2626
Langmuir 1991, 7, 2626-2629
Measurement of the Coefficient of Thermal Expansion of Uniaxial and Biaxial Lyotropic Nematics: Disks and Rods or Intrinsically Biaxial Micelles? C. P. Bastos dos Santos and A. M. Figueiredo Neto' Instituto de Fisica, Universidade de SZro Paulo, Caixa Postal 20516, 01498 S&o Paul0 (SP), Brazil Received June 28,1990. In Final Form: March 25, 1991
A theoretical study using classicalthermodynamicarguments about the coefficientof thermal expansion
(a)of lyotropic nematics is done with the models of disks and rods and with intrinsically biaxial micelles.
The behavior of a as a function of temperature is compared with experimental values measured through a micropicnometry technique in the lyotropic mixture potassium laurate, decanol, and water. The experimental values of a fit well with the theoretical expressionsobtained by use of the intrinsicallybiaxial micelles model.
Introduction Mixtures of amphiphilic molecules and water are known for exhibiting three types of nematic phases, depending upon the conditions of temperature and relative concentrations. Two of them are uniaxial' and the third one is biaxial2 (NBx). The uniaxial phases have been further classified3 as calamitic (Nc) and discotic (ND),according to the orientation of the director A, parallel or perpendicular to the magnetic field, respectively. On the other hand, X-ray4 and neutron3 diffraction experiments have been meaningful in the determination of the micellar shape. Thus, in evidence of such experiments, the micelles were first thought to be prolate and oblate aggregates in the N, and ND phases, respectively. It was then further considered that from the uniaxial to the biaxial transition the micelles actually undergo a change in shape, changing from a flat disk (ND) or an elongated cylinder (Nc) to a kind of flat ellipsoid. However,another argument has been recently proposed, mostly on the basis of X-ray diffraction patterns obtained at different temperatures in the three nematic phase^.^^^ The experimental results are shown to be related to the model of similar micellar aggregates in the three nematic phases, undergoing different orientational fluctuations. The different nematic phases were then analyzed as being the macroscopic result of orientational fluctuations of the micelles (full rotation around the director in the case of the uniaxial phase and small amplitude oscillations in the biaxial one). Therefore, the uniaxial phases are not formed within this scheme7 by cylinder or disklike aggregates, but rather by statistically similar and biaxial-shaped ones. This model is also corroborated by neutron scatterings in the N, phase. Even after the publication of the experimental results of refs 5-7 supporting the model of similar and biaxial ~
(1) Radley, K.;Reeves, L. W.; Tracey, A. J. J. Phys. Chem. 1976,80, 174. (2) Yu, L. J.; Saupe, A. Phys. Reo. Lett. 1980,#, 1oOO. (3) Hendrikx, Y.;Charvolin, J.; Ftawiso, M.; Liebert, L.; Holmes, M. C. J. Phys. Chem. 1983,87, 3991. (4) Charvolin, J.; Levelut, A. M.; Samulsky, E. T. J. Phys. Lett. 1979,
- - - - ..
40 , 5A7
(5) Figueiredo Neto, A. M.; Galeme, Y.; Levelut, A. M.; Liebert, L. J. Phys. Lett. 1986. 46. 499. 16) Figueiredo-Neto, A. M.; Levelut, A. M.; Liebert, L.; Galerne, Y. Mol. Cryst. Liq. Cryst. 1985,129, 191. (7) Galerne, Y.; Figueiredo Neto, A. M.; Liebert, L. J. Chem. Phys. 1987.87. 1861. (8) Hendrikx, Y.; Charvolin, J.; Rawiso, M. Phys. Reo. E 1986, 33, 3534.
micelles in the three nematic phases, many authors still use the model3which has different shapes for the micelles in lyotropic nematics. In this paper we have made a theoretical study, based on classical thermodynamic arguments, of the coefficient of thermal expansion9 a of lyotropic nematics using both models for the micelles: disks and rods3 and also a ~arallelepiped.~ The behavior of a as a function of temperature obtained from these considerations is compared with values of a measured by means of a micropicnometry technique specially developed for this purpose.
Theory The coefficient of thermal expansion (a)is defined as9
457
=
1/vw/anp
(1) where T i s the temperature, V is the volume, and P is the pressure (kept constant). p being the density of the sample, we have
40 = - ( 1 / P ) ( a P / a 0 (2) The lyotropic mixture is essentially composed of micelles and water. With regard to that, we define a1 and VI ( a 2 and V2) as the coefficient of thermal expansion of water (micelles) and the volume filled with water (the micelles), respectively. So the coefficient of thermal expansion of the lyotropic mixture can be written as 40 = a1CV,/v, + a,(V2/V)
(3) (a) The Model of Disks and Rods.3 Following the definitions of ref 3,111 is the dimension of the micelle parallel to the director it (thickness of the disk or length of the rod); 11 is the dimension of the micelle perpendicular to A (diameter of the disk or diameter of the rod); dll is the dimension of the volume available to one micelle parallel to A; and d l is the dimension of the volume available to one micelle perpendicular to it. In a volume available to one micelle (where there is water and a micelle) we have
VI = 7r(d,/2)2dIl - 7r(1L/2)2111 v = V' + v,
v2
= 7r(11/2)211, (4)
Introducing the numerical values given in ref 3 for the (9) Callen, H.B. Thermodynamics; John Wiley & Sone, Inc.: New York, 1960; p 54.
0743-7463/9r/2407-2626$02.5010 0 1991 American Chemical Society
Langmuir, Vol. 7, NO. 11, 1991 2627
Coefficient of Thermal Expansion of Uniaxial and Biaxial Lyotropic Nematics
lyotropic mixture of potassium laurate, decanol, and water one obtains CY
= q0.7
a = a,0.6
+ ~ ~ ~ 0 (disks) . 3
(5)
+ a20.4
(rods) (6) To evaluate a2 we assume that temperature acts to modify lI1and I*. The coefficient of thermal expansion'o (linear) of the paraffinic chains is -1.3 X OC-I. So, using expressions 4 one obtains a2 = 3.1 X 10-2(dl,/dT) - 1.3 X
(disks) ( 7 )
a2 = 1.5 X 10-2(dll,/dT) - 2.6 X
(rods)
(8)
with 111 and lL measured in angstroms. With regard to that, we assume in a first-order approximation a linear behavior of d l / d T (for both I and 11) with
T
ai/aT = CT + D (9) where C and D are constants. The coefficient of thermal expansion of the water, in the range of temperature 10 "C < T < 35 "C can be written as11 a1= 5.2229 X 10-8T
+ 2.2708 X 10"'
(in OC-') (10)
So, expressions 5 and 6 can be written as CY
= (3.7 X lo4
+ 9.3 X 10-2C) T + 9.3 X lO-'D 2.3 X 10"'
a = (3.1 x io-*
(disks) ( 1 1 )
+ 6.0 x ~ o - ~ c )+T6.0 x ~ o - ~- D 8.6 X loW4 (rods) ( 1 2 )
(b) The Model of Biaxial micelle^.^ Following definitions of refs 5 and 7 , A' and B' are the dimensions of the micelles in the plane per endicular to the bilayer; C' is the bilayer; A, B, and are the correspondent dimensions of the volume available to one micelle. In this scheme
E
VI = ABC
- V2
V2 = A'B'C' (13) Introducing the numerical values given in ref 7 for the lyotropic mixture of potassium laurate, decanol, and water, one obtains = CY10.7 + (r20.3 (14) To evaluate a2 we assume that the coefficient of thermal expansion (linear) of the dimensions A' and B' are equal, presenting a linear behavior with T (as it was assumed in expression 9 ) . Introducing the numerical values and expression 10, one can write CY
= (3.7 x io-8
+ 7.1 x ~ o - ~ c +) T7.1 x ~ o - ~- D 2.3 X
(15) (c) First Comparison between the Models. With regard to that, we can make the first comparison between the two models. We need an evaluation of the parameters C and D (eq 9 ) in order to do that. Therefore, we expect that the modulus of the coefficient of thermal expansion (linear) in a direction perpendicular to the paraffinic chain is not very different from this coefficient along the direction of (10) Huseon, F.; Muatacchi, H.; Luzzati, V. Acta Crystallogr. 1960,13,
688.
(11) Handbook of Chemistry and Physics, 52nd ed.; The Chemical Rubber Co.: Cleveland, OH, 1972.
the paraffinic chain. This hypothesis is reasonable because the 'trans-gauche" configurations of the chains are directly related to the behavior of the thermal expansion of the chains in both directions (along the chain and perpendicular to that). Typical values of C and D can be independently obtained by fitting expressions 11,12, and 15 to experimental values of CY as a function of temperature. Through this procedure we obtain 3 X 10" < C < 3 X 10-5 (A OC-2) and 10-2 < D < lo-' (A O W ) . This values will be discussed in more detail in the following sections. We assume C = 4 X 10-5 A 0C-2 and D = lo-' A O C - l to make this comparison between the models. These values give ( l / l ( d l / d 7 ' ) 10-3 OC-l (which agrees with our hypothesis discussed above). Introducing this values in expressions 11,12, and 15, we obtain
-
a
N
CY N
+9 X
(disks)
(lla)
3 x lO-'T - 3 X
(rods)
(12a)
(biaxial)
(15a)
4 X 10*T
CY E
3 x 10-7T + 5 X 10"'
Comparing eqs lla-l2a, we have noticed that the angular coefficients of both equations differ by 1 order of magnitude and the linear coefficients differ by 2 orders of magnitude (there is also a change of sign) in both phases. The model of biaxial micelles (eq 15a) gives the same behavior of CY as a function of T for the uniaxial phases as well as the biaxial phases. The differences obtained between the angular and the linear coefficients for the Nc and NDphases, considering the data of ref 3 (and typical values of C and D ) , indicate that precise measurements of a can be used to test the different models for micelles.
Experimental Section The device used for the density measurementsconsisted of a cylindric water chamber (8 cm in diameter),with a double wall. Inside the wall, water circulates by means of a pump, which is programmed for a certain temperature by digital control. The temperature inside the chamber is further measured by a thermometer (accuracy of 0.05 OC). Also inside the chamber is a little hollow metal cylinder, whose purpose is to support a micropicnometer containing the sample. The shape of this picnometer is particularly important for its attainable precision. It consists of a capillarypart of 6 cm long which goes into a cylindric bulk of 3 cm in length and diameter of 7 mm. The capillary is in turn just thin enough to allow the introduction of a needle and the placing of the material inside. At its lower end (almost at the junction with the bulk), there is a tiny control mark which makes possible the definition of a fixed volume for the picnometer. Provided that the capillaryis very thin (diameterless than 1 mm) and the bulk contains almost all the fluid, even a very slight change in the density can be detected by the level's alterationof the sampleat the control mark. All these precautions are essential to minimize the experimental errors and increase the accuracy of the density measurements. The whole structure (waterchamber,picnometer, connecting pipes) is closed and protected during the measurementsby thick insulating layers. Before each measurement, the sample is left for 30 min in the constant-temperaturebath. After that, the picnometer is taken out and the mass of the sample is determined. It is weighed on a Mettler analytical balance with a readability of 0.01 mg. The density is directly calculated. Provided that a previous calibration has been made (in that case water was used), we can consider the dilation of the glass due to the variation of temperature.
2628 Langmuir, Vol. 7, No. 11,1991 I
0
a
t
Bastos dos Santos and Figueiredo Net0
I
I
I
I
I ;
!
I
1
-
5.41
;
I;
I
-
c
1
Y
0
I
I
0
I NBX
NC
~
I
I I I
I I
I
10
15
20
25
30
35
T("C) Figure 2. Coefficient of thermal expansion (a)as a function of temperature: lyotropic mixture of potassium laurate, (28.00 wt %), decanol, and water. near the phase transition was detected. The results of Figure 2 can be fitted to a linear law as a function of T:
+ 5.3185 X 10''
aF= 2.94 X lO-'T
l.Cc4
-
I
I I (
-
I I
I
I
I ,
I
1
I
("C-') (16)
in all the nematic domain (ND, NBX,and Nc). With regard to that, one can compare these experimental results with expressions 11 and 12 of the disk and rod model, and expression 15 of the biaxial micelle model. Within the hypothesis presented in the theoretical section, both models give a linear behavior of a as a function of T. However, there is an important difference between them: in the disk-rod model, the calculation predicts different angular and linear coefficients of CY( 2') in the ND and Nc phases; the biaxial micelle model predicts the same angular and linear coefficients of a(V in all the nematic range. Fitting expression 11to the experimental results at the ND phase, one obtains
+
--= 4.3 X 10*T 1.3 X 10'' ("C-') (17) 1 dT with C = 2.7 X 10-6 A 0C-2 and D = 8.2 X lomsA O C - l . Introducing expression 17 in eq 12 one obtains = 4.8 X 10*T - 8.1 X 10"
(rods)
(18) This expression does not fit the experimental results to the Nc phase. On the other hand, fitting expression 12 to the experimental results at the Nc hase, one obtains C = 4.4 X A oC-2and D = 2.3 X 10-l O C - l . Introducing these values in (11)one obtains CY
s
a = 4.1 X 104T
+ 2.1 X lo-'
(disks)
(19) This expression does not fit the experimental results to the ND phase. Fitting expression 15 to the experimental results, one obtains
--= a1 4.3 x 1 aT
(12)Figueiredo Neb, A. M.;Liebert, L.; Galerne, Y.J. Phys. Chem.
1986,89, 3737.
+ 1.3 x 10-3
i o - 7 ~
(oc-1)
(20)
(here, 1 stands for the dimensions A' or B'). The fitting gives C = 3.6 x 10-5 A O C - 2 and D = 1.1 X 10-l A OC-1. Analysis of expression 20 shows that, in the temperature range of existence of the nematic phases (about 1060 "C),the linear coefficient of thermal expansion of the OC-'. This value has dimensions A' and B' is -1.3 X the same order of magnitude of the linear coefficient of thermal expansion of the paraffinic chains being, however, positive. This result is compatible with the increase of the surface per polar head occupied by an amphiphilic
Coefficient of Thermal Expansion of Uniaxial and Biaxial Lyotropic Nematics
molecule as a function of temperature due to the transgauche configurations of the paraffinic chain. In conclusion, the experimentalresults of the coefficient of thermal expansion of lyotropic nematics, in both uniaxial and biaxial phases, can be fitted to expression 15 of the biaxial micelles model with a linear coefficient of thermal expansion of the micelles in the plane of the bilayer given by expression 20. It is necessary to have different values of the linear coefficientof thermal expansion for the same amphiphilic molecule for disks and rods, in order to obtain a good fit
Langmuir, Vol. 7, No. 11, 1991 2629
of the experimental resulta for expressions 11 and 12 of the disk-rod model. In our point of view, there are neither topological nor thermodynamic reasons to justify this difference between the linear coefficients of thermal expansion of rods and disks.
Acknowledgment is made of the financial support of Fundaqiio Banco de Brasil. Registry No. K-L, 10124-65-9; DeOH, 112-30-1.
