2368
Ind. Eng. Chem. Res. 1994,33,2368-2373
Uniaxial Extension of a Lyotropic Liquid Crystalline Polymer Solution Yong W. Ooi and T. Sridhar' Department of Chemical Engineering, Monash University, Melbourne, Australia
Uniaxial extensional stress growth of a lyotropic liquid crystalline polymer solution consisting of 40 wt 5% (hydroxypropy1)cellulose in glacial acetic acid was measured using a filament stretching technique. The Doi molecular theory and a modified Leslie-Ericksen continuum theory were used to predict the experimental results. While the Doi theory agreed with both steady shear and steady extensional data, it failed to predict the transient stress growth data. On the other hand, the modified Leslie-Ericksen theory gave a better qualitative prediction of the transient curve, although it predicted that both steady shear viscosity and steady extensional viscosity were constants. The steady state data agrees with the results obtained by Metzner et al., who used a fiber-spinning technique.
Introduction Liquid crystal polymers are well-known for their attractive mechanical properties and unique rheological characteristics. Many theories have attempted to describe the behavior of liquid crystal polymers. The Doi molecular theory for monodomain liquid crystalline polymer solutions (Doi, 1980,1981)and the Leslie-Ericksen continuum theory (cf. Leslie, 1979) appear to be the most developed theories for describing the rheology of liquid crystalline polymers. While the Doi theory is able to predict the steady deformation behavior of the LCP solutions, it gives only a limited description of the transient behavior (Metzner and Prilutski, 1986; Larson and Mead, 1989; Wilson and Baird, 1992; Ooi and Sridhar, 1992). The development and applications of this theory to nematics are well described (Doi, 1980,1981;Metzner and Prilutski, 1986; Doraiswamy and Metzner, 1986; Ooi and Sridhar, 1992). The Doi molecular theory used the kinetic equation of Kirkwood and Auer along with a Maier-Saupe type potential. The model for concentrated liquid crystalline polymer solutions was then derived by generalizing the Doi-Edwards theory for dilute solutions (Doiand Edwards, 1978). The free energy associated with the anisotropic distribution of the polymer molecules was assumed to cause the stress in a deforming medium. Since the stress tensor is generally asymmetric in nematics, the Doi model simplifies considerably when the system possesses uniaxial symmetry. This leads to the following equations relating the orientation order tensor, Sa,,the tensors describing both the macroscopic flow field and the Brownian motion, Fa,and Gab as well as the stress tensor, cab:
Fus = -6Dr[ (1- :)Sa, - U( S,,S,, - %S,:) 3
+
0888-5885/94/2633-2368$04.50/0
The effective rotational diffusion constant averaged over orientation, D,, is related to the rotational diffusion constant in a dilute solution, Dfi, through the following equation:
where D is an adjustable constant used by Doraiswamy and Metzner (1986). c is the number of rods (with length L and diameter d ) per unit volume. K,, is the velocity gradient tensor and U, the dimensionless concentration, is expressed below:
U = v2cdL2
(6)
In the above equation, v2 is a numerical factor of an order of unity. In eqs 1-5, Supis the traceless symmetric order tensor which reduces to the following equation under equilibrium condition: 1 s,, = s( nung- -aa8) 3
(7)
Here, S is the order parameter, n, is the unit vector called "director", and is the unit tensor. In this work, equations presented by Metzner and Prilutski (1986) are closely followed and they will not be repeated in this paper. However, the adjustable parameters A and D are used, following the work by Doraiswamy and Metzner (1986) where A = 3 c k ~ T .The parameters in the Doi theory were obtained following the method detailed by Doraiswamy and Metzner (1986). U, the dimensionless concentration, can be calculated from the critical concentration at which the liquid crystalline phase emerges, c*, which then gives the order parameter at equilibrium condition, S . The ratio of parameter A to parameter D is determined by the zero shear viscosity of the solution. Both parameters A and D and the components of the orientation order tensor, Sap,are then obtained by curve-fitting the theoretical stress curve to the steady shear data. The parameters A and D obtained are used to predict the behavior of the solution under extensional flows. Doraiswamy and Metzner (1986) modified the Doi model to include the stress from rod-solvent friction and the 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2369 stress from solvent in the total stress term. This modification is significant only at high stretch rates. Since this work operated in the range of low stretch rates, the modification was not considered. The Doi equations discussed previously incorporated certain closure approximations which were discussed by Marrucci and Maffettone (1989)and Larson (1990). These investigators have found that the exact solutions of the Doi theory can predict negative values of the first normal stress difference. However, the prediction of the steady state behavior is not affected by the approximations. Magda et al. (1991) further showed that the exact solution of the Doi theory can predict the second normal stress difference of liquid crystalline polymer solutions. The Doi theory has been developed for untextured liquid crystalline polymers while the rheological behavior of liquid crystalline polymers is thought to be affected by their texture evolution. Larson and Doi (1991) attempted to overcome this deficiency by assuming that the LeslieEricksen continuum theory for nematics applies to microscopic regions that are small compared to the texture length scale and yet large enough to contain many molecules. Marrucci (1985)proposed that over some range of shear rate, the square of a characteristic length scale of the director inhomogeneities, a, is inversely proportional to the shear rate. Following the scale argument, Larson and Doi (1991) found that after start-up or cessation of shearing,
where L is inversely proportional to the square of "a",and a and p are phenomenological parameters. Equation 8 predicts that L will increase at the start-up of a flow and decrease when the flow stops. By spatially averaging the Leslie-Ericksen theory and combining the theory with eq 8, a complete set of equations was obtained (Larson and Doi, 1991):
+ a3)("6 - a51
(a2
" =a1 T-
(14)
- a31
In eq 12 uiis are constants given by Kuzuu and Doi (1984), C is c/c*, the ratio of the sample concentration to the critical concentration when the anisotropic phase emerges, and B* is the critical viscosity at c*. As shown by Larson et al. (19921, the characteristic texture length scale becomes constant after shearing for some time. Under such conditions, eq 11predicts that a t steady state the characteristic texture length depends on the deformation rate. For steady shear at a shear rate +, the following set of equations is obtained:
I = ai. q
=p
(17)
+ 2p1S*,2 + '"32 + p2 -(S + S,,) 2
1 - -(CY2 2
yy
+ as)(&+ 4)aSZy (18)
~ s , +, y s , , - 2xs,,2 - &aszy + 5h = 0 - easy,= 0 + j)hS,, 1
(1+ X)S,
- "S,,
(A - l)S,,
- 2 (sx x + - AS,
9
- &as,, = 0
-dS - X ( 1 + 2S)(1- S)k - 61s
dt
dt
uzz- urr =
dt
12
(11)
whereD,, and wap are symmetric and antisymmetric parts of the velocity gradient tensors. IIDis the second invariant of Dap and 1 is the rescaled L where 1 pL. 6 and 4 are parameters which are determined experimentally. ai's are the Leslie viscosities computed from the Doi molecular theory (Kuzuu and Doi, 1984), and p and pj's are the Ericksen viscosities:
(20) (21)
Following the same argument given by Metzner and Prilutski (19861, it is assumed that sa^ is not symmetrical for shear flows, especially at high shear rates. Note that eqs 19-21 indicate that Sa$s are not dependent on shear rate. Therefore, the shear viscosity, q, is not a function of shear rate. Under the assumption that Sapis symmetrical, eqs 9-11 are simplified to obtain the stress growth after start-up of uniaxial extension:
2 -dSa' - uraSpB+ S a p r a+ h[jDa, + Da,S,, +
dl = aII*1-
(19)
(23)
(311. + p2)i+ p2si + 1 2p1S2i- -(a2
+ CY,)(&+ 4)lS (24) 2 In the above equations, i: is the stretch rate of a flow field. At steady state, the following equations are obtained:
BE
1 = (31'2)ai
(25)
h(l + 2S)(1- S) = &(31/2)aS
(26)
31/2 = 3p + p2(1 + s)+ 2p1s2-$a2
+ (u3)(& + 4)aS
(27) Equation 27 predicts a constant extensional viscosity, BE. Experimental Technique
A lyotropic liquid crystal polymer solution, 40 wt % ' (hydroxypropy1)cellulose(HPC) in glacial acetic acid (AA),
2370 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 loo
loo
I
: o
i
Data from this work a t 19 t o 24 "C 0 Doraiswamy and Metzner (1966) Metzner and Prilutski (1986) Dol theory modified LeslieEricksen theory
/
h
i ]&$'
E
v
E w
10-1
3n
0;
6
=
.$
0.4 1,"s
/
/
0.0 0.5
lo-' 1.0 1.5 2.0 2.5 3.0 3.5 4.0
TIME
(s)
Figure 1. Variation of diameter and length of the filament during extension.
is used due to its well-definedrheological properties (Bheda et al., 1980;Metzner and Prilutski, 1986;Doraiswamy and Metzner, 1986). The HPC is Klucel L (Lot No. 41371, which was provided by Hercules Incorporated. This HPC has a typical M, of 100,000and a MJM,of 6. The glacial acetic acid was obtained from Rhone-Poulenc, with a density of about 1.05 g/mL at 20 "C and a viscosity of 0.0017 P a s at 19 "C. The sample was prepared at room temperature with very slow stirring using a helical stirrer. Both the bottle and the stirrer were carefully sealed to prevent evaporation. The critical concentration at which the anisotropic phase emerges is about 29 wt 96 (Metzner and Prilutski, 1986). The solution density is about 1.1 g/mL, and the surface tension of the solution was found to be about 30 mN/m at 20 "C. The filament stretching technique described by Tirtaatmadja and Sridhar (1993) was used to stretch the solution. The sample, held between two disks, was elongated as the disks moved apart. Measurements using a force transducer attached to one of the disks and a diameter measuring device were collected in real time by a computer. During extension the filament showed signs of necking and filament breakage which reduced the maximum length attainable. Comparing the stretching characteristics of the lyotropic liquid crystalline polymer solution to the polymer solutions used in the earlier work (Tirtaatmadja and Sridhar, 1993), we conclude that the stretching technique is not as accurate for these liquid crystalline solutions. In any case, the data which is presented here represent, in our view, a good approximation to the extensional behavior of these solutions. Figure 1shows the diameter and length of the filament at any time. Due to the necking characteristics and end effects, the stretch rate calculated from the filament diameter is much larger than that calculated from the
10-3
10-2
lo-'
100
10'
102
SHEAR RATE (l/s)
Figure 2. Steady shear stress of 40 wt % HPC in acetic acid.
filament length. In keeping with our earlier work, the stretch rate is calculated from the diameter measurements. The exponentially decreasing diameter at the midlength of the filament indicates a constant stretch rate.
Results and Discussion Steady shear data (Figure 2) was obtained from the Weissenberg Rheogoniometer R19 in the range of 0.00181.125 l / s and, they fell close to the results of Metzner et al. (Metzner and Prilutski, 1986;Doraiswamy and Metzner, 1986). Experiments were performed from two individual samples to confirm their reproducibility and at different temperatures, ranging from 19to 24 "C. The results show that the shear viscosity of the solution seems to be independent of temperature in this temperature range. A zero shear viscosity of 132 Pa s was obtained. The steady extensional results also agree with Doraiswamy and Metzner's data (1986)where a fiber-spinning technique was used (Figure 3). At low stretch rates, a constant extensional viscosity of 1000 Pa s was obtained. The filament stretching technique described above was conducted in both vertical and horizontal configurations, and both configurations produce the same outcome. The error in the axial stress is about 10%. It is worth remarking that this is, to our knowledge, the only published example of two different techniques yielding the same extensional viscosity. It occurs primarily because the solution exhibits a constant extensional viscosity and reaches steady state in a much shorter period than flexible macromolecules. In both Figures 2 and 3, theoretical predictions by the Doi and the modified LeslieEricksen theories are included for comparison. For this solution, Uis approximately 4.2 and under no flow condition S = 0.703. The zero shear viscosity of 132 Pa s was used to calculate the ratio AID. By fitting the theoretical curve to the steady state data
1
0
~
~
~
~
.
~
l , , , l l l l l l l , l l l l l l , .
~
~
V
/
10'
I
lo-'
'
'
'
' ' c , , '
10-1
10'
I
100
1'
"
' *
"
"
" "
'
"
" '
"
" "
'
" "
'
"
'
1
10'
STRAIN RATE ( l / s )
Figure 3. Steady extensional stress of 40 wt % HPC in acetic acid. Table 1. Leslie Coefficients and Other Constants for CY/@ of 1.38 a1 2825 a2 -3610 ag 109 a4 104 3308 a6 193 p 51.80 p i 236 p2 -90.9 0.9416
1o4
Table 2. Experimental Parameters Obtained from Data of Shearing Flow Measurements S,, 0.2372 &a 0.1577 Syy 0.5156 @a 0.0715 0.9065
in Figure 2, A and D of 1.8 X lo4 Pa and 0.45 l/s were obtained. These values of A and D were used to predict the behavior of the solution under extensional flows. For both steady shearing and extensional flows, the Doi theory gives a good prediction (Figures 2 and 3). The Leslie coefficients and other constants in the modified Leslie-Ericksen constitutive equations were calculated with a critical viscosity of 1000 P a s and c/c* of 1.38 (Table 1). Due to the larger number of unknowns in this theory, equations for steady shear (eqs 18-21) and steady extension (eqs 26 and 27) were solved simultaneously for Sxy,S,,, S,,, S, &a,and $a. A constant steady shear viscosity of 132 Pa s and a constant extensional viscosity of 1000 Pa s were used in the above calculations. The outcome is summarized in Table 2. Although the modified Leslie-Ericksen theory predicts constant shear and extensional viscosities, it gives a reasonably good prediction for our data in the deformation range investigated (Figures 2 and 3). The behavior of the solution at start-up of flow has also been investigated. The reproducibility of the stress growth of extension is shown in Figure 4. Comparing with the steady state data, the stress growth data clearly indicates more scattering. This is mainly due to the initial orientation of the liquid crystals before stretching and the inconsistent initial surface tension as a result of the slow
I i = 3.1 l/s
1o3 h
6
a
v
cn cn
Ecn 10' i
1
10'
0
1
2
3
4
5
TIME (s)
Figure6. Comparisonof the data from transient extension (symbol@ with the Doi theory (solid lines) and the modified Leslie-Ericksen theory (dotted lines) at three stretch rates.
relaxation of the sample after loading. The loaded sample cannot be exposed for too long before stretching to avoid
2372 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
1'
~
"
" "
~
w
loo
'
I
'
"
9
"
'
I
" "
I '
'
' I
'
"
(a) Doi Theory
1
IR IR
"
'4
n
Y4 u
rn
31
5 0.0 0.5 1.0 1.5 2.0 2 . 5 3.0 3.5 4.0
10-1
STRAIN
1
J
I ~ " ' I " " l " " 1 " " l " " l " ' ~ l " "
-
rn IR
--
w
rb
1
t
0
E rn
0.9
1.1 0 1.3 A
(b) Modified Leslie-Ericksen Theory
loo
0 1.6
3.1
I 10-21
0
"
'
'
"
1
"
"
2
"
"
3
"
'
4
STRAIN Figure 6. Normalized axial stress for 40 wt % HPC in acetic acid
at various stretch rates. The solid line is the prediction by the modified LeslieEricksen theory at a stretch rate of 1.0 U s .
evaporation of the acetic acid which will give a higher measured stress. Both the above theories were used to predict the transient behavior of the solution. In the Doi theory, the values of parameters A and D obtained from the shear data were used to predict the instantaneous axial stress. Figure 5 shows the predictions of the Doi theory for three stretch rates. The theory predicts that steady state is reached at a much earlier time compared to the results. The instantaneous axial stresses of the modified LeslieEricksen theory were predicted by eqs 22-24. To solve these equations, initial conditions are required. At equilibrium, 1 = 10 (eq 22) and S = 0 (eq 23). 10 is treated as a parameter, although it can be estimated directly from the measurement of the texture under deformation. Theoretically, the characteristic texture length scale immediately after start-up of extension should be very large so that 10 would be very close to 0. Larson et al. (1992)found that a should be less than 1in order to obtain the transient shear prediction. Comparison of the solution of eqs 22-24 with the experimental stress growth data allows the values of a and 10 to be determined. Note that since the steady state data have already determined the values of &aand 4a, G and 4 will be affected by a change in the value of a. This procedure yields a = 0.2 and 10 = 0.26. As a result, & and 4 are 0.7885 and 0.3575, respectively. Since the aim of this study has been to illustrate the application of the modified Leslie-Ericksen theory, a detailed study of the uniqueness of the parameters has not been carried out. The predictions by the modified Leslie-Ericksen theory are included in Figure 5. Considering that this theory was developed from a phenomenologicalequation derived from the shear deformation, the qualitative agreement with the transient results is very encouraging.
lo-'
1 ,I '
00
"
"
05
"
' I
" "
1.0
15
"
'
2.0
"
'
I
" "
2.5
30
"
'
3.5
"
,I
4.0
STRAIN Figure 7. Predictions of (a) the Doi theory and (b) the modified LeslieEricksen theory for the normalized transient stress growth.
In Figure 6, the data at various stretch rates was plotted as scaled stress (transient stress over steady state stress) against strain. The figure shows that the data seems to collapse into a master curve. Similar method of scaling has been found in other work for liquid crystal polymers or polymer solutions (e.g., Larson and Mead, 1989; Moldenaers and Mewis, 1990; Moldenaers et al., 1991; Wilson and Baird, 1992). Figure 7 illustrates the ability of both the above theories to predict the relationship of scaled stress and strain. While the theories do suggest an effect of stretch rate, the magnitude of the effect is small. Incorporation of texture development increases the time (strain) required to reach steady state. In Figure 6, the prediction of the modified Leslie-Ericksen theory for a nominal stretch rate of 1 l / s is shown to be capable of adequately describing the transients. The results presented here do indicate that the modified Leslie-Ericksen theory appears to afford a better description of the data than the Doi theory. However, some of these improvements must be due to the larger number of parameters in the modified Leslie-Ericksen theory. In addition, results obtained from steady shearing flow, steady extensional flow, and transient stress growth measurements are all required to completely determine the parameters. In contrast, the Doi theory uses parameters derived only from the steady shear data to predict the data of extensional flow measurements. The inability of the Doi theory to predict the transient behavior may also be due to the closure approximations used in the original theory. Conclusions
Experimental results obtained using a filament stretching technique have been used to test the theories of Doi
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2373 and Leslie-Ericksen. Although the steady extensional data was adequately predicted by these theories, the transient stress growth is better predicted by the modified LeslieEricksen theory which properly describes the growth of texture in these solutions. However, in view of the larger number of parameters in the modified Leslie-Ericksen theory which were obtained using data from both shear and extensional flow measurements, the Doi theory is considered to be more robust.
Acknowledgment This work was supported through a program grant from the Australian Research Council. The authors are delighted to join their colleaguesin this special issue to honor Professor Metzner. One of us (T.S.)is keen to acknowledge the benefits of a sabbatical at Delaware in 1986 and the stimulating discussions with Professor Metzner. Literature Cited Doi, M. RheologicalProperties of Rodlike Polymers in Isotropic and Liquid Crystalline Phases. Ferroelectrics 1980,30, 247-254. Doi, M. Molecular Dynamics and Rheological Properties of Concentrated Solutions of Rodlike Polymers in Isotropic and Liquid Crystalline Phases. J. Polym. Sci. Polym. Phys. Ed. 1981, 19, 229-243.
Doi, M.; Edwards, S. F. Dynamics of Rod-Like Macromolecules in Concentrated Solutions. J. Chem. SOC.,Faraday Trans. 2 1978, 74, 560-917,918-932.
Doraiswamy, D.; Metzner, A. B. The Rheology of Polymeric Liquid Crystals. Rheol. Acta 1986,25,580-587. Kuzuu, N.; Doi, M. Constitutive Equation for Nematic Liquid Crystals under Weak Velocity Gradient Derived from a Molecular Kinetic Equation. 11. Leslie Coefficients for Rodlike Polymers. J.Phys. SOC. Jpn. 1989,53, 1031-1040. Larson, R. G. Arrested Tumbling in Shearing Flows of Liquid Crystal Polymers. Macromolecules 1990,23, 3983-3992. Larson, R. G.; Mead, D. W. Time and Shear-Rate Scaling Laws for Liquid Crystal Polymers. J. Rheol. 1989, 33 (8),1251-1281.
Larson, R. G.; Doi, M. Mesoacopic Domain Theory for Textured Liquid Crystalline Polymers. J. Rheol. 1991, 35 (4,539-563. Larson, R. G.; Mead, D. W, Gleeson, J. T. Texture of a Liquid Crystalline Polymer During Shear. In Theoretical and Applied Rheology;Proceedings of the Eleventh International Congress on Rheology, Brussels, Belgium; Elsevier: Amsterdam, 1992; pp 6569.
Leslie, F. M. Theory of Flow Phenomena in Liquid Crystals. Adu. Liq. Cryst. 1979,4, 1-81. Magda, J. J.; Baek, S. G.; DeVries, K. L.; Larson, R. G. Shear Flows of Liquid Crystal Polymers: Measurementa of the Second Normal Stress Differenceand the Doi Molecular Theory. Macromolecules 1991,24 (15), 4460-4468.
Marrucci, G. Rheology of Liquid Crystalline Polymers. Pure Appl. Chem. 1985,57,1545-1552. Marrucci, G.; Maffettone, P. L. Description of the Liquid-Crystalline Phaaae of RodlikePolymers at Highshear Rates. Macromolecules 1989,22,4076-4082.
Metzner, A. B.; Prilutaki, G. M. RheologicalProperties of Polymeric Liquid Crystals. J. Rheol. 1986,30 (3), 661-691. Moldenaers, P.; Mewis, J. Relaxation Phenomena and Anistropy in Lyotropic Polymeric Liquid Crystals. J. Non-Newtonian Fluid Mech. 1990,34, 359-374. Moldenaers, P.; Yanase, H.; Mewis, J. Flow-Induced Anisotropy and ita Decay in Polymeric Liquid Crystals. J.Rheol. 1991,35,16811699.
Ooi, Y. W.; Sridhar, T. Extensional Stresses in a Model Liquid Crystalline Solution. In Proceedings of the Siwth Nationul Conference on Rheology, Melbourne; Australian Society of Rheology: Melbourne, 1992; pp 75-78. Tirtaatmadja, V.; Sridhar, T. A Filament Stretching Device for Measurement of Extensional Viscosity. J. Rheol. 1993, 37 (6), 1081-1102.
Wilson, T. S.; Baird, D. G. Transient Elongational Flow Behaviour of Thermotropic Liquid Crystalline Polymers. J.Non-Newtonian Fluid Mech. 1992,44,85-112. Receiued for review December 29, 1993 Revised manuscript received May 2, 1994 Accepted May 10, 1994. Abstract published in Aduance ACS Abstracts, August 15, 1994.
