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Rheological behaviour of a lamellar liquid crystalline surfactant–water system
Colloids and Surfaces A: Physicochemical and Engineering Aspects 145 (1998) 107–119
Rheological behaviour of a lamellar liquid crystalline surfactant–water system Zs. Ne´meth a,*, L. Hala´sz a, J. Pa´linka´s b, A. Bo´ta a, T. Hora´nyi a a Technical University of Budapest, Department of Physical Chemistry, H-1111, Budapest, Budafoki ut 8, Hungary b CAOLA Cosmetic and Household Industrial Ltd., H-1116, Budapest, Vegye´sz u 19–25, Hungary Received 29 January 1998; accepted 19 June 1998
Abstract The rheological behaviour of a lamellar liquid crystalline surfactant–water system was investigated using a Haake RS-100 viscometer. The repeat distance of lamellae as a function of surfactant concentration and temperature was measured by small-angle X-ray scattering. The rheological characteristics were measured at different temperatures with systems containing surfactant in different concentrations. The frequency-dependent storage and loss modulus were found to be characteristic to the lamellar phase in the linear viscoelastic region. The results are analysed on the basis of Jones–McLeish slip-plane theory. Trends of the fitted constants are discussed based on the general knowledge on the interactions in dispersions stabilized by non-ionic surfactants, and the structure of lamellar liquid crystalline samples. Time-dependent compliance was also measured. The instantaneous elasticity measured in the creep tests was compared with that predicted from oscillatory tests. Burger’s model was used to describe the time-dependent compliance. Viscosity measurements in the non-linear region were also done. A modified Carreau equation is used to describe the viscosity versus shear rate curves. The changes in sample under shear is described briefly. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Non-ionic surfactants; Lamellar liquid crystals; Rheology
1. Introduction Surfactant-based liquid crystals have a broad range of application in the cosmetic and food industries [1–6 ]. One of the most important systems are those in which bilayers are formed from the surfactants: these are the lamellar mesophase, the bicontinuous cubic phase, the liposome phase and the so-called onion phase. The rheological properties of such systems were reported in some papers [7–25]. In spite of its importance, the microscopic origin of the rheological behaviour is * Corresponding author.
not yet well understood and has been the subject of many recent investigations. Explaining the flow curves of a three-component lamellar mesophase, Bohlin [7] modelled the lamellar structure as flexible layers of water in a liquid hydrocarbon chain environment. Flow was associated with cooperative changes of conformation of flexible layers subject to frictional forces in the liquid layers. The flow coefficients were found to be dependent on the thickness of water layer, the bilayer interaction and the mobility in the hydrocarbon chain layer. Investigating the temperature dependence of flow, Oswald [8] took into consideration the
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importance of the undulation of the bilayers and in the higher temperature range the number of the dislocation loops. Other authors [9–13] believe in the importance of the orientation of lamellae under shear. The orientation of domains in the case of an unsheared sample is dependent on the domain–domain interaction. If stress is applied on the sample the orientation changes, and the competition between the interactions and the external stress defines the viscosity of the system. Mang et al. [12] measured the small-angle X-ray scattering (SAXS ) of a lamellar system under shear and found that the domains order in the direction of the stress as the strain increases. Penfold et al. have investigated the alignment under shear of lamellae of an aqueous hexaethylene glycol monohexadecyl ether (C E ) sample in a 16 6 Couette shear cell by small-angle neutron scattering experiments [13]. They have found that at low shear the lamellae are ordered parallel to the flow–vorticity plane. At high shear gradients the lamellae are ordered in the orthogonal director, parallel to the flow–shear gradient plane, consistent with the theoretical predictions of Cates and Mildner [14]. At intermediate shear a biaxial alignment was found. They have also made measurements on a sample subjected to oscillatory flow in the range of 0–180 s−1. Results were similar to those observed with the conventional Couette flow. At low values of oscillatory flow, alignment of lamellae in the flow–vorticity plane has been found. Penfold et al. propose to re-evaluate the nature of the ‘‘defective’’ lamellar phase on the basis of shear-induced ordering. The viscoelastic behaviour of a lamellar anionic surfactant, AOT–water system was investigated by Robles-Vasquez et al. [15,16 ]. They considered the lamellar mesophase as a weak gel, but quantitative descriptions were not used for the explanation of the experimental results. Viscoelastic behaviour of liposomes was intensively investigated by Hoffmann et al. [17,18], who found a frequency-independent storage and loss modulus. Onion phases have been investigated by Panizza et al. [19]. The creep compliance curves were described by Burger’s model, consisting of a
Maxwell and a Voigt element in series. In investigating the results of the dynamic tests the same model was used by Laplace transformation of the creep equation. The agreement between the calculated and measured results were good in the case of the storage modulus, but a significant difference was found in the case of the loss modulus, mainly in the high frequency region. Doi et al. [20] used a slippage model to describe the rheological characteristics of ordered block coplymer samples. With the slippage mechanism, non-linear elasticity and a definite yield stress have been explained. Jones and McLeish [21,22] introduced a slipplane theory to describe the frequency dependence of the storage and loss modulus of cubic phases. They propose their model to be used more generally to describe relaxation in weak solids. It is an alternative to the more familiar mechanism of the movement of dislocations such as that found in many crystalline solids. The present paper reports the results of careful rheological investigations which were done both in the linear and in the non-linear viscoelastic regions on non-ionic lamellar samples containing surfactant in different concentrations or at different temperature. The applicability of the McLeish theory is investigated in the linear viscoelastic region. The trends of the fitted parameters with surfactant concentration and with sample temperature are discussed based on the basic knowledge on the properties of non-ionic surfactant systems and on the results of SAXS measurements. The results of the time-dependent creep tests are also compared with the results of the tests made by using oscillatory flow. The rheological characteristics of the sample in the non-linear region are interpreted by using the well-known Carreau equation, which is a general equation of the polymer rheology to characterize gels in which a timedependent temporary network exists. 2. Experimental section 2.1. Materials We studied a lamellar liquid crystalline system consisting of a non-ionic surfactant (Synperonic A7, ICI product) and water. The surfactant was
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supplied by ICI Surfactants and used as received. It was made by the ethoxylation of Synperol alcohol which consists of 66% C and 34% C 13 15 alkyl chains. The ethoxylation process gives rise to a wide distribution of ethoxylate chains, therefore only the average ethoxylation number can be declared, which is in case of A7 is 7. Although the properties of sample should not change from batch to batch, the preparation of the lamellar crystals investigated here was always done from surfactants from the same batch. The phase diagram of the Synperonic A7–water system was established by Tadros et al. [24,26,27] by rheological, calorimetric and NMR measurements and by polarization microscopic observations. They have found normal micellar (L ), 1 hexagonal liquid crystalline (H ), lamellar liquid 1 crystalline (L ) and inverse micellar (L ) phases a 2 with increasing surfactant concentration at 23°C. At this temperature the lamellar phase exists in between 55 and 80% (w/w). At 50–55% (w/w) the coexistence of the hexagonal and the lamellar phase was observed. The borders of the lamellar phase in the phase diagram do not change markedly with the temperature, but lamellae melt at about 50°C and at higher temperature an optically isotropic liquid forms. Our samples were prepared by heating the aqueous mixtures to about 50–55°C where they were easily homogenized, and after homogenization mixtures were left to cool to room temperature. Then the samples were stored for 1 week before measurements. Special care was taken to prevent evaporation of the water content of the samples during both their preparation and their storage. Mixture were prepared with distilled water, having a surface tension of 72–73 mN m−1. The samples prepared this way were checked by observing their textures in a polarization microscope before rheological measurements. Lamellar phases were found in all cases in the concentration range of 60–80% (w/w) in good agreement with the findings of Tadros et al. 2.2. Methods Rheological measurements were made with a Haake RS 100 apparatus. A cone–plate sensor was
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used, with a diameter of 20 mm, and a cone angle of 4°. The sample thickness in the middle of the sensor was 0.134 mm. The maximum allowed variation in temperature during measurements was ±0.2°C. Samples were kept under saturated water vapour for the whole time of the measurements. Because the rheological properties of such systems are dependent on the shear-deformation history, the sample was gently inserted to the top of the plate of the sensor, and then the plate was slowly elevated to its measuring position with constant velocity. The sample squeezed out from the sensor system was then gently removed. Measurements were carried out after a 10 min waiting (relaxation– thermostation–saturation) period. Dynamic (oscillation) and static (creep– recovery) test were done on all samples. First the linear viscoelastic region was determined by measuring the complex modulus versus stress at a given frequency, and then 2.5 Pa was chosen as a stress amplitude, which was found to be in the linear viscoelastic region in all cases. Creep tests were also made on the same samples. A stress of 2.5 Pa was applied for 5 min on the samples and the compliance (J ) was monitored during both the creep and the following recovery period (with zero applied stress). Flow curves were measured in two ways: in case of the yield value determination the measurements were made by increasing the stress with a constant rate (0.166 Pa s−1). The shear rate-dependent viscosity was determined by steady state measurements. At each stress value the result was accepted if its change was lower than 0.05% s−1 with the time. For the SAXS measurements the samples were transferred to thin-walled Mark quartz capillaries (Hilgenberg, Germany) of diameter 1 mm. The capillaries were closed with a two-component synthetic resin and transferred to metal capillary holders into an aluminium block. This block was placed directly in the beamline and used as a thermal incubator for controlled annealing at different temperatures. The block was held at the desired temperatures using a thermostat. The actual temperatures were constant to within less than 0.05°C, as monitored with a thermocouple.
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The windows of the block were covered with Mylar foil. The SAX camera was a Kratky camera (Anton Paar, Graz, Austria) attached to the line-focus window of a Cu tube at a generator (Seifert, Ahrensburg, Germany) operated at 40 kV and 20 mA. A Ni filter was used to eliminate Kb radiation. The scattering data were collected with a proportional counter. The scattering data curves were normalized and desmeared for the slit geometry.
3. Results 3.1. Small-angle X-ray scattering On the experimental SAXS curves of the lamellar samples investigated a very intense first-order Bragg refraction and a second, very diffuse peak corresponding to the second-order Bragg refraction appear. The desmeared first-order peak profile is shown in Fig. 1. The scattering of the samples is characteristic of a strongly ordered structure containing lamellae with a well-defined repeat distance. The repeat distance of lamellae was measured as a function of the surfactant concentration at 25°C. The repeat distance of the samples decreases with increasing concentration. The effect of temperature on the repeat distance, measured in the case of the 70% (w/w) sample was found to be negligible in between 25 and 50°C.
Fig. 1. Typical results of small angle scattering measurements at 25°C. The calculated repeat distance is also shown on the figure for samples containing A7 in different concentrations.
3.2. Dynamic oscillatory tests Typical results of the complex modulus as a function of the stress at low frequency (0.036 Hz) can be seen in Fig. 2. It can be stated that most of the samples have linear viscoelasticity up to about 10 Pa. For subsequent dynamic experiments we have chosen the stress value of 2.5 Pa. Typical results of the storage modulus, loss modulus, and complex viscosity of the lamellar mesophase, formed from aqueous Synperonic A7, as a function of the frequency in the linear viscoelastic region can be seen in Fig. 3. The system is more elastic than viscous in the range of frequency
Fig. 2. Complex modulus (G*) of a lamellar sample of 70% (w/w) A7 solution as a function of the applied stress (t) at a constant ( low) frequency (0.036 Hz) at 25°C.
investigated, and the storage modulus is higher by about one order of magnitude than the loss modulus throughout the whole frequency range. While the storage modulus has a weak dependence on
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Fig. 3. Elastic (G∞), and loss modulus (G◊), and complex viscosity (g*) of 70% (w/w) aqueous A7 lamellar system as a function of the frequency (v) at a stress of 2.5 Pa at 25°C.
the applied frequency, the loss modulus shows a minimum. The complex viscosity is strongly frequency dependent: the higher the frequency, the lower is the complex viscosity. The picture described above is characteristic only of the lamellar mesophase; both of the other phases (hexagonal phase, and inverse micellar phase) which surround the lamellar phase in the phase diagram show a completely different behaviour as a function of frequency ( Figs. 4 and 5). Because the rheological behaviour of lyotropic mesophases is often strongly dependent on the shear history, we examined the effect of preshear on the results of the dynamic measurements. For this a 70% (w/w) sample was sheared for 5 min with a constant strain. In the first experiment
Fig. 4. Elastic (G∞), and loss modulus (G◊), and complex viscosity (g*) of 45% (w/w) aqueous A7 hexagonal system as a function of the frequency (v) measured at 2.5 Pa stress at 25°C.
111
Fig. 5. Elastic (G∞), and loss modulus (G◊), and complex viscosity (g*) of 90% (w/w) aqueous A7 inverse micellar system as a function of the frequency (v) measured at a stress of 2.5 Pa at 25°C.
0.2 s−1 and in the second one 10 s−1 preshear was applied, and soon after that dynamic measurements were carried out on the presheared sample. Results of these measurements showed that the preshear has only a negligible effect on the results of the dynamic tests. Neither the storage modulus nor the loss modulus changed strongly with the applied preshear. Only the minimum of the loss modulus became more pronounced as the strain of the preshear increased. The effect of concentration on the storage and loss moduli was examined within the lamellar concentration region at 25°C. The shape of the above-mentioned characteristics as a function of frequency does not change with the concentration of the surfactants. A summary of the results can be seen in Fig. 6, where we show the storage modulus, the loss modulus and their ratio (tan d) as a function of the surfactant concentration at a given frequency. It can be seen that both the storage and loss modulus increase with increasing concentration, but the rate of their increase is different. As the system becomes more concentrated the sample becomes more and more elastic, as can be seen from the decrease in tan d. In the case of 80% (w/w) concentration, while the loss modulus increases further, the storage modulus begins to decrease. Investigating the structure of this sample in the polarization microscope, we observed a change in texture at 80% (w/w) concen-
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Fig. 6. The concentration dependence of the elastic (G∞), the loss modulus (G◊) and their ratio (tan d) at 1 Hz, measured with stress amplitude of 2.5 Pa at 25°C.
Fig. 7. The temperature dependence of the elastic (G∞), the loss modulus (G◊) and their ratio (tan d) at 1 Hz measured with a stress amplitude of 2.5 Pa.
tration. While for the concentration range 60–75% (w/w) ‘‘Maltese crosses’’ are seen, in the case of 80% (w/w) surfactant concentration, the texture is of the oily streaked type. This change of texture indicates an orientation change [25] of the domains, which is connected to the deviation of the trend of the storage modulus. These observations are in good agreement with the findings of Tadros et al. [26,27] The effect of temperature was also examined for the 70% (w/w) A7 lamellar system. The shape of the storage and loss modulus and the complex viscosity as functions of frequency do not change with the temperature until the phase boundary. Above this temperature the system shows rheological behaviour characteristic of a system consisting of inverse micelles. This was found at the same temperature where the scattering experiments indicated the complete disappearance of the lamellae. In the lamellar region the quantitative results of dynamic tests are strongly temperature dependent (Fig. 7). Both the storage and loss modulus are decreasing with increasing temperature, but the rate of their change is different, the decrease of the elastic component being much more pronounced. This can be evaluated from tan d, which increases with temperature. The change of storage modulus is more pronounced in the lower temperature region (between about 5 and 25°C ). The elastic component of flow diminishes as the phase boundary is reached with increasing temperature.
The change of storage modulus can be characterized by two straight lines and the break point of the lines is at 25°C. This value is very close to the melting point of the water-free surfactant. Assuming that the melting point of the carbon chains is not influenced strongly by the aqueous layers which are separated from these chains with the ethoxylate groups, this break point seems to be a transition temperature between the gel and the liquid crystal state of the lamellar domains.
3.3. Creep tests Creep–recovery tests were also made on the samples under the same conditions as in the case of the dynamic tests. Typical results can be seen in Fig. 8. When the constant stress impulse value was increased, the shape of the creep curves did not change in the linear viscoelastic range and above this range the creep curves showed only a simple viscous behaviour. The stress limit where the shape of the curves changed was almost the same as that obtained from the dynamic measurement. From the curves it can be concluded that all of the samples can be described by three values of compliance: J=J +J +J 0 R V
(1)
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3.4. Steady state viscosity measurements
Fig. 8. Typical results of creep-recovery tests at 25°C. 2.5 Pa stress was applied for 5 min and the creep value was monitored during this and the subsequent 5 min period. The concentrations of the samples were: &, 60% (w/w); *, 65% (w/w); $, 70% (w/w); #, 75% (w/w); n, 80% (w/w).
Flow curve measurements were made on the same samples that had previously been tested in the linear viscoelastic region. The effect of temperature and concentration was investigated The shear viscosity versus shear rate and the complex viscosity versus frequency for different surfactant concentration are shown in Figs. 9 and 10. Both the shear viscosity and the complex viscosity are decreasing functions of the shear rate and the frequency, respectively, but the Cox–Merz rule (i.e. the complex viscosity is equal to the shear viscosity if the shear rate is the same as the frequency) does not hold. Both the shear viscosity
where J is the instantaneous elastic compliance, 0 J is the retarded elastic compliance and J is the R V viscous or remaining compliance. The retarded creep compliance can be represented by a sum of discreet number i of compliances J and timei dependent parts:
C
A BD
t J =∑ J 1−exp − R i l i i
(2)
where l is the retardation time of ith component. i From the curves of Fig. 8. the parameters of Eqs. (1) and (2) were determined and are summarized in Table 1.
Fig. 9. Shear viscosity versus shear rate at 25°C for different surfactant concentrations: &, 60% (w/w); *, 65% (w/w); $, 70% (w/w); #, 75% (w/w); n, 80% (w/w).
Table 1 The parameters of the fitted Burger’s model of creep tests which were done at 25°C Concentration (% (w/w))
J ×10−3 0 (Pa−1)
J ×10−3 1 (Pa−1)
J ×10−4 v (Pa−1)
l (s) 1
60 65 70 75 80
3.94 1.64 1.22 0.90 0.80
0.56 0.19 0.16 0.05 0.03
6 2 0.8 0.2 0.1
8.1 6.2 5.6 3.8 3.5
Fig. 10. Complex viscosity vs. frequency at 25°C for different surfactant concetrations: &, 60% (w/w); *, 65% (w/w); $, 70% (w/w); #, 75% (w/w); n, 80% (w/w).
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Fig. 11. Shear viscosity versus shear rate for 70% (w/w) surfactant concentration at different temperatures: &, 25°C; $, 40°C.
and the complex viscosity increase with increasing surfactant concentration in all region of the shear rate and frequency. The viscosity of samples decreases with increase in temperature at low shear rates, where the structure of the sample is only partly destroyed, but as can be seen from Fig. 11 the temperature has no effect on the viscosity above 10 1 s−1, where the structure of the sample is more or less completely destroyed. The yield values (t ), which were measured by 0 increasing the stress acting on the samples at a constant rate, are shown in Table 5. They show an increasing tendency with increasing surfactant concentration and a decreasing tendency with increasing temperature.
4. Discussion The rheological properties of the lamellar mesophase formed from the non-ionic Synperonic A7 were investigated both dynamically (with oscillation tests) and statically (with creep tests and steady state viscosity measurements). In the linear viscoelastic region, where the structure of the system does not change significantly during the rheological measurements, the lamellar mesophase behaves as an elastic solid, its storage modulus being about one order of magnitude higher than
Fig. 12. The slip-plane model.
its loss modulus. This elastic behaviour can be also seen from the compliance–time curves, where the curves show a large instantaneous compliance. The dynamic behaviour as a function of frequency, which can be seen in Fig. 3, is as mentioned before characteristic of lamellar systems. Other phases which can be formed from the same surfactant mixtures at other concentrations behave in a completely different way (Figs. 4 and 5). Lamellar phases formed from other aqueous surfactant mixtures have a similar viscoelastic behaviour to the system here, for example RoblesVa´squez [15,16 ] have reported a similar behaviour for lamellar mesophase of aqueous Aerosol AOT. In that case the storage modulus was nearly independent of frequency, the loss modulus showed a minimum, and the complex viscosity decreased with increasing frequency and did not obey for the Cox–Merz rule, which is the same as our results. According to this we can conclude that the type of behaviour which we observed for the lamellar phases is characteristic of lamellar phases in general. The role of preshear was negligible, only the minimum of the loss modulus curves becoming more pronounced. This means that during steady state shear the lamellar liquid crystalline structure changes and these changes recover quickly after the cessation of shear; thus the effect of preshear seems to be negligible. The results of the rheological behaviour in the linear viscoelastic range were evaluated using the
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slip plane theory developed by Jones and McLeish [21,22] for cubic phases. This theory assumes that there are planes in the sample which are aligned parallel to the shear–vorticity plane. As mentioned in Section 1, this structure of a sample under shear was experimentally found in a very similar system by Penfold et al., so it is reasonable to try to fit the slip-plane theory to our experimental results. The shear stress acting tangentially results in a displacement between the layers of the sample (Fig. 12). When we apply a fixed strain to the sample, if no slipping occurs, the stress exerted at the top surface of the sample is: t=GP+gP˙
(3)
where G is the bulk modulus, g is the bulk viscosity of the liquid crystalline, P is the gradient of displacement and P˙ is its time derivative. Eq. (3) is a simple Voigt–Kelvin equation. Once a slip has occurred, the strain in one layer is high but in all others is very low. The deformation in the slip plane is very high and mainly viscous arising from the flow along the surface. The viscous interaction disturbs the lattice potential as an elastic tension. In the case of constant strain a second viscous term is added to Eq. (3) in which h is the viscosity 2 of the slipping plane. The relaxation modulus is now:
AB A B A B
G t = G+
g
l
exp
−t l
(4)
where l=
g+Ng
2
(5)
G
is the relaxation time. The components of the complex modulus are: G∞(v)=AG
(lv)2−(l v)2 1 1+(lv)2
(6)
G"(v)=AG
lv(1+ll v2) 1 1+(lv)2
(7)
where l =g/G 1
(8)
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and Ng 2 g+Ng
(9) 2 This linear model can be extended by introducing a weak elastic force acting at the slip planes due to the connectivity of the structure. The components of complex modulus considered a series, and the higher terms were calculated by the perturbation method. It was supposed that at low frequencies there is a slip–stick phenomenon, but there is no slip below a critical stress value. In this region the two components of complex modulus are: A=
G∞=
Ge c 0 Ge
C A BD A B 1− 1− e
c 0
#c−3/2 0
(10)
#c−1 (11) 0 c 0 0 where e and c are the critical values of the strain 0 and the amplitude, respectively. Considering the shape of our modulus curves it seemed to us that the slip–stick and the linear model may describe our curves. In Table 2 we summarize the values for G∞ and G◊ calculated using the slip–stick model (assuming a ratio of 0.8 for e/c and a G value 0 which was taken as an inverse of the instantaneous creep compliance.) The agreement between the calculated and measured values is rather good. The fitted (using the results of the linear theory) modulus values are given in Table 3. The fitted curves can describe the experimental curves rather well in the case of storage modulus, but the agreement is not good in the case of the loss modulus curves, the minimum values of the of the fitted curves being much smaller and the fitted function more curved than the experimentally measured ones. When we took into consideration the effect of lattice potential by the perturbation method the results did not improve significantly. As an example we show in Fig. 13 the fitted results as a function of frequency in the case of 70% (w/w) Synperonic A7 at 25°C. Let us first compare G calculated from the o fitted results with that measured in the creep tests. The predicted results shown in Table 4 are in rather G"=
c
1−
e 2 1/2
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Table 2 Measured and calculated moduli at 10−2 Hz for samples at 25°C containing surfactant in different concentrations Concentration (% (w/w))
Storage (measured ) (Pa)
Modulus (calculated) (Pa)
Loss (measured ) (Pa)
Modulus (calculated) (Pa)
60 70 80
220 606 1250
198 636 1050
35 113 202
40 97 200
good agreement with the experimental results measured at 25°C, showing no discrepancy between the two type of results. Let we then discuss the trends of the fitted values. The density of slip planes (N ) was calculated from the fitted values of AG, l and l in 1 Eqs. (6) and (7). A#1 because l and l are 1 strongly different, so the fitted AG was kept to be equal to G. Knowing l and G, g was calculated 1 from Eq. (7), and then N could be calculated as a function of g (which is unknown) from Eq. (9). 1 Table 3 Fitting results of the linear slip plane theory for samples as a function of the surfactant concentration and sample temperature c (% (w/w))
T (°C )
AG (Pa)
l (s)
l (s) 1
v (rad s−1)
60 65 70 75 80 70 70 70 70 70 70
25 25 25 25 25 5 15 35 40 45 54.5
220 360 560 1010 1010 2040 1375 570 320 350 110
161 85 132 122 152 24 38 83 65 58 10
0.0317 0.0240 0.0184 0.0082 0.0103 0.0154 0.0086 0.0149 0.0189 0.0172 0.0346
0.424 0.7 0.8 1 0.8 1.9 1.75 0.9 0.9 1 1.7
Nevertheless, because g is the viscosity of the slip 1 planes, its value should be between g and the viscosity of the pure solvent (water) Taking these limits for g we obtain a range of values between 1 103 and 107 for N. G is increasing with increasing surfactant concentration and is decreasing with increasing temperature. It seems that G is a function of the lamellar–lamellar interactions. At a given temperature, the repeat distance of the lamellae decreases, and this means an increased interaction between lamellae which causes an increase of G. At a given surfactant concentration the repeat distance is independent of the temperature, but naturally the interactions are functions of temperature. In the case of lamellae formed from non-ionic surfactants the interactions originate from the van der Waals forces, and steric–hydration forces. The hydration repulsion is strongly temperature dependent [28], it decreases with increasing temperature. From these facts it seems reasonable to conclude that steric–hydration interactions play an important role in determining the interlamellae elastic modu-
Table 4 Instantaneous elasticity measured in creep tests and predicted from the oscillating tests for samples at 25°C Concentration (% (w/w))
G (predicted) (Pa) o
G (measured) (Pa) o
60 65 70 75 80
220 360 560 1010 1010
254 609 816 1110 1250
Fig. 13. Measured and calculated moduli. The measured data are the same as was depicted in Fig. 3. Lines represent calculated values.
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Fig. 14. Shear viscosity versus shear rate for for 70% (w/w) surfactant concentration at 30°C. The dots represent the measured values, the line is the fitted curve. The fitted parameters are also shown on the figure.
lus (G ). The same conclusion (the influence of the repulsion forces in the rheological characteristics) was drawn by Robles-Vasquez [15,16 ] l shows a maximum as a function of the temperature. It is dependent on the viscosity values, the elastic modulus, and the rheological domain size (N ) Eq. (5). While the ratio of the viscosity to elastic modulus increases with increasing temperature, N may decrease (the enhanced temperature may increase the number of slip planes, i.e. the disorder in the sample), and the superposition of these changes can cause the maximum in l. l , which is the ratio of the interlamellar viscos1 ity and elasticity ( Eq. (8) is a monotonically decreasing function of the concentration. This is in good agreement with the fact that the ratio tan d of the macroscopic elasticity (G∞) to the
viscous component (G◊) shows a monotonic increase with concentration. The data of creep tests show that the liquid crystalline structure has a bulk elastic and viscous strain component, and the elastic component in the slip plane also plays a role in the deformation process. Thus the creep curves have three types of compliances as shown in Eqs. (1) and (2). The steady state flow curves show yield value. The theory given by Doi et al. [20] predicted a yield value for block copolymer liquid crystals but their simple equation describing the steady state viscosity versus shear rate did not give a good approximation for our experimental results. In our system the shear rate causes some orientation and some change in structure. The steady state flow curves were described by a modified Carreau equation [29]: g=
g a 1+b(lc˙ )m
(9a)
where g is the viscosity extrapolated to zero shear, a l is a characteristic time, and b and m are constants. This equation is generally applicable to the description of the shear-dependent viscosity function of any interlinked samples in which the shear causes a disruption of the sample structure in the non-linear viscoelastic region. The fitted curves showed very good agreement with the experimental values ( Fig. 14). In the case of a lamellar system the change in structure can be the change of the domain size, the change of the alignment of lamellae and the change of the
Table 5 Fitted constants of Eq. (9a) as a function of the surfactant concentration and the sample temperature c (% (w/w))
T (°C )
t (Pa) o
g (Pa s) a
b
l (s)
m
60 65 70 75 80 80 80 80
25 25 25 25 25 30 40 50
4 14 28 33 39 35 25 8
19.03 80.38 144.70 220.4 250 78.6 6.5 0.6
8.988 7.277 3.926 2.109 16 0.392 2.246 8.905
0.111 0.810 1.544 0.227 0.388 48.967 321.6 985.9
0.446 0.633 0.782 0.751 0.758 0.667 0.308 0.022
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shape of the lamelllae (for example by liposome formation). Lauger et al. [30], investigating a three-component system, found reversible liposome formation due to the effect of the shear which was accompanied with the increase of the viscosity. Because in all cases with our sample the viscosity is a monotonically decreasing function of the stress the last effect may be considered to be negligible. The other mechanisms can play a role in changing the viscosity by changing the connectivity of the samples. The fitted parameters are shown in Table 5. g a is increasing with increasing concentration and decreasing with increasing temperature. The decrease of g with temperature is exponential and a can be characterized by an Arrhenius-type of equation. The activation energy determined was 2 J mol−1 for 70% (w/w) concentration. The value of m does not change markedly with the sample concentration, showing that the mechanism of the disruption of the samples is nearly independent on the surfactant concentration, although m depends strongly on the temperature.
5. Summary The complex rheological behaviour of a liquid crystalline surfactant–water system was investigated. The frequency-dependent storage and loss modulus seemed to be characteristic of lamellar samples in general. The storage modulus versus frequency was well described by the slip-plane theory, but in the case of the loss modulus the fitting process does not give a complete agreement with the experimental values. The fitted constants show trends which are qualitatively understandable based on the general knowledge on the interaction forces acting in dispersions stabilized by non-ionic surfactants. To describe the results of the creep test the elastic properties of the slip plane had to take into consideration. The values of the instantaneous elastic modulus calculated from the creep test agreed well with those obtained from the dynamic measurements. The steady state flow curves were
well described by a modified Carreau-type equation and the possibilities of a change in the structure were briefly analysed.
Acknowledgment The authors thank Judit Sebestyen for her experimental work. This work was financially supported by the CAOLA Cosmetic and Household Industrial Ltd. and was made in a cooperation with ICI Surfactants.
References [1] G.H. Brown ( Ed.), Advances in Liquid Crystals, vols. 1–5, Academic Press, New York, 1975, 1976, 1978, 1979, 1982. [2] G. Gray, P.E. Winsor, Liquid Crystals and Plastic Crystals, Wiley, New York, 1974. [3] A. Oswald, H. Huang, J. Huang, P. Valint, US Patent 4, 434 062, 1984. [4] G.H. Brown, J.J. Walker, Liquid Crystals and Biological Structures, Academic Press, New York, 1979. [5] H. Kru¨ssmann, R. Bercovici, Tenside Surf. Det. 30 (1993) 99. [6 ] T. Suzuki, H. Tsutsumi, A. Ishida, Proc. 12th. Int. Congress IFSCC, Paris, vol. 1, 1982, p. 117. [7] L. Bohlin, K. Fontell, J. Colloid Interface Sci. 67 (1978) 273. [8] P. Oswald, M. Allain, J. Colloid Interface Sci. 126 (1988) 45. [9] C. Galleos, M. Nieto, C. Nieto, J. Munoz, J. Progr. Colloid Polym. Sci. 84 (1991) 236. [10] M.R. Alcantara, J. Vanin, Colloids Surf. A 97 (1995) 151. [11] M. Valiente, Colloids Surf. 105 (1995) 265. [12] J.T. Mang, S. Kumar, B. Hammouda, Euro. Phys. Lett. 28 (1994) 489. [13] J. Penfold, E. Staples, A. Khan Lodhi, I. Tucker, G.J.T. Tiddy, J. Phys. Chem. B 101 (1997) 66–72. [14] M. Cates, S.F. Mildner, Phys. Rev. Lett. 62 (1989) 1865. [15] O. Robles-Vasquez, S. Corona-Galvan, J.F.A. Sottero, J.E. Puig, J. Colloid Interface Sci. 160 (1993) 65. [16 ] O. Robles-Vasquez, J.F.A. Sottero, J.E. Puig, O. Monero, J. Colloid Interface Sci. 160 (1994) 436. [17] H. Hoffmann, C. Thuing, P. Schmeidel, V. Munkert, Langmuir 10 (1994) 3972. [18] H. Hoffmann, C. Thuing, P. Schmeidel, V. Munkert, W. Ulbricht, Tenside Surf. Det. 31 (1994) 389. [19] P. Panizza, D. Roux, V. Vuillame, C.Y.D. Lu, M.E. Cates, Langmuir 12 (1996) 248. [20] M. Doi, J.L. Harden, T. Ohta, Macromolecules 26 (1993) 4935. [21] J.L. Jones, T.C.B. McLeish, Langmuir 11 (1995) 785.
Z. Ne´meth et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 107–119 [22] S. Radiman, C. Toprakcioglu, T.C.B. McLeish, Langmuir 10 (1994) 61. [23] S. Paasch, F. Schambil, M.J. Schwunger, Langmuir 5 (1989) 1344. [24] G.T. Dimitrova, Th.F. Tadros, P.F. Luckham, M.R. Kipps, Langmuir 12 (1996) 315–318. [25] H. Hofmann, D. Nickel, W. von Rybinski, Langmuir 11 (1995) 4250–4255. [26 ] Th.F. Tadros, G.T. Dimitrova, P. Luckham, M.C. Ftaelman, P. Loll, Euro Cosmetics 3 (1995) 17.
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[27] G.T. Dimitrova, Th.F. Tadros, P.F. Luckham, Langmuir 11 (1995) 1101–1111. [28] P.M. Claesson, R. Kjellander, P. Stenius, H.K. Christenson, J. Chem. Soc., Faraday Trans 82 (1986) 2735–2746. [29] L. Hala´sz, L. Macska´si, Plastic Rubber 33 (1996) 146 in Hungarian [30] J. Lauger, R. Weigel, K. Berger, K. Hiltrop, W. Richtering, J. Colloid Interface Sci 181 (1996) 521–529.
Rheological behaviour of a lamellar liquid crystalline surfactant–water system Zs. Ne´meth a,*, L. Hala´sz a, J. Pa´linka´s b, A. Bo´ta a, T. Hora´nyi a a Technical University of Budapest, Department of Physical Chemistry, H-1111, Budapest, Budafoki ut 8, Hungary b CAOLA Cosmetic and Household Industrial Ltd., H-1116, Budapest, Vegye´sz u 19–25, Hungary Received 29 January 1998; accepted 19 June 1998
Abstract The rheological behaviour of a lamellar liquid crystalline surfactant–water system was investigated using a Haake RS-100 viscometer. The repeat distance of lamellae as a function of surfactant concentration and temperature was measured by small-angle X-ray scattering. The rheological characteristics were measured at different temperatures with systems containing surfactant in different concentrations. The frequency-dependent storage and loss modulus were found to be characteristic to the lamellar phase in the linear viscoelastic region. The results are analysed on the basis of Jones–McLeish slip-plane theory. Trends of the fitted constants are discussed based on the general knowledge on the interactions in dispersions stabilized by non-ionic surfactants, and the structure of lamellar liquid crystalline samples. Time-dependent compliance was also measured. The instantaneous elasticity measured in the creep tests was compared with that predicted from oscillatory tests. Burger’s model was used to describe the time-dependent compliance. Viscosity measurements in the non-linear region were also done. A modified Carreau equation is used to describe the viscosity versus shear rate curves. The changes in sample under shear is described briefly. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Non-ionic surfactants; Lamellar liquid crystals; Rheology
1. Introduction Surfactant-based liquid crystals have a broad range of application in the cosmetic and food industries [1–6 ]. One of the most important systems are those in which bilayers are formed from the surfactants: these are the lamellar mesophase, the bicontinuous cubic phase, the liposome phase and the so-called onion phase. The rheological properties of such systems were reported in some papers [7–25]. In spite of its importance, the microscopic origin of the rheological behaviour is * Corresponding author.
not yet well understood and has been the subject of many recent investigations. Explaining the flow curves of a three-component lamellar mesophase, Bohlin [7] modelled the lamellar structure as flexible layers of water in a liquid hydrocarbon chain environment. Flow was associated with cooperative changes of conformation of flexible layers subject to frictional forces in the liquid layers. The flow coefficients were found to be dependent on the thickness of water layer, the bilayer interaction and the mobility in the hydrocarbon chain layer. Investigating the temperature dependence of flow, Oswald [8] took into consideration the
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importance of the undulation of the bilayers and in the higher temperature range the number of the dislocation loops. Other authors [9–13] believe in the importance of the orientation of lamellae under shear. The orientation of domains in the case of an unsheared sample is dependent on the domain–domain interaction. If stress is applied on the sample the orientation changes, and the competition between the interactions and the external stress defines the viscosity of the system. Mang et al. [12] measured the small-angle X-ray scattering (SAXS ) of a lamellar system under shear and found that the domains order in the direction of the stress as the strain increases. Penfold et al. have investigated the alignment under shear of lamellae of an aqueous hexaethylene glycol monohexadecyl ether (C E ) sample in a 16 6 Couette shear cell by small-angle neutron scattering experiments [13]. They have found that at low shear the lamellae are ordered parallel to the flow–vorticity plane. At high shear gradients the lamellae are ordered in the orthogonal director, parallel to the flow–shear gradient plane, consistent with the theoretical predictions of Cates and Mildner [14]. At intermediate shear a biaxial alignment was found. They have also made measurements on a sample subjected to oscillatory flow in the range of 0–180 s−1. Results were similar to those observed with the conventional Couette flow. At low values of oscillatory flow, alignment of lamellae in the flow–vorticity plane has been found. Penfold et al. propose to re-evaluate the nature of the ‘‘defective’’ lamellar phase on the basis of shear-induced ordering. The viscoelastic behaviour of a lamellar anionic surfactant, AOT–water system was investigated by Robles-Vasquez et al. [15,16 ]. They considered the lamellar mesophase as a weak gel, but quantitative descriptions were not used for the explanation of the experimental results. Viscoelastic behaviour of liposomes was intensively investigated by Hoffmann et al. [17,18], who found a frequency-independent storage and loss modulus. Onion phases have been investigated by Panizza et al. [19]. The creep compliance curves were described by Burger’s model, consisting of a
Maxwell and a Voigt element in series. In investigating the results of the dynamic tests the same model was used by Laplace transformation of the creep equation. The agreement between the calculated and measured results were good in the case of the storage modulus, but a significant difference was found in the case of the loss modulus, mainly in the high frequency region. Doi et al. [20] used a slippage model to describe the rheological characteristics of ordered block coplymer samples. With the slippage mechanism, non-linear elasticity and a definite yield stress have been explained. Jones and McLeish [21,22] introduced a slipplane theory to describe the frequency dependence of the storage and loss modulus of cubic phases. They propose their model to be used more generally to describe relaxation in weak solids. It is an alternative to the more familiar mechanism of the movement of dislocations such as that found in many crystalline solids. The present paper reports the results of careful rheological investigations which were done both in the linear and in the non-linear viscoelastic regions on non-ionic lamellar samples containing surfactant in different concentrations or at different temperature. The applicability of the McLeish theory is investigated in the linear viscoelastic region. The trends of the fitted parameters with surfactant concentration and with sample temperature are discussed based on the basic knowledge on the properties of non-ionic surfactant systems and on the results of SAXS measurements. The results of the time-dependent creep tests are also compared with the results of the tests made by using oscillatory flow. The rheological characteristics of the sample in the non-linear region are interpreted by using the well-known Carreau equation, which is a general equation of the polymer rheology to characterize gels in which a timedependent temporary network exists. 2. Experimental section 2.1. Materials We studied a lamellar liquid crystalline system consisting of a non-ionic surfactant (Synperonic A7, ICI product) and water. The surfactant was
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supplied by ICI Surfactants and used as received. It was made by the ethoxylation of Synperol alcohol which consists of 66% C and 34% C 13 15 alkyl chains. The ethoxylation process gives rise to a wide distribution of ethoxylate chains, therefore only the average ethoxylation number can be declared, which is in case of A7 is 7. Although the properties of sample should not change from batch to batch, the preparation of the lamellar crystals investigated here was always done from surfactants from the same batch. The phase diagram of the Synperonic A7–water system was established by Tadros et al. [24,26,27] by rheological, calorimetric and NMR measurements and by polarization microscopic observations. They have found normal micellar (L ), 1 hexagonal liquid crystalline (H ), lamellar liquid 1 crystalline (L ) and inverse micellar (L ) phases a 2 with increasing surfactant concentration at 23°C. At this temperature the lamellar phase exists in between 55 and 80% (w/w). At 50–55% (w/w) the coexistence of the hexagonal and the lamellar phase was observed. The borders of the lamellar phase in the phase diagram do not change markedly with the temperature, but lamellae melt at about 50°C and at higher temperature an optically isotropic liquid forms. Our samples were prepared by heating the aqueous mixtures to about 50–55°C where they were easily homogenized, and after homogenization mixtures were left to cool to room temperature. Then the samples were stored for 1 week before measurements. Special care was taken to prevent evaporation of the water content of the samples during both their preparation and their storage. Mixture were prepared with distilled water, having a surface tension of 72–73 mN m−1. The samples prepared this way were checked by observing their textures in a polarization microscope before rheological measurements. Lamellar phases were found in all cases in the concentration range of 60–80% (w/w) in good agreement with the findings of Tadros et al. 2.2. Methods Rheological measurements were made with a Haake RS 100 apparatus. A cone–plate sensor was
109
used, with a diameter of 20 mm, and a cone angle of 4°. The sample thickness in the middle of the sensor was 0.134 mm. The maximum allowed variation in temperature during measurements was ±0.2°C. Samples were kept under saturated water vapour for the whole time of the measurements. Because the rheological properties of such systems are dependent on the shear-deformation history, the sample was gently inserted to the top of the plate of the sensor, and then the plate was slowly elevated to its measuring position with constant velocity. The sample squeezed out from the sensor system was then gently removed. Measurements were carried out after a 10 min waiting (relaxation– thermostation–saturation) period. Dynamic (oscillation) and static (creep– recovery) test were done on all samples. First the linear viscoelastic region was determined by measuring the complex modulus versus stress at a given frequency, and then 2.5 Pa was chosen as a stress amplitude, which was found to be in the linear viscoelastic region in all cases. Creep tests were also made on the same samples. A stress of 2.5 Pa was applied for 5 min on the samples and the compliance (J ) was monitored during both the creep and the following recovery period (with zero applied stress). Flow curves were measured in two ways: in case of the yield value determination the measurements were made by increasing the stress with a constant rate (0.166 Pa s−1). The shear rate-dependent viscosity was determined by steady state measurements. At each stress value the result was accepted if its change was lower than 0.05% s−1 with the time. For the SAXS measurements the samples were transferred to thin-walled Mark quartz capillaries (Hilgenberg, Germany) of diameter 1 mm. The capillaries were closed with a two-component synthetic resin and transferred to metal capillary holders into an aluminium block. This block was placed directly in the beamline and used as a thermal incubator for controlled annealing at different temperatures. The block was held at the desired temperatures using a thermostat. The actual temperatures were constant to within less than 0.05°C, as monitored with a thermocouple.
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The windows of the block were covered with Mylar foil. The SAX camera was a Kratky camera (Anton Paar, Graz, Austria) attached to the line-focus window of a Cu tube at a generator (Seifert, Ahrensburg, Germany) operated at 40 kV and 20 mA. A Ni filter was used to eliminate Kb radiation. The scattering data were collected with a proportional counter. The scattering data curves were normalized and desmeared for the slit geometry.
3. Results 3.1. Small-angle X-ray scattering On the experimental SAXS curves of the lamellar samples investigated a very intense first-order Bragg refraction and a second, very diffuse peak corresponding to the second-order Bragg refraction appear. The desmeared first-order peak profile is shown in Fig. 1. The scattering of the samples is characteristic of a strongly ordered structure containing lamellae with a well-defined repeat distance. The repeat distance of lamellae was measured as a function of the surfactant concentration at 25°C. The repeat distance of the samples decreases with increasing concentration. The effect of temperature on the repeat distance, measured in the case of the 70% (w/w) sample was found to be negligible in between 25 and 50°C.
Fig. 1. Typical results of small angle scattering measurements at 25°C. The calculated repeat distance is also shown on the figure for samples containing A7 in different concentrations.
3.2. Dynamic oscillatory tests Typical results of the complex modulus as a function of the stress at low frequency (0.036 Hz) can be seen in Fig. 2. It can be stated that most of the samples have linear viscoelasticity up to about 10 Pa. For subsequent dynamic experiments we have chosen the stress value of 2.5 Pa. Typical results of the storage modulus, loss modulus, and complex viscosity of the lamellar mesophase, formed from aqueous Synperonic A7, as a function of the frequency in the linear viscoelastic region can be seen in Fig. 3. The system is more elastic than viscous in the range of frequency
Fig. 2. Complex modulus (G*) of a lamellar sample of 70% (w/w) A7 solution as a function of the applied stress (t) at a constant ( low) frequency (0.036 Hz) at 25°C.
investigated, and the storage modulus is higher by about one order of magnitude than the loss modulus throughout the whole frequency range. While the storage modulus has a weak dependence on
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Fig. 3. Elastic (G∞), and loss modulus (G◊), and complex viscosity (g*) of 70% (w/w) aqueous A7 lamellar system as a function of the frequency (v) at a stress of 2.5 Pa at 25°C.
the applied frequency, the loss modulus shows a minimum. The complex viscosity is strongly frequency dependent: the higher the frequency, the lower is the complex viscosity. The picture described above is characteristic only of the lamellar mesophase; both of the other phases (hexagonal phase, and inverse micellar phase) which surround the lamellar phase in the phase diagram show a completely different behaviour as a function of frequency ( Figs. 4 and 5). Because the rheological behaviour of lyotropic mesophases is often strongly dependent on the shear history, we examined the effect of preshear on the results of the dynamic measurements. For this a 70% (w/w) sample was sheared for 5 min with a constant strain. In the first experiment
Fig. 4. Elastic (G∞), and loss modulus (G◊), and complex viscosity (g*) of 45% (w/w) aqueous A7 hexagonal system as a function of the frequency (v) measured at 2.5 Pa stress at 25°C.
111
Fig. 5. Elastic (G∞), and loss modulus (G◊), and complex viscosity (g*) of 90% (w/w) aqueous A7 inverse micellar system as a function of the frequency (v) measured at a stress of 2.5 Pa at 25°C.
0.2 s−1 and in the second one 10 s−1 preshear was applied, and soon after that dynamic measurements were carried out on the presheared sample. Results of these measurements showed that the preshear has only a negligible effect on the results of the dynamic tests. Neither the storage modulus nor the loss modulus changed strongly with the applied preshear. Only the minimum of the loss modulus became more pronounced as the strain of the preshear increased. The effect of concentration on the storage and loss moduli was examined within the lamellar concentration region at 25°C. The shape of the above-mentioned characteristics as a function of frequency does not change with the concentration of the surfactants. A summary of the results can be seen in Fig. 6, where we show the storage modulus, the loss modulus and their ratio (tan d) as a function of the surfactant concentration at a given frequency. It can be seen that both the storage and loss modulus increase with increasing concentration, but the rate of their increase is different. As the system becomes more concentrated the sample becomes more and more elastic, as can be seen from the decrease in tan d. In the case of 80% (w/w) concentration, while the loss modulus increases further, the storage modulus begins to decrease. Investigating the structure of this sample in the polarization microscope, we observed a change in texture at 80% (w/w) concen-
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Fig. 6. The concentration dependence of the elastic (G∞), the loss modulus (G◊) and their ratio (tan d) at 1 Hz, measured with stress amplitude of 2.5 Pa at 25°C.
Fig. 7. The temperature dependence of the elastic (G∞), the loss modulus (G◊) and their ratio (tan d) at 1 Hz measured with a stress amplitude of 2.5 Pa.
tration. While for the concentration range 60–75% (w/w) ‘‘Maltese crosses’’ are seen, in the case of 80% (w/w) surfactant concentration, the texture is of the oily streaked type. This change of texture indicates an orientation change [25] of the domains, which is connected to the deviation of the trend of the storage modulus. These observations are in good agreement with the findings of Tadros et al. [26,27] The effect of temperature was also examined for the 70% (w/w) A7 lamellar system. The shape of the storage and loss modulus and the complex viscosity as functions of frequency do not change with the temperature until the phase boundary. Above this temperature the system shows rheological behaviour characteristic of a system consisting of inverse micelles. This was found at the same temperature where the scattering experiments indicated the complete disappearance of the lamellae. In the lamellar region the quantitative results of dynamic tests are strongly temperature dependent (Fig. 7). Both the storage and loss modulus are decreasing with increasing temperature, but the rate of their change is different, the decrease of the elastic component being much more pronounced. This can be evaluated from tan d, which increases with temperature. The change of storage modulus is more pronounced in the lower temperature region (between about 5 and 25°C ). The elastic component of flow diminishes as the phase boundary is reached with increasing temperature.
The change of storage modulus can be characterized by two straight lines and the break point of the lines is at 25°C. This value is very close to the melting point of the water-free surfactant. Assuming that the melting point of the carbon chains is not influenced strongly by the aqueous layers which are separated from these chains with the ethoxylate groups, this break point seems to be a transition temperature between the gel and the liquid crystal state of the lamellar domains.
3.3. Creep tests Creep–recovery tests were also made on the samples under the same conditions as in the case of the dynamic tests. Typical results can be seen in Fig. 8. When the constant stress impulse value was increased, the shape of the creep curves did not change in the linear viscoelastic range and above this range the creep curves showed only a simple viscous behaviour. The stress limit where the shape of the curves changed was almost the same as that obtained from the dynamic measurement. From the curves it can be concluded that all of the samples can be described by three values of compliance: J=J +J +J 0 R V
(1)
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113
3.4. Steady state viscosity measurements
Fig. 8. Typical results of creep-recovery tests at 25°C. 2.5 Pa stress was applied for 5 min and the creep value was monitored during this and the subsequent 5 min period. The concentrations of the samples were: &, 60% (w/w); *, 65% (w/w); $, 70% (w/w); #, 75% (w/w); n, 80% (w/w).
Flow curve measurements were made on the same samples that had previously been tested in the linear viscoelastic region. The effect of temperature and concentration was investigated The shear viscosity versus shear rate and the complex viscosity versus frequency for different surfactant concentration are shown in Figs. 9 and 10. Both the shear viscosity and the complex viscosity are decreasing functions of the shear rate and the frequency, respectively, but the Cox–Merz rule (i.e. the complex viscosity is equal to the shear viscosity if the shear rate is the same as the frequency) does not hold. Both the shear viscosity
where J is the instantaneous elastic compliance, 0 J is the retarded elastic compliance and J is the R V viscous or remaining compliance. The retarded creep compliance can be represented by a sum of discreet number i of compliances J and timei dependent parts:
C
A BD
t J =∑ J 1−exp − R i l i i
(2)
where l is the retardation time of ith component. i From the curves of Fig. 8. the parameters of Eqs. (1) and (2) were determined and are summarized in Table 1.
Fig. 9. Shear viscosity versus shear rate at 25°C for different surfactant concentrations: &, 60% (w/w); *, 65% (w/w); $, 70% (w/w); #, 75% (w/w); n, 80% (w/w).
Table 1 The parameters of the fitted Burger’s model of creep tests which were done at 25°C Concentration (% (w/w))
J ×10−3 0 (Pa−1)
J ×10−3 1 (Pa−1)
J ×10−4 v (Pa−1)
l (s) 1
60 65 70 75 80
3.94 1.64 1.22 0.90 0.80
0.56 0.19 0.16 0.05 0.03
6 2 0.8 0.2 0.1
8.1 6.2 5.6 3.8 3.5
Fig. 10. Complex viscosity vs. frequency at 25°C for different surfactant concetrations: &, 60% (w/w); *, 65% (w/w); $, 70% (w/w); #, 75% (w/w); n, 80% (w/w).
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Fig. 11. Shear viscosity versus shear rate for 70% (w/w) surfactant concentration at different temperatures: &, 25°C; $, 40°C.
and the complex viscosity increase with increasing surfactant concentration in all region of the shear rate and frequency. The viscosity of samples decreases with increase in temperature at low shear rates, where the structure of the sample is only partly destroyed, but as can be seen from Fig. 11 the temperature has no effect on the viscosity above 10 1 s−1, where the structure of the sample is more or less completely destroyed. The yield values (t ), which were measured by 0 increasing the stress acting on the samples at a constant rate, are shown in Table 5. They show an increasing tendency with increasing surfactant concentration and a decreasing tendency with increasing temperature.
4. Discussion The rheological properties of the lamellar mesophase formed from the non-ionic Synperonic A7 were investigated both dynamically (with oscillation tests) and statically (with creep tests and steady state viscosity measurements). In the linear viscoelastic region, where the structure of the system does not change significantly during the rheological measurements, the lamellar mesophase behaves as an elastic solid, its storage modulus being about one order of magnitude higher than
Fig. 12. The slip-plane model.
its loss modulus. This elastic behaviour can be also seen from the compliance–time curves, where the curves show a large instantaneous compliance. The dynamic behaviour as a function of frequency, which can be seen in Fig. 3, is as mentioned before characteristic of lamellar systems. Other phases which can be formed from the same surfactant mixtures at other concentrations behave in a completely different way (Figs. 4 and 5). Lamellar phases formed from other aqueous surfactant mixtures have a similar viscoelastic behaviour to the system here, for example RoblesVa´squez [15,16 ] have reported a similar behaviour for lamellar mesophase of aqueous Aerosol AOT. In that case the storage modulus was nearly independent of frequency, the loss modulus showed a minimum, and the complex viscosity decreased with increasing frequency and did not obey for the Cox–Merz rule, which is the same as our results. According to this we can conclude that the type of behaviour which we observed for the lamellar phases is characteristic of lamellar phases in general. The role of preshear was negligible, only the minimum of the loss modulus curves becoming more pronounced. This means that during steady state shear the lamellar liquid crystalline structure changes and these changes recover quickly after the cessation of shear; thus the effect of preshear seems to be negligible. The results of the rheological behaviour in the linear viscoelastic range were evaluated using the
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slip plane theory developed by Jones and McLeish [21,22] for cubic phases. This theory assumes that there are planes in the sample which are aligned parallel to the shear–vorticity plane. As mentioned in Section 1, this structure of a sample under shear was experimentally found in a very similar system by Penfold et al., so it is reasonable to try to fit the slip-plane theory to our experimental results. The shear stress acting tangentially results in a displacement between the layers of the sample (Fig. 12). When we apply a fixed strain to the sample, if no slipping occurs, the stress exerted at the top surface of the sample is: t=GP+gP˙
(3)
where G is the bulk modulus, g is the bulk viscosity of the liquid crystalline, P is the gradient of displacement and P˙ is its time derivative. Eq. (3) is a simple Voigt–Kelvin equation. Once a slip has occurred, the strain in one layer is high but in all others is very low. The deformation in the slip plane is very high and mainly viscous arising from the flow along the surface. The viscous interaction disturbs the lattice potential as an elastic tension. In the case of constant strain a second viscous term is added to Eq. (3) in which h is the viscosity 2 of the slipping plane. The relaxation modulus is now:
AB A B A B
G t = G+
g
l
exp
−t l
(4)
where l=
g+Ng
2
(5)
G
is the relaxation time. The components of the complex modulus are: G∞(v)=AG
(lv)2−(l v)2 1 1+(lv)2
(6)
G"(v)=AG
lv(1+ll v2) 1 1+(lv)2
(7)
where l =g/G 1
(8)
115
and Ng 2 g+Ng
(9) 2 This linear model can be extended by introducing a weak elastic force acting at the slip planes due to the connectivity of the structure. The components of complex modulus considered a series, and the higher terms were calculated by the perturbation method. It was supposed that at low frequencies there is a slip–stick phenomenon, but there is no slip below a critical stress value. In this region the two components of complex modulus are: A=
G∞=
Ge c 0 Ge
C A BD A B 1− 1− e
c 0
#c−3/2 0
(10)
#c−1 (11) 0 c 0 0 where e and c are the critical values of the strain 0 and the amplitude, respectively. Considering the shape of our modulus curves it seemed to us that the slip–stick and the linear model may describe our curves. In Table 2 we summarize the values for G∞ and G◊ calculated using the slip–stick model (assuming a ratio of 0.8 for e/c and a G value 0 which was taken as an inverse of the instantaneous creep compliance.) The agreement between the calculated and measured values is rather good. The fitted (using the results of the linear theory) modulus values are given in Table 3. The fitted curves can describe the experimental curves rather well in the case of storage modulus, but the agreement is not good in the case of the loss modulus curves, the minimum values of the of the fitted curves being much smaller and the fitted function more curved than the experimentally measured ones. When we took into consideration the effect of lattice potential by the perturbation method the results did not improve significantly. As an example we show in Fig. 13 the fitted results as a function of frequency in the case of 70% (w/w) Synperonic A7 at 25°C. Let us first compare G calculated from the o fitted results with that measured in the creep tests. The predicted results shown in Table 4 are in rather G"=
c
1−
e 2 1/2
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Table 2 Measured and calculated moduli at 10−2 Hz for samples at 25°C containing surfactant in different concentrations Concentration (% (w/w))
Storage (measured ) (Pa)
Modulus (calculated) (Pa)
Loss (measured ) (Pa)
Modulus (calculated) (Pa)
60 70 80
220 606 1250
198 636 1050
35 113 202
40 97 200
good agreement with the experimental results measured at 25°C, showing no discrepancy between the two type of results. Let we then discuss the trends of the fitted values. The density of slip planes (N ) was calculated from the fitted values of AG, l and l in 1 Eqs. (6) and (7). A#1 because l and l are 1 strongly different, so the fitted AG was kept to be equal to G. Knowing l and G, g was calculated 1 from Eq. (7), and then N could be calculated as a function of g (which is unknown) from Eq. (9). 1 Table 3 Fitting results of the linear slip plane theory for samples as a function of the surfactant concentration and sample temperature c (% (w/w))
T (°C )
AG (Pa)
l (s)
l (s) 1
v (rad s−1)
60 65 70 75 80 70 70 70 70 70 70
25 25 25 25 25 5 15 35 40 45 54.5
220 360 560 1010 1010 2040 1375 570 320 350 110
161 85 132 122 152 24 38 83 65 58 10
0.0317 0.0240 0.0184 0.0082 0.0103 0.0154 0.0086 0.0149 0.0189 0.0172 0.0346
0.424 0.7 0.8 1 0.8 1.9 1.75 0.9 0.9 1 1.7
Nevertheless, because g is the viscosity of the slip 1 planes, its value should be between g and the viscosity of the pure solvent (water) Taking these limits for g we obtain a range of values between 1 103 and 107 for N. G is increasing with increasing surfactant concentration and is decreasing with increasing temperature. It seems that G is a function of the lamellar–lamellar interactions. At a given temperature, the repeat distance of the lamellae decreases, and this means an increased interaction between lamellae which causes an increase of G. At a given surfactant concentration the repeat distance is independent of the temperature, but naturally the interactions are functions of temperature. In the case of lamellae formed from non-ionic surfactants the interactions originate from the van der Waals forces, and steric–hydration forces. The hydration repulsion is strongly temperature dependent [28], it decreases with increasing temperature. From these facts it seems reasonable to conclude that steric–hydration interactions play an important role in determining the interlamellae elastic modu-
Table 4 Instantaneous elasticity measured in creep tests and predicted from the oscillating tests for samples at 25°C Concentration (% (w/w))
G (predicted) (Pa) o
G (measured) (Pa) o
60 65 70 75 80
220 360 560 1010 1010
254 609 816 1110 1250
Fig. 13. Measured and calculated moduli. The measured data are the same as was depicted in Fig. 3. Lines represent calculated values.
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Fig. 14. Shear viscosity versus shear rate for for 70% (w/w) surfactant concentration at 30°C. The dots represent the measured values, the line is the fitted curve. The fitted parameters are also shown on the figure.
lus (G ). The same conclusion (the influence of the repulsion forces in the rheological characteristics) was drawn by Robles-Vasquez [15,16 ] l shows a maximum as a function of the temperature. It is dependent on the viscosity values, the elastic modulus, and the rheological domain size (N ) Eq. (5). While the ratio of the viscosity to elastic modulus increases with increasing temperature, N may decrease (the enhanced temperature may increase the number of slip planes, i.e. the disorder in the sample), and the superposition of these changes can cause the maximum in l. l , which is the ratio of the interlamellar viscos1 ity and elasticity ( Eq. (8) is a monotonically decreasing function of the concentration. This is in good agreement with the fact that the ratio tan d of the macroscopic elasticity (G∞) to the
viscous component (G◊) shows a monotonic increase with concentration. The data of creep tests show that the liquid crystalline structure has a bulk elastic and viscous strain component, and the elastic component in the slip plane also plays a role in the deformation process. Thus the creep curves have three types of compliances as shown in Eqs. (1) and (2). The steady state flow curves show yield value. The theory given by Doi et al. [20] predicted a yield value for block copolymer liquid crystals but their simple equation describing the steady state viscosity versus shear rate did not give a good approximation for our experimental results. In our system the shear rate causes some orientation and some change in structure. The steady state flow curves were described by a modified Carreau equation [29]: g=
g a 1+b(lc˙ )m
(9a)
where g is the viscosity extrapolated to zero shear, a l is a characteristic time, and b and m are constants. This equation is generally applicable to the description of the shear-dependent viscosity function of any interlinked samples in which the shear causes a disruption of the sample structure in the non-linear viscoelastic region. The fitted curves showed very good agreement with the experimental values ( Fig. 14). In the case of a lamellar system the change in structure can be the change of the domain size, the change of the alignment of lamellae and the change of the
Table 5 Fitted constants of Eq. (9a) as a function of the surfactant concentration and the sample temperature c (% (w/w))
T (°C )
t (Pa) o
g (Pa s) a
b
l (s)
m
60 65 70 75 80 80 80 80
25 25 25 25 25 30 40 50
4 14 28 33 39 35 25 8
19.03 80.38 144.70 220.4 250 78.6 6.5 0.6
8.988 7.277 3.926 2.109 16 0.392 2.246 8.905
0.111 0.810 1.544 0.227 0.388 48.967 321.6 985.9
0.446 0.633 0.782 0.751 0.758 0.667 0.308 0.022
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shape of the lamelllae (for example by liposome formation). Lauger et al. [30], investigating a three-component system, found reversible liposome formation due to the effect of the shear which was accompanied with the increase of the viscosity. Because in all cases with our sample the viscosity is a monotonically decreasing function of the stress the last effect may be considered to be negligible. The other mechanisms can play a role in changing the viscosity by changing the connectivity of the samples. The fitted parameters are shown in Table 5. g a is increasing with increasing concentration and decreasing with increasing temperature. The decrease of g with temperature is exponential and a can be characterized by an Arrhenius-type of equation. The activation energy determined was 2 J mol−1 for 70% (w/w) concentration. The value of m does not change markedly with the sample concentration, showing that the mechanism of the disruption of the samples is nearly independent on the surfactant concentration, although m depends strongly on the temperature.
5. Summary The complex rheological behaviour of a liquid crystalline surfactant–water system was investigated. The frequency-dependent storage and loss modulus seemed to be characteristic of lamellar samples in general. The storage modulus versus frequency was well described by the slip-plane theory, but in the case of the loss modulus the fitting process does not give a complete agreement with the experimental values. The fitted constants show trends which are qualitatively understandable based on the general knowledge on the interaction forces acting in dispersions stabilized by non-ionic surfactants. To describe the results of the creep test the elastic properties of the slip plane had to take into consideration. The values of the instantaneous elastic modulus calculated from the creep test agreed well with those obtained from the dynamic measurements. The steady state flow curves were
well described by a modified Carreau-type equation and the possibilities of a change in the structure were briefly analysed.
Acknowledgment The authors thank Judit Sebestyen for her experimental work. This work was financially supported by the CAOLA Cosmetic and Household Industrial Ltd. and was made in a cooperation with ICI Surfactants.
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