J. Phys. Chem. 1985, 89, 517-522
517
Viscoelastic Detergent Solutions. A Quantitative Comparison between Theory and Experiment H. Thurn,* M. Liibl, and H. Hoffmann Uniuersitat Bayreuth, Lehrstuhl fur Physikalische Chemie, Bayreuth, West Germany (Received: June 21, 1984; In Final Form: September 19, 1984)
A general equation is given that describes both the frequency and shear-rate dependence of the non-Newtonian viscosity according to the theory recently developed by Hess.' Expressions for the complex shear modulus and the extinction angle in flow birefringence are given and compared with experimental data of viscoelastic micellar solutions in a self-consistent way. The micellar solutions were mixtures of tetradecylpyridiniumsalicylate (TPS) and tetradecyltrimethylammonium salicylate (TTAS) in the presence of salt.
Introduction
Viscoelastic detergent solutions, which show nonlinear flow behavior, e.g., Occurrence of a shear-rate dependent non-Newtonian viscosity and a normal pressure difference, are examples of relatively complicated s o l ~ t i o n s . ~ -Usually ~ micellar solutions of this kind contain nonspherical particles and show both electric and flow birefringen~e.~.~ This article proceeds as follows. A short description of the theory developed by Hess is given. A generalized equation for the dependence of the viscosity on the shear rate and the frequency is derived. Expressions for the extinction angle in flow birefringence and the complex shear modulus are given. Then the experimental setup and data handling are described. Experimental data that show the extinction angle as a function of the shear rate and storage and loss moduli as a function of frequency are presented and compared in a self-consistent way.
is used.'JO t is an energy associated with the alignment of one particle influenced by the neighboring particles. It is convenient to use the following definitions' for the high-shear-rate limit of the viscosity
and the energy-dependent relaxation time 1,I
7a -
1--f_ kT Together with the ansatz (4) and the definitions ( 5 ) and (6) we get from eq 1, neglecting the influence of 7 M
Theory Fundamental Equations. The starting point is the time-dependent transport relaxation equations, which Hess derived from a Fokker-Planck e q ~ a t i o n : ' - ~ ~ *
(7) It is interesting to note that the alignment tensor ri is related to the irreducible part of the dielectric tensor z by" Z
The overhead bar denotes the symmetric irreducible part of a tensor of second rank; for mathematical details see ref 9. Pk is the kinetic pressure defined by Pk = t k T , where t is the number density of the particles, k is the Boltzmann constant, and T i s the temperature of the micellar solution. 7 M is the Maxwell relaxation time which is very small in comparison to the other relaxation times T..., which have the properties 7p > 0, 7a > 0, 7,p7pa< 7,7,,. The relaxation times 7p and 7, describe the relaxation of the pressure tensor p and the alignment tensor r i , respectively. Usually 7, is given by 7, = 1/60 where D denotes the rotational diffusion coefficient of the particles. For 7p = 0, Le., no coupling between alignment and viscous flow, (1) reduces to the equation of the Maxwell relaxation model for viscoelasticity. Furthermore, the nondiagonal relaxation times obey the Onsager symmetry = 7pa. The flow velocity v and the average rotational relation velocity Q of the particles are related by n = f/2v x v (3) The (first Newtonian) zero viscosity 7 can be written as 7 = Pk7p For the tensor z ( a ) the ansatz (4)
* Present address: Siemens AG,Bereich Bauelemente, Balanstrasse, 73, 8000 Miinchen, West Germany.
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= 4?r?(Cill - Cil)a
(8)
where alland cyI are effective polarizabilities parallel and perpendicular to the axis of rotation of a Brownian particle. Therefore in the case of small 7- the pressure tensor p is proportional to Z. In the literature this fact is called the stress optical law.I29l3 The Complex Viscosities. Fourier transforming eq 2 and 7 and solving eq 2 for the Fourier transform ii and inserting into ( 7 ) , we get in the case of a Couette flow, where r denotes the gradient of the velocity (1) Hess, S. 2.Naturforsch., A 1980, 35A, 915-9. (2) Gravsholt, S. J. Colloid Interface Sci. 1976, 57, 575-7. (3) Rehage, H.; Hoffmann, H. Faraday Discuss. Chem. SOC.1983, 76, 1-11. (4) Hoffmann, H.; Platz, G.; Rehage, H.; Schorr, W.; Ulbricht, W. Ber. Bunsenges. Phys. Chem. 1981,85, 255-66. (5) den Otter, J. L.; Papenhuijzen, J. M. P. Rheol. Acta 1971,10,457-60. Janeschitz-Kriegl, H.; Papenhuijzen, J. M. P. Rheol. Acta 1971,10, 461-6. (6) Schorr, W. Dissertation, Bayreuth, 1982. (7) Hess, S. Z . Naturforsch., A 1975, 30A, 728, 1224. ( 8 ) Hess, S. Physica A (Amsterdam) 1977, 87A, 273. (9) Has, S.;Kohler, W. 'Formeln zur Tensor-Rechnung";Palm and Enke: Erlangen, West Germany, 1980. (10) Hess, S. In 'Liquid Crystals of One- and Two-Dimensional Order"; Helfrich, W., Heppke, G., Eds.; Springer-Verlag: West Berlin, 1980; p 225. (1 1) Hess, S. Physica 1974, 74, 277-93. (12) Doi, M.; Edwards, S. F. J. Chem. Soc., Faraday Trans. 2 1978, 74, 918-32. (13) Janeschitz-Kriegl, H. "Polymer Melt Rheology and Flow Birefringence"; Springer-Verlag: West Berlin, 1983.
0 1985 American Chemical Society
518
p =
-?[
Thurn et al.
The Journal of Physical Chemistry, Vol. 89, No. 3, 1985
)I*
1
+ ( 9 - 7-1
(1 + iUTf)( 1 +
(&))
I
I
E+ Furthermore, we obtain by the definition16J7 C’(w) C’’(W)
The overhead bar and wave combination denotes the Fourier transform of the time-dependent part of the irreducible tensor of second rank. Together with v = r y e , V = e‘(d/ay), = (I’/2)eZ we obtain from (9)
P = -2)I’f
8x7- )I-f(ev- eyey)
In the case of a Couette flow I’ is constant and hence The complex viscosities q+ and )I- are defined by (
(10)
(18)
5 -)I”+w
= $+u
(19)
the equations for the storage and loss moduli C’(w) and G”(w)
where the plateau value Go is defined by ()I - ) I - ) / T , .
f = rS(w). T > )Iis always valid and hence the plateau value is practically determined by 7 , and )I; i.e. Go
It should be noted that for w = 0, eq 1 1 and 12 are identical with those derived by Hess in the stationary case r = ~ o n s t a n t . ’ ~In the same sense w # 0 leads to the complex viscosities derived by Hess. ?+
= 9-
)I - )I-
+ (1 + ius,)
Hence eq 1 1 and 12 contain the special case considered by Hess. Furthermore, it is interesting to note that, for example, the time-dependent shear viscosity is simply given by the following equations:I s
)1/7,;
)I-
> )I-,no free fit parameter is available to adjust the calculated values of G‘(o) and G“(o) to the measured quantities, because )I and 7,are already determined by other experiments, Le., by measuring the first viscosity )I and the extinction angle X. The Extinction Angle. The optical extinction angle x measured in flow birefringence experiments is determined by P ;w = 0. From eq 7 and 9 we get the following equation, which Hess derived earlier in a similar context:ls
x = l-4 r- -12 arctan (I?,)
(23)
For small gradients r and c/KT g, is always valid (see Table I). The influence of the quantity g, therefore becomes evident only in the functions G”(w) in frequency regions, where G’(w) reaches its plateau value. This is shown in Figure 3. The different frequency dependence of the two theoretical functions G”(w) in this figure is due to different values g, used in the calculation. In one case (function l), g, was chosen to fit the experimental data in the high-frequency range. In the other case (function 2), 7, is neglected. These functions clearly show that the value of g, in our case does not influence the characteristic intersection point of the functions G’(w) and G”(w), which is practically determined by the measured value 7,. It should be noted that the experimental values G’(w) and G”(o) at higher frequencies w are calculated from flow birefringence measurements by the stress optical law. In the constant-shear-rate mode the limit of the validity of this law is given by the fact that the “optical“ and the “mechanical” extinction angles X, x’become different from each other at higher shear rates due to the influence of the term 7 - 6 (see eq 7). In the oscillatory mode the same phenomenon should w u r in the functions G”(w) (see eq 21). The “optical response” of a solution is due only to the term G0(w.,)/( 1 + ( w ~ , ) ~whereas ), the mechanical behavior also depends on the product 7-w. Therefore values of G”(w) measured in rheological experiments and those calculated from flow birefringence experiments should become different a t frequency regions where the product 7-w is no longer negligible. Hence function 1 reflects the “mechanical” properties, while function 2 shows the “optical” behavior of the solution and can be described by a Maxwell element. It is very interesting that such a behavior was already predicted by a theory of Thurston and MunkZ3for oscillating flow birefringence data. Discussion Figures 3-8 show that the dynamic properties G’(w) and G”(w) of the micellar solutions are practically determined by the sta(23) Thurston, G. B.; Munk, P. J . Chem. Phys. 1970, 52, 2359-66.
d’ loo 0’ Figure 4. Same as Figure 3: 0 , 20% TPS.
ld{
IO
w)’
G’.G”/mPa
‘(21
10‘
10’
loo
lo1
IO1
IO’
Figure 5. Same as Figure 3: A, 40% TPS.
tionary values of the viscosity 7 and the extinction angle x,Le., the relaxation time 7,. First of all, this can be seen by the fact that at the intersection point of G’(w) and C”(w) within the experimental error the relation wip = 1/7, holds. A very similar experimental result for another detergent system was already published by Janeschitz-Kriegl and Wales.24 This means that the micellar interaction energy e and the rotational diffusion coefficient D control both the stationary and the dynamic aspects of the micellar systems. Furthermore, the plateau value Go. although it can be measured for some of the solutions (see Figures 6-8) only in an approximate way for experimental reasons, is (24) Janeschitz-Kriegl. H.; Wales, J. L.S. ‘Sympsoium on the Photoelastic Effect and its Applications, ottignies, Belgium, Sep 1973”; Springer-Verlag: West Berlin, 1975.
The Journal of Physical Chemistry, Vol. 89, No. 3, 1985 521
Viscoelastic Detergent Solutions
+ G’,G”/mPa
IO
j
/
lo{
100% TPS
5 0 % TPS
w/ 5-l 10’ loo Figure 6. Same as Figure 3: lo2
10’ +, 50% TPS.
IO‘
IO’ IO’ IO0 lo1 IO’ lo3 Figure 8. Same as Figure 3: X, 100% TPS, from rheological experiments; +, 100%TPS, from flow birefringence measurements.
10)
G‘ G”(1 1
G “ (21
s
+.:: m‘
7 0 % TPS
’
+.
oh-’ 10Z 16’ loo 10’ Figure 7. Same as Figure 3: V, 70% TPS.
loz
16‘
I
I I,,:,,,
1 1 ll11Il,
I 1 1 11111)
I
,
111111/
1
I ((I(’,(
Ut,
10’
described rather well by the quantity v / T , . Finally, we put our measured values on a dimensionless scale x = U T , and g’(x).= (T,/v)G’(x),which is shown in Figure 9. The full line, which may be regarded as a “master curve”, is given by g’(x) =
X2
1
+x2
As can be seen from Figure 9, all the scaled experimental values coincide within the experimental error, which is the best check for the fact that the dynamic behavior of all these solutions may in fact be determined just by measuring the two stationary quantities 7 and T , for each solution. Up to now it was not necessary to reflect upon the microscopic aspect of our micellar systems.
From kinetic measurements it is known25that micelles can be . relaxation characterized by two relaxation times T~ and T ~ The time T~ is related to the exchange of monomers between the micelle and the solution. Usually T , is very small compared with the relaxation times considered in this paper. The relaxation time T* is connected with the lifetime of the micelle and may vary from ( 2 5 ) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C . J . Phys. Chem. 1976, 80,905-22.
J . Phys. Chem. 1985,89, 522-524
522
milliseconds to hours. Therefore one has to keep in mind that 72 can influence the relaxation time of the birefringence 7,. Hence it should be interesting to compare 7 2 gained from T-jump measurements and T , gained from flow birefringence measurements. First experiments gave promising results and a detailed study will be published in a forthcoming paper. Following the concept of Hess,I4 one obtains for the relaxation times T~~ and T~~ T~~
=
T~~
= (3/5)'I27&R
(29)
of the micelles, which are considered as ellipsoids of revolution. The quantity R is defined by
32). On the other hand, we know from light-scattering experiments that the systems TPS and TTAS form rodlike micelles in the presence of salt. For rods P is proportional to L-l and therefore, as Go is approximately constant, the length of the micelles is proportional to (1 - t / k T ) , i.e., L (1 - t / k T ) . Therefore in these mixtures, the length of the micelles is influenced by their interaction energy. An increasing energy forces the micelles to become smaller, while decreasing energy enhances micellar growth. In addition to this microscopic aspect of viscoelasticity, one often likes to discuss the properties of viscoelastic micellar solutions on the basis (of the theoretical model) of entanglement networks. In this theory the plateau value Go is given by
-
Go = VkT where p is the axial ratio a / b and hence R > 0 for prolate and R < 0 for oblate ellipsoids. From definition (5) we obtain, inserting eq 29 and noting T , = 1 / 6 0 kT t) - t)- = -ER2 1OD where E is the number of particles per unit volume. This equation becomes identical with the one derived by Doi and Edwards,I2 replacing t)- by the viscosity of the solvent and setting R = 1 . Combining eq 31 with the definition of the plateau value Go, one obtains Go =
$(
1 - $)R2EkT
Hence the plateau value Gois determined by the number density of the micelles, the interaction energy E , and the axial ratio a / b . It is interesting to note that for 0% and 100% TPS the corresponding viscosity t) (and the relaxation time T J change by a factor of 60, while the plateau value Go remains nearly constant. This means that the product (1 - (c/kT))Eis nearly constant (see eq
(33)
where v denotes the number density of the elastically effective chains between the cross-links.26 Comparing eq 33 and 32 may lead to a microscopic aspect of v. According to this v is influenced by the interaction energy and the number of the micelles per unit volume. Concluding Remarks The presented theory of Hess makes it possible to relate the stationary and dynamical properties of viscoelastic solutions, studied by flow birefringence and rheological experiments. These properties can be described by the molecular dimensions of the micelles and their interaction energy. Also the influence of counterion, headgroup, etc. on the microscopic parameters E and L may be studied. Hence this theory should provide a good basis for explaining the behavior of micellar solutions on a microscopic scale.
Acknowledgment. The authors thank Prof. S.Hess for stimulating discussions. Registry No. TPS, 94070-50-5; TTAS, 86996-35-2. ( 2 6 ) Flory, P.J. J . Cfiem. Pfiys. 1950, 28, 108.
Thermal Proton Transfer in Crystalllne N,N-Dialkylated Amino Acids Mark A. Peterson and Charles P. Nash* Department of Chemistry, University of California, Davis, California 95616 (Received: May 4, 1984)
The infrared spectra of p-(NJ-di-n-buty1amino)propanoic acid and its monodeuterioderivative, @-(NJ-diethy1amino)propanoic acid, and y-(N,N-di-n-buty1amino)butanoicacid have been studied from room temperature to 10 OC above the melting points of the several solids. At room temperature each spectrum is that of the zwitterionic tautomer. At temperatures about 10 OC below the melting points, absorptions characteristic of carboxylic acid and free amine functionalities appear and then increase in intensity very rapidly up to the melting point. The melt spectra are thereafter insensitive to temperature. This behavior implies that in these compounds the spatially inhomogeneous electric field produced by the crystalline array of zwitterionic dipoles creates a slightly asymmetric, double-minimum, hydrogen-bond potential function with an intermediate barrier height. The marked thermal sensitivity of the solid spectra also implies that the potential function is itself sensitive to the degree of advancement of the proton transfer reaction.
Introduction The nature of the hydrogen-bond potential function has long been a subject of both theoretical and experiment interest. Previous studies from this laboratory have shown that in aprotic solvents N,N-dialkylated a-,p-, and y-amino acids form intramolecularly hydrogen-bonded monomers that exhibit a solventdependent tautomeric equilibrium between the zwitterionic and classical forms of the Only the classical form is (1) Horsma, D. A,; Nash, C. P. J . Pfiys. Cfiem. 1968, 72, 2351-2358. (2) Tam, J. W.-0.; Nash, C. P. J . Pfiys. Cfiem. 1972, 76, 4033-4037.
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observed in nonpolar solvents such as benzene or carbon tetrachloride, but in polar solvents such as acetonitrile or nitromethane, significant or even predominant amounts of the zwitterion occur. In the latter media a double-minimum potential function is indicated, together with the possibility for both wells to be populated at ambient temperature. Proton transfer across hydrogen bonds has been implicated in a number of phenomena of both physical and biological importance. Among these are proton conduction in solids3 and across (3) Glasser, L. Cfiem.Reu. 1975, 75, 21-65.
0 1985 American Chemical Society
