Thermal Expansion and Temp Dependece of e Moduli Orf Aromatic PA

8/3/2019 Thermal Expansion and Temp Dependece of e Moduli Orf Aromatic PA

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7198 L w k s a nd R u t l e dget e r m i n e d f r o m f i r s t p r in c i p le s b y m i n i m i zi n g th e q u a n t u mm e c h a n i c a l f r e e e n e r g y o f th e c r y s t a l c a l c u l a t e d i n thequa s i -h a r m o n i c a p p r o x i m a t i o n . These results w e r e i n g o o da g r e e m e n t w i t h t h e r e s u l ts o f e x p e r i m e n t s on h i g h l y< rystailine s a m p l e s and l e d to n e w c o n c l u s i o n s r e g a r d i n gthe t e m p e r a t u r e d e p e n d e n c e o f th e e l a s ti c m o d u l i o fp o l y e t h y l e n e and the a x i a l thermal e x p a n s i o n o f i s o t a c t i cp o l y p r o p y l e n e . In t h e p r e s e n t p a p e r w e a p p l y the s a m et e c h n i q u e s to t h e a r o m a t i c p o l y a m i d e s p o l y ( p - p h e n y l -e n e t e r a p h t h a l a m i d e ) IPPTA) an d p o l y ( p - b e n z a m i d e )i P R > 1 .

Ma c romole c u le s , Vol. 27, No. 24, 1994

11. Computational Methodin tne present s imulations, the forces between atoms arerepresented by a m olecular m echanics force field, and ther maleffects are included within the quasi-harmonic approximation.The unit cell volumes are ob tained by minimizing the free energy

r',t t h r ;pecified temperature.a. V i b r a t i o n a l Free En erg y. Th e vibra tional f ree energyand tem perature depend ence of properties can be efficientlyevaluated within t he quasi-harmonic approximation (QHA).lIIn the QHA, terms higher than second-order in the Taylorexpansion of th e poten tial energy are neglected, and the resulting

harmonic poten tial energy surface and vibrational frequenciesvary with the lattice parameters . Theref ore anharm onic effectsdue to volume changes are incorporated in the QHA, but theanh arm oni c coupling of th e vibrational m odes is neglected.Quan tum mechanical effects , which are impor tant for vibrationalmotion with frequencies Y > kT/h ( ~ 2 0 0m-l at room tempe ra-ture) , are included in th e present calculations.Th e quasi-harmonic vibrational free energy, Avl b, a t a t e m-pera ture T is given qua ntum mechanically by

4 . 2=. vib

(4)

where k s a wavevector, w,(k) is the vibrational frequency ofmode i for wavevector k , Nu s the number of a toms in the uni tcell, and t he integrals over BZ ar e integrals over the B rillouinzone. Th e quasi-harmonic heat capacity at constant volume, C,,is given by

where the w in eq 5b takes on the va lues wi(k). The e n t ropy , S ,can be calculated by

wliew E , is the vibrational energy,

J d k T 1 / 2 h w i ( k )+ hwi(k)BZE,,, =- ,.J dkBZ

Th e elastic moduli and G runeisen parameters were obtained fromeq 1 and 3 by calculating the derivatives with respect to strainnumerically.Th e equation of s tat ewas determ ined by minimizing th e quasi-harmonic free energy with respec t to unit cell lattice parameters .This method has previously been applied to ionic crystals,12minerals,13zeolites,14me ta l a lloys,15 nd po lymer c r y s t a l~ .~ J ~eexamined th e accuracy of this m ethod elsewhere by comparingthe results t o exact results obtain ed by M onte Carlo simulationand found t ha t properties ar e accurately calculated up t o half totwo-thirds of the m el ting tempera ture for the Lennard-Jonessolid.'6The equilibrium structures at a given temperature wereobtained as follows: F or a given set of lattice par ameters , thepotential energy is minimized with respect to the variablesdescribing the atomic positions in th e unit cell, using an analyticalfirst derivative technique.17 After minimization of these inte rnalcoordinates, the numerical integration of t he vibrational freeenergy over the Brillouin zone is carried out: for each integrationpoint, analytical second derivatives of the energy, and from thesethe vibrational frequencies and the vibrational free energy, arecalculated. Gauss-Legendre quad ratur e with 24 integrationpoints is used to carry out the Brillouin zone integrations. Th etotal G ibbs free energy is obtained as the sum of the potent ia lenergy and the vibrational free energy for this set of latticeparameters . Th e lattice param eters which yield th e minim umGibbs free energy are the equilibrium lattice parameters .

b. F o r c e F i e l d. The PCFF force field18 was used for thepresent s imulations. Th e partial atomic charges were computedus ing the bond increment method as implemented in the PCF Fforce field, with the m odification of setting the bond increm entpa ra m et er s t o z ero for th e Ccarbonyl-CammPtio wt,,,ny~-N, an dCmmtic-N bond s (which isconsistent with th e C FF91 force field's).The modified partial atomic charges are closer to previousestimates of th e atomic charges in PP TA derived from semiem-pirical quan tum mechanical calculation^^ and lead to a nequilibrium s tructu re in better agreem ent with experiment.Where indicated, calculations were also carried out with theDreiding force field,20 with the purpose of demonstrating thatour conclusions are indep endent, a t least qualitatively, of t hechoice of force field.In our computer program, the long range Coulombic anddispersion interactions are evaluated with Ewald sum tech-niques,1132' and th e repulsive no nbo nde d interaction is evaluatedin a direct space sum. Analytical first and second derivatives ofthe po tential energy with respect to th e atomic coordinates arecalculated for use in geometry optimizations and the calculationof vibrational frequencies.

c. C r y s t a l S t r u c t u r e s . T h e r e pe a t u n it s of P P T A a n d P B Aare shown in Figure 1. Th e crystal s tructures investigated hereare shown in Figure 2. Structures based on the experimentalcrystal s tructures were used as start ing points;22.23 no at te m ptwas made to map o ut multiple minima in th e energy surface aswas done previously.19 Bo th poly mers consist of phenyl ring sconnected by amid e groups but differ in t he orientation of everysecond amide linkage along the chain. Th e experimentallydetermined conformations of th e amid e groups are approximatelyplanar, and t he phenyl rings rotate ou t of th e plane of the am idegroups by approximately 3 5 O . Th is angle represents a compro-mise between th e repulsive nonbonded interactions of the am idehydrogen or oxygen with a phenyl hydrogen, which are minimizedwhen the phenyl rings are perpendicular to th e plane of the amidegroups, and the stabilizing x electron delocalization, which ismaximized when th e phenyl rings are in the plane of the amidegroups. In P P T A the two phenyl r ings in the repe a t uni t a rerotated in opposite directions with respect to th e am ide groups,whereas in PBA the rings are rotated in the same direction.


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Aromatic Polyamides 7199PBA, as compared to the experimental results of 0.12GPa IK for PP TA and 0.21 GPa lK for PBA.27 Althoughthe calculated changes in E3 with temperature aresignificantly smaller tha n those observed experimentally,we note that the changes in the experimental E3 withtemperature may also be influenced by factors such aschanges in crystallinity and crystallite orientation withtemperature.Th e complete set of elastic moduli are given in Tab le2. Th e precision ofthesevaluesisapproximately A 1 GPa,due t o th e error arising from numerical evaluation of thesecond derivative in eq 1; this imprecision limits theanalysis of the elastic moduli with small temperaturedependences. T he calculated elastic stiffness matrix is insubstantial agreement with previous static lattice cal-culations: th e average rms deviation between the elementsof the present stiffness matrix (at T= 0) and th at of Yangand HsuZ6 s 11 GPa, and t he rm s deviation from thestiffness matrix of Rutledge and S uter (s tructu re3of th atpaper)25 s 14 GPa. Th e (712 stiffness moduli decreasewith temperature for both PBA and PPT A, while the Cpl3and ( 7 2 3 moduli increase with temperature for PBA andremain approximately constant with temperature forPPTA.The Gruneisen parameters for PPTA and PBA areshown in Figure 5. Th e Gruneisen parameters along thea and b axes positive a nd are nearly equal for PPTA andPBA. Th e Gruneisen parameter along the c axis, 73. isnegativeover the entire temperature range, indicating tha tthe entropy increases with contraction; 7 3 is significantlylarger in magnitude for PBA th an for PPTA.IV. Discuss ion

a. T h e r m a l E x p a n si o n . 1. Axia l Thermal Expan-sion. Negative axial thermal expansion has been observedexperimentally in PPTA an d PBA.Z4.27Th e explanationpreviously given for this thermal contraction is that thecontraction allows increased vibrational motion trans verseto the chain axis while maintaining (approximately)con stan t bond lengths, or equivalently tha t the entropyincreases with axial contraction because of a decrease inth e frequelicies of the tran svers e vibrational motion.5 Inter ms of eq 2, this mechanism implies tha t y3< 0 and thej = 3 term leads to he observed 013.~Th is mechanism isthe commonly proposed explanation for negative axialtherma l contraction in other polymer cr y~tals .~,8,3 ~35urprevious simulation results for negative axial thermalexpansion in polyethylene a re also in agreem ent with thismechanism.9However, negative axial therm al expansion can also bedue to an elastic response to th e ther ma l stresses in thedirections transve rse to he ch ain axis, i.e., th e term s ylSr13and 7 2 9 2 3 in eq Z.36 We have previously shown tha t itis this effect which leads to the axial thermal contractionin isotactic polypropylene; th e entropy increases with axialexpansion for isotactic polypropylene, which alone w ouldlead top os itiu e axial therm al expansion.1 Therefo re, themechanism in which th e negative axial the rm al expansionin polymer crystals is due to an increase in entropyassociated w ith contraction does not app ear t o he "uni-~ e r s a l " . ~ ~he elastic response to transverse thermalstresses mus t be considered in a n objective analysis of th eaxial therm al expansion of polymer crystals.We therefore subdivide our results for the axial thermalexpansion of PPTA and PBA into contributions fromthermal effects both along and transverse to the chainaxis. These results are presented in Table 3. As notedabove, both PPTA and PBA have negative axial (i.e.,

Macromolecules, Vol. 27,No. 24, 19948

PPTA PBAFigure 1. Repeat units of PPTA and PBA. Key: carbon (darkgray), hydrogen (white), oxygen (black), nitrogen (ligh t gray).Hydrogen bonding occurs between amide groups on adjacentchains along the b axis. The PBA crystal lattice is orthorhombic,and the PPTA lattice is monoclinic, h u t with th e nonortbogonalangle between the a and b axes differing only slightly from 90111. Results

The equilibrium lattice parameters as a function oftemperature for PPTA and PBA are shown in Figure 3.The calculated volumes are somewhat larger than theexperimental values because the nonbonded force fieldparameters were obtained by fitting static-lattice struc-tures to experimental data taken at room temperature(some effects of both zero-point and the rma l vibrationalmotion are implicitly included in the nonb onded p otentialparameters). Th e results for the average thermal expan-sion between 300 and 500 K are given in Table 1 andcompared with the experimental results of Ii et a1.24 Inagreement with experimental results, thermal expansionis smaller along the b axis tha n along the a axis, and th etherm al expansion is negative along the c axis (i.e., a, >a*> 0 > aJ .The calculated Young's moduli E; (defined as theinverses of th e diagonal compliance moduli, E; = l lSri ; ,i = 1,2 ,3)are shown as functio ns of tem pera ture in Figure4. Th e Young's modulus is largest along the chain directiondue to the strong intramolecular bonds a nd is larger alongthe b axis than along the a axis because of th e hydrogenbonds parallel to the b axis; similar resu lts were repo rtedin previous static-lattice calculation^.^^^^^ Th e calculatedE3 at room temperature of 273 GPa for PPTA is highertha n the ex perimentally repo rted values of 15iF240.27-31Allof theYoung'smoduliarefound to begreater forPPT Athan for PBA, over the whole range of temperatures. Th elarger E3 fo r PPTA as compared to PBA disagrees withthe available experimental results,2' although it agreeswith previous calculation^.^^^^^ As noted by othe r inves-tigatom,"anall-trans conformation for PBA would exhibitalargeE3,inacco rd withthee xperim ental record; however,thisconformation ap pears to be inconsistentwith theX-raydata.For both PPTA and PBA, E2 an d E3 are found todecrease significantly with temperature, while the tem-perature dependence of E, is relatively small. T hecalculated E3 decreases between 300 and 500 K byapproximately0.07 GP aIK for PPTA and 0.13 GP a/K for


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7200 Lacks and Rutledge Macrom olecules, Vol.27,No. 24, 1994

b b

Figure 2. (a, left) Crystal struc ture of PPTA (modification 1); h, right) crystal structure of PBA. Key: carbon (dark pay ), hydrogen(white),oxygen (black),nitrogen (light gray). Based on structures from ref 23.

-58 0-im98 10 0 200 T (K ) 00 400 50 0 3" 13.021 I12.961 412.9272.900 100 200 300 40 0 50 0T (K )

z, 5 15 :;5 10

0 100 200 300 400 50005 0 100 200 300 400 500T K) T iK)Figure 3. Lattice parameters as a function of temperature. Filled symbols are PPTA, open symbols are PBA. For PBA, the angley is always 90. The experimental room tem perature results for PPTA a re a = 7.80 A, b = 5.19 A, = 12.9 A, y = 90" and for PBAare a = 1.71A, b = 5.14 A, c = 12.8 A.21

Table I. Average Thermal Expansion between300 and 500 KPPTA PBA

calculated experimental' calculated experiment&al(1WC') 1.9 8.3 7.1 7.0~ 1 2W K-1) 2.9 4.1 4.6 4.1013 (1VK-') -0.57 4 . 2 9 4.84 4.17

Ii et al., ref 24.parallel to the chain axis) Gruneisen parameters, whichcontribute to axial thermal contraction as had previouslybeen suggested.5 However, th e domina nt contribu tion toaxial thermal contraction is the elastic response to thetransverse thermal stresses, which leads to 80% of thetotal axial thermal contraction in both PP TA and PBA.Previous analyses which assumed th at th e term s yiSTi8 ( i# 3) are insignificant5 should be reconsidered.The importance of the elastic coupling on the axialthermal contraction is further demonstrated hy calcula-tions which show tha t in response t o changes in the a an db lattice parameters, elastic coupling due to potentialenergy effects alone causes the c lattice parameter to

change significantly. Tw o seta of static-la ttice calculationson PPTA were carried ou t in which the lattice parametersa and b were fixed at their calculated equilibrium valuesfo r 300 and 500 K, while the lattice parame ter c (aswellas all atomic positions) was varied to minimize thepote ntial energy. Th e lattice parameter c which gives th eminimum poten tial energy was found t o decrease in goingfrom th e equilibrium a and b parameters a t 300 K to theequilibrium a and b parameters at 500 K, by an amountcorresponding t o a therm al expansion of -0.34 XleK-'.This calculation was also carried out with the Dreidingforce field, which led to an effective therm al expansion of4.55 x l W 5K-l. We therefore believe tha t the importanceof th e elastic coupling in the axial therm al co ntraction ofPPTA is not a n art ifact of the PC FF force field.

Th e magnitude of the axial thermal contraction, olg, islarger in PBA tha n in PPTA because of both a larger axialcontraction du e to the associated entropy increase andgreater elastic coupling to the transverse thermal stress.The axial contraction due to the associated entropyincrease is larger forPBA th an for PP TA primarily becausethe entropy increase is larger for PBA (at 400 K, y3(PBA)


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Macromolecules, Vol. 27, No. 24 , 1994 Aromatic Polyamides 72013007

I0 0 00 0 0

r N m P m220A ' 0 ' 0

i

00 100 20 0 30 0 40 0 500T ( K )

Figure 4. Young's moduli as a function of temperature. Filledsymbols are PPTA, open symbols are PBA; triangles are El,squares are El, and circles are Es.Table 2. Isothermal Elastic Stiffness Constantsa

OK 100K 200K 300K 400K 500KPPTA13.3 13.5 12.4 11.9 11.8 12.141.6 40.7 36.1 32.2 27.6 23.4300.1 297.4 290.6 283.7 276.5 271.814.7 15.1 14.4 14.3 14.1 14.50.1 1.0 0.1 0.5 0.1 1.0

6.1 6.7 6.1 6.1 5.7 8.113.1 12.8 12.0 11.1 10.6 9.910.9 10.9 10.6 10.5 10.7 10.814.5 14.5 14.3 14.2 14.5 14.812.4 12.6 11.4 10.1 9.1 8.733.0 31.6 28.5 24.6 21.4 17.1287.3 284.0 276.4 267.9 258.2 246.422.9 22.6 21.7 19.0 16.4 14.30.2 0.9 1.7 0.7 0.2 0.26.9 7.6 7.5 6.2 5.7 5.112.8 12.3 11.4 10.5 9.7 8.710.3 10.2 10.4 10.5 11.0 11.014.7 15.0 15.3 15.6 15.9 16.5

PBA

a The off-diagonalelements Cr,, cTb, and cTb were all found tobe less tha n 1GP a for PPTA and are equal to zero by symmetry forPBA. The precision of these results is approximately A 1 GPa.= 4 . 4 and y3(PPTA)= 4 - 1 6 ) ,bu t also because ST33 islarger for PBA (see Figure 4,which shows E3 = 1/sT33).Th e axial contraction due to the elastic coupling to th etransverse thermal stresses is larger for PBA than forPPTA because the off-diagonal compliance moduli arelarger for PBA, while th e transverse thermal stresses aresimilar in magnitude for the two polymers.2. Transverse Thermal Expansion. The thermalexpansion is smaller along th e b axis tha n along th e a axisbecause the hydrogen bonding between am ide groups onadjacent chains along the b axis leads to STZZbeingsignificantly less tha n sT11.Simulation and experiment agree in the result thatthermal expansion along the a axis, a1, s slightly g reater

2k.

- 2 ' " " " "T (K)10 0 200 30 0 4 0 0 50 0Figure 5. Gruneisen param eters as a function of temperature.Filled symbols are PPT A, open symbols are PBA; triangles areEl, squares are Ez, and circles are Es.Table 3. Component Contributions to Axial ThermalExpansion of PPTA and PBA, at T = 400 K

a3 = (Cv/V)[YlSrl3+(C./v) E?-IY;S*Q; Y 2 f l d ( C d v ) Y@33PPTA (10-5K-l) -0.54 -0.43 -0.11PBA (10-5K-l) -0.94 -0.63 -0.31in PPTA than in PBA. Th e simulations find the b axisthermal expansion, (YZ, to be greater for PBA than forPPTA; his result is due to S T 2 2 being larger for PBA th anfor PPTA (see Figure 4, which shows EZ 1 / S T 2 2 ) . T heexperimental investigation, in contrast, reports t ha t a2 islarger for PPTAaZ4This discrepancy may be due toapproximations in either t he experimental analysis or theprese nt calculations. In the experimen tal investigation,Ii et al. used X-ray measurements to obtain the latticeparameter b based on the assumption of a pseudo-orthohombic lattice, which is exact when the crystal-lographic angle y = 90'. T h e angle y is close t o 90122*23and the errors in b due to the pseudo-orthorhombicassumption ar e small relative to b. However, since thechanges in b with temperature are small, the errorsresulting from th e neglect of changes in y with temperaturemay be significant for (YZ. Our calculations suggest tha ty will change by appro ximately 0.2' in going from 300 to500 K; even such a small change in y with temperaturecan lead to errors in pseudo-orthorhom bic estimations ofa2 which are comparable in magnitude to the reportedexperimental differences in a2 between PPT A and PBA(refertoTable 2). We note alsothata t higher tempera tures(500-700 K), Ii et al . find a2 to be larger for PBA than forPPTA .24 Another possible source of the discrepancy maybe the app roximate natur e of the force field itself. Inparticular, th e room temperature b lattice parameter isunderestimated for PPTA and overestimated for PBA.While the errors in th e equilibrium b lattice parametersare small (


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7202 Lacks and Rutledge Macromolecules, Vol. 27, No. 24 , 1994axial deformations and found that the difference arisesfrom a coupling between th e phenyl ring rotation anglesan d th e amide bond angles (Le., Cpheny1-Cmide-N an dCpheny1-N-Camide). As discussed above, althou gh th estabilization associated with delocalized a bonding ismaximized when the phenyl rings are in the plane of theamide groups, the phenyl rings are rotated out of this planein order to decrease the repulsive nonbonded interactionsbetween a phenyl hydrogen and an amide oxygen orhydrogen (see Figure 1); he resulting phenyl ring rotationangle of approximately 35" represents a compromisebetween these two opposing effects. However, f a n amidebond angle is increased, he phenyl hydrogen and the am ideoxygen or hydrogen are moved further apart and thenonbond ed repulsion is reduced; the phenyl ring rotationangle will therefore decrease to enhance t he delocalizedT bonding. Conversely, if an amide bond angle isdecreased, the phenyl ring rotation angle will increase dueto the increased nonbonded repulsions. Since the PPTAan d PBA chains deform axially primarily through changesin the amide bond angles,37 the relaxation of the phenylring rotation angles in response to these bond angle changeswill moderate t he increase in energy associated with thedeforma tion, thereby decreasing th e axial elastic modulus(specifically, h e second derivative of th e potential energy)relative to tha t in which th e phenyl ring rotation angle isfixed. The PPTA and PBA crystals have higher axialelastic moduli th an th e corresponding single chains becausethe phenyl rings are "locked" in place by the surroundingchains within th e crystal and can not rota te in response tochanges n the amide bond angles. This effect was observedwith both th e PC FF and Dreiding force fields.

With thermal expansion the polymer chains movefurther apart at higher temperatures and the crystalbecomes more like a collection of single chains. There fore,therm al expansion in PPTA and PBA leads to the phenylrings being able to rota te m ore freely, with an associateddecrease in the second derivative of th e potential energy.We note that such a change in the value of the secondderivative of the potential energy when the polymerchains move farther a par t does not occur for all poly-mers: previous calculations for polyethylene found th ecrystal and single chain axial elastic moduli to be verysimilar.6T he larger decrease in CT33 with tem peratu re for PBAas compared to PPTA can be a ttributed to th e entropiccon tributio ns to CT33. T he decrease in CT33due to thermalexpansion effects (predominantly in term 1 of eq l b ) isessentially the same as in PPTA and PBA, while thedecrease due t o th e entropic contributions t o C T 3 3 (term3 in eq lb) is twice as large in PBA.2. Transverse Stiffness Moduli. The transverse

stiffness moduli, CTll and c T 2 2 were found to decreasewith temperature, the decrease for c T 2 2 being larger tha nfor CT1l. A decrease in these elastic moduli would beexpected based on potentia l energy effects alone becausethe interchain interactions decrease as th e polymer chainsmove further ap art . T he contribution to cT22 due topotential energy alone, CTzz(PotE), is compared to thetotal C T 2 2 for PP TA in Figure 6. In this case 70% of thedecrease in elastic modulus from 300 to 400 K is due toth e effects of therm al expansion on the potential energycontr ibution, of which 90% results from a decrease in thevalue of th e second derivative of th e potential energy andonly 1096 from the increase in volume. Th e smaller effectof th e volume increase on C T 2 2 as compared to P 3 3 s dueto th e sma ller mag nitud e of cT22-an increase in volum eleads to a decrease in elastic modulus which is proportional

240 ' ' ' ' ' '0 100 2 0 0 30 0 C O O 500T i K >

45 1 ' ' I

1 5 ' " " " I0 100 20 0 300 40 0 500I i KFigure6. Comparison of P, , (PotE) o P,,. a, op) CT33(PotE)vs PSS;PTA ( fiied symbols),PBA (open symbols). (b,bottom)CT22(PotE)vs CT22; symbols same as in a. The effects thatcontribute to CT,,,other than the potential energy, are the zero-point vibrationalenergy and the entropic erm in the free energy.These effects can, in general, lead to P,,eing either greater orless than P,, (P otE ).vibrational motion on the magnitude of CT33 in thesematerials are found to be much less than previouslye ~ t i m a t e d . ~ ~pproximately 80% of the decrease in cT33from 300to 400K is due to th e effects of ther mal expansionon the p otential energy contribution, of which half is fromth e increase in volume an d half is from a decrease in thevalue of th e second derivative of th e poten tial energy. Ingeneral, the vibrational contribution to CT33 (i.e.,CT33(PotE)) ontains two contributions arising from thevibrational energy Uvib and entropy S (eq lb) . The f i rs tcontribution, dWvib/d~3~ ,an lead to either a n increase (asin PPTA) or a decrease (as in PBA) in cT33 with respectto CT33(PotE) at low tempera ture, bu t th is effect neces-sarily falls to zero at higher temperatures, since thevibrational energy is frequency-independent in th e classicallimit. Th e second contribution, -Td2S/d~32,eads to adecrease in CT33 with resp ect to CT33(PotE) which is z eroa t T = 0 K and increases in magnitude with increasingtemperature. These effects are seen in Figure 6a.

The decrease in the value of the second derivative ofthe potential energy as the volume expands (i.e., as thepolymer chains move farther apart) is related to theobservations from previous static-lattice calculations th atth e axial elastic modulus of th e PPTA or PBA crystal issignificantly larger tha n th at of th e corresponding singlechain. (T he single chain axial elastic modulus is definedusing th e same volume as or the crystal modulus, and anydifference from the crystal axial elastic modulus is dueonly to the second derivative of the potential en er g~ .~ 5, ~6 )Previous calculations are in agreement that an axialdeform ation in a single chain involves a larger change intorsional energy and a smaller change in van der Waalsenergy tha n an equivalent deformation in th e crystal. Wehave carried out furth er calculations t o rationalize thisdifference in the energetics of th e single chain an d crystal


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Macromolecules, Vol. 27, No. 24, 1994 Aromatic Polyamides 7203

I 4 l

0 100 200 300 400 5 0 0T ( K )

Figure 7. Difference between E3 and P 3 3 . Filled symbols arePPTA; open symbols are PBA.to the magnitude of the elastic modulus. Th e temperaturedependen ce of c T 2 2 for PBA is similar to th at for PPTA.3. Off-D iagonal S t i ffn ess Modul i . As wi th thetransverse elastic moduli, the off-diagonal moduli woulddecrease with temperature based on potential energyeffects alone because th e interchain interac tions decreaseas the polymer chains move further a part. For this reason,the CT1z of both PPTA and PBA decrease with tempe ra-ture. However, th e c T 1 3 and CT23 of PBA are found toincrease with temperature, as was found previously forp ~ l y e t h y l e n e . ~he reason for th e increase in these elasticmoduli with temperature is related to t he axial thermalcontraction mechanism in which chain contraction is dueto a n entropy increase caused by th e transverse vibrationalmotion: T he increase in entropy with contraction (Le.,the quant i ty dSJdt3 < 0) increases in magnitude as th epolymer chains move further a par t because the transversevibrational motion becomes less hindered by the sur-rounding polymer chains. Therefore, the qu antity d2S/( d t 3 d y )< 0 for j = 1 , 2 , and ac ts to increase the c T j 3 . Fo rthis reason, the Pi3 and ( 7 2 3 of PBA increase withtemperature. For PPTA, the cT13 and P 2 3 remainapproximately constant with temperature because thiseffect, which is smaller in PPTA than in PBA (see Table31, just compensates for the decrease in CT13and CT23dueto potential energy effects.

4. S h e ar Modu li . Th e temperature dependences ofthe shear moduli (CTii i = 4,5,6) are too small to bedistinguished from th e numerical noise in the calculation(with the exception of C T 4 4 of PBA, which decreasessignificantly with temperatu re). As with the transversean d off-diagonal stiffness moduli, a decrease in the shea relastic moduli would be expected based on potential energyeffects alone because th e interchain interactions decreaseas the polymer chains move further a part.5. Yo ung 's Moduli . The Young's moduli Ei,which

are exactly equal to th e CTii ( i = 1,2,3) when all of theoff-diagonal stiffness mo duli CTij equal zero, have tem-perature dependences which are similar to those of theCTii; slight differences arise due to the temperaturedependences of the nonzero off-diagonalPij, The quan-t i ty& - CT33 is shown in Figure 7. Th e larger temperaturedependen ce of E3 - CT33 for PBA as compared to PPTAis due to the larger temperature dependence of andCT23 for PBA, as discussed above. Th e greater tempe raturedependen ce of the E3 for PBA as compared to PP TA canthus be at tribu ted to larger entropic effects in the diagonalC T 3 3 and the off-diagonal and CT23 elastic moduli;although the effects of therm al expansion on the pote ntialenergy contribution are the dominant causes of thetemperature dependence of E3 , the magnitudes of theseeffects are similar for PPTA and PBA.

V. Conc lus ionsCalculations of th e thermal expansion an d temperaturedependen ce of the elastic moduli of PPTA and PBA werecarried out from fiistprinciples, using molecular mechanicsforce fields for th e interatomic interactions an d th e quasi-harmonic approximation for the vibrational free energy.The temperature dependent properties reported herefollow entirely from self-consistent lattice dynamic cal-culations under these constraints, without any furtherassump tions. Th e results of our calculations for thetherm al expansion coefficients are in good agreem ent withexperiment. In both PPT A and PBA, thermal expansionis larger along he a axis (between hydrogen bondedsheets)than along the b axis (within th e hydrogen bonded sheets)an d is negative along the c axis (along th e polymer chain).The calculated decrease with temperature of the axialYoung's modulus (shown in Figure 4) is somewhat smallerthan tha t found from X-ray e~ pe r i rn en t s ;~ ~owever, theuncertainties in experimentally determined crystalliteelastic moduli for polymer crystals are well-known. Forexample, for polyethylene the axial elastic modulusdetermined by different experimental methods varies by50?6 ;38 also, Northolt has shown that even slight misalign-ment of crystallites along the fiber axis significantly

reduces t he observed elastic modulus.29~30The negative axial thermal expansion for PPTA andPBA is attributed primarily to elastic coupling to thethermal stresses transverse to the chain axis. Th e increasein entropy associated with chain contraction makes a m inorcontribution (approximately 25 74 ) to th e total negativeaxial thermal expansion.Th e decreases in the axial elastic moduli of PPTA andPBA w ith tem per ature are due primarily to effects arisingfrom the thermal expansion transverse to the chainaxis-this therm al expansion leads to both an increase inequilibrium volume, which decreases th e elastic modulus(the stiffness moduli are t he second derivatives of t he freeenergy with str ain pe r unit volume),and a decrease in thelocal curvature of th e potential energy surface as th e chainsmove apart; this decrease in th e second derivative of t hepotential energy is attributed to a m ore complete relaxationof th e phenyl ring rotation angle in response to the changesin the amide bond angles accompanying axial strain. Th eeffects of the direct entropic contribution (Le., secondderivative of th e entropy with respect to strain) on th etemp erature dependence of th e elastic modulus are smallerin magnitude.Th e calculations find th at th e negative axial thermalexpansion an d the decrease in the axial Young's moduluswith temperature are larger in magnitude for PBA th anfo r PPTA, in agreement with e ~ p e r i m e n t . ~ ~ ~ ~ ~he largeraxial thermal contraction for PBA as compared to PPTAis due to both a larger increase in entropy with chaincontraction a nd increased elastic coupling o the transversethermal stresses. Th e larger temperature dependence ofthe axial elastic modulus in PBA is due to the directentro pic contribution to th e modulus, since the effects oftherm al expansion on the elastic modulus are similar forth e tw o materials.Arguments may be proposed for the design of compositematerials based on th e results of the curre nt work. Fiber-reinforced composite materials are often composed of apolymer fiber such as PPTA embedded in an amorphouspolymer matrix. W ith changes in temp eratur e, stressesdevelop at the ends of the fiber because th e fiber undergoesaxial thermal contraction while the matrix undergoesthermal expansion;39 these stresses are believed to be amajor cause of material failurea40 T he pre sent results


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7204 Lacks and Rutledgepredict th at t he development of these stresses could bemitigated by matching the fiber and matrix m aterials suchtha t am, he ma trix therm al expansion coefficient, is lessthan the transuerse therm al expansion coefficients of thefiber. Under suc h conditions external stresses from th esurrounding matrix would counteract the transversetherm al stresses of th e fiber, leading to a reduction in th enegative axial thermal expansion of the fiber an d thereforea reduction in the stress generated a t the fiber ends. Also,the present results lead to the prediction th at a smalleram would reduce the transverse ther mal expansion of thefiber a nd improve the axial stiffness a t high temperatures.

Acknowledgment. Th e authors are grateful to Texaco,Inc. and to E. I. Du Pon t de Nemours for providing thefinancial supp ort for this work in th e form of the Texaco-Mangelsdorf Chair an d a Du Pon t Young Faculty Awardto G.C.R.References and Notes

Macromolecules, Vol.27,No. 24, 1994(13) Ca tti, M.; Pavese, A.; Price, G. D. Phys. C hem. Minerals 1993,19,472. Isaak,D. G.; Cohen, R. E.; Mehl, M. J.J. eophys. Res.1900,95, 7055.(14) Jackson, R. A.; Parker, S. C.; Tschaufeser, P. Modelling ofStructure and Reactivity in Zeolites; Catlow, C. R. A., Ed.;Academic Press: London, 1992; p 43.(15) Rubini, S.; Ballone, P. Phys. Rev. B 1993,48, 99.(16) Lacks, D. J.; Rutledge, G. C. J . Chem. Phys., in press.(17) Press, W. H.; Flannery, B. P.; Teukolsky, S.A.; Vet terling , W.T. Numerical Recipes: T he Art of Scientific Computing;Cambridge University Press: Cambridge, 1986.(18) Discover Softwa re, version2.3; Biosym Technologies: San Diego,

1993.(19) Rutledg e, G. C.; Sut er, U. W. Macromolecules 1991,24, 1921.(20) Mayo, S. L.; Olafson, B. D.; Goddard, W. A. J . Phys. Chem.(21) Karasawa, N.; Goddard, W. A. J. Phys. Chem. 1989,93,7320.(22) Northolt, M. G. Eur. Polym. J. 1974, 10 , 799.(23) Tadokoro, H. Structure of Crystalline Polymers; Joh n Wiley(24) Ii, T.;Tas hiro, K.; Kobayashi, M.; Tadok oro, H. Macromolecules(25) Rutledge, G. C.; Suter, U. W. Polymer 1991, 32, 2179.(26) Yang, X.; Hsu, S. L. Macromolecules 1991,24,6680.(27) Ii, T.; Tashiro, K.; Kobayashi, M.; Tadok oro, H.Macromolecules(28) Gayman s, R. G.; Tijssen, J.; Harka ma, S.;Bantjes, A. Polymer(29) Northolt, M. G. Polymer 1980,21, 1199.(30) Northolt, M. G.; Hout, R. v. d. Polymer 1985,26, 310.(31) Kim, P. K.; Chang, C.; Hsu, S. L. Polymer 1986,27, 34.(32) Choy, C. L.; Nakafuku, C. J . Polym. Sci.: Polym. Phys. Ed.(33) Kobayashi, Y.; Keller, A. Polymer 1970, 11, 114.(34) Davis, G. T.; Eby, R. K.; Colson, J. P. J.Appl. Phys. 1970,41,4316.(35) Choy, C. L. Developments i n Oriented Polymers;Ward, I. M.,Ed.; Applied Science Publishers: London, 1982; Vol. 1,p 121.(36) Munn, R. W. J. Phys. C 1972,5, 535.(37) Tashiro, K.; Kobayashi, M.; Tadokoro, H.Macromolecules1977,10, 413.(38) Nakamea, K.; Nishino, T.; Ohkubo, H. J.Macromol. Sci.-Phys.1991, 830, 1. Holliday, L.; White, J. W. Pure Appl. Chem.1971, 26, 545.(39) See, for example: Fan, C. F.;Hsu, S.L. Macromolecules 1989,22, 1474.(40) See, for example: Termo nia, Y. J.Polym. Sci.: Polym. Phys.

E d . 1994, 32, 969.

1990,94, 8897.

& Sons: New York, 1979.1986,19, 1772.

1987, 20, 347.1976, 17, 517.

1988, 26, 921.

Tashiro, K. Prog. Polym. S ci. 1993,18, 377.Weiner, J. H. Statistical Mechanics of Elasticity; John Wiley& Sons: New York, 1983.Leibfried, G.; Ludwig, W. Solid State Physics, Seitz, F.,Turnb ull, D., Ed.; Academic Press: New York, 1961; Vol. 12 ,p 275.Tashiro, K.; Kobayashi, M.; Tadokoro, H. Polym. J. 1992,24,899.Ii, T.; Tashiro, K.; Kobayashi, M .; Tadokoro, H.Macromolecules1987, 20, 552.Karasawa, N.; Dasgupta, S.; Godd ard, W. A. J . Phys. Chem.1991,95, 2260.Chen, F. C.; Choy, C. L.; Young, K. J . Polym. Sci.: Poly m.Phys. Ed. 198 0,18 , 2313. Chen, F. C.; Choy, C. L.; Wong, S.P.; Young, K. J . Polym. Sci.: Polym. Phys. Ed . 1981,19, 971.Baughman, R. H. J . Chem. Phys. 1973,58, 2976.Lacks, D. J.; Rutledge, G. C. J. Phys. Chem. 1994, 98, 1222.Lacks, D. J.; Rutledge, G. C. Chem. Eng. Sci. 1994, 49 (17),2881.Born, M.; H uang, K. Dynam ical The ory of Crystal Lat tices;Oxford University Press: Oxford, 1954. Venkataraman, G.;Feldkamp, L. A,; Sahni, V. C. Dynamics of Perfect Crystals;MIT Press: Cambridge, 1975.Boyer, L. L. Phys. Rev. B 1981,23,3673. Hemley, R.J.;Gordon,R. G. J . Geophys. 'Res. 1985,90, 7803.

Thermal Expansion and Temp Dependece of e Moduli Orf Aromatic PA

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