Complete Set of Enthalpy Recovery Data Using Flash DSC

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Complete Set of Enthalpy Recovery Data Using Flash DSC: Experiment and Modeling Luigi Grassia,*,† Yung P. Koh,‡ Mattia Rosa,† and Sindee L. Simon‡ †

Department of Industrial and Information Engineering, Università degli Studi della Campania “Luigi Vanvitelli”, Via Roma 19, 81031 Aversa (CE), Italy ‡ Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409-3121, United States ABSTRACT: Enthalpy recovery of a single polystyrene thin film is quantified by both experiments using Flash DSC and modeling using a new modified TNM model. Experimental data include Kovacs’ three signatures of structural recovery: intrinsic isotherms after temperature down jumps, the asymmetry of approach after temperature down and up jumps of the same size, and the memory effect after a two-step history. A new modified TNM model is proposed to quantitatively fit all three signatures of structural recovery with a single set of model parameters. Here, we elucidate the detailed derivation of the new modified model and demonstrate its applicability to the experimental Flash DSC results.

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where β is the nonexponetiality parameter, varying from 0 to 1, which indicates the width of distribution of relaxation time. When β equals 1, eq 1 becomes simple exponential decay and only one relaxation time exists. The nonlinearity of the structural recovery kinetics in TNM model is expressed by the dependence of the relaxation time τ on both temperature and structure with the fictive temperature Tf being employed to represent the instantaneous state by Tool.10 An Arrhenius type of equation is generally applied to describe the temperature and structure dependence of τ:

INTRODUCTION Upon cooling, noncrystallizable glass-forming materials depart from their equilibrium liquid density and become glasses at the glass transition temperature (Tg), which depends on the experimental time scale or cooling rate.1 At any given temperature below Tg, the resulting nonequilibrium glassy structure spontaneously evolves toward equilibrium and, at long enough aging times, recovers its equilibrium density. This relaxation process is termed structural recovery and can be further classified as volume recovery or enthalpy recovery when those variables are followed.2 Kovacs investigated the volume recovery of poly(vinyl acetate) and other glass-formers, and he demonstrated the essential kinetic features of the structural recovery: nonlinearity, nonexponentiality, and memory.3 Nonlinearity was demonstrated by the asymmetry of approach experiments which showed that the kinetics of the structural recovery depend on the sign of the temperature jump (i.e., up- or down-jump) and its magnitude. The nonexponentiality was demonstrated by the stretched responses of the intrinsic isotherms as well as by two-step experiments which resulted in a memory effect or overshoot type of behavior. The latter also demonstrated the importance of thermal history or memory. These features of structural recovery are captured using phenomenological Tool−Narayanaswany−Moynihan (TNM)4−6 and Kovacs−Aklonis−Hutcheson−Ramos (KAHR)7 models of structural recovery. In the TNM model, the nonexponential decay of Tf is achieved using a continuous relaxation time distribution of the Kohlrausch−Williams−Watts (KWW) function:8,9 ⎡ ⎛ = 1 − exp⎢ −⎜ ⎢⎣ ⎝ dT

dTf

∫0

t

β dt ⎞ ⎤ ⎟ ⎥ τ ⎠ ⎥⎦ © XXXX American Chemical Society

⎡ ⎞⎤ (1 − x)Δh ⎛ 1 Δh ⎜⎜ − 1 ⎟⎟⎥ τ = A exp⎢ + ⎢⎣ RT R T ⎠⎥⎦ ⎝ Tf

(2)

where A is the pre-exponential factor, Δh is the apparent activation energy of the structural recovery, and x varies from 0 to 1 and separates the temperature and structure dependence. When x is equal to 1, the relaxation time of physical aging is linear without structure dependence. As x decreases, the response becomes more nonlinear as structure contributes more to τ. Although the TNM and KAHR models can capture all of the phenomenology associated with the glass transition and structural recovery, the limitations of these models are wellknown.11−21 In particular, a single set of model parameters is unable to quantitatively describe a broad range of structural recovery results.16−21 As shown in eq 2, the TNM model often assumes an Arrhenius temperature dependence of relaxation time, and this Arrhenius dependence is found to be applicable at Received: October 27, 2017 Revised: December 20, 2017

(1) A

DOI: 10.1021/acs.macromol.7b02277 Macromolecules XXXX, XXX, XXX−XXX

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equilibrium structure “frozen” into the glassy state. Further, it is assumed that the relaxation at T some distance ΔT below Tf is longer than the equilibrium relaxation time at Tf by the same amount as the difference in the equilibrium relaxation times at Tf and T0, the latter the same distance ΔT above Tf:

or below Tg and in a relatively narrow range of temperatures.22−34 However, for temperatures above Tg, equilibrium relaxation times over a wide range cannot be predicted by the TNM model, since the temperature dependence in the equilibrium liquid above Tg state follows a super-Arrhenius Williams−Landell− Ferry (WLF)35/Vogel−Tammann−Hess−Fulcher (VTHF)36−38 temperature dependence. Thus, the original TNM model is limited to apply only in the glassy state below Tg and/or over a limited range of temperatures.2,13,14,20 In order to extend the applicable temperature range of relaxation time expression to both nonequilibrium glassy and equilibrium liquid states, in our previous study,39 we developed a new equation for the relaxation time in the framework of the TNM model, an odd symmetric function of the WLF equation with respect to the origin of Tg. The rotational symmetry allows description of the weak temperature dependence of the glassy relaxation time below Tg and the strong dependence above Tg using one expression. As a result, this modified TNM model provided quantitative fits of the cooling rate dependence of Tg for over 5 decades as well as the evolution of fictive temperature during the enthalpy recovery up to 20 K above the nominal Tg.39 Here, we test our previously modified model against a more complete set of enthalpy recovery data, including the asymmetry of approach and memory experiments which provide a more strenuous test of nonlinearity. In addition, we propose a new modification of the TNM model in order to quantitatively describe the experimental results. The data fit includes previously published down-jump, asymmetry of approach, and memory experiment data from Flash DSC,39−41 as well as new data performed at aging temperatures as low as −30 °C. The paper is organized as follows. We first delineate the modeling effort, explaining the reasoning behind the odd-symmetric function for the relaxation time and our new approach. The fit to the data is then presented. We end with a brief discussion and conclusions.

log τ(T , Tf ) − log τeq(Tf ) = log τeq(Tf ) − log τ(T0), T0 = Tf + (Tf − T )

(4)

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MODELING As mentioned, the classical TNM model is unable to describe a wide set of experimental data with a single set of model parameters.16−21 To improve the model, we focus on the equation for the relaxation time, replacing eq 2 with a new phenomenological equation. The aim of this equation is to reduce the number of fitting parameters and at the same time to be able to describe a wider range of experimental data, including the temperature dependence of the relaxation time in equilibrium conditions, which can be successfully described by the Williams−Landel− Ferry (WLF) equation:35 log τeq = log τR −

C1(T − TR ) C2 + T − TR

(3)

where TR is an arbitrary reference temperature, τR is the equilibrium relaxation time at TR, and C1 and C2 are constants. In our previous work,39 for the glassy state characterized by fictive temperature Tf and temperature T < Tf, the relaxation time is assumed to be related to the equilibrium relaxation time at Tf and the distance from equilibrium as given by the quantity T − Tf. This is similar to the approach used in the Kovacs−Aklonis− Hutchinson−Ramos (KAHR) model7 where the relaxation time in the glassy state is assumed to be that at equilibrium with a deviation related to the departure from equilibrium δ, which is directly related to the quantity (T − Tf). Here, though, the equilibrium relaxation time at Tf, rather than at T, is the starting point, reflecting Tool’s idea10 that Tf reflects the liquid

Figure 1. (a) Schematic of the logarithmic relaxation time versus temperature. The red dashed line is the equilibrium WLF temperature dependence, and the blue line is the system path on cooling from T0 to T showing the odd symmetric of WLF function with respect to the origin of Tf. The two green lines are of equal length. (b) Temperature dependence of the logarithmic relaxation time, shown in red, the equilibrium WLF temperature dependence where Tf equals T, and in blue, the dependence in the glassy state at constant Tf, with Tf increasing from the uppermost blue curve to the bottom. (c) Schematic of the logarithmic relaxation time in the glassy state as a function of the distance from equilibrium Tf − T at constant temperatures of 95, 100, and 105 °C for the condition where Tf > T. B


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relaxation time at the end of heating (τeq(T)). In other words, the relaxation time during the heating process should be described by a curve included in the area highlighted in light blue in Figure 2a.

The idea is shown schematically in Figure 1a where the logarithmic relaxation time is plotted versus temperature. Here, the red dashed line shows the equilibrium WLF temperature dependence, whereas the blue line displays the path followed by the system during cooling from T0 to T to form a glass with fictive temperature Tf. The green arrows at T and T0 are of identical length. Thus, the relaxation time in the glass deviates from the equilibrium line during cooling with a path that is odd symmetric to the path on which the system walked to reach Tf from T0; i.e., the relaxation time depends only on the distance from Tf both in the equilibrium state and on the glass line. Rearranging eq 4, we obtain log τ(T , Tf ) = 2 log τeq(Tf ) − log τeq(2Tf − T )

for Tf > T (5)

Equation 5 yields the relaxation time in the glassy when Tf > T and does not assume the specific functional form of τeq. Assuming that the temperature dependence of the equilibrium relaxation time can be described with a WLF equation, we get from eqs 3 and 5 the specific equation for the relaxation time in the glassy state when Tf > T: log τ(T , Tf ) = log τR ⎡ ⎞⎤ ⎛ 2 1 ⎟⎥ + C1⎢− 1 + C2⎜⎜ + ⎢⎣ − C2 + T − 2Tf + TR ⎟⎠⎥⎦ ⎝ C2 + Tf − TR for Tf > T

(6)

The temperature and fictive temperature dependence of the relaxation time in the glassy state based on eq 6 are shown in Figures 1b and 1c as a function of T at constant Tf and as a function of the distance from equilibrium (Tf − T) at constant T, respectively. In Figure 1b, the red line represents the equilibrium relaxation time (Tf = T) and coincides with the standard WLF equation, and each blue line represents the relaxation time along the glass line at constant Tf. The blue lines can be thought of as cooling curves, with cooling rate and Tf increasing as one moves from the top curve to the bottom, with Tf being the temperature where a given blue curve intersects the red equilibrium curve. The model yields an apparent activation energy along the glass line that diminishes as the temperature decreases at fixed Tf and which increases at fixed temperature when Tf decreases. These features are consistent with the literature data22−34 showing a turnover from the WLF-type temperature dependence of the relaxation time to the weaker temperature dependence in the glassy state. In Figure 1c, the relaxation time from eq 6 is calculated as a function of the distance from equilibrium (Tf − T) during aging at constant temperature Ta, with data reported in terms of (log τ − log τeq). The curves of Figure 1c show that the logarithm of the relaxation time does not depend linearly on the distance from equilibrium close to the equilibrium condition when Tf − T is small. However, linearity is restored when the distance from equilibrium increases. Furthermore, the slope of the curves in the linear region is not constant as predicted by the KAHR model7 but increases as the aging temperature decreases. A glass can be also obtained by heating from the equilibrium condition if the experimental time scale characteristic of the heating process is much smaller than the relaxation time at the initial temperature of heating. The resulting glass has lower enthalpy and volume than the equilibrated system at the same temperature, and hence Tf < T. For this glass, the relaxation time cannot be larger than the equilibrium relaxation time at the beginning of heating (τeq(Tf)) and cannot be lower than the equilibrium

Figure 2. (a) Schematic of the logarithmic relaxation time versus temperature for a glass obtained by heating from the equilibrium. The red dashed line is the equilibrium WLF temperature dependence, and the light blue area indicates the feasible relaxation times during the heating process. (b) Schematic of the temperature dependence of the logarithmic relaxation time, as given by our previous modified TNM model39 and eq 9. The red line is the equilibrium WLF temperature dependence, and the blue line is the temperature dependence of the relaxation time in the glassy state as a function of temperature at constant Tf. (c) Schematic of the logarithmic relaxation time in the glassy state as a function of the distance from equilibrium Tf − T at constant temperatures of 95, 100, and 105 °C for conditions where Tf > T and Tf < T, based on eq 9

In our previous modeling work,39 we assumed that the logarithmic relaxation time in the glassy state at a given temperature is an odd symmetric function of the distance from the equilibrium (T − Tf): log τ(T , Tf 1) − log τeq(T ) = log τeq(T ) − log τ(T , Tf 2) (7)

where T − Tf1 = Tf 2 − T and Tf 2 > T > Tf1. The right-hand side of the eq 7 can be evaluated with eq 6 because Tf 2 > T, and C


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Macromolecules consequently, we can obtain from eqs 6 and 7 an equation for the relaxation time applicable for any value of Tf − T: |log τ(T , Tf ) − log τeq(T )| = log τδ+(T , T + |Tf − T |) − log τeq(T ) =

2C1C 2 |Tf − T |2 (C 2 + T − TR)(C 2 + T − TR + |Tf − T |)(C 2 + T − TR + 2|Tf − T |)

(8)

The quantity given by eq 8 should be subtracted from the logarithm of the equilibrium relaxation time when Tf > T and added in the case of Tf < T. Incorporating the WLF temperature dependence for the equilibrium relaxation time, we have log τ(T , Tf ) = log τR − ×

|Tf − T | C1(T − TR) − C 2 + T − TR Tf − T 2C1C 2 |Tf − T |2

(C 2 + T − TR)(C 2 + T − TR + |Tf − T |)(C 2 + T − TR + 2|Tf − T |)

(9)

The relaxation time expressed by eq 9 coincides with our previous modified TNM model.39 The predicted dependence of the relaxation time in the glassy state as a function of T at constant Tf is shown in Figure 2b, whereas the dependence on the distance from equilibrium (Tf − T) at constant T is shown in Figure 2c. In addition to evaluating the ability of eq 9 to describe a more complete set of recovery data, we also investigate a second way of setting the relaxation time for a glass obtained by heating from the equilibrium condition. Equation 9 is obtained assuming that at constant temperature the logarithmic relaxation time of the system in the glassy state differs from the equilibrium condition by the same amount, irrespective of the sign of Tf − T, although the sign does affect whether the glassy relaxation time is larger or smaller than the equilibrium value. A different approach is to assume that the relaxation time upon instantaneous heating from the equilibrium condition does not change with temperature and only depends on Tf, as shown by the blue line in Figure 2a, representing the upper limit of the relaxation time and yielding the maximum degree of nonlinearity during heating. With this

Figure 3. (a) Schematic of the temperature dependence of the logarithmic relaxation time, as given by our new modified TNM model and eq 10. The red line is the equilibrium WLF temperature dependence, and the blue line is the temperature dependence of the relaxation time in the glassy state as a function of temperature at constant Tf. (b) Schematic of the logarithmic relaxation time in the glassy state as a function of the distance from equilibrium Tf − T at constant temperatures of 95, 100, and 105 °C for conditions where Tf > T and Tf < T, based on eq 10.

assumption, we then get the following equation for the relaxation time:

⎧ ⎡ ⎛ ⎞⎤ 2 1 ⎪ log τ + C ⎢ −1 + C ⎜ ⎟⎥ T < Tf + R 1 2⎜ ⎪ ⎢⎣ −C2 + T − 2Tf + TR ⎟⎠⎥⎦ ⎪ ⎝ C2 + Tf − TR log τ(T , Tf ) = ⎨ ⎪ C (T − TR ) ⎪ log τR − 1 f T > Tf ⎪ C2 + Tf − TR ⎩

Figures 3a and 3b show the behavior of the relaxation time in the glassy state based on eq 10. The data are plotted as a function of T at constant Tf in Figure 3a and as a function of the distance from equilibrium (Tf − T) at constant T in Figure 3b. As can be seen by comparing Figures 2c and 3b, eqs 9 and 10 are identical for Tf ≥ T and differ when Tf < T. As a consequence, eqs 9 and 10 give identical predictions for down-jump experiments (i.e., the intrinsic isotherms) but different predictions for up-jump experiments. The τ-effective paradox and the expansion gap phenomenology3,42−44 suggest that the dependence of the relaxation time on the distance from equilibrium (namely, Tf − T) should be nonsymmetric when the equilibrium is approached from up and down jumps. The assumptions that yield eq 10, and consequently the discontinuities in the first derivative of the relaxation time reported in the Figures 3a and 3b when Tf = T, fix an upper bound to the asymmetry in terms of nonlinearity in the approaching the equilibrium. This approach is useful to establish the degree

(10)

of nonlinearity required to describe the data and to determine if having a discontinuity of the relaxation time when the equilibrium is approached from the bottom and from the top should be a characteristic of a model describing the structural relaxation of glassy materials.

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EXPERIMENTAL SECTION

Data are reported for a monodispersed high molecular weight polystyrene sample, obtained from Sigma-Aldrich, with a number-average molecular weight of 1 998 000 g/mol and a polydispersity index of 1.02. Polystyrene thin films were deposited by spin-coating polystyrene dissolved in toluene (HPLC-grade; Sigma-Aldrich) onto freshly cleaved 1 in.2 mica substrates. Film thickness is 1.1 μm, based on atomic force microscope measurements, and thus, the material should have a “bulk”like response. After spin-coating, the films were removed from the mica substrate by floating onto water and then picked up with an aluminum O-ring to maintain an unwrinkled film. In order to remove any residual solvent and/or adventitious water, the films were stored under D


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Macromolecules Table 1. Temperature Histories for the Signatures of Enthalpy Recovery signatures

T0 (°C)

t0 (s)

intrinsic isotherm asymmetry of approach

190.0 109.5 101.5 111.0 114.0 117.0 103.0 100.0 97.0 190.0 190.0

3 180 1800 180 60 60 600 1800 18000 3 3

memory effect

T1 (°C)

t1 (s)

100.5 80.5

60 780

ambient conditions for 24 h and then annealed in vacuum at 50 °C and 5 Torr for 12 h. The film preparation methodology and the resulting film thickness are also the same within experimental error as our previous enthalpy recovery studies using Flash DSC.39,40 A commercially available rapid-scanning calorimetry, Mettler Toledo Flash DSC, was used with nitrogen purge. Annealed films were cut to a square shape, approximately 0.4 mm × 0.4 mm, under a microscope prior to transfer to the Flash DSC chip. A small amount of Krytox oil is used between the film sample and the chip to facilitate heat transfer. The polystyrene sample mass is calculated from the step change in heat flow at Tg, based on the bulk polystyrene ΔCp at Tg (0.290 J/(g K)) and found to be approximately 100 ng for 1.1 μm thickness. The temperature histories for the enthalpy recovery experiment are the same as those detailed in our previous work using Flash DSC,39−41 and relevant details are briefly repeated here for the sake of clarity. All cycles begin with the sample being heated to 190 °C and held for 0.1 min to erase any previous thermal history. For down-jump experiments, samples were then cooled to the aging temperature (Ta), ranging from −30 to 100 °C, aged isothermally for aging times ranging from 0 to 1000 min. For the asymmetry of approach, the sample is first equilibrated at temperatures T0 for equilibration time of t0, and then the temperature is jumped down or up to the aging temperature Ta of either 105.5 or 107.0 at 1000 K/s. The memory effect experimental protocol is a two-step experiment: after equilibration at 190 °C, a temperature down jump to T1 is made, aging is performed at T1 for time t1, and then the temperature is jumped up to the aging temperature Ta; the aging time t1 is such that Tf ≈ Ta before the second jump. After aging at Ta, in all cases, the temperature was increased or decreased to 35 °C at 1000 K/s and held for 0.1 min, and then the fictive temperature Tf of the aged sample was then measured on a subsequent heating scan at 600 K/s. Tf is determined using the Moynihan45 or Richardson46 methods when Tf is above the onset of devitrification, whereas a simplified but equivalent version47 is used when Tf is below the onset of devitrification. Following each scan of the aged sample, the sample was cooled at 1000 K/s from 190 to 35 °C and then heated at 600 K/s without aging; this provided an unaged scan. Table 1 summarizes the experiments performed and the aging temperatures. The Flash DSC chip sensors are preconditioned and calibrated following the manufacturer recommendation. The manufacturer calibration is done statistically for selected chips in a given batch, and thus for a given chip, there may be some small error in the temperature calibration. However, the error of manufacturer calibration is found to be 0.6 ± 0.7 K in our previous work.40 Hence, no additional calibration is employed except for the manufacturer calibration.

Tf1 (°C)

type of expt

Ta (°C)

104.5 104.5

down down up down down down up up up 2-step 2-step

−30 to 120 105.5 105.5 107.0 107.0 107.0 107.0 107.0 107.0 104.5 104.5

cooling rate dependence of the limiting fictive temperature Tf ′ (which is equivalent to Tg48) for cooling rates ranging from 1 to 1000 K/s. The fit 1 parameters are similar to those of our previous work39 but differ slightly since a broader range of data were fit. The fit 2 parameters are obtained by fitting the same data sets, but C1 and C2 were fixed using the values reported from a rheological study49 for a similar molecular weight polystyrene; hence, only two fitting parameters, τR, and β, used. The two sets of fitting parameters are shown in Table 2 along with those of Table 2. Sets of Parameters for the New and Old Modified TNM Models fit 1 fit 2 old model39 a

C1

C2 (K)

log τR/s

β

12.4 13.4a 12.4

40.7 39.7a 42.3

1.51 1.63 1.41

0.600 0.595 0.550

From ref 49.

previous modeling work39 for comparison. The ability of these sets of parameters to describe the full range of enthalpy recovery experiments covering Kovacs’ three signatures of structural recovery is examined.

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RESULTS The applicability of the new model to the experimental Flash DSC results is demonstrated starting from the cooling rate dependence of the limiting fictive temperature Tf ′. As expected, Tf ′ decreases nonlinearly with decreasing cooling rate in Figure 4. The thick solid line represents the model prediction with fit

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MODELING METHODOLOGY As already noted, the modified TNM models based on eqs 9 and 10 predict the same response for cooling rate dependence of Tg and the intrinsic isotherms (or down-jumps). For this reason, these sets of experimental data were used to determine model parameters. The fit 1 parameters are obtained based on fitting the intrinsic isotherms for Ta ranging from 40.5 to 115.5 °C and the

Figure 4. Data and model predictions for the cooling rate dependence of Tg. Data for the bulk, 160 nm, and 1 μm are from previous works.39,47,50 The thick and thin solid lines are the new modified TNM model prediction with fit 1 and fit 2 parameters, respectively. E


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Macromolecules 1 parameters, whereas the thin solid line is with fit 2 parameters; this convention is used throughout. Both model predictions, with fit 1 and fit 2 parameters, capture the observed nonlinear dependence of Tf ′ on cooling rate over the entire 6 decades for which data are available. For the structural recovery intrinsic isotherms, we include here the fits of previously published down-jump data39−41 along with new data at lower aging temperatures. The evolution of the fictive temperature during isothermal aging is shown in Figure 5a for

1000 K/s cooling rate, to an equilibrium value of Tf = Ta except at the highest aging temperature of 115.5 °C. At that temperature, additional relaxation occurs during cooling such that the longtime value of Tf is several degrees lower than Ta.39 In addition, for the highest aging temperatures, the evolution proceeds immediately, but for lower aging temperatures, an initial plateau is observed where no aging occurs, the length of which increases as temperature decreases. Finally, the time required to reach equilibrium increases as Ta decreases, analogous to the dependence of Tg and Tf ′ on cooling rate. In all cases, the evolution of Tf, in the range of ta explored, is smooth and without intermediate plateaus; hence, the data can be described with a model implementing only one mechanism of relaxation. Our data are consistent with that of Cangialosi et al.,51 who showed smooth behavior at times shorter than an intermediate plateau, the onset of which occurred at 104.8 for a polystyrene with Mw = 85K and Ta = 90 °C and which would be expected at longer times for higher molecular weights. We are unable to explore such long times due to irreproducibility of the baseline if the instrument and cooling system are left on for more than a few days. We do note that multiple mechanisms could be implemented in the model, if needed, but that is not the aim of the present work. The models (solid lines) quantitatively describe the intrinsic isotherms at the higher aging temperatures, but the length of initial plateau is underestimated below Ta = 40 °C. This discrepancy is expected because the new modified TNM model is based on an odd symmetric function of the WLF equation with respect to the origin of Tg, resulting in a hyperbolic temperature dependence below Tg; as a result, an unrealistic plateauing of the temperature dependence of the relaxation time occurs approximately 80 K below Tg, i.e., at low enough Ta. Thus, the model does not well predict aging at temperatures below 40 °C. In order to examine the low aging temperature behavior in more detail, representative Flash DSC scans are shown in Figure

Figure 5. (a) Data and model predictions for the intrinsic isotherms for aging temperatures ranging from Ta = −30.5 to 115.5 °C. Data for Ta = 50.5 to 115.5 °C are from previous works;39,40 data at lower temperatures are from this work. The thick and thin solid lines are the new modified TNM model prediction with fit 1 and fit 2 parameters, respectively. The horizontal dashed lines are added for clarity. (b) Data and model predictions for the intrinsic isotherms for higher aging temperatures ranging from Ta = 100.5 to 115.5 °C. Data are from previous works.39,41 The thick and thin solid lines are the new modified TNM model prediction with fit 1 and fit 2 parameters, respectively.

Figure 6. Representative Flash DSC heating scans as a function of aging time at Ta = −30.5, 0.5, and 30.5 °C after cooling at 1000 K/s.

6 as a function of aging time at aging temperatures of −30.5, 0.5, and 30.5 °C after cooling at 1000 K/s. At the lowest two temperatures, good superposition of aged and unaged scans is observed, indicating that no significant aging takes place. At Ta = 30.5 °C, on the other hand, the development of enthalpy overshoots with aging time is clearly observed. For the Tf calculation, the Moynihan equation45 is integrated over temperature ranges of 80−160 °C for all Ta as this range shows good superposition of glassy and liquid lines, which is critical to obtain an accurate Tf.39 In fact, for Ta = 0.5 °C, Tf increases slightly (less than 2 K) as a function of ta in Figure 5a due to poorer superposition of the raw data, as shown in Figure 6; the reason for the poorer superposition is not understood, but

aging temperatures ranging from 115 to −30 °C. Data are plotted in terms of the departure from equilibrium, Tf − Ta. A close-up of the higher temperature data, with more data at other aging temperatures added, is provided in Figure 5b. During aging, Tf decreases from its initial value of 117 °C, which is a result of the F


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Figure 7. Data and model predictions for the for the intrinsic isotherms after cooling at 1000, 100, 3, and 0.3 K/s at Ta = 102.5 (a), 105.5 (b), 110.5 (c), 115.5 (d), and 120.5 °C (e). Data are from previous work.39 The thick and thin solid lines are the new modified TNM model prediction with fit 1 and fit 2 parameters, respectively.

the standard deviation of ±1.1 is not that much larger than that for unaged samples (±0.6 K). Thus, we suggest that the slight increases in Tf at low Ta in Figure 5a are due to experimental error resulting from poor superposition of glassy and liquid lines. In addition to changing the aging temperature using a fixed cooling rate of 1000 K/s, as shown in Figure 5, we also change the size of the down jumps by varying the cooling rate, which changes Tf ′. Results are shown in Figures 7a to 7e for aging temperatures ranging from 102.5 to 120 °C. At a given Ta, the fictive temperature evolves toward equilibrium from an initial value of Tf ′, which increases as cooling rate increases. Three behaviors are observed: when Ta is far below Tf ′, Tf evolves to Tf = Ta, as shown in Figure 7a for aging at the highest two cooling rates. When Ta is far above Tf ′, no evolution (aging) occurs, as shown in Figure 7e for Ta = 120.5 °C for the lowest three cooling rates. Intermediately, when Ta is within the glass transition region, Tf at the completion of aging is lower than Ta

due to relaxation that occurs during cooling, after aging. This behaviors is shown at Ta = 102.5 °C for q = 0.3 K/s, at Ta = 105.5 °C for q = 3 K/s, Ta = 110.5 °C for q = 100 K/s, and Ta = 115.5 °C for q = 1000 K/s. The models capture these three types of aging behavior, which clearly depend on the relative location of Ta and Tf ′. The enthalpic asymmetry of approach experiments is shown in Figures 8a and 8b for aging temperatures of 105.5 and 107.0 °C, respectively. Temperature-down and -up jumps of 4 K are shown in Figure 8a, whereas jumps of 4, 7, and 10 K are shown in Figure 8b. As well described in Kovacs’ original work,3 the evolution of the structure (as depicted here by Tf − Ta) for the down and up jumps are not mirror images of each other; i.e., there is an asymmetry of approach. This asymmetric response demonstrates that the kinetics of the structural recovery depend on both temperature and structure; in other words, structural recovery is nonlinear. The autoretarded shape for the temperature down jump arises due to the continuous contraction of volume during G


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To complete the test of our model, data for the memory experiment are plotted in Figure 9 along with the fit of the new

Figure 9. Data and model predictions for the memory effect at 104.5 °C after a two-step history. The blue square data have the first step of T1 = 100.5 °C and t1 = 60 s, and the red circle data have the first step of T1 = 80.5 °C and t1 = 780 s. Data are from previous work.41 The thick and thin solid lines are the new modified TNM model prediction with fit 1 and fit 2 parameters, respectively.

modified TNM model, again using the same values of model parameters as for the other fits shown. As previously mentioned, the memory experiment is a two-step temperature history, in which aging at the first temperature T1 is accomplished such that the fictive temperature after the second step and at the start of aging at Ta is approximately equal to Ta. If the structural recovery process depended only on the structure of the glass, no evolution of the structure would be expected since Tf = Ta, a condition required for but not sufficient for equilibrium. As the name indicates, a nonmonotonic response is observed, in which the departure from equilibrium (Tf − Ta) increases and then decreases back to zero. This response indicates that the glass must have a distribution of relaxation times (i.e., the behavior is nonexponential) and that the evolution of the glass depends on the thermal history and not only on the instantaneous structure. Our new modified TNM model represented by solid lines qualitatively predicts the response of memory effects; the departure from equilibrium is initially approximately zero and increases until a maximum and then returns to zero, consistent with the data. In addition, the model correctly predicts that the time of the maximum departure from equilibrium increases as the temperature of the first step (T1) increases and that the magnitude of maximum departure decreases with increasing T1. Although the new model is not quantitative in describing the memory effect, qualitative agreement is obtained with the same set of of model parameters as used to describe the previous intrinsic isotherm and asymmetry of approach experiments.

Figure 8. (a) Data and model predictions for the asymmetry of approach at Ta = 105.5 °C with a 4 K jump size. Open symbols are data from previous work;41 solid symbols are new data, with each point showing the average and standard deviation of three experiments. The thick and thin solid lines are the new modified TNM model prediction with fit 1 and fit 2 parameters, respectively. The thick and thin dashed lines are the old modified TNM model prediction39 with fit 1 and fit 2 parameters, respectively. (b) Data and model predictions for the asymmetry of approach at Ta = 107.0 °C with 4, 7, and 10 K jump sizes. Each data point shows the average and standard deviation of three experiments. The thick and thin solid lines are the new modified TNM model prediction with fit 1 and fit 2 parameters, respectively. The thick and thin dashed lines are the old modified TNM model prediction39 with fit 1 and fit 2 parameters, respectively.

the down jump, resulting in continuously decreasing mobility, whereas the autoacceleration observed for the up jump is due to the volume dilation that occurs during the up jump, resulting in an increase in mobility as structural recovery proceeds. Since the longest relaxation time for down jumps is that at equilibrium, the times required to obtain equilibrium are approximately the same for down jumps of different sizes, whereas for the up jump, the longest relaxation time is that at the start of the jump, which depends on the jump size (or equivalently on the value of T0 for a given Ta); hence, the times required to reach equilibrium for up jumps increase as the jump size increases, as shown in Figure 8b. Comparing the two models, the previous model does not include sufficient nonlinearity and, thus, does not well describe the up jump data. However, the new modified TNM model (eq 10) can more quantitatively capture the asymmetry of approach observed for the up jumps but may include too much nonlinearity. The implication is that a slightly better description may be observed by τ decreasing slightly as temperature increases in Figure 2a rather than by assuming no change in τ during the upjump. Importantly, the assumption that during an up-jump relaxation time should be confined to the shaded area in Figure 2a appears to be valid.

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CONCLUSIONS The enthalpy recovery of single polystyrene thin films is investigated under the diverse temperature histories both by experiments using Flash DSC and modeling using a new modified TNM model. Experimentally, we demonstrate all three signatures of structural recovery, including the intrinsic isotherms after down jump to Ta ranging from −30 to 120 °C, asymmetry of approach after down and up jumps to Ta of 105.5 and 107.5 °C for three different jump sizes of 4, 7, 10 K, and memory effects for two different partial aging conditions before an up jump to Ta of 104.5 °C. The signatures enthalpy recovery are consistent with those of volume recovery by Kovacs. A new H


Article

Macromolecules

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modified TNM model is able to reproduce quantitatively the experimental results for intrinsic isotherms and asymmetry of approach and qualitatively the memory effect with a single set of parameters. The model requires the fit of two parameters, in addition to the two WLF parameters which can be obtained from other experiments; these represent the equilibrium relaxation time at a reference temperature and the broadness of the relaxation time distribution. In the framework of the proposed model the relaxation time coincides with the classical WLF in the equilibrium condition, whereas in the glassy state it can be obtained from knowledge of the temperature-dependent equilibrium relaxation time without using any additional model parameters to describe the degree of nonlinearity.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (L.G.). ORCID

Luigi Grassia: 0000-0002-6796-9209 Sindee L. Simon: 0000-0001-7498-2826 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding from NSF DMR-1610614 is gratefully acknowledged by S.L.S.

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REFERENCES

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Complete Set of Enthalpy Recovery Data Using Flash DSC

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