Enthalpy relaxation and annealing effect in polystyrene

The effects of thermal history on the enthalpy relaxation in polystyrene are studied by differential scanning calorimetry. The temperature dependence of the specific heat in the liquid and the glassy states, that of relaxation time, and the exponent of the Kohlrausch-Williams-Watts function are determined by measurements of the thermal response against sinusoidal temperature variation. A phenomenological model equation previously proposed to interpret the memory effect in the frozen state is applied to the enthalpy relaxation and the evolution of entropy under a given thermal history is calculated. The annealing below the glass transition temperature produces two effects on enthalpy relaxation: the decay of excess entropy with annealing time in the early stage of annealing and the increase in relaxation time due to physical aging in the later stage. The crossover of these effects is reflected in the variation of temperature of the maximum specific heat observed in the heating process after annealing and cooling.

Figure 1

Schematic diagram of the thermal history in the annealing measurement.

Figure 2

Schematic graph of entropy vs temperature. The solid, dotted, dashed, and dash-dotted lines represent the instantaneous entropy $S\left(t\right)$, the liquid entropy $S^{\mathrm{eq}}\left(T\right)$, the hypothetical glassy entropy $S^{\mathrm{g}}\left(T\right)$ defined by Eq. (6), and the modified glassy entropy $S^{\mathrm{g}\prime }\left(T\right)$ defined by Eq. (14), respectively. At a temperature $T_2$, $S^{\mathrm{g}}$ is equal to $S^{\mathrm{eq}}$.

Figure 3

Temperature dependence of the real (upper data) and imaginary (lower data) parts of the specific heat. The symbols represent the experimental data: $◯$, $P=20$ s; $▵$, $P=30$ s; $\square$, $P=50$ s; $\diamond$, $P=100$ s; and $+$, $P=200$ s. The solid lines represent $C_{\mathrm{p}}^0$ (upper) and $C_{\mathrm{p}}^{\infty }$ (lower), respectively. The dotted, dashed, and dash-dotted lines represent $C_{\mathrm{p}}^{\prime \prime }$ calculated for $P=$ 20, 50, and 100 s, respectively. The closed circle and triangles are the data from Refs. [28, 29] by DSC, respectively.

Figure 4

Temperature dependence of relaxation time. Symbols represent the experimental results: $◯$, the results in the present study; $\square$, TMDSC data [30]; $\diamond$, TMDSC data [31]; $+$, TMDSC data [32]; and $▵$, 3$\omega$ method [30]. The solid and dotted lines represent Eq. (4) with ${\tau }_{\infty }=1.20\times {10}^{-8}$ s, $A=333$ J g${}^{-1}$, and $T_2=336$ K and the dotted line represents Eq. (4) with ${\tau }_{\infty }^{\left(3\right)}=4.35\times {10}^{-9}$ s, $A^{\left(3\right)}=550$ J g${}^{-1}$, and $T_2^{\left(3\right)}=329$ K, respectively.

Figure 5

(a) Real and (b) imaginary parts of the normalized specific heat: $◯$, $P=20$ s; $▵$, $P=30$ s; $\square$, $P=50$ s; $\diamond$, $P=100$ s; and $+$, $P=200$ s. The solid curves represent the specific heats calculated from the Fourier transform of $1-\varphi (t/{\tau }_{\mathrm{KWW}})$.

Figure 6

Specific heat vs temperature in the heating process after cooling at a constant rate. (a) Results of the cooling experiment and calculation 3 with $\Delta {\chi }_{\mathrm{F}}=-6.17\times {10}^{-4}$ J g${}^{-1}$ K${}^{-2}$, ${\sigma }_{\mathrm{F}}=23$ K, and $T_{\mathrm{F}}=371$ K. The specific heat in the cooling process for the cooling rate of 9.5 K/min are shown by circles [light green (light gray), experimental results]. (b) Calculations 1 [cyan (light gray), thick] and 2 [blue (dark gray), thin]. The symbols represent the experimental data: $▽$, ${\dot{T}}_{\mathrm{c}}=0.1$; $▵$, ${\dot{T}}_{\mathrm{c}}=0.3$; $\diamond$, ${\dot{T}}_{\mathrm{c}}=1$; $\square$, ${\dot{T}}_{\mathrm{c}}=3$; $+$, ${\dot{T}}_{\mathrm{c}}=10$; and $◯$, ${\dot{T}}_{\mathrm{c}}=30$ K/min. The short-dashed, dash-dotted, dotted, dash–double-dotted, solid, and dashed lines are the data for ${\dot{T}}_{\mathrm{c}}=0.1$, 0.3, 1, 3, 10, and 30 K/min, respectively. The results of calculation 2 for ${\dot{T}}_{\mathrm{c}}=0.3$ and 0.1 K/min are not shown in the figure because their $T_{\max }$ are beyond the range of this figure. The numbers in the figure represent the cooling rate before heating.

Figure 7

(a) Cooling rate dependence of $\delta S\left(T_{\mathrm{low}}\right)$. The symbols represent the results of cooling experiments and the curves the calculated ones. The top, middle, and bottom lines represent the results from calculations 1, 2, and 3, respectively. The line colors and thickness are the same as in Fig. 6. (b) Annealing temperature dependence of $\delta S\left(T_{\mathrm{low}}\right)$ for $t_{\mathrm{a}}$ from 1 to 10${}^4$ min. The left axis $T_{\mathrm{f}}^{\prime }$ is approximately proportional to $\delta S\left(T_{\mathrm{low}}\right)$. The symbols represent the experimental results: $▵$, $t_{\mathrm{a}}=1$; $▾$, $t_{\mathrm{a}}=10$; $\square$, $t_{\mathrm{a}}={10}^2$; $⧫$, $t_{\mathrm{a}}={10}^3$; and $◯$, $t_{\mathrm{a}}={10}^4$ min. The dash–double-dotted, dash-dotted, dotted, solid, and dashed lines are the results from calculation 3 for $t_{\mathrm{a}}=1$, 10, 10${}^2$, 10${}^3$, and 10${}^4$ min, respectively.

Figure 8

Specific heat on heating after annealing at $T_{\mathrm{a}}=95.8{}^{\circ }$C for $t_{\mathrm{a}}$ in the range from 1 to 10${}^3$ min. (a) Results of the annealing experiments and those of calculation 3. (b) Calculations 1 and 2. The symbols show the experimental results: $▵$, $t_{\mathrm{a}}=1$; $▽$, $t_{\mathrm{a}}=10$; $\square$, $t_{\mathrm{a}}={10}^2$; and $\diamond$, $t_{\mathrm{a}}={10}^3$. The curves represent the results of calculations 1 [cyan (light gray), thick], 2 [blue (dark gray), thin] and 3. The line types are the same as in Fig. 7b. The numbers in the figure represent the annealing time.

Figure 9

Specific heat on heating after annealing at $T_{\mathrm{a}}=86.1{}^{\circ }$C for $t_{\mathrm{a}}$ in the range from 1 to 10${}^4$ min. (a) Results of the annealing experiments and those of calculation 3. (b) Results of calculations 1 and 2. The symbols show the experimental results: $▵$, $t_{\mathrm{a}}=1$; $▽$, $t_{\mathrm{a}}=10$; $\square$, $t_{\mathrm{a}}={10}^2$; $\diamond$, $t_{\mathrm{a}}={10}^3$; and $◯$, ${10}^4$ min. The curves represent the results of calculations 1 [cyan (light gray), thick], 2 [blue (dark gray), thin] and 3. The line types are the same as in Fig. 7b. The results from calculation 1 are almost identical to each other within $1\le t_{\mathrm{a}}\le {10}^4$ min. The maximum of $C_{\mathrm{p}}$ from calculation 2 for $t_{\mathrm{a}}={10}^4$ min is not included in this figure. The numbers in the figure represent the annealing time.

Figure 10

Annealing time dependence of (a) $C_{\mathrm{p}\max }$ and (b) $T_{\max }$ at $T_{\mathrm{a}}=61.3{}^{\circ }$C ($\diamond$), 76.2 ${}^{\circ }$C ($▽$), 86.1 ${}^{\circ }$C ($▵$), 95.8 ${}^{\circ }$C ($◯$), and 110.6 ${}^{\circ }$C ($\square$). The arrows indicate $t_{\mathrm{a}\min }\left(T_{\mathrm{a}}\right)$. The curves represent the results from calculation 3. The solid, dashed, dash–double-dotted, dash-dotted, and dotted lines are $T_{\mathrm{a}}=61.3$ ${}^{\circ }$C, 76.2 ${}^{\circ }$C, 86.1 ${}^{\circ }$C, 95.8 ${}^{\circ }$C, and 110.6 ${}^{\circ }$C, respectively. Also shown is the annealing temperature dependence of (c) $C_{\mathrm{p}\max }$ and (d) $T_{\max }$. The symbols and the line types are the same as in Fig. 7b.

Figure 11

Variation with annealing time of (a) $\delta S_{\mathrm{ba}}\left(t\right)$ and (b) $\delta S_{\mathrm{aa}}\left(t\right)$ (thick) and $\delta S\left(t\right)$ from calculation 1 on heating at $T_{\mathrm{a}}=95.8{}^{\circ }$C and (c) $\delta S_{\mathrm{ba}}\left(t\right)$ and (d) $\delta S_{\mathrm{aa}}\left(t\right)$ (thick) and $\delta S\left(t\right)$ from calculation 2. The line types are the same as in Fig. 7b.

Figure 12

Change in (a) $\delta S\left(t\right)$ and (b) $\tau \left(t\right)$ from calculation 3 during annealing. The solid, dashed, dotted, dash–double-dotted, dash-dotted, and short-dashed curves are $T_{\mathrm{a}}=65.3$ ${}^{\circ }$C, 76.2 ${}^{\circ }$C, 86.1 ${}^{\circ }$C, 90.9 ${}^{\circ }$C, 95.8 ${}^{\circ }$C, and 100.8 ${}^{\circ }$C, respectively. The solid arrows show the experimental $t_{\mathrm{a}\min }$ and the dashed arrows the calculated $t_{\mathrm{a}\min }$. The numbers in the figure represent the annealing temperature.

Figure 13

Temperature dependence of (a) $\delta S_{\mathrm{ba}}\left(t\right)$ (upper) and $\delta S_{\mathrm{aa}}\left(t\right)$ (lower), (b) $\delta S\left(t\right)$, and (c) $\tau \left(t\right)$ on heating at $T_{\mathrm{a}}=86.1{}^{\circ }$C for $t_{\mathrm{a}}$ in the range from 1 to 10${}^4$ min. (b) Experimental $\delta S\left(t\right)$: $▽$, $t_{\mathrm{a}}=10$; $\square$, $t_{\mathrm{a}}={10}^2$; $\diamond$, $t_{\mathrm{a}}={10}^3$; and $◯$, $t_{\mathrm{a}}={10}^4$ min. The curves represent the results from calculation 3. The calculated $\delta S\left(t\right)$ and $\tau \left(t\right)$ for $t_{\mathrm{a}}=1$ min are not shown in the figures. The line types are the same as in Fig. 7b. The dashed line in (c) represents the temperature dependence of the relaxation time in equilibrium ${\tau }^{\mathrm{eq}}$. The left pointing arrows in (b) and (c) show the cooling process before heating in the annealing experiments. The numbers in the figure represent the annealing time.