Koreano r. Chem. Eng., 18(1), 46-53 (2001)

Microstructural Lattice Simulation and Transient Rheological Behavior of a Flow-aligning Liquid Crystalline Polymer under Low Shear Rates

Hansol Cho*, Mingzhe Xu, Sang Ouk Kim, Kwang Man Kim ** and In Jae Chung*

Depamnent of Chemical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Kusong, Yusong, Taejon 305-701, Korea

(Received 24 April 2000 �9 accepted I6 November 2000)

Abstract-A microstracmral lattice simulation for textured liquid crystalline polymer is carried out to predict rheo- logical behavior, especially the stress evolution after shear inception. It is based on a combination of two main con- cepts: (i) the director in each cell of a supramolecular lattice has an orientation described by the minimization of total energy of director map, and (ii) the torque balance of each director under shear flow and mlisotropic relaxational shear moduli depends on the averaged orientation of the director map. By considering the interaction between the nearest- neighbor directors, the spatial orientational con-elation is introduced mid the spatial heterogeneity, i.e., a polydomain texture, is generated simultaneously. For the start-up shear flow, the overshoot mid the steady value of shear stress in- crease and the former shins toward a shorter time as the applied shear rate increases. Also, the calculated stress evolution is compared with the experimental result of a thermotropic liquid crystalline poly(ester-imide).

Key words : Lattice Simulation, Polydomam Structure, Therlnotropic Liquid Crystalline Polymer (TLCP)

INTRODUCTION

Liquid crystalline polymers (LCP), lyotropic and/or themaotro- pic, with rigid mesogenic lmit and flexible spacers in the main chain arouse interest academically and industlially. They show a m~nber of ~uausual rheological phenomena such as long relaxation time [Wissbilm, 1980], negative first normal stress diffea-ence, damped oscillatory transient stress in the case of inception of shear flow [Guskey and Wintel; 1991; Kiln and HaIL 1993; Winter and Wed- lel; 1993; Baek et al., 1994; Han and Chang, 1994]. These phe- nomeua may be explained by applying various factors: the possi- bility of director t~nbling/wagging in a negative fast nomml stress, file textured or polydomain structtae by file inhomogeneity of di- rector field and so forth. Most themlotropic liquid crystalline poly- mer (TLCP) very often shows thennal hystea-esis [Done and Baird, 1990; Driscoll et al., 1994] in which a theological property has two different values at a certain nematic temperature on heating and cooling. It is explained by the supercooling effect. They also show the change of a theological property with time at a fixed tempera- ture, because residual crystallites accelerate the fonnation of high- temperat~e crystallites [Lin and Winter, 1989]. The negative value of first nomlal stress difference NI, first reported by Kiss and Porter [1978] for lyotropic liquid crystalline polymer (LLCP) solutions, was predicted through the exact n~nefical calculation of Doi's dif- fusion equation without deconplmg approximation by Larson [1990]. The existence of tl~nbling for TLCP is, however, conIro- versial.

One assertion is associated with tumbling/wagging of directors as in LLCP Another assertion is related to no tumbling but a tech- nical problem involved in the measua-ement of first nom~al stress, that is, the method of how to set up the base line in the rheometer

tTo whom coiiespondence should be addressed. E-mail: chualg@cais .kaist. ac .kr

46

[Langelaan and Gotsis, 1996]. According to Baek et al. [1994], the high viscosity caused by polymer-polymer dense packing and fiic- tion may be responsible for the disappearance of negative N~ val- ue. VVllen a TLCP sample is loaded in nematic state for rheological measuremenh normal stress NI is not completely relaxed and shows a residual normal stress, which is difficult to eliminate by heating the sample above the isotropization temperature because the TLCP may be degraded before reactmlg file iso~opic state. When the ura-e- laxed state is set up as a base line, a negative N~ may be observed [Kiln and Han, 1993a, 1994; Langelaan and Gotsis, 1996]. A liq- uid crystalline polymer shows three regions of a viscosity curve, Region I, Region II and Region UI in the shear rate [Asada et al., 1980]. In paticul~; it is known fl~at die shear fltinning in a low shear rate region (Region I) is ascribed to the presence of polydomain structure [Kim and Chung, 1997 ]. For the transient evolution of stress under shear flow, initial polydomain structure significantly affects the bansient rheological behavior of TLCP

The rheological properties ofLCP ~ce closely related to its sb~ac- ture fommtion on a micron scale as well as its pattern responding to the imposed flow condition, especially in the low shear rate re- gion (Region I of Onogi-Asada's three region viscosity curve). How- ever, the difficulty in rheology of LCP at a low shear rate arises for the qt~alitative as well as ~mntitative prediction of its dimensional inhonaogeneity. Recently, a simple lattice model has been suggested, which is lmsed on the heterogeneity in the director field [Bedford et al., 1991; Picken et al., 1992; Gervat et al., 1995]. Simulations of director dynmnics were also carried out on the micros~-uctural scale [I-Ian and Rey, 1993; Dlng and Yang, 1994]. Advantage of these models is that they are not necessarily of a delicate mathe- matical formulation in the complex and hierarchical smlc0are of LCP orientation.

In this paper, the micTostmctural simulation of director evolution is based on the Frank flee energy for the director distoi~ion and on the Leslie-Encksen hydrodynamic theory. It is carried out to inves-

LaNce Simulation and Tlansient Rheology of TLCP

tigate the texture evolution and the rheological chauge of a TLCP after the start-up of she~ flow. Model parameters are of typical val- ues of them~m'opic liquid ClTstalline polyrnea~ and the moduli are of poly(ester-imide).

M I C R O S T R U C T U R A L S I M U L A T I O N

1. Derivat ion of Const i tut ive Equat ion

In the Frank elasticity theory, the total energy fox" the spatial dis- tortion of liquid crystal molecules can be given as

&=~K~(divn)~ 1 2 1 +~K2(n-cm-1 n) +~K3(n• n) = (1)

where K~, K2 and K3 ~:e the elastic constants for splay, twist and bend modes, respectively, and n denotes a director vector at a local position. For TLCPs, the magnitude of elastic constants is in the following order: K~>--K3>>K2 and the smallest twist mode (out-of plane) is excluded automatically for a two-dimensional consider- ation. Typical values of K~ and K3 for TLCPs are 10-'-10-~2 N [Lee and Meyer, 1988], which are about two orders larger than those for small molecule liquid crystals.

For a discrete two-dimensional lattice model without shear im- position [Picken et al., 1992], the approximate energy of the central cell is expressed as

4 1 (ao,) ..... = ~ ( K , sin~Ok +K3 cos~,)~)sin~(~-0)

4

=~K(r sxn 20, +(1 r)cos~O~)sxn ~(O)k 0) (2) k

The angles 0 and d? k are defined in Fig. 1. a3 k is identified with d?, for the right and left lattices and r~2-1O, I for the upper and lower. Various energy ft~ctions for the discrete lattice model have been derived in the fomls of E sin:(**- 0), E(~)~-0) ~ [Bedford et al., 15;91] or ~ ( 2 - 2 c o s ( 0 k ~ ) ) [Kimum~d Gray, 1993]. The en- ergy ft~mllon fox- the interaction between the adjacent cells takes

Fig. 1. Schematic diagram of array of lattice cells. Polydomain structure of fiquid crystalline polymers tin'ough an optical microscope is modeled as directors in lattice cells. DeFini- tion of two-dimensional lattice for the microstructural simulations is shown and its dynamics is determined by the four adjacent director orientations and hydrodynamic torque exel~ed by shear flow. Dh'ectol~ in fact have no di- rection and therefore are in the range from - 9 0 ~ to 900 .

47

the form of sin2(~)k-0) in this work Although the energy function affects the fmal morphology [Assender and Windle, 1994], the above equations have a similarity in energy mitmnization procedure and slightly affect the mo@lological behavior under shear flow. The polydomain slmcture is observed by the simulation ficom the initial director distribution given arbitrarily. This texture formation corresponds to the real simataon where the nematic texture is gen- epated by cooling the isotropic melts of TLCPs below isotropiza- tion tempepature. Spontaneous texture formation by energy mini- mization is thought to be successfiJl in a sense that the order Far - a-neter value, S-0.5(3(cos20)-1), of about 0.3-0.4, coincides well with an experimental observation [Hongladarom and Bttcghardt, 1994].

In this work, director dynamics is obtained fiorn the torque bal- ance in a two-dimensional lattice model. The lance model [Picken et al., 1992] is combined with the orientation change of dkectox~ [Semenov, 1993] with time. From the theory of hydrcxlynmnics, the torque exerted on the central cell is approximated by

O(t+At)-O(t)At ~'M 2~(1 Lcos2O) (3)

where

4

M = - ~ ( r sin2~3, +(1 2 " -r)cos O,)sxn(2((~,-0)), (4)

2 " for i :2 ,4

1 o:/% Y1

K 7, c%-c6, % c5+%, K' - -

a ~

Here, 0% and 0% are Leslie viscosities and a is the characteristic length of a lattice cell size. K is the average Frank elastic constant defined as K~ 2Kr and K3 2K(1-r) where r is a sliding factor be- tween bend and splay (r 0.5 is fixed in tim simulation). Hence r sinq3~+(1 -r)cos29k 1. The values of a2 and % are rarely reported for TLCPs, so the values used for the simulation are -415--170 of o% and - 4 Pa- s of 0% obtained for semiflexible thmmob-opic liquid crystalline polyester obtained fi-om 4,4'-dioxy-2,2'-dimethyl azoxy- benzenedodecanediyl [Donald and Windle, 1992]. The lattice size a is somewhat ambiguous. However, its possible minimttal value is used, which takes the value of 1 (y7 m that corresponds to the con- tour length of aromatic polyester with degree of polymerization ca. 150. This means the minmlttal length scale cannot be smaller than the molecular level. The values of a, k and 71 determine the rate of convergence in the simulation as discussed by Picken et al. [1992].

Because the lattice size a is 10 -7 m, we have to take a 500x500 lattice to match the rheometer gap spacing of 50 l-ma (the domain size observed in experiment is order of 1-10 Jim). However, the comparison between the 50x50 and 500• lance cases shows little difference in texttCe or stress after shear inceptmrt This is due to the fact that the motion of directors does not direr for the two cases and the stress function (sm2Ocos2O) is calculated on an aver- age for all cells under shear flow. Thus, we have used the 50x50 lattice for all simulations. Perfect anchoring (0=0) boundary con- ditions are set for the upper and lower plates, and periodic bound- ary conditions are used at the right and left edges of the lattice. The

Korean J. Chem. Eng.(Vol. 18, No. 1)

48 H. Cho et al.

time interval of an iteration is set to 0.05 s. eralized and expressed as

STRESS EXPRESSION IN TERMS OF D I R E C T O R ORIENTATION

At this time, we would like to consider the Leslie-Ericksen theory for the two-dimensional case under simple shear flow. This equa- tion is usually adopted as a conslitutive equation for the small mol- ecule liqttid crystal [Semenov, 1993]:

[c~ +cq +% ~ c~ +(cq +~JsinSOcosS0[ (5) a---%.=[~ 7, ' . ~',.' l

where o;; 's are Leslie viscosities [de Gennes and Prost, 1992]. How- eve:; many molecular parametei~ (Cq's) are used inconveniently and the relaxation of director orientgtion is not calculated like a small molecule liquid crystal in this equaliort Thus, Eq. (5) is not thought to be adequate for the stress prediction of ova- TLCP. A direct ap- plication of Leslie-Ericksen theory to LCP rheology [Viola and B~-d, 1986] has been shown to be insufficient for a quantitative prediction as well as for a qualitative one of transient behavio::

To obtain a proper stress expression using the director orienta- lion resulting fiom the above simulation, we choose the theory of transverse isolropic liquid crystals developed by Larson and Mead [1989], which was derived for the theory of lyotropic LOPs. lu this theory, the final stress expression in the monodomaiu is based on the director orientation only. We assume that the flow behavior of rodlike molecules m the lyotropic system has some similarity and is applicable to that ha the themaotropic system. The stress expres- sion is given on the basis of linear viscoelasticity:

o,: =f_ G,~,(t t')y~(t')dt' (6)

An anisotropic relaxalional mtxlulus G,:~(t-t') has three distinct terms:

=3ck~T 1 ~ ( 8 ~ 8 ~ 11,i1~8~ t t'

{ t t"~ +(1 +2S)(1 S)(A S)nj]n~a ' e x p [ - ' - ~ 1 ) 2

+ i (1 S) ( t t"l] 2 ~':v:~n; exp', --~-~ ) f

c is the nmnber concentration of rods per unit vohune, k~ is the Boltzmar~ coustm~k and T is absok~te temperature. A and B are parameters depending on the scalar order parameter S and Maier- Saupe mean field potential U, respectively. There are two relax- ation times )~ and ~ that can be expressed in terms of S, U, and rotational diffusivity (D,.) of the rod molecules. Simplified to a two- dimensional case with n~=cos0 and ny=sm0, the shear component of relaxational moduli reduces to the form:

" ~ ' I Y ' sO s0" - cL~=j_~ ,isln cos ? exp (t')dt' (8)

(A} means the average over all the lattice cells. In the theoretical calculatior~ an expansion with multiple time comtants has been fie- quently perfomaed when the polydispei~ity effect is considered [Chow and Fuller, 1985; La:~on and Mead, 1991]. In spite of the fact that Eq. (8) is derived from the theory of lyolropic LCPs, we adopt the equation as a constitutive equation because it is similar to the generalized Maxwell-type integral stress equation in which the constants ~ will be determined from the experime:gal result. Sim- ilarly, first normal stress difference, N: is presented as follows:

N:=o~ % : ~ = ~(cos 0sin0 sin39cos0)exp (t')dt' -~1= i

(9)

EXPERIMENTS

1. Materials Methyl hydroquinone (MeHQ) was acetylated by meaas of ace-

tic anhydiide in boiling toluene with a catalytic amount of pyri- dine. The c0-amino acid and trimellitic anhydride were placed ha dried dm:ethylfonnamide with toluene, and fftis mixttwe was re- fluxed for 4 In in a flask. A mixture of N-(6-carboxyalkylene) tri- mellitic imides and hydroquinone diacetate with magnesium oxide was weighted in a round-bottomed cylm&ical glass reacted- equipped with a stainless steel sNre1: The reactants were condensed at 220- 270 ~ for about 7 In under the nitrogen atmosphere, and the~l under vacuan: for 1 In- at 270 ~ The poly(ester-imide) was synthesized and used for the test. Its scheruatic chemical structure is shown in Fig. 2. 2. Rheological Experiments

Polymer powder was dried ha a vacuum oven at 80 ~ for about 24 h to remove any residual solvent and moisture, and disk type samples (approximately 2 lllnl thicklaess and 25 lllnl diameter) were prepared and kept above the melting poiut of poly(ester-imide) at the pressure of 10 MPa for 5 mm in a hot press. This poly(ester- hnide) undergoes only two thermal trausitious: (i) glass trausition at ca. 68 ~ a id (ii) nematic-isotropic transition at ca. 179 ~

An ARES with cone and plate frxttu-e was used to measure (1) dynamic modulJ (G' and G") as fimction of the angaflar fi-equency, and (2) tile shear stress grouYth and first normal sh-ess difference growth after shear inception as a fimction of time for various shear rates. A cone angle of 0.1 md was used; it had a diameter of 25 mm. All gap sizes were fixed at 50 l-tm. All experiments were carried out under nitrogen atmosphere in order to preclude the oxidative

exp t - t ' - , + - , _ - - - ~ - G~m(t t ) G~y~(t t) - ~ sinS0cos~0 (7)

To apply the theory of lyotropes to themaotropes, the l:arametelz F~ ~ in Eq. (7), are inh-oduced and determined by the experiment, which includes all the parametric values (U, S, c, kB, T, and D,.) in the theory of La-son and Mead [1989]. The shear sla-ess may be gen-

Fig. 2. Chemical structm'e of thermotropic liquid ciTstalline poly (ester-imide) used in this work.

January, 2001

Lattice Simulation and Tlansient Rheology of TLCP 49

Fig. 3. Frequency sweep measurement of poly(es~er-hnide) with g = 1 0 % at 140 ~ The symbols are measured values of G' and G". The solid line and dot line are calculated Maxwell elements from non-linear least square f'flting.

degradation of samples. Temperature ccim-ol was satisfactory within • ~ In a frequency sweep, the extent of strain amplitude was well withhl the linear viscoelastic range of materials investigated. After the sample was loaded at 190 ~ for 10 mm, which is well above the isolropization temperature of poly(ester-imide), it was cooled down slowly to a predetermined temperatta-e, and held for 10 min to eqtfilibiate thenuodynamically prici to measurements

R E S U L T S A N D D I S C U S S I O N

1. h ~ e q u e n c y Sweep M e a s u r e m e n t

From data in Fig. 3, we could obtain the Maxwell elements ofre- laxational mcduli of our poly(ester-imide) by non-linear least square methcd [Bird et al., 1987; Kanlath and Mackley, 1989]. The cal- culated moduli values are listed in Table 1. The symbols are exper- hnental values; the solid line and dotted line are calculated values.

Themlal hysteresis of a viscoelastic proFeity is found for vari- ous TLCPs, such as p-hydroxybenzoic acid (PHB)/poly(ethylene

Table 1. Calculated pre-exponenlial relaxation moduli. Nine t ime constants in a range from and are selected and flleir cor- responding Mauxwcell elements are obtained

i Time constant 5~, (s) Modulus F, (Pa)

1 0.00112 8400.0 2 0.00316 8240.6 3 0.01212 8303.1 4 0.04642 1623.5 5 0.1778 550.38 6 0.6813 450.20 7 2.610 370.533 8 6.31 300.35 9 10.00 260.00

Fig. 4. Stal~-up shear sQ'ess measurement of poly(ester-imide) at 140 ~ with various shear rates.

terephthalate) (PET) and PHB/HNA copolyester [Done and Bah-z[ 1990; Driscoll et al., 1994]. It may be due to the supercooling ef- fect. hi this work, poly(ester-imide) is thought to be very adequate for the test to ex~wnine the only effect of texttue on the rheological properties of TLCPs, because the hysteresis and the formation of high temperatta-e crystallization do not exist in this sample [Kin1 et al., 2000]. This ~ique rheological behavior has already been ex- plained in tenus of the monomeric sequence disllibution compared with homopolymeis by Kim et al. [1997a]. 2. Start-up Shear F l o w M e a s u r e m e n t

Fig. 4 shows the stress growth after the staitup of shear flow. It should be mentioned that a flesh sample was employed for each shear rate. Upon staimp of shear flow, the shear stress, cy(t, 7) goes through a maximum in a veIs, short time. As the shear rate m- creases, the peak becomes higher and tends to shift towards a shorter time, although the extent of shift is seen to be rather smaU and the ratio of the maximum stress to the equiliblJtan stress ~(t, 7)/~(t, 7), is found to lie bet-vveen 6.43 and 6.74. Such a large value of~(t, 7)/ ~(t, 7) ratio is believed to be caused by the abrttpt break-up of the polydomain structure at an early stage.

The evolution of the fn~t normal stress difference N,(t, 7) with time is shown in Fig. 5 for the different shear rates. Upon stai~up of shear flow Nl(t, 3t) goes through a maxirnum, which becomes greater with increasiug 7- At a high shear rate, N,(t, 7) goes down to a negative value at a very early stage of transient shear flow, but soon goes up to a positive value and reaches a positive of steady- state. The ratio of the maximum to the eqttilibriurn value of first nonual stress diffe~-ence, Nl.,,,~(t, 7)Ml( ~176 7) lies between 5.44 and 5.55, depending on the applied shear rate y. Such large values of Nl,,,~(t, T)/NI(% T) ratio are believed to be characteristics of TLCPs m general and are attributable to the polydomaitl% which exist in the nematic state of a poly(ester-imide) before the application of sudden shear flow [Khu and Hail, 1993].

In lyotropes, the most reliable signature of tumbling is the pre- sence of negative normal stresses in s t e @ shear and stress osciUa- tions in transient shear flows. It has been reported that the poly-

Kol~an J. Chem. Eng.(Vol. 18, No. 1)

50 H. Cho et al.

Fig. 5. Start-up first normal sh'ess difference measurement of poly(ester-hnhle) at 140 ~ with various shear rates.

mer-polymer dense packing and fiiction might obsanct the tum- bling of rod-like molecules and be responsible for the disappear- ance of negative N~ values in TLCPs [Baek et al., 1994]. Measure- ments in li-ansient shear flows show no sb-ong indication of the os- cillatory responses of tumbling. Recently, using the same sort of ma- terial we found only posilive values of N l in steady shear measure- ment.

A comparison of Fig. 4 with Fig. 5 shows that N~(t, 7) has the maximt~m at a linger value of Nne and takes a longer period to reach a steady state than o(t, ~/). 3. Mierostruetural S imulat ion

As reported by many researdlei~, TLCP in the nematic state has a supra-molecular stmctt~-e [Larson and Doi, 1991 ; Knn et al., 1994; Han and Rey, 1995; Langelaan and Gotsis, 1996]. Such a supra- molecular structure is often referred to as a polydomain, which may be regarded as aggregates consisNlg of many small-sized domains (monodomams). It can be postulated that when a sudden shear flow is applied to a TLCP ha the ne~natic region, the domains will be bro- ken up and the extent of destruction depends on the intensity of shear rate. When the flow is stopped, the broken domain texture will start to reorganize and coalesce to form polydcmains, probably regain- ing the original domain texture if a sufficiently long tnne is al- lowed to elapse.

In this microsb-ucttual snnulation, polydomain sb~claz-e in a qui- escent state is developed fi-om the random configuration of initial director field Textures appear [Fig. 6(a)] between the aligned aggre- gates of directors or domains, although every lattice cell is under an equivalent dynamics. Fig. 6 shows the director evolution after shear inception for shear rate 1 s-'.

After shear inception, directors are in motion govemed mainly by viscous torque and closely aligned to the flow direction with a Leslie angle OL =tan-' ~/-~-~ [Fig. 6(b)-(d)]. In this work, the total energy of the system of LCP and relaxation mo&di decrease as di- rectoi's align in the shear direction, whereas the oiientation parame- ter is increased. This result is contrary to the result of the solid mod- el such as a calculation of an anisotropic aggregate model based on

January, 2001

Fig. 6. Evolution of dh'ector fiehl by means of the microstructural simulation. (a) Textm-ed dil-ector map tt~-ough a Frank enelgy minimize- lion. This director map is taken as an milial textme of s~rt-up shear flow. Co) Textures afka- 1 s under a shear flow wifll 3 ~ 1 s ~. (c) Tex~-e aftca- 2 s. (d) Textm-e after 25 s.

the lattice cell perpendicular to the shearing plane in that the shear mcdulus increases as the directors align in the shear or elongation direction [Gervat et al., 1995]. It has to be noted that the director onentalion out of the shear plane l~oduces no shear stress [Larson and Mead, 1989]. The fact that the post-sheared (aligned) TLCP shows lower mcduli than the pre-sheared (tmaligned) one has been known by the experimental results fiom the study about the effect of shear history on the small-oscillatory measurement [Dfiscoll et al., 1994]. Tim microstmctta-al shnulation explains a hydrodynam- ics of director in connection with polydomam texture in quiescent state and its break-up by shear imposition in a sheafing plane.

It is noted that we take into account heterogeneity of TLCPs in this work. It ks well known that there are inherent defects by the chain end In a&tition, phase sepmation between high molecular weight LC phase and rather isotropic low molecular weight LC phase is known to cause a high density of defects in TLCP [Nakai et al., 1994]. The effects of these inherent defects are very impor- tant for the transient rheological properties. For a lattice sinmlation, the defects have been modeled by means of fLxing some lattice ori- entation [Gervat et al., 1995].

Fig. 7 shows the calculated shear slress for a start-up shear flow t~om the simulation of Eq. (8). The values of 1~, in the eqt~ation are obtained by frequency sweep measurement: For the case of a sin- gle time constant, the Maxwell component F, in Table 1 can be re- garded as identical to the function of ~sin20cos20 for the initial di- rector ofientatioias. The total stress of TLCP can be calculated by using the pre-expanential part of moduli in Eq. (8) with the aver- age director oiientation at each Nne step. The ratio of the maxi- mum to the equilibfiu~n value of shear sa-ess, o(t, ~/)/o~(t, ~/), lies between 1.2 and 2.95, depending on the applied shear rate 7. First nomaal stress difference is also calculated fi-om Eq. (9) and is de-

Lattice Simulation and Tlansient Rheology of TLCP 51

Fig. 7. Transient shear stress upon a stal~-up shear flow h'om the microstructural simulation.

Fig. 8. Transient fi,~A normal stress dkfference upon a start-up of shear flow from the microstructural simulation.

picted in Fig. 8. The ratio of the maximtrn to the equilibfitan value of first nomaal stress difference, Nl. ,,~(t. 7)/N1(% 7) lies between 1.36 and 1.95, depending on the applied shear rate 7. It also shows a single overshoot. In both of the calculated results, the faster ap- pearance of ma,,amum overshoot position with increasing the shear rate seems to be responsible for the faster decrease of pre-exponen- tial moduli by the faster loosing of the polydomain siaaacture.

Comparing expenment and simulation results (Fig. 4, 5, 7, 8), after startup of shear flow, both shear stress and first normal stress difference go ttu-ough a maximum which is believed to be caused by the aN-apt break-up of the polydornain slmcture at an early stage. As the shear rate incaeases, the peak becomes higher and tends to shift towards a shorter time.

In simulation and expenment results there are differences in the maxinmm magnitude of ovei'shoot. Tttis is attributed to a distor-

Fig. 9. Calculated steady shear viscosity fi'om the microstructural simulation.

tional elasticity associated with the defonnation of disclinations in the polydomain sa-nple and defect der~sity that is ascribed to fer some directors to represent a more real situatiotl The fn~t nomml stress difference in experiment shows a negative value, which may be caused by sample relaxed incompletely, at a very early stage of transient shear flow, but it does not in simulation.

Fig. 9 shows the calculated steady state shear viscosity as a func- tion of shear rate. It shows shear t iming in the low shear rate lange and a Newtonian plateau in the large shear rate region (Region I and Region ]I of Onogi-Asada's three-region viscosity curve). Such behavior is very important because it is a cha-actetistic of TLCP, and t t~ result is weU consistent with experimental results [Chang and Han, 1997]. Shear thinning in the low shear rate region is be- lieved to be catsed by the abrupt br~k-up of the polydolnain stme- rare at an early stage.

C O N C L U S I O N S

The theological behavior ofa TLCP in the nemahc phase is very unusual and shows quite different phenomena such as thermal hy- steresis and high temperature crystallizahon. Theirnal hysteresis is caused by the supercooling effect, and high temperature crystalli- zation is caused by the existence of residual crystallites. Herein a new TLCP, poly(ester-imide), was synthesized to remove these t~l- usual phenomena.

Model prediction by means of microslructural simulation shows a decrease in relaxation modulus of TLCP because the polydomain texture disappem's as the &-ectoi's align in the shear direction. Dy- namic moduli obtained ficom the experiment are applied to the cal- culation of Imrameters in Eq. (8). In this manner the texture of TLCP is modeled fiom the proper paanete:~; for example, Frank elastic constants and the Maxwell type moduli obtained experimentally are used to predict :(t, y) and Nl(t, y) for the start-up shear flow. The results of simulation are in agreement with the e>cpei~nental results for TLCPs qtalitatively. For a start-up shear flow the de- shaaction of polydomain texture by flow and its resultant decrease

Korean J. Chem. Eng.(Vol. 18, No. 1)

52 H. Cho et al.

of shear moduli, which depend on the average orientation of di- rectors, produce a large stress overshoot_ Also, we could predict shea" thinning m a low shear rote region (Region I), which is as- cTibed to the break-up of polydomain structure. It is possible to pre- dict the flow behavior of a thermolropic polymer by nsmg the lat- tice model and the rheological equations extending from the theo- ties for lyolropic liquid crystalline polymers.

A C K N O W L E D G M E N T

The authors thank Korea Science & Engineering Foundaticn for the partial financial support though the grants-in-aid (KOSEF 94- 0300-04-01-3).

N O M E N C L A T U R E

151

(Zj

C

G' G "

K K, K2 kB M

!1

r

S T U

(3

N,

G ,j~

x, 3' co

0 OL

: a characteristic length of a lattice cell size : Leslie viscosities : the total energy for the spatial distortion of liquid crystal

molecules : a number concentration of rods per unit volume : storage modulus : loss modulus : the average Frank elastic constant : splay elastic constant : bend elastic cor~stant : Boltzmann constant : torque for the spatial distortion of liquid crystal mole-

cule8 : denotes a director vector at a local position : sliding factor : scalar order parameter : absolute temperattu-e : Maier-Saupe mean field potential : angle of neighboring directors : shear stress : steady shear stress : first normal stress difference : maxu-num first normal stress difference for starmp test : relaxafional modulus : relaxation times : shear rate : parameters which include the all parametric values (U,

S, c, kB, T, and D,.) in the theory of Larson and Mead : angle of director : Leslie angle

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