~.” x ,- ..-. .7. .:... ,.:,. .; ,.’”“.:;,C-,,::.”;*”.., ,,...— . . . . .- .- —. . .

- .- .-. . .>.:;::L6~An&.ti66..Lo;:.mo#,Nei;Mexico:545

LosAlamos NationalLaboratory. . $,..._.—:.....LM..=.,.,,+------.

ASSAffiith Actiort/Eqrsd@sMUSSityEIsSPIOYCS

DISCLAIMER

This report waspreparedas an account of work sponsoredby an agencyof the UnitedStates Government.Neitherthe UnitedSta!esGovernmentnor assyagencythereof, nor asayof Ureuemployees,makesasrywarranty,expressor implied,or assumesany legalliabilityor responsibilityfor the accuracy,completeness,or usefulnessof any information,apparatus,product, or processdisclosed,or represenfi that its usewouldnot infrusgeprivatelyownedrights. Reference hereinto any spe&Ic commercialproduct, process,orserviceby trade mme, trademark,manufacturer,or otherwise,doesnot necesamilyconstitute or imply itsendorsement,recommendation,or favoringby the UnitedStateaGovernmentor any agencythereof. Theviewsand opinionsof authors expressedhereindo not necesuily state or reflect those of the UnitedStates Covernmemor any agencythereof.

LA-1024I-Ms

UC-34Issued: September 1984

A Simple EOSfor Linear Polytetradeuteroethylene

F.Dowell

-..

- ... . .t- -

.

.,- .+

-—

.. .

,.

-- . . . .

A SIMPLE EOS FOR LINEAR POLYTETRADEUTEROETHYLENE

by

F. Dowell

ABSTRACT

A simple equation of state (EOS) for linearpolytetradeuteroethylen(PTDE), with initial state

3density p. = 1.093 g/cm , was generatedand added to-the T-4 Sesame EOS Library as material number 7230.This EOS reproducesthe experimentalshock Hugoniotdata for PTDE and for isotonicallyscaled linearpolyethylene.

INTRODUCTION

In this report, the generationof a simpleequation of state (EOS)

for linear polytetradeuteroethylene(PTDE) is presented. The EOS

presentedhere reproducesthe experimentalshock Hugoniot data for PTDE

and for isotonicallyscaled linear polyethylene,thus fulfillingthe

primary purpose of this EOS to describe the compressionregion.

METHOD

A simple EOS for PTDE was generated from various theoretical

models using the GRIZZLY1 computer code. (SeetheAppendix for a

pedigree of GRIZZLY.)

For reasons of tractability,the models used to generate the

simple EOS describedhere for PTDE do not explicitlytreat PTDE as a

polymer. However, experimentaldata for PTDE are used in these models,

and thus the polymericnature of PTDE is implicitlyincluded in at

least parts of the EOS.

I

The repeat unit in the PTDE polymer chain is CD2. Therefore,PTDE

was modeled as an average atom with an average atomic number of 8/3 and

an average atomic weight~of 5.3469. An initialdensity

Po = 1.093 g/cm3 at P = O (P - 1 bar) and T = 298.15 K was used.

The total EOS is a sum of cold curve (T = O K isotherm),thermal

electronic,and nuclear (ion) contributions.

Cold Curve:——

The cold curve in the compressionregime from v = 1 to q = 2.042

(where q = p/po, where p is the density)was calculatedfrom

experimentalshock Hugoniot data assuming a Mie-Grlineisen2EOS.

Experimentalshock Hugoniot data for PTDE3S4 and for ISM [linear

4 (~rla) that has been isotonicallyscaled] were used.polyethylene

The q of 2.042 is the largestrIfor which there are experimentalshock

Hugoniot data.3,4 AS seen in Fig. 1, the Us vs Up data for PTDE and

ISM fall on the same curve.

The followingfits to this shock Hugoniot data were used:

Us = 2.441+ 1.990U - 0.1560 UP2 , Up< 1.828P

and

Us = 2.719 + 1.606 Up - 0.02365UP2 , Up > 2.342 ,

where Us is the shock velocity and Up is the particlevelocity;Us and

5 two us VSup fits forUp are given in km/s. As with other polymers,

PTDE were made, one below and one above the slight break in the

Us Vs Up curve at Up - 2 ‘m/s= (See Fig. 1.)

2

For v > 2.042, the cold curve was calculatedusing an analytic

form6 for the Thomas-Fermi-Dirac(TFD) electronicmodel. Atq = 2.042,

the energy, the pressure,and the first derivativeof the pressurewere

required to be continuous.

For q < 1, the cold curve was calculatedwith an analytic

Lennard-Jones(LJ) fonn6 with the term FACLJ = 2 (correspondingto an

r-6 attractiveterm, where~ is the separationdistance involved in the

LJ pair potential). At q = 1, the energy, the pressure,and the first

derivativeof the pressurewere required to be continuous.

The cohesive energy Ecoh (the energy of vaporizationof the solid

at O K) used in this EOS for PTDE was calculatedto be 0.302 FLJ/kg,7 of vaporization

applying the estimationmethod of Bunn7 to the energy

of 680 cal/mole (at the normal boiling point) for the repeat unit, CH2,

of polyethylene.

At high temperatures,one expects the polymer to dissociate

chemically. Thus, a dissociationenergy (rather than a vaporization

energy) might be more appropriateat higher temperatures. A

dissociationE~oh for a CH2 repeat unit was calculatedusingbondstrengths8for a c-c singlebond and for two C-H bonds. This E~oh gave

virtually the same numericalresults for the total EOS in the

compressionregion as did the Ecoh of 0.302 MJ/kg, and gave similar

qualitativeresults--but somewhat differentnumericalresults--inthe

expansionregion, especiallyfor intermediatetemperatures.

Thermal Electronic:

The thermal electroniccontributionsto the total EOS were

calculatedusing a Tho~s-Fermi-Dirac theorywith an exchange constant

of 2/3. This particularTFD theory is the same as that describedin

Ref. 9, except that the TFD theory used here for PTDE uses a local

density approximationto the exchange.

3

Nuclear (Ion):

The nuclear (ion) contributionsto the total EOS were calculated

by a solid-gas interpolationscheme,6which reduces to a Debye model at

low temperaturesand high densities and which reduces to an ideal gas

at high temperaturesor low densities.

The Grtineisenfunctiony and the Debye temperature0 at a given p10were calculated from Y. and 00 using the Thompson formulas:

Y = yog + [2(1 - 5)2/3]

and

e = eo~2/3exp{yo(l- E) - [(3- 4E + g2)/31},

where & = polp = llq. An effectivey. of 0.4047 for PTDE was

calculatedusing the Thompson formula and values of y at variousp

calculatedfrom a combinationof off-Hugoniotdata (sound speed and

multishockmeasurementsll)and shock Hugoniot data3’4’11for PTDE and

ISM.

The 00 was calculatedin the followingmanner: The lattice

(intermolecular)contribution001 to e. was calculatedfrom

e = 444.12[(1- 26)/(1+ dll’2co[po/(wA)ll’301

with

A = 2 + [(0.5 - 6)/(1 - a)]slz

using p. = 1.093 g/crn3,the interceptc. (in km/s) of the given

quadratic fit for Us vs Up shock Hugoniot data for PTDE, and a

Poisson’s ratio a of 1/3; this equation6can be derived from the usual

relation between e and the longitudinaland transversesound speeds.

For the quadratic fit for IJp< 1.828 km/s, eol = 248.4 K; for the

quadratic fit for Up > 2.342 km/s, eol = 276.6 K.

The intrachain(intramolecular)contribution(302to (10wascalculated in the followingmanner: For q < 1.493 (correspondingto

Up < 1.828 km/s),

002 = [(e~e2e3e4eIje6e7e8eg)2/g(e10e11e12) 1/3]1/3,

where these ei’s correspondto the followingvibrational

frequencies -1):12,13 vi’s (in cm

‘1 : 2197

I

CD2 asymmetric

‘2 : 2192 stretch

V3 : 2102

1

CD2 symmetric

‘4 : 2088 stretch

‘5 : 1146

I

CD2 bend

V6 : 1093

V7 : 991

1

CD2 ro:k

V8 : 522

‘9 : 916 CD2 twist

5

‘lo : 1249

’11 : 1146

V12 : 522

The eo2 was

C-C stretch

C-C stretchand C-C-C bend

C-C deformation.

calculatedas a geometricmean14 of the ei’sj ~th each gi

weighted according to the number of chemicalbonds that involve the

atoms in that vibrationalmode vi in the repeat ~it of pTDEo

In a similarmanner for q > 1.572 (correspondingto

Up > 2.342 kdS),

e = m310e11e12) 2/3g14]l/3 ,02

where 014 corresponds ‘1 forto the vibrationalfrequencyV14 = 3053 cm

D-D, which was calculatedfrom the vibrationalfrequency15 of

= 4315 cm-l for H-H using16v the ratio of the square roots of the

masses of deuteriumand hydrogen.

For q > 1.572, the polymer is being treated as a slightly

different polymer, i.e., a crosslinkedpolymer with a repeat unit of C:

I-c-

1

Here, deuteriumatoms on neighboringpolymer chains have come off as

D2, and

chains,

bonds.

carbon-carbonsingle bonds have formed between neighboring

thus crosslinkingthe original polymer chains through side

This very simplistictreatmentis conservativelyconsistent

with analyses17 of the products recoveredfrom shock Hugoniot

experimentson various polymers,which imply that the original polymer

has undergone some chemicalmodification(which seems to involve some

extent of irreversibledissociation)on the shock Hugoniot in the

1

regime above theIreak in the Us vs Up curve. That the crosslinked

structureused here for TI> 1.572 is only slightlydifferent from the

originalPTDE structure used for q < 1.493 is highly consistentwith

the fact that, as with other saturated polymers,5,18

there is only a

rather small change in the slope of the Us vs U curve at the slightI

break at Up - 2 km/s in the curve for PTDE. (S~e Fig. 1.) By the

methods described above, 002 was calculatedto be 1680 K for T ~ 1.493

and 1956 K for q > 1.572.

The 00 was c~lculatedas the geometricmean14 of eol and 002:

Ie =

o (001002)1/2. ~

Here, three norma} modes have been assumed for the lattice vibrations

and three normal lodeshave been assumed for the intrachainvibrations,7

thus leading to a? equal weighting of O.l and 002 in O.. In this1

~nner, e. was ca culated as 646.0 K for q < 1.493 and 735.6 K for~

q > 1.572.

Final EOS:— —

Tables of pressure,energy, and Helmholtz free energy for PTDE as

a function of temperatureand density were calculatedin the manner

described above, ~irst using the input terms appropriatefor ~ < 1.493

and second using the input terms appropriatefor q > 1.572. GRIZZLY

automaticallysets!the zero of energy at P = O and T = 298.15 K. The

final tabular EOS is composed of the table values from the first EOS

for ~ < 1.493 and of the table values from the second EOS for

q > 1.572. This inalEOS has been added to theT-4 Sesame EOS Library

/under material nu ber 7230. The followingkinds of pressure, energy,

and Helmholtz free!energy tables were added: 301 (total EOS), 304

(thermalelectronic),305 (ion, including zero point contributions),

and 306 (cold curve, with no zero point contributions).

I

DISCUSSION

As seen in Fig. 1, the EOS described in this report for PTDE

3,4 for PTDE and ISM.reproducesthe experimentalshock Hugoniot data

Values of pressure,energy, and negative Helmholtz free energy for

every other isothermfrom the total EOS tables calculatedfor PTDE are

plotted vs density in Figs. 2-4.

APPENDIX

PEDIGREE OF GRIZZLY

GRIZZLY1 is a computercode put togetherby J. Abdallah

(Group T-4) with contributionsfrom J. D. Johnson (T-4) and

B. I. Bennett (T-4). GRIZZLY uses subroutinestaken, with appropriate6 EOSC~y, CANDIDE, andmodifications,from the computer codes PANDA,

CHART D.l”

PANDA is a computercode made by G. I. Kerley (then in T-4), using

ideas and some coding from the computer code EOSLTS, developedby

Bennett and modified by Bennett, Johnson,Kerley, and R. C. Albers

(then in T-4). Some of the ideas and models in EOSLTS came from the

following sources: the computer code SESAME, developedby J. F. Barnes

and G. T. Rood (both then in T-4), and the CHART D

10 of Se L. Thompson andradiation-hydrodynamiccomputer code

H. S. Lauson (both of Sandia National Laboratories).

EOSCRAY is a version of EOSLTS. CANDIDE is a TFD computercode

developedby D. A. Liberman (then in T-4) and modified by Bennett and

Johnson.

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Joseph Abdallah,Jr., “User’s Manual for GRIZZLY,” Los AlamosNational Laboratoryreport LA-10244-MS (September1984).

M. H. Rice, R. G. McQueen, and J. M. Walsh, “Compressionof Solidsby Strong Shock Waves,” Solid State Phys. ~, 1 (1958).

C. E. Morris, data tables, Los Alamos National Laboratory (1983).

J. N. Fritz, data tables, Los Alamos National Laboratory (1984).

W. J. Carter, S. P. Marsh, and R. G. McQueen, “HugoniotEquationof State of Polymers,”Los Alamos ScientificLaboratory documentLA-UR-77-2062(1977).

G. I. Kerley, “User’s Manual for PANDA: A Computer Code forCalculatingEquations of State,” Los Alamos National Laboratoryreport LA-8833-M

C. W. Bunn, “The16, 323 (1955).

(November1981)..

Melting Points of Chain Polymers,”J. Polym. Sci.

R. C. Weast, CRC Handbook of Chemistryand Physics, 51st ed. (TheChemicalRubb~Co. , Cleve~nd, 1970),RF-166.

R. D. Cowan and J. Ashkin, “Extensionof the Thomas-Fermi-DiracStatisticalTheory of the Atom to Finite Temperatures,”Phys. Rev. 105, 144 (1957).

S. L. Thompson and H. S. Lauson, “Improvementsin the CHART DRadiation-HydrodynamicCode III: Revised Analytic Equations ofState,” SandiaNational Laboratoriesreport SC-RR-710714 (March1972).

C. E. Morris, R. G. McQueen, and J. N. Fritz, data tables, LOSAlamos National Laboratory (1983).

L. Piseri and G. Zerbi, “DispersionCurves and FrequencyDistributionof Polymers: Single Chain Model,”J. Chem. Phys. 48, 3561 (1968).

M. Tasumi and T. Shimanouchi,“CrystalVibrations andIntermolecularForces of Polyethylene Crystals,”J. Chem. Phys. 43, 1245 (1965).

14. B. Wunderlichand H. Baur, “Heat Capacitiesof Linear HighPolymers,” Adv. polym. Sci. ~, 151 (1970).

15. T. L. Hill, An Introductionto StatisticalThermodynamics(Addison-Wes~y, Reading, m=., 1960), P. 153=

16. H. W. Siesler and K. Holland-Moritz,Infraredand Raman——Spectrosco~ of Polymers (MarcelDekker, New York, 1980),pp. 292-296.—

17. c. E. Morris, E. D. Loughran, R. G. McQueen, and GO F“ Mortensen>“Shock InducedDissociationof PE, PS, PTFE, Kel F, and pVF2,” LosAlamos NationalLaboratoryGroup M-6 progress report (1983).

18. S. P. Marsh, Ed., LASL Shock Hugoniot Data (UniversityofCaliforniapress,=eW1980)~ —

10

c

O*Z

TO

’TT

O“O

T0“6

0“8O

“A0“9

0“sO

“*O

“c(s/u

xq)

sn

mttioIA0.mtioOi

.l-l

11

tL(Vd!l)

WLN

kid

$-l

0cclor4

12

u)e

IkinE

+P

I

m0A

wm“13

R-2.46+06m- 1.01+07o-4. ii4+07P- 1.Ha- 1.10+09

10 9

10 8

10 7

10 6

le 5

10 4

la 3

10 2

10 1

19 ‘

10-1

&

II I I 11111 I 1I I llEI111IIIUIII IfTl131

g=

E lb J10-3 10 2 10 3 19 4

10-7

Fig. 4. sow negativeHelmholtz free energyvs density iso~~rmsfor ~DE. Helmholtzfree energy is in MJ/kg; density is

I

from th~ total EOS tablesin Mg/m5.

.

., . ,. -

., ...,

..... .,. . .. ..;Cc&@h ,.

~,. .<+ -. .1 .? .-a- ,.. . . ,, ,.,. ,

,-.,, . “.,. ,... *,.-1, -., .-, .

,4 . . . .

.,. .. . . ..... ,,. .,

.

. . . ...! ..., .-.. - . ., ....

>.,..- . ---,.-,-- . “ . “.. . . --- .- :—,:.—–-.———-42 !’-,, :: -,– . -. . ..- , - .=. , ..-

.. . . . . >.,---- -

.....: .,---

,.. ...—. .- .. .. . .... .. ..... .. .,.+..,,, .- - -,,-. . ,.

- .--— .-. . - . . . <v-,.,. , . “ .,-,

, -, ,,, .--- .,. ;, ‘.’ ‘“,- ..,,.

;7. ,. . --: ~Y ,

,.. . . . . . . . . . . . ., ,.

,., ,.

,. ,,,. .. ,.. ,#f,-! ‘“,>...,-. ,.

. ... . “

.-. .-,

‘. , , , ., ‘.,-. ..+2. s’.”-”.’””” ..,””,.?, :-“. .

.;7...

.- .

,. . . .

>.. . . - ......,–”

,,. .. -.

. ..

---—.— ---- ——-—.——. ... —.-—

. . .--- , .- .-- Printedinthe Unikd SUIICSor Anierki

Avajlabk fromNational TechnicalInformationScrvicc. . . .

“USDepc.mimt ai&”rnrnerce”5VS Pod Royal RoadSpringfield.VA 22161

Mimof&c (AOI)

NTIS NTIS

—. . .. -., ,,;-,+.,Yti -., w

‘ . . . ;, . - J

- 1.; , . . . - . .. ,

. . .

,. : n. . ,.-.

. ..,. . -: -.,,.,..,,,,., !.* .’.— -..

,.-. .,:: .f. , ., ~-:..

NT2S

. jjp ‘.

: ,JylpPnge Range Prk Cc&

.———.—— -P-C k$mge Prim code Page Range I%& Code Page Range WCC Cock. —

001425 A02 151.175 AOS 301.325 A14026.050 A03 176.200 A09 326 3S0 A15OJ1J2J5 A04 201.225 AIO 351.37s A16076.100 AOS 226.236 All 376400 A17101.125 . “X66 - : 251.2?5 A12 401.425 A18126.1S0 w . .?7+W 61?. ..426-450 ,,419

..-. . . .. . ,. --- --- . . . . . .. . - . .

WOnlact NTIS for a price quote.

. . - . . . . ., . .

.- .-

,. — —- - .- . .- -.., ,. ....——-. —....

451475 ::: A20476.500501.525

.--+–..

526.550 ;::+.!, . .U .

551.575 . A2i?76400 ,:. A2UI6ol. up” A99 \...,, . .,

I.— -—.-. -+----. .. ,!!!e---- .’ ‘ ‘ ~‘ (::::’:-‘., (.\.

.

.,ne-- ,-- - ..- , .:. , ,.. :. .... ~,.... .,. *.*.——.— -

..-.———. .———.;-7”- .“.

bwlamnlm