J . Phys. Chem. 1987, 91, 3830-3834
3830
Effect of a Cosurfactant on a Micellar Nematic Liquid Crystal Charles Rosenblatt Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: February IO, 1987)
The phase diagram and results from magnetic susceptibility measurements in the isotropic phase are presented for the micellar liquid crystal cesium perfluorooctanoate water in the presence of the cosurfactant perfluorooctanol. Based upon the evolution of the Cotton-Mouton coefficient and in the context of an Onsager entropy calculation, the average aggregation number and size of the micelle are obtained as a function of cosurfactant concentration.
+
Following the discovery of micellar nematic liquid crystals by Lawson and Flautt,' initial interest focused primarily on the determination of phase diagrams. In subsequent years, however, work quickly split along two paths. One body of literature has been concerned primarily with understanding the macroscopic phases and phase transition^,^-^ particularly as they relate to traditional thermotropic liquid crystals. Another body of literature has been concerned with micellar structure and the effects of added species such as electrolytes or alcohols.+12 These two branches of investigation are closely related, however, and must be treated as such to achieve a complete understanding of these materials. For example, the presence of a macroscopic biaxial phase in the ternary system potassium laurate, 1-decanol, and D20 derives from the shape and size distribution of the constituent micelles. This topic has been treated extensively both theoretically and experimentally.z~'3-2' In addition, the potassium laurate system also exhibits a reentrant oblate nematic-to-isotropic phase transition with decreasing t e m p e r a t ~ r e , ~a~ result - * ~ arising from the temperature dependence of the micellar anisometry, as shown by the X-ray results of Galerne et aLz3 Another interesting system for investigation is cesium perfluorooctanoate (CsPFO) and water, which forms oblate micelles4 whose symmetry axes align parallel to an applied magnetic field.3 In addition to several studies on the nature of the nematic-isotropic phase transition,3 considerable attention has also been paid to the micelle structure'0*24and nature
(1) Lawson, K. D.; Flautt, T. J. J . A m . Chem. SOC.1967,89, 5489. (2) Yu, L. J.; Saupe, A. Phys. Reu. Lett. 1980, 45, 1000 J . A m . Chem. SOC.1980, 102, 4879. (3) Rosenblatt. C . Phvs. Rev. A 1985. 32. 1 1 15. (4j Boden, N.; Jackson, P. H.; McMullen, K.; Holmes, M. C. Chem. Phys. Left. 1979, 65, 476. (5) Chen, D. M.; Fujiwara, F. Y.; Reeves, L. Can. J . Chem. 1977, 55, 2396. (6) Kuzma, M.; Saupe, A. Mol. Cryst. Liq. Crysr. 1983, 90, 349. (7) Boden, N.; Corne, S. A,; Jolley, K. W. Chem. Phys. Lett. 1984, 105, 99. (8) Photinos, P. J.; Saupe, A. J . Chem. Phys. 1986, 84, 517. (9) Hendrikx, Y.; Charvolin, J.; Rawiso, M. J. Colloid Interface Sci. 1984, 100,591. (IO) Rosenblatt, C . Phys. Reu. A 1985, 32, 1924. ( 1 1 ) Rosenblatt, C.; Zolty, N. J . Phys. Lett. 1985, 46, L-1191. (12) Hendrikx, Y.; Charvolin, J.; Rawiso, M. J . Phys. Chem. 1983, 87, 3992. ( 1 3 ) de Oliveira, M. J.; Figueiredo-Neto, A. M . Phys. Reu. A 1986, 34, 3481. (14) Figueiredo-Neto, A. M.; Galerne, Y.; Levelut, A . M.; Liebert, L. J . Phys. Lett. 1985, 46, L-499. (15) Knisely, W. N . ; Keyes, P. H. Phys. Reu. A 1986, 34, 717. (16) Galerne, Y.; Marcerou, J. P. J . Phys. 1985, 46, 589. (17) Vause, C. Phys. Rev. A 1984, 30, 2645. (18) Alben, R. Phys. Rev. Lett. 1973, 30, 778. (19) Shih, C. S.; Alben, R. J . Chem. Phys. 1972, 57, 3055. (20) Saupe, A.; Boonbrahm, P.; Yu, L. J. J . Chim. Phys. 1983, 80, 7. (21) Boonbrahm, P.; Saupe, A. J . Chim. Phys. Phys.-Chim. Biol. 1984, 81, 2076. (22) Rosenblatt, C. Phys. Reu. A 1986, 34, 3551. (23) Galerne, Y.; Figueiredo-Neto, A. M.; Liebert, L. Phys. Reu. A 1985, 31, 4047.
0022-3654/87/2091-3830$01.50/0
of the layered phase.' Trying to understand the role of additives, a recent paper reported" on the effects of an electrolyte on micelle structure, as well as on the quasi-excluded volume type interactions among micelles. The role of a cosurfactant on anisometric mesogenic micelles has only recently begun to receive attention. Until now, much of the work has centered on spherical micelles or microemulsion systems, where various techniques have been used to probe the solubilization, the location within the micelle, and the conformation of an added alcohol.2s-B In general, it was found that single-chain alcohols tend to concentrate within the micelle26rather than remain in solution and partition preferentially toward the outer part of the micelle rather than the i n t e r i ~ r , ~ especially ~-~' at low concentrations. A notable exception is benzyl alcohol, which exhibits a low concentration at the micelle-water interface.28 In addition, the effects on size and cosurfactant partitioning between water and micelle were found to depend strongly on the isotopic composition of both the aqueous environment and the surfactantsz6 Perhaps most interesting from the standpoint of a nematic phase is that at sufficiently high concentrations a cosurfactant can introduce an eccentricity into the otherwise spherical micelle.z6 This phenomenon is, of course, intimately related to the crossover from a hexagonal to lamellar phase (Le., from higher average curvature to lower curvature) where the surface charge of an ionic surfactant is "diluted" by the addition of a nonionic c o s ~ r f a c t a n t , ~ ~ assuming the tails of both species pack in a similar manner. The effects of an alcohol on nematic-forming micelles has been studied by Hendrikx et al.9 In an elegant neutron scattering experiment they employed a contrast variation method by selectively deuteriating the amphiphiles to show that the alcohol preferentially partitions to the flat core of the oblate micelle rather than to the high-curvature rim. This effect has been shown theoretically by Gelbart et aL30 for the case of finite rods with end caps. In this paper I report on magnetic susceptibility measurements for the system CsPFO and H 2 0 upon the addition of the cosurfactant perfluorooctanol, a molecule identical with CsPFO except in the head group. Two significant effects were observed: the nematic-isotropic transition temperature T,* is a strongly increasing function of molar concentration X , of cosurfactant, and the quantity ( T - P ) C [E@] increases rapidly with X,. Here T* is the supercooling temperature of the uniform isotropic phase and C is the Cotton-Mouton coefficient, defined as dAn/d@JH4, where An is the field-induced optical birefrin(24) Boden, N.; Corne, S. A,; Holmes, M. C.; Jackson, P. H.; Parker, D.; Jolley, K. W. J . Phys., in press. (25) Hayter, B.; Hayoun, M.; Zemb, T. Colloid Polym. Sci. 1984, 262, 798. (26) Zana, R.; Picot, C.; Duplessix, R. J . Colloid Interfuce Sci. 1983, 93, 43.
(27) Tabony, J. Chem. Phys. Left. 1985, 113, 79. (28) Shih, L. B.; Williams, R. W. J . Phys. Chem. 1986, 90, 1615. (29) Stilbs, P.; Rapacki, K.; Lindman, B. J . Colloid Inferface Sci. 1983, 95,583. (30) Gelbart, W. M.; McMullen, W. E.; Marteis, A,; Ben-Shaul, A. Langmuir 1985, I, 101.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 14, 1987 3831
Cosurfactant on a Micellar Nematic Liquid Crystal
I I I I 1 4 0 9
'.I
i
32
i
cc
i
'@
1.4
I .3
30[
28
t
8 0.01
0.02
0.03
0.04
(Molar) Figure 1. Nematic-isotropic supercooling limit T* vs. weight percent X, of cosurfactant perfluorooctanol. X,
gence. Based upon a Landau free energy and Onsager formulation31for the entropy of hard disks, the data suggest an increase in the micelle aggregation number s with increasing X,. The CsPFO was synthesized according to the procedure in ref 32 and recrystallized from absolute ethanol. The samples were prepared to contain a fixed amphiphile number density N A = 5.47 X lozo~ m - corresponding ~, to a molar concentration of 0.909 M. For cosurfactant concentration Xc = 0, a 40 wt ?6 sample of CsPFO in HzOwas prepared, which has a density of 1.24 f 0.02 g/cm3;" this corresponds to the desired molar concentration of 0.909 M. Subsequent samples were prepared by substituting perfluorooctanol for CsPFO on a molecule-for-molecule basis up to X, = 0.048 M, while maintaining the total amphiphile concentration at 0.909 f 0.002 M. (The perfluorooctanol was supplied by Ffaltz and Bauer and used as received.) The prepared samples were contained in a 1 cm path length stoppered cuvette housed in a temperature-controlled oven. The oven was in turn situated in an 1 1.2-T Bitter magnet possessing a transverse optical port. Details of the oven and birefringence apparatus are given elsewhere.33 The field H was swept from 0 to 11 T in 30 s, and the field-induced birefringence An was computer recorded. Over the duration of the sweep temperature control was better than 10 mK. Well above T* the birefringence was found to be linear in IP and the Cotton-Mouton coefficient C(Xc,T)= a An/a#lH-o was determined by a linear least-squares fit to the data. Closer to T* ( T - T* < 200 mK) deviations from linearity were observed; here the initial slope a An/a@lH4 was determined by a least-squares fit to a polynomial quadratic in #. Data were taken over a temperature range T,* 5 T < Tc* 1.5 K, and in all cases C1 found to be linear in temperature, indicating a mean field susceptibility exponent y = 1. Given the apparent mean field behavior in the isotropic phase, the Cotton-Mouton coefficient can be shown to bej4
+
where Ac and A x are the volume dielectric and magnetic susceptibility anisotropies for fully saturated order, ii = zl/*, and a. is the coefficient of the quadratic term in the Landau free energy were expansion. The two fitted parameters T+(Xc)and 9(Xc) obtained by a linear least-squares fit of C1vs. T . Fitted values (31) Onsager, L. Ann. N.Y.Acad. Sci. 1949, 51, 621. ( 3 2 ) Rosenblatt, C.; Kumar, S.;Litster, J. D. Phys. Reu. A 1984, 29, 1010. (33) Rosenblatt, C. J . Phys. 1984, 45, 1087. (34) Stinson, T. W.; Litster, J. D.; Clark, N. A. J . Phys., Colloq. 1972, 33, C1-69.
0.01
0.02
0.03
0.04
X, (Molar) Figure 2. The experimental quantity CP (cf. eq 1) vs. X,.
of T* vs. X, are shown in Figure 1. The quantity [=AtAx/9fiao] is plotted in Figure 2 and, as readily apparent, is a strongly increasing function of alcohol concentration. We now turn our attention to establishing a model for the quantity 9 in terms of the size and shape of the micelle. We consider an assembly of rigid, oblate, micelles monodisperse in aggregation number s. The oblate micelles consist of a flat, bilayer-like central core region of diameter dc and thickness 1 bordered by a semitoroidal rim, such that the molecules all exhibit the same average bond orientational order independent of location in the micelle and locally orient perpendicular to the surface. Moreover, we assume that the local molecular polarizability and susceptibility tensors are additive, these being good approximations for small birefringence~.~~ This simple model does not, of course, account for density considerations in the semitoroidal region, which will be discussed later. To develop a model for the At and Ax parts of 9 (cf. eq l ) , we note that both quantities are expected to scale as the amphiphile number density N A times a shape factorfCfl 1). (It is assumed that At and Ax are the same for both amphiphilic species. This question will also be addressed later.) Due to the additivity of the molecular polarizability and susceptibility tensors, the quantity averaged over all molecules in f i s defined as ( 3 / 2 cosz the micelle, where p is the angle between the molecular orientation within the micelle and the micellar symmetry axis. In the core region, where the molecules are all parallel to the symmetry axis, p is everywhere equal to zero. Thus, this region contributes the to the value o f f , where scoreis the aggregation quantity s,,/s number of the core. For the rim, however, the molecular anisotropy tensors must be orientationally averaged. Consider the 0 and d, >> I , where I is the thickness two limiting cases: d, of the bilayer (and therefore twice the interior radius of the toroidal cross section). In the first case the semitorous reduces to a sphere, where the integral involves a sin 0 "density of states" and the rim contribution tofvanishes. For the case of a large core diameter, however, the sin % density of states correction can be taken as 1 because all the rim molecules lie far from the micelle symmetry axis and thus near polar angle 0 = ~ / 2 (Note . that % is a function of d,, I , and molecular orientation 6.) Thus, in this case, both the optical and magnetic contributions to f from the rim are sIim/4s,where srim is the aggregation number of the rim. For the intermediate case (finite d J l ) the rim contribution becomes a simple problem of trigonometry, where the contribution from each molecule must be scaled by an appropriate polar density of states ?(e), analagous to the sin % correction for a sphere. It can thus be shown36that the rim contribution to the quantityf is (YS,,,,,/S, -+
(35) Rosenblatt, C . J . Phys. 1986, 47, 1097. (36) Rosenblatt, C., to be published.
3832 The Journal of Physical Chemistry, Vol. 91, No. 14, 1987 where a is between 0 and
Rosen bla tt
and is given by o'25
5
1
and
Note that a goes to 0 when dc vanishes and goes to 'I4 when d,/l >> 1, as expected. Since eq 3 actually involves only the ratio dJ1, the quantity a vs. d J l is calculated from eq 2 and 3 and is shown in Figure 3 for the relevant region of dJ1. Note that a is a weak indicating that we are function of d c / l and is nearly equal to near the limiting case d c / l >> 1 described above. Gathering together the core and rim contributions, we find that the shape factor f is given by
f=
score
+ asrim
I.2
I .6
2.4
28
/z
Figure 3. Calculated value of a for rim contribution vs. d J l (cf. eq 2
and 3).
(4)
Note, of course, that s = score + srim. The next step is to determine ,s, and srim in terms of micelle shape and size. If one assumes a fixed molecular volume V for all amphiphiles independent of location, it can be shown that3'
4rl(Y6rl3 - sV)
2.0
dc
1
/ ~ l ~- 1 / 4
50
(5)
lJZ O0
We take V to be 3.6 X cm3 (ref 24), and based upon X-ray data,z4we take 1 to be 22 A. Using these values, we then calculate dc vs. s and plot this quantity in Figure 4 (solid line, left-hand = scale). From simple trigonometry we further note3' that score 1rd2114V. Since d , vs. s has just been calculated, we can thus find ,,s vs. s, which is also plotted in Figure 4 (dashed line, vs. s. Finally, we right-hand scale), as well as s, = [= s - s,] need to obtain a vs. s. Since we have calculated both a vs. d,/l (Figure 3) and d, vs. s (Figure 4), we can obtain a vs. s for a fixed vs. s, the quantity value of 1. Combining this with s,,, and score f(cf. eq 4 ) can finally be calculated vs. s and is shown in Figure
8
50
5.
At this point we have a model for A€ and Ax in terms of N A and s; to complete eq 1 we need a model for ao. We assume steric interactions which, as is well-known, can result in a nematic phase. (Weaker, long-ranged interactions affect -Prather than ao.38) In the context of a Landau picture -Tr aoQ2represents to O(Tr Oz) a? orientatiocal entropy change S p_er unit volume between the Q = 0 and Q # 0 states, where Q is the nematic order parameter. As argued earlier,I0 this entropy has a direct orientational component Soand a translationally coupled orientational component ST.Assuming Onsager's orientational distribution function, to the level of thewsecondvirial coefficient we can obtai_n1° the entropy S to O(Tr Qz) and thus obtain a. [= -S/Tr Q2]
[
NA~B N A 2 h 1 l r Z d 3+ - X ( X 90s 1440s2 4 2
aoa---
+ 3)l& + rd12]
Figure 4. Diameter d, of core region (solid line, left-hand scale) and aggregation number,,s (dashed line, right-hand scale) vs. total aggregation number s (cf. eq 4). Calculation for d, assumes 1 = 22 A and V = 3.6 X cm3.
eq 6 remains a good approximation. Finally, note that ~i deviates by no more than 2% from the pure water value fi = 1.33 over a wide concentration range. Gathering all these factors together, we find from eq 1 that
(6)
where k B is Boltzmann's constant. Note that the first term represents Soand the second term ST.It should also be noted that, formally, the micelle dimensions d and 1 in eq 6 are for a right circular cylinder, whereas in the above calculation for fabove the shape was taken as a central bilayer core with a semitoroidal rim. One can thus equate the bare diameter d with d, + I , and (37) McMullen, W. E.; Ben-Shaul, A,; Gelbart, W. M. J. Colloid Interface Sci. 1984, 98, 523. ( 3 8 ) Rosenblatt, C . Mol. Cryst. Liq. Cryst. 1986, 141, 107. (39) Boden, N.; Holmes, M. C . Chem. Phys. Lett. 1984, 109, 76.
For the experiment at hand N A = 5.47 X IOzo ~ m - ~In. the absence of alcohol the bare micellar dimensions are given by 1 22 A (ref 24) and d 61 %, (ref 38); these are the values used in eq 3 and 5 . Note, however, that for ionic micelles the intermicellar steric interactions are governed by effective dimensions which are greater (all around the micelle) than the bare dimensions by a value of the order of the Debye screening length K - I . For this system K-I 6 %,'I and is relatively independent of alcohol concentration since only small alcohol concentrations are used. Thus, the relevant dimensions to be used in the denominator of
-
-
-
Cosurfactant on a Micellar Nematic Liquid Crystal ~.
I
l
I
l
200 v)
e
190-
a,
n
G 180-
e e
C
0
+
e
% 170?!
CT 01
a
160e
100
I50
200
250
I50 +
330
S
Figure 5. f v s . s calculated for micelles with 1 = 22 A and V = 3.6
X
cm'.
0.02
0.03
0.04
X,
eq 7 are the eflectiue dimensions. Finally, from previously reported density data" the aggregation number s is expected to be in the neighborhood of 150 at X , = 0. This aggregation number, moreover, is consistent with that obtained from X-ray results of CsPFO in D20.@ These values are taken as the "base line" values at X , = 0. Upon the addition of perfluorooctanol the quantity 9 increases rapidly (cf. Figure 2), in fact by some 48% when only 5% of the CsPFO is replaced with the alcohol. As observed by Hendrikx et al.? alcohols tend to aggregate in the lower curvature (the core) region of finite, oblate micelles. In fact, in a later study41 it was found that the addition of an alcohol converts the hexagonal phase into a lamellar phase, the latter having zero curvature. Thus, we can expect a similar phenomenon here, namely, the cosurfactant tends to reduce the overall curvature by increasing the fraction of molecules in the now larger bilayer core (smre scales as d:) at the expense of the (also larger) rim, where s,, scales as d, as d, m. Therefore as alcohol is added, the micelles continue to grow with a concomitant decrease in the number of micelles, until entropy considerations no longer permit a further reduction in micelle population. We now return to eq 7 in order to determine aggregation number s as a function of X,. With N A and I [Le., the effective micelle thickness, which includes the bare part (22 A) plus the cm-3 and Coulombic part 2K-1 (12 A)] taken to be 5.47 X 34 A, respectively, eq 7 becomes a function of s , A and effective d. But the effective d [= d, + lhre2 ~ and ~ f a~ r e 1implicit functions of s, as shown in Figures 4 and 5. 9 is thus a function of s only and can be calculated from eq 7. Thus, to determine the aggregation number s vs. X,, the measured quantity O(X,)/9(XC=O)at concentration X , is equated with the calculated quantity Cp(s)/9(s=l50)at the appropriate value of s. In this manner s is obtained as a function of X , and is plotted in Figure 6. As expected, the addition of a small amount of alcohol significantly increases the average micellar size as the cosurfactant head groups "dilute out" the Coulombic interactions of the CsPFO molecules. It is interesting to examine eq 7 for the origin of this increase. It turns out that the shape-dependent part of the denominator arising from the 8Tentropy term is an extremely weak function of X,. This is because the terms involving I and d (in parentheses) grow nearly concomitant with s. In fact, as s increases from 150 to 200, the right-hand side of the denominator decreases by only 3%. It is clear, then, that the bulk of the change
-
+
(40) Holmes, M.C.; Reynolds, D. J.; Boden, N. Presented at the 1 l t h International Liquid Crystal Conference, Berkeley, CA, 1986. (41)AlpBrine, S.;Hendrikx, Y.;Charvolin, J. J . Phys., Lett. 1985, 46,
L-21.
0.01
(Molar) Figure 6. Aggregation number s, calculated using the susceptibility data and model presented herein, vs. X,.
in 9 is due to the sf term in the numerator of eq 7. The weak dependence of the denominator on s is particular1 important for another reason as well. The choice of K - I = 6 is a rather good estimate based upon calculations3' and previous experimental results." Nevertheless, it is probably wise to place an error bar or f 3 8, on this value when it is substituted into the equation for the effective diameter, viz. d, + lbre + 2 K - I . It turns out that, owing to the weak dependence of the denominator on changes in s and therefore changes in d, the resulting uncertainty in s vs. X , remains rather small. For example, due to the uncertainty in K - I , at X, N 0.03 M the error bar in s is approximately h3, and a t X , 0.05 M the error bar is approximately f5. Moreover, although eq 6 and 7 are taken only to the level of the second virial coefficient, inclusion of higher order terms would give the same qualitative result; i.e., as alcohol is added, the larger micelle dimensions would be divided by the larger occupation number. Thus, the serendipitous result that the denominator in eq 7 is only an extremely weak function of s would still hold beyond the level of the second virial approximation. It was noted in Figure 1 that the transition temperature increases with increasing X,. Again, this is not surprising, given that the micelle anisometry is also increasing. In the ternary system potassium laurate, decanol, and D20,Galerne et al. showed that a reentrant isotropic phase occurs upon lowering the temperature, concomitant with a decrease in micelle a n i ~ o m e t r y . ~ ~ Only sufficiently anisometric micelles can support a nematic'phase, and it is expected that the transition temperature would couple to the a n i ~ m e t r y It . ~is~interesting to note that the opposite effect is observed in the purely binary mixture CsPFO in water. As the concentration of CsPFO is increased, the N I transition temperature increases, although the micelles are growing smaller.@ Here the higher density of micelles is sufficient to offset the smaller anisometry . The foregoing analysis contains a number of assumptions and approximations which tend to decrease the reliability of the results. The picture of the micelle involved uniform molecular order independent of location, which of course is a poor approximation to real it^.^*-^^ Fortunately, the rim contributes only a small amount to Ae and Ax, and thus perhaps the final result is not greatly in error. In addition, I have assumed a uniform molecular volume throughout. Presumably, the different environments in (42) Gruen, D. W.R. J. Phys. Chem. 1985.89, 146. (43)Charvolin, J.; Hendrikx, Y . Nuclear Magnetic Resonance of Liquid Crystals; Ensley, J. W., Ed.; Reidel: Dordrecht, 1985;p 449. (44) Mely, M.;Charvolin, J.; Keller,P.Chem. Phys. Lipids 1975, 15, 161. (45)Walderhaug, H.; SMerman, 0.;Stilbs, P.J. J . Phys. Chem. 1984, 88, 1655. (46)Charvolin, J. J . Chim. Phys. Phys.-Chim. Bioi. 1983, 80, 15.
3834
J . Phys. Chem. 1987, 91, 3834-3840 larger or smaller than 150 at X, = 0, one would expect a related change in Figure 6 for X, > 0. Thus, it is clear that the model involves numerous approximations to reality (as does any model used to interpret experimental data), and we must use caution in analyzing results. In summary, it was found that the addition of a cosurfactant to an oblate micellar system raised the nematic-isotropic phase transition temperature. Moreover, by use of a particular model in conjunction with susceptibility data, it was shown that the micelles grow substantially in aggregation number upon addition of cosurfactant. Work is proceeding to understand the role of cosurfactants of various lengths.
the core and rim regions would entail different values for V, although not significantly. More disturbing, however, is the assumption that the alcohol and surfactant molecules contribute identically to At and Ax. The latter approximation is not too bad, given that most of the magnetic anisotropy arises from the perfluorinated alkyl chain, which is identical for both alcohol and surfactant. For At, on the other hand, we need to worry about the effects of the ester group in the surfactant molecule as compared to the single bands in the alcohol head. Moreover, the degree of hydration and dissociation of the Cs' ion will depend on the concentration of cosurfactant and thus influence the dielectric anisotropy. Nevertheless, it should be noted that at most only 5% of the CsPFO molecules are being replaced by alcohol with a resulting 48% change in a. It is therefore unlikely that the small number of perfluorooctanol molecules could significantly affect At directly through a differential polarizability anisotropy. Finally, it must be emphasized that I needed to rely on previous data" to obtain a bare value for s; I chose s = 150. This is reasonable and is also supported by X-ray data.40 If, however, s were in fact
Acknowledgment. The author is indebted to Dr. N. Boden for supplying him with relevant preprints and X-ray data and to Dr. M. Holmes for useful discussions. This work was supported by the National Science Foundation Solid State Chemistry Program under Grant DMR-8613455 and through its Division of Materials Research under cooperative agreement DMR-85 1 1789.
A Statistical Model of the Hydrated Electron W. M. Bartczak,* M. Hilczer,* and J. Kroh Institute of Applied Radiation Chemistry, Technical University, Lddi, Poland (Received: September 10, 1986; In Final Form: February 1 1 , 1987)
A new model of the solvated or trapped electron is proposed. The model is based on the assumption of a statistical variety of electron traps in disordered media. The energy of an electronic state is expressed as a function of the random variables related to the local structure of the trap and to the bulk structure of the matrix. The distribution function of the ground-state energy of the trapped electron is derived, and the numerical calculations are performed for water and ice as the trapping matrices. The optical absorption spectra of the electron in liquid water at 300 K and in crystalline and amorphous ice at 7 7 K are calculated. The absorption maxima, shapes, and half-widths of the calculated spectra are in good agreement with the experimental data.
Introduction A deeper understanding of the electron-trapping process is one of the most important problems of the theory of electrons in disordered polar media. The existing models of the trapped or solvated electron provide valuable insight into the properties of excess electrons but still do not account for the width and shape of the electron absorption spectrum.'-8 The calculated optical spectra were as a rule more narrow than the experimental spectra. Two explanations of the abnormally large width of the spectrum of the solvated electron are usually discussed.' According to the hypothesis of the homogeneous broadening, the trapping sites are identical and the shape of the spectrum results from different electronic transitions. The hypothesis of the heterogeneous broadening assumes a statistical variety of trapping sites. The observed spectrum is thus an envelope of the different individual spectra. Most of the previous models assumed a single type of electron trap.'-1° It seems that this assumption is the reason for the failure of the theoretical predictions for the shape and width of the solvated electron spectrum. (1) Copeland, D. A.; Kestner, N. R.; Jortner, J. J . Chem. Phys. 1970, 53, 1189. (2) Kevan, L. Ado. Radiat. Chem. 1974, 4, 205 and references therein. (3) Newton, M. J . Phys. Chem. 1975, 79, 2795. (4) Hart, E. .I.Anbar, ; M. The Hydrated Electron; Wiley: New York, 1970. ( 5 ) Jortner, J. Ber. Bunsenges. Phys. Chem. 1971, 75, 696. (6) Fueki, K.; Feng, D.-F.; Kevan, L. Chem. Phys. Lett. 1971, 10, 504. (7) Kestner, N. R.; Logan, J. J . Phys. Chem. 1975, 79, 2815. (8) Kevan, L.; Feng, D.-F. Chem. Rev. 1980, 80, 1. (9) Kestner, N. R. In Electrons in Fluids; Jortner, J., Kestner, N. R., Eds ; Springer-Verlag: New York, 1973; p 1 . (10) Kestner, N . R. In Electron Solvent and Anion Solvent Interactions; Kevan, L., Webster, B., Eds.; Elsevier: Amsterdam, 1976; p 1 .
0022-3654/87/2091-3834$01.50/0
An interesting approach was developed by Funabashi, Carmichael, and other a ~ t h o r s . ~ ~In- these ' ~ papers the absorption band of the excess electron is connected with the photoinduced electron transfer from the trapping site of a single type either to neighboring sites or to more distant sites. A related but generalized model in which the band shape function was computed as the Fourier transform of the time correlation function of the electronic dipole moment operator was worked out by Simons et al.'4J5 The computations performed for some specific cases involved, however, several adjustable parameters. Solvated or trapped electrons are most often observed in pure liquids, liquid solutions, or glasses, Le., in the systems with a high degree of molecular disorder. The distance between molecules and the orientations of molecules are to some extent random. Therefore, it seems highly improbable that the structure of a trap and the electron energy are the same over the whole matrix. In the case of an amorphous matrix a large part of the spectral broadening is probably caused by a statistical variety of trapping sites. Such an approach to the problem of the trapped electron was also suggested by other authors.10,'6-2' (1 1) 2652. (12) (13) (14)
Funabashi, K.; Carmichael, I.; Hamill, W. J . Chem. Phys. 1978,69,
Hug, G. L.; Carmichael, I . J . Phys. Chem. 1982,86, 3410. Funabashi, K. J. Chem. Phys. 1982, 76, 5519. Banerjee, A.; Simons, J. J . Chem. Phys. 1978, 68, 415. ( 1 5 ) McHale, J.; Simons, J. J . Chem. Phys. 1977, 67, 389. (16) Gaathon, A.; Jortner, J. In Electrons in Fluids; Jortner, J., Kestner, N. R., Eds.; Springer-Verlag: New York, 1973. (17) Tachiya, M.; Tabata, Y.; Oshima, K. J . Phys. Chem. 1973, 77, 236, 2286.
(18) Tachiya, M. J . Chem. Phys. 1974, 60, 2275. (19) Tachiya, M.; Mozumder, A . J . Chem. Phys. 1974, 60, 3037.
0 1987 American Chemical Society
