J. Phys. Chem. 1982, 86, 4791-4798
There was no drastic change in the value of C,. (ii) To make the Pyrex wall cleaner, we heated it in vacuo with great care before a run, but no measurable changes in C, were observed. (iii) Aqueous vapor was added to the system during a run, but no detectable change could be observed after the equilibrium pressure recovered to the previous one. (iv) After a run was performed at P, = 1.60 torr, krypton gas (whose molecular weight (84)is nearly equal to that of Me& (88)) was added abruptly to the system, when the total pressure kept increasing slowly after the injection and then decreased to 1.65 torr. Changes in the two components of the response data on Me& caused by the krypton are shown in Figure 12. However, since the equilibrium pressure was identical in the various runs
4?9 1
(see Table I) within the experimental errors, the effect by a foreign gas could not explain the difference in Cn's. Though the present investigation is not complete because of the uncontrollable factor of C,, we believe that the new aspects discovered by the frequency-response method are sufficiently interesting to be reported and could stimulate both experimental and theoretical studies on the important phenomenon. Acknowledgment. We thank Miss Atsuko Morimoto for the preliminary experiments with pentane. We also thank Professor Taiji Kitagawa and Professor Yoshihiro Mori of Toyama Medical and Pharmaceutical University for their fruitful discussion.
End Effects on Longitudinal Accordion Modes: Fatty Acids and Layered Systemst Gluseppe Mlnonl and Gluseppe Zerbl Istituto di Chhnilca Industrkrk PoMecnim, 32 Ml&no. Italy (Received: Februtzry 16, 1982: I n Flnal Form: June 2 1, 1982)
Heavy substituents at either end of chain molecules, weak or strong interactions between molecular chains facing each other in fiiite systems, and interlamellar forces in crystals critically affect the so-called longitudinal accordion modes (LAM). From a general dynamical treatment on a simplifed molecular model new information has been obtained for real cases of polymethylene-type chains. General results are presented and particular attention is focussed on fatty acids, on which structural information is derived by comparison with newly recorded Raman spectra. Dispersion curves for LAM modes are calculated for one-dimensional layered crystals with interchain interactions of the type CHpCH3and OH.-O (n-alkanesand a,w-dicarboxylicacids, respectively).
1. Introduction
It has been shown1 that polymethylene molecules of finite length in the trans conformation exhibit a characteristic spectral pattern in the low-frequency Raman spectrum which has to be associated with the so-called "longitudinal accordion modes" commonly referred to as LAM. LAM modes correspond to stationary vibrational waves with one or more nodes in the profile of the longitudinal displacements of the skeletal carbon atoms. If j is the number of nodes, only modes with odd j are Raman active and the Raman intensity strongly decreases with increasing j . LAM 1 has received most of the attention since it gives rise to the strongest line in Raman spectrum, thus becoming a useful spectroscopic probe. Moreover it has been shown that its frequency can be easily related to the length of the molecular chain. Indeed Shimanouchi and Mizushima2 have shown that if the segment of a trans planar polymethylene molecule is represented by an elastic rod of length I , density p, and Young modulus E , the LAM 1 frequency (cm-') can be related to 1 by
i (E)l" p
= 21c
Various groups have taken up this idea and developed it with the main purpose of characterizing the structure of n-alkane chains or of polymer chain^.^ The approximation of the elastic rod has been retained by many authors when end effects (substituents at either ends, interlamellar 'Presented at the March Meeting of American Physical Society,
Dallas, 1982.
forces, etc.) on LAM frequencies had to be accounted A realistic approach which relates LAM modes to a molecular system is to consider the LAM frequencies of the finite n-alkane chains as points at properly chosen wavevectors on the w 5 branch of the dispersion curves of an infinite and isolated polyethylene chain. This has been accomplished by Shaufele and Shimanouchi' who have thus provided the longitudinal section of the w5 branch from observed Raman-active LAM modes of n-alkanes. Our work stems from the observation that the spectral Raman pattern so far observed for organic molecules containing n-alkyl residue (such as fatty acids etc.) shows some unusual and yet unexplained features. We thought that it was necessary to tackle again the problem of end effects on LAM modes and decided that it would have been more meaningful, and in some cases necessary, to keep the molecular model, though in a simplified version. The results by Hsu and Krimm5 from the continuous model have been reobtained by us as particular cases of a more general treatment which not only provides additional and interesting new information about other systems (1)R. F. Schaufele and T. Shimanouchi, J. Chem. Phys., 47 3605 (1967). (2)S.Mizushima and T. Shimanouchi, J. Am. Chem. SOC.,71, 1320 (1949). (3)For review articles on the subject see, for instance, T. Shimanouchi in "Structural Studies of Macromoleculea by Spectroscopic Methods", (K. J. Ivin, Ed., Wiley, New York, 1976;S.K r i i m , Znd. J.h r e Appl. Phys., 16, 335 (1978);G.V. Frazer, Zbid., 16, 344 (1978);R. G.Snyder, S. J. Krause, and J. R. Scherer, J.Polym. Sci., Polymn. Phys. Ed., 16, 1593 (1978). (4) G. Strobl and R. Eckel, J . Polym. Sci. Phys. Ed., 14, 913 (1976). (5) S.L.Hsu, S. Krimm, and G. Ford, J. Polym. Sci., Polym. Phys. Ed., 15, 1769 (1977);S.L.Hsu and S. Krimm, J . Appl. Phys., 48,4013 (1977);47,4265 (1976).
0022-365418212086479 1$01.25/Q 0 1982 American Chemical Society
4792
Minoni and Zerbi
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982
b
I
Flgure 1. Molecular model adopted for (a) finite system and (b) an infinite system.
(such as fatty acids and monolayered systems), but also allows one to obtain the dispersion of LAM vibrations in a one-dimensional crystal where the repeat unit consists of a molecular chain of finite length. 2. Theory 2.1. Finite Systems. We consider first the case of two trans planar polymethylene like molecules of equal length, joined at one end by some type of bond whose strength can be varied at will. The masses of the end groups, either the free or coupled ones, may take on any value with the restriction that the uncoupled groups be equal and, similarly, the coupled ones. In this class of molecules we can easily collect a large group of organic materials ranging from n-hydrocarbons to true dimeric systems such as fatty acids in the solid state and layered systems (phospholipids, biomembranes, penvoskite-type layered compounds, etc.). In this work we approximate the real system in the following way: groups of atoms (CH2CH,, COOH,etc.) are taken as point masses (hereafter referred to as “atoms?) and the trans planar geometry is substituted by a linear one. The interactions are taken to the first neighbor and only longitudinal displacements are considered. The geometry of the model and the labeling of the atoms are given in Figure la. In Figure l a it can be seen that all bonds are of the type C-C with the exception of the bond coupling the two chains through atoms N and N + 1. This represents the actual case related to the molecules of interest. We then proceed along the following lines: (i) From the equations of motion of the atoms of the “bulk” of the molecule a dispersion relation w = w ( 8 ) , 4 being the “molecular phase”, is derived. (ii) The equations of motion of the atoms either of the free or bound ends are used as boundary conditions in order to select the value of d which can be substituted in the dispersion relation to give the frequencies of the normal modes of the molecule. (iii) Physically reasonable elastic spring constants F and f and effective masses m, Me, and Mi are suitably chosen as discussed below in order to evaluate the perturbations on LAM. The equations of motion are the following: (1) MeX1 = F(x2 - X I ) s = 2, N - 1 mf, = F(x,-l - x , ) + F(x,+, - x,) (2)
(3) MifN = F(xN-1 - x N ) + f ( ~ N + 1- X ) (4) M $ N + ~= F ( x N +-~ X N + I ) + ~ ( X -N X N + ~ ) s = N + 2 , 2N 1 m f , = F(x,-, - x,) + F ( X , +-~ x,) (5) Mef2N = F(~2,+1- xzhi, (6) One looks for solution of the type x , = [ A sin sd + B cos ~ 4 ] e - ~ ‘ ” s~ = 1, 2 , ..,, N x , = [C sin (2N - s + 1)8 + D cos (2N - s + 1)4]e-cot s = N + 1, N + 2, ..., 2N
and from eq 2 and 5 one obtains the usual dispersion relation for the monoatomic linear chain 2F w2 = -(1 - cos 4) (7) m The linear system in the unknowns A , B, C, and D, constructed with eq 1, 3, 4, and 6, can be solved if the molecular phase d satisfies the following equation sin ( N - l)d(cos d - 1)[2KiK,- Ki - 2RKi - K, + R + cos 9(Ki + K , - 2KiK, - l ) ]+ cos ( N - 1)8 sin 4 [ ( l COS O)(Ki K, - 1) - R ] = 0 ( 8 )
+
which is obtained by equating to zero the determinant of the coefficients of A , B, C, and D in the case of centrosymmetric modes ( X N = -xN+,). In eq 8, Ki = M J m , K , = M J m , and R = f / F . By a suitable choice of the above parameters real cases can be simulated: (a) Ki = K , = 1, R = 0: two uncoupled chains of identical atoms (in this case one treats the dynamics of n-alkanes if the difference of the mass of CH2 and CH, groups is disregarded); (b) Ki = K, = 1, R = 1: a single chain of 2N identical atoms; (c) Ki (or K,) = 1, K , (or Ki) > 1,R = 0: two uncoupled chains of N atoms with an heavier mass at one end (e.g., isolated a or w monohalo-n-alkanes or free monocarboxylic fatty acids); (d) Ki = K , # 1,R = 0: two uncoupled chains of N atoms with a heavier mass at both ends (e.g., isolated a,w-dicarboxylic acids); (e) Ki = 1, K, > 1,R < 1 ( # 0): dimeric molecule with 2N atoms with heavy ends and weakly coupled internal groups (e.g., monolayered phospholipids, biomembranes, etc.); (f) Ki e 1,K , = 1,R < 1 (#O): dimeric molecule with 2N atoms with two heavy and weakly coupled internal groups (e.g., dimeric fatty acids). 2.2. Infinite Systems. In section 2.1 we have discussed the case of isolated molecules while in most real systems trans planar molecules pack in a tridimensional lattice and interact through intermolecular forces. For the dynamics of LAM modes, which is the object of this paper, only interchain longitudinal forces are to be taken into account. For simplicity we treat here only the case of a one-dimensional and infinite crystal whose repeat unit is a single and symmetric chain as shown in Figure lb. In this class of systems one can find such typical cases as solid-crystalline n-alkanes and solid-crystalline a,w-dicarboxylic acids. n-Alkane molecules are subjected to weak vander-Waals type forces while a,w-dicarboxylicacids are held together by stronger forces originating from the hydrogen bonds. Let 1 and s label the unit cell and the position of the atoms in the unit cell, respectively; in the harmonic approximation, with interactions to the first neighbor, the equations of motions are the following:
Mfl,i = F(xl,2 - XL,J mil, = F(xl.,-, - xl,J
+ g(xl-1a - X L J )
+ F(xl,s+l- XI,,)
(9)
= 2, N - 1 (10) (11) + g(Xl+l,l - x i s ) s
M f ~ a= F(xla-1 - x l a ) where M and m are the masses of the end and inner atoms, and F and g are the inter- and intramolecular force constants, respectively. Bloch’s functions = xo+ exp(ikll) (12)
+
(L= ( N - 1) d d , ) being the distance between lattice points satisfy the translational symmetry of the system. Substitution of eq 12 into eq 9-11 yields M f l = F(x2 - x , ) m f , = F(x,-, - x,)
+ g(e-ikLxN- x , )
+ F ( X , +-~ x,)
s = 2, N
(13)
-1
(14)
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982 4793
End Effects on Longitudinal Accordion Modes
M i N = F(xN-’ - X N ) + g(eikLx,- x N )
(15)
lo
Looking for solutions of the type x , = ( A sin sd + B cos sd)e-i“t for eq 14 one obtains the dispersion relation w2 = ( W / m ) ( l- cos 29) where 9 is the “molecular phase”. From the eq 13 and 15 one obtains, by equating to zero the determinant of the coefficients of A and B, the following equation for 0 : sin ( N - 1)9(cos 29 - 1)[(2K- 1 - 2 p ) cos 19 + 2@ 2K + G - 2GK] cos ( N - l ) d sin d [ ( l - 2K)(cos t9 1) - GI + G sin d cos kL = 0 (16)
220
+
I
( K and G are respectively Mlm and g/F.) From eq 16 an implicit relation is obtained between d and the lattice wavevector k. This equation has N solutions in 29 for each k. From the dispersion relation w2 = (2F/m)(l - cos 8 ) each of these solutions gives a point on the N dispersion branches w = w ( k ) of the one-dimensional crystal. The optical modes of the crystal are those with k = 0 and differ in frequency from those of the “isolated molecule” (G = 0) by an amount due to the “interlamellar” forces. 3. Numerical Calculations The theory discussed in section 2 can be summarized in a few basic concepts; the basic ingredient is knowledge of the longitudinal acoustic branch of the dispersion relation of the infinite chain. One moves with the “molecular phase” along this branch as a function of the external parameters which represent the perturbation (i.e., Ki, K,, R, K , and G). The frequency thus turns out to be function of the same parameters. The translation of the above theoretical concepts into a numerical treatment requires some decisions. Strictly speaking the dispersion relation 7 and eq 8 and 16 for d should be considered together to keep internal consistency in the treatment. In practice, we keep eq 8 and 16 as they are and, instead of assigning suitable values to F and m appearing in eq 7, we replace eq 7 with an expression for the dispersion curve w5 as given by Shimanouchi and Schaufele (SS) for polyethylene.’ We consider such an approximation useful, handy, and more than acceptable since the SS function very closely approximates eq 7 in the range of applicability. The SS function originally is given as a polymonial expression in terms of r l n , where r is an integer and n is the number of carbon atoms of the n-alkane molecules whose Raman spectra have been studied by SS. However, we consider the SS function as a continuous function of 8 = d / r . We are aware that the SS function, since it has been derived from Raman-active LAM modes of finite crystalline nalkanes, is intrinsically affected by the “interlamellar forces” which are activated during the odd-order accordion motions. Thus the S-S function deviates from the true dispersion curve of infinite polyethylene. We neglect this fact since in this work we are not interested in the detailed values of the elastic constants and the derived quantities, but we aim at disentangling some dynamical features which do not crucially depend on such small details. Within this approximation we can estimate the value of Flm by equating for small 9 eq 7 reexpressed as u (cm-’) = [l/(rc)](F/m)1/2 sin ( 8 r / 2 )with the S-S function v ( 8 ) = A 8 + Ba2 + Cd3 + Db4 Ea5 La6
+
+
A = 2495 cm-l, B = -5867 cm-’, C = 62530 cm-l, D = -348.500 cm-’, E = 732.900 cm-l, and L = -472.400 cm-l. Taking m as the point mass of the CH2 group one obtains F = 4c2A2m 5.2 mdyn/A. Such a value of F is of the order of the diagonal force constant for a C-C bond in a valence force field, thus showing that the model of a linear
-
I6O
0
t
0 O
O
O
M
-00
Flgure 2. LAM 1 frequency of monosubstituted ndecane as function of the mass of the heavy end group (-CH,X).
b)
e
e
e
e
e
e
e
e@--.@*
e
e
e
e
e
a
e
Flgure 3. Sketch of the molecular model of (a) case e of section 2.1 and (b) case f of section 2.1.
chain does not substancially change the nature of the bonding from the dynamical viewpoint. Equations 8 and 16 have been solved numerically by computer. 4. Cases Treated 4.1. Finite Molecules. The various cases listed in section 2.1 are treated below; however, some of the cases are trivial or some provide information which has been already obtained by other authors4i5on the basis of the elastic rod model. Thus we shall mainly focus on the cases of dimeric systems for which we have also obtained some experimental results in order to support the theoretical predictions. 4.1.1. Single Uncoupled Chains. Cases a and b of section 2.1 refer to sinngle uncoupled n-alkane chains of length N and 2N, respectively, for which eq 8 becomes very simple and has solutions for 0 = r ( r / n ) (where r is an integer and n the number of C atoms). Obviously, the corresponding frequencies are those observed and used by SS.’ Case c describes the effect of LAM of a heavier mass at one end. LAM frequencies decrease by increasing the mass of the end group. Figure 2 gives, as an example, the dependency of LAM I frequency for N = 10 (e.g., a or w monosubstituted decane). The asymptotic limit indicated in Figure 2 is reached when Ki(or K,) tends to infinity and corresponds to the case of a chain with one fixed end (xl or x N = 0). To our knowledge reliable experimental data are not yet available for checking such a prediction. Case d is treated in section 5 of this paper in a more general way. 4.1.2. Systems with Two Coupled Chains. We discuss in this subsection the cases given in parts a and b of Figure 3 which correspond to cases e and f of section 2.1. Some dynamical features are common to all the systems which can be collected in this class independently of the location of the heavier mass or of the strength of the spring constant of the bond joining the two chains. Thus, for sake of simplicity, let us consider first a system of two uncoupled chains of identical atoms (i.e., R = 0, Ki = K , = 1). Each of the noma1 modes of the system is doubly degenerate and each chain performs ita rigid translation, as well as its LAM modes, unperturbed. When interchain coupling is switched on, each doubly degenerate vibrational
4794
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982
Minoni and Zerbi
Eigenvectors in Internal Coordinates for LAM 1 of a Dimer Molecule (2N= 3 0 ) as a Function of R = f / F a
_-TABLE I:
c-c
external C-C
terminal
0 . 6 ~ 1.4X 2.0X 10-4 10-4 10-4
2.7X
0.6X
2.6X
1.3 X
10-3
3.3X 3.8X 10-4 10-4 10-4
1.9 X
10-3
10-3
10-3
R= 4.7X 10-4
v = 9.1 cm-’ 5 . 2 ~ 6 . 1 ~6 . 6 ~ 7 . 2 ~ 8 . 0 ~ 8 . 6 X
10-4
10-4
10-4
10-4
10-4
10-4
9.4X
1
10-4
R = lo-’, v = 27.1 cm-’ 3.8 X 4.4 X 5.0 X 5.8 X 6.3 X 6.9 X 7.5 X 8.0 X 8.6 X 0.9 10-3 10-3 10-3 10-3 10-3 10-3 10-3 10-3 10-3 10-3
3.3 X
R = lo-’, v = 61.6 cm-’ 1.7A 2.0X 2.4X 2.7x 3 . 0 ~ 3 . 3 ~ 3.6X 3.8X 4.1X 4.4x 4 . 7 ~ 0.5 10-2 10-2 lo-’ 10.’ 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2
0 . 4 ~ 0 . 8 ~1.2X 10-2
10-2
0 . 0 8 x 0 . 1 5 ~ 0 . 2 3 ~ 0 . 3 0 ~ 0.36X 10-l lo-] 10.l 10-l 10.’
R = 1, v = 78.6 cm-’ 0 . 4 4 ~ 0.50X 0.55X 0.61x 0 . 6 4 ~ 0 . 6 6 ~ 0 . 6 9 ~ 0 . 7 2 ~ 0 . 7 5 ~ 0 . 7 6 ~ 10-l 10-l 10-1 10-1 10-1 10-1 10-1 10-l 10-l 10-l
The numbering starts from the outside of the molecule and values are reported for half of the molecule; by inversion through the center of symmetry on the 15th bond the other half can be obtained.
‘ ‘
LAM 7 LAM E
=:::_I_
LAM3
deg
LAM2
deg
LAMI
deq
LAM 3 ,-I--LAM. 2 ..
deg 2
,:I.-
_.Z
