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Low-resolution low-frequency depolarized light (VH) scattering from binary mixtures of fluids and from pure fluids of nonrigid molecules
Physica 78 (1974) 527-532 © North-Holland Publishing Co.
L O W - R E S O L U T I O N L O W - F R E Q U E N C Y D E P O L A R I Z E D L I G H T (VH) S C A T T E R I N G F R O M B I N A R Y M I X T U R E S OF F L U I D S A N D F R O M P U R E F L U I D S OF N O N R I G I D M O L E C U L E S * N.D. GERSHON and I. OPPENHEIM Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva, Israel and Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA Received 22 Augustus 1974
Synopsis A simple formalism is presented here which describes low-resolution low-frequencydepolarized light scattering from binary mixtures of fluids of nonspherical molecules and from pure fluids of nonrigid molecules, each one containing two segments. It is shown that when the k dependent features are negligible these spectra can contain two lorentzians because only dissipative modes are possible. The results presented here can also be used to describe dielectric relaxation from similar systems. I. Introduction. This work presents a simple formalism that can explain lowresolution low-frequency depolarized light scattering from two types of systems: 1) binary mixtures of fluids of nonspherical molecules and 2) pure fluids of nonrigid molecules, each one containing two segments. These results can also be applied to dielectric relaxation from these systems. In this paper the coupling with hydrodynamic modes is neglected. The theory describes the results of measurements on fluids at temperatures far from the freezing point performed with low-resolution spectrometers so that the wave vector dependent fine structure caused by the coupling to the hydrodynamic modes does not show up in the spectrum. The importance of such simple experiments lies in the fact that they can supply information on the coupling of the rotational motion of two different species in a fluid, in the case of binary mixtures, and on the internal rotational motion in nonrigid molecules and its relation to the overall rotation of the molecules. * A portion of this work was supported by the National Science Foundation. 527
528
N.D. GERSHON AND I. OPPENHEIM
In the next section, the relaxation spectra for systems in which there are two important slow variables are presented. In the 3rd section the low-frequency depolarized light scattering spectra for binary mixtures and for pure fluids of nonrigid molecules are given.
2. General formulae. The system consists of N molecules having two possible slow variables A'k(t) and A'k'(t) of the same symmetry. It is useful for the applications described in section 3 to choose Ak(t) = A'k(t) + A'k'(t) and Bk(t) = Ak(t) -- + A#(t) +
as the variables of interest. These variables have the important property of being orthogonal at t = O, i.e. = O. The frequency dependence of the autocorrelation function of Ak is given by1): = ( -
1 ) --io>l + M A A,
(2.1)
where the subscript AA denotes the AA element of the 2 x 2 matrix ( - i t o / + M)- 1. ! is the unit matrix and M is the hydrodynamic matrix for the two variables Ak and Bk which is independent of frequency (~) at low frequencies when A~ and Bk are the only slow variables of the system. The elements of M are easily calculated from eqs. (3.1) and (3.2) in Selwyn and Oppenheiml). They are: MA,~ = lim [<.~lk~,,4_k> -- ] G - 1 ,
(2.3)
to~O
M s s = lira
[ - ] G - ' ,
(2.4)
to~O
Maa = lira - [ - <,4k~,B-k> ] G -1,
(2.5)
OJ---~0
Msa = lim - [ -- ] G - ' ,
(2.6)
to~O
where
G = [ -- ] + .
(2.7)
Given M it is possible to invert the matrix - i o d + M and to obtain the frequency dependence of. The resulting expression is a ratio of two polynominals in co. The numerator is of 1st order and the denominator is of 2nd order in to. In order to separate the expression into different contributions or modes the roots
LOW-FREQUENCY DEPOLARIZED LIGHT SCATTERING
529
of the quadratic expression in the denominator are obtained:
RI. 2 = -½i (MAA + MBB) -T- ½i [(MAA -- Msa) 2 + 4MAaMB*]~ -
iR1.2-
(2.8)
It is easy to show in the case considered here that all the elements of M are real, that MA,, MBB > 0, and that MAB and MBA have the same sign since MAB (B 2) = MBA (A2). This is shown in the appendix. Thus, the two roots, R1.2, are imaginary and are equal only in the trivial case MAn = MB, = 0 and MaA = MaB. The frequency dependence of (Ak~,A_ ~) is:
(AmA_~)=D
R~
+C
o,2 + ~
R2 o,~ + ~
+i{D
09 ~,~ + ~
+C
o) } ~,~ + ~ (2.9)
where D =
C
=
M.. - ~,, /~1 - R2
MBB
--
R~ (AkA-k>.
(2.10)
(2.11)
R1 -- R2 The results of this section are applied to low-frequency depolarized light scattering in the next section.
3. The low-frequency depolarized light scattering from binary mixtures and pure fluids of nonrigid molecules. In the previous section the low-frequency dependence of the autocorrelation function of the variable Ak = Ak + A;,' was given, for a system in which there are only two slow dynamical variables A;, and Ak, having the same symmetry. In the depolarized spectrum we observe correlation functions of different components of the anisotropic polarizability density ~k of the system. In the VH (vertical-horizontal) geometry 2) the active components are ~k, al and ock,12 where k is the magnitude of the wave vector k = k~ - kf,
(3.1)
where ki and kf are the wave vectors of the incident and the outgoing beam, respectively; 1, 2, 3 are the x, y, z directions of a laboratory cartesian set of axes where k is along z; and zy are in the scattering plane.
530
N.D. G E R S H O N A N D I. O P P E N H E I M
The VH spectrum, Ivn, is given in terms of the real parts of (o~k,~.31o~-k, 13)
and
(o~k~,.12o¢-/¢,21)"
Ivn oC COS2½0 Re (~to. 310¢_< 13) + sinZ ½0 Re (O~ko.,2O~_k.21),
(3.2)
where 0 is the scattering angle. If the k dependence of these correlation functions is very small, as in the cases discussed in section 1, then the formulae of the previous section can be used. When the roots R1.2 are real the spectrum has the following form: / v l t OC COS2 ½0D31
+ sin2 ½0D12
--31 R1
+
+ c o s 2 ½0C 31
*~231 +
]~]~12)2 + sin2 ½0C12 ~12 oJ+t 1 a,2+(R~2)2'
(3.3)
where the superscript ij denotes that the variables ~ . ~j and o~. ~j are used to calculate the constants D, C, -R1 and .t~2 given by eqs. (2.8), (2.10), (2.11) of section 2. From symmetry (o~k.al0~-k. 137 = (O~k.1_,~-k.21), D 31 Ol2, etc. and the two parts of Ivn can be combined with a geometrical factor of 1. In binary mixtures of two anisotropic components the primes in ~ , ~j and o~,~j refer to components 1 and 2, respectively, and thus, in these systems the low-resolution low-frequency depolarized VH spectrum has the form of two positive lorentzians. For a binary mixture of two anisotropic components a similar result has been obtained by Kivelson and Tsay a) using Mori-Zwanzig projection operator techniques. Their expresdons for M are different and less convenient for calculation than those derived in this work. In a system of a pure fluid of molecules each one containing two segments capable of slow internal rotation the slow variables are the polarizability density of the whole molecule, Ek, and the polarizability density of one of the segments, E~, or their linear combination when the hydrodynamic variables can be neglected. The two roots R1.2 are real and the low-frequency depolarized light scattering spectrum is given by eq. (3.3). The two roots are equal only when MAA = MBB and MaB = MnA = 0. This case corresponds to the case when the two variables are uncoupled, and the VH spectrum consists of one lorentzian. =
4. Discussion. In the previous sections the line shapes for low-resolution lowfrequency depolarized (VH) light scattering were developed for binary mixtures of anisotropic species and for pure fluids of nonrigid molecules with two segments. These expressions are expected to be valid when the interaction of the anisotropic polarizability density with the hydrodynamic modes is too weak to influence the results of low-resolution experiments.
LOW-FREQUENCY DEPOLARIZED LIGHT SCATTERING
531
The modes of the system are real and when they are different the spectrum consists of two positive lorentzians with different widths. The intensity of the two lorentzians depends on the constants D, C [eqs. (2.10) and (2.11)]. The apparent width of the line shape does not have the simple form as in a pure fluid (see Gershon and Oppenheim2)). By fitting the theoretical line shapes to experimental results on these systems, the constants Maa, Man, MAn and Mna can be extracted and furnish information on the orientational motions in these fluids. The results of section 2 can also be applied to dielectric relaxation from the same systems as mentioned above, where the variables are dipole moments.
APPENDIX In this appendix it is shown that the discriminant of eq. (2.8) is always positive. Thus the roots -R1.2 of eq. (2.8) are real. In order to see this point it is necessary to show that the term MaBM~a is positive. If A and B have the same symmetry in time
(A.1)
(Ak,~B_k> = (,Bk,oA-k>.
The low-frequency dependence of
(AA~'B-k) = (
(AkoB_k) and of (BkcoA_k) is given by1):
I + M ) as (BkB-k)
(Bk°A-k)=( --iogll + M )B = (AkA-k) a
--MBA det I - i t o l + MI --MAa det [ - k o / +
M[
(BkB_ k),
(A.2)
(AkA_l,).
(A.3)
As the static averages in (A.2) and (A.3) are positive the equality (A.1) leads to the result that MA~ and MBA have the same sign. The next step is to show that in this system M is real. If Ak is a vector which includes Ak and Bk as its components, then x) (A.4) Since in the limit as k ~ 0
(Ak,~Ar--k)* = (AI,-~,A~k), + k o / + M(-~o) = ion/+
tA.5)
M*(og)
(A.6)
and M(-co) = M*(o;).
(A.7)
532
N.D. GERSHON AND I. OPPENHEIM
It was mentioned before that if Ak contains all the slow dynamical variables o f the system M is independent o f frequency at low o~. A n d so, at low frequencies
(A.8)
M*=M
and therefore M is real. Thus from (A.1), (A.2), (A.3) and (A.8) (A.9)
MABMBA >10. REFERENCES 1) Selwyn, P.A. and Oppenheim, I., Physica 54 (1971) 161. 2) See e.g. Gershon, N.D. and Oppenheim, I., Physica 64 (1973) 247. 3) Kivelson, D. and Tsay, S.J., Preprint.
L O W - R E S O L U T I O N L O W - F R E Q U E N C Y D E P O L A R I Z E D L I G H T (VH) S C A T T E R I N G F R O M B I N A R Y M I X T U R E S OF F L U I D S A N D F R O M P U R E F L U I D S OF N O N R I G I D M O L E C U L E S * N.D. GERSHON and I. OPPENHEIM Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva, Israel and Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA Received 22 Augustus 1974
Synopsis A simple formalism is presented here which describes low-resolution low-frequencydepolarized light scattering from binary mixtures of fluids of nonspherical molecules and from pure fluids of nonrigid molecules, each one containing two segments. It is shown that when the k dependent features are negligible these spectra can contain two lorentzians because only dissipative modes are possible. The results presented here can also be used to describe dielectric relaxation from similar systems. I. Introduction. This work presents a simple formalism that can explain lowresolution low-frequency depolarized light scattering from two types of systems: 1) binary mixtures of fluids of nonspherical molecules and 2) pure fluids of nonrigid molecules, each one containing two segments. These results can also be applied to dielectric relaxation from these systems. In this paper the coupling with hydrodynamic modes is neglected. The theory describes the results of measurements on fluids at temperatures far from the freezing point performed with low-resolution spectrometers so that the wave vector dependent fine structure caused by the coupling to the hydrodynamic modes does not show up in the spectrum. The importance of such simple experiments lies in the fact that they can supply information on the coupling of the rotational motion of two different species in a fluid, in the case of binary mixtures, and on the internal rotational motion in nonrigid molecules and its relation to the overall rotation of the molecules. * A portion of this work was supported by the National Science Foundation. 527
528
N.D. GERSHON AND I. OPPENHEIM
In the next section, the relaxation spectra for systems in which there are two important slow variables are presented. In the 3rd section the low-frequency depolarized light scattering spectra for binary mixtures and for pure fluids of nonrigid molecules are given.
2. General formulae. The system consists of N molecules having two possible slow variables A'k(t) and A'k'(t) of the same symmetry. It is useful for the applications described in section 3 to choose Ak(t) = A'k(t) + A'k'(t) and Bk(t) = Ak(t) --
as the variables of interest. These variables have the important property of being orthogonal at t = O, i.e.
1 ) --io>l + M A A
(2.1)
where the subscript AA denotes the AA element of the 2 x 2 matrix ( - i t o / + M)- 1. ! is the unit matrix and M is the hydrodynamic matrix for the two variables Ak and Bk which is independent of frequency (~) at low frequencies when A~ and Bk are the only slow variables of the system. The elements of M are easily calculated from eqs. (3.1) and (3.2) in Selwyn and Oppenheiml). They are: MA,~ = lim [<.~lk~,,4_k>
(2.3)
to~O
M s s = lira
[
(2.4)
to~O
Maa = lira - [
(2.5)
OJ---~0
Msa = lim - [
(2.6)
to~O
where
G = [
(2.7)
Given M it is possible to invert the matrix - i o d + M and to obtain the frequency dependence of
LOW-FREQUENCY DEPOLARIZED LIGHT SCATTERING
529
of the quadratic expression in the denominator are obtained:
RI. 2 = -½i (MAA + MBB) -T- ½i [(MAA -- Msa) 2 + 4MAaMB*]~ -
iR1.2-
(2.8)
It is easy to show in the case considered here that all the elements of M are real, that MA,, MBB > 0, and that MAB and MBA have the same sign since MAB (B 2) = MBA (A2). This is shown in the appendix. Thus, the two roots, R1.2, are imaginary and are equal only in the trivial case MAn = MB, = 0 and MaA = MaB. The frequency dependence of (Ak~,A_ ~) is:
(AmA_~)=D
R~
+C
o,2 + ~
R2 o,~ + ~
+i{D
09 ~,~ + ~
+C
o) } ~,~ + ~ (2.9)
where D =
C
=
M.. - ~,
MBB
--
R~ (AkA-k>.
(2.10)
(2.11)
R1 -- R2 The results of this section are applied to low-frequency depolarized light scattering in the next section.
3. The low-frequency depolarized light scattering from binary mixtures and pure fluids of nonrigid molecules. In the previous section the low-frequency dependence of the autocorrelation function of the variable Ak = Ak + A;,' was given, for a system in which there are only two slow dynamical variables A;, and Ak, having the same symmetry. In the depolarized spectrum we observe correlation functions of different components of the anisotropic polarizability density ~k of the system. In the VH (vertical-horizontal) geometry 2) the active components are ~k, al and ock,12 where k is the magnitude of the wave vector k = k~ - kf,
(3.1)
where ki and kf are the wave vectors of the incident and the outgoing beam, respectively; 1, 2, 3 are the x, y, z directions of a laboratory cartesian set of axes where k is along z; and zy are in the scattering plane.
530
N.D. G E R S H O N A N D I. O P P E N H E I M
The VH spectrum, Ivn, is given in terms of the real parts of (o~k,~.31o~-k, 13)
and
(o~k~,.12o¢-/¢,21)"
Ivn oC COS2½0 Re (~to. 310¢_< 13) + sinZ ½0 Re (O~ko.,2O~_k.21),
(3.2)
where 0 is the scattering angle. If the k dependence of these correlation functions is very small, as in the cases discussed in section 1, then the formulae of the previous section can be used. When the roots R1.2 are real the spectrum has the following form: / v l t OC COS2 ½0D31
+ sin2 ½0D12
--31 R1
+
+ c o s 2 ½0C 31
*~231 +
]~]~12)2 + sin2 ½0C12 ~12 oJ+t 1 a,2+(R~2)2'
(3.3)
where the superscript ij denotes that the variables ~ . ~j and o~. ~j are used to calculate the constants D, C, -R1 and .t~2 given by eqs. (2.8), (2.10), (2.11) of section 2. From symmetry (o~k.al0~-k. 137 = (O~k.1_,~-k.21), D 31 Ol2, etc. and the two parts of Ivn can be combined with a geometrical factor of 1. In binary mixtures of two anisotropic components the primes in ~ , ~j and o~,~j refer to components 1 and 2, respectively, and thus, in these systems the low-resolution low-frequency depolarized VH spectrum has the form of two positive lorentzians. For a binary mixture of two anisotropic components a similar result has been obtained by Kivelson and Tsay a) using Mori-Zwanzig projection operator techniques. Their expresdons for M are different and less convenient for calculation than those derived in this work. In a system of a pure fluid of molecules each one containing two segments capable of slow internal rotation the slow variables are the polarizability density of the whole molecule, Ek, and the polarizability density of one of the segments, E~, or their linear combination when the hydrodynamic variables can be neglected. The two roots R1.2 are real and the low-frequency depolarized light scattering spectrum is given by eq. (3.3). The two roots are equal only when MAA = MBB and MaB = MnA = 0. This case corresponds to the case when the two variables are uncoupled, and the VH spectrum consists of one lorentzian. =
4. Discussion. In the previous sections the line shapes for low-resolution lowfrequency depolarized (VH) light scattering were developed for binary mixtures of anisotropic species and for pure fluids of nonrigid molecules with two segments. These expressions are expected to be valid when the interaction of the anisotropic polarizability density with the hydrodynamic modes is too weak to influence the results of low-resolution experiments.
LOW-FREQUENCY DEPOLARIZED LIGHT SCATTERING
531
The modes of the system are real and when they are different the spectrum consists of two positive lorentzians with different widths. The intensity of the two lorentzians depends on the constants D, C [eqs. (2.10) and (2.11)]. The apparent width of the line shape does not have the simple form as in a pure fluid (see Gershon and Oppenheim2)). By fitting the theoretical line shapes to experimental results on these systems, the constants Maa, Man, MAn and Mna can be extracted and furnish information on the orientational motions in these fluids. The results of section 2 can also be applied to dielectric relaxation from the same systems as mentioned above, where the variables are dipole moments.
APPENDIX In this appendix it is shown that the discriminant of eq. (2.8) is always positive. Thus the roots -R1.2 of eq. (2.8) are real. In order to see this point it is necessary to show that the term MaBM~a is positive. If A and B have the same symmetry in time
(A.1)
(Ak,~B_k> = (,Bk,oA-k>.
The low-frequency dependence of
(AA~'B-k) = (
(AkoB_k) and of (BkcoA_k) is given by1):
I + M ) as (BkB-k)
(Bk°A-k)=( --iogll + M )B = (AkA-k) a
--MBA det I - i t o l + MI --MAa det [ - k o / +
M[
(BkB_ k),
(A.2)
(AkA_l,).
(A.3)
As the static averages in (A.2) and (A.3) are positive the equality (A.1) leads to the result that MA~ and MBA have the same sign. The next step is to show that in this system M is real. If Ak is a vector which includes Ak and Bk as its components, then x) (A.4) Since in the limit as k ~ 0
(Ak,~Ar--k)* = (AI,-~,A~k), + k o / + M(-~o) = ion/+
tA.5)
M*(og)
(A.6)
and M(-co) = M*(o;).
(A.7)
532
N.D. GERSHON AND I. OPPENHEIM
It was mentioned before that if Ak contains all the slow dynamical variables o f the system M is independent o f frequency at low o~. A n d so, at low frequencies
(A.8)
M*=M
and therefore M is real. Thus from (A.1), (A.2), (A.3) and (A.8) (A.9)
MABMBA >10. REFERENCES 1) Selwyn, P.A. and Oppenheim, I., Physica 54 (1971) 161. 2) See e.g. Gershon, N.D. and Oppenheim, I., Physica 64 (1973) 247. 3) Kivelson, D. and Tsay, S.J., Preprint.