4124
J. Phys. Chem. 1985,89, 4124-4128
Effects of Mlcelle Shape and Fluctuatlons In Shape on Orlentatlonal Order in Lyotropic Liquld Crystals M. R. Kuzma Department of Physics, Temple University, Philadelphia, Pennsylvania 191 22 (Received: July 16, 1984; In Final Form: April 1, 1985)
Micelles in lyotropic nematic solutions assume anisometricshapes. Experimentallyobservable quantities, such as the birefringence or NMR splitting, measure the degree of order relative to a fmed lab direction h of surfactant molecular chains (or a specific bond direction on a molecule). The micelle shape geometrically restricts the allowed projections, at different positions in the micelle, of an arbitrary molecular axis m (or bond direction) onto a macroscopic laboratory direction. Averaging m over simple micelle shapes (disk, rod, prolate and oblate spheroid) we obtain analytic expressions connecting the average dimensions (radius, length, thickness, eccentricity) to the shape-induced chain order of the aggregate. An average lower bound on micelle size can be inferred from experiments. Inclusion of fluctuations in micelle shape increases this bound.
I. Introduction Soap molecules formed of a polar head group and a linear hydrocarbon tail aggregate, in aqueous solution, into roughly spherical micelles just above the critical micelle concentration.’ As the surfactant concentration is increased, geometric constraints of the hydrocarbon interior force the micelles to adopt anisotropic shapes.’V2 At sufficiently high soap concentration (-8 mol %) interaction between the micelles may stabilize a nematic phase in which the symmetry axes of the micelles are aligned to some extent.f5 The uniaxial nematic state may be composed of disklike micelles, No phase, or cylinder micelles, Nc phase. A biaxial nematic phase, Nbx,is also known to e x i ~ t . ~The . ~ microscopic structure of Nbxhas at least two interpretations: (1) It is composed of a mixture of finite disk and rod micelles, with the symmetry axes of the disks and rods respectively aligned along orthogonal lab direction^.^ (2) The micelles themselves have the symmetry of biaxial ellipsoids, and align in the biaxial state the axes of the ellipsoid^.^ At still higher soap concentration, and/or at different temperatures, a transition from the nematic phase to a smectic phase may occur. The smectic phase may either be a layered alternating arrangement of soap bilayers and water (neat soap) or a hexagonal packed columnar arrangement of cylindrical micelles (middle ~ o a p ) . There ~ , ~ appears to be no fundamental reason to assume the smectic states are composed of infinite bilayers or rods, and in fact recent conductivity experiments indicateE(at least for neat soap) that the smectic state can be modeled by translationally ordered finite micelles. An important physical quantity whose behavior is used to characterize the above systems is the degree of orientational order present in the system. DMR measurements of D 2 0 ordered by the micellar interface and DMR and PMR measurements of surfactant alkyl chain order have been extensively investigated in a number of lyotropic ~ y s t e m s . ~Birefringence ,~ measurements of alkyl chain order have been performed in ref 6 and 10. The (1) Tanford, C. ‘The Hydrophobic Effect”; Wiley: New York, 1973. (2) Israelachvili, J. N . ; Mitchell, D. J.; Ninham, B. W. J . Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (3) Charvolin, J. Nuouo Cimento 1984, 3, 3 . (4) Saupe, A. Nuouo Cimento 1984, 3, 16. (5) Forrest, B. J.; Reeves, L. W. Chem. Reu. 1981, 81, 1 . (6) Meuti, M.; Barbero, G . ;Bartilino, R.; Chiaranza, T.; Simoni, F. Nuouo Cimento 1984, 3, 30. (7) McMullen, W. E.; Ben-Shaul, A.; Gelbart, W. M. J . Colloid Interface Sci. 1984, 98, 523. Alben, R. J . Chem. Phys. 1973, 59, 4299. Stroobants, A.; Lekkerkerker, H. N. W. J . Phys. Chem. 1984,88, 3699. (8) Bcden, N.; Corne, S. A,; J o k y , K. W. Chem. Phys. Lett. 1984, 105, 99. Photinos, P. J.; Yu, L. J.; Saupe, A. Mol. Cryst. Liq. Cryst. 1981, 67, 277. (9) Charvolin, J. J . Chim. Phys. 1983, 80, 15
0022-3654/85/2089-4124.$01.50/0
magnitude of the degree of order varies drastically depending upon the constituents of the system one is observing. For example, the whereas the water order parameter S, is typically of order surfactant chain order S, is of order lo-’.’ However, as the temperature is varied it is generally found within the nematic phase that S, a S, and the order of each of the constituents in the system may be set proportional to a value S representing the degree of order of the micelles. Exceptions to this proportionality are known to occur in the smectic phase of decylammonium chloride/ammonium chloride/ water. The proportionality constant c relating the order of the constituents to S is difficult to calculate. One important factor that c depends upon is the shape of the micelle, irrespective of any orientational ordering of an arbitrarily chosen micellar axis. That the shape of the micelle primarily influences the birefringence and the NMR splitting through geometric constraints has been emphasized by Holmes and Charv01in.l~ In section I1 we discuss the above shape effects based on simple geometrical models for micelle shapes. In section I11 we discuss the effect of fluctuations in micelle shape on the orientational order, assuming the micelle dimensions are large enough that a continuum approximation may be applied. We also pheomenologically include effects of interactions between micelles using a method of DeGennes and TaupinI4 and He1fri~h.l~ In section IV we discuss the results in relation to experiment. 11. Model
We assume the shape of the individual micelle in equilibrium does not fluctuate wildly during the time of observation. For an order of magnitude equal to the lower bound estimate of the time of a shape change of a micelle we may take the exchange time of a single surfactant molecule between aqueous solution and miceIleI6 T,,
=
yo-]
exp(Ap/kn
-
s
(1)
vo is a typical rotation frequency for a molecule in a nematic phase; Ap is the chemical potential difference between a surfactant
molecule as a monomer and in an aggregate. This value depends on ion concentration, alkyl chain length, etc. An order of mag(10) Haven, T.; Radley, K.; Saupe, A . Mol. Cryst. Liq. Cryst. 1981, 75, 87. ( 1 1 ) Kuzma, M.; Saupe, A. Mol. Cryst. Liq. Cryst. 1983, 90, 349. (12) Fujiwara, F.; Reeves, L. W. J . Am. Chem. SOC.1976, 98, 6790. (13) Holmes, M. C.; Charvolin, J. J . Phys. Chem. 1984, 88, 810. (14) DeGennes, P. G . ;Taupin, C. J . Phys. Chem. 1982, 86, 2294. (15) Helfrich, W. Z . Naturforsch. 1978, 33a, 305. Helfrich, W.; Servuss, R. M. Nuouo Cimento 1984, 3, 137. (16) Israelachvili, J. N.; Marcelja, S.; Horn, R.G . Q.Rev. Eiophys. 1980, 13, 121.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 19, 1985 4125
Lyotropic Liquid Crystals
-
nitude estimate for the rearrangement time T~ of a micelle is TR Mrex,where M is the aggregation number of the micelle. W e use rRas an upper limit for the time it takes for the micelle to significantly change its shape. A typical NMR time scale is set by the inverse of the largest observed splitting Aw. Integrity of the average micelle shape can be assumed if T~ >> (Aw)-' which is in general satisfied. (Setting e.g., M = 100, rR lo-', >> (Aw)-' low5s). On the other hand random motions of the surfactant molecules parallel to the water-surfactant interface must be sufficiently rapid so that intermolecular dipolar interactions average to a small enough value to allow observation of a splitting. A typical translational diffusion constant for the 10"-10-7 cm2 s - ' . ~ Similar systems under consideration is D arguments can be made in the case of light scattering using the equivalence between time and space (ensemble) averages. We now focus our attention on a single micelle composed of M surfactant molecules. For definiteness we examine the orientational order of a proton pair on an alkyl chain separated by the vector r,. Writing for the average over the micelle of the nth proton pair
0
-
-
-
Figure 1. Definition of dimensions of micelles. Rotate around axis 0 for disk micelle of radius R. Rotate around axis 0' for spherocylinder micelle of length L + 2a.
M
S(")= (l/M)C(P,(r,c')-h)) i= 1
We have used the addition theorem for spherical harmonics and have assumed the distribution function factorize^'^^'^ into three statistically independent parts, the first giving the average orientation of a selected molecular bond relative to the molecule, m, the second giving the average orientation of the molecular long axis relative to the micelle axis z, and the third an average orientation of a micelle axis, relative to an arbitrary lab axis, h. For simplicity we let each of these distribution functions have uniaxial symmetry. If this assumption is not made, the ensuing algebra and notation obscures the analysis. u is the surface density of molecules in the micelle. The brackets are a thermal average over molecular motions from the factorized distribution function. We assume the average orientation m of the chain axis at position i is parallel to the micellar surface normal siat i. For untilted chains at an interface, experimentali8and t h e ~ r e t i c a l ' results ~ ? ~ ~ demonstrate that the first few methylenes in a chain show a higher orientational order parameter parallel to the surface normal than the chain ends. Uniform tilting of the chain relative to the surface normal is a complication, and for each of notation we ignore it. Again for simplicity we assume a constant surface density u = M I S . Therefore S(") = ( P2(m.r,) ) ( P2(z.h)) fls
(3)
Figure 2. Shape-induced order vs. particle size: (a) R, vs. x = R i a , (b) Rm vs. x = b / c - 1 , (c) RE vs. x = Lfa, (d) R, vs. x = cJb - 1 . See eq 5-8 of text.
of the error is obtained by assuming the micelle can be divided into k parts, each of which has a constant density uk,e.g., the edge density and the body density uwyof a disk micelle. The error is then the difference between the nonuniform surface average 1 / a s d s k "k ( k = edge, body), and the uniform surface average l/MSds uwv where we have selected the uniform surface density as u = uMY. Figure 1 gives the dimensions of the spherocylinder and disk shapes. Equation 4 is calculated for the case of oblate and prolate spheroids in the Appendix; the same procedure applies to the other shapes. The results are as follows. (i) Spherocylinder (sc)
Q,, = -y2(1
fl, = (l/S)]dS
-
P2(s.z), slim
-
= Pz(S'z) = '/(s.z)2 - t/z
(4a) (4b)
represents the dependence of S(")on micelle shape. The assumption of constant surface density warrants further discussion. Since the chain order depends upon head group area (this dependence being weaker for the methylenes near the interface'* the use of a constant areal density u is strictly incorrect. The magnitude of the error made in passing from eq 2 to eq 3 should be of O( 1 for disk micelles and O( 1/M) for cylinder micelles. The error comes mostly from the factor (P2(r,,.m))which has the strongest dependence on u.18-20The order of magnitude
(s,Z)2.
(ii) Disk (d)
-
( R / a ) [ ( R / a )+ r / 2 1 + 73
=
( R / a ) [ ( R / a )+ r/21
(6)
+2
(iii) Oblate spheroid (os); here we define a = b principal axes of the oblate spheroid
> c as the
- (1 - t2/2t) In (1 + t / l - E ) 1 + (1 - e2/2e) In (1 + t / 1 - t) I
t
1
= (1 - ( c / b ) 2 ) ' / 2
(iv) Prolate spheroid (ps); a = b
t
(17) Doane, J. W. In "NMR of Liquid Crystals, in Magnetic Resonance of Phase Transitions"; Owens, F. J., Poole, C. P., Jr., Farach, H. A., Eds.; Academic Press: New York, 1979, p 171. This reference discusses a similar factorization (or separation of time scales) of the distribution function. (18) Mely, B.; Charvolin, J.; Keller, P. Chem. Phys. Lipids 1975, 15, 161; 1977. 19, 43. (19) Cantor, R. S.; Dill, K. A. Macromolecules 1984, 17, 384. (20) Dill, K. A. J . Phys. Chem. 1982, 86, 1498.
(5)
For the rest of the shapes it is perhaps clearer to state the average -
(s,z)2d
where
+ 2a/L)-'
(7b)
> a we arrive at 3 kT 2 KK~
Q6= 1 - - - In (R/ae'/2) Equation 14a demonstrates that with no suppression of fluctuations
To discuss fluctuations in cylinder micelles we analyze the disturbed cylinder by the vector t(z) = (au,/az, du,/dz,
= (bt,, (21) Gelbart, W. M.; Ben-Shad, A,; McMullen, W. E.; Masters, A. J . Phvs. Chem. 1984. 88. 861. 122) Wegner,-F. J. 2. Phys. 1967,206, 465. Berezinski, V L. Zh. Eksp Teor. Fiz. 1970, 59; Sou. Phys.-JETP (Engl. Trans1 ) 1971, 32, 493 (23) Sheng, P. Solid State Commun. 1976, 18, 1165. (24) Marcelja, S.; Radic, N . Chem. Phys. Letf. 1976, 42, 129.
st,,
1)
1)
(25) Ornette, D.; Ostrowsky, N . J . Phys. (Paris) 1984, 45, 265. (26) Ninham, B. W.; Parsegian, V. A. J . Chem. Phys. 1970, 53, 3398. (27) Verwey, E. J. W.; Overbeek, J. Th. G. "Theory of the Stability of Lyophobic Colloids": Elsevier: New York, 1948.
The Journal of Physical Chemistry, Vol. 89, No. 19, 1985 4127
Lyotropic Liquid Crystals
TABLE I: Minimum Sizes R / n of Disk Micelles Inferred from NMR Measurements"
ref system 30 octylammonium chloride/ HzO,70130 wt %
i
A
C
Figure 3. Cross sections of disks (or cylinders): (a) rigid limit; (b)
flexible with reduced orientational order; (c) lamellar mode with layer distance constant.
parallel to the local cylinder axis. The functions ux, uy denote transverse displacements of the cylinder axis relative to the unperturbed cylinder. The free energy per unit cylinder length28 is g = yzKc[(
z ) (?)'I
(23)
+
-
By equipartition we get (we estimate the bending modulus K , = 2?ra2K33where K3, is a typical bend elastic constant 10-6. With a N 30 %.,K, N (3-5) X erg cm) (24)
We proceed as above, Setting
v2t2)) = (IW) - w)i2) = (kT/2KC)z
(25)
Averaging over a micelle of length L we obtain
This average can be rendered independent of L by again including a "steric- interaction whidh mimics confinement to a tube.
G' = '/,bfzJdz
(u:
+ u:)
Thus
So for the case of cylinder micelles S = s0Q8.cy1
( :)
=so
l---X,
Using the above h i m a t e s for K,, Kd and setting X, N &, we find the first-order corrections (TN 300 K) to the order parameter S due to micellar flexibility are less than about 2% for disk micelles, up to about 10%for cylinder micelles. The higher value obtained for the correction for cylinder micelles owes to the well-known stronger divergence of long-wavelength fluctuations in one dimension. T o be self-consistent one should include the effects of orientational ordering on K, and Kd. In the nematic phase we can write Ki = K,(o)+ K,(])S2 where Ki(0)and Ki(') (i = c, d) are constants. The Ki values do not vanish in the isotropic phase because the individual micelles
temp, phase O C lamellar 27
order parameter R / a 0.56 1
11 decylammonium chloride/ ammonium chloride/ D20,7.56/2.74/89.70 mol %
ND
67 48
0.32 0.67
0.4 1.4
5 guest stearate-& in host sodium decyl sulfate
ND
30
0.31
0.4
0.59
1.1
31 potassium palmitate/ H20,72/28 wt %
lamellar 80-110
+
'For the oblate spheroid shape we compare (R 2a)/a with b/c; b is the semimajor axis and c the semiminor axis of the uniaxial ellipsoid.
still have a resistance to deformati0n.l One can think of the K,(O) as finite due to the presence of chain order18-20induced by the interface. The effect of eq 22 and 29 is illustrateti in Figure 3 in which the fluctuations in neighboring micelles are present. Modes such as Figure 3c with qr = 0, q l n are observable in lamellar systems, because of long range translational order.28 IV. Discussion Equation 3 may be used to deduce an average lower bound on micelle size. For definiteness we select the average of eq 3 , where m is a molecular axis, z a micelle symmetry axis, and n the director.
S(")= (P2(mr,))(Pz(z.h))b, (30) h = n optic axis or H magnetic field direction. For instance, if we set the left-hand side equal to 0.7 obtained by Hendrix et al.29 as a crude estimate for orientation (order derived from neutron scattering of the paraffin chain) in an ND phase of s6dium decyl sulfate, decanol, D20; upon setting (P2(n.z))= (P2(mr,)) = 1 , we obtain the value nD= 0.7. Reference to Figure 1 gives R i a = 1.6. For the Nc-ND transition in the same system again 0.7 = 2nsc; fisc = 0.35 gives L / a = 5 . In reality however, (P2(n.z)) and (P2(mr,)) < 1 and fluctuations in micelle shape occur; therefore these estimates are only lower bounds. It is possible to study systems in which the effects of particle flexibility have been eliminated. For instance tobacco mosaic virus (TMV) can be treated as a hard rod model of an N, phase.30 Similarly suspensions of the clay bentonite when exposed to a magnetic field-" or a flow field,32display nematic order in which the consistent particles can be regarded as hard plates or A size analysis can be made with N M R data and is presented in Table I. No attempt is made to be complete. S is deduced from the PMR line width in ref 11, 12, and 33 and from the largest observed splitting, ref 5 and 34, when the surfactant chains are examined by DMR. This splitting is usually associated with the methylene group closest to the head group. It is interesting to note that the order parameters measured in the smectic phase can be obtained by translationally ordered finite micelles, as well as the usual picture of infinite bilayers. Some final remarks to be made are as follows. ( 1 ) It would be interesting to know the temperature dependence of the miceIle dimensions, in particular near the nematic isotropic and nematic smectic transitions. In a homogeneous phase, as the micelles increase in size the total number of micelles decreases due to the fixed volume fraction of surfactant. Strong flow birefringence (shear rate 10' PI)above T N I(AT = T - T N I= 2 "C) in many surfactant systems indicate the
-
(28) DeGennes, P. G. "The Physics of Liquid Crystals"; Clarendon Press: Oxford, 1975. (29) Hendrix, Y.;Charvolin, J.; Rawlso, M.; Liebert, L.; Holmes, M. C. J. Phys. Chem. 1983,87, 3991. (30) Straley, J. P. Mol. Cryst. Liq. Cryst. 1973, 22, 333. (31) Mehta, R. V. J. Colloid Interface Sri. 1973, 42, 165. (32) Langmuir, I. J. Chem. Phys: 1938, 6, 873. (33) Dijkema, C.; Berendsen, H.J. Mugn. Reson. 1974, 14, 251. (34) Vaz, N.; Doane, J. W.Phys. Rev. A 1980, 22, 2230.
4128
J . Phys. Chem. 1985, 89, 4 128-4 131
micelles are anisotropic in the isotropic phase for T 2 TNI. This phenomena is not universal in lyotropics as can be seen in the system disodium cromoglycate-H20. For AT = 2 OC the flow birefringence is much reduced compared to the surfactant syst e m ~ .This ~ ~ is probably connected to the fundamental difference in the construction of the DSCG micelle^.^^^^^ (2) Subjecting lyotropic nematic (e.g., Nc phase) solutions to longitudinal flows would be interesting particularly if the micelles dimensions were typical of polymers. At sufficiently high shear rate the micelles, upon entering the region of high gradient, will be pulled apart. The creation of a higher density of ends would lead to a decrease of birefringence within the region of strong gradient. (4) Effects of polydispersity, especially relevant for cylinder micelle^,^*^^ should be estimated. Some dynamical effect might be present since smaller cylinder micelles being more mobile can relax an imposed deformation faster than in a monodisperse solution of long rods. In summary, we have examined the effect of micelle shape on the observed orientational order of micellar systems. An average lower bound on micelle size can be deduced. Inclusion of fluctuations in micelle shape and symmetry axes increases this lower bound.
Appendix Here we sketch the derivation of eq 7 and 8 for prolate and oblate ellipsoids. We consider the surface integral of the scalar function
(S.i)*
+
s = bc sin2 8 cos 4 i
d S = IsldO d 4
> c)
For an oblate spheroid ( a = b
1sI2 = b2cZsinZO(1
/3 = b2/C2 - 1,
t
+ p cos2 8)
= (1 - ( c / b ) 2 ) " 2
and
t
Evaluating the integral and converting to e gives eq 7 in the text. The procedure is exact the same for prolate ellipsoids ( a = b < c ) ; we obtain
here E
= (1 - ( b / c ) 2 ) ' / 2
(A31
and
where S is the unit surface normal and i is the symmetry axis. The equation for the surface of a general ellipsoid is @(8,4)= a sin 8 cos 4 P b sin O sin 4 j + c cos 8 i The unnormalized surface normal is s = (a@/d8)
The differential surface area is
X
(d@/aO)
+ ac sin2 8 sin 4 j + ab sin 8 cos 8 2
(35) Kuzma, M., unpublished work. (36) Hartshorne, N. H.; Woodward, G. D. Mol. Crysr. Liq. Crysr. 1973, 23, 343. (37) Lydon, J. E. Mol. Cryst. Liq. Cryst. 1980, 64, 19. (38) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y . J . Phys. Chem. 1980, 84, 1044.
S,, = 2ab2
2~bc +sin-l
t
€
Again the integral is trivial and combining (A3) and (A4) and writing as a function of t (eq A4) we get eq 8 of the text.
Acknowledgment. I express my appreciation for the hospitality of Dr. M. M. Labes and the Temple University Chemistry Department. I also thank Prof. A. Saupe for conversations which initiated this paper. The helpful comments of the referees are appreciated. This work was supported by the National Science Foundation under Grant DMR81-07142. I thank Kathy Long for typing the manuscript.
An Observed Retationship for the Vaporization of Liquids to the Critical Temperature William L. Marshall Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 (Received: March 10, 1983; In Final Form: May 13, 1985)
A vaporization ratio for liquids, AE/(PAV), is shown to fit closely to a straight line function of (1 - T r ) zfrom the critical temperature (T,) to a value of 0.6 for T,, the reduced temperature. At the limit of T,, AE/(PAV) is considered to be a critical vaporization ratio. The simple form of the relationship would appear to require the approach to a temperature-independent proportionality of A E to PAYat (or very near) this limit. Scaling laws will, however, complicate the limiting behavior. Its reasonable compatibility with the equations of Plank and Riedel and of Pitzer et al. suggests some fundamental significance.
In searching for a structural correlation of AE with PAV, expressed as a vaporization ratio, AE/(PAV), for liquids at the approach to the critical temperature, a simple relationship for this property over a wide range of temperature has been observed. In this relationship, AE/(PAV) is closely a straight line function of 0022-3654/85/2089-4128$01.50/0
(1 - T,)2,where AE is the internal energy of vaporization, P is the vapor pressure, AYis the volume change upon vaporization, T, is the reduced temperature ( T / T , ) with T i n K, and T, is the liquid-vapor critical temperature. Numerous dissimilar liquids appear to follow this behavior at values of T , from very near the
0 1985 American Chemical Society
