Langmuir 1994,10,3419-3426
3419
Elongational Flow of Solutions of Rodl2ke Micelles Robert K. Prud'homme and Gregory G. Warr*>t Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received January 19, 1993. In Final Form: June 10, 1994@ Solutions of wormlike tetradecyltrimethylammonium salicylate fI"l"M'Sa1) micelles in the presence of electrolyteare examined in a uniaxial elongationalflow field. We show that, in contrast with their shear thinning behavior, these solutions elongationally thicken in a similar manner to polymer solutions. The dynamic nature of these micelles affects their flow behavior, causing micelle growth in the elongational field up to a criticalrate of strain, above which a reversible thinning occurs. Flow visualizationexperiments also indicate the formation of a nematic phase under elongation,and this is discussed in terms of current models for micelle dynamics in external fields.
Introduction Cationic surfactants in the presence of certain strongly binding counterions form micellar solutions with extraordinary rheological properties. Typically the surfactants are long chain quaternary ammonium or pyridinium ions, and the counterions are hydroxy- or halo-substituted These organic counterions bind nearly 100% to the micelle, suppressing the effect of electrostatic interactions in these systems. Not only are intermicellar electrostatic interactions damped, so are the interactions between surfactant molecules within a single micelle. This decreases the surfactant head group area and causes the micelles to adopt a cylindrical rather than a spherical morphology. These micelles grow and become polydisperse with increasing concentration, often achieving a certain degree of flexibility. At sufficiently high concentration the micelles overlap and become entangled with each other, in a manner analogous to semidilute polymer solutions. Similar effects have been reported for cationic surfactants in high ionic strength s ~ l u t i o n . ~ The polymer analogy has been the catalyst for recent theoretical advances in understanding the properties of anisometric micellar solutions. By incorporating the dynamic nature of micelles into polymer solution theories, Cates has been able to predict scaling laws for both equilibrium and dynamic properties of these systems which have come t o be known as living polymer^.^,^ "Livingpolymer" micelles display interesting rheological behavior at concentrations only slightly above their critical micelle concentration (cmc). This has been documented by Gravsholt4 and extensively by Hoffmann et Such dilute solutions are rheopectic on time scales of up to 1 h and may thicken by around a factor of 2 after long periods of steady shear. This may be due to growth of the micelles in the shear field or to the generation of a shear-induced state. In this concentration region the solution viscosity is only 1 to 2 times that of the solvent, yet the non~
1
.
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t Permanent address: Department of Physical and Theoretical Chemistry, University of Sydney, NSW 2006, Australia. Abstract published inAdvance ACSAbstracts, August 15,1994. (l)Hoffmann, H.; Rehage, H.; Rauscher, A. In Structure and @
Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution; Chen, S. H., et al., Eds.; Kluwer Academic Publishers: 1992. (2)Rehage, H.; Hoffman, H. Faraday Discuss. Chem. SOC.1983,76, 363-73. (3)Wunderlich, I.; Hoffman, H.; Rehage, H. Rheol. Acta 1987,26, 532-542. (4)Gravsholt, S.Polym. Colloids II. ( 5 ) Appell, J.; Porte, G.; Poggi, Y. J. Colloid Interface Sci. 1982,87, 492-99. (6)Cates, M.E. Macromolecules 1987,20, 2289. (7)Turner, M.S.; Cates, M. E. J. Phys.: Condens. Matter 1992,4, 3719-41.
Newtonian flow behavior is dramatic. These solutions are also viscoelastic,displaying characteristic bubble recoil when gently shaken4 At higher concentrations where the micelles become entangled, long term changes in viscosity are not reported, although non-Newtonian character persists on a shorter time scale. In both steady and oscillatory shear experiments micellar solutions shear thin above a shear rate or oscillation frequency which lies between 0.1 and 10 s-l, depending on both concentration and surfactant struct ~ r e . ~Typical , ~ , ~ of these rheologically interesting surfactants is tetradecyltrimethylammonium salicylate (TTASal), which is the subject of this investigation. Despite a large body of work characterizing their rheology,there is still much which is not understood about these solutions. For example, the structure of the viscous, shear-induced state is not known, and there has been no direct determination of micelle growth in flow fields. However for polymers, elongational flow has proven to be a useful probe of structure and dynamics. Elongational fields have the added advantage of being more amenable to theoretical analysis than shear fields. Elongational Flow. Elongational flow differs from shear flow in that the velocity gradient lies along the flow direction. In uni- or biaxial extension, unlike shear, the principle strain direction is parallel to the direction of the extension, so that no rotation of the fluid is induced. Elongational flow is thus irrotational. There have been numerous studies of elongational flow oflinear polymer solution^.^-^^ These samples, which are shear thinning, exhibit elongational thickening. This arises when the elongational stress is sufficient to uncoil the polymer from its usual random coil configuration into an aligned and partially o r fully extended chain. The stress distribution along the extended chain causes an increase in the viscosity of the liquid to a limiting high value. For highly flexible polymers, elongational thickening occurs abruptly at a particular elongation rate, corresponding to the uncoiling of the polymer when the flow time scale is faster than the molecular relaxation time scale.12 Rigid and semiflexible particles typically undergo more gradual thickening, as the alignment mechanism is convectiveand diffusive rotation ofthe rods. Alignment is achieved more gradually by a change in the average orientation of the particles with increasing field (8) Hofhann,H.;Lobl, H.; Rehage, H.;Wunderlich, I. Tenside Deterg.
1 - _~_ _~ ~ 2 2 . 2 9 0 - 2 9 8 . I - - , - - -
(9)Odell, J. A.; Muller, A. J.; Keller, A. Polymer 1988,29,1179. (lo) Keller, A,; Odell, J. A. Colloid Polym. Sci. 1985,263,181-201. (11)Pfeiffer, D.G.; Kim, M. W.; Lundberg, R. D. Polymer 1986,27, 493-502. (12)deGennes, P. G.J. Chem. Phys. 1974,60, 5030-5042.
0743-7463/94/2410-3419$04.50/00 1994 American Chemical Society
3420 Langmuir, Vol. 10, No. 10, 1994
Prud‘homme a n d Warr
strength. In both cases a limiting high elongation rate viscosity eventuates at “saturation”, where all particles are aligned with the field. Some studies have shown that the stress difference between chain ends in linear polymers in the stretched state is sufficient to break bonds in a polymer backbone, causing chain s c i s ~ i o n . ~This ~ J ~increases with increasing elongation rate, causing a gradual decomposition of the polymer and a corresponding decrease in the measured elongational viscosity.
Torque output to Gap setting rmcromcter
Zero displacement
f- Controlpanel
force transducer
m-&syringe Pump
Experimental Section Materials. Tetradecyltrimethylammoniumbromide (Sigma, 99%) and sodium salicylate (Flukapurun, ’99%) were used as
received. Solutions were prepared by mass in deionized water, so that all concentrations are reported as mol/kg of solution. All solutions are equimolar in the surfactant TI“M‘r) and sodium salicylate. Due to the high binding of the salicylate ion to the micelles, the bromide ions are essentially present only as electrolyte in bulk solution. We therefore denote the solutions as TTASal/NaBr. Both shear and elongational rheological measurements were carried out on the same samples. The elongational results were found to be reproducible after an interval of several days. Shear Rheology. Steady shear and dynamic oscillatory measurements were made using a Rheometrics RFSZ rheometer in the Couette configuration. Samples were investigated by both methods where possible. Strains in the oscillatorymeasurements were usually loo%, but ranged between extremes of 3% and 1000%. The strains used depended on the response ofthe system and were determined to be in the linear response region by scanning the strain. Steady shear measurements were used to extend the dynamic range from 100 up to 1000 s-1. In the steady shear experiments the viscosity is simply the shear stress divided by the shear rate. The complex viscosity is obtained from dynamic m e a ~ u r e m e n t s using l~ a sinusoidally oscillatingstrain applied at some frequencyo. This applied strain sets up a stress response which may lag behind the strain due to viscous dissipation. The (complex)stress response is usually written in terms of two quantities the storage modulus, G , and the loss modulus G . The complex viscosity is then defined as
Figure 1. Schematic diagram of the Rheometrics RFX elongational flow analyzer used in this study. The nozzles are interchangeable and their separation may be adjusted by use of a micrometer.
Z
stagnation plane
Figure 2. Extensional flow geometry in the RFX. Nozzle separation, d, and nozzle diameter, 2R, are shown together q* is approximately equal to the steady shear viscosityq in many systems.15 The Opposing Jet Rheometer. Elongational flows were studied using a Rheometrics RFX fluids extensional analyzer. The RFX has an opposing nozzle geometry through which liquid is sucked or ejected from a reservoir. For this study only the suction mode was used to generate uniaxial elongational flow. The instrument is shown schematically in Figure 1. This flow geometry with equal flow rates through both nozzles generates a symmetric flow field with a stagnation plane at the midpoint, depicted in Figure 2. Along the axis of the nozzle centers (2)the flow is purely extensional (or compressive if the fluid is being ejected), which generates a stress in the axial direction. The stress is measured as torque on the left nozzle arm (see Figure 1)by a zero displacement force transducer which maintains constant nozzle separation. The torque thus measured was output to a chart recorder so that the achievement of steady state could be ascertained. The elongation rate is varied by adjusting the total flow rate, which is controlled by syringe pumps. Elongation rate also depends on the radius of the nozzles, R, and their separation, d. In this work nozzle diameters of 2R = 4,3,2,1,and 0.5 mm were used. Average elongation rate is obtained from flow rate, Q, and geometry as ~~
(13)Atkins, E.D.T.; Taylor, M. A. Biopolymers 1992,32,911-923. (14)Bird, R.B.; Armstrong, R. C.; Hassager, 0.DynamicsofPolymeric Liquids;John Wiley & Sons: New York, 1987;Vol. 1. (15)Al-Hadithi, T.S.R.; Barnes, H. A.; Walters, K. Colloid Polym. Sci. 1992,270,40.
with the stagnation plane and idealized flow streamlines. z is the flow (and strain) axis. &=---
- 2Q
a~
nR2d
The stress is calculated from the torque T,and the lever arm length L,from
T
zzz - ,z = -
~ R ~ L
(3)
The apparent elongational viscosity is calculated directly as stress divided by elongation rate. The RFX calculates a running average of the apparent viscosity and was obtained separately or verified by measuring the stress from the chart recorder trace. The measured torque includes all stress components along the axial direction. The stress includes shear componentsaround the nozzles and also the effectof fluid inertia. The effect of inertia was corrected in the measured viscosity using the analytical result
Schunk et aZ.16have examined the stresses in this system in detail for Newtonian liquids by finite element analysis and (16)Schunk,P.R.;deSantos,J. M.; Scriven, L. E. J.Rheol. 1990,34, 387-414.
Elongational Flow of Micelle Solutions
Langmuir, Vol. 10, No. 10, 1994 3421 An 7" 0
4t
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0
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I
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G 2 20
i
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A
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.
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100
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e I d , p I s.l Figure 3. Flow curves for 15.3 mM "TASaVNaBr solutions shown as (inertia corrected) elongational viscosity versus elongation rate. Different symbols denote different nozzle diameter (U2)lgap(d/2)sizes (m, 3 m d 3 mm; 0 , 2 m d 2 mm). Also shown for comparison are shear viscosities determined from steady shear as a function of shear rate (A).
concludedthat the torque gives a good measure ofthe elongational stresswhen the nozzle diameter is equal to the nozzle separation. The magnitude of shear and normal stress effects in the axial direction depend strongly on the rheology of the fluid and have been neglected here as they cannot easily be isolated. They will however tend to cancel under the conditions of nozzle diameter equal to separation which we have used.l6 Flow Visualization. As the micelles are anisotropicand align in the flow field, the solution becomes birefringent during the e~periment.~ Solutionswere observed and photographed through crossed polarizing sheets. Polaroid photographs (107CPolaroid film, IS0 3000)were taken using a large format camera adaptor through an Olympus zoom stereo microscope (Model SZ-Tr). Samples were backlit using a white light source or electronic flash through perpendicular polarizers each oriented at approximately 45" to the axis of extensional flow.
ElS",
The elongational viscosities, ve,of 6.92,15.3,25.2,69.2, and 252 mM TTASal/NaBr solutions were measured as a function of elongation rate, 6, from lo-' to a t least lo3 s-l. For the lowest concentration solution, 6.92 mM, the measured elongational viscosity was a constant 0.03 f 0.02 Pa s over the range 1-5000 The large values for the experimental uncertainly are due to the fact that the torques a t low rates are near to the resolution limit of the instrument and a t higher rates the inertial correction becomes larger than the stress contribution due to the elongational viscosity. Therefore, subtle changes in the viscosity of the 6.92 mM solution with elongation rate could not be determined, and solutions more dilute than this were not examined. Above 7 mM the solutions become significantly more viscous, so that inertia corrections are of the order of at most a few percent in all of the following. Measured elongational viscosities,qe,are shown for 15.3, 25.2, 69.2, and 252 mM TTASaVNaBr solutions as a function of 6 in Figures 3-6 Also shown in each case are the steady shear and complex viscosities as a function of shear rate, y , or frequency, determined from the Couette measurements. In all cases v* and 7 agree very closely with each other over the range where both were measured. The elongational data shown in Figures 3-6 include viscosities recorded with different orifice sizes and gaps. The data superimpose well at low 6 , indicating that such geometrical variations are unimportant and suggesting that the experimentally obtained torque is a good measure of the elongational stress and viscosity.
0
/rads''
Figure 4. Flow curves for 25.2 mM TTASaVNaBr solutions shown as (corrected) elongational viscosity versus elongation rate. Different symbols denote different nozzle diameter (a)/ gap (d/2)sizes (W, 4 mm/4 mm; 0 , 2 m d 2 mm). Also shown for comparison are shear viscosities determined from steady shear as a function of shear rate (A)and dynamic shear as a function of oscillation frequency, rad s-l (A). 0
3
30
b
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;
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A h A A A A A A A A A A b A
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Results and Discussion
7/S-',
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1
, 0.1
; n2;
1
E/S", y / s - ' ,
A : Ao A
A;,
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0
100
lo00
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Figure 6. Flow curves for 69.2 mM TTASaVNaBr solutions shown as (corrected) elongational viscosity versus elongation rate. Different symbols denote different nozzle diameter (%)/ gap (d/2)sizes (H, 4 m d 4 mm; 0,2m d 2 mm; 0,lm d l mm; 0,0.5 md0.5 mm). Also shown for comparison are shear viscosities determined from steady shear asa function of shear rate (A)and dynamic shear as a functionof oscillation frequency, rad s-l (A).
The transient elongational stress behavior can be broadly divided into three regions. At low strain rates a steady stress is achieved shortly after the beginning of the test. This occurs roughly by one strain unit; i.e. 6t = 1. At intermediate rates a slow relaxation is in evidence during which the stress slowly climbs toward its steady state value over '100 strain units. Finally at high elongation rates the measured stress varies wildly with time, and an average stress is barely recognizable from the noise. In this region the apparent elongational viscosity decreases with increasing rate of strain. Sample stress curves are shown in Figure 7. Elongational thickening is well-documented in dispersions of anisotropic particles. The phenomenon of elongational thickening has as its origin the generation of a stress difference due to the velocity difference between the ends of aligned particles in an elongational field. The gradual thickening that occurs over a decade in elongation rate for the more dilute solutions, in contrast with the sudden effects documented for flexiblepolymers, identifies
Prud‘homme and Warr
3422 Langmuir, VoZ. 10, No. 10, 1994 120
1
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80
100
120
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Figure 6. Flow curves for 252 mM TTASaVNaBr soslutions shown as (corrected) elongational viscosity uersus elongation rate. Different symbols denote different nozzle diameter (%)I gap ( d 2 ) sizes (m, 4 m d 4 mm; 0 , 3 m d 3 mm; 0 , 2 m d 2 mm; 0 , l m d l mm). Also shown for comparison are shear viscosities determined from dynamic shear as a function of oscillation frequency, rad s-l (A).
them as fairly rigid particles which orient primarily by rotation, coupled with dynamic exchange in this case, rather than by the “snapping straight” envisaged for random coil chains. At higher elongation rates for the higher concentration solutions (Figures 4-6) the viscosity increases by as much as 4-fold over a factor of 3 to 5 in 6 . In the elongational thickening region the data for different geometries are generally not superimposable. In addition the measured elongational stress in this region is time dependent over the duration of the experiment, which varies from 15 s up t o several minutes. Hoffmann et aZ.17 also observed time dependence in the shear viscosity of such systems, although in more dilute solution, and has postulated a shear-induced phase transition to a network structure. At 15.3 mM the low 6 viscosity is 2.4 f 0.2 Pa s, which increases to a maximum of 4.0 Pa s a t 100 s-l, before gradually decreasing upon further increasing the strain rate. In the 25.2 mM solution the maximum viscosity of approximately 38.0 Pa s occurs at 10 s-l, increasing from a low 6 value of 10.0 f0.7 Pa s. At higher concentrations, the viscosity maximum occurs at around 35 s-l, which is higher than at 25 mM. These liquids are all mildly shear thinning; that is they have constant shear viscosities up to a certain shear rate or frequency, above which they are shear thinning. This viscosity fall-off corresponds to the rate of the longest relaxation of the solution, and above this point shear flow significantly modifies solution structure, which will be discussed below. The fall-off in shear viscosity occurs at the same point at which the solutions begin to thicken under elongation (see Figures 4-6). This suggests that the two phenomena arise from the same mechanism; alignment of the micellar rods with the flow streamlines. In shear this facilitates the flow, decreasing viscosity by decreasing rod-rod interacti~ns,’~ but in elongation rod alignment increases tensile stress.ls In the low elongation rate (Newtonian) limit the viscosities should obey the Trouton ratio, ~ $ 7= 3. Elongational and shear viscosities are compared with limiting low shear viscosities in Figure 8. In none of these systems is the Trouton ratio observed, even at very low shear. The deviation at the lowest concentration could be (17)Rehage, H.; Hoffmann, H. J.Phys. Chem. 1988,92,4712-4719. (18)Batchelor, G.K.J . Fluid Mech. 1971,46, 813-829.
400
300
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200
100
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40
60
Figure 7. Transient elongational stress curves for 69.2 mM solutions at & = (a, top) 2.0, 5.0, and 10.0 s-l; (b, bottom) 30.0, 50.0, and 100.0 s-l. Chart deflection (mV) is proportional to the total torque on the nozzle arm of the rheometer.
ascribed to experimental uncertainty, but the remaining values all follow the same downward trend, only intersecting the accepted value of 3 at a single, high concentration. Relaxation Dynamics in Micellar Solutions. Polymer-like micelles are labile systems which may relax by several processes. Being dynamic aggregates, it must be recalled that they have a characteristic relaxation time associated with their rate of break up and fusion to reach an equilibrium size distribution. This process has been experimentally investigated by temperature jump and typically lies in the range of 10-1000 ms.I9 It has been observed that in dilute solution the micelle breakup rate parallels the shear rate at which shear thinning sets in. For TTASal, electric birefringence measurements of the characteristic micelle break-up time are te= 6.6 s at 2.5 mM, 2.7 s a t 5 mM, 1.1s at 7.5mM, and 0.34s at 10 mM, all at 20 O C Z 1 (19)Kern,F.;Lemarechal, P.; Candau, S. J.;Cates, M. E. Langnuir 1992,8,437-440.
Elongational Flow of Micelle Solutions
o
Langmuir, Vol. 10,No. 10, 1994 3423 above equation for times shorter than the entanglement time scale, and for longer times goes approximately as D(L) #-2L-7. A characteristic diffusive rotation time may therefore be defined as
-
30
3
-8 i” 20 ; I
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0
5
Trouron Ratio
---------------------a------,
’
a-------
A
where is the mean micelle length, which varies with composition. In a solution at equilibrium, micelles can break with equal probability at any point along their length. These breaks occur with some “unimolecular”rate constant, Itl, giving rise to a characteristic breaking time
(6) Micellar solutions thus exhibit a spectrum of breaking times, of which an average will be measured by temperature jump. The decreasing average breaking times given above indicate increasing micelle length with increasing concentration. This parallels temperature jump results for the relaxation times of cetyltrimethylammonium bromide micelles, which decrease with increasing con~entrati0n.l~ In a flow field in which the micelles align, there is a second characteristic time scale associated with the rotation ofthe micelles. For rigid rods this time is a strong function of rod length and entanglement.22 For dilute slender rods the rotational diffusion coefficient is
D(L)=
3kT ln((L1b)-a)
VsL3
(7)
where L is rod length, b rod diameter, vssolvent viscosity, and a approximately constant and of order 1. In entangled networks the rotational diffusion constant is given by the (20) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W.J. Chem. Soc., Faraday Trans. 1 1976, 72, 1525.
(21) Ohlendorf, D.;Interthal, W . ;Hoffmann,H. Rheologica Acta 1986, 25,468-86.
(22) Doi, M.;Edwards, S.F. J.Chem. Soc., Faraday Trans.2 1978, 74, 918-32.
where DOis a constant and 5 >2.7t23 In micellar systems, break-up dynamics and rotational alignment are coupled, so that an angle-dependent size distribution qXL,@can develop. Turner and Cates‘ note that micelle polydispersity embodied in eq 5 necessarily gives rise to two regimes of behavior. Micelles shorter than some critical length will rotate in the field faster than they redistribute their sizes, whereas very long micelles will break into segments before rotating through a very large angle. There is a competition between two mechanisms, the final balance depending on the two characteristic times rbre& and Trot, and their differing dependences on micelle length. Sufficiently long micelles thus re-equilibrate their size distribution much more rapidly than they diffuse between angles. Short rods however will re-equilibrate slowly compared with their rotational diffusion time. In both cases, however, strong alignment of micelles is predicted in an elongational field. In addition, the relatively rapid re-equilibration of long rods leads to the prediction of a dramatic increase in length in the flow d i r e c t i ~ n . ~This , ~ ~occurs especially for elongational fields because of the absence of vorticity and the consequent tumbling of the aligned micelles. At higher concentrations where micelles become significantly entangled, rotation of rods gives way to reptation as the main relaxation mechanism. Cates6has examined the modification of stress relaxation by reptation in networks with dynamic breakup of the entangled chains. This appears to work well for small deformations, but in strong alignment the rods will behave more like a dilute dispersion, at least in the plane normal to the flow direction. The relevant experimental time scale, excluding nonsteady-state macroscopic flows, is the residence time in the elongational field. Avolume element at the stagnation point nominally takes infinite time to reach an orifice; however a mean residence time can be deduced from the time required for the flow field to suck out the volume between one orifice and the stagnation plane Zres
2Q = -1 -nR2d
rresis thus independent of the geometry selected for a particular experiment if the ratio of Rld is held constant. In order that the above considerations apply to the present system and a steady state of#(L,B)be generated, we would require that Trot and rbre&
