquiddensitv. Assuming that E is the same for each explosion, it follows that the initial velocity ofthe droplets is dependent only on the density of the liquid.

The dynamical problem for the projected droplets can now be worked outstraightforwardlv, at least in lowest order approximation. We assume that the droplets arespherical and that the size distribution is characterized by a function f(a) which gives the proba-hility of a droplet's radius falling in the range a to a+da. We take the forces acting on a dropletof radius a to he

(a) gravity: .4 ra 3 pg,and 3

(b) air resistance: -61rnav.

In those Pxprpssions g is the acceleration due to gravity, 71 is the viscosity of the medium (air)into which the droplet cloud is projected, and v is the instantaneous droplet velocity in thevertical direction. Newton's law then takes the form

dv/dt + v/T = -g, (Z)

whore for a droplet of radius a

= Zpaz/9n. (3)

.R

The solution of (Z) is just

v = v0 exp(-t/T)-gT [1 - exp (-t/T)]. (4)

Integrating this again gives the equation of motion

z(t) = v 0 T 11-exp (-t/T) + gTZ 1 --{-exp (-t/T)]. (5)

The height to which a droplet rises, z, can then be found by setting v=0 in (4) and solving for thetime:

t o = V 1( + v0 /gT) (6)

Putting z = z (t ) gives0

2Z =v 0 T- gT L( + v0 /gT). (7)

For typical values of 7, P, and a (17= 0.18 mP,Pt--1.3 g/cm 3 , a =Z5 um),rt-.O1 sec. Using vo 3x 10 = cm/sec and g = 980 cm/sec one gets t = .08 sec, which is of the correct order ofmagnitude to what is experimentally observed. avoreover v 0 >>gT so that approximately

z - v0 T. (8)

Equation (8) is accurate to better than a percent or so for the full range of liquids and experi-mental conditions studied by Gerber and Stuempfle. Using equation (3) and averaging gives

<z> = (Zv 0 /9n)p<aZ>, (9)

where < aZ> is the mean square droplet radius.

4. A PHYSICAL MODEL FOR COMMINUTION

The central issue is clear from equation (9): What are the processes - and thus theliquid characteristics - that determine the mean square droplet size? We assume that themechanism of droplet formation is mainly a mechanical (as opposed to a thermal) one. At anyinstant even an equilibrium liquid exhibits local deviations from homogeneity and isotropy. Onecharacterization of the deviations from homogeneity is the distribution of fluctuations in thedensitv. From classical statistical mechanics the probability of occurrence of a densityfluctuation in the range Apto Ap+ dp is

P(AP)dP = (Zir< APZ>) - % exp [-AP 20/2< A PZ>] dp, (10)

where < A p > is the mean square density fluctuation given by

< AP Z> = P2 (kT/vV/K, (II)

where T and p are the temperature and average density of the system, V is the volume of theregion under consideration, k is Boltzmann's constant, and K is the equilibrium bulk modulus ofthe material.

We suggest that under the influence of the large explosively generated stresses, theselocal inhomogeneities will behave as structural flaws in the material: shear stresses tend to

9

The solution of (Z) is just

v = v0 exp(-t/T)-gT [1 - exp (-t/T)]. (4)

Integrating this again gives the equation of motion

z(t) = v 0 T 11-exp (-t/T) + gTZ 1 --{-exp (-t/T)]. (5)

The height to which a droplet rises, z, can then be found by setting v=0 in (4) and solving for thetime:

t o = V 1( + v0 /gT) (6)

Putting z = z (t ) gives0

2Z =v 0 T- gT L( + v0 /gT). (7)

For typical values of 7, P, and a (17= 0.18 mP,Pt--1.3 g/cm 3 , a =Z5 um),rt-.O1 sec. Using vo 3x 10 = cm/sec and g = 980 cm/sec one gets t = .08 sec, which is of the correct order ofmagnitude to what is experimentally observed. avoreover v 0 >>gT so that approximately

z - v0 T. (8)

Equation (8) is accurate to better than a percent or so for the full range of liquids and experi-mental conditions studied by Gerber and Stuempfle. Using equation (3) and averaging gives

<z> = (Zv 0 /9n)p<aZ>, (9)

where < aZ> is the mean square droplet radius.

4. A PHYSICAL MODEL FOR COMMINUTION

The central issue is clear from equation (9): What are the processes - and thus theliquid characteristics - that determine the mean square droplet size? We assume that themechanism of droplet formation is mainly a mechanical (as opposed to a thermal) one. At anyinstant even an equilibrium liquid exhibits local deviations from homogeneity and isotropy. Onecharacterization of the deviations from homogeneity is the distribution of fluctuations in thedensitv. From classical statistical mechanics the probability of occurrence of a densityfluctuation in the range Apto Ap+ dp is

P(AP)dP = (Zir< APZ>) - % exp [-AP 20/2< A PZ>] dp, (10)

where < A p > is the mean square density fluctuation given by

< AP Z> = P2 (kT/vV/K, (II)

where T and p are the temperature and average density of the system, V is the volume of theregion under consideration, k is Boltzmann's constant, and K is the equilibrium bulk modulus ofthe material.

We suggest that under the influence of the large explosively generated stresses, theselocal inhomogeneities will behave as structural flaws in the material: shear stresses tend to

9

elongate the inhomogeneities by "organizing" the liquid structure* while the tensile stressescause the flaws to propagate at the stress intensification points. The situation is depictedpictorially in figure 1. However, in a liquid, even at the elevated pressures produced by theexplosive shock, one expects that the rate at which the flaws heal and the rate at which theypropagate via growth at their tips should be comparable; consequently, one is not dealing with aclassical brittle fracture situation, but rather one involving relatively little growth. Thus themaximum flaw dimension should be comparable with the size of the particles that result fromthe explosive comminution.

It is the processes described in this last paragraph that will depend sensitively on thenonlinear viscoelastic response characteristics of the liquid system. Because the large stressesto which the material is subjected are "switched on" essentially instantaneously, the nature ofthe behavior will be governed in large measure by the characteristic time scales of the shearard structural responses of the liquid, i.e., the shear and structural relaxation times. Thespecific picture that we have described above is an oversimplified one in which we have as-sumed that the hear relaxation time is negligibly small in comparison with the structuralrelaxation time. Only in this limiting situation is one able to portray the flaw development andliquid fracture processes without reference to the specific viscoelastic and strength parametersthat describe the material. Generally speaking these extreme conditions do not obtain; theshear relaxation time, while smaller than the structural time, is of the same order of magni-tude. Consequently, the sequential nature of the mechanisms that operate in comminution thatwas assumed above - the development of large shear strains, followed by structural reorganiza-tion (flaw development), followed by liquid fracture from tensile stresses - is not nearly soprecisely defineable. The various mechanisms do not proceed strictly serially, but rather couplein a complicated interactive and, perhaps, competitive fashion. Depending upon the relativetime scales and specific forms of the various response mechanisms, this coupling can eitherreinforce the liquid comminution process (leading to smaller particle sizes) or interfere with it(resulting in larger particle sizes). Nevertheless, the elementary picture that we are examiningis a useful one for it provides a general framework in which to analyze and compare thebehavior of different liquids acted on by explosive stresses, as well as highlighting the crucialliquid properties that govern (at least in lowest order approximation) the cloud dispersal charac-teristics.

In order to estimate the droplet size, we can employ the following sort of elementarynotions. We consider that in irder to generate droplets of characteristic dimension = a, a regionof the liquid of volume V:a must contain a flaw whose volume exceeds some critical volumeVc. The action of the explosive shear stresses is to elongate and shape the flaw[into roughly a"pillow" shape (see figure Z)] so that it can be described by dimensions a, b, and c (a>b>c). Theminimum flaw volume V = abc, that will lead to the production of droplets with dimensions onthe order of "a" will resuit if one has a local density fluctuation

AP >pVc/V = (1)

in the volume V. The liquid will comminute into particles of this size only if a fluctuation ofsuch magnitude occurs with significant probability. The condition for this to obtain is

A~ c < A 0l/:(13)

* Heyes, D. M., Kim, J. J., Montrose, C. J. and Litovitz, T. A. Time-Dependent Non-LinearShear Stress Effects in Simple Liquids: A Molecular Dynamics Study. J. Chem. Phys. 73,

3987 (1980).

10

elongate the inhomogeneities by "organizing" the liquid structure* while the tensile stressescause the flaws to propagate at the stress intensification points. The situation is depictedpictorially in figure 1. However, in a liquid, even at the elevated pressures produced by theexplosive shock, one expects that the rate at which the flaws heal and the rate at which theypropagate via growth at their tips should be comparable; consequently, one is not dealing with aclassical brittle fracture situation, but rather one involving relatively little growth. Thus themaximum flaw dimension should be comparable with the size of the particles that result fromthe explosive comminution.

It is the processes described in this last paragraph that will depend sensitively on thenonlinear viscoelastic response characteristics of the liquid system. Because the large stressesto which the material is subjected are "switched on" essentially instantaneously, the nature ofthe behavior will be governed in large measure by the characteristic time scales of the shearard structural responses of the liquid, i.e., the shear and structural relaxation times. Thespecific picture that we have described above is an oversimplified one in which we have as-sumed that the hear relaxation time is negligibly small in comparison with the structuralrelaxation time. Only in this limiting situation is one able to portray the flaw development andliquid fracture processes without reference to the specific viscoelastic and strength parametersthat describe the material. Generally speaking these extreme conditions do not obtain; theshear relaxation time, while smaller than the structural time, is of the same order of magni-tude. Consequently, the sequential nature of the mechanisms that operate in comminution thatwas assumed above - the development of large shear strains, followed by structural reorganiza-tion (flaw development), followed by liquid fracture from tensile stresses - is not nearly soprecisely defineable. The various mechanisms do not proceed strictly serially, but rather couplein a complicated interactive and, perhaps, competitive fashion. Depending upon the relativetime scales and specific forms of the various response mechanisms, this coupling can eitherreinforce the liquid comminution process (leading to smaller particle sizes) or interfere with it(resulting in larger particle sizes). Nevertheless, the elementary picture that we are examiningis a useful one for it provides a general framework in which to analyze and compare thebehavior of different liquids acted on by explosive stresses, as well as highlighting the crucialliquid properties that govern (at least in lowest order approximation) the cloud dispersal charac-teristics.

In order to estimate the droplet size, we can employ the following sort of elementarynotions. We consider that in irder to generate droplets of characteristic dimension = a, a regionof the liquid of volume V:a must contain a flaw whose volume exceeds some critical volumeVc. The action of the explosive shear stresses is to elongate and shape the flaw[into roughly a"pillow" shape (see figure Z)] so that it can be described by dimensions a, b, and c (a>b>c). Theminimum flaw volume V = abc, that will lead to the production of droplets with dimensions onthe order of "a" will resuit if one has a local density fluctuation

AP >pVc/V = (1)

in the volume V. The liquid will comminute into particles of this size only if a fluctuation ofsuch magnitude occurs with significant probability. The condition for this to obtain is

A~ c < A 0l/:(13)

* Heyes, D. M., Kim, J. J., Montrose, C. J. and Litovitz, T. A. Time-Dependent Non-LinearShear Stress Effects in Simple Liquids: A Molecular Dynamics Study. J. Chem. Phys. 73,

3987 (1980).

10

imam=

Figure l.A. A Schematic Representation of Liquid Flaws Elongated under the Action of LargeShear Stresses

Figure 1.B. Liquid Flaws That Have Grown and Intersected to Form Comminution Dropletsunder the Action of Tensile Stresses.

11

Figure 2. A Flaw of Critical Size: a is the Largest Dimension, c the Smallest.

12

that is, the critical density fluctuation must be comparable with th PMS density fluctuation.Substituting equations (11) and (Z) in equation (13) and using Va and V = abc leads to anapproximate expression for the value of a:

a -- (bc) K/kT. (14)

Recalling equation (9), this leads to an equation for the mean height of the droplets:

<z> - (2/9n) v0 p (bc)4 (K/kT)g . (15)

In order that the flaws do not heal themselves too rapidly, it is essential that the smallest flawdimension c be at least on the order of a molecular separation distance; with this in mind wehave

C -/ (16)

and recalling that vo ,P -1/z, we may collect together all the density-dependent quantities onthe right hand side of equation (15). The resulting expression takes the form

<z> c p-5/6 K2 (17)

5. COMPARISON WITH EXPERIMENT

Equation (17) is in a form that can be compared with experimental results. Thecomparison is shown in figure 3 for six liquids at 20°C. Many of the data for this figure weretaken from the work of Gerber and Stuempfle* and are presented in the table which willfollow. As is apparent there is fairly good agreement between these da and the prediction ofequation (17) (the slope of the line shown in figure 3 is 65.5 g5O cm3/ZGPa-?). This offersfairly strong support for the essential correctness of the view of the comminution mechanismthat was described above. It is also clear that the modifications of this overly simplified viewthat are required if one correctly treats the dynamical aspects of the liquid response are notunderstood in any kind of quantitative fashion. Some of the molecular dynamics results(presented in the appendix) give a fair indication of the type of phenomena that must beconsidered - especially in regard to liquid structural changes occasioned by large shear stress -but efforts to apply these results quantitatively to comminution are at this point premature.

Of the six liquids which were considered above, except for carbon tetrachloride,comminution data have also been obtained at 130 0 C. The variation that one would ex ect fromour model is given by equation (15); the major effect appears to be the factor (1/kT) . On thebasis of this onI would expect the ratio of the mean height at 20 0 C to that at 130 0 C would beabout (403/293) -1.9. The experimentally determined ratios for the liquids tha' we have con-sidered fall into the range of about 1.0 to 1.5.

*Gerher, B. V., and Stuempfle, A. D., op. cit.

13

that is, the critical density fluctuation must be comparable with th PMS density fluctuation.Substituting equations (11) and (Z) in equation (13) and using Va and V =abc leads to anapproximate expression for the value of a:

a -- (bc) K/kT. (14)

Recalling equation (9), this leads to an equation for the mean height of the droplets:

<z> - (2/9n) v0 p (bc)4 (K/kT)g . (15)

In order that the flaws do not heal themselves too rapidly, it is essential that the smallest flawdimension c be at least on the order of a molecular separation distance; with this in mind wehave

C -/ (16)

and recalling that vo ,P -1/z, we may collect together all the density-dependent quantities onthe right hand side of equation (15). The resulting expression takes the form

<z> c p-5/6 K2 (17)

5. COMPARISON WITH EXPERIMENT

Equation (17) is in a form that can be compared with experimental results. Thecomparison is shown in figure 3 for six liquids at 20°C. Many of the data for this figure weretaken from the work of Gerber and Stuempfle* and are presented in the table which willfollow. As is apparent there is fairly good agreement between these da and the prediction ofequation (17) (the slope of the line shown in figure 3 is 65.5 g5O cm3/ZGPa-?). This offersfairly strong support for the essential correctness of the view of the comminution mechanismthat was described above. It is also clear that the modifications of this overly simplified viewthat are required if one correctly treats the dynamical aspects of the liquid response are notunderstood in any kind of quantitative fashion. Some of the molecular dynamics results(presented in the appendix) give a fair indication of the type of phenomena that must beconsidered - especially in regard to liquid structural changes occasioned by large shear stress -but efforts to apply these results quantitatively to comminution are at this point premature.

Of the six liquids which were considered above, except for carbon tetrachloride,comminution data have also been obtained at 130 0 C. The variation that one would ex ect fromour model is given by equation (15); the major effect appears to be the factor (1/kT) . On thebasis of this onI would expect the ratio of the mean height at 20 0 C to that at 130 0 C would beabout (403/293) -1.9. The experimentally determined ratios for the liquids tha' we have con-sidered fall into the range of about 1.0 to 1.5.

*Gerher, B. V., and Stuempfle, A. D., op. cit.

13

MODU LUS/GPa

60001 1

400E

07

~ 200

[MODULUS /GPa] 2

Figure 3. The MIean Height to Which the Droplets Rise Multiplied by Density to the 5/6Power Plotted Versus the Square of the Bulk Modulus. The Solid Line Is DrawnAccording to Equation (17). The data shown are for the six liquids listed in table 1(whprfe the, various svvmbols are identified) at a temperature of 20'C.

14

MODU LUS/GPa

60001 1

400E

07

~ 200

[MODULUS /GPa] 2

Figure 3. The MIean Height to Which the Droplets Rise Multiplied by Density to the 5/6Power Plotted Versus the Square of the Bulk Modulus. The Solid Line Is DrawnAccording to Equation (17). The data shown are for the six liquids listed in table 1(whprfe the, various svvmbols are identified) at a temperature of 20'C.

14

Table. Data Used in Constructing Figure 3. All Data Are Given for a Temperature of 20°C.

Liquid Density <Z> Modulus

(g/cm 3 ) (cm) (GPa)aTetrachloroethvlene (o ) 1.62 64 1.8

Carbon tectrachloride (e) 1/59 96 1.39

2,2-dichloroethanol (A) 1.40 232 1.91

1,2,3-trichloropropane (A) 1.61 182 2.03

Water (0) 1.00 336 2.24

Tetrabromoethane (a) 2.91 201 2.81

(a) IGPa = 109 N/m 2 - l 10 0 dyne/cm 2 = 145.5 kpsi

This cannot he accounted for by considering the chane with temperature of the density and themodulus. It probably reflects the increasinglv important role of the viscoelastic responses, forwhich the relaxation times will be significantly shorter at the higher temperature.

6. THE EFFECT OF POLYMERIC ADDITIVES

Gerber and Stuempfle* also report measurements at 20 0 C on two samples oftetrachloroethvlene to one of which was added 1% polvisobutylene (PIB), while the othercontained 1% polystyrene (PS). In the former case <z>was increased by nearly a factor of fourwhile in the latter case the increase was less than 50%f1. Phrased another way. the effect of PIBis nearly to double the average droplet radius, whereas PS produces only about a 20% increase indroplet size. The model of comminution that we have considered would lead one to expect anincrease in droplet size on the order of 10% or less. Clearly some other mechanism is operativein determining the particle size. We suggest that the effect of the polymeric molecules is toact as agents that terminate the flaws hv inhibiting their Rrowth under the applied stresses.The PIB molecules, which were of substantially greater average chain length, are much moreeffective in this regard than the (relatively shorter) PS molecules, and thus lead to theformation of rather large droplets.

7. SUMMARY AND CONCLUSIONS

In this report we have presented a physical modl nf the explosive comminution processin liquids. The key feature of the model involves the development of flaws in the liquid underthe action of large explosively generated stresses. It is at these flaws that the material failsvia a process that we have termed "liquid fracture." Guided bv previous molecular dynamics(MD) studies that we have carried out on the nonlinear response character of liquids (a partialsummary of these is given in the Appendix), we hav proposed a highly, simplified view of this

*(;crbcr, B. V., and Stuempfle, A. I)., op. cit.

I

Table. Data Used in Constructing Figure 3. All Data Are Given for a Temperature of 20°C.

Liquid Density <Z> Modulus

(g/cm 3 ) (cm) (GPa)aTetrachloroethvlene (o ) 1.62 64 1.8

Carbon tectrachloride (e) 1/59 96 1.39

2,2-dichloroethanol (A) 1.40 232 1.91

1,2,3-trichloropropane (A) 1.61 182 2.03

Water (0) 1.00 336 2.24

Tetrabromoethane (a) 2.91 201 2.81

(a) IGPa = 109 N/m 2 - l 10 0 dyne/cm 2 = 145.5 kpsi

This cannot he accounted for by considering the chane with temperature of the density and themodulus. It probably reflects the increasinglv important role of the viscoelastic responses, forwhich the relaxation times will be significantly shorter at the higher temperature.

6. THE EFFECT OF POLYMERIC ADDITIVES

Gerber and Stuempfle* also report measurements at 20 0 C on two samples oftetrachloroethvlene to one of which was added 1% polvisobutylene (PIB), while the othercontained 1% polystyrene (PS). In the former case <z>was increased by nearly a factor of fourwhile in the latter case the increase was less than 50%f1. Phrased another way. the effect of PIBis nearly to double the average droplet radius, whereas PS produces only about a 20% increase indroplet size. The model of comminution that we have considered would lead one to expect anincrease in droplet size on the order of 10% or less. Clearly some other mechanism is operativein determining the particle size. We suggest that the effect of the polymeric molecules is toact as agents that terminate the flaws hv inhibiting their Rrowth under the applied stresses.The PIB molecules, which were of substantially greater average chain length, are much moreeffective in this regard than the (relatively shorter) PS molecules, and thus lead to theformation of rather large droplets.

7. SUMMARY AND CONCLUSIONS

In this report we have presented a physical modl nf the explosive comminution processin liquids. The key feature of the model involves the development of flaws in the liquid underthe action of large explosively generated stresses. It is at these flaws that the material failsvia a process that we have termed "liquid fracture." Guided bv previous molecular dynamics(MD) studies that we have carried out on the nonlinear response character of liquids (a partialsummary of these is given in the Appendix), we hav proposed a highly, simplified view of this

*(;crbcr, B. V., and Stuempfle, A. I)., op. cit.

I

process. Even this simplified picture is relatively successful in yielding semiquantitative agree-ment with experimental data.

In order to develop a truly quantitative picture, it is our belief that the dynamicalaspects of the liquid failure mechanisms must be more fully understood. A thorougbgoingmolecular dynamics investigation of the time-dependent structural response of liquids to high-amplitude shear and tensile stresses would prove most useful in elucidating the essentialfeatures of these nonlinear viscoelastic dynamics that operate in explosive comminution. Weare at present examining the possibilities of initiating such an effort.

16

__, I II IINNN . .. " - " H~ll~a

process. Even this simplified picture is relatively successful in yielding semiquantitative agree-ment with experimental data.

In order to develop a truly quantitative picture, it is our belief that the dynamicalaspects of the liquid failure mechanisms must be more fully understood. A thorougbgoingmolecular dynamics investigation of the time-dependent structural response of liquids to high-amplitude shear and tensile stresses would prove most useful in elucidating the essentialfeatures of these nonlinear viscoelastic dynamics that operate in explosive comminution. Weare at present examining the possibilities of initiating such an effort.

16

__, I II IINNN . .. " - " H~ll~a

APPENDIX

MOLECULAR DYNAMICS

Not only in the process of comminution but also in a rather wide variety of technologicalapplications, for example elastohydrodynamic lubrication, shock loading, solid propellantfailures, etc., liquids and amorphous solids are subjected to extremely large shearing forces.Because the microscopic mechanisms of importance are not well understood, solutions toproblems involving nonlinear behavior are quite limited.

For example in order to predict shear failure (or tearing) of a liquid under the forces ofan explosion or shock wave, one must be able to compute the time evolution of the stress in afluid element when it encounters a simultaneously applied time varying shear rate and a timevarying pressure. The response of the system under these conditions can, for low shearing rates,be parameterized in terms of the shear viscosity, Tl, a shear rigidity modulus, G., and adistribution of relaxation times. At the rather large shearing rates generally encountered, thisrelatively simple description must be modified to incorporate the nonlinear character of thedynamical str,,sq response. Some authors* suppose that this can be done by appending the ideaof a limiting shear stress to the low amplitude, i.e., linear, shear response. In order to test thish\ poilcsis and, more generally, to provide a framework on which to base analytical repre-sentations of nonlinear viscoelastic behavior, we have used Molecular Dynamics (MD) to inves-tigate a model liquid system under conditions approximating those found in at high shear rates.Even though the model system is particularly simple, we have been able to reproduce the majorfeatures of shear response of real liquids. Because an MD experiment provides one with acomplete microscopic record of the system's evolution in time, one can perform "experiments"and "measure" aspects of its behavior that are not accessible in conventional experiments, butwhich provide useful insights into the origin of the properties of liquids under extreme stresscond it ions.

The MD calculation procedure observes a portion of liquid as it is sheared. Although asimultaneously applied time dependent pressure and shear rate would more closely represent theconditions of a liquid undergoing explosive comminution, we have confined our attention up tonow to considering the response of a model fluid to a time varying shear rate at a series ofdensities (pressures).

THE MODEL

MD is a computer simulation technique for probing the microscopic behavior of a systemthat allows one to account for the many-body nature of the molecular interactions. It involvesfollowing some representative number of molecules, N, by solving the classical equations ofmotion on a high speed computer to obtain their trajectories in time steps of typically 0.01 ps.Since only rather a limited number of molecules is treated (generally less than a thousand), themolecules are confined in a cubic box which is surrounded by images of itself to avoid severeboundary effects. In the studies described below the artificialities introduced by this procedureare rathe~r insignificant. From the results of an MD "experiment" - essentially a record of thesystem's path through phase space - one can determine any measurable property of the system,as well as many that are experimentally inaccessible.

* Bair, S.. and Wincr, W. 0. A Rheological Model for Elastohydrodynamic Contacts Based on

Primary Laboratory Data, ASME J. Lub. Tech., 101, 258 (1979).

Apppnd ix

17

APPENDIX

MOLECULAR DYNAMICS

Not only in the process of comminution but also in a rather wide variety of technologicalapplications, for example elastohydrodynamic lubrication, shock loading, solid propellantfailures, etc., liquids and amorphous solids are subjected to extremely large shearing forces.Because the microscopic mechanisms of importance are not well understood, solutions toproblems involving nonlinear behavior are quite limited.

For example in order to predict shear failure (or tearing) of a liquid under the forces ofan explosion or shock wave, one must be able to compute the time evolution of the stress in afluid element when it encounters a simultaneously applied time varying shear rate and a timevarying pressure. The response of the system under these conditions can, for low shearing rates,be parameterized in terms of the shear viscosity, Tl, a shear rigidity modulus, G., and adistribution of relaxation times. At the rather large shearing rates generally encountered, thisrelatively simple description must be modified to incorporate the nonlinear character of thedynamical str,,sq response. Some authors* suppose that this can be done by appending the ideaof a limiting shear stress to the low amplitude, i.e., linear, shear response. In order to test thish\ poilcsis and, more generally, to provide a framework on which to base analytical repre-sentations of nonlinear viscoelastic behavior, we have used Molecular Dynamics (MD) to inves-tigate a model liquid system under conditions approximating those found in at high shear rates.Even though the model system is particularly simple, we have been able to reproduce the majorfeatures of shear response of real liquids. Because an MD experiment provides one with acomplete microscopic record of the system's evolution in time, one can perform "experiments"and "measure" aspects of its behavior that are not accessible in conventional experiments, butwhich provide useful insights into the origin of the properties of liquids under extreme stresscond it ions.

The MD calculation procedure observes a portion of liquid as it is sheared. Although asimultaneously applied time dependent pressure and shear rate would more closely represent theconditions of a liquid undergoing explosive comminution, we have confined our attention up tonow to considering the response of a model fluid to a time varying shear rate at a series ofdensities (pressures).

THE MODEL

MD is a computer simulation technique for probing the microscopic behavior of a systemthat allows one to account for the many-body nature of the molecular interactions. It involvesfollowing some representative number of molecules, N, by solving the classical equations ofmotion on a high speed computer to obtain their trajectories in time steps of typically 0.01 ps.Since only rather a limited number of molecules is treated (generally less than a thousand), themolecules are confined in a cubic box which is surrounded by images of itself to avoid severeboundary effects. In the studies described below the artificialities introduced by this procedureare rathe~r insignificant. From the results of an MD "experiment" - essentially a record of thesystem's path through phase space - one can determine any measurable property of the system,as well as many that are experimentally inaccessible.

* Bair, S.. and Wincr, W. 0. A Rheological Model for Elastohydrodynamic Contacts Based on

Primary Laboratory Data, ASME J. Lub. Tech., 101, 258 (1979).

Apppnd ix

17

The model system we consider here is an assembly of 108 Lennard-Jones (LJ)particles; these molecules interact through pairwise additive forces described by the LJ 6-12potential,

CO = -4u [(a/r)6 - (a/r)1Z], (1)

where r is the separation of an interacting pair, and u and a are constants with dimensions ofenergy and distance, respectively. All the computed quantities are in so-called LJ reduced unitswhich are given in terms of u, a, and m, the mass of an individual particle. Although no realliquid can be characterized by a potential of the LJ form (only the noble gas fluids Ar, Kr, etc.,are quantitatively represented in this way) their structural and dynamical behavior can beexpected to be (and we have found actually are) qualitatively similiar to that of the modelfluid. As a result one can expect that the microscopic insights derived from MD experimentswill serve as useful guidelines for understanding and modeling the behavior of real liquids andperhaps for the "molecular engineering" of improved comminution properties of liquids. For thepurpose of comparison with experiment frequent conversion from reduced to "real" units will bemade, using the parameters for Ar which are given in table A-I.

Table A-1. A Summary of the Reduced Units in Terms of Which All Ouantities Are Given.

Boltzmann's constant is denoted by kB.

Ouantity 1 Reduced Unit SI Unit for Ar

distance a 0.3405 nm

mass m 6.64 x 10 - 26 kg

energy u 1.65 x I0 -Zl J

time a(m/u) 1/ ' 2.16 ps

density a 3 4Z.1 kg mol/m 3

temperature u/kB 119.8 K

pressure, stress u/a 3 41.8 MPa

modulus u/a 3 41.8 MPa

viscosity (mu) /a 2 9.03 x 10- 5 Pa s

thermal conductivity kB (m/u) " a - Z 1.88 x 10 2 J m -1 K -1 sI

Two methods of shearing the box of molecules can be used. The results of each methopagree within statistical error; however, each method is well suited to examine a particulIraspect of the investigated phenomenon.

The first method attempts to mimic a situation in which the fluid forms a thin film.

The model system we consider here is an assembly of 108 Lennard-Jones (LJ)particles; these molecules interact through pairwise additive forces described by the LJ 6-12potential,

CO = -4u [(a/r)6 - (a/r)1Z], (1)

where r is the separation of an interacting pair, and u and a are constants with dimensions ofenergy and distance, respectively. All the computed quantities are in so-called LJ reduced unitswhich are given in terms of u, a, and m, the mass of an individual particle. Although no realliquid can be characterized by a potential of the LJ form (only the noble gas fluids Ar, Kr, etc.,are quantitatively represented in this way) their structural and dynamical behavior can beexpected to be (and we have found actually are) qualitatively similiar to that of the modelfluid. As a result one can expect that the microscopic insights derived from MD experimentswill serve as useful guidelines for understanding and modeling the behavior of real liquids andperhaps for the "molecular engineering" of improved comminution properties of liquids. For thepurpose of comparison with experiment frequent conversion from reduced to "real" units will bemade, using the parameters for Ar which are given in table A-I.

Table A-1. A Summary of the Reduced Units in Terms of Which All Ouantities Are Given.

Boltzmann's constant is denoted by kB.

Ouantity 1 Reduced Unit SI Unit for Ar

distance a 0.3405 nm

mass m 6.64 x 10 - 26 kg

energy u 1.65 x I0 -Zl J

time a(m/u) 1/ ' 2.16 ps

density a 3 4Z.1 kg mol/m 3

temperature u/kB 119.8 K

pressure, stress u/a 3 41.8 MPa

modulus u/a 3 41.8 MPa

viscosity (mu) /a 2 9.03 x 10- 5 Pa s

thermal conductivity kB (m/u) " a - Z 1.88 x 10 2 J m -1 K -1 sI

Two methods of shearing the box of molecules can be used. The results of each methopagree within statistical error; however, each method is well suited to examine a particulIraspect of the investigated phenomenon.

The first method attempts to mimic a situation in which the fluid forms a thin film.This thin film is achieved in the model system by employing periodicity in the x and y directionsonly; the external shear forces are applied by two fluid layers translating with the desired wallvelocities, U1 and U2. Molecules from the three regions (i.e., the two wall regions and theliquid itself) are not allowed to mix and are kept from doing so by reflection boundaryconditions in the z direction. The fluid walls (FW) are maintained at a preset temperature byadjusting their velocities at each time step of the integration scheme.

Append i x

18

The second method is a modification of the Homogeneous Shear technique* (MRS), whichis better suited to study the viscoelastic behavior of the fluid because it employs periodicity inall directions so that a bulk material is sheared. It also enables a desired strain rate to beinstantaneously achieved throughout the liquid. The molecular trajectories are disturbed fromthose governed by equilibrium fluctuations by imposing a shear strain rate, , on the system.This is accomplished by displacing (after each time step, At) the x-coordinate of the i'th mole-cule in the MD cell by an amount 6x. = zAt, where z. is the z coordinate of particle i and Atis the length of the time step, for as long as the shearing is required. The results of thistechnique, in particular those relating to viscoelasticity (derived using a time dependent strainrate), complement those of the FW method.

We have used non-equilibrium MD experiments to investigate the linear and nonlinearshear response of the LJ system under a variety of thermodynamic conditions. The tempera-ture, T, and number density, p, initially studied were lose to the normal freezing temperature,T , of the LJ liquid, that is 0.722 k T/u and 0.844Z Na /V, respectively. Densifications of 10%,"6o, and 5 % relative to the starting density were achieved by suddenly (within one time step)ctmpressit ' system at constant temperature. The densifications were achieved so rapidlythat the systems were kinetically prevented from crystallizing.

MD MEV xSTiLVENT" OF SHEAR VISCOSITY OF HIGH STRAIN RATES

We first rcnsider the dependence of shear viscosity on shear rate in the nonlinear re-gion. In the Hom,< nous Shear method a uniform x velocity profile was set up instantaneouslywhereas a period of several time units was required for the fluid walls to drive the centralregion into this state. An x velocity gradient was achieved by maintaining the upper and lowerwalls at velocities of U and -U, respectively. An example of a v (z) so d2rived is shown iniur~e A-1. The value of corresponds to an extremely large shear rate - on the order of1l0 s- , depending somewhat upon the particular values of u, a, and m that are chosen. Suchlarge values of were chosen since it is desirable to explore the system's behavior at shearrates on the order of the reciprocal of the viscoelastic relaxation times. For the UJ fluids theseare approximately unit- (about 10-12 s). The shear stress needed to define a shear viscosity isreadily derived from the velocities and positions of the molecules. The aB component of thestress tensor is defined below,

NN1Jg =-(l/V) imv vo - 1j ( ij /rj)T_

where V is the volume of the MD celi, v? is the o velocity component of molecule i relative tothe average veloritv and a i is the a component of the separation between molecules i and j.The subscripts for the shear stress component, orxz, are dropped in further 8iscussion.

At low shear rates plots of shear stre3s versus strain rate, e , are linear; however, athigh stress levels (- G /100) the shear stress increases less rapidly than e. In other words theshear viscosity 1= o /e' decreases from the equilibrium for zero shear rate) value of ?7o withincreasing shear rate. The MI4S method was used to obtain 77 versus ; for a variety ofdensities. The forms of '71no versus stress in figure A-? resemble closely those obtained fromtwin-disk experiments for the fluids rP4E, and the lubricants L63/1271 and Oxilube 85/140 aftermaking the appropriate conversions to real units. The ratio ,i/no descends markedly from unityin the region - SOMPa. The decrease in 17 with ; is not due to thermal heating because thecalculations wore conducted isothermally.

*ltcyCs, I). \I. et al., op. cit.

Appondix

19

2

Vx [Z]

0

-2

0.6 0. 0.6z/h

I igitro A-1. The z Dependence of the Average x Velocity for the p 0.8442 and i 0.3373State Using the FW Model. The z coordinate system is measured from therenter of the film.

Appendix 20

2

Vx [Z]

0

-2

0.6 0. 0.6z/h

I igitro A-1. The z Dependence of the Average x Velocity for the p 0.8442 and i 0.3373State Using the FW Model. The z coordinate system is measured from therenter of the film.

Appendix 20

10 100

1.0 LIOVA

q 08A A

00

0.6-

0U0.4- Q0~

02 V.

0.01I I I I0.1 1.0 10.0

GFigurfe A-2. The Nonlinear Behavior of: A13P L63/1271 (P =0.8 GPa, T 27 0 C, U -1.12

m/s), 0 5P4E (P = 0.45 GPa, T = 27 0 C, U = 0.6 m/s) and rOxiltihe 85/140 (P=1.2 GPa, T =30 0 C, U 2.2 m/s) from Twin-Disk Experiments. The MHSModel: A p 0.8442, vp =0.92862, 0 p 1.01304 and open square p1.2663.

Appendix

10 100

1.0 LIOVA

q 08A A

00

0.6-

0U0.4- Q0~

02 V.

0.01I I I I0.1 1.0 10.0

GFigurfe A-2. The Nonlinear Behavior of: A13P L63/1271 (P =0.8 GPa, T 27 0 C, U -1.12

m/s), 0 5P4E (P = 0.45 GPa, T = 27 0 C, U = 0.6 m/s) and rOxiltihe 85/140 (P=1.2 GPa, T =30 0 C, U 2.2 m/s) from Twin-Disk Experiments. The MHSModel: A p 0.8442, vp =0.92862, 0 p 1.01304 and open square p1.2663.

Appendix

CALCULATION OF SHEAR INDUCED STRUCTURAL CHANGES

The results in the previous section suggest that the model and real liquids might share awinnii mechanism of "failure" at high levels of shear stress. With the support of the previous

retlts we now consider associated structural changes produced by the large shear rates in thetluids in order to better understand the failure mechanism.

A directional number density function, n(a), has been calculated for the purpose ofinvestigating structural anisotropy. The function n(x), for example, is the average number ofii)(l, cales from a reference particle in the distance range jxj to jxj +Ajx, where 4x was taken as

I Q00, selected from the N-1 particles in the re-orientated MD cell. Figure A-3 shows the n(a)A t 1wtinsheared liquid subtracted from that of the sheared system, 5n(a). Although the maxi-,,. di,,p'aement is 10% at most, there is evidence of a tendency of the molecules to re-order!h, iselves into xv layers (layers normal to the velocity gradient). The xz shearing increasest,. prdoahility cf the molecules in an xz plane being found at multiples of the intermolecularihaniter from ,a, , other in the z direction. It is energetically favorable for these layers to betaggered in the y direction. Although not so well defined, these trends are present at the

lower densities. The (grossly simplified) picture that emerges is one of corrugated sheets ofatoil .lilJi aSt one another. A pictorial representation of this suggested structure is given intigure A 1. Th structural reorganization results from the action of the imposed velocityi,rofile wiich, at the shear rates considered, dominates the molecular motion. Major move-ments art, confined to the line of shear.

Another aspect of the shear induced structural distortion is presented by the time,tependehnt pair radial distribution function, g (r,t), which is conveniently decomposed into thefollowing component configurational averages:

g(r,t) = go(r) + (xZ/rZ)uxx(r,t) + (v 2 /r)Uvv(r,t)

+ (zZ/r 2 )Uzz (r,t) + (xz/r 2 )Uxz (r,t)

. . , (3)

where go(r) is the pair radial distribution function of the unsheared medium.

If i,-t (r,t) is the average of < eij i/rij > in the radial element r---r + dr about amolecule, i at time t, then

go,,jr,t) = 15 f ot(r,t) / (4 7r p rZdr),

.ind

gaj(r,t) = 15 f (r,t) / (4rpr2dr), (4)

;here #:Bx

,The fa (r,t) have been calculated at various times during the segment. The differencebetween the g ,,(r,t) of the steady state sheared system and the unsheared system at the samedonsity, 6g,,or,t), is shown in figure A-5. It reveals that shearing produces a net movement ofparticles in cach coordination shell towards the origin molecule.

Tb, hear rate, dependent gxz(r,t) at steady state of the r = 0.84 system are comparedvith the. sph-ricallv averaged p(r,t) from these samples in figure A-6. They give convincing.vidence (if angular distortion in each coordination shell such that in the positive xz quadrants

Al'pndix 22

CALCULATION OF SHEAR INDUCED STRUCTURAL CHANGES

The results in the previous section suggest that the model and real liquids might share awinnii mechanism of "failure" at high levels of shear stress. With the support of the previous

retlts we now consider associated structural changes produced by the large shear rates in thetluids in order to better understand the failure mechanism.

A directional number density function, n(a), has been calculated for the purpose ofinvestigating structural anisotropy. The function n(x), for example, is the average number ofii)(l, cales from a reference particle in the distance range jxj to jxj +Ajx, where 4x was taken as

I Q00, selected from the N-1 particles in the re-orientated MD cell. Figure A-3 shows the n(a)A t 1wtinsheared liquid subtracted from that of the sheared system, 5n(a). Although the maxi-,,. di,,p'aement is 10% at most, there is evidence of a tendency of the molecules to re-order!h, iselves into xv layers (layers normal to the velocity gradient). The xz shearing increasest,. prdoahility cf the molecules in an xz plane being found at multiples of the intermolecularihaniter from ,a, , other in the z direction. It is energetically favorable for these layers to betaggered in the y direction. Although not so well defined, these trends are present at the

lower densities. The (grossly simplified) picture that emerges is one of corrugated sheets ofatoil .lilJi aSt one another. A pictorial representation of this suggested structure is given intigure A 1. Th structural reorganization results from the action of the imposed velocityi,rofile wiich, at the shear rates considered, dominates the molecular motion. Major move-ments art, confined to the line of shear.

Another aspect of the shear induced structural distortion is presented by the time,tependehnt pair radial distribution function, g (r,t), which is conveniently decomposed into thefollowing component configurational averages:

g(r,t) = go(r) + (xZ/rZ)uxx(r,t) + (v 2 /r)Uvv(r,t)

+ (zZ/r 2 )Uzz (r,t) + (xz/r 2 )Uxz (r,t)

. . , (3)

where go(r) is the pair radial distribution function of the unsheared medium.

If i,-t (r,t) is the average of < eij i/rij > in the radial element r---r + dr about amolecule, i at time t, then

go,,jr,t) = 15 f ot(r,t) / (4 7r p rZdr),

.ind

gaj(r,t) = 15 f (r,t) / (4rpr2dr), (4)

;here #:Bx

,The fa (r,t) have been calculated at various times during the segment. The differencebetween the g ,,(r,t) of the steady state sheared system and the unsheared system at the samedonsity, 6g,,or,t), is shown in figure A-5. It reveals that shearing produces a net movement ofparticles in cach coordination shell towards the origin molecule.

Tb, hear rate, dependent gxz(r,t) at steady state of the r = 0.84 system are comparedvith the. sph-ricallv averaged p(r,t) from these samples in figure A-6. They give convincing.vidence (if angular distortion in each coordination shell such that in the positive xz quadrants

Al'pndix 22

0

0 12

Figure A -3. The Excess Directional-Number-Densitv Functions, 6n(a), for the P =1.2663

and i X 0.7721 State.

App c~n dix 23

0

0 12

Figure A -3. The Excess Directional-Number-Densitv Functions, 6n(a), for the P =1.2663

and i X 0.7721 State.

App c~n dix 23

Figuf-~ A-A. A Pictorial Representation of the Structural Changes That Take Place on Goinefrom an Unsheared (a) to a Sheared (b) State. The arrows denote the line andplane of shear.

\PPerAIX

214

4. ......

Fi~oir 2 A-,Tegr x/)adg ()o h hae iu h nhaePorios f te =1.030 ad y 6. i6- Sgmete Clcuaton

I-.

4. ......

Fi~oir 2 A-,Tegr x/)adg ()o h hae iu h nhaePorios f te =1.030 ad y 6. i6- Sgmete Clcuaton

I-.

the i,.idle of eali h shell is on average depleted of particles when compared with the unsheare diqud. In contrast, the outside of each shell has an excess of molecules in the positive xz quad-

s when compared with the unsheared medium. The opposite changes take place in theaa .laive ,!tjadrants.

FI I lF SOF ',TPIlTUIU Al. REORGANIZATION ON MOLECULAR MOBILITY

It ib t, lie expected that the effect of the shear should not be confined to structural* Ti t uy, fut must also alt, r the self-diffusional dynamics of the molecules. To assess this, the.x,. and z caomponents of the mean square displacements, r ft)), which exclude the impose-df 1 . have been ca lcu lated, i.e.,

(-t)) - [ t v ) (t)dt' a. (0) 2 2>0

!, dL,, ti,,na! diffusion coefficients,

• 2 dr. 9

,L- ction of t, states are presented table A-2. The value of D, 0.032 for the unsht-aredSl.i, oear the triple point compares favorably with the equilibrium value of 0.033. The 1), -f

to, iii.<heared liquid It', r-ase hy approximately an order of magnitude on increasinjR the densitytr- an p 0.844" to 1.2 tl. However, at larger shear rates the densitv has a diminishi;.g influ-* - n th, diffusion coefficients. The high shear stresses enhance the fluidity and create a

juid which has directionalitv. All the diffusion coefficient components rise with shear rat(,'t a particular density bocause the structural reorganization that takes plact, on shearing appar-ently (reates paths along which particles can more readily move. The D. are typically two-t~ird , ( f the D X and )., indicating that the self-diffusion is favored in the sbearinp, plane.'I td, ally, although structural aspects of the the lionids under -Lear present a m,)r,,solid-likom appearance, other more dynamically related properties such as self-diffusion andshear vi, osity manifest changes which are associated with enhancel fluiditv.

Thus ,,- so-e tl.at the high stresses cause oxtreme distortions of the. system's strut tur,..I ne taruleculr forces are not sufficiently strong to support the enormous stress levt5ls thatdevelop: consequently there is a rupturing and reorganization of the structure resulting in,nhanced flow behavior, due to the alignment of molecules into "glide" planes along the line ,f

" hese strut torail rearrangements are also manifest in changes in the so-called n'rnI,pr,*iure components, Tv ( - O Typically, Pzz > P > P V ' although their differenc,-, ar ,

small when comparcd with each components change on shearing. In figure A-7 the time depen-t, o ft the inerase in PI U is shown. It is found that the time scale of the development )Ithe n-lrmal pressur, , -,mpononts is longer than that of the shear stress.

C! ,itterence in the viscoelastic and structural reorganzation times can he bettor, st, , t b\ fortir consid,.ration of the radial distribution functions.

a' e tlao+ d,,p, nuence of g(X (rt) and g ct(r,t) reveal that sructural reorganization;,A, 7ut tho, sym iiit rv 4, shear, <xz/r" >, is much faster than that of <0t /r >symme-try. Within

S i atI'll <x-r2> and <z 2 /rZ>evolve more rapidly to their steady state valuesv 2 r e',)L oxample-, when P = 1.01304 and Cxz - 0.7168. the former three averages have

- pl, ,d I X 26

the i,.idle of eali h shell is on average depleted of particles when compared with the unsheare diqud. In contrast, the outside of each shell has an excess of molecules in the positive xz quad-

s when compared with the unsheared medium. The opposite changes take place in theaa .laive ,!tjadrants.

FI I lF SOF ',TPIlTUIU Al. REORGANIZATION ON MOLECULAR MOBILITY

It ib t, lie expected that the effect of the shear should not be confined to structural* Ti t uy, fut must also alt, r the self-diffusional dynamics of the molecules. To assess this, the.x,. and z caomponents of the mean square displacements, r ft)), which exclude the impose-df 1 . have been ca lcu lated, i.e.,

(-t)) - [ t v ) (t)dt' a. (0) 2 2>0

!, dL,, ti,,na! diffusion coefficients,

• 2 dr. 9

,L- ction of t, states are presented table A-2. The value of D, 0.032 for the unsht-aredSl.i, oear the triple point compares favorably with the equilibrium value of 0.033. The 1), -f

to, iii.<heared liquid It', r-ase hy approximately an order of magnitude on increasinjR the densitytr- an p 0.844" to 1.2 tl. However, at larger shear rates the densitv has a diminishi;.g influ-* - n th, diffusion coefficients. The high shear stresses enhance the fluidity and create a

juid which has directionalitv. All the diffusion coefficient components rise with shear rat(,'t a particular density bocause the structural reorganization that takes plact, on shearing appar-ently (reates paths along which particles can more readily move. The D. are typically two-t~ird , ( f the D X and )., indicating that the self-diffusion is favored in the sbearinp, plane.'I td, ally, although structural aspects of the the lionids under -Lear present a m,)r,,solid-likom appearance, other more dynamically related properties such as self-diffusion andshear vi, osit v manifest changes which are associated with enhancel fluiditv.

Thus ,,- so-e tl.at the high stresses cause oxtreme distortions of the. system's strut tur,..I ne taruleculr forces are not sufficiently strong to support the enormous stress levt5ls thatdevelop: consequently there is a rupturing and reorganization of the structure resulting in,nhanced flow behavior, due to the alignment of molecules into "glide" planes along the line ,f

" hese strut torail rearrangements are also manifest in changes in the so-called n'rnI,pr,*iure components, Tv ( - O Typically, Pzz > P > P V ' although their differenc,-, ar ,

small when comparcd with each components change on shearing. In figure A-7 the time depen-t, o ft the inerase in PI U is shown. It is found that the time scale of the development )Ithe n-lrmal pressur, , -,mpononts is longer than that of the shear stress.

C! ,itterence in the viscoelastic and structural reorganzation times can he bettor, st, , t b\ fortir consid,.ration of the radial distribution functions.

a' e tlao+ d,,p, nuence of g(X (rt) and g ct(r,t) reveal that sructural reorganization;,A, 7ut tho, sym iiit rv 4, shear, <xz/r" >, is much faster than that of <0t /r >symme-try. Within

S i atI'll <x-r2> and <z 2 /rZ>evolve more rapidly to their steady state valuesv 2 r e',)L oxample-, when P = 1.01304 and Cxz - 0.7168. the former three averages have

- pl, ,d I X 26

4K2K

4l I x Z7I

4K2K

4l I x Z7I

yy

FiueA6.Cnd

App, nd1

2K

0

-2

-4 I I I0 1 2

r

C

Figure A-6. Contd.

Ap 1 e1(! ix 29

2z

Fiur -6 Fnd

-2, 3

-4 I I I

2z

Fiur -6 Fnd

-2, 3

-4 I I I

10-

l~igurf. A 7. Tith- Tiime-Dependent Normal Pressure Components for the Strain Rate History.

0.7168, P = 1.01304. P, _ px,(t),. . pyy(t), and ... pz(t)

App,-nd ix 31

10-

l~igurf. A 7. Tith- Tiime-Dependent Normal Pressure Components for the Strain Rate History.

0.7168, P = 1.01304. P, _ px,(t),. . pyy(t), and ... pz(t)

App,-nd ix 31

n,2arly reached their steady state values by t = 025 whereas that of the <yZ/r2> summation

hardly differs from zero by that time. Thus we find that when a liquid is sheared, the shearingforces alter its structure so that the steady state stress attained corresponds to a new lowerviscosity, not that characterizing the starting liquid. At high rates of shear the structuralevolution necessary to attain this new viscosity is significantly slower than the viscoelasticshear relaxation. Viewed simply this means that the original viscosity can remain for a suffi-ciently long time to enable the shear stress to exceed its steady state value.

Tahle A -Z. The Density and Shear Rate Dependence of the Directional Diffusion Coefficients,T)a , Obtained from the MD Experiments.

SD -xz Dx Dy

0.84 0.0 0.032 0.032 0.0320.84 0.67 0.052 0.045 0.05Z0.93 0.0 0.018 0.018 0.0180.93 0.69 0.043 0.037 0.045

It has come to be increasingly recognized that the process of fracture involves timedofp indent nonlinear viscoelastic behavior. In order to better understand the time dependentnonlinear viscoelasticity, the MD calculations were conducted with a time varying strain rate.In what follows we describe the viscoelastic MD "experiments" and the results. What is mostencouraging is that in every case the behavior of real liquids is mimicked by the MD results.

We consider the dynamical response of the LJ material to an imposed shear that isswitched on" suddenly using the Homogeneous Shear method. Two types of MD experiments

are Ai interest: (a) the response of the system to a steady shearing rate (uniform velocitygradient) i = o / ) z that is initiated at time t = 0; and (b) the response of the system to aconstant shear strain (a pulse of shearing rate) imposed at time t = t'. Linear response theoryprovido, us with relations between the two types of responses: the shear stress response to astep of shear strain can be written as

a (t)/ =G.0 (t), (6)

where q) (t) is the normalized ( (0) = 1) shear stress relaxation function. In terms of these pa-rameters the shear stress response to a "small" steady shear rate imposed at t = 0 is,

tC(t)/ = G ,. o- d t ' (t ) . (7)

Since the steady state (t . ) value of this is just the viscosity of the material, yo , we have,

no 0: G fo dt' It') = G t. (8)

The last equality serves to define the shear realization time r.

Appendix 32

n,2arly reached their steady state values by t = 025 whereas that of the <yZ/r2> summation

hardly differs from zero by that time. Thus we find that when a liquid is sheared, the shearingforces alter its structure so that the steady state stress attained corresponds to a new lowerviscosity, not that characterizing the starting liquid. At high rates of shear the structuralevolution necessary to attain this new viscosity is significantly slower than the viscoelasticshear relaxation. Viewed simply this means that the original viscosity can remain for a suffi-ciently long time to enable the shear stress to exceed its steady state value.

Tahle A -Z. The Density and Shear Rate Dependence of the Directional Diffusion Coefficients,T)a , Obtained from the MD Experiments.

SD -xz Dx Dy

0.84 0.0 0.032 0.032 0.0320.84 0.67 0.052 0.045 0.05Z0.93 0.0 0.018 0.018 0.0180.93 0.69 0.043 0.037 0.045

It has come to be increasingly recognized that the process of fracture involves timedofp indent nonlinear viscoelastic behavior. In order to better understand the time dependentnonlinear viscoelasticity, the MD calculations were conducted with a time varying strain rate.In what follows we describe the viscoelastic MD "experiments" and the results. What is mostencouraging is that in every case the behavior of real liquids is mimicked by the MD results.

We consider the dynamical response of the LJ material to an imposed shear that isswitched on" suddenly using the Homogeneous Shear method. Two types of MD experiments

are Ai interest: (a) the response of the system to a steady shearing rate (uniform velocitygradient) i = o / ) z that is initiated at time t = 0; and (b) the response of the system to aconstant shear strain (a pulse of shearing rate) imposed at time t = t'. Linear response theoryprovido, us with relations between the two types of responses: the shear stress response to astep of shear strain can be written as

a (t)/ =G.0 (t), (6)

where q) (t) is the normalized ( (0) = 1) shear stress relaxation function. In terms of these pa-rameters the shear stress response to a "small" steady shear rate imposed at t = 0 is,

tC(t)/ = G ,. o- d t ' (t ) . (7)

Since the steady state (t . ) value of this is just the viscosity of the material, yo , we have,

no 0: G fo dt' It') = G t. (8)

The last equality serves to define the shear realization time r.

Appendix 32

Experiments of the type described as (a) above were then carried out as a function offor values of C well beyond the range for which linear behavior can he expected. A represen-

tative selection of the results is displayed in figures A-8 and A-9 in which the density (con-stant I and shear rate (constant density) dependence of 17 are shown, respectively.

S ,.eral features of these figures are worthy of comment. At short times the systembehaves largely as an elastic solid in that the stress increases linearly with time,

a(t) = G , t. (9)

The shear rigidity modulus rises linearly with pressure at roughly the same rate as more compli-cated fluids, as is revealed in figure A-10. Liquids cannot permanently support a shear strainaod (nsequcntly the stress levels off to a value determined by the shear viscosity of the-hested statc. There is evident "shear thinning," or a decrease of viscosity with increasing-hear rate. The predicted value of the stress build-up for the p = 0.9Z862, = 0.6963 stateusing ecluati,n (7), shown in figure A-8, clearly overestimates the actual stress obtained.

kis, , a ma-dmum in che stress versus time curve for those simulations in the high0 rltv/sb ar :-ate regime is observed. This indicates that the large shear deformations are

r~diin ;itr,:tiral distortions which lag in time somewhat behind the growth of the shearatrss but follow rather closely the observed time variation of the normal pressure componentsin thtt materia! This relatively slower development of what can be thought of as transientstress-induced glide planes is consistent with the observed shear thinning and stress "overshoot"behavior presented. It is also consistent with the observation that the stress relaxation functiondecays more rapidly in a highly sheared system than in an equilibrium system (see discussion of9,O Wt ahve). A comparison of the two types of behavior is presented in figure A-il. The formof thse t't) consists of a rapidly (t < 0.2) decaying portion which is presumably inertial inoriLin and due to small motions of the molecules within the random network structure. Furtherstress relaxation requires the cooperative rearrangement of larger molecular groups and is, :nsequentlv slower. The attenuation of 0(t) with increasing shear rate is indicative of the,u..iaced stress relieving properties of the structurally reorganized system. The stress over-shoot observed in nonlinear viscoelastic experiments on polymers has a form that is quitesimilar to the curves obtained for the LJ system. While the microscopic mechanisms for thetwo systems are obviously quite different, the general principle that associates the stress over-sh ot phenomena (as well as shear thirning) with a time dependent structural reorganizationprocess rciains quite valid.

In this context it is perhaps relevant to note that a more rapid decay of the stressrelaxation function can be thought of as both a suppression of the long relaxation timesasso( iated with stress relaxation and also a reduction in the shear rigidity modulus, at lowtreqtien,.ies, ;, (a rather well known phenomenon in polymeric systems).*

Thu, we see that the viscoelastic studies at high shear rates reflect the structural reor-ganization that is ultimately responsible for failure or fracture of the liquid. The fact thatecery property "measured" at high shear rates are found in real and far more complicatedlhquds suggests strongly that MD studies can be utilized to investigate the comminution processin those liquids.

* V (;. Yanovski in Relaxation Phenomena in Polymers, ed. G. Bartenev and Y. V. Zelenev(Khinii~a, lfeningrad, 1972), p. 113.

Appendix 33

-. - -~ q z'

Experiments of the type described as (a) above were then carried out as a function offor values of C well beyond the range for which linear behavior can he expected. A represen-

tative selection of the results is displayed in figures A-8 and A-9 in which the density (con-stant I and shear rate (constant density) dependence of 17 are shown, respectively.

S ,.eral features of these figures are worthy of comment. At short times the systembehaves largely as an elastic solid in that the stress increases linearly with time,

a(t) = G , t. (9)

The shear rigidity modulus rises linearly with pressure at roughly the same rate as more compli-cated fluids, as is revealed in figure A-10. Liquids cannot permanently support a shear strainaod (nsequcntly the stress levels off to a value determined by the shear viscosity of the-hested statc. There is evident "shear thinning," or a decrease of viscosity with increasing-hear rate. The predicted value of the stress build-up for the p = 0.9Z862, = 0.6963 stateusing ecluati,n (7), shown in figure A-8, clearly overestimates the actual stress obtained.

kis, , a ma-dmum in che stress versus time curve for those simulations in the high0 rltv/sb ar :-ate regime is observed. This indicates that the large shear deformations are

r~diin ;itr,:tiral distortions which lag in time somewhat behind the growth of the shearatrss but follow rather closely the observed time variation of the normal pressure componentsin thtt materia! This relatively slower development of what can be thought of as transientstress-induced glide planes is consistent with the observed shear thinning and stress "overshoot"behavior presented. It is also consistent with the observation that the stress relaxation functiondecays more rapidly in a highly sheared system than in an equilibrium system (see discussion of9,O Wt ahve). A comparison of the two types of behavior is presented in figure A-il. The formof thse t't) consists of a rapidly (t < 0.2) decaying portion which is presumably inertial inoriLin and due to small motions of the molecules within the random network structure. Furtherstress relaxation requires the cooperative rearrangement of larger molecular groups and is, :nsequentlv slower. The attenuation of 0(t) with increasing shear rate is indicative of the,u..iaced stress relieving properties of the structurally reorganized system. The stress over-shoot observed in nonlinear viscoelastic experiments on polymers has a form that is quitesimilar to the curves obtained for the LJ system. While the microscopic mechanisms for thetwo systems are obviously quite different, the general principle that associates the stress over-sh ot phenomena (as well as shear thirning) with a time dependent structural reorganizationprocess rciains quite valid.

In this context it is perhaps relevant to note that a more rapid decay of the stressrelaxation function can be thought of as both a suppression of the long relaxation timesasso( iated with stress relaxation and also a reduction in the shear rigidity modulus, at lowtreqtien,.ies, ;, (a rather well known phenomenon in polymeric systems).*

Thu, we see that the viscoelastic studies at high shear rates reflect the structural reor-ganization that is ultimately responsible for failure or fracture of the liquid. The fact thatecery property "measured" at high shear rates are found in real and far more complicatedlhquds suggests strongly that MD studies can be utilized to investigate the comminution processin those liquids.

* V (;. Yanovski in Relaxation Phenomena in Polymers, ed. G. Bartenev and Y. V. Zelenev(Khinii~a, lfeningrad, 1972), p. 113.

Appendix 33

-. - -~ q z'

101

PINi

V..

0 1 2345

Vkiire A-8. The Time Depenidence of the Reduced Stress, a(t)/ : - - - P 0 .8442, O 0.6745,.(a) Actual..(b) Predicted Using Equation (4) P = 0.92862, O

0.6963;----p J .01304, o 0.7168; _P P1.2663, eO 0 .7721.

Appciidix 3

101

PINi

V..

0 1 2345

Vkiire A-8. The Time Depenidence of the Reduced Stress, a(t)/ : - - - P 0 .8442, O 0.6745,.(a) Actual..(b) Predicted Using Equation (4) P = 0.92862, O

0.6963;----p J .01304, o 0.7168; _P P1.2663, eO 0 .7721.

Appciidix 3

20

ot] 15-

0

10

' C\

A

0- o - _ I I I I I

0 1 2 3 4 5

tFigure A-9. The Time Dependence of the Reduced Stress, a(t)/e 0 , for the p = 1.01304

State: 0 = 0.0896, _ _0 =0.3584,. 0.7168.

Append ix 35

20

ot] 15-

0

10

' C\

A

0- o - _ I I I I I

0 1 2 3 4 5

tFigure A-9. The Time Dependence of the Reduced Stress, a(t)/e 0 , for the p = 1.01304

State: 0 = 0.0896, _ _0 =0.3584,. 0.7168.

Append ix 35

PIG Pa

00.5 1.0 1.5

5

100- 4

3

:50 2

1

0 00 10 20 30 40

P

Figrtc, A 10. The Pressure Dependence of G,,: LSantotrac 40, 0 Di(2-ethylhexyl) Phthalatefrom a Twin-Disk Apparatus. Solid line C7,,, (0.293 + 1.70P) GPa di(Z-e,)phthalate-. The filled in 0 is obtained from MHS MD.

Ix 36

PIG Pa

00.5 1.0 1.5

5

100- 4

3

:50 2

1

0 00 10 20 30 40

P

Figrtc, A 10. The Pressure Dependence of G,,: LSantotrac 40, 0 Di(2-ethylhexyl) Phthalatefrom a Twin-Disk Apparatus. Solid line C7,,, (0.293 + 1.70P) GPa di(Z-e,)phthalate-. The filled in 0 is obtained from MHS MD.

Ix 36

0.0

1.0

C[t]

0.5

0.00.0 Q.2 0.4 0.6

t

Figure A-11. The Shear Stress Relaxation Functions for the p =0.844Z ___=0 and . =

0.6745. The step in strain (~=3.6 10-3) at t =0 is illustrated above this figure.

Appendix 3p

0.0

1.0

C[t]

0.5

0.00.0 Q.2 0.4 0.6

t

Figure A-11. The Shear Stress Relaxation Functions for the p =0.844Z ___=0 and . =

0.6745. The step in strain (~=3.6 10-3) at t =0 is illustrated above this figure.

Appendix 3p

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ATTN: DRDAR-CLC-E 1 ATTN: DACS-FM, MG R. Anson

ATTN: DRDAR-CLJ-R 2 Room 1A871, Pentagon

ATTN: DRDAR-CLJ-L 3 Washington, Dc 20310

ATTN: DRDAR-CLJ-M 1

ATTN: DRDAR-CLN I Federal Emergency Management Agency

ATTN: DRPAR-CLW 1 Office of Mitigation and Research

ATTN: DRDAR-CLW-C 1 ATTN: David W. Bensen

ATTN: DRDAR-CLW-P 1 Washington, DC 20472

ATTN: DRDAR-CLW-E 1

ATTN: DRDAR-CLB-C I Deputy Chief of Staff for Research,

ATTN: DRDAR-CLB-P 1 Development & Acquisition

ATTN: DRDAR-CLB-PO 1 ATTN: DAMA-CSS-C I

ATTN: DRDAR-CLB-R 1 ATTN: DAMA-ARZ-D I

ATTN: DRDAR-CLB-T 1 Washington, DC 20310

ATTN: DRDAR-CLB-TE 1

ATTN: DROAR-CLY-A I Department of the Army

ATTN: DRDAR-CLY-R 6 Headquarters, Sixth US Army

ATTN: AFKC-OP-NBC

COPIES FOR AUTHOR(S): Presidio of San Francisco, CA 94129

Research Division 2CLB-A(RecordSet) I US Army Research and Standardization

DEPARTMENT OF DEFENSE Group (Europe)

ATTN: DRXSN-E-SC

Defense Technical Information Center Box 65, FPO New York 09510

ATTN: DTIC-DDA-2 12

Cameron Station, Building 5 HQDA (DAMI-FIT)

Alexandria, VA 22314 WASH, DC 20310

Director Commander

Defense Intelligence Agency HQ 7th Medical Command

ATTN: DB-4G1 1 ATTN: AEMPM

Washington, DC 20301 APO New York 09403

Special Agent In Charge Commander

ARO, 902d Military Intelligence GP DARCOM, STITEUR

ATTN: IAGPA-A-AN 1 ATTN: DRXST-STI

Aberdeen Proving Ground, MD 21005 Box 48, APO New York 09710

Commander Commander

SED, HQ, INSCOM US Army Science & Technology Center-

ATTN: IRFM-SED (Mr, Joubert) 1 Far East Office

Fort Meade, MO 20755 ATTN: MAJ Borges

APO San Francisco 96328

39

Director

U0ornia nd ar Human Engineering Laboratory2d Ittantry Division ATTN: DRXHE-SP (CB Defense Team)1E]Tr4 EAIDCOM 1 Aberdeen Proving Ground, MD 21005

Commandertmnn auor US Army Foreign Science & Technologyth Infe.,try olvlslon (Mach) CenterfrTN: Olvi~i.,n hamical Otticer 1 ATTN: DRXST-MT3c:;rt Foik , tA 71459 220 Seventh St., NE

Charliottesvi lie, VA 22901:FF-;E iiE SURGEON GENERAL

Director

US Army Materiel Systems Analysis Activity,!,dl0 biouigqlneuring ATTN: DRXSY-MP1

--.drcn & Development Laboratory ATTN: DRXSY-TN (Mr. Metz) 2A: R UBCR-A, 1 Aberdeen Proving Ground, MD 21005

F 1,-rItk, 13109 568

I'MD 21101 Commander

US Army Missile Command

-;j ter Redstone Scientific Information Center

y :aI ,barh and 1 ATTN:DRMI-RPR (Documents)1

,.rcMD 271Director

DARCOM Field Safety Activity-nder ATTN: DRXOS-C 1

-riorieuical Laboratory Chariestown, IN 47111*;I Su*(P-UV-L I!roeeri Proving Ground, MD 21010 Commander

US Army Nat ick Research andiA[MY HEALTH SERVICE COMMAND Development Command

ATTN: DRDNA-O 1nteidntATTN: DRDNA-VC

A iulemy of Heaith Sciences ATTN: DRDNA-VCCkaS Army ATTN: DRDNA-VM IA I FP: 'iSA-CDH 1 ATTN: DRDNA-VR 1liVIN: H!A-IPM 1 ATTN: DRDNA-VT I

t Sam Housto~i, TX 78234 Nat ick, MA 01760

'+11y ,AIERIEL DEVELOPMENT AND US ARMY ARMAMENT RESEARCH ANDDL Al. I;LSS COMMAND DEVELOPMENT COMMAND

111. Ju CommanderW orrry Hateriel Development and US Army Armament Research and

-eajiriess Command Development CommandAT i l. uPCL DC I ATTN: DRDAR-LCA-L 1ATN r D[RCSF-P I ATTN: DRDAR-LCU-CE I

I sennowor Ai, ATTN: DRDAR-PMA (G.R. Sacco) 1Ak-iiidria, VA 22333 ATTN: DRDAR-SOA-W 1

ATTN: DRDAR-TSS 5ATTN: DRCPM-CAWS-AM 1

ATTN: DRCPM-CAWS-Si 1

Dover, NJ 07801

40

Director

U0ornia nd ar Human Engineering Laboratory2d Ittantry Division ATTN: DRXHE-SP (CB Defense Team)1E]Tr4 EAIDCOM 1 Aberdeen Proving Ground, MD 21005

Commandertmnn auor US Army Foreign Science & Technologyth Infe.,try olvlslon (Mach) CenterfrTN: Olvi~i.,n hamical Otticer 1 ATTN: DRXST-MT3c:;rt Foik , tA 71459 220 Seventh St., NE

Charliottesvi lie, VA 22901:FF-;E iiE SURGEON GENERAL

Director

US Army Materiel Systems Analysis Activity,!,dl0 biouigqlneuring ATTN: DRXSY-MP1

--.drcn & Development Laboratory ATTN: DRXSY-TN (Mr. Metz) 2A: R UBCR-A, 1 Aberdeen Proving Ground, MD 21005

F 1,-rItk, 13109 568

I'MD 21101 Commander

US Army Missile Command

-;j ter Redstone Scientific Information Center

y :aI ,barh and 1 ATTN:DRMI-RPR (Documents)1

,.rcMD 271Director

DARCOM Field Safety Activity-nder ATTN: DRXOS-C 1

-riorieuical Laboratory Chariestown, IN 47111*;I Su*(P-UV-L I!roeeri Proving Ground, MD 21010 Commander

US Army Nat ick Research andiA[MY HEALTH SERVICE COMMAND Development Command

ATTN: DRDNA-O 1nteidntATTN: DRDNA-VC

A iulemy of Heaith Sciences ATTN: DRDNA-VCCkaS Army ATTN: DRDNA-VM IA I FP: 'iSA-CDH 1 ATTN: DRDNA-VR 1liVIN: H!A-IPM 1 ATTN: DRDNA-VT I

t Sam Housto~i, TX 78234 Nat ick, MA 01760

'+11y ,AIERIEL DEVELOPMENT AND US ARMY ARMAMENT RESEARCH ANDDL Al. I;LSS COMMAND DEVELOPMENT COMMAND

111. Ju CommanderW orrry Hateriel Development and US Army Armament Research and

-eajiriess Command Development CommandAT i l. uPCL DC I ATTN: DRDAR-LCA-L 1ATN r D[RCSF-P I ATTN: DRDAR-LCU-CE I

I sennowor Ai, ATTN: DRDAR-PMA (G.R. Sacco) 1Ak-iiidria, VA 22333 ATTN: DRDAR-SOA-W 1

ATTN: DRDAR-TSS 5ATTN: DRCPM-CAWS-AM 1

ATTN: DRCPM-CAWS-Si 1

Dover, NJ 07801

40

US ARMY ARMAMENT MATERIEL READINESS Commander

COMMAND USA Combined Arms Center and

Fort Leavenworth

Commander ATTN: ATZL-CA-COG

US Army Armament Materiel Readiness Command ATTN: ATZL-CAM-IM

ATTN: DRSAR-ASN I Fort Leavenworth, KS 66027

ATTN: DRSAR-SF 1

ATTN: DRSAR-SR 1 Commander

Rock Island, IL 61299 US Army TRADOC System Analysis Activity

ATTN: ATAA-SL

Commander White Sands Missile Range, NM 88002

USA ARRCOMATTN: SARTE US ARMY TEST & EVALUATION COMMAND

Aberdeen Proving Ground, MD 21010

Commander

Commander US Army Test & Evaluation Command

US Army Dugway Proving Ground ATTN: DRSTE-CT-T

ATTN: Technical Library (Docu Sect) 1 Aberdeen Proving Ground, MD 21005

Dugway, UT v&022

DEPARTMENT OF THE NAVY

US ARMY TRAINING & DOCTRINE COMMAND

Chief of Naval Research

Commandant ATTN: Code 443

US Army Infantry School 800 N. Quincy Street

ATTN: NBC Division 1 Arlington, VA 22217

Fort Banning, GA 31905

Commander

Commandant Naval Explosive Ordnance Disposal Facility

US Army Missile & Munitions Center ATTN: Army Chemical Officer (Code AC-3) 1

and School Indian Head, MD 20640

ATTN: ATSK-DT-MU-EOD I

Restone Arsenal, AL 35809 Commander

Naval Surface Weapons Center

Commandant Code G51

USAMP&CS/TC&FM Dahlgren, VA 22448

ATTN: ATZN-CM-CDM 1

Fort McClellan, AL 36205 Chief, Bureau of Medicine & Surgery

Department of the Navy

Commander ATTN: MED 3C33

US Army Infantry Center Washington, DC 20372

ATTN: ATSH-CD-MS-C 1

Fort Banning, GA 31905 Commander

Naval Weapons Center

Commander ATTN: Technical Library (Code 343)

US Army Infantry Center China Lake, CA 93555

Directorate of Plans & Training

ATTN: ATZB-DPT-PO-NBC I US MARINE CORPS

Fort Banning, GA 31905 Director, Development Center

Commander Marine Corps Development and

USA Training and Doctrine Command Education CommandATTN: ATCD-Z 1 ATTN: Fire Power Division

Fort Monroe, VA 23651 Quantico, VA 22134

41

US ARMY ARMAMENT MATERIEL READINESS Commander

COMMAND USA Combined Arms Center and

Fort Leavenworth

Commander ATTN: ATZL-CA-COG

US Army Armament Materiel Readiness Command ATTN: ATZL-CAM-IM

ATTN: DRSAR-ASN I Fort Leavenworth, KS 66027

ATTN: DRSAR-SF 1

ATTN: DRSAR-SR 1 Commander

Rock Island, IL 61299 US Army TRADOC System Analysis Activity

ATTN: ATAA-SL

Commander White Sands Missile Range, NM 88002

USA ARRCOMATTN: SARTE US ARMY TEST & EVALUATION COMMAND

Aberdeen Proving Ground, MD 21010

Commander

Commander US Army Test & Evaluation Command

US Army Dugway Proving Ground ATTN: DRSTE-CT-T

ATTN: Technical Library (Docu Sect) 1 Aberdeen Proving Ground, MD 21005

Dugway, UT v&022

DEPARTMENT OF THE NAVY

US ARMY TRAINING & DOCTRINE COMMAND

Chief of Naval Research

Commandant ATTN: Code 443

US Army Infantry School 800 N. Quincy Street

ATTN: NBC Division 1 Arlington, VA 22217

Fort Banning, GA 31905

Commander

Commandant Naval Explosive Ordnance Disposal Facility

US Army Missile & Munitions Center ATTN: Army Chemical Officer (Code AC-3) 1

and School Indian Head, MD 20640

ATTN: ATSK-DT-MU-EOD I

Restone Arsenal, AL 35809 Commander

Naval Surface Weapons Center

Commandant Code G51

USAMP&CS/TC&FM Dahlgren, VA 22448

ATTN: ATZN-CM-CDM 1

Fort McClellan, AL 36205 Chief, Bureau of Medicine & Surgery

Department of the Navy

Commander ATTN: MED 3C33

US Army Infantry Center Washington, DC 20372

ATTN: ATSH-CD-MS-C 1

Fort Banning, GA 31905 Commander

Naval Weapons Center

Commander ATTN: Technical Library (Code 343)

US Army Infantry Center China Lake, CA 93555

Directorate of Plans & Training

ATTN: ATZB-DPT-PO-NBC I US MARINE CORPS

Fort Banning, GA 31905 Director, Development Center

Commander Marine Corps Development and

USA Training and Doctrine Command Education CommandATTN: ATCD-Z 1 ATTN: Fire Power Division

Fort Monroe, VA 23651 Quantico, VA 22134

41

L)U'ARTMENT OF THE AIR FORCE

HQ Foreign Technology Division (AFSC)ATTN: TQTR1Wright-Patterson AFB, OH 45433

Commander

Aeronautical Systems Division

ATiN: ASD/AELA 1

ATTN: ASO/AESD 1Wright-Patterson AFB, OH 45433

tiQ AFLC/LOWMM1

Wright-Patterson AFBI, OH 45433

HQ,, F SL/, GNL1Andrews AF8, MD 20334

HQ, AMDI/RL

ATTN: Chemical Defense OPR1Brooks AF-B, TX 78235

NORAD Combat Operations Center

ATTN-: DOUN1

Cheyenne Mtn Complex, CO 80914

Air Force Aerospace Medical Research

LaboratoryATTN; AFAMRL/HE ( Dr. C.R. Replogle) I

Wright-Patterson AFB, OH 45433

HQ AFTEC/SGB1

Kirtland AFB, NM 87117

OUTSIDE AGENCIES

Battelle, Columbus Laboratories

ATTN: TACTEC 1505 King AvenueColumbus, OH o201

Toxicology Information Center, WG 1008National Research Council 12101 Constitution Ave., NW

Washington, DC 20418

US Public Health ServiceCenter for Disease ControlAITN. Lewis Webb, Jr. 1

Building 4, Room 232

Atlanta, GA 30333

42

L)U'ARTMENT OF THE AIR FORCE

HQ Foreign Technology Division (AFSC)ATTN: TQTR1Wright-Patterson AFB, OH 45433

Commander

Aeronautical Systems Division

ATiN: ASD/AELA 1

ATTN: ASO/AESD 1Wright-Patterson AFB, OH 45433

tiQ AFLC/LOWMM1

Wright-Patterson AFBI, OH 45433

HQ,, F SL/, GNL1Andrews AF8, MD 20334

HQ, AMDI/RL

ATTN: Chemical Defense OPR1Brooks AF-B, TX 78235

NORAD Combat Operations Center

ATTN-: DOUN1

Cheyenne Mtn Complex, CO 80914

Air Force Aerospace Medical Research

LaboratoryATTN; AFAMRL/HE ( Dr. C.R. Replogle) I

Wright-Patterson AFB, OH 45433

HQ AFTEC/SGB1

Kirtland AFB, NM 87117

OUTSIDE AGENCIES

Battelle, Columbus Laboratories

ATTN: TACTEC 1505 King AvenueColumbus, OH o201

Toxicology Information Center, WG 1008National Research Council 12101 Constitution Ave., NW

Washington, DC 20418

US Public Health ServiceCenter for Disease ControlAITN. Lewis Webb, Jr. 1

Building 4, Room 232

Atlanta, GA 30333

42