J. Phys. Chem. 1983, 87,2929-2935
2929
Optical Anisotropies of Aromatic Esters and of Oligomers of Poly(p-oxybenzoate) Peter A. Irvlne,+ Burak Elman,$ and Paul J. Flory" IBM Research Laboratory, San Jose, California 95 193 and Department of Chemistry, Stanford Universify, Stanford, California 94305 (Received: February 25, 1983)
Optical anisotropies y2of phenyl acetate (PA), methyl benzoate (MB), phenyl benzoate (PB), and of the trimer (n = 3) and tetramer (n = 4) of the p-oxybenzoate series C6H5CO(-OC~H4C0),-2-oc6H5 have been determined from the depolarized Rayleigh scattering of their solutions in CC14or, in the case of the trimer and tetramer, in dioxane. Molar Kerr constants 8of PA and of the trimer have been evaluated from the electric birefringences of their solutions in CC14 and dioxane, respectively; the K , for MB and PB have been taken from the work of Le FBvre and Sundaram. The observed values of y2 and K , are well reproduced by calculations based on anisotropy tensors & for the optical polarizabilities of the homologous aromatic esters formulated from a plausible set of mutually consistent polarizability parameters for the ester group and for the phenyl and phenylene groups, with allowance for inductive effects of substitution. The validity of a constitutive scheme for constructing the polarizability tensors, and for treating properties related thereto, is thus demonstrated. The molar Kerr constant K , is very sensitive to the direction of the dipole moment in the plane of the ester group. From the values found for K , for the aromatic esters, we obtain T~ = 123 f 3' for the angle between the dipole moment and the C-CO axis, in excellent agreement with previous determinations by other means.
Introduction Our interest in nematic fluids and the molecular features responsible for stability of the nematic liquid crystalline state has led us to examine the anisotropies of the optical polarizabilities of typical nematogens, i.e., substances that occur as nematic liquid crystals under suitable conditions. The anisotropy of the intermolecular dispersion forces, which is directly related to the optical anisotropies of the interacting molecules or structural units, appears often to be a major factor conferring stability on a nematic phase;',* it is second in importance only to asymmetry of molecular ~ h a p e . ~In . ~other investigations, we have determined the optical anisotropies of p,p'-azoxyanisole (PAA), p-methoxybenzylidene-p-n-butylaniline (MBBA), and 4,4'-di-nbutyldiphenylacetylene, all of which yield nematic phases, and of homologues of the p-phenylene series H(C6H4),H, the higher members (n = 5 and 6) of which exhibit nematic behavior above their melting points. Results of these studies are in preparation for p~blication.~ In this paper, we present the results of optical measurements on simple aromatic esters and on lower oligomers of the poly@-oxybenzoate) series having the formulae I. The homologue with n = 4 melts to a nematic C6Hg-CO[O
0 c 0 1 n - 2 0 - c 6 H 5
I
phase at 194 OC with a nematic range extending to 255 0C.5 Analogues consisting of two or three oxybenzoate residues terminated with aliphatic groups also yield nematic phases upon melting.6 Copolymers of the p-oxybenzoates exhibit the mesomorphic characteristics of nematic fluids. Besides their relevancy to nematic liquid-crystalline systems, the results presented here and their analysis afford the opportunity to examine further the degree to which the anisotropic polarizability tensors for constituent groups in molecules comprising a plurality of anisotropic groups are additi~e.~-~ 'Present address: Central Research and Development Department, Experimental Station, E. I. d u Pont de Nemours and Co., Wilmington, D E 19898. Permanent address: School of Engineering, Bogazici University, Bebek, Istanbul, Turkey.
*
0022-3654/83/2087-2929$0 1.50/ 0
The optical anisotropy of a molecule is embodied in the traceless (Le., anisotropic) part & of the polarizability tensor a. The depolarized Rayleigh scattering (DRS) and electric birefringence may serve to characterize & in whole or in part, depending on the symmetry of the molecule. The intensity of the DRS yields the invariant y2 = (3/2) tr (&&) (1) or its configurational average (3/2)(tr (&&)). If the molecule possesses an electric dipole moment p, then the electric birefringence provides the quantity 0 = pT&p, or its configurational average (pT&p) in the case of nonrigid molecules; pT is the transpose of p. In good approximation, the molar Kerr constant K , evaluated from measurements of electric birefringence is related to p and y2 according to m K = (2nN~/15lzn[0(lz77-~ + (2/3)r21 (2) where NA is Avogadro's number, lz is Boltzmann's constant, and T is the absolute temperature. The quantity p is a measure of the excess polarizability (relative to the mean) in the direction of the dipole moment. Optical measurements of the kind indicated in the preceding paragraph have been used to elucidate the structure and conformations of chain molecules. Central to this procedure is the assumption that the traceless polarizability tensor & can be expressed as the tensor sum of contributions from the molecular constituents, or groups, each such contribution being considered locally invariant within the group it represents. According to this premise & must reflect the spatial relationships between the groups comprising the molecule. (1) Maier, W.; Saupe, A. Z. Naturforsch. A 1959,14,882. Ibid. 1960, 15, 287.
(2) Flory, P. J.; Ronca, G. Mol. Cryst. Liq. Cryst. 1979, 54, 311. (3) Flory, P.J.; Ronca, G. Mol. Cryst. Liq. Cryst. 1979,54, 289. (4) Irvine, P. A.; Flory, P. J., manuscript in preparation. ( 5 ) Volksen, W., work in preparation for publication. Also, M. Ballauff, private communication. (6) Demus, D.; Demus, H.; Zaschke, H. "Flussige Kristalle in Tabellen"; VEB Deutacher Verlag fur Grundstoffindustrie: Leipzig, 1976. (7) Flory, P. J.; Saiz, E.; Erman, B.; Irvine, P. A.; Hummel, J. P. J. Phys. Chem. 1981,85,3215. (8) Erma, B.; Manin, D. C.; Irvine, P. A.; Flory, P. J. Macromolecules 1982, 15, 664. (9) Erman, B.; Wu, D.; Irvine, P. A.; Marvin, D. romolecules 1982, 15, 670.
0 1983 American Chemical Society
C.; Flory, P. J. Mac-
2930
The Journal of Physical Chemistry, Vol. 87,No. 15, 1983
The choice of constituent groups for this purpose is of some importance. A bond polarizability scheme based on anisotropies Acr = cy1 - CY^ of the polarizabilities of individual bonds is generally unsatisfactory owing to the commensurability of the cube root of ACYwith the bond length. Inductive effects between bonds are relatively large, hence the assumption of local independence of bond anisotropies in diverse conformations may be quite inaccurate. Through the choice of larger entities such as groups comprising substantial numbers of bonds as the constitutive members considered to contribute independently to d for the molecule as whole, this difficulty may be overThe validity, for the purpose stated, of a given set of constituent groups can only be assessed a posteriori, however, from the consistency of the scheme in accounting for the optical anisotropies of a family of compounds such as those of a homologous series. For the series of compounds considered in this paper, the entities of choice are the phenyl, phenylene, and ester groups. The anisotropies of the aromatic groups, in which electron delocalization is comparatively large, are strongly affected by substituents.l'J2 These effects must be taken into account in any scheme devised for the purpose indicated. Experimental Section Materials. Phenyl acetate (PA) and phenyl benzoate (PB) were purchased from the Aldrich Chemical Co. They were certified to be of 99% purity or better; methyl benzoate (MB) was obtained from Matheson Coleman and Bell. The solvents p-dioxane and carbon tetrachloride (both from the Baker Chemical Co.) were reagent and spectroscopic grades, respectively. Our samples of the trimer (mp = 136 "C) and tetramer (mp = 194 O C ) were synthesized by Dr. W. Volksen of the IBM Research Laboratory. Depolarized Rayleigh Scattering. The light scattering photometer originally designed by Pattersonlo was used for measurement of depolarized Rayleigh scattering (DRS) at 90°. A 15-mW Spectraphysics Model 124 He-Ne laser (A = 632.8 nm) with vertical polarization ratio better than 1OOO:l served as the light source. The beam incident at the scattering cell was partially focused to reduce its girth and, hence, the possibility of extraneous scattering. Narrow band-pass filters with bandwidths of 1.2 and 3.5 nm, respectively, were inserted in the scattered beam, preceding its entrance into the photomultiplier tube, as a means of separating collision-induced anisotropic scattering from that due to the intrinsic molecular anisotropy. Their transmission spectra .(A) were determined with a double monochromator using white light, the wavelength calibration being provided by the He-Ne line. Absolute values of ~(632.8nm) were assigned from the intensity of VV scattering from carbon tetrachloride.'O The required attenuation factors gl(wo) and gz(oo)expressing the fraction of the collision-induced depolarized radiation transmitted by the respective filters were evaluated according to the prescription implicit in eq 13-19 of Carlson and F10ry.l~ For the required numerical integrations T( A) was sampled directly from the measured transmission curves at fixed wavelength intervals. (10) Patterson, G. D.; Flory, P. J. J. Chem. SOC.,Faraday Trans. 2 1972. 68. 1098. 1111. (11) Suter, U. W.; Flory, P. J. J. Chem. Sac., Faraday Trans. 2 1977, 73, 1521. (12) Saiz, E.; Suter, U.W.; Flory, P. J. J. Chem. SOC.,Faraday Trans. 2 1977, 73, 1538. (13) Carbon, C. W.; Flory, P. J. J. Chem. SOC.,Faraday Trans. 2 1977, 73, 1505.
Irvine et al.
Measured depolarized intensities were analyzed according to the following procedure devised by Carlson and F10ry.l~ The molecular scattering intensity IsGpion was separated from the scattering intensities Il and I2 observed through the 1.2- and 3.5-nm filters, respectively, by use of the equation p9tion
=
[I1 - k1/g2)I2J/[f1 - f2Wg2)l
(3)
where gl and g, are the aforementioned attenuation factors; fl and f 2 are the corresponding factors for the central Lorentzian representing the molecular depolarization. For carbon tetrachloride (oo= 12 cm-l) and dilute solutions therein, g,/g, = 0.54; values of fl and f2are effectively equal to unity for this solvent. The intensity of depolarized scattering attributed to the solute is pokyte
= I d y t i o n - (1- u2)poyent
(4)
where u2 is the volume fraction of solute in the solution. Intensities ImO1 were converted to absolute Rayleigh ratios , intensity of scattering RHVFol by comparison with Ec14the from carbon tetrachloride observed through the 3.5-nm filter, i.e. RsOhte HV,mol
=
Sol te
[Imoy
/ F 2C l ]RfIv?2(fi/fiCC14)2
(5)
where ii is the refractive index of the solution and iicc14 is that of carbon tetrachloride. The absolute value of the Rayleigh ratio R$V$ for HV scattering of carbon tetrachloride observed through the 3.5-nm filter was previously determined to be (7.4 f 0.1) X cm-' at 25 OC.l0 The absolute Rayleigh ratio for any other solvent (e.g., dioxane) is given by RWV,2 = R ~ ~ ~ I ~ / ~ c 1 4 ( f i / f i c c 1 4 ) 2(6) The apparent solute optical anisotropy was calculated from the Rayleigh ratio obtained from eq 5 according to yap: = (15/p)(X/2~)~[3/(fi' + 2 ) 1 2 R ~ ~ ~ (7) ol
where p is the solute number density in the solution and X is the wavelength in vacuo. Extrapolation of yap: to infinite dilution gives the intrinsic anisotropy y2 of the solute. Electric Birefringence. Measurements required for , were carried out evaluation of molar Kerr constants K with the cell and apparatus described elsewhere.' A lowpower He-Ne laser served as light source. The retardation due to the birefringence was nulled with a Babinet compensator (Karl Lambrecht D/B/A, serial no. 170). The null point was detected with a photomultiplier tube connected to a lock-in amplifier, the latter in turn connected to a 1-kHz chopper inserted in the incident light beam preceding its entrance into the cell. Contributions of the cell windows to the observed retardation were deducted as described previ~usly.~ From the corrected retardation 6 at voltage V, the Kerr constant B in cm statvolt2 was calculated according to the relation
B = (6/2~1)(300d/V)~
(8)
where d (=0.35 cm) and 1 (=14.5 cm) are the spacing between electrodes and the optical path length inside the cell, respectively. Molar Kerr constants were calculated according to',
(9)
where t is the solvent dielectric constant, fi, is its refractive
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
Optical Anisotropies of Aromatic Esters
2931
TABLE I : Optical Anisotropies, Dipole Moments, and Molar Kerr Constants
PA 30.3 MB 60 PB (dimer) 119 trimer 370 t 50 tetramer 965 i. 100 a ( 7 2 ) and 7 calculated with A O E = 1.5, Acutph = and @ = 58". % Mean values from ref 17. From moment of the trimer.
1.71 i. 0.06b 1.88+ 0.02 1.84 i. 0.02 3.00 * 0.05
-3.95 7.68c 6.63c 61.5
30.5 111 60 121 119 121.5 369 124.5d 740 3.5, ~oi",,h = 5.4,A a ' r ' p h = 6.9, ~ a f E = 1.0,and A"fph = 3.0, all in A J , Le FBvre and Sundaram, ref 16. 125" calculated from the dipole
index, AB is the difference between B for the solution and Bo for the solvent, m is the solute molarity, and V, is its molar volume. Dipole Moments. Dielectric constants E of solutions of the esters were measured at 25 "C with a WTW dipolmeter (Model DM-01) at a fixed frequency of 2.0 MHz. Measurements were performed on solutions of methyl benzoate in CCI,, of phenyl benzoate in benzene, and of the "trimeric" homologue I with n = 3, also in benzene. Measurements were carried out at solute mole fractions of the first two compounds in the range ca. 0.005-0.05; the range for the trimer was ca. 2 X 10-4-9 X lo-,. Refractive indices of the solutions fi and pure solvents ii, were measured with an Abbe refractometer. Dipole moments were computed from the raw data (c,fi,fiJ following the procedure devised by Smith14 and Guggenheim.15 The contribution of the term dependent on the refractive indices amounted to 9-13% of the calculated dipole moment.
Results Apparent values of y2 determined as described above from DRS measurements carried out at room temperature are plotted against the volume fraction u2 of solute in Figure 1. Carbon tetrachloride was the solvent for phenyl acetate (PA), methyl benzoate (MB), and phenyl benzoate (PB); the trimer (I, n = 3) and tetramer (I, n = 4) were dissolved in dioxane. Intercepts of the least-squares lines through the points are identified with the configurational averages of the molecular optical anisotropies ( yz), the angle brackets being appropriate in those instances where the molecule may assume a diversity of conformations. These values are listed in the second column of Table I. Experimental errors are 5-1070 unless otherwise stated. Results of electric birefringence measurements on solutions of PA in carbon tetrachloride and of the trimer in dioxane conducted at room temperature are shown in Figure 2. The values of Bo determined for carbon tetrachloride and dioxane were 6.8 X and 9.7 X cm statvolt-2, respectively. Molar Kerr constants ,K calculated from the intercepts of the least-squares straight lines in Figure 2 and the quoted values for Bo are given in the fourth column of Table I. Also used for this purpose were the wavelength X = 632.8 nm in vacuo and, for CCl, at 25 OC, ii, = 1.458 and t = 2.227; for dioxane at 25 OC, ii, = 1.422 and E = 2.209. Experimental errors are f5-10%. Results for methyl benzoate (in CCl,) and phenyl benzoate (in benzene) are taken from the work of Le FBvre and Sundaram.16 Dipole moments obtained for the various esters are summarized in the third column of Table I. The value quoted for phenyl acetate (PA) is the mean of results cited in ref 17. Our results for MB and PB are consistent (14) Smith, J. W. Trans. Faraday SOC.1950, 46, 394. (15) Guggenheim, E.A. Trans. Faraday SOC.1949,45,714. 1951,47, 573. (16)Le FBvre, R.J. W.; Sundaram, A. J. Chem. SOC.1962, 3904.
lO0Ot Tetramer 9501
0.003
0.001
0.005 4
400L
Trimer
350,
0.02
0.0 1
0.10
0.20
68
/
v
64' 60
1
0.1
0.3
0.2
0.1 0.2 0.3 Volume Fraction of Solute, v p
0.4
Flgure 1. Apparent optical anisotropies yw* plotted against the solute volume fraction v 2 for phenyl acetate (PA) in CCi,, methyl benzoate (MB) in CCI,, phenyl benzoate (PB) in CCI,, trimer in dioxane, and tetramer in dioxane.
3.9
2
0.02
0.04
0 06
0.08
0.1
0.2
0.3
0.4
-0.261
c
-0.23 Volume Fraction of Solute, vp
Flgure 2. Ratios of the difference AB = B - 8 , between electric birefringences of solution and solvent to the molarity m (in mol L-') plotted against the ester volume fraction v 2 for PA in CCI, and trimer in dioxane.
The Journal of Physical Chemistry, Vol. 87,No. 15, 1983
2032
Irvine et al.
R(4,6) = Rz(S)Rx(+)
(12)
Rz(6) and Rx(4) being the transformations for rotations 6 about Z and 4 about X, respectively (see Figure 3); e.g. H,C-C,
/O
[' o.]
+
,
0-CH,
(13)
R.y(O)= 0 o sOc c
MB
Flgure 3. Hypothetical processes for forming PA and MB from benzene and methyl acetate (MA) which provide the basis for formulating the anisotropy tensors. Coordinate systems for the phenyl (X'Y'Z') and ester ( X Y Z ) groups are shown. The dipole moment pE is represented by an arrow. Its direction is specified by the angle rE.
where c = cos 4 and s = sin 4. The optical anisotropy y2 for PA is readily calculated from tipA according to eq 1. The parameter 4 required for calculation of the molar Kerr constant according to eq 2 is furnished by m
P = PE'WE = I.rE*[cOs T E , sin
TE,
01,
rsTj sin
TE
(14)
with literature valued7 for these compounds. Analysis of Experimental Results Phenyl Acetate (PA). The anisotropy tensor for PA may be formulated by considering the hypothetical process whereby benzene and methyl acetate (MA) are condensed through elimination of methane as indicated in Figure 3. Formation of MB may be considered to occur similarly, as shown in this figure, by elimination of methane from the methyl group of the acetate residue in MA. Since &CHI = 0 and the numbers of C-C and C-H bonds are conserved in these processes, it follows that & = &MA
+ &ph
(10)
where & is the molecular anisotropy tensor for either PA or MB, and &MA and &ph are the tensors for MA and benzene, respectively, mutually oriented in accordance with the molecular structure and conformation of the aromatic ester, all tensors being expressed in a common reference frame. The tensor 6 p h attributed below to the phenyl group formally includes the C6H5-H bond8," and hence would be more accurately written Gph-H. As applied below, this tensor must include effects of substitution, which often are substantial.8J1J2 The anisotropy tensor &E attributed to the ester group is identified with &MA, the tensor for methyl a ~ e t a t e .It~ is considered arbitrarily to be unaffected by neighboring substituents. This artifice is of no consequence inasmuch as any effect of substitution on the tensor for the ester group is incorporated in the tensor &ph ascribed to the adjoining phenyl group according to the scheme adopted. The tensor &ph for the phenyl group in PA is expressed initially in the X'Y'Z' coordinate system shown in Figure 3. The X'axis is taken along the direction of substitution, the Y' axis is in the plane of the ring, and the 2' axis is perpendicular theretoela The reference frame XYZ for the ester group also is shown in Figure 3. The axes X'and x of the reference frames in which &ph and &E, respectively, are expressed (for PA) differ by 6 = 6' owing to the difference between fCC*O and fC*OCph, the carbonyl and phenyl carbons being denoted by C* and by Cph. The anisotropy tensor for PA expressed in the ester group reference frame XYZ is therefore & = &E 4-
R(#,6)&,,RT(4,6)
(11)
where (17) McClellan, A. L. 'Tables of Experimental Dipole Moments"; Rahara Entrp.: El Cerrito, CA, 1974; Vol. 11. (18) The Y (Y? and 2 (29 axes are interchanged compared with the reference frame used in ref 11 and 12. Consequently, the signs of the Am' defined by eq 15 are reversed.
where pE is the magnitude of the dipole moment of the ester group and T~ is the angle specifying its direction, as defined in Figure 3. Whereas X', Y', and Z'dre principal axes of the phenyl group, the principal axes of &E in the XY plane depart somewhat from X and Y of the XYZ system. Analysis of y2 from DRS and p from electric birefringence of aliphatic diesters' yields -6 and -13' for this rotation, depending on the particular diester. Values found' for the components of &E in XYZ are & E x x = 1.05 f 0.05 A3,&E,W = 0.02 f 0.07 A3,and &E,XY = &E,YX = a E , X Y = -0.2 f 0.1 A3, with &E,ZZ = -(&XX
+ GYY).
Approximate values of the corresponding parameters for the phenyl gr0up*9~3"J~ in PA (see below) are &ph,xrx = 2.3 A3, & p h , y y = 0.3 A3, and, of course, &phX'y = 0. According to estimates of the conformational energy of PA as a function of 4, supplemented by crystallographic data on analogous phenyl esters, minima occur at 141 = 58 f Identification of ( sin2 4) required for the calculation with sin2 4 for the discrete value q5 = 58' entails an error that is negligible compared with the range of uncertainty, *lo', in the location of the minimum. Exploratory calculations of ( y 2 ) p A carried out according to eq 1, 11,and 12 by using various values of the parameters covering ranges approximating those indicated above lead to the following conclusions: (1)The experimental value of ypA2 given in Table I is well reproduced for plausible values of the parameters. (2) The contribution of a E X Y to ypA2is negligible; its omission lowers ?PA2 by ca. 0.4%. (3) Replacement of 6 = 6' by 6 = 0 raises y2 by only ca. 0.4%. The values reported for the dipole moment of PA cover the range 1.65-1.77 D.17 A "best" value appears to be ppA = 1.71 D. Its direction was previously foundz0to be TE = 120-125' with rE defined as in Figure 3. Exploratory calculations of molar Kerr constants for PA according to eq 2 with 4 determined by using eq 14 and the quoted values of p p A and TE together with the tensor & for PA evaluated from parameters in the ranges indicated above show K , also to be comparatively insensitive to a E x y and to 6. It depends critically on rE,however. In light of the results cited, it is expedient to set aE,Xy= 0 and to take 6 = 0, whereupon XYZ becomes the principal axis system for tiphand the axes X and X' are taken to be collinear. The practicability of these approximations is enhanced by the fact that the contribution of &E to ti for P A is overshadowed by that of &ph. The same holds for the other aromatic esters to be considered. (19) Hummel, J. P.;Flory, P. J. Macromolecules 1980, 13, 479. (20) Saiz, E.; Hummel, J. P.; Flory, P. J.; Plavsic, M. J. Phys. Chem. 1981,85, 3211.
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
Optical Anisotropies of Aromatic Esters
2933
With abandonment of the off-diagonal elements of &E, this tensor may be resolved into cylindrical and acylindrical parts according to &E
= AaEJ
+ AatEJt
(15)
where J = diag
(2/,-1/,-f/3)
Jt = diag (0,y2,-y2) A a = axx - (ayy + azz)/2 Aat =
ayy
- azz
(16) (17) (18) (19)
(The cylindrical and acylindrical parameters A a and Aat are equivalently defined by the components GXx, etc., of the anisotropic part of the polarizability tensor.) Rounding the values above, one obtains A ~ =E 1.5 A3and ACJE = 1.0 A3.
The tensor for the phenyl group expressed in the coordinate system X'YZ' may be similarly rendered; i.e. &,h
= h,hJ
+ AfftphJt
(20)
No approximation is involved in this representation of &,h since X', Y', and 2' are principal axes. In the approximation 6 = 0, eq 11 reduces to
+ RX(4)&,hRXT(@)= (A~+ E haph)J+ A a t ~ J+ t Aat,hRX(4)JtRXT(4) (21)
& = &E
Substitution in eq 1 yields
+ Affph)' + (3/4)(Afft~+ Afft,h)z -
yz = (ACYE
3ACXt~Afftph ( S 2 ) (22) where (9)is the configurational average of sin2 4. Substitution of eq 21 in eq 14 gives
+ (1/2) X Sill2T E - Afft,h(Sz) Sin2 731 (23)
fl = ~E'[(AcuE + AaPh)(2/3- sinz T E )
+ AcYt,h)
(Afft~
For a wide variety of phenyl and phenylene comp 0 ~ n d ~ , 4 AatPh , ~ ~ =~ 3.0 , ~ f~ 0.5 , ~ A3. ~ This parameter appears to be little affected by substitution, the principal effect of which is manifested in A a . Hence, we take Aat,h = 3.0 A3 in all calculations and ad$st AaPhas required to reproduce y2. Using the rounded values of AaE and A d E given above together with Aat,h = 3.0 A3 and 4 = 58', we reproduce the experimental result y p =~30.3 ~ A6 given in Table I by taking ha' h = 3.48 fi3 = 3.5 A3. (The rounded value Aff',h = 3.50 AP yields y p ~ '= 30.5 As.) The prime is appended to a p h specifically to identify the phenyl group bonded to oxygen of the ester group. The value thus deduced is consistent with results for other compounds C6H5-X."J2 With pE = 1.71 D and TE = 120', we obtain ,KpA = -1.1 x 10-lo statvolt-z cm5mor1 according to eq 2 and 23, which is smaller than the observed value (Table I). As already indicated, the disparity is little affected by any reasonable alteration in the values assigned to CYEJ;~and to 6. Increase in 4 raises the magnitude of ,K, agreement with experiment would require 4 = 90°, however. A torsion angle of this magnitude would be incompatible with other evidence.19 The value calculated for K , is most sensitive to T E . The experimental result for K , is reproduced by taking C#J = 58' and T E = 111'. This value of 73 is inconsistent with the dipole moments of aromatic diestersz0and with the Kerr constants for the other esters treated below. We conclude that ypA2is well reproduced by calculations based on additivity of polarizability tensors for the groups
I 2
120
124 I
T E p
128
120
deg.
Flgure 4. Calculated molar Kerr constants, configurationally averaged, plotted against the directlon of the dipole moment T~ in the plane of the ester group for the molecules indicated. Horizontaldashed segments denote experimentally observed values. T = 298 K.
comprising this molecule. The required value of AaPh is in the range expected. The calculated and observed values of ,K for PA are disparate, however. Only by choosing an unacceptably low value for the angle T E defining the direction of the dipole moment can reasonable agreement be achieved. Methyl Benzoate ( M B ) . The anisotropy tensor & for MB is formulable exactly as for PA; see Figure 3. Equation 11 is directly applicable. In this case, however, the axes X and x'of the reference frames in which &E and &,h are presented are collinear. Moreover, the preferred value of the torsion angle $, defined in Figure 3, is zero according to conformational energy calculations and evidence from crystal structures of esters of aromatic acids.lQ Hence, eq 11 may be replaced by
where &" h denotes the tensor for the phenyl group attached to tke carbonyl carbon of an ester group. Trial calculations demonstrate that the off-diagonal element &E,XY contributes negligibly to y2 and to fl (and hence to ,K) for MB. As in the case of PA, therefore, we neglect this element. Equations 22 and 23 are then applicable to the calculation of these quantities for MB, subject to simplifications resulting from (s2) = 0. From the values for AaE, A a t ~ and , AatPh used in the calculations pertaining to PA and the experimental result ymz = 60 A6,we obtain Ad'ph = 5.43 A3 i= 5.4 A3 according to eq 22. The larger value of compared to A d p h is consistent with the greater electron delocalization in the benzoyl group. Values of the molar Kerr constant calculated according to eq 2, 22, and 23 from the Ads quoted above and the measured dipole moment given in Table I are plotted against TE in Figure 4. As in PA, the molar Kerr constant is markedly dependent on the direction of the dipole moment. This sensitivity is a consequence of large terms of opposite signs entering into fl. Comparison with the experimental result for 3 given in Table I yields T E = 121 f 2O, which is in good agreement with the previous valuez0 of 121' deduced from the dipole moment of dimethyl
2934
The Journal of Physical Chemistty, Vol. 87,NO. 15, 1983
MB
PB
Trimer
Figure 5. Hypotheticalprocesses for forming PB from MB and benzene and of trimer from MB and PB pursuant to formulation of anisotropy tensors.
terephthalate. The angle calculated from K , is comparatively insensitive to other parameters; e.g., alteration of h by f0.5 A3 changes 73 by F3’. Pienyl Benzoate (PB). The anisotropy tensor for PB may be constructed from &mand &ph (or from &PA and &p) according to the scheme in Figure 5. We thus obtain &PB
= (ACdph
+ A d r p h + A a E ) J + (Afftph + AatE)Jt+ AatPhRx(4’)J’RxT(4’)
(25)
with approximations corresponding to those adopted above. The phenyl groups attached to oxygen and to carbonyl are identified by prime and double prime, respectively, as above. It will be apparent from comparison of eq 25 with 21 that eq 22 and 23 may be adapted to PB by substituting Adph for hffph, ACYE+ A d f p hfor ACUEand AatE + Aatph for A d E . Again taking ACYE= 1.5 A3, A a t ~ = 1.0 A3,AatPh = 3.0 A3, and 4’ = 58O, and assigning to A d and the values found above, namely, 3.5 and respectively, we obtain ypB2 = 119 A6 in exact agreement with experiment (see Table I). The molar Kerr constant calculated according to eq 23 adapted to PB in the stated manner is plotted in Figure 4 against 73. Agreement with experiment requires ?E = 121.5 f lo.This result is in close agreement with the angle deduced above for MB. It agrees also with previous values deduced from the rms dipole moments of aromatic diesters.20 Trimer and Tetramer. The square of the dipole moment of the trimer in the conformation specified by torsion angles c$~ and about the central phenylene group is given by ptr:
= 2pE2[1
+ cos 6(cos273 + sin2 7 E cox x) + sin 6 sin
73
cos
?E(1 -
cos x)] (26)
where x = d 2 + $2 is the torsion of one ester group with respect to the other. Dismissing effects of dipole-dipole interactions, which should be negligible in the preferred conformation with 4’ ;= f58O, the states $ = 0 and a are equally probable and x is twofold symmetric. Hence, (cos X ) = 0. It follows that (ptrf)
Irvine et al.
= 2pE2(1 + cos 6 cos2 7 E
+ sin 6 sin 7 E cos iE) (27)
Taking pE = 1.84 D and 6 = 6 O , one finds 73 = 128 f 3’ to be required to reproduce the measured dipole moment within the limits quoted in Table I. Assignment of 6 = 0 raises the calculated value of ( h2)1/2 about 2%. The value of 7 E calculated on this basis is 125O. The anisotropy tensor for an n-mer (see formula I), formulated as illustrated in Figure 5 for the trimer ( n = 31, is given generally by
here is the cylindric part of the anisotropy of a p-phenylene group bearing both carbonyl and oxygen substituents from adjoining ester groups. This expression, like its simpler antecedents, is written in the approximation that the axes of successive units are parallel. Evaluating ( y2) on the basis of this equation by extension of the procedures above, we find that agreement with the observed value, 370 f 50 A6,of the optical anisotropy of the trimer, requires = 6.9 f 1.3 A3 in conjunction with other parameters used above. If we adopt the mean value of h ” ‘ p h , then the effects of the two substituents on the benzene ring will be seen to be approximately additive. This follows from the near equality of A C Y ” ’-~Aa!”ph ~ = 1.5 A3 for para substitution of ester oxygen on the benzoate phenyl group and A d p h - Affbenzene= 3.5-1.8 = 1.7 A3for attachment of ester oxygen to benzene. Additivity of inductive effects of para substitutions on the cylindrical part of the polarizability tensor has been noted previously.8J2 The molar Kerr constant of the trimer calculated on the basis of eq 28 from pE = 1.84 D together with quoted values of other parameters and averaged over all conformations is shown as a function of 73 in Figure 4. Agreement with experiment requires 73 = 124.5 f 2O, in satisfactory agreement with results above and with previous deductions of ?E from dipole moments of aromatic diesters.20 The error of ca. f 1 0 % in K , imparts an uncertainty of only f1.2O in 7E. Change of over the maximum per, by f 8 % , which missible range quoted above alters K translates to a further uncertainty of f1° in ?E. The acylindrical terms in eq 28 contribute only ca. 3% to ( r2)for the trimer. Their contribution being proportionately smaller for the tetramer, they may be neglected. The values of the cylindrical parameters given above give (ytetra2) = 740 A6,which is substantially lower than the value observed; see Table I. The discrepancy may be due to higher oligomers present as impurities in the tetramer. As little as 10% of pentamer would raise ( ytetr2)to the value observed. Alternatively, the discrepancy can be brought within the range of uncertainty in the measurements by increasing Aa’”ph to its upper limit stated above, namely, 8.2 A3. The value of (ytetr2)calculated on this basis is ca. 890 A6.
Discussion Results of the calculations presented above are summarized in the last two columns of Table I. Because of the exceptional sensitivity of K , to the direction of the dipole moment, the value of 73 required to reproduce the observed molar Kerr constant is recorded instead of a value calculated for the latter quantity. The consistency of the results is gratifying. Values required to be assigned to the parameters (see footnote Q in Table I) are reasonable. Those for AaPh fall in ranges to be expected according to previous work.*J1J2In particular, A d p h is comparable to the value, 4.0 f 0.15 A3,found for the phenyl group in diphenyl carbonate.8 The carbonate group resembles the ester group in its planarity and constitution, but differs from it in details of structural geometry. The torsion angle 4 is smaller, being ca. 460e8 This is indicative of greater electron delocalization, which may account for the somewhat larger value of A d p h in the carbonate. The much
J. Phys. Chem. 1983, 87, 2935-2940
larger value of for the benzoate residue, with IC/ = 0, is consistent with this interpretation, as noted above. The invariance of the transverse anisotropy Aatph of the phenyl group to substitution is compatible with the present results, which therefore substantiate this inference from previous studies.8J’J2 The additivity of group contributions to the anisotropic polarizability is supported by the success of constitutive schemes in evaluating molecular anisotropy tensors ?i for the oligomeric poly@-0xybenzoate)’s and for the polycarbonate~.~ Separation of the highly anisotropic phenyl and phenylene groups by ester or carbonate groups, with which they are noncoplanar, may be assumed to interrupt conjugation between them, such as occurs in the poly@phenylene)'^.^ Torsional rotations in the latter are only 20-30’. Directions of the dipole moment of the ester group deduced from the molar Kerr constants for the various aromatic esters we have investigated are in excellent agree-
2935
ment with previous results20 with a single exception, namely, phenyl acetate. The discrepancy of ca. 10’ in this instance is unexplained. Otherwise, the present results provide compelling, independent confirmation of the earlier work.20 Acknowledgment. We acknowledge with gratitude the assistance of Dr. W. Volksen, of the IBM Research Laboratory, who generously provided samples of the trimer and tetramer of the polyester series I, and of Dr. M. Plavsic who kindly carried out the dielectric measurements for determination of the dipole moments. This work was supported in part by the National Science Foundation, Polymers Program, Grant DMR-80-6624-A01to Stanford University. Registry No. I ( n = 3), 58607-86-6; I ( n = 4), 85800-06-2; phenyl acetate, 122-79-2;methyl benzoate, 93-58-3; phenyl benzoate, 93-99-2.
Eyring Rate Theory in Excitable Membranes: Application to Neuronal Oscillations Teresa Ree Chay Lbpartment of Biological Sciences, University of Pittsburgh, Pittsburgh, Pennsylvania 15280 (Received January 25, 1983; In Flnal Form: April 11, 1983)
The Eyring multibarrier rate theory has been useful in the analysis of membrane ionic currents, since it is based on a microscopic theory and consequently has wider applicabilities than the “electric circuit” model usually used in describing transport processes of ions across the excitable cell membranes. The models based on the Eyring rate theory that have been worked out so far, however, consider only steady-state cases. We have extended Eyring’s multibarrier kinetic model to a nonsteady-state model and applied it to the oscillatory phenomena involved in the neurons of gastropod mollusks. Our model is seen to give a very good quantitative account of the many of experimental results reported in the these neurons.
Introduction One of the most fascinating fields in biophysics concerns the basis of ion permeation through exitable membranes. Excitable membranes contain several types of ionic channels, each of which consists of macromolecular pore and exhibits selectivity in the charge and size of the ions. Hodgkin and Huxley (HH) treated the problem using an equivalent electrical circuit where current can be carried through the membrane, either by charging the membrane capacity or by movement of ions through the resistance in parallel with the capacity.’ The ionic current is divided into components carried by sodium and potassium ions and a small “leakage current” made up of chloride and other ions. The linearity for the ionic fluxes assumed in HH has become increasingly less satisfactory as more refined observations about transport through channels have accu(1) Hodgkin, A. L.; Huxley, A. F. J. Physiol. 1952, 117, 500-44.
0022-3654/83/2087-2935$0 1.50fO
mulated.2 In recent years, workers have treated the movements of ions through ion specific channels using the Eyring rate theory3 or the electrodiffusion t h e ~ r y . ~The ,~ Eyring theory represents the membrane channel as multibarriers, in which the permeating ions jump from one site to an adjacent site with a rate that decreases exponentially with the height of the energy barrier over which the ion must jump. Selectivity of the channel arises, in this view, from the different well depths and barrier heights through which ions of the various species must jump.2 Eyring and his colleagues derived an equation for current flowing across a nerve membrane in steady states, which becomes (2) “Membrane Transport Processes”; Stevens, C. F., Tsien, R. W., Ed.; Raven Press: New York, 1979; Vol. 3. (3) Glasstone, S.K.; Laidler, J.; Eyring, H. “The Theory of Rate Processes”; McGraw-Hill: New York, 1941; Chapter X. (4) Plonsey, R. “ElectricPhenomena”;McGraw-Hill: New York, 1969; Chapter 3. (5) Goldman, D. E. J. Gen. Physiol. 1943, 27, 37-60.
0 1983 American Chemical Society
