Flow birefringence and rheological measurements on shear induced micellar structures

Rheologica Acta Rheol Acta 26:532-542 (1987)

Flow birefringenee and rheological measurements on shear induced micellar structures *) **)

I. Wunderlich, H. Hoffmann and H. Rehage

Institut ffir Physikalische Chemic der Universit~it Bayreuth (F.R.G.)

Abstract: Aqueous solutions of cationic surfactant systems with strongly binding counterions show the striking phenomenon of shear induced phase transitions. At low shear rates or angular frequencies, the solutions exhibit Newtonian flow. At high rates of shear, however, the rheological properties change dramatically. Above a well defined threshold value of the velocity gradient, a supermolecular structure can be formed from micellar aggregates. This shear induced structure (SIS) behaves like a gel and exhibits strong flow birefringence. The formation of the shear induced structure is very complicated and depends on the specific conditions of the surfactant system. In this paper we discuss new results which have been obtained from rheological measurements and from flow birefringence data. We examine the stability of the shear induced state as a function of temperature, surfactant concentration and salt concentration and we analyse the effect of solubilisation of alcohols and hydrocarbons. The results are interpreted in terms of a kinetic model which accounts for the observed behavior.

Key words." _Surfactant s_olution, micelle, viscoelasticity, drag reduction, _flow birefringence

I. Introduction

The frictional resistance of aqueous solutions can be considerably reduced by adding small amounts of surfactants to the water [ 1 - 11]. This phenomenon, which is known under the name "drag reduction", is due to the presence of large aggregates or supermolecular structures, which can be formed during flow [2]. In former investigations we have shown that these supermolecular structures can be formed when the shear rate exceeds a well-defined threshold value [8-10]. The critical shear rate depends very much on the specific conditions of the surfactant system like temperature, concentration or the ionic strength of solution. The formation of the supermolecular structure can already occur in the laminar flow regime, but sometimes it is only observed at high velocities, so that turbulent flow arises [8, 10]. In general, the phenomenon of shear induced structures (SIS) is limited to well-defined ranges

*) Dedicated to the 60. birthday of Prof. H. Harnisch, Hoechst AG

**) Partly presented at the 2nd Conference of European Rheologists, Prague, June 17- 20, 1986 227

of the shear rate [7, 8]. At very high values of the velocity gradient, the supermolecular structures are no longer stable and the flow induced state decreases [7, 8]. The formation and destruction of the SIS can easily be monitored by measuring the rheological and optical properties during flow. When the turbulent flow is stopped, the shear induced structure decays and the solution relaxes to its original state [10]. In previous papers we have shown that in the unsheared solutions rod-like micelles exist [7-10, 12, 13]. The solution at rest exhibits only sol properties, but at high velocities the supermolecular structures can be formed and the surfactant solutions are transformed into highly viscoelastic gels [9, 10, 12]. When the flow process is stopped, the gel state decays and the quiescent state is reforming again [10]. Flow induced phase transitions are processes which are, at all conditions, completely reversible [10].

The formation of shear induced structures (SIS) has a high potential significance for various technical ap- plications. In processes, where large amounts of water have to be circulated with high speeds, the effect of SIS can cause a drastic decrease of the turbulent drag and leads hence to a gain in energy. In technical ap- plications, the solution can come in contac t with

Wunderlich et al., Flow birefringence and rheological measurements on shear induced micellar structures 533

different additives and it is therefore of interest to investigate the effect of these materials on the formation of flow induced structures. In the framework of these problems we have studied the influence of salt concentration and various other additives on the SIS. The measurements have been carried out on the surfactant system Tetradecyltrimethylammoniumsalicylate where the striking phenomenon of SIS can easily be detected.

2, Experimental

2.1 Materials and Methods

Tetradecyltrimethylammoniumsalicylate (TTAS) is produced by Hoechst Company. The surfactant has been prepared as previously described by ion exchange procedure from TTACl-solutions or by dissolving TTAS which has been synthesized before [9, 10, 12, 14]. Both methods gave identical results. The solutions were left standing for two days in order to reach equilibrium. The rheological properties of the surfactant solutions were measured with the Rheometrics Fluid Rheometer RFR 7800 (cone-and-plate geometry). The experimental equipment for flow birefringence measurements consists of a concentric cylinder apparatus which has been described in detail in [9, 15, 16].

2.2 Theory

2.2.1 Rheology

The transient behavior of viscoelastic surfactant solutions can be investigated by measuring time dependent theological properties. In this type of experiment, a step-wise shear rate is suddenly applied to the solution and the shear stress and the first normal-stress difference are measured as a function of time. When the flow process is suddenly stopped, the shear induced structure decays and the upertubed state is reforming again. This relaxation process can be observed by measuring the shear stress and the first normal-stress difference after cessation of steady state flow.

Alternatively, the rheological properties can be investigated if periodic or dynamic experiments are performed. In this way it is possible to obtain informations on the equilibrium state of the surfactant solutions without disturbing internal supermoleculat structures. In a general case, a sinusoidal deformation is applied to the solution. If the sample behaves as a viscoelastic liquid, the stress is out of phase with the strain. From the phase angle between stress and strain and the amplitude of these quantities, the storage modulus G', the loss modulus G" and the magnitude of

the complex viscosity ~t/*] can be calculated. G' describes the elastic properties of the solution and G" is proportional to the energy dissipated during flow. The magnitude of the complex viscosity is related to these quantities according to:

I~* I -- ( G ' + G")/~o (1) The simplest mechanical model which can describe the

dynamic behavior of a viscoelastic surfactant solution is called the Maxwell model. It consists of a spring (elastic element) and a dashpot (viscous element) connected in series. The spring corresponds to a shear modulus Go and the dashpot to a viscosity ~/0. The behavior of the Maxwell material under harmonic oscillations can be obtained from the following equations:

G' (c.o) = G O o 2 r2/(1 + o 2 T 2 ) , (2)

G" (co) - tlsCO = Goo)r/(1 + 092"62) . (3)

In these equations q~ denotes the solvent viscosity and Go is the shear modulus, r is called the stress relaxation time and it is given by:

~0 = G0~. (4)

We see immediately that for ~or> 1 the modulus G' approaches a constant limiting plateau value. Under such experimental conditions solutions behave as elastic bodies. At low frequencies co ~ ~ 1 and G' becomes proportional to co 2. This region is called the terminal zone where the surfactant solution behaves as a simple liquid. The viscoelastic properties of the sample are functions of the angular frequency.

2.2.2 Flow birefringence

Anisometric colloidal particles can be aligned in the streaming solutions under the orienting forces of a velocity gradient. Typical examples for such particles are the tobacco mosaic virus, long chain polymeric macromolecules and surfactant solutions containing rod-like micelles. Experiments in flow birefringence are usually interpreted in terms of the two measured quantities extinction angle X and birefringence A n. The extinction angle is defined as the smallest angle b e - tween the direction of flow and one of the two mutu- ally perpendicular extinction positions. This quantity describes the average orientation of the anisometric particles.

In the constant shear rate mode the measured values of the extinction angle X and the flow birefringence A n of a solution containing rigid, colloidal rodlike particles is given by the theory of Peterlin and Stuart [17]:

Z = ~z/4 - 9~12Dr, (5)

A n = (2n fo/n) (gl - g 2 ) b / 1 5 . (6)

These equations are valid in the zero shear rate limit at infinite small concentrations. Dr is the rotational diffusion constant, 9 the shear rate, ~o the volume frac-

534 Rheologica Acta, Vol. 26, No. 6 (1987)

tion of the dispersed particles, n the refractive index of the isotropic solution, g l - g 2 the optical anisotropy factor and b a geometric factor, depending on the axial ratio of the particles. Measurements of the extinction angle and the flow birefringence thus permit the determination of the rotational diffusion constant of the dispersed particles. In this way it is possible to obtain informations of the shape and the size of the micelles at concentrations, where interactions between the par- tides can be neglected. The formation of the shear induced structure is associated with the build-up of flow birefringence. The optical anisotropy of the solutions can easily be monitored by a light beam passing through the sample and two crossed polarizers. The amount of birefringence is determined by introducing a suitable compensator - e.g. a Senarmont or a Babinet compensator - between the measuring cell and the analyser. The flow birefringence A n is given by the difference of the refractive indices of the extinction positions.

A general way to interpret flow birefringence data is to compare them with rheological measurements. It is well known that for solutions of flexible macromolecules and for three-dimensional networks the so called "stress-optical law" is supposed to be valid. Such a law was first proposed by Lodge and also postulated in recent years by Doi and Edwards for solutions containing rigid, overlapping rods [18, 19]. The stress- optical law describes a simple relationship between the stress tensor and the refractive index ellipsoid. The proportionaly factor C between these two quantities is designed the stress-optical coefficient. In polymer systems, it depends on the anisotropy of the polarizability of a monomer unit of the macromolecule and it has therefore a characteristic value for each system. From the stress-optical law thus defined the following equations can be derived:

A n sin 2 X = 2 Cp21 , (7)

An cos2x = C (Pl l --P22) • (8)

P21 is the shear stress, Pll -P22 is the first normal-stress difference and Z denotes the extinction angle. The va- lidity of the stress-optical law can simply be testet by plotting measured values of 3 n sin 2 Z as a function of the shear stress or by plotting Ancos2 Z as a function of the first normal-stress difference.

3. Results

At very low shear rates the supermolecular structure cannot be formed and the solutions exhibit only sol-

10 5-

1¢-

10 3 -

10 2

lO ~

qo / rnPa .s

.r °/

/ c/mM

I0 o ~ e J I ' ' ' I , 1

1 '0 -" ' 1'0 ° 1 ' 2

Fig. 1. Zero-shear viscosity t/0 versus surfactant concentrations (T= 20 °C, TTAS)

properties. In figure 1 the zero-shear viscosity of the TTAS-system is given as a function of the concentration.

The phenomenon of shear induced structures can only be observed in the highly dilute concentration regime, where the zero shear viscosity is very near to the one of water. Above the transition point where the viscosity starts to rise, the solutions are elastic already at rest and exhibit strong gel-properties. The theoretical understanding of the sharp rise of the viscosity at such low volume fractions of surfactant is of funda- mental importance for the interpretation of the shear induced structures. In previous papers we reported that in the highly dilute concentration regime rodlike micelles are present [7, 10, 12, 13]. The length of these particles increases linearly with the surfactant concentration. At the transition concentration c* the rods start to overlap and the free motion of a single particle is restricted by the presence of other aggregates. In theories on the viscosity of rodlike particles, the viscous resistance is determined by the rotational relaxation time of the anisometric particles [19]. The sharp increase of the experimental curves therefore indicates a slowing down of the rotation times by several orders of magnitude within a small concentration range. Ex- perimental investigations of other colloidal systems containing rodlike particles show, that in contrast to the theoretical predictions the viscosity increase does not begin exactly at conditions, where the rotational volumes of the particles start to overlap [20]. In general, the transition region is shifted to higher concentrations.


We have to conclude, therefore that the drastic increase of the relaxation time must be due to some ad- ditional cause which starts beyond the overlap. It seems that the micellar rods can stick together and form a three-dimensional network. The network is of temporary nature as is clearly evidenced from figure 2, where the dynamic properties of the supermolecular structure have been investigated.

The storage modulus G' and the loss modulus G" were measured as a function of the angular frequency co. ~7s denotes the solvent viscosity. The dynamic char- acteristics of this solution exhibits some typical features. At high frequencies the storage modulus approaches a constant value. In polymer systems such a region is called the rubber plateau. Under these experimental conditions the solution behaves as an elastic body. At low frequencies G' becomes proportional to 032 and the loss modulus is large in comparison to the storage modulus. This region where the surfactant solution behaves as a liquid is called the terminal zone. The dynamic properties mentioned above are typical for all surfactant solutions. At high frequencies they behave as elastic solids and at low frequencies the viscous properties dominate. The intermediate frequency range is characterized by an ambivalent behavior.

Just as a micelle constantly exchanges monomers with the bulk solution, so the network exchanges micellar rods which are in equilibrium with free unbound particles. When we perturb the equilibrium conditions, the system relaxes with a characteristic time constant to the new conditions. After mechanical perturbations this time constant is given by the stress relaxation time constant. In figure 2 the intersection point of G' and G" corresponds to ~ = 1/co where r is time constant for stress relaxation. It is interesting to note that the process of stress relaxation can be described by only one time constant, indicating that this process follows first-order kinetics. The filled lines in figure 2 are theoretical curves according to eqs. (2) and (3). From the fit of the curves we get: T = 8 s.

After a mechanical perturbation the system relaxes with this time constant to the new condition. The stress relaxation time describes the dynamic processes of the colloidal system. For concentrations not too high above the sol/gel transition, there will always be a consid- erable fraction of rods in the unbound state. These rodlike micelles can be oriented in an electric field and will desorient again after the field has been switched oil'. The time constant of this process is much faster than the stress relaxation time and of the same order of magnitude as the stress relaxation time in the highly dilute concentration regime [21].

10~ i G'.G"/PQ Iq'l / PQ.s

lq*l

1 0 2 . o. ° "O.ck

o-o. ~' ~-~ -~¢~ .~y~ ~ ~-~-~-~-~-~-~-~-~.~-~

i o ~. ~ e ~ % ~ ~'o-%.o. ° o~" ~ o. "%.

<3.~ y zx/ ~ °- o

:c- ~ ~'~" "°'° o 1 O° A ~ ~ ~ 0 0 G" " "°'°'°'o.

10 -1 o

~j /s "1 I0"2 ' ' 'I , L 'I ~ ' '

;-' # ' " ' ' ,'e Fig. 2. Storage modulus G', loss modulus G" and magnitude of the complex viscosity ] t/*] versus angular frequency for a solution of 50 mmol TTAS (T= 20 °C). Full lines are theoretical curves according to the Maxwell model

The transient behavior of the shear induced phase transition can be investigated by measuring of time dependent rheological properties. In this type of experiment, a step-wise shear rate is suddenly applied to the solution and the shear stress and the first normal-stress difference are measured as a function of time. In former investigations we have pointed out that the formation of the flow-induced stucture is accompanied by only a small increase of the shear stress, but there are large effects concerning the first normal-stress difference and the flow birefringence [10]. Typical results of these measurements are represented in Figure 3.

In this experiment the step-wise shear rate is applied at t = 0. Immediately after the onset of flow, the first normal-stress difference is equal to zero, indicating that the solution behaves still as a Newtonian fluid. After a short time of shearing, however, the rheological properties change dramatically and the first normal- stress difference rises steeply to a plateau value. The occurence of normal forces is usually attributed to the presence of finite strains in viscoelastic materials [22]. This definition implies that only the flow-induced state is viscoelastic whereas the solution at rest behaves like a Newtonian liquid. Below the critical shear time of 10 s only viscous properties can be detected.

In analogy to the transient rheological experiments, time dependent flow birefringence measurements can be performed. We observe a curve similar to that of the first normal-stress differences (Figure 3). After a critical shearing time, the flow birefringence increases steeply, and after about 20 s a steady-state value is reached. In this range the new structure, the shear in-

~nllff 6] duced phase, is formed and strong birefringence appears. It is evident that the flow-induced structure has anisotropic optical properties.

It is interesting to note that the extinction angle Z, which describes the dynamic orientation process of particles under shear conditions, is equal to zero for the shear induced structure [10]. That means the flow induced phase is completely aligned in the direction of flow. Under these conditions eq. (7) is reduced to

An = C (Pu - P22). (9)

From this equation it is clear that there exists a simple relationship between first normal-stress differences and flow birefringence. When the flow process is stopped, the shear induced phase decays and the unpertubed state is reforming again. This process can be observed by measuring the time dependent functions P u - P 2 2 and An after cessation of steady-state flow. Some results of this type of measurement are summarized in figure 4.

In this experiment the shear rate was suddenly reduced to zero at t = 0. In the semi-logarithmic plot we obtain a single straight line indicating a mono-expo- nential decay of the flow induced state. During the whole process of relaxation the extinction angle remains at exactly zero, and from eq. (9) the analogy between flow birefringence and first normal-stress difference can easily be explained. It is interesting to note, that the relaxation process can be characterized by a single time constant, indicating that this decay follows first-order kinetics. In this case we obtain r = 3.4 s and we conclude therefrom that the decay of the shear induced phase proceeds with this time constant. After the decay of An, however, some finite elastic and viscous properties are still present in the solution. This is clearly demonstrated in Figure 5, where storage and the loss moduli have been measured after cessation of steady-state flow.

When the shear induced state decays with a relaxation time of several seconds, as is implied by the results of the flow birefringence measurements and the data of the first normal-stress difference, we should expect that storage and the loss moduli decrease with the same time constant. Figure 5, however, shows that there exists a broad spectrum of relaxation times which are of the order of 20 minutes. It is evident, that the dynamic measurements after cessation of steady-state flow are in particular useful for the determination of such long relaxation processes. The experimental results indicate that the decay of the flow induced state proceeds with time constants of the order of several minutes.

2-

P11 -P22 IPa_

40-

I I I I I I I 20 40 60 80 100 120 140

I I 160 180 200 t/s

30.

20-

10-

10 o

5


\ 20 /~0 60 810 I00 120 I~0 160 180 200 tls

L r

0 5 10 15

= ,

2

10 -~

5

3 2

102 t _ _ A n { t ) p11{t) - P22(U

An(t=0) i pl l l t --0)- P22(t=0]

Fig. 4. Relaxation function of flow birefringence and first normal-stress difference after cessation of steady flow (TTAS, c = 2.5 mmol, T= 20°C, ?= 200 s -l)

20 t/s T

~: An

" : P~I- P22

v

Fig. 3. First normal-stress difference Pu-P22 and flow birefringence d n as a function of the shear time after a step in shear rate (TTAS, c = 2.5 mmol, T= 20 °C, ~ = 200 s -1)


102 -

101 .

10 °

G;G"/mPa

l 12 I

11

10

~ o 7-

~°'~'°------o____._~____ G' o 6- o o t / ra in

' ' ' I ' ' ' ' I ' ' ' ' I m 5 -

0 50 100 150 4-

3-

2-

1-

Fig. 5. Storage modulus G' and loss modulus G" after cessation of steady flow (TTAS, c=2.5mmol, T=20°C, ~= 200 s -1)

The formation of the shear induced state depends upon the actual surfactant concentration. Typical results of flow birefringence measurements are summarized in figure 6.

It is interesting to note that for all concentrations the optical anisotropy effect can be observed at the same characteristic threshold value of the shear rate. At high values of the velocity gradient the curves approach a saturation value. Some curves show a slight inflection point. While this might not seem worth mentioning in these plots and is in the limit of the experimental error, it nevertheless seems to be significant. The S-shape character could be an important hint for the qualitative explanation of the phenomena. The saturation values of An increase more or less linearly with the total surfactant concentration. A plot of the flow birefringence against the concentration c is shown in figure 7.

The experimental results indicate that there are no shear induced states below a surfactant concentration of 1.5 mmol. This phenomenon could be caused by a sphere-rod transition of the micellar aggregates and we could argue that the globular micelles do not form a SIS. This, however, is in contradiction to actual drag reduction measurements in which at concentrations below 1.5 mmol this effect is still observed. The exact determination of this threshold concentration depends on the accuracy of experimental results and may therefore be different for diverse experimental techniques. It is also conceivable that the disappearance of the birefringence at this particular concentration is due to a fortunate compensation of form and intrinsic birefringence. Such compensations have been observed in electric birefringence measurements when parameters like the counter-ions, the temperature or the concentra-

L-An C t ~ ' l ~ 6 • .5mM

• • / *

. /

/. mM

3 mM × × - - - -

_ _ _ _ n [ ] [ ]

o 2 rnM

o.oo.~OO o~O o ~ o -- o . o-- --

2.5 mlM

Is ~1

0 I00 500 1000 1560 I~

Fig. 6. Flow birefringence versus shear rate for different surfactant concentrations (TTAS, T = 20 ° C)

_z~n/lO -6

15 / /

/ //

/' 10' //

/

S-- / e

./ / o c /mM

0 I I I I I 0 1 2 3 /, 5

Fig. 7. Saturation values of the flow birefringence versus surfactant concentration (T = 20 ° C)

tion were changed [21]. The saturation behavior with increasing shear rate is probably due to the complete alignment of all surfactant micelles. Very similar saturation values were also obtained in electric birefringence measurements for high electric fields [21].


0 -

0

-an ( t - - ~ 1 ~ 6 ~.

if/° /" /

/ /

/

T : 20°C

T = 25°C

.4:X ~

f ~ # T:30°C

Z~/ x ~ x - - x - - x - - x - - x - - X x

/ /= ~ - x

a/ /x T=35oc

500 1000 1500 2000 "~/s 4

Fig. 8. Flow birefringence versus shear rate at different temperatures (TTAS, c = 2.5 mmol)

The formation of the SIS depends on the temperature of the surfactant solution. This is clearly seen in figure 8 where the flow birefringence is given as a function of the shear rate for several temperatures. With increasing temperature, the critical shear rate where the SIS can be detected, is shifted to higher values. At temperatures above 40 °C the flow induced state disappears. This indicates a rod-sphere transition at ele- vated temperatures. Globular micelles, which are formed under these conditions, are not able to form shear induced phases.

The absolute values of flow birefringence decrease with increasing temperature, even though a plateau value is reached for each temperature. We have associated the plateau value with the complete alignment of the SIS. In view of this model it might be surprising that the flow birefringence values decrease so much with the temperature. The experimental results, however, are in very good agreement with our assumption. The lengths of the rod-like micelles in the isotropic solutions decrease with increasing temperature and the SIS are therefore formed with shorter particles at higher temperatures [9, 10]. The intrinsic birefringence of the rod-like particles depends upon the order pa- rameter of the hydrocarbon chains inside the rods, the anisotropy of the polarizability of the hydrocarbon chains and upon the lengths of the rod-like particles. At rod-sphere transitions the contribution of the intrinsic birefringence must disappear. It is this what is experimentally observed.

The formation of the SIS depends in a complicated way upon the ionic strength of the solution. This is demonstrated in figure 9, where flow birefringence data are plotted versus the shear rate at different NaBr concentrations.

The plateau values of n pass over a maximum and decrease at high salt concentrations. Previous investiga-

5 / i 4

0.7 mM

/ " +

/ ..i_,,11-- .--*~ o.5 mM

t - - ' / ~ ~ - -OmM

/ A , / ' " x / x _ _ × 1.SmM /2y ~ / 2 5mM

2- / / / × / ....~n j ~ n ~ a

~t I / >'P" / " 7~M°_ /f / / / / / # / / / " /" o/°

0 500 1000 1500 2000 2500

Fig. 9. Flow birefringence of TTAS-solutions versus shear rate for different concentrations ofNaBr (T = 20 °C, c = 2.5 mmol)

tions on the influence of salt upon the length of the rods had shown that the anisometric particles become longer at first and that the size approaches a constant value at high concentrations of excess electrolyte [23, 24]. The small increase of An at low salt concentrations probably reflects the increase in size of the rods. The decrease of An is more difficult to realize. It is conceivable that the rods become more flexible with increasing salt and that their radius of curvature finally becomes smaller than their contour length. In this situation the rods begin to coil and are no longer capable


of building up dynamic networks. As a result of this situation, the flow birefringence disappears.

Micellar solutions can always solubilize a certain amount of hydrocarbons [25]. The amount which can be enriched in the hydrophobic core of the micellar aggregates depends very much upon the particular conditions and may vary for the same surfactant ion more than one order of magnitude as a function of salt concentration or temperature. In the case of globular micelles, the amount of single chain hydrocarbons which can be solubilized by a single chain surfactant of about equal length is very limited. Usually the molar ratio between the hydrocarbon and the surfactant is below 0.1. A rather high solubilization capacity for single chain hydrocarbons have rod-like micelles. In such systems solubilisation ratios of the order of one can be observed. In order to investigate the effect of solubilization on the SIS, we have carried out measurements of the flow birefringence of these solutions. Typical results of our measurements are shown in figure 10.

With increasing solubilization the critical shear rates are shifted to higher values while the plateau level of the flow birefringence decrease. It is obvious that the solubilization effects the SIS in the same way as an increase of the temperature. This suggests a decrease of the axial ratio of the anisometric particles with increasing oil concentration. The SIS disappears at conditions where the rods are transformed into globular micelles. From geometrical constraint follows that the available area for the surfactant headgroup at the micellar interface is about twice as large as that of the cross section of the chain for the rod-like micelles and about three times for the globular micelles. The strongly adsorbed counterions need at least as much area as the headgroups themselves, It is thus likely that the

whole interface of the cylinders is completely covered with the headgroups and the counter-ions while at the end caps there is some free area available in which the hydrocarbon core is in direct contact with water. It is conceivable that these sites are responsible for the con- tacts between different rods. In the globular particles, which form when the hydrocarbon is solubilized in the interior, these areas disappear because the surfactant molecules can now be packed more tightly at the micellar interface. If cosurfactants like short chain alcohols are added to the surfactant solutions, the shear induced state becomes unstable. This is clearly demonstrated in figure 11 where increasing amounts of pentanol are added to the surfactant solutions.

Short chain alcohols have, in general, a strong influence on the dynamics of micellar systems. The relaxation time r2 with which a micellar equilibrium relaxes after a perturbation can be shortened by several orders of magnitude by the addition of increasing amounts of alcohols. Usually the micellar kinetics is strongly affected before any change on the size or shape of the aggregates can be detected. The reason for this unusual behavior lies in the fact that the micellar kinetics is mainly due to the change of the nuclei concentration. The pentanol molecules increase the chance of nuclei formation because they are uncharged and the formation of small aggregates is thus favored. Both globular and rod-like micelles usually become smaller with increasing concentrations of pentanol. In this re- spect the cosurfactant molecules behave quite differ- ently to hydrocarbon molecules. The polar OH-group of the alcohol has a strong tendency to remain at the micellar interface, and it stays always in an energeti- cally favorable contact with water. As a consequence of this situation the molecules cannot form a core inside

OmM

z~ /

/ ~ . " .--008mM D~ • - f

2 / / / _ _ × ~ X x - - X - - x - - O 1 6 m M / , / /

? / / ~ o - 023ram

[ o/I /× o - / ~ °

g~ ~/ i ~ i _ _ i i - - i - - 0 3 1 r n M

0 500 1000 1500 2000

l- Fig. 10. Flow birefringence of TTAS-solutions versus shear rate for different concentrations of decane (c = 2.5 mmol, T= 20 °C)


-~ n ( t ~ ) ~ 1 ~ 6 0 + lOmM

° ~ ° - ~ - ~ ",r 20m~4

- - & &

/ ../ . / " / o J

500 1000 1500 2000 2500

Fig. 11. Flow birefringence of TTAS-solutions versus shear rate for different concentrations of pentanol (c = 2.5 mmol, T= 20 °C)

the micellar aggregates and the radius remains nearly constant. Based upon these observations the effect of pentanol on the formation of the SIS is easy to explain. On close inspection we find that the threshold shear rate is already affected at conditions where An is still the same as in the case of the pure surfactant system. Obviously the addition of alcohol affects first the structural relaxation time before the dimensions of the rods are also influenced. With continuous increase of alcohol, however, the size of the micelles is reduced which leads to a decrease of the flow birefringence and finally to the disappearance of the SIS.

4. Discussion

The main objective in this part is to explain the formation of the shear induced structures in a qualitative way and to form a basis for a more quantitative treat- ment which shall follow in a forthcoming paper. If we look at other phenomena in which systems can be switched between two completely different states, it is apparent that such phenomena occur only in highly co- operative processes. Typical examples are the melting of a crystal, the helix-coil transition or the binding of surfactants on polyelectrolytes. In all these systems an abrupt transition occurs when only one variable is varied. The crystal melts at a particular well-defined temperature and a micelle can only be formed above a certain critical surfactant concentration. In these systems the chemical potential is varied as a function of a certain variable until the new state is formed. Ob- viously in our case of SIS the shear rate plays the role of a thermodynamic variable. With these considera- tions we can now proceed and compare the formation of the SIS with the kinetics of micelle formation. The

thermodynamics and the kinetics of micelle formation are well understood and these systems are therefore well suited for direct comparisons with the formation of shear induced structures.

Micellar aggregates are always formed, when the chemical potential of the monomers is continuously increased until the value of the monomers in the micelle is reached. This is usually achieved by increasing the surfactant concentration. Close to the critical micelle concentration (cmc), micelle formation can already be induced by a small change in temperature, pressure or by a change of the salt concentration. In the shear induced structures, the sol-gel transition plays the role of the cmc. The gel state is formed from clusters of micelles and corresponds to the globular aggregates in the normal process of micellisation. In this simple model we assume that the rod-like micelles can form dimers, trimers, tetramers, etc. as a function of the shear rate. This process is described by the reaction scheme:

A1 +AI =A2, (10)

A2 +AI =A3, (11)

A3 +A1 =A4, (12) . . .

As-I+ A1 =As. (13)

In order to show qualitatively the principal features of this multistep equilibrium system, it is sufficient to assume that all equilibrium constants are identical and have the value Ki. The concentration As is then given by:

As= (KiA1)S-I AI . (14)

This equation shows that the concentration As is small when s is a large number and the product KI A1 < 1. At this condition only a few higher aggregates are present


in the quiescent state. This situation has its analogue in the formation of micelles from monomers below the value of the critical micelle concentration (cmc). In the thermodynamics of micelle formation A~ would corre- spond to a micelle with the aggregation number s. The same kinetic model can be applied to the formation of SIS by introducing the shear rate dependence in this theory. Supermolecular structures and higher aggregates can be formed by collisions which are due to the Brownian motion or due to the influence of a velocity gradient. We thus have two contributions for the formation of the oligomers and we can write:

dA2/dt = k+A 2 + 9 ~ A2 - k -A2 . (15)

In these equations k + denotes the chemical rate constant for the formation of dimers, k - is the dissociation rate constant and { is a constant with the dimensions of a reciprocal concentration. The value of { depends upon the effective cross-section of the rod-like micelles.

At steady-state conditions for the dimers and oligomers we obtain:

analogy to the fast process in micellar kinetics we can therefore write:

1/7.SIS = k-/a2. (20)

In this equation a describes the variance of the size dis- tribution of the different clusters. We thus obtain

7-Si s _____ (a2/k-) ,.~ 0.2 .CD (21)

and

7-SIS Y ~ 0"2 "CO "~ 0"2' (22)

Furthermore it follows that the SIS is only stable at conditions where 9 7.sis >> 1. At such high velocity gra- dients any network structure will be completely stretched. It is not surprising that under these drastic conditions the aggregates of the SIS are completely aligned in the direction of flow. The angle of extinction for a dilute solution of rod-like particles is 45 ° for 9 r <~ 1 and approaches zero for 9 r > 1.

5. Conclusions

(k+/k-) A2I + (9 ~) ~k-A2 = A2,

(k+/lc -) + (9 ¢ ) / k - = A2/A ] = K2,

o r

K2 = K° q- (9 ~ ) / k - .

(16)

(17)

(18)

The equilibration constant K2 is thus given by a shear- rate independent term K ° and a shear-rate dependent term ()~ ~ ) / k - which is equal to 9 ~ ~D when z D = 1/k-. From eq. (18) and (14) we finally obtain:

As = [(K ° + 9 ~ ZD)All s- lA1. (19)

This equation shows that the SIS will be formed when the product (K ° + 9 ~ zD) A1 approaches unity. The shear rate at which this occurs is the critical threshold value. Furthermore it is shown that this threshold value will depend very much on the value of rD. At short relaxation times rz) the critical shear rate has to be high enough in order to shift the equilibrium to- wards the SIS. This simple kinetic model is also capable to explain another characteristic feature of the shear induced state. All the experiments show that the product of the relaxation time of the SIS and the critical shear rate is constant and a large number of the order of magnitude 100. The decay of the SIS state is, in analogy to its formation, a multistep process. After the cluster of rods has been formed, the new structure can also disappear by a sequence of steps. The situation is similar like in micellar kinetics where all steps do not automatically go into the same direction. In

Solutions of cationic surfactants with strongly binding counterions show the striking phenomenon of sol- gel transition. For concentrations above the gel point the solutions exhibit strong viscoelastic effects and their rheological properties can be explained by the presence of temporary networks. For concentrations below the sol-gel-transition the solutions are shear- sensitive. At low values of the velocity gradient the solutions behave as Newtonian liquids with constant viscosities. For shear rates above the characteristic threshold value large supermolecular structures are formed during flow. The shear induced structures (SIS) are completely aligned during flow with an extinction angle close to zero. The complete orientation of the supermolecular structure, the flow birefringence and the strong normal forces are striking phenomena which have also been observed in other colloidal systems. Glass spheres which are suspended in ~a viscoelastic liquid [26, 27] and some polymer solutions which are at concentration regions not too far away from a sol-gel transition point have rheological properties, which are very similar to the phenomena described above [28]. From our experimental results we conclude that the SIS consists of small clusters of rod-like micelles. A kinetic model is presented which can qualitatively explain the formation of SIS and in particular the threshold value of the shear rate. According to this simple model the rods have the tendency to form higher aggregates. In the quiescent state the equilibrium is still on the side of the individual rods. With


increasing shear the constants describing the individual equilibrium are continuously increased. Under these conditions the higher oligomers grow rapidly and finally all surfactant material is bound into the shear induced state.

The decay of the SIS is a multistep process and can be compared with the fast kinetic mechanism in micellar solutions. The time constant (rsis) of this process is much longer than the kinetics of breaking a single dimer and also longer than the structural relaxation time of the dynamic network at the sol-gel transition. The product of shear rate and relaxation time of the SIS is therefore much larger than one and the complete orientation of the SIS is a simple consequence of this condition. It is conceivable that some of the rods coalescence into larger aggregates during flow. This could be the reason for the fact that the solutions have still viscoelastic properties after a long period of rest. The viscoelastic properties decay with a time constant which seems to be kinetically controlled.

Acknowledgement

Financial support of this work by a grant of the "Deutsche Forschungsgemeinschaft" (SFB 213, Project C1) is gratefully acknowledged. The authors are also indebted to Farbwerke Hoechst Company for the free of charge supply with TTAS.

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Authors' address:

(Received March 12, 1987)

Dr. I. Wunderlich, Prof. Dr. H. Hoffmann, Dr. H. Rehage Institut fiir Physikalische Chemie I Universit~it Bayreuth Postfach 10 12 51 D-8580 Bayreuth

Documents

Flow birefringence and rheological measurements on shear induced micellar structures