- Email: [email protected]
The relaxation of a model system of interacting dipoles
Journal of Non-Crystalline Solids 235±237 (1998) 154±159
The relaxation of a model system of interacting dipoles V. Halpern
*
Department of Physics, J. & P. Resnick Institute of Advanced Technology, Bar-Ilan University, 52900 Ramat-Gan, Israel
Abstract We consider a simple model system of interacting dipoles which can be either free to rotate or trapped, and in which the rotation of a dipole can change the state of neighboring dipoles from trapped to free or vice versa, with probabilities that ensure that the fraction of dipoles that are free remains constant, as is required for systems in dynamic equilibrium. The system was studied in a mean ®eld approximation and also by computer simulations. The results show that the non-exponential decay of the system's response function, and hence of its relaxation function, is associated with the fact that initially the decay is due mainly to the depolarization of the free dipoles, while the later stages are associated with the freeing of the polarized dipoles that were trapped and their subsequent depolarization. This process could be a quite general mechanism for non-exponential dielectric relaxation. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The frequently observed non-exponential decay of the dielectric polarization following the removal of an applied ®eld [1], especially in disordered systems, has given rise to a large number of theories that attempt to explain this phenomenon. One class of theories [2] attributes it to the superposition of numerous independent processes such as the relaxation of individual dipoles each of which does relax exponentially, but with a distribution of relaxation times as a result of the disorder. Another class of theories attributes it to time-dependent transition rates (TDTRs) for the dipoles in the system which can arise, for instance, from the migration of a defect that is required to trigger the relaxation of a dipole [3,4] or from interactions between the dipoles [5]. One problem with these TDTRs, which we have considered elsewhere [6],
* Tel.: +972-3 5318433; fax: +972-3 5353298; e-mail: [email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 5 0 3 - 1
is the choice of the origin of time, or equivalently of the state of the system at time t 0. In this paper, we examine a model system of interacting dipoles for which the initial state of the system is very clearly de®ned. The results of our calculations suggest a very simple and quite general mechanism that will lead to non-exponential decay. 2. The model system 2.1. Description of the system We consider a set of interacting dipoles, which for convenience we assume are located on the sites of a simple cubic lattice. At least for systems that are not ferroelectric, a small applied electric ®eld will aect mainly the orientation of these dipoles, and can have only a minor eect on their transtion rates between dierent states. Accordingly, we make a clear distinction between the orientations of the dipoles and their environments and associated transition rates. In this paper, we consider the
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
simplest possible system of this type, namely one in which the dipoles can have just two possible orientations, with moments m, the transitions between which will be called a ¯ip, and where they can be in just one of two possible environments or states, A or B, where only the dipoles in state A are free to ¯ip, while those in state B are immobile. The novel feature of our model system is in the choice of the following interaction between the dipoles. When a dipole in state A ¯ips, it does not change its own state but has a certain probability of changing the state of each of its neighboring dipoles from A to B or vice versa. A possible scenario for such an interaction, not to be taken too literally, is that when a dipole in state A ¯ips from a cis to a trans orientation, which we will call from right to left, it can free space for its righthand neighbor to ¯ip, and so transfer it from state B to state A, and can block any ¯ips of its lefthand neighbor, and thus transfer it from state A to state B. However, the dipole that has just ¯ipped can still ¯ip back from the trans to the cis orientation, and so is still in state A. This process can also be regarded, in somewhat more general terms, as the migration of local free volume [7] when a dipole ¯ips. Our choice of the above form of interaction was not motivated by an attempt to describe real systems but rather by the important property that there is a clear distinction for it between the relaxation function F(t) and the response function f(t), which we will now discuss. As is well known, [1], these two functions are related by f t ÿdF =dt:
1
The response function describes the response of a system to a pulse excitation. While such an excitation is, of course, a theoretical idealization, its meaning for our model system is that the ®eld is applied for long enough for some of the dipoles in state A to be polarized (i.e. for more to ¯ip from the orientation with moment )m to that with moment +m than vice versa) but not long enough for an appreciable number of these polarized dipoles to be transferred to state B. Thus on the removal of the ®eld the system starts to relax from a state in which only some of the dipoles in state A are polarized. For the relaxation function, on the
155
other hand, the ®eld has been applied for a long time, so that the exchange of polarized dipoles between states A and B has reached equilibrium. The reason that it is important to distinguish between these two functions is that, as we have pointed out elsewhere [6], in the case of non-exponential relaxation the choice of the origin of time is of critical importance. For instance, only if b 1 is b exp ÿt s proportional to exp ÿtb , so that only in this case is it unimportant from what instant one measures the time. Otherwise, the basic non-exponential relaxation is that which starts following a pulse excitation, i.e. the response function, and the relaxation function follows from the sum of the relaxations to a continuous set of these pulses applied at dierent times. Now that we have described the system qualitatively, we can de®ne it formally. Let 1=sA be the transition probability for a ¯ip of a dipole in state A in the absence of an applied ®eld. When an electric ®eld E is applied, it lowers the energy of a dipole parallel to the ®eld and raises that of a dipole anti-parallel to the ®eld by mE. As a result, the transition probabilities 1=s A for transitions out of these orientations (for thermally activated transitions at temperature T) will be multiplied by a Boltzmann factor exp mE=kB T , where kB is the Boltzmann constant and the upper sign is for dipoles parallel to the ®eld. Thus in the small ®eld approximation, when mE kB T , these transition probabilities are given by 1=s A 1=sA 1 cE;
2
where c m=kB T . In addition, we denote by pAB be the probability that such a ¯ip will change the state of an adjacent dipole from A to B, and by pBA the probability that the ¯ip will change it from B to A. 2.2. Analysis of the system Since each dipole can be in either orientation and in either state, the exact analysis of this model system for a set of N dipoles requires the solution of 4N simultaneous linear rate equations for the probability of the system being in each of its possible 4N states. While an exact solution of these equations is not really practicable for N > 3, the
156
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
problem can be solved analytically in the following mean ®eld approximation. We assume that each dipole has exactly the same number Z of dipoles adjacent to it, and that of these zA ZfA are in state A and so are free to ¯ip while zB ZfB are frozen in state B. Let n A and nB denote the number of dipoles with moments m in states A and B, respectively; for a system in dynamic equilibrium ÿ with respect to the states, their sums nA n A nA ÿ and nB nB nB will be constant. The quantities of interest are the net dipole moment, or polarization, of the dipoles in states A and B, MA ÿ ÿ m n A ÿ nA and MB m nB ÿ nB , respectively, and the net polarization M MA MB
3
of the system. The rate equations for n A and nB are, to ®rst order in the applied ®eld E, dn A =dt ÿ nA =sA nA =sA zA =sA pBA n B ÿ pAB nA ;
4
and dn B =dt
moval of the applied ®eld can be expressed in the form M t c1 exp ÿs1 t c2 exp ÿs2 t;
where the exponents s1 and s2 are the roots of the equation s2 ÿ 2 pAB pBA =sA s 2zA pAB =s2A 0
ÿ
pAB n A :
5
It follows from Eq. (5) that dnB =dt ÿ zA =sA pBA nB ÿ pAB nA
6
and so, since in the steady state nB must be constant, that nB =nA pAB =pBA :
7
It follows from Eqs. (2) and (4) after simple calculations that dMA =dt ÿ 2MA =sA zA =sA pBA MB ÿ pAB MA c0 E;
8
0
where c 2cmnA =sA , and from Eq. (5) that dMB =dt ÿ zA =sA pBA MB ÿ pAB MA : dn B =dt
9
and so for dMB =dt do The equations for not involve the ®eld E since in state B the dipoles cannot change their orientation. Eqs. (8) and (9) are a pair of coupled linear dierential equations with constant coecients, which can readily be solved by use of the Laplace transform and taking the inverse transform of the solution. One ®nds that the polarization M t at time t after the re-
11
and the coecients cj are readily found as the solution of a pair of simultaneous equations, while for MA t and MB t we replace the coecients cj by aj and bj , respectively. In particular, the coecients aj and bj , and hence cj , depend linearly on the polarizations MA 0 and MB 0 at the time t 0 when the applied ®eld was removed, and so are proportional to the applied ®eld E that produced these polarizations. As explained above, for the response function MB 0 0, while for the relaxation function the initial state is the steady state in the presence of the ®eld, so that, in view of Eqs. (7) and (9), MA 0=MB 0 pBA =pAB nA =nB :
ÿ zA =sA pBA n B
10
12
We also performed computer simulations for a number of systems with N 253 , and the results of these were found to be very similar to those of the mean ®eld approximation. While the system could have been analyzed in terms of correlation functions [8] rather than in terms of the polarizations, such an analysis would not be any simpler, and would be far less informative. In order to calculate the correlation functions, one has to know the initial state of the system, and as we have pointed out elsewhere [6] these are different for the response function and for the relaxation function. The appropriate initial conditions can only be found by use of the rate equations with a term involving the ®eld, which is the only point in our analysis in which we use the applied ®eld E. The subsequent relaxation can be described either by our rate equations without an applied ®eld or by a study of the correlation functions, but the latter can only be de®ned for the total polarization and not for the partial polarizations of the dipoles in states A and B, while the results presented and discussed below show that important physical insight can be gained from studying these partial polarizations.
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
3. Results of the calculations While in experiments one can, of course, only measure the total polarization, in our simulations and calculations we can separate the contribution to the polarization of the dipoles in states A from that of the dipoles in state B, a point that is of great value for the analysis of the results that we now present and for our discussion of their significance in the next section. In particular, while Eq. (1) provides a simple relation between the total polarizations given by the relaxation and response functions, this equation does not apply to the partial polarizations of the two sets of dipoles. The ®rst signi®cant point in the results of our analysis is that if one de®nes a partial relaxation function FA t for the dipoles in state A (whose identity is not ®xed, since the dipoles are continually undergoing transitions between states A and B) by MA t FA tMA 0
13
then one ®nds from the formulae for the coecients aj and cj that FA t kA f t;
14
i.e. that, apart from the normalization, the response function is just the relaxation function of the dipoles in state A. This means that the initial polarization of the dipoles in state B, which is present in the relaxation function of the system but not in the response function, does not aect the decay of the polarization of the dipoles in state A. Such a result is not entirely obvious, but occurs because
157
the dipoles do not change their orientation when becoming trapped or untrapped while in the mean ®eld approximation the transitions between the two states occur at exactly the same rate in each direction. The other features of our results can be deduced from Table 1, in which we present, for some typical values of the parameters fA and pAB , the values of the exponents sj and of some of the coecients. For the response function we present the coecients of kA f t rather than those of f t since kA f 0 1, so that M t=M 0 kA f t, and since MB 0 0 we can write MB t=M 0 kA b2 exp ÿs2 t ÿ exp ÿs1 t: 15 The other coecients can readily be deduced from those listed in the table by means of Eqs. (3), (10)± (15). Most of the results presented in this table can readily be understood qualitatively. For instance, in the absence of any interactions the polarization of the dipoles in state A would decay as exp ÿ2t=sA . Since the interactions introduce an extra path for the decrease of this polarization, whose eciency is proportional to zA pAB , we can understand why s1 sA is always greater than 2 and increases as fA pAB increases. With regard to the coecients, the maximum of MB t=M 0 for the response function is expected to increase as the rate of transfer of dipoles from state A to state B increases, and this explains why kA b2 increases as fA pAB increases. The other results of our calculations can readily be understood qualitatively in a similar manner.
Table 1 Typical results of the mean ®eld calculations fA
pAB
s1 sA
s2 sA
C1
kA c1
kA b2
0.5 0.4 0.5 0.6 0.5 0.5
0.1 0.2 0.2 0.2 0.3 0.4
2.34 2.55 2.77 3.10 3.24 3.76
0.256 0.251 0.434 0.696 0.555 0.638
0.36 0.24 0.24 0.21 0.17 0.12
0.84 0.76 0.67 0.54 0.54 0.44
0.14 0.21 0.26 0.30 0.33 0.38
The systems have Z 6 and fraction fA of the sites occupied by dipoles in state A, with probability pAB for transition from A to B. The quantities listed in the other columns are the exponents and coecients de®ned in and after Eq. (10), with C1 being the coecient c1 for the relaxation function and kA c1 and kA b2 coecients for the response function, while kA is de®ned by Eq. (14). The rows of the table are arranged in increasing order of the product zA pAB .
158
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
4. Discussion It is easier to understand what processes are involved in our systems from an examination of the graphs of the total and partial polarizations than from Table 1, and so we present a typical example of these graphs, for the relaxation function in Fig. 1 and for the response function in Fig. 2. From Fig. 1, we see that for the relaxation function, the polarization MB of the dipoles in state B decreases much more slowly than that, MA , of those in state A, as one would expect since the change in MB occurs only as a result of the transfer of dipoles between the two states. It is dicult to obtain much more information from this ®gure, so we now turn to Fig. 2, for the response function, which immediately clari®es the processes involved. Here, at time t 0 only the dipoles in state A are polarized, so that initially MB increases, while MA decreases rapidly as a result both of the spontaneous depolarization and of the transfer of dipoles to state B. As time proceeds, the rate of transfer of polarization from A to B decreases as MA decreases, while the rate of transfer of polarization from B to A increases as MB increases, and so MB reaches a maximum and then starts to decrease. From this stage onwards, the polarization and its rate of decrease is determined primar-
Fig. 1. The total polarization M (solid line) and the partial polarizations MA (dotted line) and MB (dashed line) of the dipoles in states A and B for M(0) 1 for the relaxation on removal of a steady ®eld of a system with Z 6, fA 0.5 and pAB 0.2, as a function of the time t.
Fig. 2. The total polarization M (solid line) and the partial polarizations MA (dotted line) and MB (dashed line) of the dipoles in states A and B for M(0) 1 for the response to a pulse excitation of a system with Z 6, fA 0.5 and pAB 0.2, as a function of the time t.
ily by MB , rather than by MA as in the initial stages. The above results suggest a very general mechanism for non-exponential relaxation, which will occur in any system for which some of the dipoles are (relatively) free to make transitions and others are not. A pulse ®eld will only polarize the free dipoles, and these can either become depolarized or be trapped, much as proposed by Ngai [5]. These trapped dipoles can only depolarize by being transferred back to state A, and this mechanism can also be the dominant one even if the trapped dipoles are not completely trapped but can slowly depolarize in state B. Similarly, in systems where dipoles require the presence of a defect in order to make a transition, only those dipoles where the defect is present are free, and the migration of the defect corresponds to the transition of dipoles between the free and trapped states. In all cases, the initial decay of the response function is associated with the decay of the polarization of the free dipoles, while the ®nal stage of the decay is associated mainly with the depolarization of dipoles that were initially free, became trapped, and are subsequently released. These two stages in the response function will lead, of course, to a non-exponential decay of both the response function and of the relaxation function. In the general case,
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
these will not be just the sum of two exponentially decaying functions as for our simple model system. 5. Conclusions From our results, we can derive two important conclusions about the non-exponential relaxation of systems of dipoles. Firstly, in order to understand the physics of the relaxation process, it is important to consider the system's response function rather than its relaxation function. Secondly, a quite general mechanism for the decay of the response to a pulse ®eld is that initially some of the dipoles that were free at time t 0 depolarize while others become trapped, while in the later
159
stages the decay is associated with the freeing of the trapped dipoles and their subsequent depolarization. References [1] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics, London, 1983. [2] J.R. MacDonald, J. Appl. Phys. 62 (1987) R51. [3] S.H. Glarum, J. Chem. Phys. 33 (1960) 639. [4] M.F. Shlesinger, Ann. Rev. Phys. Chem. 39 (1988) 269. [5] K.L. Ngai, Comment Solid State Phys. 9 (1979) 127. [6] V. Halpern, J. Phys.: Condens. Matter 6 (1994) 9451. [7] M.H. Cohen, D. Turnbull, J. Chem. Phys. 31 (1959) 1164. [8] G. Williams, Chem. Rev. 72 (1972) 55.
The relaxation of a model system of interacting dipoles V. Halpern
*
Department of Physics, J. & P. Resnick Institute of Advanced Technology, Bar-Ilan University, 52900 Ramat-Gan, Israel
Abstract We consider a simple model system of interacting dipoles which can be either free to rotate or trapped, and in which the rotation of a dipole can change the state of neighboring dipoles from trapped to free or vice versa, with probabilities that ensure that the fraction of dipoles that are free remains constant, as is required for systems in dynamic equilibrium. The system was studied in a mean ®eld approximation and also by computer simulations. The results show that the non-exponential decay of the system's response function, and hence of its relaxation function, is associated with the fact that initially the decay is due mainly to the depolarization of the free dipoles, while the later stages are associated with the freeing of the polarized dipoles that were trapped and their subsequent depolarization. This process could be a quite general mechanism for non-exponential dielectric relaxation. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The frequently observed non-exponential decay of the dielectric polarization following the removal of an applied ®eld [1], especially in disordered systems, has given rise to a large number of theories that attempt to explain this phenomenon. One class of theories [2] attributes it to the superposition of numerous independent processes such as the relaxation of individual dipoles each of which does relax exponentially, but with a distribution of relaxation times as a result of the disorder. Another class of theories attributes it to time-dependent transition rates (TDTRs) for the dipoles in the system which can arise, for instance, from the migration of a defect that is required to trigger the relaxation of a dipole [3,4] or from interactions between the dipoles [5]. One problem with these TDTRs, which we have considered elsewhere [6],
* Tel.: +972-3 5318433; fax: +972-3 5353298; e-mail: [email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 5 0 3 - 1
is the choice of the origin of time, or equivalently of the state of the system at time t 0. In this paper, we examine a model system of interacting dipoles for which the initial state of the system is very clearly de®ned. The results of our calculations suggest a very simple and quite general mechanism that will lead to non-exponential decay. 2. The model system 2.1. Description of the system We consider a set of interacting dipoles, which for convenience we assume are located on the sites of a simple cubic lattice. At least for systems that are not ferroelectric, a small applied electric ®eld will aect mainly the orientation of these dipoles, and can have only a minor eect on their transtion rates between dierent states. Accordingly, we make a clear distinction between the orientations of the dipoles and their environments and associated transition rates. In this paper, we consider the
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
simplest possible system of this type, namely one in which the dipoles can have just two possible orientations, with moments m, the transitions between which will be called a ¯ip, and where they can be in just one of two possible environments or states, A or B, where only the dipoles in state A are free to ¯ip, while those in state B are immobile. The novel feature of our model system is in the choice of the following interaction between the dipoles. When a dipole in state A ¯ips, it does not change its own state but has a certain probability of changing the state of each of its neighboring dipoles from A to B or vice versa. A possible scenario for such an interaction, not to be taken too literally, is that when a dipole in state A ¯ips from a cis to a trans orientation, which we will call from right to left, it can free space for its righthand neighbor to ¯ip, and so transfer it from state B to state A, and can block any ¯ips of its lefthand neighbor, and thus transfer it from state A to state B. However, the dipole that has just ¯ipped can still ¯ip back from the trans to the cis orientation, and so is still in state A. This process can also be regarded, in somewhat more general terms, as the migration of local free volume [7] when a dipole ¯ips. Our choice of the above form of interaction was not motivated by an attempt to describe real systems but rather by the important property that there is a clear distinction for it between the relaxation function F(t) and the response function f(t), which we will now discuss. As is well known, [1], these two functions are related by f t ÿdF =dt:
1
The response function describes the response of a system to a pulse excitation. While such an excitation is, of course, a theoretical idealization, its meaning for our model system is that the ®eld is applied for long enough for some of the dipoles in state A to be polarized (i.e. for more to ¯ip from the orientation with moment )m to that with moment +m than vice versa) but not long enough for an appreciable number of these polarized dipoles to be transferred to state B. Thus on the removal of the ®eld the system starts to relax from a state in which only some of the dipoles in state A are polarized. For the relaxation function, on the
155
other hand, the ®eld has been applied for a long time, so that the exchange of polarized dipoles between states A and B has reached equilibrium. The reason that it is important to distinguish between these two functions is that, as we have pointed out elsewhere [6], in the case of non-exponential relaxation the choice of the origin of time is of critical importance. For instance, only if b 1 is b exp ÿt s proportional to exp ÿtb , so that only in this case is it unimportant from what instant one measures the time. Otherwise, the basic non-exponential relaxation is that which starts following a pulse excitation, i.e. the response function, and the relaxation function follows from the sum of the relaxations to a continuous set of these pulses applied at dierent times. Now that we have described the system qualitatively, we can de®ne it formally. Let 1=sA be the transition probability for a ¯ip of a dipole in state A in the absence of an applied ®eld. When an electric ®eld E is applied, it lowers the energy of a dipole parallel to the ®eld and raises that of a dipole anti-parallel to the ®eld by mE. As a result, the transition probabilities 1=s A for transitions out of these orientations (for thermally activated transitions at temperature T) will be multiplied by a Boltzmann factor exp mE=kB T , where kB is the Boltzmann constant and the upper sign is for dipoles parallel to the ®eld. Thus in the small ®eld approximation, when mE kB T , these transition probabilities are given by 1=s A 1=sA 1 cE;
2
where c m=kB T . In addition, we denote by pAB be the probability that such a ¯ip will change the state of an adjacent dipole from A to B, and by pBA the probability that the ¯ip will change it from B to A. 2.2. Analysis of the system Since each dipole can be in either orientation and in either state, the exact analysis of this model system for a set of N dipoles requires the solution of 4N simultaneous linear rate equations for the probability of the system being in each of its possible 4N states. While an exact solution of these equations is not really practicable for N > 3, the
156
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
problem can be solved analytically in the following mean ®eld approximation. We assume that each dipole has exactly the same number Z of dipoles adjacent to it, and that of these zA ZfA are in state A and so are free to ¯ip while zB ZfB are frozen in state B. Let n A and nB denote the number of dipoles with moments m in states A and B, respectively; for a system in dynamic equilibrium ÿ with respect to the states, their sums nA n A nA ÿ and nB nB nB will be constant. The quantities of interest are the net dipole moment, or polarization, of the dipoles in states A and B, MA ÿ ÿ m n A ÿ nA and MB m nB ÿ nB , respectively, and the net polarization M MA MB
3
of the system. The rate equations for n A and nB are, to ®rst order in the applied ®eld E, dn A =dt ÿ nA =sA nA =sA zA =sA pBA n B ÿ pAB nA ;
4
and dn B =dt
moval of the applied ®eld can be expressed in the form M t c1 exp ÿs1 t c2 exp ÿs2 t;
where the exponents s1 and s2 are the roots of the equation s2 ÿ 2 pAB pBA =sA s 2zA pAB =s2A 0
ÿ
pAB n A :
5
It follows from Eq. (5) that dnB =dt ÿ zA =sA pBA nB ÿ pAB nA
6
and so, since in the steady state nB must be constant, that nB =nA pAB =pBA :
7
It follows from Eqs. (2) and (4) after simple calculations that dMA =dt ÿ 2MA =sA zA =sA pBA MB ÿ pAB MA c0 E;
8
0
where c 2cmnA =sA , and from Eq. (5) that dMB =dt ÿ zA =sA pBA MB ÿ pAB MA : dn B =dt
9
and so for dMB =dt do The equations for not involve the ®eld E since in state B the dipoles cannot change their orientation. Eqs. (8) and (9) are a pair of coupled linear dierential equations with constant coecients, which can readily be solved by use of the Laplace transform and taking the inverse transform of the solution. One ®nds that the polarization M t at time t after the re-
11
and the coecients cj are readily found as the solution of a pair of simultaneous equations, while for MA t and MB t we replace the coecients cj by aj and bj , respectively. In particular, the coecients aj and bj , and hence cj , depend linearly on the polarizations MA 0 and MB 0 at the time t 0 when the applied ®eld was removed, and so are proportional to the applied ®eld E that produced these polarizations. As explained above, for the response function MB 0 0, while for the relaxation function the initial state is the steady state in the presence of the ®eld, so that, in view of Eqs. (7) and (9), MA 0=MB 0 pBA =pAB nA =nB :
ÿ zA =sA pBA n B
10
12
We also performed computer simulations for a number of systems with N 253 , and the results of these were found to be very similar to those of the mean ®eld approximation. While the system could have been analyzed in terms of correlation functions [8] rather than in terms of the polarizations, such an analysis would not be any simpler, and would be far less informative. In order to calculate the correlation functions, one has to know the initial state of the system, and as we have pointed out elsewhere [6] these are different for the response function and for the relaxation function. The appropriate initial conditions can only be found by use of the rate equations with a term involving the ®eld, which is the only point in our analysis in which we use the applied ®eld E. The subsequent relaxation can be described either by our rate equations without an applied ®eld or by a study of the correlation functions, but the latter can only be de®ned for the total polarization and not for the partial polarizations of the dipoles in states A and B, while the results presented and discussed below show that important physical insight can be gained from studying these partial polarizations.
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
3. Results of the calculations While in experiments one can, of course, only measure the total polarization, in our simulations and calculations we can separate the contribution to the polarization of the dipoles in states A from that of the dipoles in state B, a point that is of great value for the analysis of the results that we now present and for our discussion of their significance in the next section. In particular, while Eq. (1) provides a simple relation between the total polarizations given by the relaxation and response functions, this equation does not apply to the partial polarizations of the two sets of dipoles. The ®rst signi®cant point in the results of our analysis is that if one de®nes a partial relaxation function FA t for the dipoles in state A (whose identity is not ®xed, since the dipoles are continually undergoing transitions between states A and B) by MA t FA tMA 0
13
then one ®nds from the formulae for the coecients aj and cj that FA t kA f t;
14
i.e. that, apart from the normalization, the response function is just the relaxation function of the dipoles in state A. This means that the initial polarization of the dipoles in state B, which is present in the relaxation function of the system but not in the response function, does not aect the decay of the polarization of the dipoles in state A. Such a result is not entirely obvious, but occurs because
157
the dipoles do not change their orientation when becoming trapped or untrapped while in the mean ®eld approximation the transitions between the two states occur at exactly the same rate in each direction. The other features of our results can be deduced from Table 1, in which we present, for some typical values of the parameters fA and pAB , the values of the exponents sj and of some of the coecients. For the response function we present the coecients of kA f t rather than those of f t since kA f 0 1, so that M t=M 0 kA f t, and since MB 0 0 we can write MB t=M 0 kA b2 exp ÿs2 t ÿ exp ÿs1 t: 15 The other coecients can readily be deduced from those listed in the table by means of Eqs. (3), (10)± (15). Most of the results presented in this table can readily be understood qualitatively. For instance, in the absence of any interactions the polarization of the dipoles in state A would decay as exp ÿ2t=sA . Since the interactions introduce an extra path for the decrease of this polarization, whose eciency is proportional to zA pAB , we can understand why s1 sA is always greater than 2 and increases as fA pAB increases. With regard to the coecients, the maximum of MB t=M 0 for the response function is expected to increase as the rate of transfer of dipoles from state A to state B increases, and this explains why kA b2 increases as fA pAB increases. The other results of our calculations can readily be understood qualitatively in a similar manner.
Table 1 Typical results of the mean ®eld calculations fA
pAB
s1 sA
s2 sA
C1
kA c1
kA b2
0.5 0.4 0.5 0.6 0.5 0.5
0.1 0.2 0.2 0.2 0.3 0.4
2.34 2.55 2.77 3.10 3.24 3.76
0.256 0.251 0.434 0.696 0.555 0.638
0.36 0.24 0.24 0.21 0.17 0.12
0.84 0.76 0.67 0.54 0.54 0.44
0.14 0.21 0.26 0.30 0.33 0.38
The systems have Z 6 and fraction fA of the sites occupied by dipoles in state A, with probability pAB for transition from A to B. The quantities listed in the other columns are the exponents and coecients de®ned in and after Eq. (10), with C1 being the coecient c1 for the relaxation function and kA c1 and kA b2 coecients for the response function, while kA is de®ned by Eq. (14). The rows of the table are arranged in increasing order of the product zA pAB .
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V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
4. Discussion It is easier to understand what processes are involved in our systems from an examination of the graphs of the total and partial polarizations than from Table 1, and so we present a typical example of these graphs, for the relaxation function in Fig. 1 and for the response function in Fig. 2. From Fig. 1, we see that for the relaxation function, the polarization MB of the dipoles in state B decreases much more slowly than that, MA , of those in state A, as one would expect since the change in MB occurs only as a result of the transfer of dipoles between the two states. It is dicult to obtain much more information from this ®gure, so we now turn to Fig. 2, for the response function, which immediately clari®es the processes involved. Here, at time t 0 only the dipoles in state A are polarized, so that initially MB increases, while MA decreases rapidly as a result both of the spontaneous depolarization and of the transfer of dipoles to state B. As time proceeds, the rate of transfer of polarization from A to B decreases as MA decreases, while the rate of transfer of polarization from B to A increases as MB increases, and so MB reaches a maximum and then starts to decrease. From this stage onwards, the polarization and its rate of decrease is determined primar-
Fig. 1. The total polarization M (solid line) and the partial polarizations MA (dotted line) and MB (dashed line) of the dipoles in states A and B for M(0) 1 for the relaxation on removal of a steady ®eld of a system with Z 6, fA 0.5 and pAB 0.2, as a function of the time t.
Fig. 2. The total polarization M (solid line) and the partial polarizations MA (dotted line) and MB (dashed line) of the dipoles in states A and B for M(0) 1 for the response to a pulse excitation of a system with Z 6, fA 0.5 and pAB 0.2, as a function of the time t.
ily by MB , rather than by MA as in the initial stages. The above results suggest a very general mechanism for non-exponential relaxation, which will occur in any system for which some of the dipoles are (relatively) free to make transitions and others are not. A pulse ®eld will only polarize the free dipoles, and these can either become depolarized or be trapped, much as proposed by Ngai [5]. These trapped dipoles can only depolarize by being transferred back to state A, and this mechanism can also be the dominant one even if the trapped dipoles are not completely trapped but can slowly depolarize in state B. Similarly, in systems where dipoles require the presence of a defect in order to make a transition, only those dipoles where the defect is present are free, and the migration of the defect corresponds to the transition of dipoles between the free and trapped states. In all cases, the initial decay of the response function is associated with the decay of the polarization of the free dipoles, while the ®nal stage of the decay is associated mainly with the depolarization of dipoles that were initially free, became trapped, and are subsequently released. These two stages in the response function will lead, of course, to a non-exponential decay of both the response function and of the relaxation function. In the general case,
V. Halpern / Journal of Non-Crystalline Solids 235±237 (1998) 154±159
these will not be just the sum of two exponentially decaying functions as for our simple model system. 5. Conclusions From our results, we can derive two important conclusions about the non-exponential relaxation of systems of dipoles. Firstly, in order to understand the physics of the relaxation process, it is important to consider the system's response function rather than its relaxation function. Secondly, a quite general mechanism for the decay of the response to a pulse ®eld is that initially some of the dipoles that were free at time t 0 depolarize while others become trapped, while in the later
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stages the decay is associated with the freeing of the trapped dipoles and their subsequent depolarization. References [1] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics, London, 1983. [2] J.R. MacDonald, J. Appl. Phys. 62 (1987) R51. [3] S.H. Glarum, J. Chem. Phys. 33 (1960) 639. [4] M.F. Shlesinger, Ann. Rev. Phys. Chem. 39 (1988) 269. [5] K.L. Ngai, Comment Solid State Phys. 9 (1979) 127. [6] V. Halpern, J. Phys.: Condens. Matter 6 (1994) 9451. [7] M.H. Cohen, D. Turnbull, J. Chem. Phys. 31 (1959) 1164. [8] G. Williams, Chem. Rev. 72 (1972) 55.