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Mechanical and Dielectric Relaxation Phenomena and their Molecular-Physical Interpretation.

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Mechanical and Dielectric Relaxation Phenomena and their Molecular-Physical
Interpretation
BY DR. WERNER ZEIL
INSTITUT F m PHYSIKALISCHE CHEMIE UND ELFKTROCHEMIE DER
TECHNISCHEN HOCHSCHULE KARLSRUHE (GERMANY)
Measurement of mechanical and dielectric relaxation phenomena provides some information
concerning intermolecular interactions. These have been thoroughly elucidated for gases,
whereas with liquids and amorphous solids (high polymers), we are limited to simplified
representations by models. Although at present the main emphasis in applied technology is
placed on relaxation investigations of high polymers, it seems appropriate to present here
the general physical principles of mechanical and dielectric relaxationphenomena. Furthermore, a review of the relaxation behavior of gases, liquids, and amorphous solids (high
polymers) is given. These problems were the main topic of the Sixtieth Congress of the
“Deutsche Bunsen Gesellschaft fur Physikalische Chemie” [11. Relaxation phenomena
dependent upon chemical equilibria and the field of magnetic relaxation phenomena are not
discussed.
1. Basic Theory
to a new length, then the stress corresponding to Hooke’s
Law,
GO=
In physics the elongation of a body as the result of a
stress is described by Hooke’s Law:
A1
B
=
strain, i.e. the relative change in length -
cs
=
stress, expressed as force per unit area
M
=
modulus of elasticity
I
Equation (1) holds rigorously only for the static case and
includes no information regarding the interim between
the moment a mechanical stress is applied to a body and
the moment the strain process is completed. That is, the
equation contains no statement about the speed with
which mechanical strain occurs.
In practically all cases, a measurable period of time
must elapse before the above values are attained. If one
applies a definite stress cr to a body, then the final strain
BO is attained only after a definite time. An analogous
phenomenon is as follows: if one stretches a piece of
rubber very quickly from its originally tension-free state
[I] 60. Hauptversammlung der Deutschen Bunsen-Gesellschaft
fur physikalische Chemie, May 1 1 th.-14th. 1961, in Karlsruhe,
with the main topic “Relaxation Behavior and the Molecular
Structure of Matter”, together with the Spring 1961 joint meeting
of the technical committees on “Acoustics” and “High Polymers”
of the Verband Deutscher Physikalischer Gesellschaften. See
particularly the main lectures :
H . 0. Kneser: Fundamentals of Dielectric and Mechanical
Relaxation Phenomena. (Grundlagen der dielektrischen und mechanischen Relaxationserscheinungen)
W. Muier: Dielectxic and Mechanical Relaxation in Liquids. (Die
dielektrische und mechanische Relaxation in Flussigkeiten)
K. A . Wolf: Relaxation Investigations of Molecular Processes in
High Polymers. (Relaxationsuntersuchungen zum Studium der
molekularen Vorgange in Hochpolymeren)
For a report see Z . Elektrochem. 65, 718 (1961).
246
MO*Bo
(2)
is reached only after a certain time. Immediately after
stretching, a maximum stress crm is observed. In time,
this stress decreases and approaches the constant value
00 which is given by Hooke’s Law. This specific process
whereby the original maximum stress decreases to the
value corresponding to Hooke’s Law is termed relaxation. The above described process delays the
attainment of a deformation or elongation after the
suddenapplicationof astressandiscalledretardation.
However, it is common usage to call relaxation phenomena proper, as well as retardation phenomena, by the
common name of relaxation phenumena. This isjustified,
since both phenomena are causally connected and can
be mutually inter-converted if necessary. A measure of
the time of adjustment to the final state is the so-called
relaxation time, or retardation time.
In Figure l a , it is shown how a retarded deformation
changes with time after application of a stress, whose
course time dependence is given. Similarly, the relaxation
behavior is represented in Figure 1 b. The curve is drawn
so that the value of the final stress corresponds to the
value of the stress applied for the retardation case.
Here, and in what follows, the mechanical stress cr may
be replaced by the electrical field strength E. In an analogous manner, the strain B may be replaced by the electric
polarization P, or the electric displacement D, so that
Eq. (1) can be written:
D = EE
(3a)
or
P = XE
E = electric field strength
D = electric displacement vector
P = electrical polarization vector
c = dielectric constant
x = susceptibility, given as
.
~
(3b)
4x
Angew. Chern. internat. Edit. Vol. I (1962) No. 5
Similar phenomena apply also in the field of paramagnetism,
but these will not be treated in this article. The meaning
which the quantities a, B, and M can assume are shown in
Table 1.
Table 1. Variables in Mechanical and Dielectric Relaxation
External force
Stress, u
1
Reaction of system
Strain, B
I
Modulus
Recipr6cal of the
1
elastic modulus-
M
Electric field
strength, E
Electric
displacement, D
[*I
tardation time and relaxation time and are - as
already mentioned - a measure of the delay in attaining
the final state. For the following analysis, both these
times are still distinct; in the course of this paper we shall
speak only more generally of relaxation times and analogously - of relaxation phenomena.
Equations (4) and (5) resemble equations in chemical
kinetics. If, for example, we replace a by a reaction rate
dc
(00-a) by a concentration c, and l/-r by the reaction
dt ’
rate constant k, then we obtain the equation for a fastorder reaction. This similarity in formalism is no accident. It is also observed on comparison of relaxation
phenomena with the rate of approach to a thermodynamic equilibrium. Almost all theories of relaxation phenomena are based on this analogy. We can therefore formulate relaxation phenomena as gradual adjustments,
Dielectric
constant, E
”t
d t
t
Lam
Fig. 1. Stress and strain versus time a) for retardation behavior,
b) for relaxation behavior
With the aid of some simple considerations which are
based on experimental facts, basic equations for relaxation behavior can be formulated thereby making
plausible the frequency dependence of the modulus observed experimentally. We assume that the rates of approach to the final value for the strain in the case of
retardation and to the final value of stress in the case of
relaxation, increase with the deviation from the final
value. That is, the rate of change of the strain under
constant stress, or that of the stress under constant
strain, is set proportional to the deviation of the strain
or the stress from the final value (linearization of the
problem). This can be expressed by the following
equations:
(4)
(5)
The equations are in agreement with the exponential
variation of and B observed experimentally. The proportionality constant must then be expressed in units of
reciprocal time; these times
and T~ are called reAngew. Chem. internal. Edit. / Vol. I (1962)/ No. 5
after a disturbance of a previously existing state of
thermodynamic equilibrium, to a new state of equilibrium caused by the disturbance.
We now assume that the final values of the strain Bo,
and the stress a0 are proportional, the proportionality
constant being the elastic modulus of Hooke’s Law. In
this case, it is designated by Mo,because the operation is
allowed to occur only once, i.e. no periodic change of
state is involved. The quantity MOis therefore a characteristic constant for each material. This assumption
in mechanical cases implies applicability of Hooke’s
Law. If we confine ourselves to small elongations (linearity), this is always attained to a good approximation.
In dielectrics, also, the assumption of proportionality is
always valid.
This assumption, as well as the assumption of proportionality of the rate and the deviation from the final
state, are the conditions for the linear relaxation
behavior, to which we will limit ourselves here. With
the aid of equation (2), we can now eliminate the quantities BOand a0 from Eqs. (4) and ( 5 ) and thereby obtain
equation (6).
This is the basic equation for describing simple relaxation
behavior or - using Kneser’s terminology - a “simple
247
relaxation body.” The characteristic of a simple relaxation body is that it possesses only a single relaxation
or retardation time.
In general, for technical reasons, experiments [2] for determining relaxation time should, be carried out by allowing
the change of state to vary periodically :
o(t) = a0
cos
(7)
w = 2xv
at
or, in complex notation [3] :
where:
(10 = amplitude
BO = amplitude
6 = phase shift between a and B
As is well known:
cia* = cos wt + i sin wt
and analogously B
- are complex quantities.
Calculation using complex quantities permits a mathematical
treatment of the phase shift between applied stress and strain.
With the aid of these quantities and equation (6), the following relationship is obtained:
60(1 + ioTB)eiat
-
M@O(I+ iwTo)ei(a‘-s)
For very high frequencies, i.e. for w
(8)
1
-
TB
and w 3
1
-
,
TG
dence or - expressed differently - the dispersion of the
modulus or the reciprocal dielectric constant, as derived by
experiment. We see that M , > Mo and ,E <EO. This
can be generalized, as can be shown theoretically, by
1
1
The difference A M =
saying that M , > MOand - >
-.
Em
€0
M,-Mo,
which is always positive, is called by Kneser the
> 0.
“Relaxationsbetrag”. Analogously: Ac = €0-E,
Furthermore, it follows from equation (9) that rG > r g , i. e.
the retardation time is greater than the relaxation time. If the
amount of relaxation (Relaxationsbetrag) is small, as for
example in the case of dilute solutions of dipolar molecules
in nonpolar solvents, the difference between retardation and
relaxation time can be neglected. Then, as shown later, the
general expression for ,,relaxation time” can be used.
In the course of our simplified considerations on the existence
of a dependence of the elastic modulus on the frequency of
the induced deformation or strain, we have considered only
the limiting cases of static behavior and the behavior at
extremely high frequencies as well. For arbitrary frequencies,
we must apply complex values to the elastic modulus or the
corresponding reciprocal dielectric constant, as is usually done
when dealing with a phase shift between an external influence
and the attainment of its corresponding value [*I.
M‘ represents the real part, M” the imaginary part of the
complex modulus.
By a separation of the real and imaginary parts it follows
from equations, ( l l ) ,(7a), (7b), (8) and (9),that
Equation (8) becomes
(9)
In the dielectric case we obtain
From this, it follows that
D
E
=-=
We see that Mca, i.e. the modulus for very high frequencies,
is different from the modulus for the limiting case of
zero frequency. Figs. 2 and 3 show the frequency depen-
“I
I /
IA18331
I
(13)
and Ac = cO-cm/.-----
It is characteristic of the equations, that both the real
and the imaginary parts depend on the relaxation value.
The relaxation behavior is therefore described by the relaxation time, the amount of relaxation, and the modulus for the static case. A lucid representation of this relation is obtained by plotting the modulus or the dielectric constant at different frequencies in the complex
plane. In both cases the result is a semicircle whose
centre lies on the real axis. This representation was first
chosen for dielectric behavior by Cole and Cole [S]. The
description was introduced by Kneser [6] for mechanical
relaxation. Such circles are called “Cole circles.”
Elimination of OT from equations (12) or (14) gives
log w
Fig. 2. (above) and Fig. 3. (below). Frequency dependence of the real
part M’ and imaginary part M” of the modulus of elasticity
[2] For methods of measurement see, e. g. R. Nitsche and V. A.
Wov: Kunststoffe. Springer, Heidelberg 1962, Vol. 11.
[31 Regarding the application of calculations with complex
quantities to the treatment of vibrations, see e.g. DIN (German
Industrial Standards) 5483, “Character of Complex Numbers,”
and DIN (German Industrial Standards) 1302.
248
+ is’’
whereby
I
’
e’
[*] Cf. e.g. the complex impedances in alternating-current and
high-frequency circuits which permit treatment of the correlation
of voltage and current as well as their phase shift. The impedance
representation of relaxation behavior depends on this analogy [41.
[4] J. Meixner, Nederl. Tijdschr. Natuurkunde 26,259 (1960).
K. S. Cole and R. IT. Cok, J. chem. Physics 9, 341 (1941).
[6] H. 0. Kneser, Ann. Physik 43,465 (1943).
[5)
Angew. Chem. internat. Edit. 1
Vol. I (1962)/ No. 5
This is the equation of a circle obtained by plotting M"
as a function of M'. The diameter of the circle is given
by the amount of relaxation.
From the theories mentioned above, we also expezt to be
able to interpret the phenomenon of dispersion, i.e. the
frequency dependence of the elastic modulus or the dielectric constant, and the energy absorption always
associated with it.
Thermodynamic Theory
Fig. 4. Circular plot of _M= M'
+iM
In Figure 4,the complex representation is shown as a vector
diagram. Above all, it may be noted from the figure that the
phase-shift 6 between the corresponding quantities increases,
and later decreases, with frequency. As is known, such a
phase shift leads to the irreversible conversion of a fraction
A W of the expended energy W into heat during each cycle.
It follows that:
-A
w - 2 xsins
where 6 cannot exceed a definite value. This value depends
upon the amount of relaxation.
It can be shown, furthermore, that the imaginary part of M
or E always has its maximum at the relaxation frequency
1
:B
7E
w=-or-.
The differential of the free energyfof a system subject to
a one-dimensionaldeformstioa leading to a stress c as a
consequence, is
df = a . dR - A . dc
(17)
o
W
1
We shall now derive the dynamic equation of state for a
simple relaxation system with the aid of the thermodynamic theory of relaxation. Using an example taken
from a paper by Meixner [4], we will simultaneously become acquainted with the method of the thermodynamic
theory and its application to relaxation processes.
=
the stress resulting from a strain B. It is equal to the
(2)c
partial derivative A
=
affinity, in the sense of irreversible thermodynamics [lo].
It is given by the partial derivative
- -
3$I:(
an intrinsic variable; it can signify the degree of advancement of the reaction [lo] or the concentration of some
component in the system subject to a change of state.
=
On the other hand, the frequency at which the angled and,
therefore, its tangent, tan 6, is greatest also depends esFigure 5 shows the free energy f a5 a function of the
sentially on the amount of relaxation. This means that the
quantity tan 6, generally taken as a measure of the loss,
intrinsic variable at constant strain B. At the minimum
reaches its maximum not at the relaxation frequency l / ~ ~ ,
of the curve, i.e. ($)B = 0, the affinity is zero. The
but at a frequency o =
(see also Fig. 14). In most
hli.m
minimum in f means that the system is in a state of
cases this distinction is admittedly insignificant, but it
should always be kept in mind in more precise discussions of
relaxation curves.
5
1
2
t
Two general theories apply to the treatment of relaxation
phenomena. Neither contains any assumption concerning
the molecular-physical event forming the basis of the
phenomena. Conversely, by means of these theories.
there is no possibility of obtaining information about
the molecular event from relaxation measurements.
These theories are:
1. Thethermodynamic theory of relaxation phenomena, based on irreversible thermodynamics and due to
Meixner [7].We will meet some principles of this theory
by way of an example.
2. The after-effecttheory of relaxation phenomena. This
theory is more general, but also more formal than the
thermodynamic theory; it will not be discussed in detail
here (cf. e.g. Meixner [8]).
A third representation is the kinetic, or statistical,
theory, developed in its general form by De Kronig [9].
It utilizes a statistical approach and can thus tender
some knowledge of molecular-physical events from relaxation measurements. However, it is necessary to use
models because it is impossible to draw a direct conclusion concerning these events from relaxation phenomena alone.
[71 J. Meixner, Z. Naturforsch. 4a, 594 (1949).
[ 8 ] J. Meixner, Kolloid-Z. I 3 4 , 3 (1953).
[91 R . De Kronig, Z. techn. Physik 19, SO9 (1938).
Angew. Chern. internat. Edit. 1 Vol. I (1962) No. 5
1
iA183.51
I
I
z
*E
Fig. 5. Free energyfas a function of the intrinsic variable 4at constant
strain B
thermodynamicequilibrium. 5 represents the equilibrium
value cf the intrinsic variable 6.The affinity is equal to
- tan a,where u represents the slope of the tangent tc the
curve describing the dependence of the free energy J'on
the variable The affinity is therefore a measure of the
deviation of the system from equilibrium. We now make
the assumption that, for sufficiently small values of B
E.
dE,
and 5, the variation in 6with time, i.e. the derivativedt
is proportional to the affinity A. Thereby, the affinity A
simultaneously becomes a measure of the reaction rate
[lo] with which the system restores itself to thermo[lo] UZich-Jost: Kurzes Lehrbuch der Physikalischen Chemie.
9th. Ed., Steinkopff, Darmstadt 1956; S. R. De Groot: Thermodynamik irreversibler Prozesse. Hochschultaschenbucher. Bibliographisches Institut, Mannheim 1960, Vol. I.
249
dynamic equilibrium after a disturbance - in our case
after a deformation. We designate the proportionality
constant by 1/C.Therefore the followingequation applies
to the relation between affinity and the reaction rate v:
equation of dynamic state with which we are already
acquainted. The ratio of the relaxation time at constant
,
strain T~ to the relaxation time at constant stress T ~ i.e.
the ratio of relaxation time and retardation time, is
MOO
equal to the fraction ~.
MQ
We assume isothermal behavior, i.e. the temperature remains constant during the experiment, and we expand
the free energy (which then depends only on the strain
B and the intrinsic variable 6) in powers of B and&
Furthermore, we define E such that, at thermodynamic
equilibrium and, therefore, in the initial state, when the
stress B is equal to zero, 5 is also equal to zero. As a
consequence, there are no linear terms the exponential
series. The expansion for small B and E then reads as
follows :
f ( B ; c )=f(O;O)
+ 11 M ,
Bz
+ bBf + 5 c.Z2
-
M,‘B
+6.4
(20)
By analogy, for the affinity we get:
- A = b * B +c.E
(21)
Equations (20) and (21) describe the thermodynamic
properties of matter, assuming constant temperature.
For A = 0, i.e. for the equilibrium state, we obtain the
following relationship between stress and strain:
This relation holds - since it was derived for A = 0 only under the assumption of continuous establishment
of equilibrium conditions. Under this assumption, however, Hooke’s Law applies :
(23)
a=Mo.B
We therefore obtain a relationship between the modulus of elasticity in the case of equilibrium MOand the
modulus of elasticity for very fast deformation, the
“instantaneous modulus of elasticity.”
Mo=M,-
The after-effect theory (“Nachwirkungsdarstellung”)of relaxation phenomena obtained by integration of the dynamic
equation of state, shows how the stress o(t) depends on the
instantaneous values of the deformationc(t) and on the previous history before deformation. For our special case compare Meixner [4]. Muller [ll] has presented a treatment of
the after-effect theory for dielectric relaxation phenomena.
Statistical Theory
We have thus derived the dynamic equation of state
using thermodynamics. The frequency dependence (dispersion) of the modulus can be derived from this equation. We shall now derive directly, with the help of the
statistical theory of De Kronig, the equations given above
concerning the frequency dependence of the real and
imaginary parts of the modulus of elasticity or the dielectric constant. This theory is not as free from postulates as the thermodynamic theory; on the basis of its
assumptions and on this basis alone can we make some
statements concerning special molecular-physical processes. De Kronig’s theory attempts to describe relaxation
phenomena from a common molecular-physical point
of view. In a publication which appeared in 1938, De
Kronig 191 formulates the problem as follows:
“A total system consists of a large number of mostly
similar partial systems which are independent of one
another to a first approximation, and each of which is
capable of a series of states 1 with energy W,. At thermodynamic equilibrium, the partial systems are distributed
with respect to the energy value W, according to Bottzmann’s Law, and the population corresponding to each
state is
bz
-
From further considerations, which will not be examined
here, it follows that M , > 0, c > 0 and cM, > 62.
From this we obtain M , > Mo. It follows from
equation (20), by solving for E and differentiating with
dS .
respect to time, that for -.
By elimination af 6 , A, E and dF
- from equations (21) to
dt
(25)
If we now replace
C
by -cB and
where N is the total number of partial systems, k is the
Boltzmann constant, and T is the absolute temperature.
It follows then that
N = ~ N *
dt
C Ma0
c
Mo
while T is unequivocally defined through fixation of the
total energy
W = W,N,
If the distribution at a given instant is made to differ
from Bolizmann’s distribution, then the problem
reduces to a matter of determining the order of
by
T ~ then
,
equation (26) is transformed into equation (6), the
250
dynamic equation of state thermodynamically.
This same approach is applicable to all other linear relaxation processes, as well as in caSes involving several
intrinsic variables. A more complete treatment is given
by Meixner [7, 81.
(19)
M , is the modulus of elasticity for very rapid deformation. A comparison of Eq. (19) with Eq. (17) leads to the
following relation for the stress:
a
We have thus derived the
[I I ] F. H . Miiller and Chr. Schmelzer, Ergebn. exakt. Naturwiss.
25, 357 (1951).
Angew. Chem. internat. Edit. I Vol. 1 (1962)I No. 5
magnitude of the time in which a suitably defined mean
of the deviation from the population N, decreases from
its equilibrium value to a certain fraction of its original
value, e.g. to the e-th part." [*I
The De Kronig theory therefore makes special assumptions about the energy distribution in the system exhibiting relaxation phenomena. De Kronig also explzins
what he means by partial systems: these are the atoms
or molecules of the gases, liquids, or solids in question.
The energy values of which De Kronig speaks consist of
contributions from translational, rotational, and vibrational energy, whereby the first fluctuates continuously while the rotational and vibrational energies. because of quantum condition, can assume only a series
of discrete values. These statements hold particul+rlyfor
the problem of mechanical-(acoustical) relaxation phenomena. In dielectric relaxation phenomena, the partial
systems are the permanent electrical dipoles of the
molecules. Their energy is the interaction energy with
the electricfield, which in general will be an external field.
However, we must keep in mind that there are also perturbations in this field, which may originate from neighboring molecules or ions. In this case, the orientation is
assumed to be continuously changeable. The energy of
interaction can also assume continuously any value between the limiting values of parallel and antiparallel
orientations. It is therefore a prerequisite for the establishment of a new thermodynamic equilibrium after
applying a disturbance that transitions between the
various states of the partial systems are possible, i.e.
there must be a finite probability of transition A , between the states 1 and rn. This probability of transition
may be dependent not only on the energy of the various
states, but also on pressure, temperature, and external
electric fields. De Kronig formulates the following differential equation which specifies the rate of change of
population number NI.
[*I Original text from reference [9]: ,,Ein Gesamtsystem besteht
aus einer groBen Anzahl von meistens gleichartigen Teilsystemen,
die in erster Naherung voneinander unabhbgig sind und deren
jedes einer Reihe von Zustanden I mit der Energie W1 fiihig ist.
Im thermodynamischen Gleichgewicht sind die Teilsysteme uber
die Energiewerte Wl nach dem Boltzmannschen Gesetz verteilt
mit den Besetzungszahlen
This equation indicates that the rate of change of Ni with
time has two causes:
1. the influx of partial systems from state m with a transition
probability Am1 and
2. the outflow of partial systems from the state l with the
probability of transition A[,,,.
De Kronig then assumes that the external disturbance, which
results in the relaxing change in state, is small and that the
population, which alters as a result of the external disturbance,
varies linearly with the disturbance F. The respective difference of population is designated by ni. The distribution after
the disturbance is then
N1 = N;
f
nI
(29)
N; signifies the population of the states at equilibrium. As
already mentioned, the probability of transition Aim also
depends on F. We expand Arm in terms of F and break off
the series after the linear term:
A!,,, then signifies the value of the probability of transition
in the absence of a disturbance F. In the equilibrium state, the
following must apply:
In this case the number of transitions, which is equal to the
product of the numerical distribution and the probability
of transition, is by definition equal in both directions. In
consideration of equation (27), we then obtain the following
relationship :
We now differentiate with respect to F and get
(33)
Considering again equation ( 2 3 , then we obtain
Referring to equations of (31) and (34) and substituting
equations (29) and (30) in equation (28), we then get the
following system of linear differential equations:
If F is a known function of time, then nl can be derived as a
function of t. If the external disturbance F is periodic in
nature, which for technical reasons is almost always the
case, then, by analogy to the assumption made above concerning the linear relationship between n and F,
wo N die Gesamtzahl der Teilsysteme, k die Eoltzmannsche Konstante, T die absolute Temperatur bedeuten. Es gilt dann
N
=
~
N
~
wahrend durch Vorgabe der Gesamtenergie
T eindeutig festgelegt wird. Bewirkt man irgendwie, daB die Verteilung in einem gegebenen Augenblick von der Boltzmannschen
verschieden ist, so handelt es sich d a m , die GroBenordnungder
Zeit zu bestimmen, in der ein geeignet definiertes Mittel der Abweichungen der Besetzungszahlen NI von ihren Gleichgewichtswerten auf einen gewissen Bruchteil seines urspriinglichen Betrages, z. B. auf den e-ten Teil, abnimmt."
Angew. Chem. internat. Edit. 1 Vol. I (1962)/ No. 5
We shall now assume, for the sake of simplicity, that I can
assume only two values, viz, the values 1 and 2. This means
that there are only two states between which participate in
equilibrium. For example this is valid to a very close approximation in the case of vibrational relaxation of a diatomic
molecule, (considered later), where only the vibrational
ground state and the first excited state are occupied to an
appreciable extent. In this case, the solution consists of two
linear differential equations :
25 1
where
(39)
If we now consider nl as a complex quantity
n1 = n '1- i n ; '
initial disturbance is the cause of relaxation phenomena.
Up to now, in both the thermodynamic and the statistical-kinetic theories, we have spoken of the free
energy and. its dependence on variables or of the distribution of atomic or molecular units in the various
states of free energyf. On the basis of the Gibbs-Helmholtz equation we can now divide the free energy into
an internal energy term and an entropy term. We can
therefore also write for the population of a given state
then we obtain the following relationship for its real and
imaginary parts:
If we compare the equations regarding the frequency
dependence of the population of a state, derived from
the statistical postulate using Boltzmann's Law, with the
equations obtained from the phenomenological approach, then we see that the relaxation time corresponds
to the reciprocal of the sum of the transition probabilities
in the equilibrium state:
(43)
The amount of relaxation corresponds to the quantity
C in the De Kronig thwry. The real part n' of the additional numerical distribution related to the disturbance
and
of magnitude 1 corresponds to M'-M,, or E'-E,,
the imaginary part n" corresponds to M" or E".
To summarize the first section regarding the theoretical
treatment of the relaxation phenomena: The experimental retardation or relaxation curves, which lead to
equations (4) and ( 5 ) and from which both the basic
equation for a simple relaxation body, (i.e. the dynamic
equation of state) as well as the frequency dependence of
the real and imaginary parts of the modulus of elasticity
or dielectric constant can be ascertained, are obtainable both by a purely thermodynamic approach, i.e.
using a continuous-process theory, and also from a
statistical approach based on the distribution of energy
in different states according to Boltzmann's ratio and
finite probabilities of transition between these states. The
thermodynamic theory is more general and is based on
fewer assumptions. This freedom from assumptions,
however, is connected with the fact that no statements
can be made about the molecular mechanism effecting
relaxation behavior. The statistical-kinetic theory is
less general but is based on atomic and molecularphysical principles regarding the existence of partial
systems and the distribution of energy associated with
these partial systems. With adequate certainty of the
identity of these partial systems with atomic or molecular parts of the total system, and of the possible states in
which these systems can exist, it is possible to draw conclusions from relaxation measurements regarding the
molecular processes involved and their energetic relationships.
It is common to both theories, to assume that the establishment of a thermodynamic equilibrium after an
252
From Equation (44) we find that, in order to investigate
relaxation phenomena, the system can be disturbed in
principle in three ways: either we change the temperature
or the energy of the possible states or we increase the
order of the system, i.e. we decrease the entropy. The
experimental possibilities are:
An increase in temperature can be brought about by
subjecting a gas or a liquid to a rapid adiabatic compression. This is easily done by passing a sound wave of
sufficiently high frequency through the substance. A
change in the entropy is thought to be possible only by
mechanical deformation. On the other hand the increase in internal energy can be caused by mechanical
stress as well as by electric or magnetic fields (the latter
will not be discussed here). The most important relaxation phenomena to be discussed are enumerated below:
1. Acoustic relaxation phenomena in gases and liquids,
known as sound dispersion.
2. Mechanical relaxation phenomena in plastics (high
polymers);
3. Dielectric relaxation phenomena in liquids and plastics (high polymers).
We will not go into the broad range of relaxation phenomena in which a chemical equilibrium is disturbed
and whose investigations,currently knowli as the kinetics
of rapid reactions, form a specific field of research in
chemical reaction kinetics [*I. Furthermore - as already mentioned at the beginning - magnetic relaxation
phenomena and the closely connected phenomenon of
nuclear relaxation will not be discussed, since, for their
comprehension, extensive details concerning the interaction of spin-systems and neighboring fields must be
included. This would be far beyond the scope of this
paper.
2. Mechanical Relaxation Phenomena in Gases
(Sound ReIaxation)
In gases, mechanical relaxation phenomena are perceptible in the dispersion of sound and in the corresponding absorption of sound. Kneser [12] summarized experimental methods for investigating molecular ab-
[*I See e.g. the report on the Discussions of the Deutsche Bunsen-Gesellschaft fur Physikalische Chemie concerning rapid
reactions in solutions (Z. Elektrochem. 64, No. 1 (1960)).
[12] H. 0.Kneser, Handbuch der Physik. Springer, Berlin-Gottingen-Heidelberg 1961, Vol. XI/l, p. 129.
Angew. Chem. internat. Edit. 1 Vol. I (1962)/ No.5
sorption and dispersion of sound in gases and liquids.
Measurements are made of the dependence of the velocity of sound on frequency (sound dispersion) and the
dependence of the absorption of sound on frequency.
The equation for the velocity of sound is
M
P
,2 = _
Transformation of Equation (47) produces the formula
for the propagation of sound, where co represents the
propagation for the limiting case of zero frequency and
the new relaxation time T’ thus introduced is slightly different from the relaxation time T defined above:
(45)
Here again Mrepresents the modulus of elasticity which,
in the special case of the propagation of longitudinal
waves in gases, equals the ratio of specific heats at
constant pressure and volume multiplied by the reciprocal isothermal compressibility. For an ideal gas,
this is equal to l/p. It follows then that
The quantity
by
E
represents the “relaxationbetrag”. It is given
A - Ci
(484
E=
c 2 =P ’ Y
(‘a
~
+ Ci) (‘a
+ A)
P
where y
=
CP
Y
cv
‘isotherm
-, M = - -
One can now assume the following as the cause of sound
dispersion and, therefore, of relaxation phenomena [121:
on adiabatic compression of a portion of the volume of
a gas, caused by the passage of a sound wave, the temperature of this portion of the volume increases. If all the
degrees of freedom of the energy are newly distributed
in a way corresponding to this rise in temperature, as is
the case at low sound frequencies, we always have an
equilibrated system. At very high sound frequencies a
portion of the degrees of freedom cannot assume the
energy corresponding to the equilibrium distribution in
the short time available. This is equivalent to a decrease
in specific heat, which is composed additively of the
translational, rotational, and vibrational energies per
unit degree. It will now be assumed that at very high
frequencies, a certain number of the degrees of freedom
is always excited according to an equilibrium state. The
specific heat corresponding to these degrees of freedom
we call C,;the specific heat which is caused by degrees
of freedom whose adjustment require a certain time we
call Ci. In the statistical case then
cv=ca+ci
The equation for the dependence of the complex speed
of sound, c, on the sound frequency now reads [12]:
(Confusion of this quantity E with the dielectric constant
appearing later seems not likely). On examination of equation
(48) we find that the complex speed of sound assumes real
values for the limiting cases of frequency = 0 and a : in the
former case the value CO, in the latter case the value C a = CO/
From the last relationship we can obtain a new
definition for E:
icL
Since the complex sound velocity in the case of a plane wave
is connected with the phase velocity c and the absorption
coefficient per centimeter u by the relationship
then by separating real and imaginary parts and neglecting
second- and third-order terms of E, in the formula for sound
propagation, a n approximate dependence of the sound
velocity on the frequency, the relaxation time t’ and the
relaxation value E is obtained:
Without neglecting terms the absorption coefficient then
reduces to:
1 c E02T’
==-A(52)
2 c2 1 + 02T’Z
0
u is therefore a function of the amount of relaxation (“relaxa-
tionsbetrag”), relaxation time, frequency, and sound velocity.
Sometimes the absorption coefficient will be represented by
a dimensionless coefficient for each wavelength. This leads to
with A = Cp-C, (= R in the case of an ideal gas). The
relaxation time corresponds, in De Kronig’s sense, to the
reciprocal sum of the transition probabilities:
r=
1
Aiz
+ Azi
In our case
It is possibleto represent relaxation behavior in the sound
propagation in four ways as shown in Figs. 6 (a), (b),
(c) and (d). A circular (complex plane) plot, first
suggested by Kneser [6],is also possible in the case of
*
A12
~A21
< 1 [12]. That is, T is approximately
equal to 1/&. In this case, the relaxation time is equal
to the transition time of vibrational energy illto translational energy on collision. This corresponds to the lifetime of a “vibrational quantum” and must not be confused with the lifetime of an “excited vibrational state”.
Angew. Chem. internat. Edit. / Vol. I (1962)/ No. 5
sound absorption. The real part is given by
2
c’
-O
C2
and
the imaginarypart by - !! 3 . This circular plot is analox
B
gous to the Cole-Cole plot for the dielectric constant.
The problem evolved here following Kneser’s method
and by application of the De Kronig’s concept can also
be treated with the aid of thermodynamic theory. This
253
I
I
log wr'
I
P
culated from spectroscopic measurements (valence vibration in the Raman spectrum) of oxygen by means of
characteristic temperature. A well-defined correlation of
relaxation phcnomena to excitation of vibrationalenergy
is thereby obtained. The absorption curves of COZtoo,
are explained by the relaxation of vibrational energy.
Also in this case only one relaxation time appears. A relaxation of rotational energy has been found un-
I
t
2.01
17
6
T
7
8
Fig. 8. Sound dispersion in HZgas resulting from rotational relaxation.
Valves given by E. S. Stewart, Physic. Rev. 69, 632 (1946). (Kneser [15],
p. 151.)
equivocally in the case of HZ (Fig. 8). Rotational relaxation can also be observed in oxygen as well. Since
the relaxation times are quite differentfor the excitation
of vibrational and rotational energy, two separate circles
wd=?
rn
Fig. 6. Representation of relaxation behavior in sound propagation
(Kneser 1121, p. 146)
has been shown by Meixner, who also made allowance
fcr viscosity, thermal conductivity and diffusion [131.
The question is: Which molecular-physical processes
take place during the occurrence of relaxation phenomena ingases? Inmost cases,relaxation of vibrational
energy is observed. Figure 7 shows measurements of
"n l
LQ&g2
I
I
ZOO00
05
I
I
M
15
$
I
20
,
25
- , P o )
iW30
Fig. 7. Dependence of the sound absorption on its velocity:
(Kneser [151, P. 149)
0 2
gas.
sound absorption in oxygen, given in a circular plot. The
behavior required by theory is very nearly attained. This
means that there exists only one single relaxation time,
i.e. one simple relaxation body. From the radius of the
circle, the value of the relaxing portion of the specific
heat Ci can be determined using Eq. (48). This value
agrees well with the vibrational energy of oxygen cal[13] J. Meixner, Ann. Physik 43,244 (1944).
254
Fig. 9, Circular plot of vibrational and rotational relaxation for
(Kneser [121, p. 178).
0 2
can be established in the circular plot in the case of
oxygen. Figure 9 shows the smaller circle, which corresponds to the vibrational relaxation, as well as a
portion of a larger circle corresponding to the rotational
relaxation.
Angew. Chem. internat. Edit. 1 Vol. I (1962)/ No. 5
What conclusions can now be drawn from measurements of sound absorption? We observe immediately
that the excitation of vibrational energy requires a certain
time lapse. Since an exchange of energy between gas
molecules can occur only in the relatively very short
period of time involved in collision of two moleculeswith
one another, we are led to the question, how often must
one gas molecule collide with another until vibrational
energy is exchanged? This means that by measwing the
relaxation time and from a knowledge of the collision
number (given by the kinetic theory of gases) we can
obtain the number of collisions and thereby the answer
to the above question through the relation
(54)
T=lt’Z
where t represents the reciprocal of the collisions number,
Le., the average duration of impact, and z is the number
of collisions survived by a vibrational quantum before
being transformed into translational energy. As an
example of the order of magnitude of z some values
may be indicated :
Clz
o2
N~
zoSc.= 4.2 x 104
zoSc.= 2 x 107
zoSc.= 2.7 x 103
(2OoC)
(200~)
(200~)
For rotational relaxation, z lies between 3 and 10 for
diatomic molecules. Investigations to date have shown,
furthermore, that neither rotational nor vibrational
maxima show great deviations from the theoretically
expected behavior. This means that only one relaxation
time occurs, i.e. only one relaxing process is observed.
In the case of molecules with several degrees of freedom*,
of the vibration energy, experiments have shown that the
excitations which occur practically always involve the
vibrational degree of freedom corresponding to the
frequency.
Here we may consider briefly the relaxation phenomena
involved in the exchange of rotational and vibrational
energy, factors which today have gained great importance. Hot combustion gases stream from rocket
combustion chambers through a Laval-nozzle so quickly
that, to a great extent, no exchange of energy can
occur, i.e. the temperature adjustment of the individual
degrees of freedom can no longer follow the changing
pressure conditions in the nozzle. These processes are of
importance for the calculation of thrust data.
Investigations in the gaseous state can be facilitated if
instead of varying the frequency over a large range (this
requires rather complicated apparatus), the relaxation
time can itself be altered by increasing the density of the
gas or its pressure. Since vibrational relaxation is essentially a question of the collisions number required to
exchange vibrational energy, the relaxation time can be
decreased by increasing the collisions number (as is always possible by increasing the pressure). It is quite
easy, to accumulate extensive experimental data which
can be converted to standard pressure. Strangely enough,
the relationship, that the reciprocal relaxation time is
proportional to the pressure, holds time not only in the
gaseous state, but also in the liquid state. In this sense at
least, the liquid can be interpreted as highly compressed
gases. We thus arrive at a discussion of relaxation phenomena in liquids.
Angew. Chem. internat. Edit.
Vol. I (1962) No. 5
3. Mechanical and Dielectric Relaxation
Phenomena in Liquids
In this section we will discuss first of all the results of
mechanical relaxation phenomena and then deal with
dielectric relaxation phenomena in liquids. The experimental methods for investigating mechanical relaxation proporties of liquids are .analogous to those
used for gases [12].
Whereas relaxation phenomena in gases generally involve simplerelaxation processes, which can be described
by a single relaxation time, or in which the individual
relaxation processes were independent of one another
and correspond to a definite relaxation time (e.g. in the
case of oxygen where rotational and vibrational relaxation correspond to separate relaxation), in liquids
it is very frequently observed, in mechanical and more
particularly in dielectric relaxation, that the relaxation
curves are no longer characterized by a single relaxation
time. Here we have in general to deal with phenomena
which cannot be interpreted without assuming a large
number of relaxation times; even with a continuous distribution. In this case, we can no longer postulate with
certainty that the relaxation times and the corresponding
processes are independent of one another. Although
these phenomena are actually treated as if this were the
case, in more detailed treatments of the distribution of
relaxation times, the question of independence of the
individual processes should either be examined or the
assumption of independence should always be borne in
mind.
The relaxation curves for liquids can be represented in a
manner analogous to those for the gaseous state. After absorption measurements have been made, the relaxation value
can be determined expediently from the maximum value of
the absorption p, and the relaxation time can be determined
from the frequency of the maximum of the p curve. The
analogous equations for the case of relaxation phenomena in
gases are valid [*I.
We will now discuss some examples in which it was possible to find a well-defined correlation between the
molecular physical processes and the observed relaxation
curves and the relaxation times calculated from the latter.
Disregarding processes in which chemical reactions
(such as dissociation and solvation processes) OCCUT, and
which in general will not be treated here, essentially two
processes have been observed. In the one case, analogous
to the processes in the gaseous state, there is the process
of relaxation of vibrational energy (vibrational relaxation). Following a proposal by Herzfeld [141, liquids,
in which vibrational relaxation is observed and which
can be treated by analogy to relaxation phenomena in
the gaseous state, can be designated as “Kneser-liquids,”
since Kneser was the first to apply the theory developed
for gases to the case of liquids [15]. The theoretical treatment, similar to the theory of vibrational relaxation in
gases, proceeds from the assumption that only bimole1
~-
(55)
[*I pmax=r2 . E
Vmax = 2 X
T
(56)~
Vl-z’
[I41 K . F. Herzfeld and T. A . Litovitz: Absorption and Dispersion of Ultrasonic Waves. Academic Press, New York 1959.
[15]H . 0 . Kneser, Ergebn. exakt. Naturwiss. 22, 121 (1949).
255
~
~
cular and to a limited extent, trimolecular collisions, are
involved in the exchange of vibrational energy. Thus,
the collision number in the liquid is considered proportional to the number of collisions in the gas times
the density of the liquid. Cooperative behavior of a
large group of molecules does not appear to be essentially connected with relaxation behavior. Litovifz [16]
carried out a theoretical treatment of the temperature
and pressure dependence of relaxation behavior; this
stands in very good agreement with experiment.
Carbon disulfide, for example, has been thoroughly
investigated. Figure 10 shows the absorption of sound
in C S 2 at two temperatures. Table 2 shows the qualitative
dition for the occurrence of ultrasonic relaxation in
these isomers is related to the fact that both forms possess a noticeable difference in energy. Actually it is a
025,
a20
Fig. 12. Rotational isomers of triethylamine (schematic); (Davies and
Lamb [17] p. 153)
0
I
I
I
2
5
70
mm
20
l
l
50 100 200
I
500
v[lo6cps]
Fig. 10. Sound absorption in CSz. A at 25 “ C ;B at 63 “C (Davies and
Lamb 1171. p. 155)
agreement of the relaxation times for vibrational relaxation in the liquid and in the gaseous state, taking
into account the ratio of densities, and simultaneously
confirms the idea that a retarded attainment of the vibration temperature is the cause of sound relaxation
phenomena in liquids.
Table 2. Relaxation times and density ratios of several compounds in
the gaseous state (Tg) and liquid state (TI)(after Herzfeldd)
Compound
I
T ~ lOa[sec]
X
7.5
1.35
2.1
0.5
5
72.7
I
q x 1010 [secll ig/r1
1.26
1.3
1.26
0.495
2.7
28.3
600
105
170
I00
180
260
I
pl/pg
460
320
260
320
280
420
The other cause for the appearance of relaxation phenomena is observed in liquids whose molecules can
occur in two or more forms. This is the case, for example, with cyclohexane derivatives, which can exist in two
different chair forms (Fig. 11) or for rotational isomers,
e.g. triethylamine (Fig. 12) (x 2-methylbutane. The con-
IA183.111
CH,
Fig. 1 1 . The two chair forms of methylcyclohexane
[I61 T.A. Litovitz, J. chem. Physics 26, 469 (1957).
256
phenomenon completely analogous to vibrational relaxationexcept that the “vibration” corresponds to a “flopover” of the one form into the other, involving the crossing of a potential barrier, associated with a hindrance
potential, or, (in the language of reaction kinetics) an
activation energy. This causes an important difference in the temperature dependence of the collision
number necessary in order to accomplish intramolecular
conversion or inversion. In general one expects an exponential temperature dependence of the collision number z necessary for conversion.
- VlRT
z=zO*e
(57)
Here V is the hindrance potential. In the case of the
cyclohexane derivatives or the indicated rotational isomers it amounts to the order of magnitude of some
kcal/mole The difference compared with vibrational relaxation, for which the same process is principally
responsible, lies in the fact that here the quantity V is
generally very much smaller than the thermal energy
RT. This results in the weak temperature dependence
of the collisions number necessary for vibrational excitation. Whereas pure cyclohexane shows no relaxation
phenomena, since both chair forms possess the same
energy, relaxation investigations of methylcyclohexane
reveal an energy difference of 1.6 kcal./mol between the
two forms. In the case of triethylamine, shown in Fig.
12, three rotational isomers between which transitions
can occur are possible. Figure 13 shows the temperature
dependence of ultrasonic wave absorption.
We will not discuss here the results which can be obtained
from mechanical relaxation measurements on solutions and
the various types of relaxation curves which thereby result.
The reader is referred to reviews by Davies and Lamb [17]
and Sette [18]. With regard to liquids, which contain larger
molecular aggregates, and to the consequent temperaturedependent structural effects and their influence on mechanical
[17] R. 0. Duvies and f. Lamb, Quart. Rev. (Chem. SOC., London) 11, 134 (1957).
[18] D . Sette: Handbuch der Physik. Springer, Berlin-GottingenHeidelberg 1961, Vol. XI/], p. 275.
Angew. Chem. internat. Edit. 1 Vol. I (1962)/ No.5
fw
10
9-
4
87-
I
I
6-
I
I
I
54 3 -
I
I
27 70
20
50
700
ZOO
500
I
I
I
I
t
I
I
L
I
!&
I
loa w
v [106cpsI
Fig. 13. Ultrasonic wave absorption in triethylamine: A at 25 “C,
B at 35 “C, C at 45 “C. (Davies and Lamb [17], p. 153)
relaxation behavior, reference should be made to Sette’s
review [18]. In the same paper, results of studies of the
pressure-dependent relaxation behavior of water are to be
found. Investigations concerning sound absorption in
electrolytes, have been summarized by Tamm [19]. This topic
is of significance in the theory of electrolyte solutions.
The first theoretical treatment [*] of dielectricrelaxation
phenomena of liquids is connected with the name of
Debye [20]. He derived the following equations for the
frequency dependence of the real and imaginary parts
of the dielectric constant:
Fig. 14. Dependence of the real and imaginary parts of the dielectric
constant as well as tan S on the frequency (Bdttcher 1331)
(59)
Figure 14 shows typical curves for the dependence of the
real and imaginary parts of the dielectric constant on the
frequency. Furthermore, Figure 14 shows the dependence of
tan 8 on the frequency.
Here it can be seen that the maximum of the tan 8 curve does
not coincide with the maximum of the E” curve. Eqs. (58) and
(59) can also be obtained from the general relaxation equation
(14), indicated in the first section, by a simple transformation.
The Debye equations can also be represented in the Cole-Cole
plot by eliminating at. The equations are only valid for
systems which possess a single relaxation time and are,
therefore characterized by a single relaxation process.
Debye has already given a molecular physical interpretation of the dielectric relaxation phenomena. A
sphere (with its solvation shell) possessing a dipole
moment and moving within a viscous medium in accordance with Stoke’s Law serves as a model. Here, the
viscosity of the solvent is used for the viscosity term.
(However, Debye always speaks of an intrinsic or microscopic viscosity, in contrast to the macroscopic viscosity
measured by the usual viscosimetric methods. This problem is treated by Smyth [21]). The relation between the
[19] K. Tamm, Handbuch der Physik. Springer, Berlin-GottingenHeidelberg 1961, Vol. XIj1, p. 202.
[*I Concerning methods of measurement in the high-frequency
and microwave range for the determination of the real and
imaginary parts of the dielectric constant, see: Discussions of the
Faraday Society,“Dielectrics’’(1946); F. W.Miiller and C. Schmelzer, Naturwissenschaften 25, 359 (1951); Handbuch der Physik.
Springer, Berlin-Gottingen-Heidelberg 1956, Vol. XVII; C. G.
Montgomery: Technique of Microwave Measurements. McGrawHill 1947; and other publications cited.
[20] P. Debye: Polare Molekeln. Hirzel, Leipzig 1929.
[21] C. P. Smyrh: Dielectric Behavior and Structure. McGrawHill, New York-Toronto-London 1955; J. physic. Chem. 58,
580 (1954).
Angew. Chem. internat. Edit./ VoI. I (1962) / No. 5
relaxation time T, the viscosity q of the solvent, and the
dimension of the sphere, i.e. of the dipole molecule is
given, according to Debye by the following relationship
(a = radius of the sphere). This model has been used
many times for the interpretation of dielectric relaxation
measurements. However, it soon became evident that
the resulting molecular diameters very often did not
coincide with values for molecular diameters obtained
by other ways. Nevertheless, Meakins [22] was able to
show that the Debye equation acquired a higher degree
of validity when the dipole molecule is larger in comparison to the solvent molecule.
Eyring and Kauzman [23] have proposed a quite different approach to the molecular-physicalinterpretation
of dielectricrelaxation times. They regard the relaxation
process as being caused by a “jump” process of orientation in the sense of the theory for absolute reaction
rates developed by Eyring.
The Eyring-Kauzman concept is as follows: The molecular
dipole occurs in a potential trough. A “rotation” into the
direction of an applied electric field, or reorientation into the
position corresponding to the thermodynamic state of
equilibrium after removing the field, is only possible by
overcoming a potential barrier AG#. The height of this
potential barrier is regarded as the activation energy for the
“reaction” rotation of the dipole. The dipole oscillates back
and forth in the potential trough with a certain frequency
kT
which, according to Eyring-Kauzman, is equal to -. At
h
~
[22] R . I . Meakins, Trans. Faraday SOC. 54, 1160 (1958).
[23] S. Glasstone, K. J. Laidler, and H. Eyring: Theory of Rate
Processes. McGraw-Hill, New York 1941; W. Kauzman, Rev.
mod. Physics 14, 12 (1942).
257
each eAG#/RT-th oscillation, it is able to jump over the potential barrier AG# and assume a new orientation. The
potential trough, which corresponds to a directed orientation, lies lower than that of the random orientation. As
a consequence, the potential barrier to be overcome is greater for the deorientation in the case of an applied field and
thus the probability of deorientation is smaller when the
electric field is applied. On the average, there results an
excess of orientated molecules and thus causing polarization
of the dielectric. After turning off the field, reorientation is
caused only by the difference in the population at each
position, since the potential barrier is equal in both directions.
The temperature dependence of the relaxation time then
follows from the Eyring-Kauzman theory as:
corresponding equationfor circular plot in the following
way :
The parameter d~ is meant to be a measure of the distribution
of relaxation times, and can be derived from the circular plot.
Therefore, the broadening of the absorption curve is interpreted by the assumption of a distribution of the relaxation
times. So far, the Cole-Cole interpretation of the distribution
of relaxation times has been applied mostly to the evaluation
of relaxation curves. However, there are a whole series of other
proposals for interpretating the deviation from the Debye
behavior. Wagner [27a] was the first who attempted to explain
deviations from normal dispersion behavior by considering
the distribution of relaxation times about a focal point. His
distribution function for the relaxation times is
Taking logarithms, and by transposing, we get:
where
From which it is evident that from a knowledge of the temperature dependence of the relaxation time, not only does the
quantitiy A G # correspond to the free enthalpy of activation
of the orientation process and is a measure of the height of
the potential barrier, but also that the activation enthalpy
and the activation entropy for the orientation process can
be calculated. The Eyring-Kauzman concept contains two
hypotheses, whose validities are doubtful :
z = In t/q
Corresponding to this distribution function, or to the parameter b determining the extent of distribution, we obtain
(Figs. 16 and 17) the dependence of the real and imaginary
parts of the dielectric constants on the logarithm of the
frequency. The distribution of the relaxation time itself is
given in Fig. 18.
1. The heights of the potential barriers may not fluctuate
during an orientation process. The basis of a theory which
allows for the fluctuation of the potential trough has been
attempted by Holzmiiller [24].
k. T
2. The validity of the frequency factor - without which
h
calculation of the activation entropy is impossible. This
factor, which arises from the Eyring theory for calculating
rate constants of gas reactions, was adopted in the theory
of an orientation process in which an initially unknown
factor x was set equal to unity [23]. Up till now, no agreement
has been reached regarding the validity of this frequency
factor for an orientation process, although SchaNamach [25]
argues for its validity.
In many cases, the experimentally observed relaxation
curves, i.e. the frequency dependence of the real and
imaginary parts of the dielectric constant, cannot be
described by the Debye equations. A leveling off of the
dispersion curve and a broadening of the absorption
curve is observed. In the circular plot (Fig. 15), the
qFig. 16. Graphical representation of the dependence of the leveling off
of the dielectric constant on the distribution constant b (Wagner and
Yager [27 hJf
t
Fig. 15. Cole circle for n-octylhromide at 25 "C (E. J . Hennelly, W . M .
Heston, and C. P. Smyfh, 5. Amer. Chem. SOC.70, 4102 (1948))
values still lie on the arc of a circle, but its centre no
longer lies on the abscissa. Cole and Cole [26],who deal
with this behavior, modify the Debye equations and the
[24] W . Holzmiiller, Physik. Z. 41,499 (1940).
[25] A. Schallamaeh, Trans. Faraday SOC. 42A, 166 (1946).
[26] K. S. Cole and R . H . Cole, J. chem. Physics 9, 341 (1941).
258
W 'tn-
Fig. 17. Graphical representation of the dependence of the leveling off
of the angle of loss on the distribution constant b (Wagner and Yager
"27 bl)
[27] (a) K.W.Wagner, Ann. Physik 40, 817 (1913);
(b) K . W. Wagner and W.A . Yager Physics 7, 434 (1936).
Angew. Chem. internat. Edit. 1 Vol. I (1962)/ No.5
d
16
t
b=2.0
II
221
0
0
0
0.
07 02 0 3 0 4 05 016 07 08 09 10
El-&,
ELl
7
LUG-’
Fig. 18. The influence of the distribution constant b on tbe density of
distribution of relaxation times (Wagner and Yager [27 bl)
Gevers [28] and Frohlich [29] suggest a different distribution,
in which the distribution function has a practically constant
value between two z values, but has the value zero outside
this interval. If, in the sense of the Eyring theory, the potential barrier related to t~ is designated by Ho, then the z
values for homogeneous distribution, disregarding dipole
interactions, vary between zl and t o :
- Ecn
Fig. 19. Normalized Frohlich curves for a) Vo = 0 kT, (Debye circle)
b) Vo = 4 kT C) Vo= 10 kT
An additional distribution function for relaxation times was
specified by Fuoss and Kirkwood [31], whose equation reads:
max
cos h In
d-Function
Fuoss and Kirkwood
t
0
log T&
m
- arc tan O
T ~
1
]
Emax = (SO
v,
1
- arc tan e
‘<v‘:
-2
vo
iZ] (69)
The ratio of both limiting relaxation times can be determined
from the equation:
arc tan
E“, I (w) --
%ax
-
am
arc tan
f;
2
-arctan-
-
(68)
T ~
vu
kT arc tan e
- Em)
E
Fig. 20. Various distribution functions. Density of the relaxation times
on a logarithmic scale (Miiller and SchmeOer [I I], p. 384)
The deviation range of the potential barrier, i.e.Vo, can be
obtained from the value of the maximum of E” using the
following equation:
,I
w
fM
In this distribution, the Debye equations now have following
forms :
arc tan W
~
It leads to:
0.5
The height of the potential barrier therefore fluctuates between HO and NO+ VO.The distribution function is then:
Omax
3
am T l
_
‘
I
_
-
(70)
The various distribution functions proposed are compiled in
Fig. 20. P o k y [32] related the distribution parameter p of the
Fuoss-Kirkwood equation and the parameter a. of the ColeCole distribution. It holds to a f i s t approximation [33]:
p .1/2
=
1 - c(/ cos
(1 -a) * x
4
(73)
A further attempt to describe the deviation from Debye behavior was undertaken by Perrin [34]. He considers nonspherical molecules as ellipsoids and introduced three relaxation times corresponding to the three major axes of an
ellipsoid. According to Fischer [35], the relationship between
the relaxation time and molecular dimensions is given by
- arc tan
It is interesting that this representation is only affected outside the range of main absorption by changes in the pattern
of the barrier height H. Figure 19 shows the normalized E”
curve as a function of E’ for three values of VO.
Bergmann [30a] was able to show that the Frijhlich curve can
be approximated closely by an ellipse. On this assumption,
Higasy, Bergmann, and Smyth [30b] derived a relationship
between the Frohlich parameter VOand the Cole parameter a.
[28] M. Gevers, Philips Research Rep. I, 279 (1946).
[29] H. Frohlich: Theory of Dielectrics. Oxford University Press,
Oxford 1958.
[30a] K. Bergmann, Ph. D. Thesis, Universitat Freiburg 1957.
Angew. Chem. internat. Edit. 1 Vol. I (1962)/ No. 5
where a, b and c are the semiaxes of the ellipsoid, f is a
structure coefficient and q* = 0.36 q. In the case of a,wdibromoalkanes, the dielectric relaxation behavior has been
[3 0 b] K. Higasy, K. Bergmann, and C. P. Smyth, J. physic. Chem.
64, 880 (1960).
[31] R. M. Fuoss and J . G. Kirkwood, J. h e r . chem. SOC.63,385
(1941).
[32] J. Ph. P o k y , quoted after C. J. F. Bdrtcher [33], p. 371.
[33] C. J. F. B6ttcher: Theory of Electric Polarization. Elsevier
Publ. Co., Amsterdam 1952.
[34] F. Perrin, J. Physique Radium 5, 497 (1934).
[35] E. Fischer, Physik. 2.40, 645 (19393.
259
described by Price with the aid of a Fuoss-Kiricwooddistribution [36].
Having now discussed the general attempts for the treatment of dielectric relaxation phenomena in liquids, some
experimental results will now be considered. Evaluation
of the temperature dependence of the relaxation times
obtained from dielectric measurements according to
E r r i n g and Kauzman almost invariably indicates negative
k.T
activation entropy (this holds only if the factor
is
valid). Since, in line with the model, one associates the
orientation process in the Eyring-Kauzman concept
with a liberation of the dipole molecule from its solvation shell, the occurrence of negative activation entropy
(corresponding to an increase in the order of the activated state) was hard to understand. Miiller [37] and
independently Levi [38] have given a qualitative explanation. They assume that in order that the dipole can
rotate, the solvent molecules they have to make room by
“crowding together.” This “crowding together” corresponds to a state of higher order and therefore furnishes
an explanation of the occurrence of negative activation
entropy.
Although at present, the Eyring-Kauzman theory offers the
best basis for a model interpretation of the dielectric relaxation processes in liquids, it has not been possible to
obtain results of general validity on the basis of this theory.
The primary reason for this may be that the modal is oversimplified and that, so far, no theory of liquids exists which
involves all physical phenomena. Despite the difficulties,
however, in individual cases, model interpretation of .t-values
resulting from dielectric relaxation measurements have been
successful.
In dilute solutions, in which a dipole-dipole interaction
is probably largely non-existent, Zeil and co-workers [39]
were able to point out qualitative relationships between
the activation quantities derived from the temperature
dependence of the relaxation time and the intermolecular
forces between solvent molecules and dipole molecules
on the one hand, and the state of order, described by the
free volume of the solvent, on the other hand. The activation enthalpy is augmented with an increase in the
London and Debye forces between solvent molecule and
dipole molecule. The magnitude of the negative activation entropy increases with increasing free volume of
the solvent. The following model representation was
proposed: In the activated state, the dipole molecule is
desolvated and the solvate molecules occupy free space
(holes) in the solvent. The energy necessary for the liberation of the solvate molecules, reduced by a value
involving by the inclusion of solvate molecules in the
holes, corresponds to the activation enthalpy. The increme in order derived from incorporation of solvate
molecules into the holes of the solution will be all the
greater, the more holes available. This model holds for
densely packed solvents and for dipole molecules which
are very large compared to the solvent molecules in
1361 A . H. Price, Bull. Groupment AMPERE (C. R. IX Coll.)
1960, 71.
[37] F. H . Miiller, Kolloid-Z. 134, 215 (1953).
[38] D. L. Levi, Trans. Faraday SOC.42A, 152 (1947).
[39] W . Zeit, H. Fischer, W. Metzger, K. Wugner, and J. Hame,
2. Elektrochem. 63, 1110 (1959); W.Zeil, J. Huuse, and 0 .Sfiefvater, ibid., 65, 616 (1961).
260
the Debye model of viscous rotating spheres, in which
only the translational process in the solvent, and thus
the viscosity, play decisive roles.
Klages et al. [40]succeeded in various cases in satisfactorily interpretinga broadening of the relaxation maxima
by assuming two discrete relaxation times. By superposition of two Debye curves and thus of two discrete
relaxation times, they were able to satisfactorily explain
measurements made on aliphatic chlorinated hydrocarbons. Experimental results with aniline derivatives
were also explained by this method. Figure 21 shows
how the data can be interpreted by superposition of two
Debye terms. One relaxation time corresponds to an
intrinsic mobility of the NH2 group, the greater relaxation time to overall rotation of the molecule. In addition,
Fig. 21. Approximation to the absorption behavior of p-toluidine by a
-) or by two Debye terms -(
Debye-term () (Kramer
WI, P. 976)
----
the dipole moment can be resolved into partial moments,
i.e. the dipole component closely associated with the
rigid part of the molecule can be separated from the
dipole component of the mobile group.
A very interesting attempt to interpret the extremely short
relaxation times and the deviation from Debye behavior of
dilute solutions of diphenyl compounds through a superposition of a Debye term by a Frohlich term has been reported
by Hufnagel[411.
A very comprehensive analysis of a relaxation time distribution originated with Schroeder [42]. Meckbach [43]
investigated pure a-bromonaphthalene and interpreted
the deviation from Debye behavior by superposition of
several relaxation times, conditioned by the ellipsoid
structure of the molecule according to Perrin [34], but
Schroeder showed that the broadening of the E” curves
for wbromonaphthalene in carbon tetrachloride is dependent on concentration. His extensive measurements
embrace practically the whole dispersion region. The
results (Table 3) can be very well represented by a Frohlich distribution.
Table 3. Concentration dependence of the width of the
well as the VO values of a-bromonaphthalene in C C 4
Concenlog (WOIOI)
tration,
weight
1.14
percentage
1
5
10
20
1 . 0 2 i 0.02
1.04 f 0.02
1 . 0 5 i 0.01
1.09 f 0.03
“E”
71/70
1 . 5 1 0.5
2.2 f 0.5
2.5f 0.2
3.5 f 0.7
curves as
vo
0.4kT
0.8 kT
0.9 kT
1.2 kT
Angew. Chem. internat. Edit. 1 Vol. I (1962)
/ No. 5
Table 3 contains the relative broadening compared with a
Debye curve. The value log ( o o / q ) is a measure of the width
of an E” curve. In the Debye case, it is 1.14. The table also
contains the ratio of the two relaxation times, between which
a continuous spectrum of relaxation times exists, as well as
the range of variation VOof the height of the potential barrier.
It shows that with increasing concentration, the variation of
the potential barrier increases. Schroeder explains this as
being due to an increasing interaction between dipole molecules without of association, since in such a case, the value
of z/q (q = viscosity) would have to increase with increasing
concentration. Experimentally, the contrary was observed.
We will not go into individual investigations of systems which
involve hydrogen bonding. One should refer to works by
FrohZich [29] or Brown [44] regarding the possibility of the
occurrence of resonance absorption. The publications of
M i l k [37] and Gross [45] are recommended regarding the
difficulties confronting analysis of relaxation curves where no
Debye behaviour exists.
If the extent of relaxation in the mechanical or dielectric region is investigated then it is evident - in
contrast to gases and many low-molecular weight
liquids - from the order of magnitude of the loss M ’
or E” compared with the frequency, that a Debye curve
is not obtained. The half-width of a Debye absorption
curve amounts to about 1.14 frequency-decades. With
high polymers, the absorption curves are considerably
broader. Thus, for example, with polyoxymethylene,
[-CH2-0--],, a half-width two to two-and-a-half times
as large is observed [49]. Figures 22 and 23 show the
The topic of dielectric relaxation in electrolytic solutions will
not be described here. In this case there occurs a further loss
through ohmic conduction and the conductivity also shows
dispersion. A summary is given by FaZkenhagen [46]. More
recent measurements in this area are due to Weber [47].
Regarding the structure of anisotropic liquids, there are
interesting investigations by Maier and co-workers [48] o n
the dielectric behavior of liquid crystals (anisotropic liquids).
v 1106 CPSI
4. Mechanical and Dielectric Relaxation
Phenomena in High Polymers [*I
Dielectric and mechanical relaxation phenomena in
plastics (high polymers) will be treated together since, in
contrast to the case of liquids, there are no completely
different mechanisms to account for both relaxation
both the diphenomena. It is generally assumed
electric as well as the mechanical losses are caused by
the same molecular physical phenomena.
[441 W. Fuller Brown jr.: Handbuch der Physik. Springer, BerlinGottingen-Heidelberg 1956, Vol. XVII, p. 131.
[45] B. Gross, Kolloid-Z. 134, 65 (1953).
[461 H. Falkenhagen: Elektrolyte. 2nd. Ed., Hirzel, Leipzig 1953.
[471 G. Weber, Ph. D. Thesis, Universit’dtTiibingen 1961.
I481 W. Maier and G. Meier, Bull. Groupment AMPERE (C. R.
1X Coll.) 1960, 38.
[‘I For the methods of measurement consult the original lit-
erature.
Angew. Chem. internal. Edit. 1 VoI. I (1962) 1 No. 5
Fig. 22. Frequency curves for the dielectric loss factor (tan 8 ) of
polyoxymethylene at various temperatures (“C)(Thurn [491, p. 324)
dielectric loss factor tan 6, measured with polyoxymethylene, plotted against the frequencyfor various temperatures and the temperature for various frequencies.
One finds such more or less broad absorption curves for
almost all high polymer materials. At an early stage,
Fig. 23. Temperature curves for the dielectric loss factor (tan 6) of
polyoxymethylene at different frequencies (Thurn 1491, p. 324)
Kuhn [50] and Smekal[51] reached the conclusion that
the cause of this is a continuous distribution of relaxation times covering a certain range, i.e. a continuous
relaxation time spectrum. Moreover materials of more
[49] H . Thurn: Festschrift C. Wurster. Ludwigshafen 1960, p. 321.
[50] W. Kuhn, 2. physik. Chem. 42, 1 (1939); Helv. chim. Acta
30,487 (1947); 31, 1259 (1948).
[51] A. Smekal, 2. physik. Chem. 44, 461 (19391.
26 I
complex structure, show two, or in many cases even
more, maxima in the loss curves obtained by dielectric
as well as mechanical relaxation measurements ; we then
speak of the appearance of a primary maximum and a
secondary maximum. It can be concluded, therefore,
that different processes, caused by a continuous distribution of relaxation times, account for this relaxation
behavior. The focal points must therefore lie sufficiently
far apart. If one the half-width of the loss curve of polyoxymethylene is plotted against temperature, then it is
seen that it becomes smaller at increasing temperatures.
This can only be explained by a temperature-dependent
distribution of the spectrum of relaxation times.
If the measurements cover the total frequency range of
relaxation, the distribution function for the relaxation
times can be determined, in principle, using complex
methods of calculation which we will not described here.
Such a procedure involves great experimental difficulties
in many cases. Whereas a relatively large range of frequencies can be applied in dielectric relaxation, this is
scarcely possible in the case of mechanical relaxation.
Comparison of the curves obtained at constant temperature and varying frequencies with those obtained at
varying temperatures but at constant frequency now
shows, however, that in the first case, about 8 to 12
frequency decades must be covered experimentally in
order to gain a rather complete understanding of a relaxation curve, while in the second case a variation in
temperature of only about 100°C is necessary. The
large frequency intervals required to evaluate the spectrum of relaxation times are necessary, because the wings
of the absorption curves enter very strongly into the calculation of the distribution of relaxation times. Since
only in rare cases can more than a two-decade range
of frequencies be covered using the same apparatus, it
is normal to investigate experimentally, the more easily
accessible temperature variation of relaxation at constant frequency. However, theoretical treatment shows
that for a quantitative interpretation, only the frequency
variation at constant temperature may be chosen as the
starting point; here, of course, for a complete understanding of the relaxation phenomena, the dependence
of these curves on the temperature must also be known.
Ferry et al. I521 developed a procedure whereby, e.g.
from measurements of the stress relaxation or of the
dynamic modulus, one can convert mathematically from
an experimentallylimited range of frequencies at varying
temperatures to a much more extensive range of frequencies at constant temperature. This method is frequently used to deduce the relaxation-time spectrum
from temperature-variation curves. It is only applicable,
however, if the relaxation-time spectrum does not
change with temperature. In the case of polyoxymethylene (Fig. 23), this does not apply. Furthermore, the
method is not applicable if the regions of different relaxation mechanisms overlap. For example, the results
of Miiller and Broens [53]could not be evaluated meaningfully by Ferry’s conversion procedure.
[52] J . D . Ferry et al., J. appl. Physics 22, 717 (1950); J. Amer.
chem. SOC. 77, 3701 (1955).
[53] F. H . Miiller and 0. Broens, Kolloid-2.140, 121 (1955); 141,
20 (1955).
262
Because of experimental and theoretical difficulties encountered in the interpretation of the loss maxima of
high polymers, another, more qualitative, method has
therefore been adopted and has already indicated significant correlations between the constitution of high polymers (structure, degree of branching, polarity, partial
crystallinity, etc.) and relaxation phenomena. Relaxation phenomena, and the underlying relaxation times
corresponding to them, are considered to be caused by
delayed orientation movements of the molecules, or of
parts of the molecules, of high polymers. Molecules, or
parts thereof, therefore follow the oscillation of an external field with a phase lag in both dielectric and mechanical cases. Causes of the delay in the adjusting motion are steric effects and internal fields, due to partial
dipole moments or bonding momegts, since the loss
maxima always occur when the duration of the periods
of applied field or of the applied mechanical strain becomes comparable to the relaxation time of molecular
motion. One considers, therefore, the appearance of a
loss maximum as an indication of a possible molecular
motion [54]. From this viewpoint, measurements at
constant frequency and varying temperature can be interpreted quite clearly. At a definite temperature, parts of
molecules or entire molecules become mobile and can
follow the applied alternating field. The temperature at
which this mobility is sufficient to follow the oscillating
field will depend on the frequency. The higher the frequency of the oscillating field, the more mobile the
molecules or parts of molecules must be, i.e. the loss
maxima shift to higher temperatures with increasing
measurement frequency.
From the foregoing, it follows that generally, loss-maximum curves can-be interpreted, although, in principle,
the real parts (elastic modulus and dielectric constant)
can also be utilized for evaluation. Frequently the results
Fig. 24. Three-dimensional representation of mechanical loss of
poly(methy1 methacrylate) (Schmieder and Wolf [551)
[541 H. Hendus, G. Schnell, H. Thurn, and K . A . Wolf, Ergebn.
exakt. Naturwiss. 31, 220 (1959).
Angew. Chem. internat.
Edit. 1 Yo/. I (1962) 1 No. 5
are recorded as a three-dimensional graph, e.g. the
measurements of mechanical loss of poZy(methy1 methacrylate) by Schmieder and Wolf in Fig. 24 [55]. The
results can also be interpreted in part by a sort of contour diagram: see Fig. 25.
to correspond to the motion of small parts of the molecule e.g. side-chains. Figure 26 shows how steric influences affect the temperature at which the primary
maxima occur and records the measurements of Schmieder and Wolf [56] on poZy(isobuty1ene) (I), poZy(styrene)
(11) and poZy(vinylcarbazo1e) (111).
150
10"
I
I
1
I
I
I
100
g
?
50
0
-50
I I
ro-'
-100
7
702
70
lzma
103
.7
ViCPSl
Fig. 26. Steric influence on the temperature pattern of the primary
maxima, measured in poly(isobuty1ene) I, polyfstyrene) 11,
and poly(vinylcarbaso1e) 111 (Schmieder and Wolf 1561)
__
0. 740
150
r
c H 3 1
a 100
0.080
_I! _
060
700
0.050
0.040
0.030
50
0 02%
10
[--CHz-yH-]
]
70
J
1
10
LaGm
10
105
ro6
ViCPSl
Fig. 25. Contour diagram of mechanical loss (a) and dielectric loss
(b) in poly(methy1 methacrylate); (J. Heijboer, P. Dekking, and A . J .
Sfaverman, Kolloid Z. 148, 36 (1956))
We now wish to consider several examples, with the aid
of models, where it was possible to correlate individual
loss maxima with the mobility of specific parts of
molecules by varying the chemical structure of the high
polymers systematically.
The results obtained so far exhibit the following regularities. The shift of the loss maxima in relation to
temperature and frequency shows that there are two
groups of maxima: the group of so-called primary
maxima has a slight temperature and frequency shift;
the group of secondary maxima, which always appears
at lower temperatures or at higher frequencies as the
primary maxima, show a large temperature and frequency shift. In general, the primary maxima are attributed to the mobility of the main chains of the high
polymers, whereas the secondary maxima are supposed
[55] K.Schmieder and K . Wolf, Kolloid-2. 1.27, 65 (1952).
[56] K . Schmieder and K.W o z Kolloid-2. 148, 136 (1956).
Angew. Chem. internat. Edit.
I VoI. 1 (1962) 1 No. 5
It is seen that, with increasing extension of the side
groups, the mobility of the molecular chain, which is
responsible for the loss maxima, is hindered more and
more; this leads to a displacement of the maxima to
higher temperatures at constant measuring frequency. At
constant temperature a corresponding shift of the
maxima to lower frequencies is observed.
The influence of polarity on the mobility of the
main chain is seen in Table 4.
Table 4. Influence of polarity on the mobility of the main chain
p&(vinyl
methyl
ether)
Polyaisobutylene)
Dipole moment
(debyes)
0
Mechanical
measurement Tmax
( " C )at about 2 cps
-48 [571
Dielectric
measurement Tmax
( " C )at 2 x 106 cps
unpolarized
no
1
lI
1.2
-10 [57]
Poly(vinyl
chloride)
p0/p
(vinyl
acetate)
1
l1
1.8
+33 1571
I
2.0
+90 [571
+ 139 I591
With increasing dipole moment, the maxima shift to
higher temperatures, corresponding to a decrease in
[57] K. Schmieder and K . Wolf,Kolloid-Z. 134, 149 (1953).
[58] H.Thurn and K. Wolf,Kolloid-2. 148, 16 (1956).
[59] H.Thurn and F. Wiirstlin, Kolloid-2. 156, 21 (1958).
263
mobility with increasing polarity. From the last two
examples, one can conclude that with increasing steric
hindrance and increasing polarity, the mobility of the
main chain is shifted to higher temperatures. For practically equal dipole moments, the influence of the length
of the side-chains is such that a shift of the maximum to
lower temperature occurs with increasing length of the
side-chains1601. The literature should be consulted for information regarding cross-linked high polymers; e.g.
Wolfet al. [54]. There the molecular weight effect is related to the temperature dependence of the primery loss
maxima.
In some cases it has been possible to assign definite molecular motion to individual loss maxima in materials
with several loss maxima. Besides its primary maximum
of mechanical loss, poly(methy1 methacrylate) possesses
a secondary maximum lying at a lower temperature. It
is interesting that in dielectric measurements, the height
of the primary and secondary peaks are just the opposite
of those found in mechanical measurements. This is
related to the fact that in dielectric experiments the
height of the maxima, i.e. the magnitude of loss, is dependent on the polarity (dipole moment) of the moving
parts of molecules whereas in the mechanical case the
mobile mass determines the height of the maximum.PoIy(methyl methacrylate) has the main chain with the
methyl groups as the nonpolar part and the methyl ester
groups as the polar part. The dielectricprimary maximum
must therefore be due to the motion of the methyl ester
groups, while the mechanical primary maximum must
be associated with the motion of the primary chain. It
turns out that the mechanical secondary maximum possesses an activation energy of 18 kcal./mole and, in good
agreement, the corresponding dielectric primary maximum possesses an activation energy of 21 kcal./mole,
thus confirming the correlation. Several maxima were
also observed f orpoty(isobuty1ene)1541. The low temperature maximum is attributed to a motion of the CH3
group. Its activation energy amounts to about 4 kca1.l
mole. An internal rotation or torsional vibration of the
CH3 group is considered to be the cause. A maximum at
intermediate temperatures is explained by the onset
mobility of the segments of the main chain. A third
maximum observed at high temperatures, occurs only for
1601 8. L. Funr and Th. Sutherland, Canad. J . Chemistry 30, 940
(1952); Th. Sutherland and B. L. Funt, J. Polymer Sci. 11, 177
(1953); reviewed in [54], p. 236.
very large molecular weights. It has a relatively small
activation energy.
As a last example of analysis of relaxation processes in
high polymers we will now consider poly(ethy1ene)
Thurn and Wolf[61] measured the dielectric loss of unoxidized poly(ethy1ene) from -190 "C to + 160 "C, at
frequencies between 5 x 103 and 1 x lo5 cps. With the aid
of a special method, they were able to measure loss
factors down to 1x 10-5. This was necessary because
only very slight partial dipole moments occur in poly(ethylene), the dielectric loss thus becoming very small.
At low temperatures, a loss maximum is observed in
linear as well as branched poly(ethy1ene). It is not large
and is attributed to a motion of the CH2 groups. In
addition, for branched poIy(ethy1ene) a very intense
maximum is observed at 0 "C. that is strongly dependent
on molecular weight. This maximum is caused by double
bonding, especially at the side chains. An important contribution can be attributed to small polarities associated
with molecular assymetry, particularly at the branch
points. Thurn and Wolf have also measured mechanical
loss maxima ofpoIy(ethy1ene) at a comparable frequency.
Corresponding loss maxima were found; in addition, at
the same frequency, the temperature patterns of the
dielectric and mechanical loss maxima were largely
consistent.
The examples cited were intended to show that relaxation investigations on high polymers are able to provide
clues as to the molecular structure of plastics (high polymers). Furthermore, these investigations afford valuable knowledge concerning the relation between the
molecular structure of materials and their technically
interesting mechanical and dielectric properties.
Dielectric relaxation phenomena in crystals cannot be discussed here. A theory of dielectric relaxation in molecular
crystals has, however, been developed [62]. Additional work
has been done on relaxation phenomena in crystals, semiconductors and in absorbed water, q.v., Conference Reports of
the Groupment AMPERE for recent years. The phenomenon
of molecular rotational relaxation in solids is under investigation by Davies [63].
Received, September 19th. 1961 [A 183129 IE]
[61] H.Thurn and K . Wolf, Paper No. 241, Meeting of the Verband Deutscher Physikalischer Gesellschaften, Vienna, October
20th., 1961.
[62] J. D . Hoflman, Maxwell-Ampere-Conference, VIII. Coll.
Ampere, Kundig, Genf 1959, p. 36.
I631 M . Duvies et al., Bull. Groupment AMPERE (C. R. IX.
Coll.) Kundig, Genf, 1960 p. 77.
COMMUNICATIONS
N-Azidodimethylamine [ 1I
By Dr. H. Bock and cand. chem. K. L. Kompa
lnstitut fur Anorganische Chemie der Universitat Munchen
In the course of experiments directed towards a new synthesis
of tetrazene according to the general equation
2 RzNCl+ 2 MeN3
+
RzN--N=N-NRz
+ 2 NZ+ 2 MeCl,
sodium azide was added to N-chlorodimethylamine in an
inert solvent. After stirring the solution for several days at
[ I 1 1st. Communication on inorganic azides.
264
room temperature, we were able to isolate a 25 % yield of
N-azidodimethylamine (1,l -dimethyltetrazadiene),which was
originally expected be an unstable intermediate
8
e
8
(CH3)2N=F--fT=N
-t)(CH3)2%N=N=N
-
9
- 1
2 @
t)( C H , ) ~ ~ - N - N E N I
-
The highly explosive compound (b.p. 32 "C/l1 mm.) was
characterized by its molecular weight (in the gaseous state),
its azide absorption bands in the infrared spectrum [2110
Cm-l (VS) and 1210 cm-l (VS)], and by an approximate
nitrogen
Received, October 30th. 1961
Publication delayed at the author's request,
[Z 251/86 IE]
Angew. Chem. internat. Edit. Vol. I (1962) / No. 5
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