Structure and Flow in Surfactant Solutions - ACS Publications

Chapter 21 Relation Between Rheology and Microstructure of Lyotropic Lamellar Phases

Downloaded by EAST CAROLINA UNIV on September 22, 2015 | http://pubs.acs.org Publication Date: December 9, 1994 | doi: 10.1021/bk-1994-0578.ch021

Didier Roux, Frederic Nallet, and Olivier Diat Centre de Recherche Paul Pascal, Centre National de la Recherche Scientifique, Avenue Doctor Schweitzer, 33600 Pessac, France

Rheological behavior of lyotropic lamellar phases is studied as a function of the membrane repeat distance. The steady state rheology is described as a consequence of the so-called orientation diagram described previously. Three distinct regions of different orientations are described that are separated by two out-of-equilibrium transitions. We show that these transitions can be either discontinuous (subcritical) or continuous. In one of these transitions, one can go continuously from one regime to the other through a bifurcation point.

The effect of shear on systems having a large characteristic length allows us to describe the effect of shear on the microstructure. Typical systems studied are either near a second order phase transition (1,2), or colloidal systems (3,4). One basic issue of these works is to understand the viscoelastic behavior of fluids in terms of microstructure in the same way that statistical mechanics allows us to describe the stability and thermodynamics of equilibrium systems. We have studied the effect of shear on a lyotropic lamellar phase. We have been able to show that the orientation taken by a lyotropic lamellar phase under shear can be described as steady states that are separated by transitions as a function of the characteristic distance between membranes and the shear rate (5). Three different states of orientation of the lamellar phase have been described: an isotropic state, where the membranes form onion-like structures (multilayer spherical objects of size R much larger than the repeating distance d) exists at intermediate shear rates in between two other states (at either lower or higher shear rates) where layers are mainly parallel to the flow. The location of these regions in the shear rate/smecticperiod plane has been called the orientation diagram (5). We also have studied the consequences of the so-called orientation diagram on the rheological properties of a lyotropic lamellar phase. Studying the shear rate as

0097-6156/94/0578-0300$08.00/0 © 1994 American Chemical Society In Structure and Flow in Surfactant Solutions; Herb, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.


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ET A L .

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a function of the stress, we have shown that the passage from one state to the other corresponds to out-of-equilibrium transitions. The first transition can be either discontinuous or continuous, depending upon the repeating distance, while the second transition seems to be always discontinuous. Upon approaching the bifurcation point where the first transition goes from continuous to discontinuous we show that oscillation in time may be observed. We interpret this behavior as due to a coupling between transitions and emphasize the fact that the rheological behavior has to be described within the framework of dynamic transition rather than static transitions. A lyotropic lamellar phase made of water, Sodium Dodecyl Sulfate (SDS), pentanol and dodecane exhibiting a lamellar phase whose repeat distance lies between 60Â to 400Â (6) has been studied. This phase corresponds to layers of water surounded with surfactant separated with dodecane and is stabilized by undulation interactions (7). A previous work, using Couette cells and different techniques, has shown that an orientation diagram can be described corresponding to different orientations of the smectic layers with respect to the flow field (5). This diagram exhibits three states of orientation as shown in ref. 5. At very low shear rates (7 < 1 s ) and high surfactant concentrations, the membranes are mainly parallel to the flow with the smectic director parallel to the velocity gradient direction (region 1). In this state, many defects (probably dislocations) persist in the two directions perpendicular to the director and are presumably similar to the ones described by Oswald and Kléman for thermotropic systems (8). At a higher shear rate or for more dilute systems, a new state appears where the smectic layers (membranes) form multilayer spherical droplets of a well defined size, controlled by the shear rate, ranging typically from 10 μπι to less than 1 μπι (region 2) (5,9). At even higher shear rates, a state where the membranes are parallel to the flow with the smectic director parallel to the gradient of velocity direction is stable. This state has some similarities with the first one but no defects remains in the direction of the flow (region 3). When observations are made in a cell where the shear rate is fixed, regions 2 and 3 are separated with a region where the two states coexist (region 2+3). In order to get rheological information, experiments have been carried out with a Rheometer Cammed 100 that fixes the stress and measures the velocity (shear rate). We have used a Mooney cell corresponding to a Couette cell terminated by a cone/plate at the bottom in order that the shear rate is uniform throughout the cell. We have measured for different repeating distances the shear rate (7) as a function of the stress (S). Three regimes are described corresponding to 3 different power laws (S ~ 7 ). Regions 1 and 3 correspond to an exponent χ = , and region 2 corresponds to χ = 0.2. One sees that regions 1 and 3 correspond to a Newtonian behavior (S 7 , viscosity: η = constant, see Figure 1) but exhibit very different viscosities. Region 2 corresponds to a continuous shear thinning (viscosity decreasing with the shear rate • -0.8 η 7 ). We also see that the passage from regions 2 to 3 corresponds to a jump x

In Structure and Flow in Surfactant Solutions; Herb, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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Figure 1. Viscosity as a function of the stress for different samples (71 % of oil volume fraction, la) and as a function of the shear rate (lb samples 50% and 69%).


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in the shear rate for a given value of the stress, while the passage from region 1 to 2 corresponds to either a jump in the stress (or viscosity) at constant shear rate or to a continuous process depending upon the dilution. Let us first study the second transition between regions 2 and 3. When the stress at which the transition occurs is reached (around 4 Pa for the 69% of oil sample and 50 Pa for the 50 %), a very small increase of the stress leads to a jump in the shear rate value. For the 69% sample the jump corresponds to an evolution of the


shear rate from typically 200 s * to 900 s 1000 s

For the 50% sample the jump starts at

and ends above the maximum shear rate that the apparatus can measure

(1200 s ). If measurements are made relatively rapidly (not waiting for the complete steady state equilibrium) one observes an hysteresis cycle. The first transition is more complex (between states I and II). Indeed, for concentrated samples ( φ ^ < 68%) the transition is discontinuous in stres, but it 0

becomes continuous for φ ^ > 68%. The transition between discontinuous (Δη Φ 0) to continuous (Δη = 0) transition is a bifurcation point (10). The fact that these transitions are out-of-equilibrium in nature leads to a behavior that is expected only from a description of stationary states. Indeed, in these transitions, either feedback effects or coupling between transitions can lead to time evolutions that are no longer stationary but oscillations or chaotic behavior as a function of time are expected in certain cases. We may expect to find such states in the rheological behavior of these systems (11). Let us come now to a microscopic explanation of the observed phenomenon. As already pointed out (5-9), under shear the preferred orientation is given by the velocity in the plane of the layer and the gradient of velocity perpendicular to the layers, as observed at very low and high shear rates. Within this orientation the gap in which the lamellar phase is moving is of the order of 1mm with a precision that is at best of the order of 1/100 mm. Consequently, the lamellar phase develops defects that are only slightly anisotropic at a low shear rate (13). In this observed state, the system flows by moving dislocations (14) and is very viscous but Newtonian (η « 1000 mPa s). For intermediate shear rates, the sample is forced to move faster and the dislocations cannot follow. The fact that the plate movements move faster than the rate of displacement of the dislocations creates a pressure perpendicular to the layer and the smectics develop an undulation instability corresponding to a lattice of 0

dislocations at a length L = ( X D ) ^ (D being the distance between the plates and λ the penetration length of the smectic phase which is of the order of the d-spacing) (8). These dislocations forbid the system to flow and instead, the system bifurcates to another orientation which consists of small spheres rolling on each other to allow the flow to proceed. At a higher shear rate the lattice of dislocations corresponding to an undulation instability is so anisotropic (14) that no dislocations are left in the direction of the velocity and the oriented state beciomes again the most stable phase with a viscosity of the order of a few mPa s (η « 3-5 mPa s).


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In the intermediate region, we can calculate the characteristic size of the onions balancing two forces: an elastic one f j and the viscous force f j ( 15j.The elastic force needed to maintain a lamellar phase at a size R can be expressed by: e

f

el

v

s

= 4 π ( 2 κ + Κ)Μ

(1)

where κ and Κ are respectively the mean and Gaussian elastic constant of the membrane. The viscous force that a droplet experiences in a flow is (15):


f

2

v i s

2

^R 7 = R S

(2)

with 7 being the shear rate, η the viscosity and R the onion size and S the stress. Balancing (1) and (2) we can calculate the equilibrium size for the steady state:

D

R=

/ 4π (2 K+ κ)

V

Sd

=

V

/ 4π (2 κ+ κ) ηάγ

3

Equation 3 has been quantitatively checked (5,9). A more microscopic model has been proposed leading to basically the same scaling (16). In order to decide whether the transition between the states 1 and 2 is continuous or not we have to compare two lengths, L (at which the dislocation array forms) and R. If L > 2R, we may understand that as soon as the dislocation array wants to form, the system has no problem to bifurcate to the onion state. However, if L < 2R, the viscous force is not large enough to form onions of size at least equal to L . The system then stays at the limiting velocity below which no dislocation array is developed and builds up stress until it reaches the value needed to form onions of size L . The system undergoes a discontinuous transition with a jump in stress. Since the elastic constants are known for this system, we can put in some quantitative numbers: we find that (6): 1/2

L = (8/47CK/kTdD) (4) As expected, L is an increasing function of d and R a decreasing one. The dilution (corresponding to a value d* of the d-spacing) at which we expect the transition to go from continuous to discontinuous can be quantitatively estimated. With κ being of the order of 1 kT and îc = -1 kT estimated from the variation of the onion size with the shear rate (7), we get:

d* = nkT

Υ κηγϋ

(5)

and is of the order of 30 nm which is effectively very close to the experimentally observed value. In conclusion, one would like to stress that the complicated behavior exhibited by the rheology of the lyotropic smectic phase can be interpreted at the microscopic level as different states of orientation separated by out-of-equilibrium transitions.


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Literature cited


1. 2.

Beysens, D.; Gladamassi, M. J. Phys. Lett. 1979, 40, 565 Safinya, C. R.; Sirota, Ε. B.; Plano, R.J.; Bruinsma, R. J. Phys. 1991, C2, 365 3. Pieranski, P. Contemp. Phys. 1983, 24, 25 4. Ackerson, B. J.; Pusey, P. N. Phys. Rev. Lett. 1988, 61, 1033 5. Diat, Ο.; Roux, D.; Nallet, F. J. Physique II France 1993, 3 , 1427 6. Bellocq, A. M. ; Roux, D. In microemulsions: structure and dynamics, Friberg, S. E.; Bothorel, P. Eds CRC Press, Boca Raton, F. L., USA 1987, p. 33 7. Roux, D.; Safinya, C. R. J. Phys. France. 1988, 49, 307 8. Oswald, P. ; Kléman, M. J. Physique Lett. 1983, 43, L411-L415 9. Diat, O.; Roux, D. J. de Phys. II 1993, 3, 9 Roux, D.; Diat, O. French patent number 92-04108 Roux, D., Diat, O. and Laversanne, R. PCT, FR 93-00335 10. Guggenheim, J.; Holmes P. In Non linear oscillations dynamical systems and bifurcations of vectorfields(Springer, New York 1983). Nicolis, G.; Prigogine, I. In Self-organization in nonequilibrium systems (Wiley, New York 1977) 11. This will be discussed in a forthcoming publication. 12. de Gennes, P.G. In The Physic of Liquid Crystals, (Clarendon Oxford 1974) 13. Clark, N.; Meyer, R.; Appl. Phys. Lett. 1973, 22, 493 14. Oswald, P.; Ben-Abraham, S. I. J. Physique 1983, 43, 1193-1197 15 . Taylor, G. I. Proc. Roy. Soc. 1932 , A 138, 41-48 and Proc. Roy. Soc. 193 , A 146, 501-523 16. Prost, J.; Leibler, L.; Roux, D. to be published RECEIVED July 24, 1994


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