A. Temperature simulation of the rolled pure W under EPT

The middle area (<3 mm) of the samples, which is very short compared with the total length of the narrow waist (20 mm), is the hottest zone during EPT. The temperature evolution in this zone can be modeled and simulated. For simplifying the calculation, an assumption is made: there is no significant thermal conduction between the concerned area and its adjacent areas. An equation of the energy conservation during EPT can be then built:

(1)$$<![CDATA[$$\Delta Q_J - \Delta Q_C - \Delta Q_R = \Delta Q_A ,$$

where ∆Q_J is the input energy in the form of Joule heat resulting from electropulsing; ∆Q_C and ∆Q_R are the convection and radiation between the heated sample and the ambient environment, respectively; and ∆Q_A is the heat absorbed by the W sample. According to Ref. Reference Zhou, Zhu and Xie14, we get:

(2)$$<![CDATA[$$\Delta Q_J = I_e ^2 \rho _e {l \over S}\Delta t,$$

(3)$$<![CDATA[$$\Delta Q_C = 2(w + d)h_a (T - T_0 )l\Delta t,$$

(4)$$<![CDATA[$$\Delta Q_R = 2(w + d){\rm{\varepsilon }}_0 {\rm{\sigma }}_b (T^4 - T_0 ^4 )l\Delta t,$$

(5)$$<![CDATA[$$\Delta Q_A = S{\rm{\rho }}_m C_p {{\partial T} \over {\partial t}}l\Delta t,$$

where I_e is the current RMS value of EPT, ρ_e the electrical resistivity, l the length of the middle area of the EPT sample, w the width of the sample, d the thickness of the sample, S the cross-sectional area of the sample, h_a the transfer coefficient between the sample and the ambient air, ε₀ the radiancy, σ_b the Stefan-Boltzmann constant, ρ_m the density, C_p the specific heat, T ₀ the ambient temperature, t the EPT time, and T the absolute temperature of the sample. According to Eqs. (1)–(5), an equation between the temperature T and the EPT time t could be established as:

(6)$$<![CDATA[$$S{\rm{\rho }}_m C_p {{\partial T} \over {\partial t}} - {{{\rm{\rho }}_e } \over S}I_e^2 + 2(w + d)h_a (T - T_0 ) +\, 2(w + d){\rm{\varepsilon }}_0 {\rm{\sigma }}_0 (T^4 - T_0^4 ) = 0,$$

which could be solved by 4th order Runge-Kutta algorithm.Reference Butcher¹⁵ The parameters used for calculation were w = 2 × 10⁻³ m, d = 1.5 × 10⁻³ m, ρ_m = 19.3 × 10³ kg/m³, S = 3 × 10⁻⁶ m², T ₀ = 298 K, I_e = 227.67 A, h_a = 127 W/(m²·K), ε₀ = 0.3,Reference Djilkibaev¹⁶ ρ_e = [9.9 + 0.01 × 2.9 × (T− 473)] × 10⁻⁸ Ω m,Reference Dekker and Lodder¹⁷ C_p = 135.165 − 6.494 × 10⁵T ⁻² + 9.078 × 10⁻³T + 23.029 × 10⁻¹⁰T ³ J/(kg·K).Reference Wang, Tang, Feng and Wu¹⁸ The calculated temperature curve is shown in Fig. 3. The calculated values are consistent with the measured values at low temperatures (see Table I). The validity of the calculated values for a higher temperature range is discussed in Sec. IV. C. According to the temperature and hardness curves in Fig. 3, the onset and finish temperature of recrystallization of the rolled pure W under EPT are about 2272 and 2510 K, respectively.

B. Grain refining effect of EPT on the rolled pure W

Figure 5 indicates that EPT can generate much finer recrystallized grains than CHT. Generally, recrystallization contains two stages: nucleation and nuclei growth. They are both related to the final size of recrystallized grains. Higher nucleation rate can result in much smaller grains.Reference Humphreys and Hatherly¹⁹ From the data of recrystallization fraction and mean grain size based on the EBSD images of EPT and CHT samples shown in Fig. 5, one can find that the grain size increases much slowly from partial recrystallization to full recrystallization under EPT. It indicates that EPT can significantly increase the nucleation rate compared with CHT. Nucleation process normally connects with the coalescence of subgrains, which can lead to the decrease of low-angle misorientations as shown in Fig. 6, and dislocation glide and climb play an important role in this process.Reference Humphreys and Hatherly¹⁹ Vacancy diffusion is considered to be a controlling factor in dislocation climb.Reference Yu and Mao²⁰ During EPT, the flux of vacancies associates with two effects: thermal and electromigration effects of EPT. They can be written asReference Xu, Tang and Jiang¹⁰:

(7)$$<![CDATA[$$j_{th} = {{D_0 C_0 {\rm{\tau }}_p Q_v {\rm{\rho }}_e t_0 N_l I_e^2 } \over {2(R_0 - r_0 )C_P {\rm{\rho }}_m S^2 RT^2 }}\exp \left( { - {{Q_d } \over {RT}}} \right),$$

(8)$$<![CDATA[$$j_{em} = {{2D_0 Z^* e{\rm{\rho }}_e {\rm{\tau }}_p fN_l J_m } \over {{\rm{\pi }}\kappa T}}\exp \left( { - {{Q_d } \over {RT}}} \right),$$

(9)$$<![CDATA[$$Q_d = Q_m + Q_v ,$$

where j_th and j_em are the flux of vacancies contributed by the thermal fluctuation and electromigration effects respectively, D ₀ the diffusion preexponential factor, C ₀ the vacancy migration preexponential factor, τ_p the current pulse duration, Q_v the activation energy for vacancy formation, Q_d the activation energy for vacancy diffusion, Q_m the activation energy for vacancy migration, t ₀ the unit time, N_l the atom density, R ₀ and r ₀ the distances of the boundaries conditions, R the gas constant, Z^* the effective valence, e the electron charge, f the frequency of the pulse, J_m the amplitude current density, and κ the Boltzmann constant.

For CHT, the flux of vacancies contributed by the normal temperature field, j_n, can be determined byReference Xu, Tang and Jiang¹⁰:

(10)$$<![CDATA[$$j_n = {{D_0 C_0 N_l } \over {R_0 - r_0 }}\exp \left( { - {{Q_d + Q_m } \over {RT}}} \right).$$

To investigate the differences in magnitudes of vacancy fluxes between EPT and CHT, two ratios q ₁ and q ₂are introduced as:

(11)$$<![CDATA[$$q_1 (T_1 ,T_2 ) = {{j_{th} (T_1 )} \over {j_n (T_2 )}},$$

(12)$$<![CDATA[$$q_2 (T_1 ,T_2 ) = {{j_{em} (T_1 )} \over {j_n (T_2 )}},$$

where T ₁ and T ₂ are the temperatures at which recrystallization occurred under EPT and CHT, respectively. Combining Eqs. (7)–(10), the parameters used for Eqs. (11) and (12) are: Q_v = 6.82 × T + 3.32 × 10⁵ J/mol,Reference Satta, Willaime and de Gironcoli²¹Q_m = 31.2 × T + 1.14 × 10⁵ J/mol,Reference Satta, Willaime and de Gironcoli²¹t ₀ = 1 s, R ₀ = 6.04 × 10⁻⁶ m, r ₀ = 2.74 × 10⁻¹⁰ m, R = 8.314 J/(mol·K), Z^* = 10,Reference Nabarro²²C ₀ = 24.5,Reference Park, Huang, Siegel and Balluffi²³e = 1.6 × 10⁻¹⁹ C, κ = 1.38 × 10⁻²³ J/K, f = 700 HZ, τ_p = 70 μs, and J_m = 4.27 × 10⁸ A/m².

Based on the experimental results, T ₁ is in the range of 2272–2510 K, and the value of T ₂ is taken to be the conventional recrystallization temperature, 1603 K, for the sake of simplification. The calculated values of the two ratios are illustrated in Fig. 7. One can see that j_th and j_em are about 6 orders larger than j_n in magnitude, and additionally, j_th is much larger than j_em. It suggests that much larger vacancy flux can be produced under EPT due to the thermal and electromigration effects of electropulsing when compared with CHT, and between the thermal and electromigration processes within the electropulsing, the thermal effect is more dominant in this study. The huge vacancy flux under EPT leads to significant promotion of dislocation rearrangement and consequently enhancement of the nucleation rate, which results in finer recrystallization grains.

FIG. 7.

The ratios of the vacancy fluxes contributed by the thermal effect (q ₁) and electromigration effect (q ₂) of EPT (2272–2520 K) to the one of CHT (1603 K).

C. Accelerated recrystallization of the rolled pure W under EPT at high temperatures

The misorientation distributions of the EPT samples shown in Fig. 6 indicate that low-angle grain boundaries are gradually transformed to high-angle grain boundaries with increasing recrystallization fraction due to the rearrangement of dislocations, and they almost disappeared when full recrystallization was achieved. Grain boundary migration by means of self-diffusion plays a key role in recrystallization.Reference Humphreys and Hatherly¹⁹ Thus the flux of vacancies accumulated during the whole EPT process might be a potential parameter to evaluate the extent of recrystallization. Eqs. (7)–(9) imply that temperature is an important factor affecting the vacancy diffusion. Since the EPT samples were air-cooled after EPT, the samples would maintain relatively high temperatures for a while. This could also contribute to the fluence of vacancies. Taking into account the cooling process, a model for the whole EPT process is built as follows.

There was no input energy in cooling process, namely, ∆Q_J = 0, the second term on the left-hand side of Eq. (6) should be neglected. Consequently, temperature during the whole EPT process can be determined by:

(13)$$<![CDATA[$$\left\{ \matrix{ S{\rm{\rho }}_m C_p {{\partial T} \over {\partial t}} - {{\rho _e } \over S}I_e^2 + 2(w + d)h_a (T - T_0 ) + 2(w + d){\rm{\varepsilon }}_0 {\rm{\sigma }}_0 (T^4 - T_0^4 ) = 0,\,\,\,\,{\rm{ }}0 \,\,\le \,t \,\le\, t_1 , \hfill \cr S{\rm{\rho }}_m C_p {{\partial T} \over {\partial t}} + 2(w + d)h_a (T - T_0 ) + 2(w + d){\rm{\varepsilon }}_0 {\rm{\sigma }}_0 (T^4 - T_0^4 ) = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{ }}t_1 \,\lt \,t \,\le \,t_2 , \hfill \cr} \right.$$

where t ₁ is the EPT time; t ₂ is the time that the samples are almost cooled. According to Eq. (13), the temperature curves with different EPT time can be calculated as illustrated in Fig. 8. The fluence of vacancies under EPT, J ₁, can be calculated by:

(14)$$<![CDATA[$$J_1 = \int_0^{t_1 } {(\,j_{em} + j_{th} )dt} .$$

FIG. 8.

The calculated temperature evolution of heating and cooling processes during EPT with different EPT time. Peak temperatures of the four EPT processes are listed in the legend.

For the cooling process, only normal temperature field exists. Thus, the fluence of vacancies during cooling, J ₂, is calculated by:

(15)$$<![CDATA[$$J_2 = \int_{t_1 }^{t_2 } {j_n dt} .$$

The fluence of vacancy of the whole process J is,

(16)$$<![CDATA[$$J = J_1 + J_2 .$$

Taking t ₂ = 20 s, we calculate J ₁, J ₂, and J when t ₁ = 4.5, 5.4, 5.8, and 6.0 s, respectively, as listed in Table II. In the table, the vacancy fluences are shown as ratios to a reference: J/J_ref. The reference J_ref is set as the vacancy fluence for t ₁ = 6.0 s, which is the critical point for full recrystallization. J ₂ turns out to be much smaller than the corresponding J ₁. This means the air-cooling process following EPT made a negligible contribution to the recrystallization of the rolled pure W. When t ₁ is 4.5 s with peak temperature T_P of 1800 K, the ratio of J/J_ref is almost zero. It indicates that little vacancy fluence is produced and tiny microstructure changes occur in the W sample. This is consistent with the little hardness change noted (i.e., the recrystallization fraction f is ∼0) at 4.5 s as shown in Fig. 3. When t ₁ is 5.4 s, the peak temperature T_P rises to 2272 K with the ratio of J/J_ref = 16.8%, indicating some valid vacancy fluence is formed. When t ₁ slightly rises to 5.8 s with T_P of 2437 K, J/J_ref rapidly reaches 55%. The variation mode of the calculated values of J/J_ref are basically in accordance with that of the measured values of recrystallization fraction f based on the EBSD images Figs. 5(b)–5(d). This implies that, on the one hand, the vacancy diffusion model is proper for the evaluation of microstructure evolution during the EPT-induced recrystallization in our study, and on the other hand, the calculated temperature values which are important for the vacancy diffusion model are reasonable. Moreover, the data listed in Table II suggests the W sample is very sluggish during the short period at the temperatures below 1800 K. When the temperature rises to a much higher level, substantial vacancies are activated and the vacancy fluence rises sharply with further increasing temperature.

TABLE II.

The calculated values of peak temperature T_p, vacancy fluence and recrystallization fraction f for different EPT time t ₁.

Note:

J ₁ is the vacancy fluence under EPT; J ₂ is the vacancy fluence during cooling; J is the vacancy fluence for the whole process. J ₁, J ₂, and J are shown as ratios to a reference J_ref which is the total vacancy fluence for EPT at t ₁= 6.0 s when full recrystallization is achieved.

It is very clear that both temperature and time noticeably contribute to the thermally activated recrystallization behavior under EPT. The two factors are linked up by the heating rate. In this study, high heating rate was generated by the huge Joule thermal effect of EPT (see Fig. 3). Thus it took a very short time to go through the low-temperature range, where W was very inertial, and approach to quite high temperatures. With the aid of the great thermal and electromigration effects of EPT in the high-temperature range, EPT induced full recrystallization within only several seconds. The view of higher heating rate resulting in higher recrystallization temperature has been confirmed by many investigations.Reference Denissen, Denissen, Liebe and van Rijswick^24, Reference Muljono, Ferry and Dunne²⁵ Our pervious studyReference Yuan, Greuner, Böswirth, Krieger, Luo, Xu, Fu, Li and Liu²⁶ also showed that a very high heating rate (∼2000 K/s) generated by high heat flux loads could significantly elevate the recrystallization temperature of W. However, our results in this article is in contrast with some previous studiesReference Zhou, Xiao and Guo^2, Reference Xu, Tang, Ding, Tian and Tian^6, Reference Jiang, Tang, Shekc and Liu⁷ in which EPT decreased the recrystallization temperature of some Mg alloys and Cu alloys. From the Eqs. (7) and (8), we can see that the thermal and electromigration effects of EPT are closely correlated to the properties of materials, especially to the activation energy for vacancy diffusion Q_d which is in the numerator of exponent. Q_d of W is about five times higher than that of Mg and about three times higher than Cu.Reference Satta, Willaime and de Gironcoli^21, Reference Janot, Mallejac and George^27, Reference Foiles, Baskes and Daw²⁸ It is possible to assume that the active temperature range of W is much higher than that of Mg and Cu under the rapid heating process of EPT. Additionally, probably due to the different relative contributions of the thermal and electromigration effects of EPT on the recrystallization behavior of different materials (thermal effect is more dominant for W in this work), opposite phenomena are observed. The exact relationship between the thermal and electromigration effects of EPT on the recrystallization will be clarified in our future work.