Rheological complexity in simple chain models

Transcription

1 Rheological complexity in simple chain models Taylor C. Dotson, Julieanne V. Heffernan, Joanne Budzien, Keenan T. Dotson, Francisco Avila, David T. Limmer, Daniel T. McCoy, John D. McCoy, and Douglas B. Adolf Citation: The Journal of Chemical Physics 128, (2008); doi: / View online: View Table of Contents: Published by the AIP Publishing Articles you may be interested in Temperature dependent micro-rheology of a glass-forming polymer melt studied by molecular dynamics simulation J. Chem. Phys. 141, (2014); / Molecular dynamics of ionic liquids as probed by rheology J. Rheol. 55, 241 (2011); / The rheology of solid glass J. Chem. Phys. 132, (2010); / Structural relaxation and rheological response of a driven amorphous system J. Chem. Phys. 125, (2006); / Dynamics and rheology of a supercooled polymer melt AIP Conf. Proc. 519, 151 (2000); /

2 THE JOURNAL OF CHEMICAL PHYSICS 128, 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 and Douglas B. Adolf 2 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 Received 4 March 2008; accepted 27 March 2008; published online 13 May 2008 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 KWW 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 0 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 i.e., the Debye Stokes Einstein relation, of motion on different length scales i.e., the Stokes Einstein relation, 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 American Institute of Physics. DOI: / 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 i.e., log 1/T, 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 reviews 1 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 i.e., of temperature and pressure. Complexities arise from the apparent divergence of at nonzero temperature, from nonequilibrium states at low temperature and the a Author to whom correspondence should be addressed. Electronic mail: mccoy@nmt.edu. 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 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 /2008/ /184905/9/$ , American Institute of Physics

3 Dotson et al. J. Chem. Phys. 128, 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 e.g., the analysis of physical aging, and even the extrapolation from one relaxation function to another e.g., the analysis of mechanical relaxation from dielectric relaxation. 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 FJ or freely rotating FR with a bond angle of 120. The intermolecular interactions were Lennard Jones potentials clipped either at 2.5 denoted as attractive systems, A or clipped at the minimum denoted as repulsive systems, R, 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 A table of the state points investigated is included in the Appendix. The primary measure of rotational relaxation used 8 in the present investigation is the second Legendre polynomial of the chain end-to-end vector, E t = 3 2 e t e , 1.1 where the angle brackets denote ensemble average and e t is the unit vector along the end-to-end vector at a time t. Inthe intermediate time regime, this function is well described by the Kohlrausch or Kohlrausch Williams Watts KWW function, 13 E KWW t = Ze t/ KWW, 1.2 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 These values were shown to be in keeping FIG. 1. Color online Analysis of the second Legendre polynomial of the end-to-end vector E t for ten-site FR-R at =1.06, T=1.6, and = A E t vs ln t. 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. B 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 = C The modified Lindsey Patterson plot. The points are the data. The dashed line shows a slope of one; and the solid, a slope of = D The Lindsey and Patterson cross-plot. The points are the data. The solid line has a slope of one and an intercept of ln with = 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 t 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 as seen in Fig. 1 a, 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

4 Rheological complexity in chain models J. Chem. Phys. 128, expected in, and the difficulty in identifying the intermediate time regime. The curve fit of the data in Fig. 1 a is reproduced slightly to the right to illustrate the switch from Kohlrausch solid line in the intermediate time regime to a single-exponential function dashed line at longer time. A graphical procedure suggested by Lindsey and Patterson 14 LP is useful in the analysis. The logarithm of the negative logarithm of E t is plotted against the logarithm of t. IfE t were identically of the Kohlrausch form, the procedure would result in ln ln E KWW t = ln t/ KWW +ln 1 ln Z t/ KWW, 2.1 where for large time, ln ln E t versus ln t becomes linear with a slope of. In Fig. 1 b, a typical LP plot is shown with a clear linear regime. The Kohlrausch fit is shown as a solid line offset for clarity 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 t becomes single exponential the offset dashed line. It is only in the intermediate time regime that the slope of the curve in Fig. 1 b can be identified with solid line. Identifying this regime is the primary challenge to achieving accuracy in to three significant figures the region eventually decided upon is indicated by vertical dashed lines for all panes in Fig. 1. Previously, 8 we suggested a modification to the LP procedure. By graphing the logarithm of the derivative of the negative logarithm of E t against the logarithm of t, the parameter Z is eliminated. In particular, ln d ln E KWW t = ln t/ KWW +ln. 2.2 d ln t An example of this is shown in Fig. 1 c. 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 ln E t against ln d ln E t /d ln t. For the Kohlrausch function, one finds that ln ln E KWW t =ln d ln E KWW t ln d ln t ln Z +ln 1 t/ KWW. 2.3 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. An example of this is shown in Fig. 1 d, where the Kohlrausch fit has a = FIG. 2. Color online Examples of the determination of the KWW stretching exponent for a number of packing fractions for FR chains. The T,,, are from top to bottom 1.6, 0.612, 0.318, R; 2.2, 0.944, 0.471, R; 2.0, 1.06, 0.536, R; 1.6, 1.06, 0.551, R; 1.6,1.06, 0.563, A. In A are shifted Lindsey and Patterson crossplots with lines of a slope of 1. In B 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 d. 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 b and 1 c. Finally, Z can be identified from the plot of E t versus ln t, as in Fig. 1 a. The three values of and two values of KWW permit the estimate of errors. For the case illustrated in Fig. 1, these are = and KWW = 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 theory 16,17 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-

5 Dotson et al. J. Chem. Phys. 128, ture range has been noted in a number of instances. Previously, 8 we demonstrated that for FJ-A chains, a = , 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 Andersen 20 mixture of Lennard Jones 6 12 atoms. The self-intermediate scattering function was found to be described by a single = across a wide range of temperatures; however, it was only approximately possible to extract a temperature independent stretched exponent using a Kohlrausch Williams Watt KWW 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. See Figs. 4 and 5 of Ref. 19. Other simulations indicating a changing are not uncommon, 18 d,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 see Fig. 6 of Ref. 24. 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 papers 1 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-workers 1,40 have fitted the dielectric spectra of a wide range of systems to Kohlrausch functions. 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-workers 39 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 KWW 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 c The limit of as KWW diverges is found by plotting as a function of 1/log KWW / 0, 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 see Fig. 5 of Ref. 27 c. The relaxation time can be represented as a single valued function of static quantities. Caslini, Roland, and co-workers 41 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-workers 10,11 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 a and 3 b, 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 c and 3 d, 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 d, are sensitive to numerical noise in the Kohlrausch region. Consequently, E t 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 t 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 a, where is found from fitting Eq. 1.2 over the restricted domain illustrated in Fig. 1. In addition, to good approximation, is identical for both attractive and repulsive systems. The packing frac-

6 Rheological complexity in chain models J. Chem. Phys. 128, FIG. 3. Color Chain center of mass diffusion coefficient D for freely jointed chains in A and C and for freely rotating chains in B and D. 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 A and B, D in Lennard Jones units is plotted in Arrhenius fashion against inverse temperature. In C and D, the reduced diffusion coefficient, D*=DN/ dt 1/2, is plotted against packing fraction = d 3 /6. In C and D, the repulsive and attractive results have been shifted, as indicated by the arrows. The lines are powerlaw fits 9 to D*: For FJ, D*= ; for FR, D*= Logarithms are base 10. tion 0, where the reduced diffusion coefficient extrapolates to zero, is different 9 for the FJ and FR systems: for FJ and for FR indicated by vertical lines. Clearly, the for FJ and FR do not collapse to the same line as a function of. The behavior of 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 and the FR chains, of 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 c 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 b and 4 c. 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 FIG. 4. Color Variation of the KWW stretching exponent as a function of packing fraction in A and as a function of the inverse of the logarithm of the KWW relaxation time KWW in B and C. The circles correspond to FJ-R, the squares to FJ-A; the inverted triangles to FR-R, and the triangles to FR-A. In A, 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 0 FR =0.577 and 0 FJ = In B is shown the results of Paluch et al. Ref. 39 c for the fragile glass former poly bisphenol A-co-epichlorohydrin, glycidyl end capped at a variety of pressures symbols representing different pressures and the line is a guide to the eye. The boxed area in B is reproduced in C ; the line is duplicated and the simulation results are plotted instead of experimental results. The relaxation times 0 used to reduce the data in B and C are s for experiment, and in Lennard Jones time units, for the FR and 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 / 0 with 0 selected to be an atomic time scale. We have followed the suggestion of Alegria et al. 27 c and choose 0 =10 12 s for the experimental results. The simulation 0 s are selected to overlay experiment in Lennard Jones LJ units, these are 0 FJ = and 0 FR =10 5. 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 b, 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 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,

7 Dotson et al. J. Chem. Phys. 128, KWW to translational T 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 simulation 44,45 and experimental studies 4,46 48 as shown, for instance, in Fig. 8 in Ref. 4. FIG. 5. Color Variation of relaxation times. In A, both KWW and Tail for FJ chains are plotted against the distance to the glass transition 0 where 0 = Circles are KWW -R; squares, Tail -R; triangles, KWW -A; and inverted triangles, Tail -R. The line has a slope of 2.2, In B the ratio of KWW to Tail is plotted against. Circles are FJ-R; squares, FJ-A; triangles, FR-A; and inverted triangle, FR-R. C is identical to A except for FR chains, 0 =0.577, and the line has slope 3.0. In D 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 b would require runs 10 6 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 a for FJ chains, both the relaxation times KWW and the relaxation time of the single exponential tail, TAIL, are seen to obey KWW 1/ 0, where =2.2 and 0 = Similar behavior is seen for FR chains shown in Fig. 5 c with =3.0 and 0 = Another indicator of the broadening of the relaxation function E t 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 b 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 the Stokes Einstein law, and the same way for rotational and translational motion the Debye Stokes Einstein law. Previously, 9 we considered the effect of particle size and the validity of the Stokes Einstein law, and here we investigate the Debye Stokes Einstein law. In Fig. 5 d, the ratio of rotational IV. DISCUSSION 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, 1 the translational relaxation time T for both chain center of mass and penetrant, 2 the Kohlrausch relaxation time KWW, and 3 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

8 Rheological complexity in chain models J. Chem. Phys. 128, 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 t 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 T g 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 experiment 39 where the results at different pressures and temperatures are seen to collapse to a single curve. An advantage of representing our results as KWW 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 Fig. 3 of Ref. 9, is a constant of Fig. 2. The reduced diffusion coefficient for the chain center of mass and the single-bead penetrant are proportional to each other Figs. 4 and 5 of Ref. 9. The rotational relaxation times for bond vector and chain end-to-end vector are proportional to each other Fig. 6 of Ref. 8. The relaxation times for translational chain center of mass motion and rotational E t motion are proportional to each other Fig. 5. 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 linearly with packing fraction Fig. 2. The reduced diffusion coefficients for the chain center of mass and the single-bead penetrant are not proportional to each other Figs. 4 and 5 of Ref. 9. The rotational relaxation times for bond vector and chain end-to-end vector are not proportional to each other Fig. 6 of Ref. 8. The rotational and translational times for chain center of mass and E t are not proportional to each other Fig. 5. 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 Figs. 5 and 6 of Ref. 8. 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-AC04-94AL 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 K.T.D., F.A., D.T.L., and D.T.M.. APPENDIX: MODEL AND SIMULATION DETAILS The simulation methodology is similar to that previously used. 10,11,51 The LAMMPS molecular dynamics code 52 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, U LJ r, clipped either at the minimum r=2 1/6 for the repulsive case or at 2.5* for the attractive case where r is the site separation, is the length scale and is the energy scale, U LJ r =4 r 12 r 6. A1 In addition, bonded sites interact through a finitely extensible nonlinear elastic FENE 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 U A, U A = K 0 2, A2 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 freely jointed chains and K=500 freely rotating chains. 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

9 Dotson et al. J. Chem. Phys. 128, TABLE I. State points simulated. Temperature in units of /k B. Density in units of 1/ 3. Pressure P in units of / 3. Freely jointed repulsive Temperature = , 0.4, 0.6, 0.9, 2.0 = , 1.0, 1.2 = , 0.9, 1.0 = , 0.8, 1.0, 2.0 P= , 0.7, 1.0 P= P= , 0.7 Freely jointed attractive = , 2.0, 2.2 = = , 0.7, 1.0, 2.0 P=2 0.7, 1.0 P= Freely rotating repulsive = , 1.2, 1.6, 2.0, 2.2, 2.4 = , 1.8, 2.0 = , 2.0, 2.2 P=2 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 P= , 1.0, 1.2, 1.6, 1.8 P= , 1.0, 1.2, 1.4, 1.6, 1.8, 2.0 Freely rotating attractive = , 1.0, 1.2, 1.4, 1.6, 1.8 = , 2.0, 2.2, 2.4 = , 1.8, 2.0, 2.2 P= , 1.8 P= , 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 thermostat 54 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 t, must decay below Typically, a time step of m 2 / reduced Lennard Jones units was used; however, for small, a time step of m 2 / was adopted. We have expanded our analysis to a greater range of packing fractions. We simulate state points requiring on the order of time steps from several hours to approximately 2 weeks of CPU time on a 3 Ghz processor for freely jointed and freely rotating systems. A freely jointed simulation required roughly 20 CPU hours on a 1.25 Ghz processor using time steps to equilibrate whereas a freely rotating simulation required roughly 40 CPU hours using time steps to equilibrate. Striving to improve the quality of our data as much as possible, we used the time for E t 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 t decays in approximately one hundredth of the total simulation time. The production data accrued under NVT conditions using this procedure was found to extend well into the Fickian diffusion regime and resulted in an E t curve containing significantly less error. After equilibration, production data were accrued under NVT conditions 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 = 6 d3, A3 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 used 56 d = 0 1 exp U r dr, A4 where the potential U r is the Lennard Jones potential clipped at the minimum and shifted up until the minimum has a value of zero. 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