J . Phys. Chem. 1987, 91, 1842-1850
1842
3 FA. For this particular device, the ID-VG curves of Figure 9 are displaced to slightly lower potentials. The response was reproducible through the course of the 6-h experiment. The cyclic voltammetry of the Ni(OH)2-derivatized microelectrodes the following day was identical with that recorded prior to the experiment. At high pH, Ni(OH)z-based transistors are quite rugged. The long-term durability of Ni(OH)z in basic solution is one reason for its utility as a rechargeable battery material.’-5 The important point from the data in Figure 10 is that Ni(OH)z-based transistors can reproducibly sense small changes in pH and can give fairly reproducible response under actual operating conditions. The sharpness of the redox wave which modulates Ni(OH)z conductivity, illustrated by the data in Figures 2 and 3, indicates that it might be possible to sense much smaller changes in pH in basic solutions.
Conclusions The operation of pH-sensitive microelectrochemical transistors based on cathodically deposited Ni(OH)z has been demonstrated. The Ni(OH)z on microelectrodes behaves in a manner similar to Ni(OH)2 used in electrochromic and rechargeable battery applications. The cathodically deposited films connect adjacent microelectrodes, allowing the resistance of Ni(OH), as a function of potential to be measured. The resistance of Ni(OH)2-connected microelectrodes decreases by over 3 orders of magnitude upon electrochemical oxidation. Ni(OH)z-based microelectrochemical transistors are poor power amplifiers compared to previously characterized redox conducting materials, and furthermore, response time is slow due to sluggishness of Ni(OH)* electrochemistry. The prospect of substantial
improvements in both amplification and response time is realistic based on improved microelectrode design and increased diffusion coefficients for protons. Simultaneous measurement of IG, I D , and V, allows the conclusion that excess anodic charge in the cyclic voltammetry of Ni(OH),, a consequence of OH- oxidation, has no effect on the conductivity of the material. The integrated cathodic charge, which corresponds to the charge involved in Ni(OH)z N i O ( 0 H ) interconversion, correlates well with I D . However, it should be emphasized that IG # 0 when a Ni(OH)z-based transistor is held in the “on” state. Though the steady-state value of I D is always much greater than ZG, this “leakage” current is much greater than in solid-state transistor^'^ or other microelectrochemical transistor^.'^-'^,^^ pH-dependent transistor properties observed are in agreement with predictions based on the pH dependence of the electrochemical oxidation of Ni(OH)2. The useful pH regime has been demonstrated to be 10-14. Ni(OH),-based microelectrochemical transistors are able to reproducibly respond to a change from pH 12 to 13 in a flowing stream for a period exceeding 6 h. The response time is adequate for sensor applications, and the durability of such devices in basic solutions is impressive. Acknowledgment. We thank the Office of Naval Research and the Defense Advanced Research Projects Agency for partial support of this research at MIT. We thank D. A. Corrigan for helpful discussions and R. A. Conell for assistance with optical measurements. Daniel Belanger also acknowledges le Fonds pour la Formation de Chercheurs et 1’Aide 2 la Recherche of QuCbec for partial support as a postdoctoral fellow (1 986). Registry No. Ni(OH)2, 12054-48-7; Au, 7440-57-5; Pt, 7440-06-4.
Flexible Solutes in a Uniaxial Field: A *H NMR Study of +Alkanes in a Nematic Solvent B. Janik,t E. T. Samulski,* Department of Chemistry & Institute of Materials Science U-136, University of Connecticut, Storrs, Connecticut 06268
and H. Toriumi Department of Chemistry, College of Arts and Sciences, University of Tokyo, 3-8- 1 Komaba, Meguro, Tokyo 153, Japan (Received: September 26, 1986)
2H NMR spectra for homologous series of perdeuteriated n-alkanes solubilized in nematic solvents are reported. These flexible solutes acquiesce to the uniaxial environment of the solvent and thereby reflect the nature of the nematic mean field. Quantitative simulations of the quadrupolar splittings exhibited by the alkanes are carried out using a parametrized potential of mean torque in conjunction with an ensemble average over alkane conformers. Two parametrizations were selected in order to gauge the relative importance of attractive (dispersion) forces and repulsive (excluded volume) forces. A detailed examination of the resulting angular dependence of the potential is shown for hexane along with a critical evaluation of the rotational isomeric state approximation itself. The findings suggest that while the latter approximation is adequate, a more elaborate specification of the orientational potential of mean torque for solutes is required-one that explicitly and rigorously couples attractive and repulsive intermolecular interactions.
I. Introduction The anisotropy of the orientational potential energy of a molecule in the average field generated by its neighbors distinguishes liquid crystals from ordinary liquids. Since the seminal contribution of Maier and Saupe,’ there have been many efforts to model this potential and thereby account for the thermodynamic stability of fluid phases with long-range orientational order. In ‘Permanent address: Institute of Nuclear Physics, Krakow, Poland.
0022-365418712091- 1842$01.5010
most of the extant models, the qualitative features of the intermolecular interactions are obscured by approximate averaging schemes, and there is, for example, a continuing debate about whether or not nematics (uniaxial melts composed of prolate molecules) are stabilized primarily by attractive forces or repulsive Herein we present new nuclear magnetic resonance ( 1 ) Maier, W.; Saupe, A. Z . Naturforsch., A : Asfrophys., Phys. Phys. Chem. 1959, 14A, 882; 1960, 15A, 287.
0 1987 American Chemical Society
n-Alkanes in a Nematic Solvent
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 1843
(NMR) data on how alkane solutes are accommodated in nematic solvents. The data are modeled with a truncated potential of mean torque by using two parametrizations, one based on attractive interactions and the second based on repulsive interactions. The results suggest that it is necessary to combine these two types of interactions in a careful way to more realistically model the orientational potential energy in nematics. In particular, closer examination should be given to the way in which the effective one-body potential used to describe these unusual fluids is related to averages over intricately coupled attractive and repulsive intermolecular interactions. Part of the difficulty associated with addressing even a qualitative inquiry into the nature of the dominant intermolecular interactions in liquid crystals comes from the molecularstructural complexity of typical mesogens, which has led to the rather extreme idealizations employed in theoretical modeling. Recently, there have been experimental programs designed to circumvent these inherent difficulties through studying probe molecules dissolved in mesophases. The primary experimental technique in this indirect approach has been NMR; the N M R probe technique is an outgrowth of solute structure determination via incompletely averaged, intramolecular, nuclear spin interaction^.^^^ Such probe studies yield observations that may be simulated with an accurate, effective, one-body potential. Particularly relevant studies involve measurements of the degree of ordering of structurally simple, rigid solutes solubilized in a nematic solvent.”l To date, however, the N M R method has yielded ambiguous results because, with few exception^,^ the probe molecules commonly used were capable of chemical interactions with the nematic solvent (e.g., hydrogen bonding, charge transfer, dipolar associations, aromatic a-electronic associations, etc.). Such specific interactions could very well mask the general forces acting on a solute probe and confuse efforts to extract information about the nature of the intermolecular interactions responsible for the long-range orientational order in liquid crystals-the so-called nematic mean field. For this reason we decided to use the normal alkanes as N M R probes in an attempt to assess the relative importance of anisotropic dispersion forces and angular-dependent excluded-volume interactions to the nematic mean field. These probes offer two advantages over previously studied solutes: (1) the alkanes are relatively inert chemically, hence specific interactions between the probe and the solvent should be minimal; (2) the inherent flexibility of these probes enables them to conformationally acquiesce to the constraints characterizing the nematic mean field. Hence, the influence of the solvent mean field appears as both an orientational bias and a conformer probability bias;I2 Le., each conformation of the solute exhibits a preferred average orientation (relative to the solvent director) and a perturbed conformer probability (relative to its probability in an isotropic medium). In addition, we use the well-developed 2H N M R technique,I3 which also has advantages. For one, all of the 2H N M R parameters that reflect molecular motion and orientation are governed by the coupling between the nuclear quadrupole moment of the deuteron and the local electric field gradient (efg) tensor q, and in alkanes, q is essentially symmetric having its principal value collinear with the C-D bond vector. Hence, incompletely averaged
quadrupolar interactions quantitatively determined with 2H N M R reflect the efficacy of the reorientational motion of this vector. Also, nematic liquid crystal solvents are ideally suited to 2H NMR investigations; the rapid, anisotropic molecular motion in these fluids averages the second rank efg tensor along the liquid crystal director n (the symmetry or optic axis of the ordered phase). In a macroscopically aligned, uniaxial phase, the residual quadrupolar interaction appears as a resolved quadrupolar splitting in the 2H N M R spectrum. Thus, the magnitude of observed quadrupolar splittings yields information about how anisotropic molecular motions average the orientation of the various C-D bonds relative to n. The particular problem of simulating the 2HNMR quadrupolar splittings exhibited by partially oriented alkanes-solutes with internal degrees of freedom-has been considered before. Samulski, using a phenomenological model depicting each alkane conformer in the nematic as if it were subjected to an ad hoc cylindrical constraint, succeeded in calculating the relative quadrupolar splittings of octane.12 The qualitative features of the ZHN M R spectra of partially oriented alkanes were also described by Creamer et al.14aand Ben-Shad et al.14busing a lattice description of the chain with a phenomenological applied field acting on the chain segments. Here we have attempted a more quantitative description of the alkane ’H N M R data. We have extracted information about the nematic mean field from the solute 2H N M R spectra by modeling the subtle variations in the observed quadrupolar splitting patterns of perdeuteriated alkane solutes as a function of the chain length. For such labeled, nonrigid molecules, the relevant molecular motion is complex: it involves a convolution of intra- and extramolecular contributions. Fortunately, the internal degrees of freedom of such solutes can be adequately handled with the rotational isomeric state ( R E ) approximation15wherein a limited number of discrete dihedral angles (conformers) are available to the alkane. And, given a particular conformation, its average orientation may be modeled with an approximate potential of mean We discuss here how this combined internal and extramolecular motion would average the efg tensor at various positions along the chain with a parametrized potential of mean torque. In section 11, we describe the model potential of mean torque, its parametrization, and the methodology employed to simulate and observe the 2H N M R quadrupolar splitting patterns. In section 111, the experimental 2HN M R data for the perdeuteriated n-alkane solutes are presented. In section IV, we examine the simulations for the homologous series of alkane solutes and we contrast in detail the different parametrizations of the potential for the solute hexane. The implications of our findings viewed against the context of earlier and less thorough modeling are summarized there. Also, our efforts to evaluate the RIS approximation itself are described. Section V reconsiders the use of an approximate potential of mean torque to model the nematic mean field. The Appendix describes the alkane molecular geometry and details about the RIS approximation used in our calculations. The parameters in the latter approximation are evaluated by computing conformationally averaged dipole moments of dibromoalkanes.
(2) Chandrasekhar, S.; Madhusudana, N. V. Acta Crystallogr.,Secf. A : Cryst. Phys., D i f f , Theor. Gen. Crystallogr. 1971, A27, 303. ( 3 ) Luckhurst, G. R.; Zannoni, C.; Nordio, P.L.; Segre, U. Mol. Phys. 1975, 30, 1345. (4) Wulf, A. J. Chem. Phys. 1976, 64, 104. (5) Gelbart, W. M. J . Phys. Chem. 1982,86,4298. ( 6 ) Cotter, M. A. Mol. Cryst. Liq. Crysf. 1983, 97, 29. (7) Saupe, A,; Englert, G. Phys. Rev. Lett. 1963, 11, 462. (8) Emsley, J. W.; Lindon, J. C. N M R Spectroscopy Using Liquid Crystal Solvents; Pergamon: Oxford, 1975. (9) Snijders, J. G.; de Lange, C. A,; Burnell, E. E. Isr. J . Chem. 1983, 23, 269. (10) Catalano, D.; Forte, C.; Veracini, A.; Zannoni, C. Zsr. J . Chem. 1983, 23, 283. (11) Emsley, J. W.; Hashim, R.; Luckhurst, G. R.; Rumbles, G. N.; Viloria, F. R. Mol. Phys. 1983, 49, 1321. (12) Samulski, E. T. Ferroelectrics 1980, 30, 83; Polymer 1985, 26, 117. (13) Hsi, S.; Zimmermann, H.; Luz, Z. J . Chem. Phys. 1978,69,4126and
XI. Methods
references cited therein.
i. Model Potential of Mean Torque. As a starting point of our description of ’H N M R observations on alkane solutes in a nematic solvent, we use an expression for the potential of mean torque acting on a rigid, biaxial solute at infinite dilution in a uniaxial nematic solvent:’OJ1
This equation comes from truncating to second rank a general (14) (a) Creamer, D. B.; Pathria, R. K.; Rabin, Y. J. Chem. Phys. 1985, 84,476. (b) Ben-Shaul, A.; Rabin, Y.; Gelbart, W. M. J. Chem. Phys. 1983, 78, 4303. (15) Flory, P.J. Statistical Mechanics of Chain Molecules; Wiley: New York. 1969.
1844
Janik et al.
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987
expansion of an effective potential that a biaxial solute experiences in a uniaxial solvent. (This potential would also apply to an isolated conformer of a flexible solute.) The din are reduced Wigner rotation matrices; /3 and y are polar and azimuthal angles that define the director in a molecular fixed frame on the solute. Expression 1 assumes that the magnitude of the so-called nematic order parameter of the solvent, P2 = is much larger than the biaxial order parameter of the nematogen, ( d i 2 ) N . If ( d $ 2 ) Nis neglected, the coefficients cZop,which represent the strength of the solute-solvent interaction, may be written as a product of two components: one component is dependent only on solvent properties, and the other (uZp)depends only on solute properties. In this way, very concise expressions for the coefficients in eq 1 may be defined: ~ 2 o o / k T=
t ~ 2 0
czo*/kT = tu22
(2)
The coefficients uZOand uz2are explicitly defined in terms of the solute properties (the solute interaction tensors), i.e., on the type of the interactions assumed to be responsible for the ordering of solute molecules in a uniaxial solvent. The two parametrizations of the ordering potential under consideration here are as follows: In case A the shape of the solute (conformer) is assumed to control the ordering of the solute; Le., short-range, angular-dependent, repulsive interactions dominate. (Note, as emphasized by Gelbart and Gelbart,I6 such angulardependent hard-core repulsions produce a net anisotropic attractive interaction; the repulsive, nonspherical shapes of an interacting pair of molecules introduce a correlation between their relative orientation and intermolecular separation, and this correlation dominates the average of the isotropic dispersion (r4) interactions.) In case B the anisotropic molecular polarizability itself is assumed to control the degree of solute (conformer) ordering; Le., U(@,y,e) is a function of intrinsic, anisotropic, attractive interactions that originate in the electronic structure of the solute (conformer). In both cases, the interaction tensor is defined in its principal axes system (PAS) yielding simple expressions for uzp(see section 1I.iii) and, concomitantly, for U(P,y,e). For both cases, the potential of mean torque is parametrized in terms of a single adjustable parameter, e , that depends on the strength of the solute-solvent interaction (and is proportional to the solvent order parameter p,),lO.lI
The orientational partition function Q(e) is defined in the standard way
When modeling rigid solutes employing the truncated potential of mean torque, theory and experiment can be compared by examining the temperature dependence of the order parameters in a plot of A vs. Szz. In this way, the universal behavior predicted for such a plot is specified by the ratio of the coefficients in the . ~ ~ * ~ truncated potential of mean torque, X = ~ 2 2 / ~ 2 o . ~However, in the case of flexible solutes, an ensemble average over all of the solute conformers must be considered. The additional average precludes this mode of contrasting theory and experiment and mandates the direct comparison of experimental and calculated quadrupolar splittings. The method for conducting the ensemble average generally follows that outlined in ref 18 and 19, but it is reproduced here in consistent notation. For each of the conformers of a flexible solute there will be a corresponding order matrix S ( n ) defined in the nth conformer's PAS. After obtaining each S ( n ) via the prescription outlined above, it is convenient to transform S ( n ) to local Cartesian frames (the 1,2,3-frames) where the efg tensor at the ith methylene unit of the chain can be defined so that it is independent of chain conformation. The ith local frame is defined in the following way: I-axis, the vector between the two geminal D atoms; 2-axis, the bisector of the D-C-D angle; 3-axis, vector linking carbon atoms CL-land C,+l. We assume uniaxial symmetry for the efg tensor with the principal value of q along the C-D bond, q C D = 168 kHz.20 The corresponding local order matrix s(n) is related to S(n) via the direction cosines, l,a, relating the local 1,2,3-frame to the PAS: XY+
s$(n) =
/,a(n) ad
Sao(n) l k o ( n )
(6)
In order to carry out the ensemble average over chain conformers, the total energy of each conformation is required, or, in corresponding to the nth other words, the internal energy qnt conformer plus the potential energy of the nth conformer in the nematic mean field, C&&3,7,e): The second term is defined by eq 3 with an additional index n denoting the conformer. The first term is readily obtained in the RIS approximation by summing the dihedral angle energy E D A ( n ) and the nonbonded interactions E N B ( n ) for the nth conformation of the chain: The conventional methodology employed to obtain E D A and E N , is described in the Appendix. Having specified the internal energy, we can now express the probability Pn(t) for finding the nth conformer (subjected to the nematic constraint and parametrized by e )
and the corresponding definition of average values of functions
f(P,r)applies:
(5)
In the three preceding equations we emphasize their parametric nature by explicitly including the additional variable e . By use ) A = S,, of these definitions, the elements S,, = ( d & ( @ )and Syy = (v'6di2(/3) cos 2 7 ) of the Saupe order matrix" can be readily computed by numerical integration of eq 5 . i i . Simulation of Quadrupolar Splittings. The order matrix S describes the time-averaged (over solute motion) orientation of the solute principal axis system (PAS; see Appendix) relative to the director. Once the value of S is known, it is possible to simulate the 2H N M R quadrupolar splittings by expressing the efg tensors in the PAS and performing the scalar product q-S. (16) Gelbart, W . M.; Gelbart, A. Mol. Phys. 1977, 33, 1387. (17) Saupe, A. Z Naturforsch., A: Astrophys., Phys. Phys. Chem. 1964,
19A. 161.
where @(e) is defined in eq 4 and the partition function Z ( E )is given by
Now a complete expression may be given for the quadrupolar splitting anticipated at the ith methylene segment of a solute chain that exhibits N discrete conformations: 1A3
Ad(€) =
y3 j , k
N
q$kxP"(t)sjk(n)
(1 1)
n
All alkane conformers are sequentially generated, and from this (18) Emsley, J. W.; Luckhurst, G. R.; Stockley, C . P.Proc. R. SOC.London 1982, A381, 117. (19) Samulski, E. T.; Dong, R. Y . J . Chem. Phys. 1982, 77, 5090. ( 2 0 ) Dong, R.Y . Can. J . Phys. 1978, 56, 678.
The Journal of Physical Chemistry, Vol. 91. No. 7, 1987 1845
n-Alkanes in a Nematic Solvent equation the Ad(€) can be calculated adjusting t to yield optimal agreement with experimental Av’. In the case of hexadecane (CmD2m+2, m = 16) a Monte Carlo method is used to generate the conformations since the large number of conformations (3m-3 = 1 594 323) makes individual generation too time-consuming. The Metropolis procedure2’ is used to determine the acceptability of a particular conformation in the nth Monte Carlo step with a relative probability pn
Pn = exP(-~to,,I(n)/~T)
EXPERIMENT
CASE A
CASE B
(12)
where
-kUL 50 kHz
50 kHz
~
5OkHz
Figure 1. 2H N M R spectra of partially oriented perdeuteriated n-alkanes CmDZm+2:(i) experimental spectra in the nematic solvent Merck Phase The average value of s$) in the local 1,2,3-frame of the ith CD segment is calculated from N
sj);) = ( l / N m p ) o n= 1
(14)
where N is the number of Monte Carlo steps and (s$), is the j k element of the ith local order matrix calculated for the nth Monte Carlo step according to eq 6. We used 80000 Monte Carlo steps in the calculations. For the shorter alkane chains ( m = 5 , 6, 7 , 8, IO), the Monte Carlo method gives results identical with those of the exact method (Le., when each possible conformer is examined). Moreover, 80000 Monte Carlo steps in the case of hexadecane reproduced the symmetry of the quadrupolar splitting pattern along the chain length: within reasonable uncertainty the calculated A d are the same for methylene groups i and m 1 - i (see Table I). Several different splittings occur for a perdeuteriated alkane solute in a nematic solvent as the efg tensor is averaged uniquely at the various sites in the Thus, a reasonably sensitive evaluation can be made of the two solute interaction tensors that are used to construct the coefficients in the potential of mean torque (see section 1I.iii). In the comparison of these two different modes of parametrizing qx,(P,r,c), c is optimized to bring the largest Av& (that corresponding to the central methylene segment(s)) into coincidence with the largest experimental quadrupolar splitting for each of the solute chain lengths. And, on the basis of the agreement between calculated and experimental Avf we try to ascertain if anisotropic attractive interactions or angular-dependent excluded-volume interactions dominate the solute-solvent interactions when the effective one-body potential is modeled with the truncated potential of mean torque. iii. Parametrization of the Potential of Mean Torque. 2H N M R spectral simulations are derived from eq 11 by using the two parametrizations of the potential of mean torque described above: case A corresponds to the angular-dependent excluded) volume parametrization of the coefficients u2pin q x t ( P , ~ , efirst suggested by S t r a l e ~ case ; ~ ~ B defines the uZpin terms of the anisotropy of the solute molecular p ~ l a r i z a b i l i t y . ~ ~ ~ ~ Case A. Alkane conformers are approximated as parallelepipeds with L , B, and W representing their length, breadth, and width, respectively. The dimensions of each conformer ( L 1 B IW) are identified with the semiaxes of the inertia ellipsoid of each conformer-an ellipsoid of uniform density having the same inertia tensor, I, as the alkane solute. The ellipsoid semiaxes a, are obtained from the principal moments of inertia of a given con-
+
f o r m e r via
ai = (I]]+ I k k - If!)s/zm
5 a t Trd = 0.95 for the solutions, 13.8 MHz; (ii) simulated spectra for case A, the excluded-volume parametrization of the potential; (iii) simulated spectra for case B, the anisotropic polarizability parametrization of the potential.
(15)
(21) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (22) Samulski, E. T. Isr. J . Chem. 1983, 23, 329. (23) Emsley, J . W. In Nuclear Magnetic Resonance of Liquid Crystals; Emsley, J. W.. Ed.; D. Reidel: Dordrecht, 1985. (24) Straley, J . P. Phys. Rev.A 1974, 10, 1881.
where m is the mass of the alkane. L, B, and Ware coincident with the principal axes of the conformer inertia ellipsoid; Le., the PAS diagonalizes I and we use the convention 15112,Bib, and Wllx. Following S t r a l e ~the , ~ ~coefficients u20and u22are defined as
u20 = [-2B(W2
+ L2) - 2W(L2 + B2) + L(W + B2) + 6WBL]/3 u22 = (L2 - B w ) ( B -
w)/G
(16)
Note that our expressions for u2pdiffer slightly from those of S t r a l e ~(and ~ ~ as transcribed by Luckhurst et aL3), but their validity is corroborated by Barboy and gel bar^^^ Case B . The potential acting on each alkane conformer is assumed to be determined by the anisotropic polarizability of the conformer. Then the coefficients uZpdefined in eq 2 contain only the solute polarizabilities and are given in the PAS (the frame that diagonalizes a) by
When associating principal molecular polarizabilities with directions in the PAS, we use the convention a,, 2 cyvy Ia,,. In turn, the molecular polarizability tensor of each alkane conformer is defined as a sum of bond polarizability tensors, as described in ref 26. In our calculations we adopted the C-H polarizabilities of Denbigh2’ (see Table I11 for aCD)which are in good agreement with the values of Bunn and Dauberry.28 With the resulting value of AacH = all- alr we use the empirical relations between AacH and Aacc as reported by Patterson and F 1 0 r y ~to~ get Aacc = 0.95. These authors tabulate computed Aacc values that suggest that the bond polarizabilities of Bunn and Dauberry are best. Combining the value of the mean polarizability ( a , ,+ 2al)/3 = 0.5O2* with the value of hacc, we find the C-C bond polarizabilities reported in Table 111. iv. Experimental Technique. 2H N M R spectra of perdeuteriated n-alkanes (number of carbons m = 5 , 6, 7 , 8, 10, 16, 36) dissolved in nematic solvents (0.5 mol % alkane) were recorded at 13.8 MHz (Bruker WH-90 F T N M R spectrometer). Degassed samples, thoroughly mixed in the isotropic phase, were inserted into the spectrometer (nonspinning), and after equilibration to the desired reduced temperature approximately 25 000 free induction decays were averaged. (25) Barboy, B.; Gelbart, W. M. J . Stat. Phys. 1980, 22, 685. (26) Toriumi, H.; Samulski, E. T. Mol. Cryst. Liq. Cryst. 1983, 101, 163. (27) Denbigh, K. G. Trans. Faraday Sac. 1940, 36, 936. (28) Bunn, C. W.; Dauberry, R. P. Trans. Faraday Soc. 1954,50, 1173. (29) Patterson, G. D.; Flory, P. J. J . Chem. Soc., Faraday Trans. 2 1972, 68, 1098.
Janik et al.
1846 The Journal of Physical Chemistry, Vol. 91 No. 7, 1987 ~
k
I
Figure 2. *H NMR spectra of partially oriented, perdeuteriated n-alkanes C,Dm2 in Merck ZLI-1167 at T , = 0.95 for the solutions. Note the scale differs from that in Figure 1 by a factor of 2.
111. Observations Figure l(i) shows the ,H N M R spectra of a series of alkane solutes in Merck Phase 5 at 300 K ( Tred= T / T N I= 0.95, where TNIis the nematieisotropic transition temperature of the solution.) Each spectrum of the labeled alkane homologues (CmD2m+2) displays a number of discrete quadrupolar doublets, Av', one for each of the inequivalent C-D bonds (Le., in addition to the doublet exhibited by the terminal CD3 groups, there are (m- 2)/2 and (m - 1)/2 doublets corresponding to the methylene groups for m even or odd, respectively). For some of the CD, units the dipolar coupling between the pair of deuterons is observed as a dipolar fine structure superposed on the quadrupolar d o ~ b l e t s ; ' ~the *'~ magnitude of this fine structure varies with position i in the chain and is not resolved for the methyl resonances. A sequential examination of the quadrupolar splitting patterns (from pentane to hexadecane) enables an inductive assignment of the A d . The methyl doublet is the most intense and has the smallest splitting; the rotation about the C3 axis averages the efg tensor by an additional factor P,(cos 70.5') = (tetrahedral geometry is assumed). The remaining doublets are assigned as follows: the smallest methylene splitting is from the (equivalent) penultimate CD, units, the next smallest doublet is from the antepenultimate CD2 units, and so on; the central internal methylene unit(s) exhibits the largest A d . For a given chain length, the A$ converge to the limiting values exhibited by the methylenes near the center of the chain. In the case of hexadecane the doublets corresponding to the internal CD, units start to overlap at the extrema of the spectrum (Figure 1(i)). This is more striking in the spectrum of n-hexatriacontane, a 36-carbon chain, dissolved in the aliphatic nematic solvent ZLI-1167 (Figure 2). With the exception of the first four of five CD2 units near the chain termini, the quadrupolar interactions for most of the CD2 units are equivalent. Figure 2 also shows the spectrum of octane and hexadecane in the ZLI-1167 solvent to enable a comparison of the behavior of the 36-carbon chain with that of the shorter chains in Figure l(i). (The larger alkane is not soluble in Merck Phase 5.) The magnitudes of the quadrupolar splittings are reduced in the aliphatic solvent (Figure 2) because the sign of the diamagnetic anisotropy of Z L I - 1 1 6 7 is negative and the director aligns perpendicular to the field.30 Note, however, that the features of the Octane and hexadecane spectra are independent of the solvent (Figures l(i) and 2), confirming that specific chemical interactions between these flexible probes and the liquid crystal are minimal; the features of the spectra are a consequence of the general orientational constraints associated with the nematic mean field. Additional points to note in the experimental data are that the spectra at the same reduced temperature, i.e., at comparable degrees of nematic order, show a tendency for increased solute orientational order (30) de Jeu, W. H. Physical Properties of Liquid Crystalline Materials; Gordon and Beach: New York, 1980; p 33.
50 kHz
I
I
Figure 3. *H NMR spectra of octane-d18in the nematic solvent Merck Phase 5 vs. temperature. TABLE I: Quadrupolar Splittings for Alkanes CmD2m+2 (kHz) at 7'4 = 0.95
expt
case A
case B
m = 5 8.0 24.7 26.8 0.0107 3.6
9.2 26.5 26.8 1.387 6.9
R,%
m = 7 8.2 28.0 35.2 37.6 0.00702 7.5
11.4 33.0 35.3 37.6 1.554 5.1
10.9 35.1 41.2 44.6 46.0
m = 10 6.8 25.7 35.8 42.5 46.00
16.3 36.7 42.0 44.8 46.0
0.004 18 16.5
1.678 8 .o
7.6 25.6 26.8 €0
R,b%
9.2 32.0 35.6 37.6 €
€
R, 70
expt 9.4 28.4 32.0 c
R,% 10.4 33.0 38.3 40.6 t
R, % 11.6 38.2 45.02 49.50 52.61 54.84 55.59 56.34 €
R,%
case A
case B
m = 6 8.4 26.9 32.0 0.00854 5.9
13.5 28.8 32.0 1.458 13.7
m-8 7.8 27.7 36.4 40.6 0.00581 12.0
15.3 33.9 38.7 40.6 1.610 9.6
m = 16 4.0 (4.4)' 16.6 (16.8) 26.3 (26.8) 35.6 (37.7) 43.6 (44.2) 50.1 (48.9) 54.2 (53.9) 57.8 (56.7) 0.00213 28.0
17.5 (17.7) 44.2 (43.3) 48.9 (47.2) 52.4 (50.5) 53.4 (50.9) 53.9 (55.1) 55.6 (54.7) 55.4 (56.9) 1.870 8.0
"Adjustable parameter in potential of mean torque in eq 3. *The degree of agreement between experimental and calculated Au': R (%) = 1 0 0 ( [ ~ ( A v ~ , i, d A u ~ ~ p l , ) z ] / ~ ( A ~ ~ ~ p lc ~( ))zcorresponds )i~z. to the calculated (by Monte Carlo method) A d at opposite chain ends (large AY') with increasing chain length. There is also a hint of an euen-odd alternation of the magnitude of the methyl quadrupolar splitting (smallest A d ) in the first four members of the homologous series. Figure 3 shows a typical example of the temperature dependence of the quadrupolar splittings. The splittings (and dipolar fine structure) of the octane solute exhibit the generally anticipated and well-documented changes in the nematic order of the solvent with temperature. IV. Results and Discussion It is clear from the simulated spectra in Figure 1 and the tabulated R values where R (%) = 100([C(Avbalcd Avt,p,,)2]/C(Av:,,,,)2)'/2 is used as a coarse indicator of the agreement between theory and experiment (Table I), that neither the excluded-volume parametrization (case A) nor the anisotropic dispersion force parametrization (case B) gives a satisfactory description of the data at all chain lengths. The former underestimates the mean field constraints at the chain ends for long chains, and the latter overestimates them at all chain lengths. The excluded-volume parametrization of the potential of mean torque uses rather coarse expressions for the u2p. These coefficients are
n-Alkanes in a Nematic Solvent
TABLE II:
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 1847
Properties of n-Hexane: Conformer Probabilities, Polarizabilities, a d Dimensions
P(4,95 n 1 2 3 4 5 6 7
conformation ttt
ttg+ tg+t u+g+ g+tg+ g+tg8+8+8+
2"
a,
aw
arx
L
B
W
1 4 2 4 2 2 2
10.74 10.97 10.94 11.39 11.42 10.97 11.42
11.48 11.67 11.42 11.48 11.42 11.69 11.50
12.68 12.25 12.54 12.02 12.06 12.24 11.97
10.23 9.16 9.30 8.00 8.39 8.81 7.49
2.77 3.55 3.31 3.99 3.36 3.49 3.47
1.96 2.26 2.24 2.83 3.04 2.24 3.39
free chain
case A
case B
2 1.49 8.29 8.36 3.16 3.24 3.43 1.32 5.9
23.12 7.85 8.97 3.02 3.06 3.38 1.29 13.7
19.05 8.23 8.23 3.56 3.56 3.56 I .54 Rb 167
"Number of equivalent conformations. bSee legend of Table 1.
derived by considering the six mutual orientations of a pair of tangent parallelepipeds which leave their respective PAS'S collinear. However, within this idealization of the angulardependent excluded volume, the only ambiguity in our prescription for computing the ub of each alkane conformation is the assignment of the parallelepiped dimensions. The algorithm we use to derive L, B, and W from the solute inertia ellipsoid seems innocuous; the results are not significantly altered for the alkanes if we use more sophisticated methods to derive the solute dimensions (see for example ref IO). Therefore, this parametrization would appear to fail to reproduce the alkane *HNMR data because of the coarse idealizations employed to express the uZpin terms of L, B, and
CASE B
CASA
W. By contrast, the expressions for uZPin terms of the solute polarizability tensor appear to be rigorously exact.3s' Difficulties may creep in when a is derived by summing bond polarizabilities: internal field considerationscast doubt on simple additive schemes, and reliable values for bond contributions are not available. However, when we simulate the *HNMR spectra using case B, we find that changes in bond polarizabilities in Table 111 by &IO% result in optimized calculations having R values values that differ by less than 1%. Moreover, using a C C and aCH as adjustable parameters in the computation fails to significantly improve agreement between experimental and calculated quadrupolar splitting patterns (seealso ref 23). It appears therefore that these additional results indicate that the shortcomings of case B are inherent in the form of the truncated potential of mean torque and are independent of the bond polarizabilities that we employ to calculate the ub. In order to gain a better understanding of the differences between the two parametrizations of the potential of mean torque, we examined the results for n-hexane in detail. We derived the principal polarizabilities of hexane (auI aw and azz)from a sum of C-C and C-H bond polarizability tensors and the dimensions of the hexane conformers (L, B, and W ) from the corresponding inertia ellipsoid semiaxes. These quantities shown in Table I1 were used to compute the uzp coefficients in KXt(f?,y,c). We also ) hexane (Table calculated the conformer probabilities P ( cfor 11). When the free chain (&(f?,y,c=O) = 0 in eq 7) probabilities are constrasted with probabilities derived for the constrained chain, there is an increase in the P(c)for the all-trans conformer-by 13% for excluded-volume (case A) and 2 1% for dispersion (case B) parametrizations. That is, when hexane is exposed to the potential of mean torque, the elongated all-trans conformer is favored. We observe similar findings for all chain lengths. Apart from small differences between case A and case B for the remaining p ( ~ )the , stronger selection of the all-trans conformer in case B seems to be the most conspicuous difference between the two parametrizations. In fact, a detailed comparison of the angular dependence of Vat(f?,y,e)shows that the two cases are quite similar. Figure 4 illustrates the f? and y dependence of & ( & y , ~ ) by using the optimal c values (Table I) for selected hexane conformers; results for both cases A and B are shown on the same energy scale. Generally speaking, it appears that the y dependence of the potential is more exaggerated for the an(31) Gray, C. G.; Gubbins, K. E. Thcory of Moleculrv Fluids;Clarendon: Oxford, 1984; Vol. 1, pp 498-500.
16+
P -ripre -.i n e angular . aepenaence . . or
a
..
.
4. the potential of mean torque UmI(&y)for selected conformers of hexane (all scales are identical): (i) case A, the excluded-volume parametrization; (ii) case B, the anisotropic 1
polarizability parametrization. CASE B
0
ib
r
s z ~ o . ~
7
0.0
A
b0.31 1
I
I
I
111
ttg
tgt
tgg
I
tg+g-
I
I
gtg
g*tg-
ggg
Figure 5. A histogram showing the computed order parameters S , and A = S, - Syyfor selected conformers of hexane: (i) case A, the excluded-volume parametrization of VmI(@,y)(shaded); (ii) case B, the anisotropic polarizability parametrization of VmI(&y) (open).
isotropic dispersion parametrization than for the excluded-volume parametrization. We completed our comparison of the two parametrizations by examining the calculated order parameters derived from numerically integrating eq 5 using the optimal c values. The results for each distinct kind of hexane conformer are shown in Figure
1848 The Journal of Physical Chemistry, Vol. 91, No. 7, 1987
5. Apart from the detailed differences predicted for S,, and A = S,, - S,,, both cases A and B exhibit similar general trends. (Note the negative sign for A is a mere consequence of the convention we employ in the definition of the PAS3*) These observations facilitate an understanding of the success of the phenomenological modeling we have used in the past to describe relative quadrupolar splittings. Although this modeling was limited (S was independent of temperature and conformation and assumed to be diagonal in the principal inertial frame), it worked reasonably well for solutes and for deuterium-labeled m e s o g e n ~ . ~ ~ . ’ In ~ , retrospect, ~ ~ , ~ ~ the success of this earlier modeling can be attributed to the fact that the location of the PAS in flexible molecules dominates the appearance of computed quadrupolar splitting pattern. The detailed dependence of the S,, and A on molecular properties (conformer shape or polarizability) plays a subsidiary role. In other words, the angular excursions that the C-D vectors experience relative to the PAS (on traversing through the conformations) control the effectiveness of the averaging of the efg tensors. In the alkanes, irrespective of the parametrization, the central methylenes collectively dictate the location of the PAS and their C-D vectors do not change orientations relative to the PAS very much from conformer to conformer. Closer to the chain ends there is an increase in accessible RIS’s, and for these terminal methylenes, the C-D bond vectors experience a larger variety of orientations relative to the PAS. Correspondingly, one observes reduced quadrupolar splittings near the chain ends. Is the complex angular averaging required to compute the Av‘ being computed by restricting alkane dihedral angles to three discrete values? We have examined the RIS approximation with respect to the 2H N M R simulations in two ways: We have systematically increased the number of dihedral angle states, and we have varied the primary RIS parameter E, in the three-state RIS approximation. We summarize the results of this examination next. For hexane we sysetmatically increased the number of rotational isomeric states from 3, the nominal value used for alkanes, to 24 in conjunction with an analytical expression for ED*. Only marginal changes in the computed quadrupolar splittings are observed, suggesting that the three-state RIS approximation is not a serious source of discrepancy between 2HNMR experiments and simulations. In fact, increasing the number of dihedral angle states does not substantially change the direct dipolar couplings between each pair of protons in hexane, a considerably more sensitive test of the RIS approximation and potentially a more discriminating test of effective one-body potential^.^^ For the three-state case we do find a significant improvement in the calculations when E, is increased from its nominal value E, = 2.092 kJ mol-’ to E, = 3.765 kJ mol-’ using the excluded-volume parametrization. For example, the R value we use to quantify the fit between experiment and calculation (Table I) decreases to 2.3%, 3.7%, and 7.8% for hexane, octane, and decane, respectively, when E , = 3.765 kJ mol-’. The increased E, value in effect lowers the probability of nonlinear conformers. Thus, conformers with extreme shape anisotropy (e.g., the all-trans conformation) become increasingly important in the ensemble average. Improved agreement between calculations and experiment have been observed when the high value of E, is employed in related computations involving deuterium-labeled mesogens.I8 It would appear, however, that there is little evidence to support this practice of enhancing E,. While there is a large range of values for E, reported in the early literature,’’ the most recent experimental investigations of alkane trans-gauche energy differences in gas and liquid phases support the nominally accepted values E, = 1.673-2.092 kJ m ~ l - l .Additionally, ~~ the enhanced E , fails to reproduce the mean square dipole moments of u,wdibromoalkanes as a function of chain lengths (see Appendix, (32) Faber, T. E. Philos. Trans.R.Soc. London 1983, A309, 115. (33) Samulski, E. T.; Toriumi, H. J . Chem. Phys. 1983, 7 9 , 5194. (34) Janik, B.; Samulski, E. T., in preparation.
(35) Kanesaka, I.; Snyder, R. G.; Strauss, H. L. J . Chem. Phys. 1986, 84, 1.
Janik et al. 121
04 0 75
0 85
095
Figure 6. The apparent temperature dependence of the nematic solvent order parameter P2(= c-Ik7‘) derived from simulating spectra of the type shown in Figure 3: (-) case A; (- - -) case B.
I7
I
t
3
4
5
6
7
8
9
IO
II
m
Figure 7. The calculated rms dipole moments of dibromoalkanes using the RIS approximation; ( 0 )experiment; (-) 9,* = 1120°, E, = 2.092 kJ mol-’; ( - - - ) O,, = 1112.S0, E , = 1.673 kJ mol-’; = 1 1 1 2 . 5 O , E, = 3.765 kJ mol-’. In all cases aP2 = 9, = 80° with E, = 0 at these bonds.46 (-e-)
Figure 7), a classical criterion for evaluating the RIS approximation. (Recall that the energetically unfavorable dyad sequences g f g r (the “pentane effect”) are excluded by the nonbonded interactions in our calculations.) These findings therefore suggest that the improved agreement between experiment and calculation achieved by arbitrarily increasing E, is fortuitous and probably compensates for other shortcomings in the model. The temperature dependence of the quadrupolar splittings of the octane solute in the nematic solvent (Figure 3) may be analyzed in terms of the parametrized potential of mean torque. Following the same procedure we used to simulate the 2H N M R data in Table I (Le., optimizing t to give agreement between experimental and calculated Ad, i = 4), we obtained the apparent temperature dependence of t, t being proportional to the nematic For illustrative purposes order parameter of the solvent P2.10.11 only, we choose the proportionality constant c so that, at T N I P, , = c-’kTt = 0.43, the critical order parameter of the Maier-Saupe theory.] With this proportionality, the optimal t values found to fit octane quadrupolar splittings (e.g., data of the type shown in Figure 3) yield the temperature dependence of the principal order parameter of the solvent. The P2 values obtained in this way are shown in Figure 6 plotted vs. T r d . The naivete of the implicit assumption here and el~ewhere~.~~-that the strength of the averaged solute-solvent interaction is independent of temperature (subsumed in the definition of is apparent: The derived (36) Emsley. J. W ; Hashim, R.; Luckhurst, G. R.; Shilstone, G. R. Lip. Cryst. 1986, 1 , 437.
n-Alkanes in a Nematic Solvent temperature dependence of pzis too strong; P2 even exceeds unity for Trd < 0.8.
V. Concluding Remarks Our new data on partially oriented alkanes presented here delineate one extremity of the range of oriented alkyl chain systems amenable to study by 2H NMR. These systems span a reasonably continuous series of constrained chains; (i) the isolated, relatively unencumbered alkane solutes studied here; (ii) chains anchored at one end such as mono-substituted alkane solutes,22amphiphiles in lyotropic phases,37 as well as thermotropic mesogens (prolate nematogens13 and oblate disco tog en^^^) with pendant chains; (iii) the more severely restricted "spacer" chains connecting mesogenic "cores" in dimer and in linear polymeric liquid crystal^.^^^^ With the obvious exception of spacer chains, all of these chains exhibit the same general quadrupolar splitting pattern (2H N M R "~ignature"~~): the Ad decrease rapidly over the last 4-5 methylene units near the methyl terminus of the chain. Qualitative changes in this 2H N M R signature reflect local geometrical aspects of how the chain is attached to its substituent. However, more refined modeling and spectral simulations will be necessary to understand quantitatively subtle changes in the 2H N M R spectra of alkyl chains in uniaxial environments. Our starting point for the computation of the quadrupolar splitting patterns of deuterium-labeled alkanes was the truncated potential originally proposed by Luckhurst et aL3 to describe nematics composed of biaxial mesogens and more recently extended to model a rigid biaxial solute dissolved in an ideal nematic solvent (composed of uniaxial mesogens).l03" Although there are different approaches to such modeling:,42 we limited our attention to the truncated potential because it may be parametrized in terms of distinct and readily computed solute attributes which in turn seemed capable of reflecting the relative importance of repulsive and attractive contributions to the effective nematic mean field. The modeling is further complicated by convolving the potential with the rotational isomeric state approximation of the alkane solute and the subsequent conformational ensemble average. These additional complications are largely compensated for by the absence of specific solutesolvent interactions and by the large body of work in polymer physics validating the RIS appr~ximation.'~ Nevertheless, we did examine this approximation carefully in the context of the more discriminating N M R data and confirmed that the R I S description of alkanes is not the source of disagreement between calculated and experimental Ad. Both the angular-dependent excluded-volume parametrization and the anisotropic dispersion energy parametrization of the truncated potential of mean torque fail to quantitatively simulate the 2H NMR data. In fact, these two parametrizations give results that suggest that more faithful simulations may require combining these two kinds of interactions. One such combination of repulsive and attractive interactions would accrue naturally by following the procedure advocated by Gelbart et al.5*16,25 in their generalized van der Waals (GVDW) model of liquid crystals, namely, to develop an expression of the effective one-body potential-the "attractive mean field"-by a rigorous average of the pair potential, V(Qi,Q,)derived from dispersion (r4) forces. When the hard-core repulsions are explicitly considered in the average over the dispersion forces, the resulting (attractive) angular-dependent pseudopotential $(Q,)includes the delicate interplay between (37) Charvolin, J.; Hendrix, Y. In Nuclear Magnetic Resonance of Liquid Crystols; Emsley, J. W., Ed.; D. Reidel: Dordrecht, 1985. ( 3 8 ) Goldfarb, D.; Luz,Z.; Zimmerman, H. J. Chem. Phys. 1983, 78,7065 and references cited therein. (39) Samulski, E. T.; Gauthier, M. M.; Blumstein, R. B.; Blumstein, A. Macromolecules 1984, 17, 419. (40) Griffin, A. C.; Samulski, E. T. J. Am. Chem. SOC.1985, 107, 2975. (41) Samulski, E. T. In Proceedings of the International Symposium on Physics of Complex Supermolecular Fluids, Exxon Research and Engineering, 17-21 June 1985, in press. (42) Snijders, J. G.; de Lange, C. A.; Burnell, E. E. J. Chem. Phys. 1983, 79, 2964. van der Est, A. J.; Barker, P. B.; Burnell, E. E. Mol. Phys. 1985, 56, 161.
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 1849 TABLE III: Structural Data for Alkyl Chainsa
c-c
atoms
all
C-D C-Br
c-c-c D-C-D (methylene) D-C-D (methyl) C-C-Br "Distances are in
1.53 1.10 2.14 112.50 109.00 109.47 112.50
1.14 0.79
ffl
0.19
0.58
A, angles in deg, and polarizabilities in
cm3.
TABLE I V Parameters for Nonbonded Potential Functions ENB= A / r 6 B / r ' * in kJ mol-' united atom uairs 106A 106E
+
CD,---CD, CD2-- -CD3 CD2-- -Br Br- - -Br
8.309 10.004 11.577 37.740
-4.724 -5.686 -9.991 -21.45 1
molecular shape and molecular electronic structure that distinguishes liquid crystals from ordinary liquids:
Equation 18 could be numerically evaluated for the usual form of the pair potential describing a biaxial particle (a solute ellipsoid with semiaxes L, B, and W) interacting with a uniaxial particle (a solvent ellipsoid of revolution) with semiaxes L'and B'. (The primed integration in eq 18 indicates that the average rigorously respects the mutual excluded volume of the two particles at relative A suitable truncated expansion of $(Qi) orientations 0, and would be the corresponding analogue of the truncated potential of mean torque we used in this study. Note that in order to evaluate $(al),an expression for the orientational distribution function,f(Q,), is required and f(Q,) will in turn determine the nematic order parameter P2.43 We recognize that replacing Uex,(QJwith $(QJ would be tantamount to shifting the parametrization of the potential from the czOpcoefficients to the specification of f(Q,). (We expect that Vat,,would be adequately expressed by the molecular polarizability tensor and molecular dimensions derived herein without introducing additional parameters.) Nevertheless, it would appear that oonsideration of the GVDW model is the next logical step in an effort to quantitatively simulate NMR spectra of flexible solutes in nematic solvents and flexible nematogens themselves. Acknowledgment. This research is supported in part by the N I H (AM17497) and by DARPA/ONR Contract No. NOO1486-K-0772. We thank S. Berger for programming the numerical integrations and M. Poliks for help with the computations and continued constructive discussions. E.T.S. thanks the numerous colleagues with whom he has discussed this general problem.
Appendix Alkane Geometry and RIS Parameters. The alkane bond lengths and valence angles used in our calculations are summarized in Table 111; this table also includes the bond polarizabilities used to compute the molecular polarizability tensor of each alkane conformer according to the procedure described in ref 26. The parameters of the RIS scheme are c ~ n v e n t i o n a l :a~three-state ~ approximation with the trans state at E, = 0 and two gauche states (agh= *120'). The latter states have a dihedral angle energy E, that is 2.092 kJ mol-' higher than that of trans state, E,. We have also considered the RIS parametrization wherein +gL = A1 12.5' with Eg- E, = 1.674 kJ Negligible differences are observed in the calculated 2H N M R data for these two sets (43) The distributionfln,) will shift with temperature, and the temperature dependence of computed quantities such as quadrupolar splittings would be more complicated than that resulting from simple proportionality between P , and the coefficients of the potential of mean torque assumed in eq 1 and 2.
1850
J . Phys. Chem. 1987, 91, 1850-1856
of RIS parameters, and we report results with the former. For alkanes in isotropic media ("free chains") the conformational energy consists of two contributions, E D A and E N B . The dihedral angle energy E D A is a sum of the local RIS energies, E, and E,, and the nonbonded energy E N B is calculated for all atom pairs separated by four or more bonds; a Lennard-Jones 6-12 potential with united atoms44is used (Table IV). E N 8 ensures that energetically unfavorable local sequences are appropriately weighted (e.g., g f g F pairs are prohibited, the so-called "pentane effect") and removes longer ranged (excluded volume) overlaps. As the ensemble average that we employ to simulate quadrupolar splittings exhibited by the alkane solutes involves an average over complex angular variables, we first demonstrate that the alkane geometry and RIS parameters used herein are adequate for reproducing conformationally averaged properties. In order to evaluate them, we repeat a classic test of the RIS approximation: we compute the dipole moment of the apdibromo-n-alkanes, Br-(CH2-)m-Br.4s In these substituted alkanes, the RIS parameters differ from the conventional values at the dihedral angles a2and a,,,, and the reported values for the dihedral angle energy suggest E, - E, = O.& In recent calculations of the Kerr constant and dipole moment of these dibromoalkanes, Khanarian and T ~ n e l l conclude i~~ that ag*at the antepenultimate bonds are in the range of 80-100°; we employ a value of @gf = 80°. Following earlier modeling of the dipole moments, the energy of each conformation U(n)consists of three internal energy
u(n)= EDA(n) + ENB(n) + Edd(n) ('41) The third term E d d is the electrostatic interaction energy between (44) Gibson, K. D.; Scheraga, H. A. Proc. Nail. Acad. Sei. U.S.A. 1967, 58, 420. (45) Leonard, N. J.; Jernigan, R. L.; Flory, P. J. J. Chem. Phys. 1965,43, 2256. (46) Khanarian, G.; Tonelli, A. E. J . Chem. Phys. 1981, 75, 5031.
the intramolecular C-Br bond dipoles (and induced dipoles). As suggested in ref 46, we assume that the C-Br bond dipole is accompanied by an induced dipole moment along the nearestneighbor C-C bonds with magnitudes for C-Br and C-C bond dipoles p = 1.73 D and p = 0.49 D, respectively. The point dipole approximation with each dipole located at the midpoint of the respective bonds gives a value for Edd(n) via
where t = 2.274 is the dielectric constant of the solvent (benzene) used in e x p e r i m e n t ~and ~ ~ ,a~is~the vector connecting the point dipoles at bonds i = 1, 2 and j = m, m 1. The rms dipole / 2 calculated by introducing the squared magmoment ( p u 2 ) 1 is nitude of the dipole moment ( p ( n ) = pl + p2 + pm + p m + l ) into the general expression for a statistical mechanical average:
+
= Z - l U n ) exp[-U(n)/kT] n
(A3)
Figure 7 shows the calculated rms dipole moments together with experimental values. The agreement is good both in the magnitude and the subtle variation in the conformationally averaged dipole moment with alkyl chain length (Le., the steep increase from m = 3 to m = 5 and the smoother changes in the higher homologues are reproduced). The calculations clearly show sensitivity to RIS parameters, in particular, to the value of E,. Calculations employing an enhanced value of E underestimate ( p 2 ) . These results indicate that the geometry and the conventional RIS parameters we are using (agf= 120°, E, = 2.092 kJ mol-') reproduce the observed averaged dipole moments of the dibromoalkanes with a reasonable degree of accuracy. Registry No. D2. 7782-39-0. (47) Hyaman, H. J. G.; Eliezer, I. J . Chem. Phys. 1961, 35, 644.
Study of Cu2+ Location in Zeolites Na-A and K-A by Electron Spin Resonance and Electron Spin Echo Spectroscopies Michael W. Anderson and Larry Kevan* Department of Chemistry, University of Houston, Houston, Texas 77004 (Received: September 29, 1986)
Electron spin resonance (ESR) and electron spin echo (ESE) spectroscopies are used to determine the cation site location of Cu2+ in hydrated and dehydrated forms of Cu2+-dopedzeolites Na-A and K-A. This is achieved by partial exchange of the Na-A and K-A zeolites by Cs+ cations which reside in well-defined crystallographicsites. Monitoring the weak hyperfine interaction between the Cuz+ and Cs+ cations using ESE makes it possible to determine the cation siting of Cu2+relative to the matrix of Cs' cations. At low Cuz+loadings the Cu2+favor sites on the threefold axis close to a zeolite six-ring. The larger the number of water molecules coordinated to the Cu2+the more the displacement of the Cu2+from the center of the six-ring face-except in the case of a trigonal-bipyramidalCu2+complex coordinated to one water molecule in the a-cage, one in @-cage,and three equatorial lattice oxygens which is located in site S2. An octahedrally coordinated Cu2+ complex in both zeolites, Na-A and K-A, coordinated to three H 2 0 molecules is displaced 0.09 nm into the @-cagewhile a tetrahedrally coordinated Cu2+complex bound to only one water molecule in K-A is displaced 0.02 nm into the @-cage. After complete dehydration the Cu2+ moves almost into the plane of the six-ring to give the strongest possible cation-zeolite interaction with a Cu2+-Oz bond length of 0.23 nm (0, = zeolite oxygen).
Introduction Transition-metal-exchanged zeolites are useful for a wide variety of catalytic reactions.' In particular Cu2+ has received much attention2-+' which, being paramagnetic, may be studied by electron (1) Maxwell, I. E. Aduan. Coral. 1982, 31, 1. (2) Mochida, I.; Hayata, S.;Kato, A.; Seiyama, T. J . Caral. 1970, 19,405. (3) Tsuruya, S.; Tsukamoto, M.; Watanak, M.; Masai, M. J. Catai. 1985, 93, 303.
0022-3654/87/209 1- 1850$01.50/0
spin resonance (ESR) technique^.^-^ ESR provides information concerning the number of different species and, from the g values and hyperfine splittings, their stereochemistry. When ESR is (4) Benn, F. R.; Dwyer, J.; Estahami, A.; Evmerides, N. P.; Szczepura, A. K. J . Catal. 1977, 48, 60. ( 5 ) Hathaway, B. J.; Billing, D. E. Coord. Chem. Rev. 1970, 5, 143. (6) Herman, R. G. Inorg. Chem. 1979, 18, 995. (7) Conesa, J. C.; Soria, J. J . Chem. Soc., Faraday Trans. 1 1978, 74,406. (8) Herman, R. G.; Flentge, D. R. J. Phys. Chem. 1978, 82, 720.
0 1987 American Chemical Society
