Molecular motions near the glass transition in diethylene glycol bis (ally1 carbonate) as studied by dielectric relaxation spectroscopy Ian K. Smith," Stuart R. Andrews,' Graham Williams"" and Paul A. Holmesb 'Department of Chemistry, University College of Swansea, Singleton Park, Swansea, UK, SA2 8PP bPilkington Technology Management Limited, Hall Lane, Lathom, Orrnskirk, Lancashire, UK L40 5 UF The monomer diethylene glycol bis(ally1 carbonate), used commercially to produce CR39 resin for optical lenses and safety apparatus, has been studied by dielectric relaxation spectroscopy in order to characterise fully the component dipolar relaxations. Various theoretical functions have been used to fit the dielectric relaxation spectra obtained above the glass transition temperature.The principal relaxation (a-process) which is associated with the main glass transition of the monomer arises from the co-operative motions of dipoles. It was found to behave in a non-Arrhenius manner, and indicates that at -95 "C and below the monomer behaves as a glass, at higher temperatures up to -60 "C it is a viscoelastic solid, and at temperatures above -60 "C the sample is a supercooled liquid. The origins of the dielectric a-relaxation process are discussed in terms of recent approaches including mode-mode coupling theory, models of dynamic heterogeneity and MD computer simulations. Cured polymers of diethylene glycol bis(ally1 carbonate), or CR39, are produced by a radical-initiated thermopolymeris- ation process.The polymer is important since it possesses a high optical clarity and impact resistance, making it useful for applications such as prescription optical lenses and for safety equipment. Many studies have been undertaken to investigate the extent of cure and the mechanical characteristics of the fully cured materials. Experimental methods used include titration of double bonds,' dilatometry,2 mid-range IR spec- tro~copy,~ Raman spec- solution-precipitation of p~lymer,~ tro~copy,~fracture testing6 and, recently, dielectric relaxation spectroscopy (DRS).',' Although these studies have been thor- ough in facilitating an understanding of the properties in the cured polymer system, only a DRS study' has given a basic insight into the dipole relaxation processes that occur in the uncured monomer, and how these might change with cure as the polymer material is formed.Motions of dipolar groups in these systems give rise to multiple dielectric relaxations which may be studied over wide ranges of frequency and temperature. In order to understand more fully these relaxation processes in the CR39 polymer, it is the aim of the present investigation to characterise fully the relaxation processes which occur within the monomer itself using DRS over a wide frequency and temperature domain. These processes can be understood in terms of relaxation phenomena in the glass transition (T,) range which arise from the micro-Brownian motions of the molecules and local motions of the dipoles. The structure of the CR39 monomer is shown in Fig.1. It is expected that the principal peak in the dielectric loss spectrum will be observed for the a-relaxation process (micro-Brownian motions of dipoles)' that gives rise to the glass transition of the material. The ester and ether groups in the molecule are dielectrically 0 'CH2-CH2-0-C-O-CH2-CH-CH2II active, yielding the r-process and, in addition, a b-relaxation process' due to the sub-% segmental motions of the carbonyl units. The dielectric data, presented as complex permittivity E*(w), can then be represented in terms of theoretical functions, from which comparison with other glass-forming liquids can be made in terms of the theoretically determined fit parameters associated with these functions.The spectral line-shape of permittivity E'(o) and loss factor E"(w)are determined over a wide range of frequency f(= co/27c) and temperature leading to the frequency-temperature locations of the a-and P-processes. Experimenta1 A sample of the pure monomer was provided by Akzo Chemicals, and was stored, when not in use, in a refrigerator below -10°C. At these temperatures the sample is a super- cooled liquid. Dielectric measurements were performed using a Solartron SI 1260 Schlumberger Impedance/Gain-phase Analyser with a Chelsea Dielectric Interface, enabling the ac frequency range 10-3-104 Hz to be covered with a high degree of precision. A full schematic representation of the measuring system is given in Fig. 2. Temperature control of the sample was achieved using a Novocontrol Quatro temperature con- troller unit that utilises a liquid-nitrogen cryostat system.This enabled heating and cooling of the sample using N, gas in the range -150 to 400°C with a precision of 0.1 C. The dielectric instrumentation and Quatro temperature controller system were controlled by a central computer that used the -I sample holder Fig. 1 Structure of diethylene glycol bis(ally1 carbonate) (CR39) monomer Fig. 2 General overview of the dielectric apparatus used J. Muter. Chem., 1996, 6(4), 539-546 539 Novocontrol ‘WinDETA’ software package WinDETA is a Windows-based software package that enables the user to set up and control a complete dielectric measurement automati- cally or interactively The sample holder consisted of two brass discs, the first was a cup-like disc (a flat circular disc of exterior diameter 40 mm, with a small raised edge to prevent sample spillage, and an inside diameter of 36 mm), which acted as the bottom sample electrode The second is a flat circular disc of diameter 30 mm which acted as the top sample electrode The top disc sat in the cup of the bottom disc, the two being separated from one another by two thin (120 pm) Teflon strips This electrode sample holder was then centred in the Novocontrol BDS1200 sample cell, and electrical contact was made to the brass discs via two circular brass electrodes forming part of the BDS1200 holder The BDS1200 discs consisted of a height-adjustable top disc of diameter 30 mm and a fixed bottom disc of diameter 40 mm diameter containing a small thermocouple that acted as the temperature sensor for the sample Tight adjustment of the BDS1200 top disc ensured good electrical contact between corresponding discs and sample The WinDETA software instructed the Solartron SI 1260 to measure the impedance of the parallel plate capacitor formed by the sample From this, the capacitance C and resistance R of the sample cell were obtained directly In order to calculate the dielectric constant E‘ and relative loss factor E” for a sample, the software used eqn (la) and (lb) E‘ =(cf-C,)/C0 + 1 (14 E” =G/coCo ( 1b) where Cf is the capacitance of the sample-filled capacitor, C, is the capacitance of the empty cell, co is the angular frequency (27cf)and C, is the active capacitance given by eqn (2) Co=AEO/d (2) where d is the perpendicular distance between the parallel electrodes, E~ is the permittivity of free space and A is the active area available to the sample given by the area of the top electrode less the area of the spacers Using the WinDETA software, frequencies of between 0 01 Hz and 10 kHz were measured for the sample CR39 monomer over the temperature range of -50 to -140°C This was sufficient to characterise fully the relaxation processes observed in the DRS experiment and to allow a good theoreti- cal analysis of the results Theoretical Considerations For a single relaxation time process, the dielectric relaxation time z is defined by eqn (3) 7 = 1Pdrnax (3) wheref,,, is the frequency at which the maximum loss occurs In practice, however, it is found that most systems do not have a single relaxation process and therefore we must define an average relaxation time, (z), given by eqn (4) (7) =1/2nfmax (4) Thus the average dipole mobility at a given temperature is measured by the frequency at which the maximum loss occurs A feature of glass-forming systems is that they exhibit common behaviour in the glass transition, or a-relaxation, range The a-relaxation process has a number of characteristic features The temperature dependence of the average relaxation time for this process is non-Arrhenius, unlike a /?-relaxation process which follows an Arrhenius relation [eqn (5)] (7) =To exp (%) where (z) is the average relaxation time given by eqn (4), zo is an empirically determined constant, Qapp is the apparent Arrhenius activation energy and R is the gas constant The Arrhenius relation has a linear relation of log fmaX vs 1/T whereas the a-relaxation process has an activation energy that increases with decreasing temperature This was first described by the empirical Vogel-Fulcher (VF) relation’ l1 [eqn (6)] wherez;, B and T,are empirically determined constants with typically lying 30-50 K below the experimentally determined apparent glass transition temperature The VF expression was put into a related form by Williams, Landel and Ferry (WLF)12 who recognised that this behaviour was common for a wide variety of materials [eqn (7)] (7) where T and & are the temperature and reference temperature, respectively, and C1 and C2 are empirically determined param- eters, given by B/(T -To)and (T-&), respectively The VF and WLF equations can then be used to calculate the apparent activation energy, Qapp, of the a-relaxation process at each sample temperature using eqn (8) (T>To) (8) This non-Arrhenius relation is curved with Qapp increasing with decreasing temperature and can be compared with the simple Arrhenius relation for broad secondary /?-relaxation processes, where (9) However, molecular glass-forming liquids all give a primary a-relaxation which appears to follow a non-Arrhenius function due to the cooperative nature of the relaxation process The complex permittivity &*(a) -~”(co)of a material =&’(a) can be expressed as eqn (10) using the Fourier-transform relation where 4(t) is a decay function for polarisation that describes the relaxation of a material following the step-removal of an electric field For many materials the approach to equilibrium following a small perturbation is non-exponential or ‘stretched exponential’ and for such materials #(t) can be expressed by the Kohlrausch-Williams-Watts (KWW) fun~tionl~-~’ [eqn (11)l The frequency dependence of the average relaxation time is based on the work by Williams and Watts14 l5 using analytical transformation of eqn (1 1) to the frequency domain according to eqn (10) When ff= 1, eqn (1 1) becomes the Debye equation which assumes a single relaxation time for all molecular species Havriliak and Negami16 have provided an expression to describe the broad asymmetric relaxation curves in the fre- quency domain [eqn (12)] 540 J Mater Chem, 1996, 6(4),539-546 where 2 and p are parameters that relate to the skew and broadness, respectively, of the implied distribution of relaxation times. Similarly to the KWW function [eqn.(ll)] the Havriliak-Negami (HN) function becomes the single relax- ation-time equation when the parameters a and /? are equal to 1. Williams and co-w~rkers~~,~~ have shown that the KWW function, using suitable values of can be used to fit, approxi- mately, the asymmetrical loss peaks in the frequency domain associated with the a-relaxation process of many glass-forming materials.They have obtained p values for the a-relaxation process of several glass-forming polymers with values ranging between 0.38 and 0.56.15,17,1sIt is expected that most glass- forming systems wzuld fall within this range. Further, they found values of /? for the dielectric a-process in viscous molecular liquids” to lie in the range 0.52-0.55. Numerical values of normalised real permittivity and loss factor as a function of oz have been tabulated by Koizumi and Kita.20 Utilising these tables, the KWW function can be used as a fitting function to data enabling values of p to be determined over a range of temperature for a particular system. Following the work by Williams and co-~orkers,~~-~~ many dielectric studies have been made with different polymeric and non-polymeric glass-forming systems, which have been reviewed extensively (see for example refs. 2 1-28 and references therein).These studies showed that the KWW function, although divergent from experimental data in the short time and/or high frequency regions, produced the shape of the asymmetric a-relaxation process successfully over a large part of the relaxation range. Results Fig. 3(a) and (b)show how the values of E’ and E” for the CR39 monomer vary with the frequency and temperature of measure- ment. Fig. 3(u) shows the decrease in E’ that occurs at frequen- cies in the range 10 mHz to 10kHz as the temperature is decreased.Fig. 3(b) shows the loss factor E” arising from both the ionic conductivity and the dielectric relaxation of the material. The ionic conductivity is seen in Fig. 3(b) as a rising loss at high temperatures and low frequencies. Above -50°C the dielectric loss values were dominated by ionic conduction losses. The loss peaks occur at increasingly higher frequencies as the temperature is raised and it is evident from Fig. 3(b) that the peak will occur at microwave frequencies at room temperature. The prominent dipole process in Fig. 3(a) and (b) is the a-process and is due to the large-scale micro-Brownian motions of the molecules. Fig. 3(c) enlarges the low-tempera- ture region in Fig. 3(b) and a small shoulder at the low- temperature (< -95 “C)side of the loss peak is seen to occur.This small, broad process is thought to be due to the localised segmental motions of the monomer dipoles which have been shown to be active in this temperature range for this7 and related materials.25 Fig. 4(u) and (b)show the behaviour of F’ and E”, respectively, as a function of temperature. Both figures show that as the frequency is decreased the region in which dielectric relaxation occurs, characterised by the decrease in E’ from high to low values and the corresponding loss curves in F”, shifts rapidly to lower temperatures. Fig. 4(a) shows that E’ falls in the relaxation range with a sharper decrease occurring at lower measuring frequencies.This is reflected in the corresponding loss curves in Fig.4(b) as a narrowing of the peaks as the measuring frequency is decreased, At 1kHz the loss curve reaches its maximum value at -72.5 “C, which compares favourably with the data of De Meuse7 who reported a loss peak at -78 “C for this frequency for the same monomer containing 3 mass% benzoyl peroxide. Fig. 5(u)and (b)show the behaviour of E’ and E”, respectively, as a function of frequency at temperatures in the range -70 to -90 “C. It can be seen that a decrease in temperature does w Fig. 3 Evolution of (a)the permittivity and (b)the loss for the monomer shown in three dimensions; (c) shows the loss for the sub-T, relaxation not alter appreciably the relaxation strength AE=E~- E, or the peak height of the loss curves.The main effect of a decrease in temperature is seen as a large shift of the dispersion region in E’ and the corresponding loss peak in E” to lower frequencies. These variations are due to the marked increase in (T) as the glass-transition temperature of the CR39 monomer is approached. Fig. 6 shows the plots of logfus. l/Tmax and logf,,, us. 1/T derived from Fig. 4(b) and 5(b),respectively. Here T,,, is the temperature of maximum loss for a measuring frequency f and fmax is the frequency of maximum loss for a sample temperature T. The loci of the two curves for the two representations are the same. The plot is curved, as expected for an a-process, showing that the apparent activation energy increases with decreasing temperature.As a consequence of this behaviour, the curve of Fig. 6 was fitted using the VF equation [eqn. (6)] and is shown as the solid line in Fig. 6. Table 1 gives the values of B, To and log(f,’= l/.r<) used for these fits. The quality of the fit shown in Fig. 6 is remarkable and shows that the VF equation gives an excellent representation of the frequency- temperature location of the a-process in CR39 monomer. We note that these data predict that the dielectric loss peak for this process occurs at 10l0Hz at 298 K, well above our measurement range. Our calculations show, based on the VF J. Muter. Chem., 1996, 6(4), 539-546 541 55r /a\ 50 45 w 40 35 30 -100 -90 -80 -70 -60 -50 -100 -90 -80 -70 -60 -50 TIT Fig.4 Dielectric spectra, measured at every decade of frequency in the range -2 <log( f/Hz) <4 for the monomer, showing (a) the permittivity and (b)the loss us temperature 0,-2 log(f/Hz) (curve on extreme left), A, 4 log(f/Hz) (curve on extreme right) 06 --05 04-b 03-02-01 -no-I.I*I.l.l~"' -2 1 0 1 2 3 4 log (f/Hz) Fig. 5 Evolution of (a)permittivity and (h)loss for the monomer with frequency at temperatures of -70°C (curve on extreme right) to -90 "C (curve on extreme left) at every 2 "C 48 49 50 51 52 53 54 55 lo3 KIT Fig. 6 Activation-energy plots for the monomer, showing non-Arrhenius behaviour for log fus l/Tmax(0)and log f,,, us 1/T(0)data Table 1 Parameters of the Vogel-Fulcher relation fitted to the tempera- ture and frequency domain of the dielectric a-relaxation process in CR39 monomer 147 4 1236 13 09 fit in Fig 6 and eqn (8), that Qappincreases from 140 kJ mol-' at 203 1 K to 265 kJ mol-' at 183 1 K Clearly Qappcannot be interpreted as being a true activation energy barrier for the reorientational motions of the dipoles, since these values approach those for the dissociation of chemical bonds It is more reasonable to relate the variation offmax with temperature to relaxation models involving the configurational entropy of the relaxing system As temperature is reduced towards the configurational entropy decreases markedly and this leads to a marked increase in the structural relaxation time in accord- ance with VF behaviour, as has been described in the relaxation model of Adam and Gibb~~~ and has been further discussed by Wong and Ange1125 and Matsuoka 30 Shape of the Relaxation Curves Fig 7 shows the experimental dielectric loss spectra for the CR39 monomer at four temperatures The data are fitted using the HN function [eqn (12)], and are represented as the unbroken lines in Fig 7 The values of a and p thus determined are given in Table2 It can be seen that the HN function describes the experimental data very well over most of the loss peak (4 decades of frequency) but predicts lower losses at higher frequencies than those observed This is possibly due to an additional higher-frequency dielectric process being 07r 02 01 00 -2 -1 0 1 2 3 4 log (f/Hz) Fig.7 Havriliak-Negami analysis of the loss of CR39 monomer at -80 (O),-82 (H),-84 (0)and -86 "C (0)The fitted HN functions are plotted as unbroken lines 542 J Muter Chem, 1996, 6(4),539-546 Table2 Parameters of the HN and KWW functions fitted to the experimental data for the CR39 monomer Havriliak-Negami fit KWW fit T/"C a P P -76 --0.55 -78 --0.55 -80 0.435 0.894 0.55 -82 0.434 0.895 0.56 -84 0.435 0.895 0.56 -86 0.435 0.895 0.56 -90 --0.56 present at the lower temperatures of measurement, due to the secondary sub-glass transition process described above, associ- ated with the local dipole motions of the monomer.Table 2 shows that over the temperature range studied, the a parameter, which is associated with the skew of the loss peak, and the p parameter, associated with broadness of the loss peak, remain approximately constant, showing that the shape of the relax- ation function is hardly changing with temperature over the range shown in Fig.7. The permittivity and loss curves of Fig. 5(a) and (b) were also fitted using the KWW function [eqn. (1l)]. For a single relaxation time process we may write: A logf= 1.14 (13) where A logf is the full half-width of the loss curve in a plot of E" us. log6 Curves that do not obey the single relaxation time model are broader by comparison and we may write as an approximation for the KWW function the relati~n:'~ Dz 1.20/A log f (14) where pis the KWW parameter introduced above.The curves illustrated in Fig. 5(b) all have A logfz2.0, thus we expect gz0.6. Using this value as an indicator, the tables of Koizumi and Kita2' were used to fit the KWW function to the E' dispersion curves and associated loss curves shown in Fig. 5(a) and (b). Fig. 8 shows representative fits to the loss data in the range -80 to -86 "C. Similar plots were also obtained for all temperatures studied and the results of this analysis are included in Table 2. For each loss curve, at each temperature, the KWW function was found to fit the data very well at the lower frequencies but fell slightly below the E' dispersion and E" loss curves at the higher frequencies. This behaviour has been found to be common to the dielectric a-relaxation of many other small-molecule glass-forming systems as reviewed by Williams.21 Again the difference may be due to a higher- frequency process being present as indicated above.Never- theless the KWW function proves a reasonable fit to the data 0.7r 0.6 0.5 0.4 w 0.3 0.2 0.1 0.0 I.I.1.I.I.I -2 -1 0 1 2 3 4 log (I/Hz) Fig. 8 KWW analysis of the loss of CR39 monomer at -80 (0),-82 (m), -84 (0)and -86 "C (0).The fitted KWW functions are plotted as unbroken lines over most of the loss peak (3 decades of frequency) and we can describe the relaxation of the CR39 monomer as being non-exponential or stretched-exponential following a pertur- bation by an ac field. Discussion It is well known that the dielectric a-relaxation in small- molecule glass-forming liquids is characterised by (i) broad dielectric dispersion and absorption features that conform, approximately, to HN, KWW or Davidson-Cole28 functions and (ii) an average relaxation time that follows, approximately, the Vogel-Fulcher relation [eqn.(6)]. Our comprehensive dielectric data for the CR39 monomer shows that this glass- forming liquid follows this pattern of behaviour and exhibits model behaviour for such a liquid. We have seen that (z) obeys, with good accuracy, the VF equation in the range studied above Tgin which the fictive temperature Tf25.26*31and the actual temperature T of the sample are equal. If measure- ments were performed in and below the glass transition region the non-equilibrium effects would become apparent since T+ Tf and the VF equation would not be obeyed.The temperature T,, which for our data for the CR39 monomer is 145.2K, is interpreted to be the temperature at which the configurational entropy S, of a material in its equilibrium state would approach zero.25 Since T,< this cannot be achieved in practice so T, is anticipated (premonitory behaviour) from our measurements on the system in equilibrium above Tg. The theory of Adam and Gibb~~~ (see also ref. 30) makes the assumption that (z) and S, are related, and this leads to the VF equation under certain circumstance^.^'^^^-^^ However, it should be pointed out that (z) is a transport coeficient, being an integral, over time, of a time-correlation function CJt) for the reorientational motions of dipole^,^^,^^,'^ while S, is an equilibrium thermo- dynamic quantity that has no obvious connection to the time- scale of dynamic events.While it is plausible to relate (z) to S, it is not evident that a dynamic property and an equilibrium property for a stationary thermodynamic system have to be related in this way. We note that To may be estimated using equilibrium statistical mechanics for a dense system of polymer chains, as was shown originally by Gibbs and diMar~io~~ (see also ref. 35). The spectral line-shape for the dielectric a-process in the CR39 monomer is seen to be fairly well represented by the KWW function over about four decades of frequency (see Fig. S), but lower values than those observed experimentally in the higher frequency range are predicted.The value of p (ca. 0.55) used to obtain these fits is the same as that found for the a-process in the glass forming liquid 0-terphenyl and solutions of dipolar solutes in that solvent,21. 36-39 de spite the different chemical structures of the CR39 monomer and o-terphenyl. Comparison of the fit of the a-relaxation loss curve for anthronelo-terphenyl (Fig. 7 in ref. 39) with the fits shown in Fig. 8 for the present material shows that the excess absorp- tion at high frequencies over that calculated from the KWW function is far higher for the CR39 monomer than it is for the anthronelo-terphenyl solution. This suggests that in the present material, local motions, primarily for the ether groups, occur prior to the main a-process and to a larger spatial extent than local motions of anthrone or o-terphenyl molecules.The HN function, with two adjustable shape parameters, provides a better fit to the data (see Fig. 7). The form of the spectral line shape of the dielectric a-process in small-molecule glass-forming liquids has been considered recently by Wu et aL4' who have proposed an empirical scaling law to reduce loss data, in the frequency domain, to normalised master curves. The applicability of this scaling law4' has been critically e~arnined.~'-~' In parallel, Gotze and co-~orkers~~-~~ have proposed a mode-mode coupling theory for relaxation in glass-forming liquids that embraces the primary (a) and J.Muter. Chem., 1996, 6(4),539-546 543 further (short-time) relaxation processes. In our case an additional absorption peak at short-time/high frequencies to the primary a-process is predicted, which we term a P-process. These processes in the mode-mode coupling theory follow from the assumption that there is a second-order memory function that takes on an assumed form which is a function of the time-correlation function being elucidated within the Such theories take no account of the kind of molecular motion being studied, which for dielectric relaxation is the reorientational motions of dipoles expressed by the time- correlation functi~n~~?~~ [eqn. ( 15)]: C(CLl(O).PJ(t)) C,(t)= (15)l,’ C(P*(O).PJ(O)> l.3 where the sum extends over all dipoles contained in a macro- scopic volume V.For the CR39 monomer, both ester and ether dipoles are involved in the sum and we note that eqn.(15) contains auto- and cross-correlation functions (both dynamic and equilibrium). Thus the memory function approach that is explicit in the mode-mode coupling theories is interesting for application to our data but requires that such memory func- tions actually exist physically for reorientational motions of dipoles in viscous molecular liquids. As we have discussed re~ently’~,~~a weighted sum of parallel processes each having no memory can be expressed in terms of virtual memory functions that have no physical significance. An alternative approach to the a-relaxation in glass-forming liquids and amorphous polymers is that of Schmidt-Rohr and Spie~s’’-’~ who have demonstrated, using multi-nuclear multi- dimensional NMR studies of the a-relaxation in amorphous poly(viny1 acetate) and poly(styrene), that these systems behave as if there were present a ‘dynamic heterogeneity’ of the moving units (chain segments for polymers).Starting at an arbitrary time t=O, an ensemble appears to comprise units finding themselves in different local states. As time progresses, units in fairly free environments reorientate quickly and partially relax the ensemble while at the other extreme units in con- strained environments hardly relax in the time-scale of the reorientation of the ‘free’ units. As a result of the broad range of environments (states) occurring at t =0, a broad relaxation function emerges for the motions of the ensemble, as a weighted sum of essentially parallel independent processes, within the total time-scale for the p-and overall a-relaxations, giving a KWW- or HN-type relaxation for the a-proce~s.’~-~~ Such a dynamic heterogeneity appears to be the source of the broad a-relaxations observed by Schmidt-Rohr and Spiess in their NMR experiments.Further experiments” showed that contin- ual exchanges occur between the local states experienced by a representative group, hence ensuring the system is ergodic, i.e. time-averaged and spatially averaged time-correlation func- tions of the motion are identical,33 and thus units behave on average in an equivalent manner in their dynamic properties.Memory functions do not appear to be involved necessarily in such an overall relaxation process. In parallel with these, Ediger et u1.60-64have used fluorescence methods and angular- dependent photoselection in optical bleaching to monitor the a-process of dyes in o-terphenyl. Their studies suggest strongly that the broad a-relaxation (KWW- or HN-type) arises from dynamic heterogeneity, as described above. Further dielectric experiments by Bohmer et uL6’ involving special pulse sequences and non-linear ‘hole-burning’ with large E fields of the dipole orientation-distribution in glass-forming liquids such as propylene carbonate and glycerol yield results for the dielectric a-relaxation that are consistent with a model of dynamic heterogeneity that is intrinsic to the amorphous state and is general for glass-forming liquids.As a result of these complementary st~dies’~-~~ we consider 544 J. Muter. Chem., 1996, 6(4), 539-546 that the dieletric a-relaxation for the CR39 monomer may be given by a weighted sum of individual reorientational processes each with its own relaxation function which is obtained by averaging over all trajectories in time for a given initial state for a reference dipole. The autocorrelation function for the motion of a reference dipole z is then given by eqn. (16): whereA(E)dE is the probability of obtaining the dipole i in the range of environments (in configuration space) between s” and (s”+ dE).Hence [(pL(O).pl(t))=]is the correlation function for a dipole that is initially in environment E and reorients in time through a range of trajectories. ()= indicates the average taken over all those trajectories starting in configuration Z. In its simplest form the terms [(pl(0).p,(t)),-] in eqn. (16) each have a single exponential decay with relaxation time z(E)so [(pl(0).pt(t))] involves a simple distribution of relaxation times with a distribution function A(..”), i.e. a simple weighted sum of parallel processes where each process has a single exponential decay in time. Such a description is precisely that used from the earliest descriptions of broad relaxation pro- cesses in polymers as reviewed by Ferry66 and McCrum et ~1.~’ However, it is entirely possible that [(pl(0).pL(t))=]is not a single exponential decay function.For example, it may contain a short-time part (p-process) and a long-time part (a-process) and their relative magnitudes, their individual relaxation times and their functional forms may vary with S. This will be the general situation and its internal structure is not easily investi- gated experimentally. A reasonable model for glass-forming liquids is that the fast components in [(pl(0).pl(t))Z]are similar in time-scale for all initial configurations (partial relax- ation of dipoles, P-process) but the slow components that contribute to the overall a-process, vary markedly with E. This results in a fast process (p)and a slow process (a) where the latter is a weighted sum over a broad distribution, f(E), involving individual relaxation times z(E).For the CR39 monomer, the secondary (p)process appears to be far smaller than the a-process. However, as we have indicated above, the ‘excess absorption’ for the a-process at high frequencies over that calculated using the KWW or HN functions makes us suggest limited motions of dipoles (mainly ether dipoles) par- tially relax [(pl(0).pt(t))3]at short times, but do not provide necessarily a distinct /I-process. We note that the loss curves for the CR39 monomer and for related glass-forming liquids2’ having pz0.55 (or HN ~1~0.42,pz0.91) are indicative of dynamic heterogeneity at the mesoscopic level in molecular liquids. It is possible to increase the coarseness of the heterogeneity by forming mix- tures of glass-forming liquids or of amorphous polymers.For example, Shears and Williams37 examined the dielectric a- relaxation in homogeneous (optically transparent) mixtures of (polar) di-n-butylphthalate (DBP) with (non-polar) o-terphenyl and found that the half-width of the a-loss curve increased from 2.0 to 2.5 on going from 0 to 30% DBP and then decreased monotonically to 1.8 for 100% DBP. This was strong evidence for heterogeneity of a dynamic kind, on a wavelength scale shorter than visible light in which o-terphenyl- rich regions relax slowly and DBP-rich regions relax quickly, compared with the average dielectric relaxation time for the mixture. The concentration fluctuations in o-terphenyllDBP mixtures occur on a time-scale shorter than that for dielectric relaxation of the individual molecules, so a weighted sum of relaxations, each of KWW-type weighted over the distribution of local concentrations of DBP molecules, contributes to the overall relaxation of the mixture.Ergodicity is maintained in such a system through the slow evolution of concentration fluctuations throughout the liquid. Thus the line shape for the loss factor for a pure glass-forming liquid (as in Fig. 7) is a limiting one based on fluctuations in local environments (ie dynamic heterogeneity that is intrinsic to such a liquid) While we have rationalised our data for the CR39 monomer in terms of the dynamic heterogeneity model, we note that real-time simulations of molecular dynamics have been made for realistic models for o-ter~henyl~~ and molten bulk poly- ethylene 69 Thus Lewis and Wahn~trom~~ showed that o-terphenyl molecules exhibited ‘libration’, ‘floppy librations’ plus ‘excursions from equilibrium’, ‘Jump motions’ and ‘very floppy diffusion’, on different time-scales during molecular reorientations Roe6869 showed that the correlation functions (P,[cos O(t)]) (where 1= 1,2,3 n, P indicates the Legendre 17 18 19 20 21 22 23 24 G Williams and D C Watts, in Dielectric Properties of Polymers, ed F E Karasz, Plenum, New York, 1971 G Williams, M Cook and P J Hains, J Chem SOC Faraday Trans 2,1972,68,1045 G Williams, J Non-Cryst Solids, 1991,131, 1 N Koizumi and Y Kita, Bull Inst Chem Res Kyoto Univ, 1978, 56,300 G Williams, in Dielectric and Related Molecular Processes, ed M Davies, Specialist Periodical Report, The Chemical Society, London, 1975, vol 1, p 151 G Williams, IEEE Trans Electr Insul, 1985, E1-20, 843 Relaxations in Complex Systems, ed K L Nargai and G B Wright, US Government Printing Office, Washington DC, 1985 K L Ngai, R W Rendall, A K Rajagopal and S Teitler Ann N Y polynomial and O(t) is the orientation of a bond vector at time t) contain a short-time part (local partial motions, 0 01-10 ps) and a longer-time a-process that is described by a KWW function with /3 =0 56 & 0 04 Thus this MD simulation for the molecular dynamics of an amorphous bulk polymer predicts behaviour of the kind observed for the CR39 monomer, which 25 26 27 28 Acad Sci , 1987,484,150 J Wong and C A Angell, Glass Structure by Spectroscopy, Marcel Dekker, New York, 1976 C T Moynihan et al, Ann NYAcad Sci, 1976,279,15 J M Pochan, H W Gibson, M F Froix and D F Hinman, Macromolecules, 1978, 11, 165 N G McCrum, B E Read and G Williams, Anelastic and emphasises the similarity in the a-relaxation for amorphous polymers and glass-forming liquids as discussed by us pre- vlous~y17 19 21 36 39 Stretched exponential behaviour is ubiqui- tous for the a-relaxation in amorphous systems as observed by different techniques 70 71 29 30 31 Dielectric EfSects in Polymer Solids, Dover Publications , New York, 1991 G Adam and J A Gibbs, J Chem Phys, 1965,43,139 S Matsuoka, Relaxation Phenomena in Polymers, Oxford University Press, New York, 1992 I M Hodge, J Non-Cryst Solids 1994,169,211 32 G Williams, Chem Rev 1972,72, 55 Conclusions 33 34 G Williams, Chem SOC Rev 1978,7,89 J H Gibbs and E A Di Marzio, J Chem Phys 1958,28,373 The dielectric a-relaxation in the CR39 monomer has been studied over ranges of frequency and temperature and is shown 35 36 37 G Williams, Trans Faraday SOC 1963,59,1397 G Williams and P J Hains, Chem Phys Lett 1971,10, 585 M F Shears and G Williams, J Chem SOC Faradav Trans 2 to be well represented by the Vogel-Fulcher equation (for average relaxation time) and fairly well represented by the KWW and HN functions (for spectral line-shape) The behav- lour is rationalised in terms of the dynamic heterogeneity model proposed originally by Schmidt-Rohr and Spiess and it appears that the dipole moment correlation function contains internal structure involving equilibrium and dynamic quantities that may only be further elucidated using techniques that give spatial information on the overall a-process The importance of such work in providing an understanding of the physical properties of the CR39 monomer and other glass-forming liquids that have widespread applications in their own right or as precursors to optically transparent glasses is self-evident 38 39 40 41 42 43 44 45 1973,69,608 M F Shears and G Williams, J Chem SOC Faraday Trans 2 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