Langmuir 1988, 4 , 817-821
I - 1 4?
2.0
I I
1 -1 5 log c
-1 0
Figure 8. Plot of log,Q versus log c from the SANS data for CI6C8NMe2Br; at c 2 0.04 M,,,Q 0: C O . ~ .
diffusion constant of 7400 s-l, the value expected for these 1000-Arods. Comparison of Fluorescence and SANS Data: Do the Micelles Continue To Grow above c *? A turnover in apparent ti with increasing ClsCBNMezBrconcentration is obtained from SANS curves when the scattering is assumed to arise from a dilute solution of individual micelles interacting according to a spherically symmetrical potential. This turnover occurs when ( r ) (evaluated as 27r/Q,) becomes less than twice the micelles' semimajor axis. Can the concentration dependence of Q, also be consistent with continued micellar growth above c*, which is the inference drawn from the fluorescence quenching data? The answer is yes if above c* 2a/Q,, is related instead to the mesh size ( l )of a network of overlapping micelles; decreases as c increases. This mesh size, or correlation length,25has been used to describe cylinderlike polymers in the semidilute concentration region, c > c*; the extension to overlapping micelles is straightforward. Just as for polymers, 5 is independent of cylinder length and is equal to that length at c = c*. At higher concentrations 5 is a decreasing
817
function of c. Approximations to the scattering lawz5 suggest a peak in S(Q),and hence in I(Q), proportional to F-l. Thus the micellar aggregation number may stay constant (the analogue of a polymer solution), increase, or diminish with increasing concentration without affecting Q, above c*. For polyelectrolytes in solutions of low ionic strength, 4-l a c1/2(ref 25-27). Figure scaling laws predict Q, 8 shows that for C16C8NMezBrsolution at c 1 0.04 M, just such a power law behavior is observed with Q, a Hoffmann and co-workerss have found an exponent of 0.43 in a related surfactant system. Conclusions Micelles of ClsC8NMeZBrin water grow from spheres to prolate ellipsoids or cylinders with increasing concentration. The increase in aggregation number, determined by using fluorescencequenching, continued up to the MSL (ca. 0.07 M). Below 0.04 M SANS and fluorescence yield excellent agreement between derived ti values. Above 0.04 M, the SANS data are consistent with the presence of elongated micelles, which are entangled to form a network. We have shown that the correlation peak in the data is then related to the mesh size of the network and is independent of micelle size. Acknowledgment. We acknowledge support from the
U.S. Army Research Office (contract DAAG-29-82-K-0115) and the National Science Foundation (CHE-861186, L. J.M.; VPW-8600285, L.J.M. and G.G.W.). Registry No. g-ma, 779-02-2;dba, 613-29-6; [RuL8].2C10,, 71883-13-1;Ru(bpy)32+,15158-62-0;C16C8NMe2Br, 107004-19-3; pyrene, 129-00-0. (25)De Gennes, P.G.; Pincus, P.; Velasco, R. M.; Brochard, F. J. Phys. 1976,37,1461. (26)Drifford, M.;Dalbiez, J.-P. J . Phys. Chem. 1984,88, 5368. Boue, F.; Lapp, A.; Oberthiir, R. Colloid Polym. Sci. (27)Nierlich, M.; 1985,263,955.
Quadrupoles on the Triangular Two-Dimensional Lattice: A Simple Model of N2 on Graphite Herringbone Transition A. Kloczkowski and A. Samborski" Institute of Physical Chemistry, Polish Academy of Sciences, 01 -224 Warsaw 42, Poland Received October 10, 1986. I n Final Form: September 25, 1987 The two-dimensional triangular lattice of ideal linear quadrupoles was studied with classical statistical mechanics in the mean field approximation. We have considered two cases: when the quadrupoles on a 2D lattice can rotate freely in 3D and the case of quadrupoles constrainedto the plane. The orientational phase transition connected with herringbone ordering in two sublattices at sufficiently low temperatures w a studied ~ by using the bifurcation analysis of the system of self-consistent equation for order parameters. We have compared the results with experimental data for N2 on graphite herringbone transition and with earlier Monte Carlo calculations. The influence of the interaction of the higher order neighbors on the results was studied. The temperatures of the herringbone transition are approximately twice as large as the results of Monte Carlo simulation for the same model.
It is well-known from experiment1 that N2 molecules adsorbed on graphite form a 31J2 X 3lI2 commensurate structure with herringbone ordering in two sublattices (Figure 1). There is a rotational phase transition some(1) Kjems, J. K.; Passel, L.; Taub, D.; Dash, T. G.; Noyaco, A. D. Phys. Rev. B: Condens. Matter 1976,13,1446. Eckert, J.; Ellenson, W. D.; Hastings, J. B.; Passel, L. Phys. Rev. Lett. 1979,43,1329.
0743-7463/88/2404-0817$01.50/0
where between 22 and 29 K. The molecular dynamics simulation of Talbot, Tildesley, and Steele2gives B transition temperature of 33 K, which is in good agreement with experiment. They have modeled the N2molecule as two Lennard-Jones centers with partial charges repre(2)Talbot, J.; Tildesley, D. J.; Steele, W. A. Mol. Phys. 1984,51,1331.
0 1988 American Chemical Society
Kloczkowski and Samborski
818 Langmuir, Vol. 4, No. 4, 1988
have used an approach different from the method of Harris and Berlinsky, based on bifurcation analysis of the system of self-consistent equations for order parameters. This method, like the Landau expansion of the free energy, does not necessitate the assumption of a particular sublattice structure. The energy of interaction of two quadrupoles is 1 / 1
1
1 / 1
I
1x1
20T E . .= 6Q2 (70n)'/'C C(224; M N ) " 9 25R$ MN
X
Y2M(QJ Y2N(Qj) Y4*M+N(?ij) (1)
Figure 1. Herringbone structure of the N2 molecules adsorbed on graphite.
where Q is the quadrupole moment, Rij is the distance between centers of quadrupole i and j t4.26 A for nearest neighbors), and C(224; M N ) are Clebsh-Gorgon coefficients. The one-particle distribution function, i.e., the probability that a particle at point i will have orientation Qi in the mean field approximation is
senting quadrupolar moment 1.52 B (1 B = esu) and used the potential of graphite of Steele.3 The recent exp[ -flF dQjEij(Qij, Qj)fj(Qj)] results of these authors give an even better temperature of transition 25 K,4 because of the improved value of the fi(Qi) = (2) quadrupole moment of N2 (the so-called X1 model5),1.17 dQi e x p [ - f lJ Z I dQjEij(Qi, Qj)fj(Qj)] B. The Monte Carlo simulation of O'Shea and Klein6 of and the free energy as a function of fi is linear quadrupoles without Lennard-Jones interaction on a 2D triangular lattice indicates that the quadrupolar inF = C I dQi dQj Eij(Qi,Qj)fi(Qi)fj(Qj) + ij teractions are mainly responsible for the herringbone transition. Using the experimental value of the quadrukT Ci S dQi fi(Qi) ln [ W i ( Q i ) l (3) polar moment of N2, 1.4 f 0.1 B, they found the temperature of the herringbone transition to be T, II 28 f 5 K, Instead of the spherical harmonics Y2M(Q)we use their in the case of planar quadrupoles, and T, CI! 19 K for symmetric combinations TM(Q), M = 1...5 (so-called symquadrupoles rotating in 3D and placed on the 2D lattice. metry adapted function'O): Mouritsen and Berlinsky' have performed Monte Carlo T , ( Q )= '/z [Y2'(Q) - Yz-'(Q)l simulations of the same system of planar quadrupoles on a triangular lattice using an anisotropic-planar rotor Tz(Q)= -i/gYz'(Q) + Yz-'(Q)] Hamiltonian. For a large number of molecules used in simulation ( N = 10000) they have found that the phase TdQ) = Y2O(Q) transition is weakly first order. The order of the phase transition in a Monte Carlo T4(Q) = 1/2WZ2(Q) + Y2-2(Q)l simulation may depend on the boundary conditions.s A T&J) = -$$[Y&Q) - Yz-2(Q)] (4) simple mean field theory which accounts for the qualitative behavior of boundary effects was also proposed.8 A mean These functions are real and also orthonormal: field theory for hydrogen molecules H2 and D2 adsorbed on graphite was formulated by Harris and Berlin~ky.~ dQ TN(Q)TM(Q) =~MN (5) They studied the model of linear quadrupoles on a triangular lattice in an external crystal field of the P2(cos 0 ) With order parameters type. Because the temperature of the orientational phase SjM= (TM(Qj)) = dQjfj(Qj)TM(aj) (6) transition for Hzand D2 is very low (- 1K), they used the quantum statistical mechanical approach. They studied a set of self-consistent equations for order parameters the phase diagram of the system as a function of the follows: strength of the crystal field using three different methods: numerical solution of the system of self-consistent equaSiL = tions for order parameters, Landau expansion of the free energy, and a method based on analytical formulas for free energies of phases under study. In this paper we have studied the herringbone transition of nitrogen molecules adsorbed on graphite, modeled by linear quadrupoles placed on a 2D triangular lattice. The temperature of the herringbone transition for N2 is 30 K so that quantum effects may be ignored, and we have used the classical statistical mechanics. In our calculation we The matrix C M : = CNM(Fij)is symmetric and depends on the direction of the vector i i i joining sites i and j at the lattice and is given in Chart I. We have used the fact that (3)Steele, W.A. Surf. Sci. 1973,36,317. quadrupoles lie on a plane so that eij = 8ij = (n/2)4ij. (4)Steele, W.A., private communication. (5) Murthy, C. S.; Singer, K.; Klein, M. L.; McDonald,I. R. Mol. Phys. We are looking for bifurcation of the nontrivial SjM# 1980,41,1387. 0 solution of eq 7 and we follow the reporting of ref 11 and (6)O'Shea, S. F.; Klein, M. L. Chem. Phys. Lett. 1979,66, 381.
J
J
J
J
(7)Mouritsen, 0.G.;Berlinsky, A. J. Phys. Reu. Lett. 1982,48,181. (8) Evans, H.; Tildesley, D. J. Sluckin, T. J. J. Phys. C 1984,17,4907. (9)Harris, A. B.; Berlinsky, A. J. Can. J.Phys. 1979,57,1852.
(10) James, H. M. Phys. Reu. 1968,167,862.
Langmuir, Vol. 4, No. 4, 1988 819
Quadrupoles on the Triangular 2 0 Lattice Chart I
M=l
M=2
M-3
M=5
M=4
Chart I1 M=1
M=2
M=3
N
L=4 0
0
M=4
M=5
0
0
0
0
-5/431/2 CZ(i)(rl/rJ6
r=l
N
L=5 0
F
0
-5/431/2 Sl(i)(rl/ri)6
r=l
Table I
3 4
6 6 6 12
5
6
1 2
1
0
3112 2
*I6
7112
3
0 arcsin (3/28)'12 arcsin (3/7)'12 0
12. In the case where the phase transition is second order, the bifurcation point is the point of the phase transition. Linearizing eq 7 we obtained
Using the Fourier transformation
sN(&) = CSiN&i i
(9)
we obtain
Figure 2. Shells of subsequent neighbors of a given molecule. The angle displacement doof the nth shell with respect to the first shell and the base vectors SI and ii2of the triangular lattice
are shown.
Table I1 ~~
with
CL&
1 ri
= EgCLM(9i)el%.ii 1
(11)
N 1 2 3
~min
N
xmin
-18.25 -19.421 -19.350
4
-19.103 -19.178
5
One of the quadrupoles is in the center of the coordinate system, so Sij = SOi = Zi. The summing over i corresponds to summing up over subsequent neighbors in subsequent shells (see Figure 2). We sum up over N,, neighbors in the nth shell. The subsequent neighbors are listed in Table I. Here, rJr1 is the relative distance of the nth shell with respect to the first shell (rl = a with a = 4.26 8, lattice constant) and $ J ~is the angle displacement of the nth shell with respect to the first shell (see Figure 2). We treat shell no. 4 with 12 neighbors as two independent shells with 6-fold symmetry so we have two different values of 9ofor this shell. At the given shell we have
the inversion symmetry and 3-fold symmetry so that summing over these six neighbors we have
(11) Bruno, J.; Giri, M. R.Phys. Reu. B: Condens. Matter 1982,25, 6711. (12) James, H. M.; Keenan, T. A. J. Chem. Phys. 1959, 71,1392.
with n = 0 , 2 , 4 and &i = 40,+ (i - 1 ) ( ~ / 3 ) Here . f$k is the direction of the vector & (k = k[cos &, sin &I). The
820 Langmuir, Vol. 4, No. 4 , 1988 temperature of the phase transition in the mean field approximation is determined by-the lowest negative eigenvalue Xmin of the matrix CLM(k),given in Chart 11. We consider the interaction up to the Nth neighbors. The coefficients C 2 ) , Cz(i),C4(i),Sz(i),and S," are defined by eq 12. The additional subscript i numbers shells and is connected with the fact that the angle displacement c#+, and the distance r vary with shell. The values of doand the relative distance ri/rl for subsequent shells-are given in Table I. The eigenvalues of the matrix CLM(k)depend on the number of neighbortincluded in the calculations and the value of the vector k. We have calculated these for djfferent values of k,varying q ! ~between ~ 0 and r/3 and lkl between 0 and 4~ra, and found that t4e lowest negative eigenvalue of CLM(k)always occurs for k = 2r/a(1,0),so that & = 0 (and also for & = */3,2~/3, *, 4*/3,and 5*/3 because of the 6-fold symmetry). The results are given in Table I1 as a function of increasing number of shells N in the calculations. We see that the inclusion of the higher order neighbors does not change the value of the lowest negative eigenvalue significantly. The temperature of the transition T, is connected-with-the lowest negative eigenvalue Amin of the matrix CLM(k)through the relation
where Q is the quadrupolar moment, k B the Boltzmann constant, and a = rl the first neighbors distance. In the calculations we use the value of A- = -19.2 and two values of the quadrupolar moment of N2-the experimental value Q = 1.4B and the value Q = 1.17B from the X1 model,5where an effective quadrupole is designed to include other elements of the overall charge distribution such as exchange forces and higher electric moments. Usin the experimental value of the lattice constant a = 4.26 for N2on graphite we obtain temperatures of the herringbone transition for quadrupolar moments rotating in 3D with centers on 2 D triangular lattice: for Q = 1.4 B T, = 38.9 K
x
T , = 27.2 K
for Q = 1.17 B (X1 model)
These temperatures are in quite good agreement with the experimental value of the temperature of the herringbone transition (-25 K) and with Monte Carlo simulation.6 OShea and Klein6 simulated classical quadrupoles on 2 D triangular lattice in two cases: (1) when quadrupoles with centers on a 2 D lattice may have arbitrary orientations in 3D; (2)when the orientations of quadrupoles are constrained to the plane. In the first case the result of the MC simulation for Q = 1.4 B,including only first neighbors interaction, is -19 K whereas the result of the mean field approximation is -37 K (Arnh = 18.25 for first neighbors interaction). We can also study the problem of planar quadrupoles on a 2D triangular lattice. We assume that quadrupoles lay in a plane due to the external graphite field and that the interaction of quadrupoles is still described by eq 1 with Qi = Oi = ~ / 2vi. , One can easily check that T1(O= 712,p) = TZ(O = ~ / 2p), , 5 0,Td(0 = ~ 1 2p) , = 151167 COS 2v,and T5(6= ~ / 2q) , = 15116~sin 2p and that the temperature of the transition is 15/8 times higher for planar quadrupoles than for 3 D case, so that T, = 72.9 K for Q = 1.4 B
T,= 51 K
for Q = 1.17 B (Xl model)
The result of the MC simulation of O'Shea and Klein6 is -28 K. In the mean field theory, including only first-
Kloczkowski and Samborski
Figure 3. Types of bifurcation of the herringbone-likesolutions S? # 0 of eq 7 from the isotropic solution S t = 0 curve a corresponds to the second-orderphase transition and curve b to the first-order phase transition with T,the temperature of the transition determined by the equality of the free energies (eq 3)
of both phases.
neighbor interactions, T, = 69 K; the mean field theory thus overestimates the temperature of the phase transition. This is because it neglects fluctuations, which play an important role here. Generally, mean field theory does not work well for two-dimensional lattices because of the small number of neighbors surrounding each lattice point. The inclusion of fluctuations in the theory would reduce the temperature of the phase transition and could give better agreement with the experimental results. Another problem appears because of the linearization of the mean field equations (eq 7). The bifurcation analysis of eq 7 gives the exact value of the temperature of the phase transition in the theory (which is determined by equality of free energies of orientationally ordered and disordered phase) only if the transition is of the second order. The real transition is probably weakly first order. However for the first-order phase transition we can expect that the temperature of the transition will be higher than the temperature of the bifurcation of the orientationally ordered phase from the disordered phase (see Figure 3). Inclusion of the Lennard-Jones interaction in addition to the quadrupolar interaction will also increase the temperature of the phase transition, i.e., will worsen the results. This is because the Lennard-Jones potential provides additional stability to the herringbone structure in the case of N2 molecules. The inclusion of the two-center Lennard-Jones potential changes the perpendicular orientation of molecules only slightly, but the resulting potential energy becomes lower. This is not true in the case of molecules with small quadrupolar moments (i,e., 02).The inclusion of Lennard-Jones interaction, which is much stronger then the quadrupolar interaction, destroys the herringbone ordering, and the parallel ordering of molecules due to the Lennard-Jones interaction prevails. One could improve the theory by inclusion of N,-graphite interaction and by assuming that centers of molecules can move around the lattice points and there is a coupling between orientational and translational ordering. We can expect that the change of the orientational ordering at the transition is accompanied by a slight change of the translational ordering. We plan to study these problems in the future. Although the mean field theory in two dimensions is not quantitatively good-the resulting temperatures are almost twice as high-it is very simple
Langmuir 1988,4, 821-826 and useful and gives qualitatively good results. It shows that the quadrupolar interaction is responsible for the herringbone ordering of N2molecules on graphite.
Acknowledgment. We are indebted to Professor J.
821
Stecki for suggesting a related problem and Professor W.
A. Steele from the University of Pennsylvania for a discussion. Registry No. Nz, 7121-31-9; graphite, 1182-42-5.
Adsorption at the Liquid Surface Studied by Means of Specular Reflection of Neutrons J. E. Bradley, E. M. Lee, R. K. Thomas,*'+ and A. J. Willatt Physical Chemistry Laboratory, Oxford University, Oxford, England
J. Penfold and R. C. Ward Rutherford-Appleton Laboratory, Didiot,England
D. P. Gregory Unilever Research, Port Sunlight, Wirral, England
W. Waschkowski FRM Reaktorstation, Garching, Munich, Germany Received August 13, 1987. I n Final Form: November 2, 1987 The technique of specular reflection of neutrons has been used for the first time to study adsorption at the surface of water. Four systems were studied. The reflectivity profile of an insoluble monolayer of fully deuteriated butyl arachidate on water was measured as a function of surface pressure. The changes of reflectivity with surface concentration were found to be easily measurable. The thickness of the layer was found to be 35 f 5 A, about 15% larger than expected for the fully extended molecule. The surface of a solution of partially deuteriated decyltrimethylammonium bromide at about half the critical micelle concentration,where the monolayer is complete, gave a reflectivity profiie corresponding to a layer thickness of 20 f 8 A and an area per molecule of 45 A2. The layer thickness i s again slightly larger than expected for the fully extended molecule. The head group area agrees with independent estimates. Sodium dodecyl sulfate was studied as the protonated form in D20 and the deuteriated form in Hz0/D20 mixtures, demonstrating the potential of contrast variation in the specular reflection technique. The thickness of the layer was found to be 20 A in both cases, and estimates of the distribution of water in the layer were made at different concentrations. The deuteriated surfadant contained about 10% dodecanol as impurity. At low concentrations the layer was found to be 100% dodecanol, but, as the concentration increased above M), dodecanol was increasingly replaced about one-tenth of the critical micelle concentration (8.1 X in the layer by sodium dodecyl sulfate.
Introduction Hayter et al.' have shown how the specular reflection of neutrons may be used to study inhomogeneity at the surface of a liquid. They have demonstrated the sensitivity of neutron reflection to an interface by measuring the reflectivity of deuteriated Langmuir-Blodgett films on glass substrates.2 It has proved difficult to obtain the necessary sensitivity to study the liquid interface because special arrangements have to be made to do specular reflection from horizontal samples. Three general procedures can be identified. The simplest in concept, although not in execution, is to use neutron mirrors or monochromators to deflect the beam from the horizontal down on to the surface of the liquid. The angle of incidence is varied by raising or lowering the liquid sample and simultaneously adjusting the angle of tilt of the mirror. The detector also has to be moved to receive the specularly reflected beam. The second method Current address: Physical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ. Current address: Physical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ.
uses a fixed angle of incidence of a white neutron beam onto the sample, and the reflectivity is measured as a function of the incident wavelength by doing a time of flight analysis of the neutrons. This is the principle of the new CRISP reflectometer at ISISa3 A third method has been employed for some years on the gravity mirror spectrometer at the FRM, GarchingS4p5In this machine the neutron beam passes through a horizontal 100-m evacuated flight tube. Under the influence of gravity neutrons of a given wavelength fall through a distance determined by their time of flight. If the liquid is placed at this height at the end of the flight tube, neutrons of the given wavelength may be reflected off the liquid surface into a detector placed immediately behind the sample. (1) Hayter, J. B.; Highfield, R. R.; Pullman, B. J.; Thomas, R. K.; McMullen, A. 1.; Penfold, J. J. Chem. Soc., Faraday Trans. I 1981, 77, 1437. (2).Highfield, R. R.; Thomas, R. K.; C u m i n s , P. G.; Gregory, D. P.; Mingins, J.; Hayter, J. B.; Schaerpf, 0. Thin Solid Films 1983,99, 165. (3) Penfold, J.; Ward, R. C.; Williams,W. G. J. Phys. E 1987,20,1411. (4) Koester, L. In Springer Tracts in Modern Physics; Springer-Verlag: Berlin, 1977; Vol. 80, p 24-26. ( 5 ) Koester, L.2. Phys. 1966, 122, 328.
0743-7463/88/2404-0S21$01.50/00 1988 American Chemical Society
