THE JOURNAL OF CHEMICAL PHYSICS 128, 184905 共2008兲
Rheological complexity in simple chain models Taylor C. Dotson,1 Julieanne V. Heffernan,1 Joanne Budzien,2 Keenan T. Dotson,1 Francisco Avila,1 David T. Limmer,1 Daniel T. McCoy,1 John D. McCoy,1,a兲 and Douglas B. Adolf2 1
Department of Materials and Metallurgical Engineering, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801, USA 2 Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
共Received 4 March 2008; accepted 27 March 2008; published online 13 May 2008兲 Dynamical properties of short freely jointed and freely rotating chains are studied using molecular dynamics simulations. These results are combined with those of previous studies, and the degree of rheological complexity of the two models is assessed. New results are based on an improved analysis procedure of the rotational relaxation of the second Legendre polynomials of the end-to-end vector in terms of the Kohlrausch–Williams–Watts 共KWW兲 function. Increased accuracy permits the variation of the KWW stretching exponent  to be tracked over a wide range of state points. The smoothness of  as a function of packing fraction is a testimony both to the accuracy of the analytical methods and the appropriateness of 共0 − 兲 as a measure of the distance to the ideal glass transition at 0. Relatively direct comparison is made with experiment by viewing  as a function of the KWW relaxation time KWW. The simulation results are found to be typical of small molecular glass formers. Several manifestations of rheological complexity are considered. First, the proportionality of ␣-relaxation times is explored by the comparison of translational to rotational motion 共i.e., the Debye–Stokes–Einstein relation兲, of motion on different length scales 共i.e., the Stokes–Einstein relation兲, and of rotational motion at intermediate times to that at long time. Second, the range of time-temperature superposition master curve behavior is assessed. Third, the variation of  across state points is tracked. Although no particulate model of a liquid is rigorously rheologically simple, we find freely jointed chains closely approximated this idealization, while freely rotating chains display distinctly complex dynamical features. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2912054兴 I. INTRODUCTION
Glasses elude simple descriptions. As temperature T is lowered, the viscosity increases as do other measures of the ␣-relaxation time ␣. If such increases were Arrhenius 关i.e., log共␣兲 ⬃ 1 / T兴, glassy dynamics could be interpreted as a simple activated process. Most glasses, particularly polymeric glasses, are strongly nonArrhenius and, consequently, nonsimple. Relaxation functions, such as stress relaxation functions, are more detailed probes of glassy behavior. If the “shape” of a relaxation function is independent of temperature, the material is said to be rheologically simple. This too is rarely found to be the case. We refer the reader to a number of excellent reviews1–7 of nonsimple behavior in glasses and supercooled liquids. In the current paper, the degree of rheological complexity in bead-spring chain models is investigated. As indicated, two aspects of the dynamical slowing down in supercooled, glassy liquids are of interest here. First is describing the characterization time ␣ as a function of state point 共i.e., of temperature and pressure兲. Complexities arise from the apparent divergence of ␣ at nonzero temperature, from nonequilibrium states at low temperature and the a兲
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associated physical aging of the glass, and from differing behaviors of ␣’s extracted from different measurements. For the simple model systems in the present study, the behavior of ␣ was addressed in previous papers.8–11 The second aspect of interest is the description of the shape of relaxation functions in terms of time and of state point. Complexities arise for many of the same reasons: The functional form may depend on the specific relaxation function studied; nonequilibrium effects come into play at low temperature and at high frequency; and the relaxation time distribution appears to diverge at finite temperature. Because functions, as opposed to individual points, are being considered, varying levels of regularity are imposed on the relaxation functions so that trends can be explored. The most extreme regularization procedure is time-temperature superposition where it is assumed that the relaxation function depends only on the ratio t / ␣, where t is time. Consequently, if ␣ is known as a function of temperature and the relaxation function is known as a function of time at a single temperature, then the relaxation function at any other temperature can be easily predicted. Moreover, if, as is often the case, the relaxation function can only be extracted over a narrow time range, a full “master curve” of the relaxation function can be constructed by shifting partial relaxation functions from different temperatures along the time axis. The justification of time-temperature superposition is a
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demonstration of master curve behavior in a log-log plot of relaxation functions from many state points against t / ␣. In many cases, this is quite convincing with the only deviation from master curve behavior being a slight thickening of the composite curve. From a pragmatic perspective, the assumption of time-temperature superposition permits the extrapolation of high temperature behavior to low temperature 共e.g., the analysis of physical aging兲, and even the extrapolation from one relaxation function to another 共e.g., the analysis of mechanical relaxation from dielectric relaxation兲. For the model systems in the present study, the analysis of relaxation functions with a time-temperature superposition spirit was reported in a previous paper.8 These results were incorporated in an analysis of physical aging.12 On the other hand, the above mentioned thickening of the master curve is believed to contain information relating to the underlying physics of the glass transition. In particular, changes in the shape of the relaxation function reflect changes in the underlying distribution of relaxation times of which ␣ is only an average. Because changes in the relaxation function are so slight, the function is often fit to a simple functional form, and the variations in the parameters of the functional form are tracked. In the present study, we have pursued this approach. Building on previous work,8–11 the relaxation behaviors of four simple molecular systems were investigated. These were ten-site bead-spring chains with backbones that were either freely jointed 共FJ兲 or freely rotating 共FR兲 with a bond angle of 120°. The intermolecular interactions were Lennard–Jones potentials clipped either at 2.5 共denoted as “attractive” systems, A兲 or clipped at the minimum 共denoted as “repulsive” systems, R兲, where is the Lennard–Jones length scale. The four system types were then FJ-A, FJ-R, FR-A, and FR-R. In addition, each system contained a few single-bead “penetrants” having the same intermolecular interactions as the chains. These are standard, coarse-grained molecular models. Model and simulation details are reported in the Appendix as well as in previous papers.8–11 A table of the state points investigated is included in the Appendix. The primary measure of rotational relaxation used8 in the present investigation is the second Legendre polynomial of the chain end-to-end vector, E共t兲 =
3 2
共具关e共t兲 · e共0兲兴2典 − 31 兲 ,
共1.1兲
where the angle brackets denote ensemble average and e共t兲 is the unit vector along the end-to-end vector at a time t. In the intermediate time regime, this function is well described by the Kohlrausch 关or Kohlrausch–Williams–Watts 共KWW兲兴 function,13 
EKWW共t兲 = Ze−关共t/KWW兲 兴 ,
共1.2兲
where KWW is a relaxation time,  is the “stretching exponent,” Z is a constant approximately equal to 1, and the value of  is between 0 and 1. In order for time-temperature superposition to strictly hold for this relaxation process,  must be independent of state point. Previously,8 we reported that, based on a master-curve analysis, this was indeed the case. The  of the FJ systems was roughly 0.75 and  of the FR systems was 0.68. These values were shown to be in keeping
FIG. 1. 共Color online兲 Analysis of the second Legendre polynomial of the end-to-end vector E共t兲 for ten-site FR-R at = 1.06, T = 1.6, and = 0.551. 共A兲 E共t兲 vs ln共t兲. The lower thick line is really the overlapping points of the raw data which would obscure the curve passing through them. This curve is reproduced shifted to the right where the solid line is the Kohlrausch function; and the dashed, the single-exponential tail. 共B兲 The Lindsey–Patterson plot. The points are the data. The dashed line is the single exponential tail; and the solid line has a slope of  = 0.656. 共C兲 The modified Lindsey– Patterson plot. The points are the data. The dashed line shows a slope of one; and the solid, a slope of  = 0.663. 共D兲 The Lindsey and Patterson cross-plot. The points are the data. The solid line has a slope of one and an intercept of −ln共兲 with  = 0.657. In all cases, the vertical dashed lines denote the intermediate time regime. Logarithms are base e.
with literature values. We also discussed the difficulty of fitting the Kohlrausch function to simulation results. In the current paper, we revisit the problem of fitting E共t兲 with the Kohlrausch function. A new analysis procedure is developed that permits  to be calculated to greater accuracy, and permits Kohlrausch fits to be performed for rapidly relaxing systems. Viewed at this higher level of accuracy and wider range of state points,  is seen to drift to lower values as the glass transition is approached and, consequently, both FJ and FR systems violate detailed time-temperature superposition. However, over a limited range of state points near the glass transition, the FJ systems approximate timetemperature superposition. The remainder of the paper is organized as follows. In Sec. II, the new analysis method and related background materials are reported. Results are presented in Sec. III and are discussed in Sec. IV. In the Appendix, details of the molecular models and simulations are reported. II. BACKGROUND AND METHODS
The fitting of the Kohlrausch function to experimental or simulation data is a delicate procedure. The function itself contains three parameters and, given the relatively featureless nature of the relaxation function 关as seen in Fig. 1共a兲兴, it is difficult to determine the parameter values to a level of precision permitting comparisons across state points. Complicating factors are the small changes, which, if any, are
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expected in , and the difficulty in identifying the intermediate time regime. The curve fit of the data in Fig. 1共a兲 is reproduced slightly to the right to illustrate the switch from Kohlrausch 共solid line兲 in the intermediate time regime to a single-exponential function 共dashed line兲 at longer time. A graphical procedure suggested by Lindsey and Patterson14 共LP兲 is useful in the analysis. The logarithm of the negative logarithm of E共t兲 is plotted against the logarithm of t. If E共t兲 were identically of the Kohlrausch form, the procedure would result in
冋
ln关− ln关EKWW共t兲兴兴 =  ln共t/KWW兲 + ln 1 −
册
ln共Z兲 , 共t/KWW兲 共2.1兲
where for large time, ln关−ln关E共t兲兴兴 versus ln共t兲 becomes linear with a slope of . In Fig. 1共b兲, a typical LP plot is shown with a clear linear regime. The Kohlrausch fit is shown as a solid line 共offset for clarity兲 with  = 0.656 and KWW = 7292. For moderate accuracy in , this is a robust procedure. At small time, the plot is nonlinear both because of non-KWW decay processes and because Z is not necessarily equal to 1. At large time, E共t兲 becomes “single exponential” 共the offset dashed line兲. It is only in the “intermediate” time regime that the slope of the curve in Fig. 1共b兲 can be identified with  共solid line兲. Identifying this regime is the primary challenge to achieving accuracy in  to three significant figures 共the region eventually decided upon is indicated by vertical dashed lines for all panes in Fig. 1兲. Previously,8 we suggested a modification to the LP procedure. By graphing the logarithm of the derivative of the negative logarithm of E共t兲 against the logarithm of t, the parameter Z is eliminated. In particular,
冋
ln −
册
d ln共EKWW共t兲兲 =  ln共t/KWW兲 + ln共兲. d ln共t兲
共2.2兲
An example of this is shown in Fig. 1共c兲. with  = 0.663 and KWW = 7302. Although the value of Z is no longer a consideration, the short and long time deviations from the Kohlrausch form are still difficulties and, in addition, the numerical derivative introduces noise not present in the LP graph. Consequently, although the modified-LP is more reliable than the LP procedure, it is still only accurate to two significant figures. In the current paper, we demonstrate that the intermediate time regime can be identified by cross-plotting ln关−ln关E共t兲兴兴 against ln关−共d ln关E共t兲兴 / d ln共t兲兲兴. For the Kohlrausch function, one finds that
冋
ln关− ln关EKWW共t兲兴兴 = ln −
冋
册 册
d ln共EKWW共t兲兲 − ln共兲 d ln共t兲
+ ln 1 −
ln共Z兲 . 共t/KWW兲
共2.3兲
In the intermediate time regime and for sufficiently large t that the last term is negligible, this results in a line with unity slope and an intercept of −ln共兲. An example of this is shown in Fig. 1共d兲, where the Kohlrausch fit has a  = 0.657.
FIG. 2. 共Color online兲 Examples of the determination of the KWW stretching exponent  for a number of packing fractions for FR chains. 关The 共T , , , 兲 are from top to bottom 共1.6, 0.612, 0.318, 0.758兲-R; 共2.2, 0.944, 0.471, 0.681兲-R; 共2.0, 1.06, 0.536, 0.660兲-R; 共1.6, 1.06, 0.551, 0.656兲-R; 共1.6,1.06, 0.563, 0.650兲-A兴. In 共A兲 are shifted Lindsey and Patterson crossplots with lines of a slope of 1. In 共B兲 are unshifted, modified Lindsey– Patterson plots with lines of slope .
The fitting procedure employed in the remainder of the paper is as follows. First, the LP cross-plot is performed and  and the intermediate time regime is identified, as illustrated in Fig. 1共d兲. The LP and modified-LP plots are then restricted to the intermediate time regime identified in the cross-plot. From both of these graphs, values of  and KWW are extracted, as illustrated in Figs. 1共b兲 and 1共c兲. Finally, Z can be identified from the plot of E共t兲 versus ln共t兲, as in Fig. 1共a兲. The three values of  and two values of KWW permit the estimate of errors. For the case illustrated in Fig. 1, these are  = 0.659⫾ 0.004 and KWW = 7298⫾ 6. In brief, it is the identification of the intermediate time regime through the LP cross-plot that increases the accuracy of the  determination. Figure 2 shows examples from several state points. For a number of reasons, it would be convenient if  did not change with thermodynamic state. First, a constant  is necessary in order for detailed time-temperature superposition to be applicable. In particular, time-temperature superposition is used to extend the frequency range of a few decades available in mechanical spectroscopy measurements to as much as 18 decades.15 Second, mode coupling theory16,17 predicts a constant , providing a theoretical justification of a simple behavior of this otherwise difficult to fit parameter. Finally, for reasons discussed elsewhere,8,18 the inherently short relaxation times studied in computer simulations cause  to be difficult to fit and, consequently, to be convenient to treat as a constant. The ability of a constant-, Kohlrausch function to describe atomic relaxation functions over a restricted tempera-
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ture range has been noted in a number of instances. Previously,8 we demonstrated that for FJ-A chains, a  = 0.75⫾ 0.02, along with only a moderately accurate identification of the Kohlrausch region, permitted a good description of rotational relaxation functions with a constant . Similar results were found for simple atomic models by Bordat et al.19 Model II in that study was a Kob–Andersen20 mixture of Lennard–Jones 6–12 atoms. The self-intermediate scattering function was found to be described by a single  = 0.81⫾ 0.1 across a wide range of temperatures; however, it was “only approximately possible to extract a temperature independent stretched exponent  using a Kohlrausch– Williams–Watt 共KWW兲 fit of the master curve in a restricted temperature range.” When they reanalyzed the data with a temperature dependent , this parameter was found to decrease with decreasing temperature from 1 at high temperature to 0.65 at low temperatures. 共See Figs. 4 and 5 of Ref. 19.兲 Other simulations indicating a changing  are not uncommon,18共d兲,21–23 albeit, often with large error bars. Similar conclusions are also reached by Vallee et al.24 in a simulation study of diatomic “probe” molecules in a matrix of ten-site FJ chains. Master curve behavior is observed for some relaxation functions and not for others, although numerical noise is clearly a problem 共see Fig. 6 of Ref. 24兲. The onset of nonmaster curve behavior appears to be associated with the development of orientational “hopping” motion. Experimental measurements also encounter this problem. Consider, for example, the work of Shi et al.25 The stress relaxation of glycerol and two other small molecule glass formers were measured over a range of temperatures. In all cases, a “time-temperature master curve can be constructed even though the ’s for the individual response curves at each temperature vary systematically.” Figure 3 of Ref. 25 illustrates this dualism in the case of glycerol. It is observed that “the KWW function gives a reasonable fit to each individual response at different temperatures without any constraint on the shape factor . This procedure gives a systematic change in , which could be interpreted to mean that the shape of the underlying relaxation spectrum changes. As a consequence, thermorheological simplicity would not work if the KWW shape parameter is taken literally. However, we successfully constructed master curves by manual shifting in spite of the changing shape factor  for the individual curves.” In brief, it is not clear if the KWW shape parameter should be taken literally. Perhaps, the best-suited experimental technique for studying variations in  is dielectric spectroscopy.26 It is fast and reliable and, most importantly, an extremely wide frequency range can be explored at each temperature. Consequently, there is at least the possibility that  will be temperature dependent. A large number of experimental papers1–5,26–39 have investigated  as a function of temperature. Most have shown  decreases with decreasing temperature; however, others have observed a constant . Cases where  increases with decreasing temperature are usually dismissed as anomalies.5 Ngai, Paluch, and co-workers1,40 have fitted the dielectric spectra of a wide range of systems to Kohlrausch func-
tions. These fits tend to agree well at low frequency and less well at higher frequency where contributions from short time relaxation processes contribute to the loss spectrum. Representative examples are given in Ref. 40. It is informative to view the behavior of  as a function of KWW. In a work on small molecule glass formers, Paluch and co-workers39 found  to be a single valued function of KWW. In particular, when the results of different pressures and temperatures are represented on a single  versus log共KWW兲 graph, the data collapse onto a single curve. This intriguing interrelation between relaxation time and the shape of the relaxation function is also seen in our simulations. This analysis procedure is taken a step further by Alegria et al.27共c兲 The limit of  as KWW diverges is found by plotting  as a function of 1 / log共KWW / 0兲, where 0 is “the inverse of a typical phonon frequency.” It is found that for a number of polymeric systems and for a number of model spin glass systems,  approaches 1 / 3 in the large KWW limit 关see Fig. 5 of Ref. 27共c兲兴. The relaxation time can be represented as a single valued function of static quantities. Caslini, Roland, and co-workers41 have demonstrated that experimental relaxation times collapse to a single valued function of 1 / TV␥CR, where T is temperature, V is the specific volume, and ␥CR is chosen to collapse the data. The implication is that this quantity is inversely proportional to the occupied volume fraction. Budzien and co-workers10,11 demonstrated that the packing fraction directly calculated from molecular parameters functions in much the same manner as 1 / TV␥CR and collapses the reduced diffusion coefficient calculated from simulation results. In Figs. 3共a兲 and 3共b兲, the diffusion coefficients of the chain center of mass of FJ and FR systems are plotted against inverse temperature, illustrating the inability of temperature alone to collapse the data. In Figs. 3共c兲 and 3共d兲, the reduced diffusion coefficient is plotted against packing fraction, showing collapse of the data. III. RESULTS
Following the procedures outlined above, our previous results have been reanalyzed and additional state points were evaluated. In particular, cross-plots, such as the one shown in Fig. 1共d兲, are sensitive to numerical noise in the Kohlrausch region. Consequently, E共t兲’s lacking a well-defined “slopeof-one” range were run for longer times, and/or with smaller “dump frequencies.” This, perhaps, excessive care in generating and fitting the E共t兲 functions permits the tracking of changes in  over the entire simulation range. Although presented in a concise manner, this effort represents a large expenditure in both human and computer time. First, it was found that  varies with temperature and pressure for both FJ and FR systems and for both attractive and repulsive interactions. In keeping with the spirit of our earlier work,8–11 it is found that  is a single valued function of packing fraction , as seen in Fig. 4共a兲, where  is found from fitting Eq. 共1.2兲 over the restricted domain illustrated in Fig. 1. In addition, to good approximation, 共兲 is identical for both attractive and repulsive systems. The packing frac-
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FIG. 3. 共Color兲 Chain center of mass diffusion coefficient D for freely jointed chains 关in 共A兲 and 共C兲兴 and for freely rotating chains 关in 共B兲 and 共D兲兴. The squares are for attractive FJ chains and the circles are for repulsive FJ chains. The triangles are for attractive FR chains and the inverted triangles are for repulsive FR chains. In 共A兲 and 共B兲, D in Lennard–Jones units is plotted in Arrhenius fashion against inverse temperature. In 共C兲 and 共D兲, the reduced diffusion coefficient, D* = DN / 共dT1/2兲, is plotted against packing fraction = d3 / 6. In 共C兲 and 共D兲, the repulsive and attractive results have been shifted, as indicated by the arrows. The lines are powerlaw fits9 to D*: For FJ, D* = 0.88共0.643− 兲2; for FR, D* = 3.5共0.577− 兲2.9. Logarithms are base 10.
tion 0, where the reduced diffusion coefficient extrapolates to zero, is different9 for the FJ and FR systems: 0.643 for FJ and 0.577 for FR 共indicated by vertical lines兲. Clearly, the  for FJ and FR do not collapse to the same line as a function of . The behavior of 共兲 for both systems is similar; as approaches 0,  decreases smoothly, although leveling off for the FJ chains. It appears that the FJ chains have a  at the ideal glass transition of 0 ⬃ 0.69 and the FR chains, of 0 ⬃ 0.65. Of course, this does not preclude the possibility that  rapidly changes near the glass transition. This is a point of some debate. Although  must be greater than zero, it is not clear if, in general,  approaches 0.5 at 0, as has been suggested,5 if it approaches 0,5,42 if it approaches 1 / 3,27共c兲 or if it approaches a system specific value. Indeed,  as measured at the experimental glass transition is commonly used as a classification system of the dynamics of glass formers.43 Our results suggest that the last of these possibilities, a system specific , is correct. Both  and KWW are single valued functions of and, consequently,  can be viewed as a single valued function of KWW. This is shown in Figs. 4共b兲 and 4共c兲. In all cases, simulation results over a range of pressures and temperatures collapse to a single curve. A couple of results are worth particular notice. First,  decreases as the relaxation time of the system increases. To the extent that  is a measure of the distribution of relaxation times, this indicates that the distribution of relaxation times broadens as the glass transition is approached. Second, the ideal glass transition corresponds to
J. Chem. Phys. 128, 184905 共2008兲
FIG. 4. 共Color兲 Variation of the KWW stretching exponent  as a function of packing fraction in 共A兲 and as a function of the inverse of the logarithm of the KWW relaxation time KWW in 共B兲 and 共C兲. The circles correspond to FJ-R, the squares to FJ-A; the inverted triangles to FR-R, and the triangles to FR-A. In 共A兲, a representative error bar of ⫾0.01 is shown to the left and vertical lines at the location of the packing fractions at the ideal glass transition are shown to the right 关0共FR兲 = 0.577 and 0共FJ兲 = 0.643兴. In 共B兲 is shown the results of Paluch et al. 关Ref. 39共c兲兴 for the fragile glass former 关poly共bisphenol A-co-epichlorohydrin兲, glycidyl end capped兴 at a variety of pressures 共symbols representing different pressures兲 and the line is a guide to the eye. The boxed area in 共B兲 is reproduced in 共C兲; the line is duplicated and the simulation results are plotted instead of experimental results. The relaxation times 0 used to reduce the data in 共B兲 and 共C兲 are 10−12 s for experiment, and in Lennard–Jones time units, 2 ⫻ 10−5 for the FR and 5 ⫻ 10−6 for the FJ simulations. Logarithms are base 10.
the limit of KWW = ⬁. Here, again, it is difficult to deduce the value of  in this limit; however, a system dependent limiting value is the most reasonable interpretation. This last conclusion is easier to draw if the abscissa is 1 / log共 / 0兲 with 0 selected to be an atomic time scale. We have followed the suggestion of Alegria et al.27共c兲 and choose 0 = 10−12 s for the experimental results. The simulation 0’s are selected to overlay experiment 关in Lennard–Jones 共LJ兲 units, these are 0共FJ兲 = 2 ⫻ 10−6 and 0共FR兲 = 10−5兴. Consequently, the LJ time scale is roughly 10−7 s. The collapse of data when  is plotted against KWW is experimentally well known.29,33,37–39 In Fig. 4共b兲, the results of Paluch et al.39 for a fragile epoxy glass former are plotted. The general trends are the same in the simulation and experimental cases:  decreases with increasing and appears to approach a nonzero value of  in the large limit. Including the results of both experiment and simulation in the same plot makes clear the limited time regime explored by simulation as compared to experiment. The experiment spans nine orders of magnitude while the simulation spans only four. Additional illustrations of 共兲 for small molecular glass formers can be found in the work of Wang and Richert.38 A limited time regime is typical of simulation studies, and while it would be possible to extend the range to shorter ’s,
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共KWW兲 to translational 共T兲 relaxation times is plotted, where T is deduced from the translational diffusion coefficient.9 This ratio drifts, although for large , the FJ ratio is nearly constant. This effect has been seen in other simulation44,45 and experimental studies4,46–48 as shown, for instance, in Fig. 8 in Ref. 4.
IV. DISCUSSION
FIG. 5. 共Color兲 Variation of relaxation times. In 共A兲, both KWW and Tail for FJ chains are plotted against the distance to the glass transition 共0 − 兲 where 0 = 0.643. Circles are KWW-R; squares, Tail-R; triangles, KWW-A; and inverted triangles, Tail-R. The line has a slope of 2.2, In 共B兲 the ratio of KWW to Tail is plotted against . Circles are FJ-R; squares, FJ-A; triangles, FR-A; and inverted triangle, FR-R. 共C兲 is identical to 共A兲 except for FR chains, 0 = 0.577, and the line has slope 3.0. In 共D兲 KWW / T is plotted against . Circles are FJ-R; squares, FJ-A; triangles, FR-A; and inverted triangle, FR-R. Logarithms are base 10.
the region of interest is the long relaxation times. Consequently, a simulation study of a range comparable to the experimental study in Fig. 4共b兲 would require runs 106 times longer than our longest runs which were on the order of months. Many of the variations seen on the approach to the glass transition can be described as power laws in the “distance” to the ideal glass transition. In Fig. 5共a兲 for FJ chains, both the relaxation times KWW and the relaxation time of the single exponential “tail,” TAIL, are seen to obey KWW ⬃ 1 / 共0 − 兲␥, where ␥ = 2.2 and 0 = 0.643. Similar behavior is seen for FR chains shown in Fig. 5共c兲 with ␥ = 3.0 and 0 = 0.577. Another indicator of the broadening of the relaxation function E共t兲 is the ratio of the inherent time scales of the Kohlrausch function and the single-exponential tail: KWW / Tail. This is shown in Fig. 5共b兲 as a function of packing fraction. If time-temperature superposition held, this ratio would be constant. In both the FJ and FR cases, it is not clear if the ratio drifts as the glass transition is approached. Finally, we revisit the Debye–Stokes–Einstein law. The underlying picture is that resistance to particle motion can be modeled by a simple continuum frictional bath without memory effects. In such a case, relaxation times should scale the same way for different particle sizes 共the Stokes–Einstein law兲, and the same way for rotational and translational motion 共the Debye–Stokes–Einstein law兲. Previously,9 we considered the effect of particle size 共and the validity of the Stokes–Einstein law兲, and here we investigate the Debye– Stokes–Einstein law. In Fig. 5共d兲, the ratio of rotational
Although the divergence of the ␣-relaxation time ␣ is the defining feature of the glass transition, it is natural to inquire if other significant features of system dynamics also change in a systematic manner as the glass transition approaches. The term “significant” is used here to denote features in some sense associated with the ␣-relaxation process, and to exclude -relaxation processes associated with intramolecular relaxations. To answer this inquiry in the negative, that is, to claim that no significant features other than the ␣-relaxation time change as the glass transition is approached, is equivalent to claiming that time-temperature superposition is rigorously true. To answer in the affirmative, to claim that other significant features do change, is to state that time-temperature superposition is violated and to invite questions concerning the nature of the violation. Systems that obey time-temperature superposition are said to be rheologically simple. This is an idealization based on a model where a particle experiences its medium as a continuum, frictional bath. While no liquid entirely fits this idealization, rheological simplicity can be manifested in a number of ways. A system can be found to be simple by some measures and complex by others. We have applied a number of measures of rheological simplicity to our systems. First is master curve behavior where relaxation functions at different state points can be shifted one onto the other by scaling the time axis by the ␣-relaxation time. The degree that FJ and FR chains have master curve behavior across state points was shown in Figs. 5 and 6 of Ref. 8. Both FJ and FR were found to show master curve behavior; however, this is an insensitive measure of rheological complexity that tends to classify as “simple” that which would be classified as “complex” by other measures. Second, rheological simplicity implies that all ␣-relaxation times scale with each other. In particular, the ␣’s calculated in our simulations were, 共1兲 the translational relaxation time T for both chain center of mass and penetrant, 共2兲 the Kohlrausch relaxation time KWW, and 共3兲 the relaxation time of the single exponential tail TAIL. A log-log plot of any two of these would yield a straight line with slope of 1 if time-temperature superposition were obeyed. To good approximation, this is often found to be the case; however, as is also the case experimentally,4,44,49 closer examination can show more complex behavior. Plots of the ratio of relaxation times often show behaviors difficult to see in a log-log plot. The proportionality of ␣’s across length scales is a feature of the Stokes–Einstein Law. The degree to which this is obeyed for the FJ and FR systems is shown in Figs. 4 and 5 of Ref. 9. The proportionality of ␣’s between translational and rotational motion is a feature of the Debye–Stokes–Einstein
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Law. The degree to which this is obeyed for the FJ and FR systems is shown in Figs. 4 of Ref. 9 and Fig. 5 of the present paper. Third, rheological simplicity implies that when the relaxation functions are fitted to the Kohlrausch function, the value of  is a constant for all state points. If master curve behavior is rigorously obeyed,  will, of course, be constant. However,  is a more sensitive measure of change than master curve behavior since log-log master curve plots tolerate a small amount of change in . Consequently, master curve plots emphasize sameness, while plots of  versus state point emphasize change. The primary difficulty in using  as a measure of rheological simplicity is the sensitivity of the curve fitting procedure, the topic of the current study. In a previous study,8 it was demonstrated that E共t兲 for FJ and FR chains displayed the characteristic Kohlrausch structure often associated with an ␣-relaxation process. Moreover, a master curve motivated analysis indicated that timetemperature superposition was consistent with the simulation results. In the current study, we introduce an analysis methodology of greater accuracy and apply it over a wider range in . The value of  for both FJ and FR is found to vary in a consistent manner as the glass transition is approached. When plotted as a function of packing fraction,  collapses to a single valued function, indicating that the value of  is “determined” by the distance to the ideal glass transition. Consequently, there is the possibility of extrapolating the behavior of the relaxation function below Tg in order to analyze physical aging from a non-time-temperature superposition perspective.50 In a related manner,  is seen to be a single valued function of the relaxation time KWW. This is in agreement with experiment39 where the results at different pressures and temperatures are seen to collapse to a single curve. An advantage of representing our results as 共KWW兲 is that relatively direct comparisons with experiments are possible. When the appropriate time scale for the simulation is identified, our results are in good agreement with experimental ones for small molecule glass formers. This also highlights the restricted range of time scales accessible to simulation: 4 decades contrasted to the 14 plus decades seen in experiment.38 To a good approximation, the FJ systems are found to be rheologically simple. Once the chain center of mass translational motion becomes caged at a ⬃ 0.52 共Fig. 3 of Ref. 9兲,  is a constant of 0.69⫾ 0.01 共Fig. 2兲. The reduced diffusion coefficient for the chain center of mass and the single-bead penetrant are proportional to each other 共Figs. 4 and 5 of Ref. 9兲. The rotational relaxation times for bond vector and chain end-to-end vector are proportional to each other 共Fig. 6 of Ref. 8兲. The relaxation times for translational chain center of mass motion and rotational E共t兲 motion are proportional to each other 共Fig. 5兲. Consequently, for both FJ-R and FJ-A systems, master curve behavior is seen, the Stokes–Einstein Law is obeyed and the Debye–Stokes–Einstein Law is obeyed. On the other hand, the FR systems are rheologically complex. As the glass transition is approached,  approaches
0.65⫾ 0.01 linearly with packing fraction 共Fig. 2兲. The reduced diffusion coefficients for the chain center of mass and the single-bead penetrant are not proportional to each other 共Figs. 4 and 5 of Ref. 9兲. The rotational relaxation times for bond vector and chain end-to-end vector are not proportional to each other 共Fig. 6 of Ref. 8兲. The rotational and translational times for chain center of mass and E共t兲 are not proportional to each other 共Fig. 5兲. Consequently, for both FR-R and FR-A systems, the Stokes–Einstein Law is not obeyed and the Debye–Stokes–Einstein Law is not obeyed. The FR system is rheologically complex. However, because  is changing slowly in the liquid region and does not show signs of rapid change as the glass transition is approached, master curves can be constructed with a great deal of overlap 共Figs. 5 and 6 of Ref. 8兲. Consequently, even in the case of the FR system, time-temperature superposition is approximately obeyed as long as investigations are restricted to the high packing fraction regime. ACKNOWLEDGMENTS
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract No. DE-AC04-94AL85000. J.D.M. and T.C.D. thank Brian Borchers for useful discussions. We also thank the Materials Engineering Department at New Mexico Tech for partial support of the undergraduate research assistants 共K.T.D., F.A., D.T.L., and D.T.M.兲. APPENDIX: MODEL AND SIMULATION DETAILS
The simulation methodology is similar to that previously used.10,11,51 The LAMMPS molecular dynamics code52 was run for systems of 80 ten-site, bead-and-spring chains with five single bead penetrants. All the sites interacted through a Lennard–Jones potential, ULJ共r兲, clipped either at the minimum 共r = 21/6 for the repulsive case兲 or at 2.5* 共for the attractive case兲 where r is the site separation, is the length scale and is the energy scale, ULJ共r兲 = 4 ⴱ
冋冉 冊 冉 冊 册 r
12
−
r
6
.
共A1兲
In addition, bonded sites interact through a finitely extensible nonlinear elastic 共FENE兲 potential.53 The length scale of the FENE bond is slightly different from that of the Lennard– Jones potential and, as a result, the system does not crystallize. The backbone stiffness was varied through a bond-angle potential UA共兲, UA共兲 = K共 − 0兲2 ,
共A2兲
where is the bond angle, 0 is 120°, and K sets the strength of this potential. Two values of K are considered here: K = 0 共freely jointed chains兲 and K = 500 共freely rotating chains兲. Four system types are considered: Freely jointed and freely rotating chains, and attractive and repulsive potentials for each case. Each system has two types of diffusing species: Penetrants and chains. Runs of high accuracy were performed for each system at many state points as denoted in
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TABLE I. State points simulated. Temperature in units of / kB. Density 共兲 in units of 1 / 3. Pressure 共P兲 in units of / 3. Temperature Freely jointed repulsive
= 0.84 = 1.00 = 1.06 = 1.033 P = 2.0 P = 3.2 P = 6.5
0.3, 0.4, 0.6, 0.9, 2.0 0.35, 1.0, 1.2 0.5, 0.9, 1.0 0.5, 0.8, 1.0, 2.0 0.5, 0.7, 1.0 1.0 0.4, 0.7
Freely jointed attractive = 0.84 = 1.00 = 1.033 P=2 P = 3.2
1.5, 2.0, 2.2 1.4 0.5, 0.7, 1.0, 2.0 0.7, 1.0 1.0
Freely rotating repulsive = 0.944 = 1.06 = 1.033 P=2 P = 3.2 P = 6.5
0.7, 1.6, 1.6, 0.8, 0.8, 0.7,
1.2, 1.8, 2.0, 1.0, 1.0, 1.0,
1.6, 2.0 2.2 1.2, 1.2, 1.2,
2.0, 2.2, 2.4
1.4, 1.6, 1.8 1.6, 1.8 1.4, 1.6, 1.8, 2.0
NVT conditions兲 using this procedure was found to extend well into the Fickian diffusion regime and resulted in an E共t兲 curve containing significantly less error. After equilibration, production data were accrued 共under NVT conditions兲 for a period roughly four times the length of the equilibration run and extending well into the Fickian diffusion regime. In this study the packing fraction is defined as
=
3 d , 6
共A3兲
where is the site density and d is the effective hard sphere diameter. To find d for the repulsive systems, the Barker– Henderson equation is used56 d=
冕
⬁
关1 − exp共− U共r兲兲兴dr,
共A4兲
0
where the potential U共r兲 is the Lennard–Jones potential clipped at the minimum and shifted up until the minimum has a value of zero. The effective hard sphere diameter for the attractive systems is dA = d共rA* / rR*兲, where r* is the location of the first peak in the pair correlation function g共r兲. The values for rA* and rR* are taken from the same density and temperature. K. L. Ngai, J. Non-Cryst. Solids 353, 709 共2007兲; K. L. Ngai, ibid. 275, 7 共2000兲, and references therein. 2 Non-Debye Relaxation in Condensed Matter edited by T. V. Ramakrishnan and M. R. Lakshmi 共World Scientific, New Jersey, 1987兲, and references therein. 3 Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter edited by J.-L. Barrat, M. Feigelman, J. Kurchan, and J. Dalibard 共Springer-Verlag, New York, 2003兲, and references therein. 4 C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, and S. W. Martin, J. Appl. Phys. 88, 3113 共2000兲, and references therein. 5 J. C. Phillips, Rep. Prog. Phys. 59, 1133 共1996兲, and references therein. 6 R. H. Boyd and G. D. Smith, Polymer Dynamics and Relaxation 共Cambridge, London, 2007兲, and references therein. 7 L. Luca and T. M. Nieuwenhuizen, Thermodynamics of the Glassy State 共Taylor & Francis, New York, 2008兲, and references therein. 8 J. V. Heffernan, J. Budzien, F. Avila, T. C. Dotson, V. J. Aston, J. D. McCoy, and D. B. Adolf, J. Chem. Phys. 127, 214902 共2007兲. 9 J. V. Heffernan, J. Budzien, A. T. Wilson, R. J. Baca, V. J. Aston, F. Avila, J. D. McCoy, and D. B. Adolf, J. Chem. Phys. 126, 184904 共2007兲. The translational relaxation time was denoted as R instead of T as in the present work. 10 J. Budzien, J. D. McCoy, and D. B. Adolf, J. Chem. Phys. 119, 9269 共2003兲. 11 J. Budzien, J. D. McCoy, and D. B. Adolf, J. Chem. Phys. 121, 10291 共2004兲. 12 D. B. Adolf, R. S. Chambers, J. Flemming, J. L. Budzien, and J. D. McCoy, J. Rheol. 51, 517 共2007兲. 13 R. Kohlrausch, Ann. Phys. 91, 56 共1854兲; G. Williams and D. C. Watts, Trans. Faraday Soc. 66, 80 共1970兲. 14 C. P. Lindsey and G. D. Patterson, J. Chem. Phys. 73, 3348 共1980兲. 15 J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed. 共Wiley, New York, 1980兲. 16 W. Gotze and L. Sjogren, Rep. Prog. Phys. 55, 241 共1992兲, and references therein. 17 U. Bengtzelius, W. Gotze, and A. Sjolander, J. Phys. C 17, 5915 共1984兲. 18 共a兲 C. Bennemann, J. Baschnagel, and K. Binder, Eur. Phys. J. B 10, 323 共1999兲; 共b兲 J. Baschnagel and F. Varnik, J. Phys.: Condens. Matter 17, R851 共2005兲; 共c兲 M. Aichele and J. Baschnagel, Eur. Phys. J. E 5, 245 共2001兲; 共d兲 G. D. Smith, D. Bedrov, and W. Paul, J. Chem. Phys. 121, 4961 共2004兲; 共e兲 E. Milotti, J. Comput. Phys. 217, 834 共2006兲. 19 P. Bordat, F. Affouard, M. Descamps, and K. L. Ngai, J. Non-Cryst. Solids 352, 4630 共2006兲. 20 W. Kob and H. C. Andersen, Phys. Rev. E 51, 4626 共1995兲; W. Kob, C. 1
Freely rotating attractive = 0.944 = 1.06 = 1.033 P = 3.2 P = 6.5
0.8, 1.6, 1.6, 1.2, 1.4,
1.0, 1.2, 1.4, 1.6, 1.8 2.0, 2.2, 2.4 1.8, 2.0, 2.2 1.8 1.8
Table I. All simulations were performed in the NPT or NVT ensemble, in which the temperature is kept constant using a Nosé–Hoover thermostat54 and pressure is kept constant using an Andersen barostat.55 For each state point, equilibrium conditions need to be reached before starting dynamic measurements. In addition to the standard MD conditions, the chain center of mass must diffuse by the chain’s radius of gyration and the autocorrelation function of the vector linking the chain ends, E共t兲, must decay below 0.01. Typically, a time step of 2 ⫻ 10−3冑m2 / 共reduced Lennard–Jones units兲 was used; however, for small , a time step of 1 ⫻ 10−3冑m2 / was adopted. We have expanded our analysis to a greater range of packing fractions. We simulate state points requiring on the order of 106 – 109 time steps 共from several hours to approximately 2 weeks of CPU time on a 3 Ghz processor兲 for freely jointed and freely rotating systems. A freely jointed simulation required roughly 20 CPU hours on a 1.25 Ghz processor using 3 ⫻ 107 time steps to equilibrate whereas a freely rotating simulation required roughly 40 CPU hours using 5 ⫻ 107 time steps to equilibrate. Striving to improve the quality of our data as much as possible, we used the time for E共t兲 to decay to 0.01 during the equilibration as a guide for obtaining production data. The dump frequency and the simulation length was adjusted such that E共t兲 decays in approximately one hundredth of the total simulation time. The production data accrued 共under
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