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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
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ABOUT THIS REPORTThis official electronic version was created by scanning the best available paper or microfiche copy of the original report at a 300 dpi resolution. Original color illustrations appear as black and white images.
For additional information or comments, contact: Library Without Walls Project Los Alamos National Laboratory Research LibraryLos Alamos, NM 87544 Phone: (505)667-4448 E-mail: [email protected]
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 and5 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 givenquadratic 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)~ —
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