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

код для вставкиСкачатьMechanical 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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