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On electromagnetic radiation scattering in liquid glassy matrices
Physica B 162 (1990) 181-187 North-Holland
ON ELECTROMAGNETIC RADIATION SCATTERING IN LIQUID GLASSY MATRICES S.O. GLADKOV Institute of Chemical Physics, USSR Academy of Sciences, 4 Kosygina str., Moscow,
Received 22 June 1989 Revised manuscript received 27 December
USSR
1989
The dielectric constants of liquid glass materials have been calculated with the help of the free volume concept and it is shown that the imaginary part of susceptibility a”(w) has a strongly oscillating nature at low frequencies. The calculations are based on the classical relaxation equation of the Boltzmann type. The approach is used to explain a number of experiments devoted to measurements of light scattering cross-sections in polymers.
1. Introduction In essence, the question we would like to discuss here is closely related with the experimental studies of electromagnetic radiation absorption by liquid glassy substances similar in properties both with protein compounds and polymers [l]. The experiments [2-41 devoted to measurements of the incident radiation absorption cross-section in proteins showed that a(w) rapidly increases at low frequencies and then, beginning from a certain frequency, approaches the saturation point. In this connection it is to be noted that the “conventional” behavior of a(w) at low frequencies is proportional to w*~i, where 3-0is a certain relaxation time typical of a given material, and does not rapidly approach zero at these frequencies. In fig. 1 the relationship is plotted by a dotted line. Below we demonstrate that the free volume model [5-71 may effectively be applied to the problem at hand to explain the sharp decrease of electromagnetic radiation absorption by liquid glassy matrices in the frequency range w < 1 /T,,. In essence the model can be described as follows (cf. ref. [5]). When melted virtually all kinds of solid crystals increase in volume. This excessive or, as it is frequently called, free volume is randomly distributed in the bulk of the material in the form of variably sized cavities. The size of
the cavities is determined by the interaction energy between the liquid molecules and is macroscopically characterized by surface tension and viscosity. Note, a propos, that the cavity size versus viscosity relationship is proportional. Investigation of viscosity of glassy matrices as a function of temperature and pressure was carried out in refs. [&lo] where the authors used the free volume concept and assumed a classical Boltzmann volume distribution of these “vacancies” to calculate the basic thermodynamical functions of substances with the ultimate objec-
Fig. 1. Absorption cross-section as a function of incident radiation frequency for a Lorentzian size distribution of the free volumes.
0921-4526/90/$03.50 (Q 1990 - Elsevier Science Publishers B.V. (North-Holland)
S. 0.
182
Gladkov
I Radiation scattering in liquid glassy matrices
tive of explaining the results of some earlier experiments [ll, 121. Our problem belongs to the class dealing with nonequilibrium effects and therefore must involve a nonsteady relaxation equation for describing the size distribution of free volumes to allow us to calculate the imaginary part of the dielectric susceptibility tensor (Y”(W). For convenience this paper will be set forth in two sections: in one the cu”(o) will be calculated and in the second the light scattering cross-section of the free volumes will be determined.
the correction
Sf,:
af, rt -$l+Sj”=--.
Sf” 7”
Putting Sf,(t) = Sf,(O) e-‘“’ and taking into account (2), we obtain the wanted:
af,&5'(t)iw
Sf”(f) = z
p
cos 8
iw + 1 /T, ’
Now since
(d,) = ii + a(o)u,E’(t) 2. The calculation
Imagine a liquid in a constant electric field E,. The direction of this field will be the z-axis. In this case each of the “vacancies” will increase in size in the direction of the field and will be a dipole with polarization P (cf. ref. [13]). If we now also apply a variable electric field E: = with an amplitude Ei G E, along the G e?’ z-axis, it will clearly cause the free volume to oscillate with a certain frequency and thereby absorb the energy of the variable field E:(t). To calculate the zz-component of the dielectric susceptibility CX’:,(W) = (U’(W)due to this absorption we will employ the relaxation equation and apply it to the size distribution of the free volumes (compare with ref. [14]):
af,_ at --
(5)
of CW”(CO)
we, in order to find a(o), calculate (d,) with the help of the distribution function f, = f, + Sf, . As a result we have:
(4) =
d(jcos f,(04u,)
du)
(6)
0
where (. . .) means thermodynamic over angles 8:
averaging
?i
I
(. . .) em
dE,,cosBlT
sin
8
do
K. 4) = O ?i
I
-Sf”
e
(7) -d&,cosfIIT
sin
e
de
0
7, ’
where Sf, = f, - f, , and the calculated in the appendix. tribution function f, will be Since the current volume given by:
relaxation time 7” is The equilibrium disdiscussed below. in a periodic field is
n(uo) is the density of the size distribution of free volumes. Normally, n(uo) = 1 lu,. Comparing eqs. (5) and (6) and using the solution (4), we finally arrive at a(w) =
v= u _ d(&, + E’(t)) cos 0 P
9
where p is the pressure in the liquid, d is dipole moment, 8 is the angle between d and substituting the distribution function f, in form f, = f, + Sf, gives the following formula
(2)
d2
-I
m af, . n(uo) au $?&
PUOo
the
u
(cos2~)
du
. (8)
E,,
the for
Calculating, further, the average square cosine with help of (7), we get for the susceptibility:
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Gladkov
183
I Radiation scattering in liquid glassy matrices
In view of (14) the asymptotes $ i
where
*yw)
-
(C - In 0~~) 0
d’e(B) PUO
p(P)=l+lfr-$cthp.
when wr(, % 1 @TO
/3 = dE,IT,
The function cp(p) behaves as: when p P 1,
[l-2//3
f+iW2
(11)
when p < 1.
To obtain an analytical formula for (Y(W) we first have to prescribe a certain form for the distribution function f, and, second, find out the functional dependence of the relaxation time on the volume u. Consider two special cases: (a) Let f, = e-“‘“O )
n(uo) = 1 lu,
1 )
7
=
UO y
1 ; 0
”
(12)
(for details of the calculation of the time rU see the appendix). In this case calculating integral (9) we obtain the following expression for (Y”:
a”(o)
-
(15)
2sin _z when wro < 1
(10)
cp(P)=
of (13) are:
where C is the Euler constant. From the above formula we see that at high frequencies the susceptibility behaves as (l/w7,)(C - In wrO), whereas in the low frequency range considerable oscillations occur (see fig. 2). From this nonstandard behavior of (Y” one may conclude that the absorption intensity is shifted in the liquid dielectrics towards lower frequencies and has a strongly oscillating nature. The reason for such susceptibility behavior consists, in our opinion, of the following. During the time 7. (which is much shorter than the field variation period o) the free volume which is an “inert” system “shoots” over the position of equilibrium and, to compensate for this effect, the dipole moment instantaneously, within a time St G r,), changes its direction to the opposite in an attempt to make the system return to the equilibrium. Yet the equilibrium is again missed in view of the “heaviness” of the free volume (this mechanism may perhaps be called fluctua-
d”(P) UOP~~O
x
1
1
1
1
1
cos ci - sin si I ON0 WTo ~7 0 07 O (13)
-0
where the integral cosine and sine are:
ci(x) = -
si(x) = -
i x
I x
ydt,
(14) ydt.
Fig. 2. Susceptibility versus Boltzmann size distribution Debye susceptibility versus the predicted oscillating a”
field frequency relationship for a of the free volumes. Curve 1: the frequency relationship; curve 2: behavior.
S. 0. Gladkov
184
I Radiation
scattering
in liquid glassy matrices
tion) and the situation is thus repeated periodically against the background of large period l/o. The order of magnitude of the frequencies for which the above treatment is valid can be evaluated as follows. Since the inverse relaxation time is (see appendix, eq. (30)): 1
-=u 7”
(Y= AT
0
-----
KPAV
(Y= T K/T’
where CY= 1 /7So is the coefficient of heat transfer from the free volume to the liquid phase, So is the free volume surface area, and the coefficient of heat conductivity K = C”U~T, and assuming that T = 300 K, p = 10” erg/cm3, ur = lo5 cm/s, 7 = lo-” s, So = lop8 cm=, C” = 10” l/cm3 we find that l/7, = lo4 Hz. This means that the frequency range Aw of interest must extend all the way from the upper limit of electromagnetic fre uencies to infralow frequen? cies of less than 10 Hz. (b) Now let the distribution function have a Lorentzian form: 1 n(uo)f, = ; &2
.
The relaxation time will naturally be the same as before. Then the substitution of (16) and time 7” from eq. (12) into the integral (9) gives, after the necessary calculation,
Fig. 3. Behavior of (Y”for Lorentzian size distribution of the free volumes.
uted in the matrix and their interactions with each other may be neglected. Besides, the incident radiation wavelength must be greater than the maximum size of free volume. If the free volume concentration increases, the dipoledipole interactions between them begin to play an important role. This in turn means that we can no longer make use of eq. (2) in the calculation of (Y”because it does not take into account the dipole interaction; instead we will have to use a general formula for v” including all the “exchange” components. In the majority of the cases, however, the free volumes occupy much less space in a sample than the substrate, which we believe justifies the approach used in this paper.
(17) 3. Calculation The function (Y”(O) is plotted in fig. 3 for this case. Note that as A increases the peak of function (17) sharply shifts towards zero. Using the above approach based on a phenomenological kinetic equation it is possible not only to calculate (Y”but also to predict the rather nonstandard (oscillating at low frequencies) absorption of the energy of applied electromagnetic variable fields by liquid dielectrics, proteins and polymers also. The calculations in this text are valid only in case the free volumes are quite sparsely distrib-
of the cross-section’s
absorption
Let us now calculate the electromagnetic radiation absorption cross-section by an ensemble of free volumes. To do this we will use a known relationship for a(w) (cf. ref. [15]), which may be written as follows in our case: u(w, u) = u
4TrTTw
4w,
7
u>
(18)
2
where, according to (9): 2 a”(Cd,
u)
=
-
iv0
-;t
1 +m;2T2
v(P). ”
S.O.
Gladkov
I Radiation scattering in liquid glassy matrices
185
Substituting this into (18), multiplying by n(u,,) and integrating both parts with respect to u, we obtain the wanted cross-section of absorption in the form: a(w) = - -
-
4uo>u
*
(19)
If we substitute first the most likely equilibrium distribution function e
Fig. 4. The oscillating behavior section. w, = 1 /TVIT~,n % 1.
cross-
In this case calculation of a(w) gives:
-u/q
4Jo)L= ug
4d2
a’(~’ - 1 - 2 In a)
CPTO A
(a’ - 1)2
a(o) = in (19) and integrate, we find the desired formula for the absorption cross-section: hd2
(22)
where a=-
a(w) = -
Wroh uo
PCUO 70
x[l+&(sin-$ci&+cos--&si&())) The asymptotic behavior of a(o)
is:
4nd2 u(w) = PCUO 70
1--&
X
of the absorption
1
0
In or0 I+2 > for wro % 1 ,
(
1 - cos $
070
It is seen from fig. 1 that in case of a broad scatter in free volume sizes and at low frequencies (07~ 4 l), the a(w) curve shows a tendency for a sharp increase, which was indeed observed experimentally, when A 4 u. the increase of o(w) is insignificant (curve 2, fig. 1) and quite expectable, since for small vacancy dimensions and long wavelengths the scattering is very weak.
for ~7~ -G1 . 0
4. Conclusion (21)
It is seen that at low frequencies, (T(W) like (Y” has an ostensibly oscillating nature. It should be noted that at very low frequencies the oscillation period also becomes very small. This is shown clearly in fig. 4. If we now take into account that the free volumes may have any size, then, to calculate g(w) we must substitute into eq. (19) a Lorentzian form instead of an exponential form of the distribution function f,, i.e. assume:
The approach based on a phenomenological kinetic equation used in this paper allows one to: (1) calculate the amount of external variable electromagnetic field energy absorbed in a liquid dielectric matrix by a macroscopic system of free volumes, (2) find the absorption cross-section as a function of the incident radiation frequency, (3) explain some experiments devoted to the study of the properties of polymers in weak low-frequency fields; and (4) predict the oscillating nature of absorption at low frequencies.
186
S.O. Gladkov
I Radiation
scattering
Appendix To calcuate the time T” one must have a clear idea of the problem to be solved. Imagine a solitary free volume and suppose that its size will decrease as the ambient temperature decreases (see fig. 5). On this assumption it is possible to write down an energy balance equation immediately: (23) which, in a spherical system of coordinates and on the condition that the sphere (free volume) radius varies little, leads to:
in liquid glassy matrices
coefficient of heat exchange from the free volume to the matrix, To is a certain temperature which is defined below. On the conditions that:
TI,=,,= T, TI,,, = T,, and
so that r* c r,(t) c f, the solution gives:
T(t) = T,, -
2 x
+$
=
T-
(24)
where liquid in the On
x is the temperature conductivity of the in the vicinity of a “vacancy”; p is pressure liquid, So is the free volume surface area. the other hand we have a condition:
aT =
K dr
r=r”(t)
where
K =
xc,,
-a(Tc,
To)
(25)
is the heat capacity, (Yis the
_
1
To)e-%,(‘)-‘*’ ,S(r’-‘)
0
of eq. (25)
T,,
(1
ex(“-‘)
+ @_
PV,
(26)
-1
.
(27)
Substituting solution (27) into eq. (24) and assuming that r,(t) is close to r*, we find, after the necessary expansions: i, = - -
1
(r. - r*> .
7”.
(28)
Omitting further the asterisk at r and passing on to the volume u, we finally obtain the desired equation: d = 1 (UC)- u) )
(29)
7”
where the relaxation is CY~ITo-
1
-=u 7”
’
TI
-
K
p(U
time used in the main text
-
6)
(30)
Fig. 5. Fluctuating variation of the free volume dimensions.
Thus, as it follows from the last equation, the relaxation time is a linear function of the free volume size, whereby our use of eq. (12) in the main text is justified.
S. 0.
Gladkov
I Radiation scattering in liquid glassy matrices
References [l] M.V. Volkenshtein, Biophysics (Nauka, Moscow, 1988) (in Russian). [2] H. Keller and P. Debrunner, Phys. Rev. 45 (1980) 68. [3] D.G. Champeney, Rep. Progr. Phys. 42 (1979) 1017. [4] F. Pavak, E.W. Knapp and D. Kucheida, J. Molec. Biol. 161 (1982) 177. [5] Ya.1. Frenkel, Kinetic Theory of Liquids (Nauka, Moscow, 1975) (in Russian). [6] M.H. Cohen and G.S. Grest, Phys. Rev. Lett. 45 (1980) 1271.
187
[7] M.H. Cohen and G.S. Grest, Solid State Commun. 39 (1981) 143. [8] M.H. Cohen and D. Turnbull, J. Chem. Phys. 31 (1959) 1164. [9] S.O. Gladkov, Physica B 160 (1989) 211. [lo] S.O. Gladkov, Phys. Lett. 140 (1989) 108. [ll] H. Vogel, Phys. Zur. 22 (1921) 645. [12] G.S. Fulcher, J. Amer. Ceram. Sot. 8 (1925) 339. [13] A.N. Gubkin, Izv. Vuzov, Ser. Phys. 1 (1979) 56. [14] S.O. Gladkov, Phys. Rep. 132 (1986) 277. [15] L.D. Landau and E.M. Livshits, Fluid Electrodynamics, Vol. 8 (Nauka, Moscow, 1982) (in Russian).
ON ELECTROMAGNETIC RADIATION SCATTERING IN LIQUID GLASSY MATRICES S.O. GLADKOV Institute of Chemical Physics, USSR Academy of Sciences, 4 Kosygina str., Moscow,
Received 22 June 1989 Revised manuscript received 27 December
USSR
1989
The dielectric constants of liquid glass materials have been calculated with the help of the free volume concept and it is shown that the imaginary part of susceptibility a”(w) has a strongly oscillating nature at low frequencies. The calculations are based on the classical relaxation equation of the Boltzmann type. The approach is used to explain a number of experiments devoted to measurements of light scattering cross-sections in polymers.
1. Introduction In essence, the question we would like to discuss here is closely related with the experimental studies of electromagnetic radiation absorption by liquid glassy substances similar in properties both with protein compounds and polymers [l]. The experiments [2-41 devoted to measurements of the incident radiation absorption cross-section in proteins showed that a(w) rapidly increases at low frequencies and then, beginning from a certain frequency, approaches the saturation point. In this connection it is to be noted that the “conventional” behavior of a(w) at low frequencies is proportional to w*~i, where 3-0is a certain relaxation time typical of a given material, and does not rapidly approach zero at these frequencies. In fig. 1 the relationship is plotted by a dotted line. Below we demonstrate that the free volume model [5-71 may effectively be applied to the problem at hand to explain the sharp decrease of electromagnetic radiation absorption by liquid glassy matrices in the frequency range w < 1 /T,,. In essence the model can be described as follows (cf. ref. [5]). When melted virtually all kinds of solid crystals increase in volume. This excessive or, as it is frequently called, free volume is randomly distributed in the bulk of the material in the form of variably sized cavities. The size of
the cavities is determined by the interaction energy between the liquid molecules and is macroscopically characterized by surface tension and viscosity. Note, a propos, that the cavity size versus viscosity relationship is proportional. Investigation of viscosity of glassy matrices as a function of temperature and pressure was carried out in refs. [&lo] where the authors used the free volume concept and assumed a classical Boltzmann volume distribution of these “vacancies” to calculate the basic thermodynamical functions of substances with the ultimate objec-
Fig. 1. Absorption cross-section as a function of incident radiation frequency for a Lorentzian size distribution of the free volumes.
0921-4526/90/$03.50 (Q 1990 - Elsevier Science Publishers B.V. (North-Holland)
S. 0.
182
Gladkov
I Radiation scattering in liquid glassy matrices
tive of explaining the results of some earlier experiments [ll, 121. Our problem belongs to the class dealing with nonequilibrium effects and therefore must involve a nonsteady relaxation equation for describing the size distribution of free volumes to allow us to calculate the imaginary part of the dielectric susceptibility tensor (Y”(W). For convenience this paper will be set forth in two sections: in one the cu”(o) will be calculated and in the second the light scattering cross-section of the free volumes will be determined.
the correction
Sf,:
af, rt -$l+Sj”=--.
Sf” 7”
Putting Sf,(t) = Sf,(O) e-‘“’ and taking into account (2), we obtain the wanted:
af,&5'(t)iw
Sf”(f) = z
p
cos 8
iw + 1 /T, ’
Now since
(d,) = ii + a(o)u,E’(t) 2. The calculation
Imagine a liquid in a constant electric field E,. The direction of this field will be the z-axis. In this case each of the “vacancies” will increase in size in the direction of the field and will be a dipole with polarization P (cf. ref. [13]). If we now also apply a variable electric field E: = with an amplitude Ei G E, along the G e?’ z-axis, it will clearly cause the free volume to oscillate with a certain frequency and thereby absorb the energy of the variable field E:(t). To calculate the zz-component of the dielectric susceptibility CX’:,(W) = (U’(W)due to this absorption we will employ the relaxation equation and apply it to the size distribution of the free volumes (compare with ref. [14]):
af,_ at --
(5)
of CW”(CO)
we, in order to find a(o), calculate (d,) with the help of the distribution function f, = f, + Sf, . As a result we have:
(4) =
d(jcos f,(04u,)
du)
(6)
0
where (. . .) means thermodynamic over angles 8:
averaging
?i
I
(. . .) em
dE,,cosBlT
sin
8
do
K. 4) = O ?i
I
-Sf”
e
(7) -d&,cosfIIT
sin
e
de
0
7, ’
where Sf, = f, - f, , and the calculated in the appendix. tribution function f, will be Since the current volume given by:
relaxation time 7” is The equilibrium disdiscussed below. in a periodic field is
n(uo) is the density of the size distribution of free volumes. Normally, n(uo) = 1 lu,. Comparing eqs. (5) and (6) and using the solution (4), we finally arrive at a(w) =
v= u _ d(&, + E’(t)) cos 0 P
9
where p is the pressure in the liquid, d is dipole moment, 8 is the angle between d and substituting the distribution function f, in form f, = f, + Sf, gives the following formula
(2)
d2
-I
m af, . n(uo) au $?&
PUOo
the
u
(cos2~)
du
. (8)
E,,
the for
Calculating, further, the average square cosine with help of (7), we get for the susceptibility:
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Gladkov
183
I Radiation scattering in liquid glassy matrices
In view of (14) the asymptotes $ i
where
*yw)
-
(C - In 0~~) 0
d’e(B) PUO
p(P)=l+lfr-$cthp.
when wr(, % 1 @TO
/3 = dE,IT,
The function cp(p) behaves as: when p P 1,
[l-2//3
f+iW2
(11)
when p < 1.
To obtain an analytical formula for (Y(W) we first have to prescribe a certain form for the distribution function f, and, second, find out the functional dependence of the relaxation time on the volume u. Consider two special cases: (a) Let f, = e-“‘“O )
n(uo) = 1 lu,
1 )
7
=
UO y
1 ; 0
”
(12)
(for details of the calculation of the time rU see the appendix). In this case calculating integral (9) we obtain the following expression for (Y”:
a”(o)
-
(15)
2sin _z when wro < 1
(10)
cp(P)=
of (13) are:
where C is the Euler constant. From the above formula we see that at high frequencies the susceptibility behaves as (l/w7,)(C - In wrO), whereas in the low frequency range considerable oscillations occur (see fig. 2). From this nonstandard behavior of (Y” one may conclude that the absorption intensity is shifted in the liquid dielectrics towards lower frequencies and has a strongly oscillating nature. The reason for such susceptibility behavior consists, in our opinion, of the following. During the time 7. (which is much shorter than the field variation period o) the free volume which is an “inert” system “shoots” over the position of equilibrium and, to compensate for this effect, the dipole moment instantaneously, within a time St G r,), changes its direction to the opposite in an attempt to make the system return to the equilibrium. Yet the equilibrium is again missed in view of the “heaviness” of the free volume (this mechanism may perhaps be called fluctua-
d”(P) UOP~~O
x
1
1
1
1
1
cos ci - sin si I ON0 WTo ~7 0 07 O (13)
-0
where the integral cosine and sine are:
ci(x) = -
si(x) = -
i x
I x
ydt,
(14) ydt.
Fig. 2. Susceptibility versus Boltzmann size distribution Debye susceptibility versus the predicted oscillating a”
field frequency relationship for a of the free volumes. Curve 1: the frequency relationship; curve 2: behavior.
S. 0. Gladkov
184
I Radiation
scattering
in liquid glassy matrices
tion) and the situation is thus repeated periodically against the background of large period l/o. The order of magnitude of the frequencies for which the above treatment is valid can be evaluated as follows. Since the inverse relaxation time is (see appendix, eq. (30)): 1
-=u 7”
(Y= AT
0
-----
KPAV
(Y= T K/T’
where CY= 1 /7So is the coefficient of heat transfer from the free volume to the liquid phase, So is the free volume surface area, and the coefficient of heat conductivity K = C”U~T, and assuming that T = 300 K, p = 10” erg/cm3, ur = lo5 cm/s, 7 = lo-” s, So = lop8 cm=, C” = 10” l/cm3 we find that l/7, = lo4 Hz. This means that the frequency range Aw of interest must extend all the way from the upper limit of electromagnetic fre uencies to infralow frequen? cies of less than 10 Hz. (b) Now let the distribution function have a Lorentzian form: 1 n(uo)f, = ; &2
.
The relaxation time will naturally be the same as before. Then the substitution of (16) and time 7” from eq. (12) into the integral (9) gives, after the necessary calculation,
Fig. 3. Behavior of (Y”for Lorentzian size distribution of the free volumes.
uted in the matrix and their interactions with each other may be neglected. Besides, the incident radiation wavelength must be greater than the maximum size of free volume. If the free volume concentration increases, the dipoledipole interactions between them begin to play an important role. This in turn means that we can no longer make use of eq. (2) in the calculation of (Y”because it does not take into account the dipole interaction; instead we will have to use a general formula for v” including all the “exchange” components. In the majority of the cases, however, the free volumes occupy much less space in a sample than the substrate, which we believe justifies the approach used in this paper.
(17) 3. Calculation The function (Y”(O) is plotted in fig. 3 for this case. Note that as A increases the peak of function (17) sharply shifts towards zero. Using the above approach based on a phenomenological kinetic equation it is possible not only to calculate (Y”but also to predict the rather nonstandard (oscillating at low frequencies) absorption of the energy of applied electromagnetic variable fields by liquid dielectrics, proteins and polymers also. The calculations in this text are valid only in case the free volumes are quite sparsely distrib-
of the cross-section’s
absorption
Let us now calculate the electromagnetic radiation absorption cross-section by an ensemble of free volumes. To do this we will use a known relationship for a(w) (cf. ref. [15]), which may be written as follows in our case: u(w, u) = u
4TrTTw
4w,
7
u>
(18)
2
where, according to (9): 2 a”(Cd,
u)
=
-
iv0
-;t
1 +m;2T2
v(P). ”
S.O.
Gladkov
I Radiation scattering in liquid glassy matrices
185
Substituting this into (18), multiplying by n(u,,) and integrating both parts with respect to u, we obtain the wanted cross-section of absorption in the form: a(w) = - -
-
4uo>u
*
(19)
If we substitute first the most likely equilibrium distribution function e
Fig. 4. The oscillating behavior section. w, = 1 /TVIT~,n % 1.
cross-
In this case calculation of a(w) gives:
-u/q
4Jo)L= ug
4d2
a’(~’ - 1 - 2 In a)
CPTO A
(a’ - 1)2
a(o) = in (19) and integrate, we find the desired formula for the absorption cross-section: hd2
(22)
where a=-
a(w) = -
Wroh uo
PCUO 70
x[l+&(sin-$ci&+cos--&si&())) The asymptotic behavior of a(o)
is:
4nd2 u(w) = PCUO 70
1--&
X
of the absorption
1
0
In or0 I+2 > for wro % 1 ,
(
1 - cos $
070
It is seen from fig. 1 that in case of a broad scatter in free volume sizes and at low frequencies (07~ 4 l), the a(w) curve shows a tendency for a sharp increase, which was indeed observed experimentally, when A 4 u. the increase of o(w) is insignificant (curve 2, fig. 1) and quite expectable, since for small vacancy dimensions and long wavelengths the scattering is very weak.
for ~7~ -G1 . 0
4. Conclusion (21)
It is seen that at low frequencies, (T(W) like (Y” has an ostensibly oscillating nature. It should be noted that at very low frequencies the oscillation period also becomes very small. This is shown clearly in fig. 4. If we now take into account that the free volumes may have any size, then, to calculate g(w) we must substitute into eq. (19) a Lorentzian form instead of an exponential form of the distribution function f,, i.e. assume:
The approach based on a phenomenological kinetic equation used in this paper allows one to: (1) calculate the amount of external variable electromagnetic field energy absorbed in a liquid dielectric matrix by a macroscopic system of free volumes, (2) find the absorption cross-section as a function of the incident radiation frequency, (3) explain some experiments devoted to the study of the properties of polymers in weak low-frequency fields; and (4) predict the oscillating nature of absorption at low frequencies.
186
S.O. Gladkov
I Radiation
scattering
Appendix To calcuate the time T” one must have a clear idea of the problem to be solved. Imagine a solitary free volume and suppose that its size will decrease as the ambient temperature decreases (see fig. 5). On this assumption it is possible to write down an energy balance equation immediately: (23) which, in a spherical system of coordinates and on the condition that the sphere (free volume) radius varies little, leads to:
in liquid glassy matrices
coefficient of heat exchange from the free volume to the matrix, To is a certain temperature which is defined below. On the conditions that:
TI,=,,= T, TI,,, = T,, and
so that r* c r,(t) c f, the solution gives:
T(t) = T,, -
2 x
+$
=
T-
(24)
where liquid in the On
x is the temperature conductivity of the in the vicinity of a “vacancy”; p is pressure liquid, So is the free volume surface area. the other hand we have a condition:
aT =
K dr
r=r”(t)
where
K =
xc,,
-a(Tc,
To)
(25)
is the heat capacity, (Yis the
_
1
To)e-%,(‘)-‘*’ ,S(r’-‘)
0
of eq. (25)
T,,
(1
ex(“-‘)
+ @_
PV,
(26)
-1
.
(27)
Substituting solution (27) into eq. (24) and assuming that r,(t) is close to r*, we find, after the necessary expansions: i, = - -
1
(r. - r*> .
7”.
(28)
Omitting further the asterisk at r and passing on to the volume u, we finally obtain the desired equation: d = 1 (UC)- u) )
(29)
7”
where the relaxation is CY~ITo-
1
-=u 7”
’
TI
-
K
p(U
time used in the main text
-
6)
(30)
Fig. 5. Fluctuating variation of the free volume dimensions.
Thus, as it follows from the last equation, the relaxation time is a linear function of the free volume size, whereby our use of eq. (12) in the main text is justified.
S. 0.
Gladkov
I Radiation scattering in liquid glassy matrices
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187
[7] M.H. Cohen and G.S. Grest, Solid State Commun. 39 (1981) 143. [8] M.H. Cohen and D. Turnbull, J. Chem. Phys. 31 (1959) 1164. [9] S.O. Gladkov, Physica B 160 (1989) 211. [lo] S.O. Gladkov, Phys. Lett. 140 (1989) 108. [ll] H. Vogel, Phys. Zur. 22 (1921) 645. [12] G.S. Fulcher, J. Amer. Ceram. Sot. 8 (1925) 339. [13] A.N. Gubkin, Izv. Vuzov, Ser. Phys. 1 (1979) 56. [14] S.O. Gladkov, Phys. Rep. 132 (1986) 277. [15] L.D. Landau and E.M. Livshits, Fluid Electrodynamics, Vol. 8 (Nauka, Moscow, 1982) (in Russian).