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Determination of molecular optical anisotropy in solutions liquids by depolarized light scattering
Determination of Molecular Optical Anisotropy in Solutions and Liquids by Depolarized Light Scattering Applications to the Study of N-Alkanes P. B O T H O R E L
[nstitut de Magndto-Chimie, "Brivazac," 33-Pessae, France Received February 1, 1968 The molecular optical anisotropies (./2) of n-alkanes were calculated using the theory of "optical bonds." A model which considered the trans and gauche isomers of rotation was used• The comparison of the theoretical values with those obtained experimentally in pure liquid and solutions by a recent technique, viz., depolarized light scattering, permitted the determination of the differences in energy between the isomers with satisfactory precision. This comparison suggests that there exists a correlation of orientation between elongated chains ("zig-zag") in these liquids; this ordering between the chains at short distances could play an important role in determining the properties of compounds in which they are found, for example fats. INTRODUCTION AND EXPERIMENTAL METHODS D u r i n g the last few years, we have developed a technique for measuring molecular optical anisotropy of pure liquids and their solutions. W e are going to review briefly the principal elements of our technique and indicate certain details of the technique which up to now were unpublished• I n order to do this, it is necessary to use the classical system presented in Fig. 1. A m o n o c h r o m a t i c , parallel, linearly-polarized light b e a m traverses the cell C which contains a sample of liquid. T h e wave scattered by the liquid in the direction OP, with the same wavelength as the incident b e a m (Rayleigh effect), can be decomposed into two vibrations, perpendicular to one another, of intensities i and I. One can isolate a depolarized vibration of intensity i, with the aid of a Glazebrook prism (G). T h e intensity i of this light b e a m scattered b y the sample is called the "depolarized" intensity. T h e electric field E of the exciting wave creates in each molecule an induced m o m e n t 529
M ( 1 ) ; the two vectors can be referred to the Cartesian coordinate s y s t e m X Y Z (Fig. 2). Molecules in general are anisotropie, in other words M and E are not parallel. If the electric field is n o t too strong 1 and if the wavelength h is small c o m p a r e d to the electronic absorption bands of the molecules being studied, 2 one can say t h a t the induced m o m e n t is proportional to the exciting field,
M = 1~ JE
Ill
where ] a ] is the molecular polarizability tensor,
loll
=
Olxx
OLxy OLxz
O~yx
Ogyy O~y~
Otzx
Otzy
OLzz
OQj --~ O~ji
• (i ~ j ) ;
[2]
1 This is the case if we use a classic source of light (mercury arc, xenon arc), a gas laser, or a Q switched laser of medium power (2). 2 Chemical compounds which are transparent to visible light can be studied for example with the aid of light from a mercury arc (k = 5.461 A). Journal of Colloid and Interface Science, Vol. 27, No. 3, July 1968
530
BOTHOREL
[V
A
p is t h e n u m b e r of molecules in each u n i t of s c a t t e r i n g v o l u m e V. M o l e c u l a r o p t i c a l anis o t r o p y 3,2 is a n i n v a r i a n t p r o p e r t y of t h e t e n s o r of p o l a r i z a b i l i t y , a n d t h e r e f o r e i t does n o t d e p e n d u p o n t h e o r i e n t a t i o n of t h e molecule. T h e 72 is expressed b y :~
c
2
= ½[(~= - ~ ) ~ + ( ~
- ~)2
+ ( ~ - ~=)~] 2
[5] 2
+ 3(~L + . ~ + .z~). If we call R~ t h e d e p o l a r i z e d i n t e n s i t y scatt e r e d b y a u n i t of v o l u m e a n d i n c i d e n t p h o t o m e t r i c i l l u m i n a t i o n , one sees t h a t
FIG. 1. "Depolarized" scattered intensity.
Z
i Ri -
VE2/2 -
32~ 4 p ( 2 + 135~ 4
2)2 2
One can easily show t h a t R i has t h e s a m e v a l u e w h e t h e r t h e i n c i d e n t light b e a m is p o l a r i z e d o r u n p o l a r i z e d . I f t h e fluid is a solution, one can w r i t e t h e general equation (7),
Y
i~oj~tio~ -
~x
16~
1~
4
~ 2
~ p~n
+ 2)23,2VE 2
co {2 + 2 )2
[7]
+ .oOYo\~o~ 7~ ~ '
A
(j)
[6]
or
~IG.
2.
M is sinusoidal a n d has t h e s a m e w a v e l e n g t h as E. A s s u m i n g t h a t t h e molecules are H e r t z oscillators, one can show t h a t t h e periodic i n d u c e d m o m e n t s c r e a t e scatt e r e d w a v e s of t h e s a m e f r e q u e n c y as t h e i n c i d e n t l i g h t (3). A d m i t t i n g t h a t t h e electric field F w h i c h effectively excites t h e molecules is a L o r e n t z field d e s c r i b e d as,
F = E (n~
+ 2) 3
[3]
w h e r e n is t h e r e f r a c t i v e i n d e x of liquid a t w a v e l e n g t h X, one c a n easily r e l a t e t h e dep o l a r i z e d i n t e n s i t y i s c a t t e r e d b y t h e ens e m b l e of t h e molecules to t h e p o l a r i z a b i l i t y of t h e molecules ( 4 ) . I f t h e fluid is pure, t h e d e p o l a r i z e d i n t e n s i t y is d e s c r i b e d as, i -
16~ 4
~
~ 2
p(n
+ 2)Ly2VE2;
[4]
Journal of Colloid and Interface Science, Vol. 27, No. 3, J u l y 1968
Risolution
32~r4 i 2 - - _l ~_4 p ( n -+ 2)~/2
c0 (n2 + 2 ~ 2 + Rio° ~ \ ~ / ,
IS]
In Eq. [5] the choice of the directions of the coordinate system X Y Z is unimportant. For a particular reference system (xyz) we have: a]~ = 0 (i # j ) ; in this case, the molecular optical anisotropy ~2 takes a simpler form: 73 = ½ [ ( ~ _ ~ ) 2 + ( ~ _ ~,o)2 + ( ~ _ ~ ) ~ ] ; in this equation a ~ , ay~ , and a~ are the principal optical polarizabilities of the molecule. This equation has been used for a long time to define 73; it is less general than Eq. [5]. This Eq. [5] has many advantages in the interpretation of molecular optical anisotropy with the aid of the theory of optical bonds, as shown below. In our previous work we used a parameter (A2RM2) which resembles 73 the following equation (4-6):A2RM = (16~2N272)/81; N is Avogadro's number.
MOLECULAR OPTICAL ANISOTI~OPY AND N-ALKANES p is the number of molecules of the solute per unit of scattering volume V; i0° is the intensity scattered by the pure solvent under the same conditions of study as the solution. The expression between brackets multiplied by i0° includes the variations of the number of molecules of the solvent and of the Lorentz field when one passes from the pure solvent to the solvent in the solution. The Co is the number of grams of solvent per milliliter of solution; do is the density of the pure solvent; no is the refractive index of the pure solvent. The Eqs. [6] and [8] show that it is possible to determine the optical anisotropy ~2 of a compound in liquid form or in solution by measuring the intensity Ri scattered by these liquids. 4 In practice these measures are relative. One actually measures the ratio ( R i / R i o = i/io) of the depolarized intensities scattered under the same experimental conditions for the sample (i) and for a standard liquid (i0). We have chosen cyclohexane as a standard and determined the intensity Rio in the following way: The Rayleigh constant R is defined as the total intensity scattered per milliliter and unit of incident photometric illumination (2). The absolute Rayleigh constant for benzene in natural light has been measured numerous times ( l c ) . The majority of recent experimental values approximate the value of Carr and Zimm (8): R~ = 16.3 X 10-6 --1 cm (X = 5.461 A). Therefore we adopted this value for our work. Measuring the total scattered intensities under the same experimental conditions for benzene (I~) and for a standard liquid (I0), the Rayleigh constant can be determined for that standard Remember that the anisotropy, 7~, of a perfect gas can be determined by measuring the polarization ratio (o = i/I) and the molecular refraction. This mode of determining v2 is not applicable to dense fluids, pure liquids or solutions (la) [in the case of pure liquids, it is also necessary to know the coefficient of isothermal compressibility, for example].
(R0):
(,0)
R0 = R~ ~r~
531
[91
We have measured the ratio Io/I~ numerous times with the aid of the light-scattering-photometer of Wippler and Scheibling (9). The value which we obtained for cyclohexane was Io/I~ = 0.271, after the scattering volume correction was made. After measuring the depolarization ratio, po, of the light scattered by cyelohexane with an unpolarized incident beam, with a wave length of 5461 A (po = 0.0553), one can calculate the constant Rio (4): R/° - 1 + p0
1 + p0
[10]
= 23.15 X 10-s cm-l(h = 5461A) Using this value, Ri can be determined for any liquid simply by measuring ?the ratio of the depolarized intensities, i and i0, scattered under the same conditions b y the liquid and by cyclohexane. Then R i = Rio(i/io)
[11]
The molecular optical anisotropy of the compound can then be calculated using Eqs. [6] or [8]. The precision obtainable for v 2 depends mainly on the precision of the measurement of the ratio i / i o . The systematic study of the molecular optical anisotropy of chemical compounds by this technique is not practical unless 72 is known with an experimental error of a few percent. With this end in view we conceived and built apparatus for making relative measurements of depolarized scattered intensity (5, 10, 11). The precision of i is 1-2%, this gives an error of about 3-4 % on v 2 in the majority of cases where compounds studied were liquids or compounds in dilute solution (in certain cases one can use solutions of a concentration of 10 4 tool/liter). Thus we studied a variety of organic compounds such as simple or substituted alkanes, polyethers, f a t t y acids, pyrimidines and purines, a number of important aromatic compounds, and others (12). Journal of Colloid and Interface Science, Vol. 27, No. 3, J u l y 1968
532
BOTHOI~EL
expressed by convention as
Z
%- = a , j -
Y
X
The measure of depolarized scattered intensity i and the interpretation of the optical anisotropy 2 as determined from i, we have called Depolarized Rayleigh Scattering (in French: Diffusion Rayleigh D6polaris6e or DRD). Interpretation of Molecular Optical Anisotropy with the Aid of the Theory of "Optical Bonds." Since the work of Meyer and Otterbein (13a), one attaches to each bond j in a molecule a principal tensor of optical polarizability (see Fig. 3). The molecular polarizability tensor is obtained simply by combining the principal optical polarizaJ bilities of the bonds ( o J.~, ayy, a ~Y ) :
o
0
0
J
= d zz
=
Ol± J,
"
5=jx ~vix
Z
5~ r ~v.~Y
~.j z ~v+z
One obtains each component of the molecular tensor using the projections of the principal optical polarizabilities of the bond on X Y Z
:xx = ~ ~j ~ + : x + ~ ~ ~y+:X J
J
+ ~J ~+~zj:X
[15]
Using the relations between cosine directions:
= 1;
and equation [14]:
;
~ ~ £~xx
Y
~,x + ~+~ + ~
These polarizabilities can be projected on the reference system X Y Z attached to the molecule according to the rules applicable to a second degree tensor. If the bonds of the molecule have a symmetry of revolution about their axis, the following simplification is possible: ~ O~yy
xj YJ
o l :z
[14]
If the molecule studied contains only bonds with a symmetry of revolution, the molecular optical anisotropy can be calculated very easily using the principal optical polarizabilities of the bonds or rather, their optical anisotropies (14, 15). Taking the orientation of the bond j with reference to the coordinate system X Y Z one can with the aid of the table of cosines determine their directions.
FIG. 3.
<=
aft'. 6
(EiJ +,
O!XX =
E
2
~
*"
[16]
Then one obtains analogous expressions for a r t and azz ; and also: J
+~
J
;
J a±~.jxS.jr
[17]
[12]
and the optical anisotropy of the bond be-
= Z ~J ~xjx ~ Y .
comes "Tj2 =
(Or[I~ - - a ± ~2.~ ] ,
[13]
T h e d e f i n i t i o n of m o l e c u l a r o p t i c a l a n i s o t r o p y
given by Eq. [5] applies also to an isolated bond. Journal of Colloid and Interface Science, Vol. 27, No. 3, July 1968
T h e a bonds p r e s e n t a case of t h i s s y m m e t r y of revolution a n d h a v e in general a m a x i m u m of polarizability along t h e l e n g t h of t h e i r axis a n d c o n s e q u e n t l y h a v e an optical a n i s o t r o p y ~/j > 0.
MOLECULAR OPTICAL ANISOTROPY AND N-ALKANES CHa
CH3
/
cf Hs H3 FIG. 4. Isomers of rotation of n-butane.
U
tropy ofn-Alkanes. In our study of n-alkanes, we assumed that all bond angles are tetrahedral. We also supposed, according to the theory of isomers of rotations, that each molecule of alkane can exist in various conformations at a given temperature. Each conformation corresponds to a minimal value of internal energy U of the molecule. It is known, for example, that three stable conformations of normal butane exist (see Figs. 4 and 5) ; there is one trans form (9 ~ 0 °) and two gauche forms g+ and g- (9 ~ -4-120°) (16). The probability of existence of the optical antipodes, g+ and g-, is the same, but it has a value less than that of the trans isomer. If, b y an appropriate technique, one determines the percentage of trans isomer (xt) and gauche (xa+ = xg- = xa) at absolute temperature T, one can write xo _ exp
120 180
(g-) (r)
(g+)
F~G. 5. Potential energy of isomers of rotation of n-butane.
2 • ~ / i ( ~2
-
~2) ] 2
+ [E ~ i ( ~ + [ E ~i ~
- ~,~)]~I
i
i
AG
=
AN
i
+ (~ ~ ~
#~,~)~].
TAS. 7
/kU = AU0°;
xj _ exp
ac00~ .
z~
R-T
[19]
~)~
--
[211
[22]
AU0° is the difference of internal energy of the two isomers at 0°K. Under these conditions
÷ 3[(~--~ ~,i ~x~x fixer) ~
+ (~2 ~ ~ i
"
The difference of free enthalpy AG between the isomers can be written in the form:
/kS = 0;
~~) ]
-
[20] ~
Since the structures of the two isomers are very similar, one normally supposes that their partition functions are equal, that is to say
and for equation [5], 2= 1
-
Xt
Au:
533
i
Therefore it suffices to know the orientation of each bond and its optical anisotropy in order to calculate the molecular optical ~aisotropy. The normal alkanes con_~in only simple C - C and C - H bonds and can be studied utilizing Eq. [19].
Interpretation of Molecular Optical Aniso-
[23]
/
One can determine A U0° by quantitative analysis of isomers of rotation. As a first approximatioa we assume t h a t the conformations of the alkanes-other than 7 If normal butane is studied in solution,-u t e r m of free enthalpy of solution must added to the equatioa, If the interactions between solvent and solute are weak, this t e r m is obviously negligible. We have assumed t h a t this interaction is indeed negligible in the case of normal alkanes of chain length longer t h a n b u t a n e in carbon tetrachloride and cyclohexane solutions. Journal of Colloid and Interface Science.
Vol. 27. No. 3, July 1968
534
BOTHOREL
butane can be obtained by rotation through the position trans (~ = 0 °) or gauche (~ = ± 1 2 0 °) about each C-C bond (12a). In order to pass from a conformation entirely trans, which we call Z chain (zig-zag), to a gauche form, r rotations are necessary, and the ratio of percentages of the two forms is equal to (12f) x(gauche chain)
x(Zchain)
( = exp
~o) - r--
TABLE I COSINE DIRECTIONS FOR CARBON-CARBON ]~ONDS IN ALKANES No.
X
Y 1
1
1
1
]
1
1
.
-~/~
[24]
This relation is valuable only if one ignores Van Der Waals interactions between methylene groups in the same chain. If one considers these interactions, it is necessary to add a supplementary term to r A U0°.
Molecular Optical Anisotropy of an Isomer of Rotation. If an isomer of rotation of an alkane satisfies the preceding conditions (tettrahedral bond angles, ~ = 0, q-120°), it is easy to determine the orientation of all the bonds. As demonstrated by Smith and Mortensen (15, 17), the bonds are necessarily parallel to the four diagonals of a cube, take for an example the diamond crystal. Take the coordinate system X Y Z of Fig. 6. If one orients it in such a way t h a t the bonds between three consecutive methylene groups are parallel to the demi-diagonals 1 and 4, then all the C-C and C - H bonds of the molecule are oriented parallel t o the directions 1, 2, 3, a n d / o r 4. The direction cosines are then given by Table I.
+~
1 ~/~
4
Z
1
1
-~/~
1 ~/~
q-~
1
Then Eq. [19] can be used to obtain the molecular anisotropy of the rotational isomer under consideration (12a, 17). 3".
2
=
4 2 [4/~(nl + --
l~(n
n2
2
--
+
n3
2
1)2]r2 =
+
n42)
[25]
A- ~ P2 ,
where n is the number of atoms of carbon in the alkane; nl, n2, m , and n4 are the number of C-C bonds parallel to the directions 1, 2, 3, and 4. The 17 is the only parameter of the optical anisotropy actually necessary for calculating 3'2. I t is expressed by: P = 3"c -- 23"~,
[26]
where 3"e is the optical anisotropy of the C - C bond, and 3", is the optical anisotropy for the C - H bond. The A~2 (Eq. [25]) depends only upon the conformation of the rotational isomer.
Average Molecular Optical Anisotropy 3"~2 of an Alkane, General Formula C~H2,,+2. An n-alkane exists at normal temperatures as a mixture of isomers of rotation. The experimental molecular optical anisotropy calculated from Eqs. [6] and [8] is therefore an average value @7n2). If one considers the isomers of a particular molecule as independent scatterers, the scattered intensities are additive s and each of the isomers con-
FIG. 6. D i r e c t i o n s of b o n d s in an alkane.
Journal of Colloid and Interface Science, Vol. 27, No. 3, Ju]y 1968
s Since t h e v a l u e s of i, a n d t h e r e f o r e v 2, are additive, one can calculate t h e i r a v e r a g e v a l u e if one t a k e s into a c c o u n t t h e s t a t i s t i c a l w e i g h t of each c o n f o r m a t i o n of t h e molecule. On t h e o t h e r h a n d , t h e p o l a r i z a t i o n r a t i o p is n o t a n a d d i t i v e
MOLECULAI~ OPTICAL ANISOTlZOPY AND N-ALKANES • (A~)
[ 1,.~
I
I
o'
I
~)o o
I I
5
10
"15
ooo
20
o n ~'
FIG. 7. 3,Iolecular optical anisotropies, %2, divided by number of "chainons" (n -- 1). Experimental curves: ~--~ , pure liquids; ~ , cyclohexane solutions; > <, carbon tetrachloride solutions; theoretical curves: (O); (the length of the bars of arrows indicates experimental error).
535
n-alkanes (4 =< n =< 24) as pure liquids or in solution with various solvents at r o o m temperature (12a) (see Figs. 7 and 8). O t h e r s u p p l e m e n t a r y values have been o b t a i n e d of late (12e). T h e experimental error is 3-4 %. If one chooses the proper values for AU0° and r with the aid of Eqs. [25] and [28] one can a t t e m p t to a p p r o x i m a t e theoretically the curves of the experimental variation of if2 as a function of n. We h a v e m a d e simple calculations of this sort only with alkanes f r o m b u t a n e to decane (12a). U n f o r t u n a t e l y the n u m b e r of calculations, which is proportional to the n u m b e r of conformations, (there are 3 ~-3 isomers of r o t a t i o n per alkane)
tributes to .~2 as a function of its own optical anisotropy, ~2, and its percentage xj (see Eq. [24]). --2
2
~[n = "Yn2(t) ~- E Xj "Yn (gi) J
7~2(t) of the optical bining
is the molecular optical anisotropy Z chain; %~(gj) is the molecular anisotropy of a gauche isomer. ComEqs. [25] and [27] one obtains,
-;J = r ~ Z
x(~XUo°, T)A~ ~ = r ~ X J .
[2s]
isomers
After calculation of the molecular optical anisotropies of all the rotational isomers present, the average molecular optical anis o t r o p y can be calculated for a n y n-alkane at t e m p e r a t u r e T as a function of the two parameters r and A U0°. RESULTS AND DISCUSSION
Interpretation of Experimental Molecular Optical Anisotropies of n-Alkancs. I n the past we have determined the molecular optical anisotropies of a n u m b e r of i m p o r t a n t function, the "average," ~, calculated in this manner does not correspond to experimental values of o [see the article of Stein (15)].
FIG. 8. Experimental molecular optical anisotropies if2: (L) pure liquids; ($I) carbon tetrachloride solutions, (S~) cyclohexane solutions; ($3) pentane solutions; ($4) heptane solutions. Theoretical curves: (T1) AU0° = 600 cal tool-1 (model of "permitted conformations"); (T~) AU0° = 600 cal tool-1 (model of "all conformations"); (T~) AU0° = ~o (Z chains independent). Journal of Colloid and Interface Science, ¥o]. 27, :No. 3, J u l y 1968
536
BOTHOREL
becomes unwieldy by the time n reaches 10. Even if one uses a computer, the time required for the calculations becomes prohibirive when n exceeds 15. It seemed, therefore, that this method could only be utilized for short chain alkanes. However using the IBM 7044, we have been able to make the calculations for chMns containing up to 50 carbon atoms in a reasonable time after taking into account the rules of recurrence (12e) (see "Appendix"). We have calculated if2 considering all permitted conformations (the model of "all conformations"). However in the case of those isomers which have two gauche forms of opposite sign (g+, g- and g-, g+) when one makes a rotation about two consecutive C-C bonds, the probability of existence of these forms is very low due to the fact that in order for them to exist it would be necessary to have a superposition of CH2 groups (the distance between the two CH2 groups in this isomeric form would be 2.5 A, whereas, the normal Van Der Waal's distance would be 4 A). It is for this reason that we have eliminated these conformations from our calculations (model of "permitted conformations"). In other words, to eliminate these conformations is the same thing as saying that there exists a very large A U between them and the Z chains. This model gives good agreement between the theoretical and experimental values of if2 for n-alkanes with up to 13 carbon atoms, if the theoretical values for G and AU0° are r = 0.87 ~ 0.02 ~3 ; hU0° = 600 ± 50 cal tool-1. The optical anisotropy parameter r appears not only in alkanes but also in a number of other organic compounds. That is the reason why its determination has been the object of much work (18). The value of F determined in the course of this study approaches the values proposed recently by other authors but is more precise. The utilization of this parameter in calculation of molecular optical anisotropy of orJournal of Colloid and Interface Science, Vol. 27, No. 3, July 1968
ganic compounds other than n-alkanes ought only to be attempted with caution. The value given above for F is for a straight chain alkane and is related to the C-C and C-H bonds of such a molecule. The values for branched and cycloalkanes are different (12a, 19). The value for A Uo° which determined by DRD also agrees well with those values obtained by various authors (400-700 cal tool-1) whether obtained experimentally (12a, 20, 21) or theoretically by using empirical potential energies of interaction (22, 16b). Note that the theoretical %2 of long chain molecules is proportional to n(12f) (see Fig. 8) and can therefore be approximated by the following expression: _
2
7~ = 0.9773 n -- 2.4432 ~6 (permitted conformations) 9
[29]
The molecular optical anisotropy calculated using the above relation differs less than 1% from that obtained by rigorous calculations in which n is greater than 8 and by less than 10/00 when n is greater than 15 (F 2 = 0.75). When the number of carbon atoms exceeds 13, there is a discrepancy between the theoretical and calculated values for ~.~ in carbon tetrachloride and cyclohexane solution (see Fig. 7). This discrepancy grows with each increment in chain length. One is tempted to blame the discrepancy oll the configurational model chosen. Some author (16b, c, 21, 22) give to the conformations containing the double rotation g+, g- and g-, g+ a statistical weight different from zero (AU = 2.500-2.800 cal tool-1 instead of infinity). On studying theoretically the potentials of interaction between all atoms in the alkanes with chain length in excess of C4, these authors (see above) have admitted the existence of two 9 Certain authors (23-25) have calculated the molecular optical anisotropy of a polyethylenic chain, but t h e y considered the statistical orientation of each C-C bond in the molecule. The m o s t recent works allow a comparison between these theoretical values and our experimental values.
MOLECULAR OPTICAL ANISOTROPY AND N-ALKANES supplementary conformations (~ = 4180-95°1 and also ~ = 0 4- 120°; these 2 new gauche forms have the same statistical weight: AU = 2000 cal mole -~) (26). Unfortunately these modifications bring about a statistical model which we used accentuates the discrepancy between the theoretical and experimental ve~lues in solution of %2 as a function of n. In effect these modifications tend to diminish the average molecular optical anisotropy beeause they a-L~ribu~e to these gauche forms which have small anisotropies, a probability of existence superior to that which we have attributed to them. However if one hypothesizes an organization of Z chains in solution of long chain alkanes, one can on the contrary explain the apparent discrepancy. Now we are going to discuss in detail this hypothesis in the case of pure liquids. Pure Liquids. Since the molecular optical anisotropy of an alkane is very different in the liquid state and in solution, the values of r and AUo° which we gave before do not predict the correct experimental curve for _
2
537 co
1500
10
800 600 400
0 cal. mole -1
=
Experimentally it was noted that: (i) values of ~7~2 arc systematically higher in the pure liquids than in solution. (ii) no value of the parameter AUo° permits explanation of the experimental variation of ~Tn2 (see Fig. 9 : the theoretical ratio ~7~2/~42 = X ~2/I A 42 depends only upon AU0°). The average slope of the experimental curve is greater than that of the theoretical curves. (iii) the experimental curve (L) of pure liquids of Fig. 8 falls between the theoretical curve (T,) for a statistical mixture of isomers and the theoretical curve (T3) which one would observe if the pure liquid contained only Z chains which are not in any way oriented one to the other. The above observations bring us to the conclusion that in pure liquids the apparent percentage of isomers which are highly anisotropie is greater than that which one would predict from the above statistical model. In other words there is a higher percentage of Z chains, which are the only isomers with an
5
10
15
n
FIG. 9. R a t i o of a v e r a g e e x p e r i m e n t a l m o l e c u l a r optical a n i s o t r o p i e s , ~7~2/ff~~, of alkanes to b u t a n e in liquid s t a t e (L) a n d t h e t h e o r e t i c a l ratios (-7,~/ff4 2 = X.~/5_~2) as a f u n c t i o n of v a r i o u s v a l u e s of AUo °.
anisotropy sufficiently superior to the average to give the observed effect on ~ experimental as opposed to ~2 theoretical. Consequently we have proposed the following model: In liquid n-alkanes, the gauche isomers have no correlation of orientation, they simply arrange themselves according to the simple statistical equilibrium described above. However Z chains have a tendency to orient themselves parallel one to another. This state of orientation will have a probability of existenee superior to that of Z chains with noneorrelation of Journal of Colloid and Interface Science, ¥ol. 27, No. 3, July 1968
538
BOTHOREL
orientation one to another. These Z chains have an optical polarizability a~ in the direction of elongation superior to that of other isomers. And also there is possibility of the chains approaching one another at a short distance d. These two conditions, al and d, enable the London forces between two associated Z chains to be maximum [the London forces are proportional to the ratio 2 7 al/d (16)]. Therefore the probability of existence of two associated Z chains is greater than the probability of existence of the two chains in the free state. In the crystalline form of normal alkanes only the Z chains are oriented parallel one to the other (27-29). In compounds such as f a t t y acids, soaps, and crystallized fats there are carbon chains bound to groups which encumber rotation. These attractive forces are sufficiently strong to cause organization of the alkane portion of the molecule. I t is obvious that the order observed in the crystalline form of alkanes is not preserved in the liquid form (28). However the broad band centered at 4.5 A in X-ray diffraction patterns of liquid alkanes has been interpreted as a tendency toward organization in which the chains are spaced approximately 5 A apart (28, 30). And also the absorption of ultrasonic waves by these liquids identities them as being "associated" (31). Bhattacharyya et al. (32) have studied thermodynamically mixtures of n-alkanes. In order to explain their results they have proposed the existence of packing in the liquid on the trans-gauche population. We think also that the abnormal variation of the anisotropy qn2 of decane liquid as a function of temperature could easily be explained by the existence of ordering at short distances between Z chains which diminishes or is destroyed by increase in temperature (10). If we make the assumption that in liquid alkanes Z chains are organized and gauche forms are free, then by comparing the values of q 2 of the curves (T1) and (L) in Fig. 8, it is possible to estimate in a crude way the percentage of Z chains actually in conJournal of Colloid and Interface Science, Vol. 27, No. 3, J u l y 1968
formity with the model 10 (i.e., assumption above). In order to explain experimental results in the pure liquids of carbon chain length greater than 5, a percentage of 1015 % of Z chains is sufficient. In other words the proportion of molecules of alkanes which are organized is greatly inferior to the number of molecules which are in a disordered state. I t is also possible to explain the divergences observed between the values of ~2 of the theoretical curve (T1) of Fig. 8 and the experimental values in various solvents with the aid of the model described above. I t seems that when the solvent is a normal alkane, one has an ordering at short distances between Z chains of the solvent and of the solute. The effect will be greatest when the chains of solvent and solute are of approximately the same length. Therefore, since the alkanes studied were for the most part of chain length > 5 one would expect a decrement of ~2 (i.e., a decrease in the percentage of Z chains). When one goes from a solvent system C7 to C5, one can see this effect (see Fig. 8). Also when one looks at the curves of alkanes in cyclohexane and carbon tetrachloride and compares them with the theoretical curves, one can see that a partial ordering still exists in these solutions. An estimated percentage of 1% of the Z chain is sufficient to explain the variation of experimental values of ~n2 from the theoretical in these solutions. One can explain the decrement of if2 when one goes from eyelohexane to carbon tetraehloride as a solvent by the fact that carbon tetraehloride is a spherical symmetrical isotropie molecule whereas eyelohexane is anisotropic and this provides a better environment for ordering of Z chains. In the course of this discussion we have assumed that all of the C - - C bonds have the same apparent optical anisotropy, ~'c, and 10To evaluate these percentages it is necessary to know the statistical correlation factor for the Z chain. When Z chains are organized with respect *.o others, their molecular optical anisotropy is multiplied by this factor. The values of percentages which follow in the text are obtained when one has a correlation fact between 1 and 3.
539
M O L E C U L A R O P T I C A L A N I S O T R O P Y AND N - A L K A N E S
that they have all been exposed to a Lorentz electrical field. Obviously, one could t r y to refne the theoretical interpretation of our results by introducing a more complex internal field (33). However we do not see how this refinement could modify our principal conclusions. Since, in effect, the molecular optical anisotropies of short alkanes are the same, within the range of experimental error, in eyelohexane or carbon tetraehloride solution, the approximation of the Lorentz field seemed justified. True, one might think that two C - - C bonds in the trans position would not have the same electrical inter-action as two C - - C bonds in the gauche position and that, therefore, their apparent anisotropy -y~ would not be the same in these two conformations. An approximate ealeulation shows that the variation is negligible (12 i). In addition, this effect cannot account for the strong variation of the anisotropy, ~,2, of long alkanes on passing fl'om a solution in cyelohexane to the pure liquid.
statistical basis to a molecule more elongated. However even in the pure liquid the number of molecules in this ordered form would be small. The situation would surely be different if you had two or more saturated carbon chains bound to the same residue in proximity to one another (for example triglycerides). One might imagine the alkane portions of a triglyceride, for example, taking a number of possible orientations yielding basically two different states of the molecule, one chaotic, the other ordered with chains in a parallel state. The manner in which the transformation from the chaotic to ordered form is effeeted could well be important to the explanation of the biological phenomena in which they are involved. APPENDIX
Rules of Recurrence Relative to the Calculation of "fn2. First we will take as an ensemble all isomers of a molecule containing n-atoms of carbon and having gone through r gauche rotations. Let us take
SUMMARY
With the aid of apparatus constructed in our laboratory, the study of "depolarized Rayleigh scattering" permitted the deterruination of molecular optical anisotropies 2 of compounds in liquid state or in solutions with a precision of 3-4 %. In the process of our work on n-alkanes, we have deretrained by the comparison between experimental and theoretical values of 7 2 the potential energy difference AU0° = 600 ± 50
cal tool for passage between one trans form to one gauche form by rotation about a single C C bond. When we compared the molecular optical anisotropies between n-alkanes as pure liquids and in solution, our results suggested the existence of parallel ordering at short distances of Z chains ("zigzag") of the alkanes. Apparently when one diminishes the average distance between molecules of n-alkanes (that is when one passes from alkanes in solution to alkane in liquid state) one favors this ordering, i.e., one passes from the average form of the molecule which one would expect from a
G(r, n) = ~ A~~(gi),
[30]
J
where one takes the sum over all isomers j having n-atoms of carbon and having passed through r gauche rotations. Using Eq. [30], Eq. [27] becomes n--3 _ 2
r~0
a(r, n) exp (-~'~Uo°/RT)
"Y~ = ~-3
. P 2, [31]
N ( r, n) e x p ( - rA Uo°/ R T ) r~O
where N ( r , n) is the number of individual conformations of a molecule having n-atoms and r gauche rotations. In the model of "all conformations" (see text for description) one calculates N(r, n) using the rules of recurrence. Take a molecule having n -- 1 carbon atoms. Then there exist two ways in which to obtain a molecule having n-atoms and r gauche rotations: (i) If the molecule with n -- 1 carbon atoms exists in a form having r gauche rotations, one can then add the additional carbon in the plane parallel to that formed by the first two carbon bonds Journal of Colloid and Interface Science, V o l . 27, N o . 3, 3"uly 1968
540
BOTHOREL
in the molecule. (ii) If the molecule with n - 1 carbon atoms exists in a form having r - 1 gauche rotations, then in order to have r gauche rotations it is necessary to place the additional carbon atom in a direction out of the plane formed by the last two carbon bonds. Then it follows that N ( r , n ) = N ( r , n - 1) [32] + 2N(r1, n 1). If this molecule having n - 1 carbon atoms is oriented in such a way that the additional C - C bond formed when the n t h carbon is added lies in the direction 111 (see Fig. 6), only the quantity n~ of Eq. [25] is modified. Then Eq. [25] becomes
a(r, n) = G(r, n -
1)
8
1)
+ ~Nl(r, n -
+ (7-
[34] 1, n -
1)
-[- 5 N 3 ( r -- 1, n -
1)
+ 2mG(r-
8
+ (7 - 2n) N ( r _
2
8
1)21
[331
A
7 -- 2n
= A~_~ + ~ n~ + ~ - -
Let N l ( r , n ) be the total number of C - C bonds in the direction 1 in all possible isomers of a molecule having n-atoms of carbon and having passed through r gauche rotations. Then N2(r, n ) , N3(r, n ) , and N d r , n) signify the same quantity relative, respectively, to directions 2, 3, and 4. I t follows that the contribution to G(r, n ) of all the conformations of a molecule having n - 1 atoms of carbon and r gauche rotations (r, n - 1) when one adds an additional carbon atom is expressed by 8 G ( r , n - 1) + ~ N l ( r , n 1) 7 - 2n 1). 3 Then going through the same process for the isomers formed by adding the additional bond in direction 3 and 4 one can by use of the rules of recurrence calculate the quantity
+ --N(r,n-
G(r, n ) n To calculate this expression, the first bond of the molecule is oriented in direction 2 and t h e second in direction 1. E a c h additional carbon atom a t o m is joined to the first carbon atom of the molecule. Journal of Colloid and Interface Science, Vol. 27, No. 3, July 1968
1)[
-
1)
N l ( r , n ) = N2(r, n -
N2(r,n) = Nl(r,n-
l(n_ 3
1, n
3 The relations which permit the calculations of the parameters N d r , n ) [i = 1, 2, 3, 4]:
+ 2N2(r -
An2 = ~4 I ( n ~ + l ) ~ + n ~ 2 + m 2 + n ~ 2
1)
2n) N ( r , n _
3
1, n -
1),
1) + N ( r , n - 1)
+ 2N3(r-
1, n -
1)
+ 2N(r-
1, n -
1),
N 3 ( r , n ) = N3(r, n -
[36]
1)
-k Na(r - 1, n - 1) + Nt(r-
[35]
1, n - -
[37]
1)
The relations of recurrence [32], [34]-[37] permit the calculation of the average molecular optical anisotropy of any carbon chain of length greater than n = 3. For a given value of n, the number of calculations necessary is given by the ratio [5(n - 2 ) ( n - 112 (r varies from 0 to n - 3). The number of calculations necessary when one does not use the rules of recurrence is 3 n-3. The time saved in calculations by using the rules of recurrence is obviously considerable. I t is also possible to determine the rules of recurrence for the model of "permitted conformations," but the operation is considerably more difficult. In this case the number of calculations varies as n 2, instead of 3~-3. ACKNOWLEDGMENTS The author is pleased to acknowledge Miss H. Lindsey for the translation of this article. REFERENCES l. (D~) CABANNES, J., " L a Diffusion Mol~culaire de la Lumi~re. Presses univ., Paris (1929); (b) KAUZMANN, W., " Q u a n t u m C h e m i s t r y . "
MOLECULAR OPTICAL ANISOTROPY AND N-ALKANES Academic Press, New York (1957); (c)
I~RATOHVIL,J. P., Anal. Chem. 36, 517-526 (1966). 2. LALANNE, J. R., AND BOTIcIOREL, P., Compt. Rend. 9.63, 693-696 (1966); J. CAirn. Phys. 1538-1548 (1966). 3. ARZELIES, I{., Electricit6 Macroscopique et Relativiste," p. 6 2 7 . Gauthiers-Villars, Paris (1963). 4. BOTHOREL, P., Ann. CAirn. (Paris) 669-712 (1959). 5. PIAUD, J. J., J. Chim. Phys. 59, 215-231 (1962). 6. UNAN~, A., AND BOTHOREL, P., Bull. Soc. Chim. France 1966, 1640-1643. 7. CLEMENT, C., AND BOTHOREL, P., Co,opt. Rend. 251, 2323-2325 (1960). 8. CARR, C. I., AND ZIMM, B. H., or. Chem. Phys. 18, 1616 (1950). 9. WIPPLER, C., AND SCHEIBLING, G., J . Chim. Phys. 201-205 (1954). 10. MAELSTAF, P., and BOUVlER, M., Compl. Rend. 265, 1072-1075 (1967). 11. BOTHOREL, P., AND PIAUD, J. J., France patent (C.N.R.S.) n °. 102 348, (1967); Manufacturer: Electronique Appliqufie-33Gradignan. ]2. (a) CLEMENT, C., .~NDBOTHOREL, P., or. CAirn. Phys. 878-883, 1262-1270, 1282-1288, (1964); (b) UNANUE, A., AND BOTHOREL, P., Bull. Soc. Chim. France 1964, 573-578; 1965, 2827-2832; 1966, 1640-1643. (c) LUONG THE MAN, CLEMENT, C., AND BOTHOREL, P., Compt. Rend. 260, 2755-2758, (1965). (d) FOIJRCYIE, G., PACAt~LT, A., BOTtIOREL, P., AND HOAR&U, J., Compt. Rend. 269., 1813-1816 (1966). (e) BOTHOREL~ P., CLEMENT, C., AND 1V[ARAVAL,P., Compt. Rend. 267, 568-570 (1967). (f) BOTHOREL, P., M#-RAVAL, P., A_ND CLEMENT, C., Corapt. Rend. 265, 1068-1071 (1967). (g) CRAIG, H. R., ANn BOTHOREL, P., a paper submitted for publication to the Compt. Rend. 1967. (h) FOURCHE, G., a paper submitted for publication to the J. CAirn. Phys. (i) CARISTAN, A., AND BOTHOREL, P., to be
published. 13. (a) MEYER,
E. H. L., AND
OTTERBEIN,
G.,
Physik. Z. 32, 290 (1931); (b) S&CESSE, G., Physil~. Z. 36, 357 (1935); (c) CL&RK, C. H. D., Nature 138, 126 (1936); (d) WANG, S. N., J. Chem. Phys. 7, 1012~ (1939); (e) DENmGH, K. G., Trans. Faraday Soc. 36, 936-947 (1940); (f) LE FI~VRE, C. G., AND LE FEVRE, R. J. W., J. Chem. Soc. 1956;
541
3549-3563; and (g) VOLKEINSTEIN,M. ¥., "Configurational statistics of polymeric chains." Wiley (Interscience), New York
(1963). 14. CHALAM, E. V., Pro¢. Indian Aead. Sci. 115, 190-194 (1942). 15. SMITE, R. P., AND MORTENSEN, E. M., J. Chem. Phys. 32, 502-507 (1960); STEIN, l~. S., J. Chem. Phys. 9.1, 1193-1198 (1953). 16. (a) BOP~ISOVA, N. P., AND VOLKEINSTEIN, M. V., Zh. Strukt. Khim. 2, 346, 469 (1961); (b) SCOTT, R. A., AND ET SHERAGA, I-I. A., J. Chem. Phys. 42, 2209-2215 (1965); 44, 3054 (1966); (c) ABE, A., JERNIGAN, R. L., AND FLOI~Y, P. J., J. Am. Chem. Soc. 88, 631 (1966). 17. SMITa, R. P., AND ~/[ORTENSEN, E. M., Or. Chem. Phys. 35, 714-721 (1961). 18. LE FEVRE, R. J. W., 0nn, B. J., AND RITCHIE, G. L. D., J. Chem. Soc. 1966, 273-280. 19. FOULANI, P.. AND CLEMENT, C., Bull. Soc. CAirn. France, unpublished. 20. LEwis, 0. G., J. Chem. Phys. 43, 2693-2696 (1965); BARTELL, L. S., AND KOHL, D. A. J. Chem. Phys. 39, 3097 (1963). 21. HOEVE, C. A. J., or. Chem. Phys. 35, 1266 (1961). 22. BonlsovA, N. P., Vysokomolekul. Soedin. 6, 135 (1964). 23. BENOIT, I-I., Compt. Rend. 236, 687-689 (1953) ; GOTLIB, Y. Y., Dissertation, Gertsen Leningrad Pedagogical Institute (1956); Zh. Te~hn. Fiz. 27, 707 (1957); 24. JERNIGAN, R. L., AND FLORY, P. J., or. Chem. Phys. 47, 1999-2007 (1967). 25. 1NAG&I,K., J. Chem. Phys. a paper submitted for publication. 26. PTITSYN, O. B., "Spectroscopy of Polymers," summer school, Prague 1967. 27. KITAIGORODSKI, A. I., "Organic Chemical Cristallography," p. 321. Consultants Bureau, New York (1961). 28. KAVAN&~r,J. L., "Structure and Function in Biological Membranes," Vol. 1, p. 75. HoldeR-Day, London (1965). 29. CHAPMAN, D., "The structure of lipids," p. 232. Methuen, London (1965). 30. WARREN, B. E., Phys. Rev. 969-973 (1933). 31. MICrIELS, B., J. CAirn. Phys. 1123-1128 (1966). 32. BHATTACH&RYYA,S. N., PATTERSON, D., &ND SOMCYNSKY,W., Physica. 1279-1292 (1964). 33. STEIN, R. S., &ND ROWELL, R. L., J. Chem. Phys. 47, 2985-2989; SICOTTE, Y., or. Chim. Phys. 58~-590 (1967); KIELICH, S., or. Chem. Phys. 4090-4099 (1967).
Journal of Colloidand Interface Science, Vol. 27, No. 3, July 1968
[nstitut de Magndto-Chimie, "Brivazac," 33-Pessae, France Received February 1, 1968 The molecular optical anisotropies (./2) of n-alkanes were calculated using the theory of "optical bonds." A model which considered the trans and gauche isomers of rotation was used• The comparison of the theoretical values with those obtained experimentally in pure liquid and solutions by a recent technique, viz., depolarized light scattering, permitted the determination of the differences in energy between the isomers with satisfactory precision. This comparison suggests that there exists a correlation of orientation between elongated chains ("zig-zag") in these liquids; this ordering between the chains at short distances could play an important role in determining the properties of compounds in which they are found, for example fats. INTRODUCTION AND EXPERIMENTAL METHODS D u r i n g the last few years, we have developed a technique for measuring molecular optical anisotropy of pure liquids and their solutions. W e are going to review briefly the principal elements of our technique and indicate certain details of the technique which up to now were unpublished• I n order to do this, it is necessary to use the classical system presented in Fig. 1. A m o n o c h r o m a t i c , parallel, linearly-polarized light b e a m traverses the cell C which contains a sample of liquid. T h e wave scattered by the liquid in the direction OP, with the same wavelength as the incident b e a m (Rayleigh effect), can be decomposed into two vibrations, perpendicular to one another, of intensities i and I. One can isolate a depolarized vibration of intensity i, with the aid of a Glazebrook prism (G). T h e intensity i of this light b e a m scattered b y the sample is called the "depolarized" intensity. T h e electric field E of the exciting wave creates in each molecule an induced m o m e n t 529
M ( 1 ) ; the two vectors can be referred to the Cartesian coordinate s y s t e m X Y Z (Fig. 2). Molecules in general are anisotropie, in other words M and E are not parallel. If the electric field is n o t too strong 1 and if the wavelength h is small c o m p a r e d to the electronic absorption bands of the molecules being studied, 2 one can say t h a t the induced m o m e n t is proportional to the exciting field,
M = 1~ JE
Ill
where ] a ] is the molecular polarizability tensor,
loll
=
Olxx
OLxy OLxz
O~yx
Ogyy O~y~
Otzx
Otzy
OLzz
OQj --~ O~ji
• (i ~ j ) ;
[2]
1 This is the case if we use a classic source of light (mercury arc, xenon arc), a gas laser, or a Q switched laser of medium power (2). 2 Chemical compounds which are transparent to visible light can be studied for example with the aid of light from a mercury arc (k = 5.461 A). Journal of Colloid and Interface Science, Vol. 27, No. 3, July 1968
530
BOTHOREL
[V
A
p is t h e n u m b e r of molecules in each u n i t of s c a t t e r i n g v o l u m e V. M o l e c u l a r o p t i c a l anis o t r o p y 3,2 is a n i n v a r i a n t p r o p e r t y of t h e t e n s o r of p o l a r i z a b i l i t y , a n d t h e r e f o r e i t does n o t d e p e n d u p o n t h e o r i e n t a t i o n of t h e molecule. T h e 72 is expressed b y :~
c
2
= ½[(~= - ~ ) ~ + ( ~
- ~)2
+ ( ~ - ~=)~] 2
[5] 2
+ 3(~L + . ~ + .z~). If we call R~ t h e d e p o l a r i z e d i n t e n s i t y scatt e r e d b y a u n i t of v o l u m e a n d i n c i d e n t p h o t o m e t r i c i l l u m i n a t i o n , one sees t h a t
FIG. 1. "Depolarized" scattered intensity.
Z
i Ri -
VE2/2 -
32~ 4 p ( 2 + 135~ 4
2)2 2
One can easily show t h a t R i has t h e s a m e v a l u e w h e t h e r t h e i n c i d e n t light b e a m is p o l a r i z e d o r u n p o l a r i z e d . I f t h e fluid is a solution, one can w r i t e t h e general equation (7),
Y
i~oj~tio~ -
~x
16~
1~
4
~ 2
~ p~n
+ 2)23,2VE 2
co {2 + 2 )2
[7]
+ .oOYo\~o~ 7~ ~ '
A
(j)
[6]
or
~IG.
2.
M is sinusoidal a n d has t h e s a m e w a v e l e n g t h as E. A s s u m i n g t h a t t h e molecules are H e r t z oscillators, one can show t h a t t h e periodic i n d u c e d m o m e n t s c r e a t e scatt e r e d w a v e s of t h e s a m e f r e q u e n c y as t h e i n c i d e n t l i g h t (3). A d m i t t i n g t h a t t h e electric field F w h i c h effectively excites t h e molecules is a L o r e n t z field d e s c r i b e d as,
F = E (n~
+ 2) 3
[3]
w h e r e n is t h e r e f r a c t i v e i n d e x of liquid a t w a v e l e n g t h X, one c a n easily r e l a t e t h e dep o l a r i z e d i n t e n s i t y i s c a t t e r e d b y t h e ens e m b l e of t h e molecules to t h e p o l a r i z a b i l i t y of t h e molecules ( 4 ) . I f t h e fluid is pure, t h e d e p o l a r i z e d i n t e n s i t y is d e s c r i b e d as, i -
16~ 4
~
~ 2
p(n
+ 2)Ly2VE2;
[4]
Journal of Colloid and Interface Science, Vol. 27, No. 3, J u l y 1968
Risolution
32~r4 i 2 - - _l ~_4 p ( n -+ 2)~/2
c0 (n2 + 2 ~ 2 + Rio° ~ \ ~ / ,
IS]
In Eq. [5] the choice of the directions of the coordinate system X Y Z is unimportant. For a particular reference system (xyz) we have: a]~ = 0 (i # j ) ; in this case, the molecular optical anisotropy ~2 takes a simpler form: 73 = ½ [ ( ~ _ ~ ) 2 + ( ~ _ ~,o)2 + ( ~ _ ~ ) ~ ] ; in this equation a ~ , ay~ , and a~ are the principal optical polarizabilities of the molecule. This equation has been used for a long time to define 73; it is less general than Eq. [5]. This Eq. [5] has many advantages in the interpretation of molecular optical anisotropy with the aid of the theory of optical bonds, as shown below. In our previous work we used a parameter (A2RM2) which resembles 73 the following equation (4-6):A2RM = (16~2N272)/81; N is Avogadro's number.
MOLECULAR OPTICAL ANISOTI~OPY AND N-ALKANES p is the number of molecules of the solute per unit of scattering volume V; i0° is the intensity scattered by the pure solvent under the same conditions of study as the solution. The expression between brackets multiplied by i0° includes the variations of the number of molecules of the solvent and of the Lorentz field when one passes from the pure solvent to the solvent in the solution. The Co is the number of grams of solvent per milliliter of solution; do is the density of the pure solvent; no is the refractive index of the pure solvent. The Eqs. [6] and [8] show that it is possible to determine the optical anisotropy ~2 of a compound in liquid form or in solution by measuring the intensity Ri scattered by these liquids. 4 In practice these measures are relative. One actually measures the ratio ( R i / R i o = i/io) of the depolarized intensities scattered under the same experimental conditions for the sample (i) and for a standard liquid (i0). We have chosen cyclohexane as a standard and determined the intensity Rio in the following way: The Rayleigh constant R is defined as the total intensity scattered per milliliter and unit of incident photometric illumination (2). The absolute Rayleigh constant for benzene in natural light has been measured numerous times ( l c ) . The majority of recent experimental values approximate the value of Carr and Zimm (8): R~ = 16.3 X 10-6 --1 cm (X = 5.461 A). Therefore we adopted this value for our work. Measuring the total scattered intensities under the same experimental conditions for benzene (I~) and for a standard liquid (I0), the Rayleigh constant can be determined for that standard Remember that the anisotropy, 7~, of a perfect gas can be determined by measuring the polarization ratio (o = i/I) and the molecular refraction. This mode of determining v2 is not applicable to dense fluids, pure liquids or solutions (la) [in the case of pure liquids, it is also necessary to know the coefficient of isothermal compressibility, for example].
(R0):
(,0)
R0 = R~ ~r~
531
[91
We have measured the ratio Io/I~ numerous times with the aid of the light-scattering-photometer of Wippler and Scheibling (9). The value which we obtained for cyclohexane was Io/I~ = 0.271, after the scattering volume correction was made. After measuring the depolarization ratio, po, of the light scattered by cyelohexane with an unpolarized incident beam, with a wave length of 5461 A (po = 0.0553), one can calculate the constant Rio (4): R/° - 1 + p0
1 + p0
[10]
= 23.15 X 10-s cm-l(h = 5461A) Using this value, Ri can be determined for any liquid simply by measuring ?the ratio of the depolarized intensities, i and i0, scattered under the same conditions b y the liquid and by cyclohexane. Then R i = Rio(i/io)
[11]
The molecular optical anisotropy of the compound can then be calculated using Eqs. [6] or [8]. The precision obtainable for v 2 depends mainly on the precision of the measurement of the ratio i / i o . The systematic study of the molecular optical anisotropy of chemical compounds by this technique is not practical unless 72 is known with an experimental error of a few percent. With this end in view we conceived and built apparatus for making relative measurements of depolarized scattered intensity (5, 10, 11). The precision of i is 1-2%, this gives an error of about 3-4 % on v 2 in the majority of cases where compounds studied were liquids or compounds in dilute solution (in certain cases one can use solutions of a concentration of 10 4 tool/liter). Thus we studied a variety of organic compounds such as simple or substituted alkanes, polyethers, f a t t y acids, pyrimidines and purines, a number of important aromatic compounds, and others (12). Journal of Colloid and Interface Science, Vol. 27, No. 3, J u l y 1968
532
BOTHOI~EL
expressed by convention as
Z
%- = a , j -
Y
X
The measure of depolarized scattered intensity i and the interpretation of the optical anisotropy 2 as determined from i, we have called Depolarized Rayleigh Scattering (in French: Diffusion Rayleigh D6polaris6e or DRD). Interpretation of Molecular Optical Anisotropy with the Aid of the Theory of "Optical Bonds." Since the work of Meyer and Otterbein (13a), one attaches to each bond j in a molecule a principal tensor of optical polarizability (see Fig. 3). The molecular polarizability tensor is obtained simply by combining the principal optical polarizaJ bilities of the bonds ( o J.~, ayy, a ~Y ) :
o
0
0
J
= d zz
=
Ol± J,
"
5=jx ~vix
Z
5~ r ~v.~Y
~.j z ~v+z
One obtains each component of the molecular tensor using the projections of the principal optical polarizabilities of the bond on X Y Z
:xx = ~ ~j ~ + : x + ~ ~ ~y+:X J
J
+ ~J ~+~zj:X
[15]
Using the relations between cosine directions:
= 1;
and equation [14]:
;
~ ~ £~xx
Y
~,x + ~+~ + ~
These polarizabilities can be projected on the reference system X Y Z attached to the molecule according to the rules applicable to a second degree tensor. If the bonds of the molecule have a symmetry of revolution about their axis, the following simplification is possible: ~ O~yy
xj YJ
o l :z
[14]
If the molecule studied contains only bonds with a symmetry of revolution, the molecular optical anisotropy can be calculated very easily using the principal optical polarizabilities of the bonds or rather, their optical anisotropies (14, 15). Taking the orientation of the bond j with reference to the coordinate system X Y Z one can with the aid of the table of cosines determine their directions.
FIG. 3.
<=
aft'. 6
(EiJ +,
O!XX =
E
2
~
*"
[16]
Then one obtains analogous expressions for a r t and azz ; and also: J
+~
J
;
J a±~.jxS.jr
[17]
[12]
and the optical anisotropy of the bond be-
= Z ~J ~xjx ~ Y .
comes "Tj2 =
(Or[I~ - - a ± ~2.~ ] ,
[13]
T h e d e f i n i t i o n of m o l e c u l a r o p t i c a l a n i s o t r o p y
given by Eq. [5] applies also to an isolated bond. Journal of Colloid and Interface Science, Vol. 27, No. 3, July 1968
T h e a bonds p r e s e n t a case of t h i s s y m m e t r y of revolution a n d h a v e in general a m a x i m u m of polarizability along t h e l e n g t h of t h e i r axis a n d c o n s e q u e n t l y h a v e an optical a n i s o t r o p y ~/j > 0.
MOLECULAR OPTICAL ANISOTROPY AND N-ALKANES CHa
CH3
/
cf Hs H3 FIG. 4. Isomers of rotation of n-butane.
U
tropy ofn-Alkanes. In our study of n-alkanes, we assumed that all bond angles are tetrahedral. We also supposed, according to the theory of isomers of rotations, that each molecule of alkane can exist in various conformations at a given temperature. Each conformation corresponds to a minimal value of internal energy U of the molecule. It is known, for example, that three stable conformations of normal butane exist (see Figs. 4 and 5) ; there is one trans form (9 ~ 0 °) and two gauche forms g+ and g- (9 ~ -4-120°) (16). The probability of existence of the optical antipodes, g+ and g-, is the same, but it has a value less than that of the trans isomer. If, b y an appropriate technique, one determines the percentage of trans isomer (xt) and gauche (xa+ = xg- = xa) at absolute temperature T, one can write xo _ exp
120 180
(g-) (r)
(g+)
F~G. 5. Potential energy of isomers of rotation of n-butane.
2 • ~ / i ( ~2
-
~2) ] 2
+ [E ~ i ( ~ + [ E ~i ~
- ~,~)]~I
i
i
AG
=
AN
i
+ (~ ~ ~
#~,~)~].
TAS. 7
/kU = AU0°;
xj _ exp
ac00~ .
z~
R-T
[19]
~)~
--
[211
[22]
AU0° is the difference of internal energy of the two isomers at 0°K. Under these conditions
÷ 3[(~--~ ~,i ~x~x fixer) ~
+ (~2 ~ ~ i
"
The difference of free enthalpy AG between the isomers can be written in the form:
/kS = 0;
~~) ]
-
[20] ~
Since the structures of the two isomers are very similar, one normally supposes that their partition functions are equal, that is to say
and for equation [5], 2= 1
-
Xt
Au:
533
i
Therefore it suffices to know the orientation of each bond and its optical anisotropy in order to calculate the molecular optical ~aisotropy. The normal alkanes con_~in only simple C - C and C - H bonds and can be studied utilizing Eq. [19].
Interpretation of Molecular Optical Aniso-
[23]
/
One can determine A U0° by quantitative analysis of isomers of rotation. As a first approximatioa we assume t h a t the conformations of the alkanes-other than 7 If normal butane is studied in solution,-u t e r m of free enthalpy of solution must added to the equatioa, If the interactions between solvent and solute are weak, this t e r m is obviously negligible. We have assumed t h a t this interaction is indeed negligible in the case of normal alkanes of chain length longer t h a n b u t a n e in carbon tetrachloride and cyclohexane solutions. Journal of Colloid and Interface Science.
Vol. 27. No. 3, July 1968
534
BOTHOREL
butane can be obtained by rotation through the position trans (~ = 0 °) or gauche (~ = ± 1 2 0 °) about each C-C bond (12a). In order to pass from a conformation entirely trans, which we call Z chain (zig-zag), to a gauche form, r rotations are necessary, and the ratio of percentages of the two forms is equal to (12f) x(gauche chain)
x(Zchain)
( = exp
~o) - r--
TABLE I COSINE DIRECTIONS FOR CARBON-CARBON ]~ONDS IN ALKANES No.
X
Y 1
1
1
1
]
1
1
.
-~/~
[24]
This relation is valuable only if one ignores Van Der Waals interactions between methylene groups in the same chain. If one considers these interactions, it is necessary to add a supplementary term to r A U0°.
Molecular Optical Anisotropy of an Isomer of Rotation. If an isomer of rotation of an alkane satisfies the preceding conditions (tettrahedral bond angles, ~ = 0, q-120°), it is easy to determine the orientation of all the bonds. As demonstrated by Smith and Mortensen (15, 17), the bonds are necessarily parallel to the four diagonals of a cube, take for an example the diamond crystal. Take the coordinate system X Y Z of Fig. 6. If one orients it in such a way t h a t the bonds between three consecutive methylene groups are parallel to the demi-diagonals 1 and 4, then all the C-C and C - H bonds of the molecule are oriented parallel t o the directions 1, 2, 3, a n d / o r 4. The direction cosines are then given by Table I.
+~
1 ~/~
4
Z
1
1
-~/~
1 ~/~
q-~
1
Then Eq. [19] can be used to obtain the molecular anisotropy of the rotational isomer under consideration (12a, 17). 3".
2
=
4 2 [4/~(nl + --
l~(n
n2
2
--
+
n3
2
1)2]r2 =
+
n42)
[25]
A- ~ P2 ,
where n is the number of atoms of carbon in the alkane; nl, n2, m , and n4 are the number of C-C bonds parallel to the directions 1, 2, 3, and 4. The 17 is the only parameter of the optical anisotropy actually necessary for calculating 3'2. I t is expressed by: P = 3"c -- 23"~,
[26]
where 3"e is the optical anisotropy of the C - C bond, and 3", is the optical anisotropy for the C - H bond. The A~2 (Eq. [25]) depends only upon the conformation of the rotational isomer.
Average Molecular Optical Anisotropy 3"~2 of an Alkane, General Formula C~H2,,+2. An n-alkane exists at normal temperatures as a mixture of isomers of rotation. The experimental molecular optical anisotropy calculated from Eqs. [6] and [8] is therefore an average value @7n2). If one considers the isomers of a particular molecule as independent scatterers, the scattered intensities are additive s and each of the isomers con-
FIG. 6. D i r e c t i o n s of b o n d s in an alkane.
Journal of Colloid and Interface Science, Vol. 27, No. 3, Ju]y 1968
s Since t h e v a l u e s of i, a n d t h e r e f o r e v 2, are additive, one can calculate t h e i r a v e r a g e v a l u e if one t a k e s into a c c o u n t t h e s t a t i s t i c a l w e i g h t of each c o n f o r m a t i o n of t h e molecule. On t h e o t h e r h a n d , t h e p o l a r i z a t i o n r a t i o p is n o t a n a d d i t i v e
MOLECULAI~ OPTICAL ANISOTlZOPY AND N-ALKANES • (A~)
[ 1,.~
I
I
o'
I
~)o o
I I
5
10
"15
ooo
20
o n ~'
FIG. 7. 3,Iolecular optical anisotropies, %2, divided by number of "chainons" (n -- 1). Experimental curves: ~--~ , pure liquids; ~ , cyclohexane solutions; > <, carbon tetrachloride solutions; theoretical curves: (O); (the length of the bars of arrows indicates experimental error).
535
n-alkanes (4 =< n =< 24) as pure liquids or in solution with various solvents at r o o m temperature (12a) (see Figs. 7 and 8). O t h e r s u p p l e m e n t a r y values have been o b t a i n e d of late (12e). T h e experimental error is 3-4 %. If one chooses the proper values for AU0° and r with the aid of Eqs. [25] and [28] one can a t t e m p t to a p p r o x i m a t e theoretically the curves of the experimental variation of if2 as a function of n. We h a v e m a d e simple calculations of this sort only with alkanes f r o m b u t a n e to decane (12a). U n f o r t u n a t e l y the n u m b e r of calculations, which is proportional to the n u m b e r of conformations, (there are 3 ~-3 isomers of r o t a t i o n per alkane)
tributes to .~2 as a function of its own optical anisotropy, ~2, and its percentage xj (see Eq. [24]). --2
2
~[n = "Yn2(t) ~- E Xj "Yn (gi) J
7~2(t) of the optical bining
is the molecular optical anisotropy Z chain; %~(gj) is the molecular anisotropy of a gauche isomer. ComEqs. [25] and [27] one obtains,
-;J = r ~ Z
x(~XUo°, T)A~ ~ = r ~ X J .
[2s]
isomers
After calculation of the molecular optical anisotropies of all the rotational isomers present, the average molecular optical anis o t r o p y can be calculated for a n y n-alkane at t e m p e r a t u r e T as a function of the two parameters r and A U0°. RESULTS AND DISCUSSION
Interpretation of Experimental Molecular Optical Anisotropies of n-Alkancs. I n the past we have determined the molecular optical anisotropies of a n u m b e r of i m p o r t a n t function, the "average," ~, calculated in this manner does not correspond to experimental values of o [see the article of Stein (15)].
FIG. 8. Experimental molecular optical anisotropies if2: (L) pure liquids; ($I) carbon tetrachloride solutions, (S~) cyclohexane solutions; ($3) pentane solutions; ($4) heptane solutions. Theoretical curves: (T1) AU0° = 600 cal tool-1 (model of "permitted conformations"); (T~) AU0° = 600 cal tool-1 (model of "all conformations"); (T~) AU0° = ~o (Z chains independent). Journal of Colloid and Interface Science, ¥o]. 27, :No. 3, J u l y 1968
536
BOTHOREL
becomes unwieldy by the time n reaches 10. Even if one uses a computer, the time required for the calculations becomes prohibirive when n exceeds 15. It seemed, therefore, that this method could only be utilized for short chain alkanes. However using the IBM 7044, we have been able to make the calculations for chMns containing up to 50 carbon atoms in a reasonable time after taking into account the rules of recurrence (12e) (see "Appendix"). We have calculated if2 considering all permitted conformations (the model of "all conformations"). However in the case of those isomers which have two gauche forms of opposite sign (g+, g- and g-, g+) when one makes a rotation about two consecutive C-C bonds, the probability of existence of these forms is very low due to the fact that in order for them to exist it would be necessary to have a superposition of CH2 groups (the distance between the two CH2 groups in this isomeric form would be 2.5 A, whereas, the normal Van Der Waal's distance would be 4 A). It is for this reason that we have eliminated these conformations from our calculations (model of "permitted conformations"). In other words, to eliminate these conformations is the same thing as saying that there exists a very large A U between them and the Z chains. This model gives good agreement between the theoretical and experimental values of if2 for n-alkanes with up to 13 carbon atoms, if the theoretical values for G and AU0° are r = 0.87 ~ 0.02 ~3 ; hU0° = 600 ± 50 cal tool-1. The optical anisotropy parameter r appears not only in alkanes but also in a number of other organic compounds. That is the reason why its determination has been the object of much work (18). The value of F determined in the course of this study approaches the values proposed recently by other authors but is more precise. The utilization of this parameter in calculation of molecular optical anisotropy of orJournal of Colloid and Interface Science, Vol. 27, No. 3, July 1968
ganic compounds other than n-alkanes ought only to be attempted with caution. The value given above for F is for a straight chain alkane and is related to the C-C and C-H bonds of such a molecule. The values for branched and cycloalkanes are different (12a, 19). The value for A Uo° which determined by DRD also agrees well with those values obtained by various authors (400-700 cal tool-1) whether obtained experimentally (12a, 20, 21) or theoretically by using empirical potential energies of interaction (22, 16b). Note that the theoretical %2 of long chain molecules is proportional to n(12f) (see Fig. 8) and can therefore be approximated by the following expression: _
2
7~ = 0.9773 n -- 2.4432 ~6 (permitted conformations) 9
[29]
The molecular optical anisotropy calculated using the above relation differs less than 1% from that obtained by rigorous calculations in which n is greater than 8 and by less than 10/00 when n is greater than 15 (F 2 = 0.75). When the number of carbon atoms exceeds 13, there is a discrepancy between the theoretical and calculated values for ~.~ in carbon tetrachloride and cyclohexane solution (see Fig. 7). This discrepancy grows with each increment in chain length. One is tempted to blame the discrepancy oll the configurational model chosen. Some author (16b, c, 21, 22) give to the conformations containing the double rotation g+, g- and g-, g+ a statistical weight different from zero (AU = 2.500-2.800 cal tool-1 instead of infinity). On studying theoretically the potentials of interaction between all atoms in the alkanes with chain length in excess of C4, these authors (see above) have admitted the existence of two 9 Certain authors (23-25) have calculated the molecular optical anisotropy of a polyethylenic chain, but t h e y considered the statistical orientation of each C-C bond in the molecule. The m o s t recent works allow a comparison between these theoretical values and our experimental values.
MOLECULAR OPTICAL ANISOTROPY AND N-ALKANES supplementary conformations (~ = 4180-95°1 and also ~ = 0 4- 120°; these 2 new gauche forms have the same statistical weight: AU = 2000 cal mole -~) (26). Unfortunately these modifications bring about a statistical model which we used accentuates the discrepancy between the theoretical and experimental ve~lues in solution of %2 as a function of n. In effect these modifications tend to diminish the average molecular optical anisotropy beeause they a-L~ribu~e to these gauche forms which have small anisotropies, a probability of existence superior to that which we have attributed to them. However if one hypothesizes an organization of Z chains in solution of long chain alkanes, one can on the contrary explain the apparent discrepancy. Now we are going to discuss in detail this hypothesis in the case of pure liquids. Pure Liquids. Since the molecular optical anisotropy of an alkane is very different in the liquid state and in solution, the values of r and AUo° which we gave before do not predict the correct experimental curve for _
2
537 co
1500
10
800 600 400
0 cal. mole -1
=
Experimentally it was noted that: (i) values of ~7~2 arc systematically higher in the pure liquids than in solution. (ii) no value of the parameter AUo° permits explanation of the experimental variation of ~Tn2 (see Fig. 9 : the theoretical ratio ~7~2/~42 = X ~2/I A 42 depends only upon AU0°). The average slope of the experimental curve is greater than that of the theoretical curves. (iii) the experimental curve (L) of pure liquids of Fig. 8 falls between the theoretical curve (T,) for a statistical mixture of isomers and the theoretical curve (T3) which one would observe if the pure liquid contained only Z chains which are not in any way oriented one to the other. The above observations bring us to the conclusion that in pure liquids the apparent percentage of isomers which are highly anisotropie is greater than that which one would predict from the above statistical model. In other words there is a higher percentage of Z chains, which are the only isomers with an
5
10
15
n
FIG. 9. R a t i o of a v e r a g e e x p e r i m e n t a l m o l e c u l a r optical a n i s o t r o p i e s , ~7~2/ff~~, of alkanes to b u t a n e in liquid s t a t e (L) a n d t h e t h e o r e t i c a l ratios (-7,~/ff4 2 = X.~/5_~2) as a f u n c t i o n of v a r i o u s v a l u e s of AUo °.
anisotropy sufficiently superior to the average to give the observed effect on ~ experimental as opposed to ~2 theoretical. Consequently we have proposed the following model: In liquid n-alkanes, the gauche isomers have no correlation of orientation, they simply arrange themselves according to the simple statistical equilibrium described above. However Z chains have a tendency to orient themselves parallel one to another. This state of orientation will have a probability of existenee superior to that of Z chains with noneorrelation of Journal of Colloid and Interface Science, ¥ol. 27, No. 3, July 1968
538
BOTHOREL
orientation one to another. These Z chains have an optical polarizability a~ in the direction of elongation superior to that of other isomers. And also there is possibility of the chains approaching one another at a short distance d. These two conditions, al and d, enable the London forces between two associated Z chains to be maximum [the London forces are proportional to the ratio 2 7 al/d (16)]. Therefore the probability of existence of two associated Z chains is greater than the probability of existence of the two chains in the free state. In the crystalline form of normal alkanes only the Z chains are oriented parallel one to the other (27-29). In compounds such as f a t t y acids, soaps, and crystallized fats there are carbon chains bound to groups which encumber rotation. These attractive forces are sufficiently strong to cause organization of the alkane portion of the molecule. I t is obvious that the order observed in the crystalline form of alkanes is not preserved in the liquid form (28). However the broad band centered at 4.5 A in X-ray diffraction patterns of liquid alkanes has been interpreted as a tendency toward organization in which the chains are spaced approximately 5 A apart (28, 30). And also the absorption of ultrasonic waves by these liquids identities them as being "associated" (31). Bhattacharyya et al. (32) have studied thermodynamically mixtures of n-alkanes. In order to explain their results they have proposed the existence of packing in the liquid on the trans-gauche population. We think also that the abnormal variation of the anisotropy qn2 of decane liquid as a function of temperature could easily be explained by the existence of ordering at short distances between Z chains which diminishes or is destroyed by increase in temperature (10). If we make the assumption that in liquid alkanes Z chains are organized and gauche forms are free, then by comparing the values of q 2 of the curves (T1) and (L) in Fig. 8, it is possible to estimate in a crude way the percentage of Z chains actually in conJournal of Colloid and Interface Science, Vol. 27, No. 3, J u l y 1968
formity with the model 10 (i.e., assumption above). In order to explain experimental results in the pure liquids of carbon chain length greater than 5, a percentage of 1015 % of Z chains is sufficient. In other words the proportion of molecules of alkanes which are organized is greatly inferior to the number of molecules which are in a disordered state. I t is also possible to explain the divergences observed between the values of ~2 of the theoretical curve (T1) of Fig. 8 and the experimental values in various solvents with the aid of the model described above. I t seems that when the solvent is a normal alkane, one has an ordering at short distances between Z chains of the solvent and of the solute. The effect will be greatest when the chains of solvent and solute are of approximately the same length. Therefore, since the alkanes studied were for the most part of chain length > 5 one would expect a decrement of ~2 (i.e., a decrease in the percentage of Z chains). When one goes from a solvent system C7 to C5, one can see this effect (see Fig. 8). Also when one looks at the curves of alkanes in cyclohexane and carbon tetrachloride and compares them with the theoretical curves, one can see that a partial ordering still exists in these solutions. An estimated percentage of 1% of the Z chain is sufficient to explain the variation of experimental values of ~n2 from the theoretical in these solutions. One can explain the decrement of if2 when one goes from eyelohexane to carbon tetraehloride as a solvent by the fact that carbon tetraehloride is a spherical symmetrical isotropie molecule whereas eyelohexane is anisotropic and this provides a better environment for ordering of Z chains. In the course of this discussion we have assumed that all of the C - - C bonds have the same apparent optical anisotropy, ~'c, and 10To evaluate these percentages it is necessary to know the statistical correlation factor for the Z chain. When Z chains are organized with respect *.o others, their molecular optical anisotropy is multiplied by this factor. The values of percentages which follow in the text are obtained when one has a correlation fact between 1 and 3.
539
M O L E C U L A R O P T I C A L A N I S O T R O P Y AND N - A L K A N E S
that they have all been exposed to a Lorentz electrical field. Obviously, one could t r y to refne the theoretical interpretation of our results by introducing a more complex internal field (33). However we do not see how this refinement could modify our principal conclusions. Since, in effect, the molecular optical anisotropies of short alkanes are the same, within the range of experimental error, in eyelohexane or carbon tetraehloride solution, the approximation of the Lorentz field seemed justified. True, one might think that two C - - C bonds in the trans position would not have the same electrical inter-action as two C - - C bonds in the gauche position and that, therefore, their apparent anisotropy -y~ would not be the same in these two conformations. An approximate ealeulation shows that the variation is negligible (12 i). In addition, this effect cannot account for the strong variation of the anisotropy, ~,2, of long alkanes on passing fl'om a solution in cyelohexane to the pure liquid.
statistical basis to a molecule more elongated. However even in the pure liquid the number of molecules in this ordered form would be small. The situation would surely be different if you had two or more saturated carbon chains bound to the same residue in proximity to one another (for example triglycerides). One might imagine the alkane portions of a triglyceride, for example, taking a number of possible orientations yielding basically two different states of the molecule, one chaotic, the other ordered with chains in a parallel state. The manner in which the transformation from the chaotic to ordered form is effeeted could well be important to the explanation of the biological phenomena in which they are involved. APPENDIX
Rules of Recurrence Relative to the Calculation of "fn2. First we will take as an ensemble all isomers of a molecule containing n-atoms of carbon and having gone through r gauche rotations. Let us take
SUMMARY
With the aid of apparatus constructed in our laboratory, the study of "depolarized Rayleigh scattering" permitted the deterruination of molecular optical anisotropies 2 of compounds in liquid state or in solutions with a precision of 3-4 %. In the process of our work on n-alkanes, we have deretrained by the comparison between experimental and theoretical values of 7 2 the potential energy difference AU0° = 600 ± 50
cal tool for passage between one trans form to one gauche form by rotation about a single C C bond. When we compared the molecular optical anisotropies between n-alkanes as pure liquids and in solution, our results suggested the existence of parallel ordering at short distances of Z chains ("zigzag") of the alkanes. Apparently when one diminishes the average distance between molecules of n-alkanes (that is when one passes from alkanes in solution to alkane in liquid state) one favors this ordering, i.e., one passes from the average form of the molecule which one would expect from a
G(r, n) = ~ A~~(gi),
[30]
J
where one takes the sum over all isomers j having n-atoms of carbon and having passed through r gauche rotations. Using Eq. [30], Eq. [27] becomes n--3 _ 2
r~0
a(r, n) exp (-~'~Uo°/RT)
"Y~ = ~-3
. P 2, [31]
N ( r, n) e x p ( - rA Uo°/ R T ) r~O
where N ( r , n) is the number of individual conformations of a molecule having n-atoms and r gauche rotations. In the model of "all conformations" (see text for description) one calculates N(r, n) using the rules of recurrence. Take a molecule having n -- 1 carbon atoms. Then there exist two ways in which to obtain a molecule having n-atoms and r gauche rotations: (i) If the molecule with n -- 1 carbon atoms exists in a form having r gauche rotations, one can then add the additional carbon in the plane parallel to that formed by the first two carbon bonds Journal of Colloid and Interface Science, V o l . 27, N o . 3, 3"uly 1968
540
BOTHOREL
in the molecule. (ii) If the molecule with n - 1 carbon atoms exists in a form having r - 1 gauche rotations, then in order to have r gauche rotations it is necessary to place the additional carbon atom in a direction out of the plane formed by the last two carbon bonds. Then it follows that N ( r , n ) = N ( r , n - 1) [32] + 2N(r1, n 1). If this molecule having n - 1 carbon atoms is oriented in such a way that the additional C - C bond formed when the n t h carbon is added lies in the direction 111 (see Fig. 6), only the quantity n~ of Eq. [25] is modified. Then Eq. [25] becomes
a(r, n) = G(r, n -
1)
8
1)
+ ~Nl(r, n -
+ (7-
[34] 1, n -
1)
-[- 5 N 3 ( r -- 1, n -
1)
+ 2mG(r-
8
+ (7 - 2n) N ( r _
2
8
1)21
[331
A
7 -- 2n
= A~_~ + ~ n~ + ~ - -
Let N l ( r , n ) be the total number of C - C bonds in the direction 1 in all possible isomers of a molecule having n-atoms of carbon and having passed through r gauche rotations. Then N2(r, n ) , N3(r, n ) , and N d r , n) signify the same quantity relative, respectively, to directions 2, 3, and 4. I t follows that the contribution to G(r, n ) of all the conformations of a molecule having n - 1 atoms of carbon and r gauche rotations (r, n - 1) when one adds an additional carbon atom is expressed by 8 G ( r , n - 1) + ~ N l ( r , n 1) 7 - 2n 1). 3 Then going through the same process for the isomers formed by adding the additional bond in direction 3 and 4 one can by use of the rules of recurrence calculate the quantity
+ --N(r,n-
G(r, n ) n To calculate this expression, the first bond of the molecule is oriented in direction 2 and t h e second in direction 1. E a c h additional carbon atom a t o m is joined to the first carbon atom of the molecule. Journal of Colloid and Interface Science, Vol. 27, No. 3, July 1968
1)[
-
1)
N l ( r , n ) = N2(r, n -
N2(r,n) = Nl(r,n-
l(n_ 3
1, n
3 The relations which permit the calculations of the parameters N d r , n ) [i = 1, 2, 3, 4]:
+ 2N2(r -
An2 = ~4 I ( n ~ + l ) ~ + n ~ 2 + m 2 + n ~ 2
1)
2n) N ( r , n _
3
1, n -
1),
1) + N ( r , n - 1)
+ 2N3(r-
1, n -
1)
+ 2N(r-
1, n -
1),
N 3 ( r , n ) = N3(r, n -
[36]
1)
-k Na(r - 1, n - 1) + Nt(r-
[35]
1, n - -
[37]
1)
The relations of recurrence [32], [34]-[37] permit the calculation of the average molecular optical anisotropy of any carbon chain of length greater than n = 3. For a given value of n, the number of calculations necessary is given by the ratio [5(n - 2 ) ( n - 112 (r varies from 0 to n - 3). The number of calculations necessary when one does not use the rules of recurrence is 3 n-3. The time saved in calculations by using the rules of recurrence is obviously considerable. I t is also possible to determine the rules of recurrence for the model of "permitted conformations," but the operation is considerably more difficult. In this case the number of calculations varies as n 2, instead of 3~-3. ACKNOWLEDGMENTS The author is pleased to acknowledge Miss H. Lindsey for the translation of this article. REFERENCES l. (D~) CABANNES, J., " L a Diffusion Mol~culaire de la Lumi~re. Presses univ., Paris (1929); (b) KAUZMANN, W., " Q u a n t u m C h e m i s t r y . "
MOLECULAR OPTICAL ANISOTROPY AND N-ALKANES Academic Press, New York (1957); (c)
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Journal of Colloidand Interface Science, Vol. 27, No. 3, July 1968