1.Guangdong Provincial Key Laboratory of Functional Soft Condensed Matter, Guangdong Provincial Research Center on Smart Materials and Energy Conversion Devices, School of Materials and Energy, Guangdong University of Technology, Guangzhou 510006, China 2.Dongguan South China Design Innovation Institute, Dongguan 523808, China
Abstract:As the electric field can affect the polarization and dielectric constant, the phenomenological coefficient a0 is an implicit function of electric field. The phenomenological coefficient a0 is determined by the polarization and the reciprocal of permittivity, and a nonlinear function of electric field in the ferroelectric phase regime. In the paraelectric phase regime, however, a0 is merely subjected to the reciprocal of permittivity, and also a nonlinear function of electric field. In this paper, we investigate the electric field dependence of phenomenological coefficient in ferroelectric copolymers, terpolymers and Ba0.85Ca0.05Sr0.1TiO3 ceramics. It is indicated that the phenomenological coefficient increases with the increasing electric field, the maximum value is obtained to be about 2 times the original value. Moreover, the electrocaloric strength is used to measure the magnitude of electrocaloric effect of electrocaloric materials in an external electric field. It can be used to find out novel and efficient electrocaloric materials through studying the electrocaloric strength. Based on the thermodynamic theory, the analytical expression of electrocaloric strength is deduced. It is found that the phenomenological coefficient, phase transition, specific heat capacity, and permittivity versus temperature peak value at the phase transition temperature have a clear influence on the electrocaloric strength. The expression can be applied to 1st order, 2nd order phase transition materials and relaxor ferroelectrics. Keywords:ferroelectrics/ phase transition/ electrocaloric effect/ electrocaloric strength
表1用于理论模拟的钛酸钡的参数[15] Table1.Parameters of BaTiO3 used for theoretical simulation[15].
图 1 钛酸钡晶体极化强度(a)和介电常数倒数(b)在外电场作用下与温度的关系 Figure1. (a) Polarization and (b) reciprocal of permittivity of BaTiO3 as a function of temperature at various external electric fields.
其中$ {a}_{0}' $是与温度和电场有关的参量. 当E = 0时, $ {a}_{0}'={a}_{0} $, 即方程(4). 图2是外加电场时由介电常数倒数与温度关系的线性拟合得到的$ {a}_{0}' $与温度和电场的关系. 很明显, 在电场作用下, 参数$ {a}_{0}' $在相变温度附近达到最大值. 此外, 注意到在适当的电场下可以得到最大的$ {a}_{0}' $, 意味着在该电场下可以得到较大的电卡效应. 图 2 从图1(a)得到的$ {a}_{0}' $与温度和电场的关系 Figure2.$ {a}_{0}' $ derived from Fig. 1(a) as a function of electric field and temperature.
3.实验结果和讨论对聚(偏氟乙烯-三氟乙烯) 55 mol%/45 mol%共聚物的介电常数与温度和电场关系的实验结果进行了分析[16]. 图3(a)给出了聚(偏氟乙烯-三氟乙烯) 55 mol%/45 mol%共聚物在不同电场下的介电常数与温度的关系. 升温降温测试和剩余极化强度与温度的关系清晰地表明, 发生在聚(偏氟乙烯-三氟乙烯) 55 mol%/45 mol%共聚物内的铁电-顺电相变是二级相变. 通过介电常数温度关系和极化强度的温度关系可以得到唯象系数a0, b以及$ {a}_{0}' $[16]. 图3(b)为介电常数倒数与温度和电场的函数关系. 根据方程(9)可以得到唯象系数$ {a}_{0}' $与电场的关系(图3(c)). 在图3(c)中, 外电场为0时, $ {a}_{0}' $ = 2.43 × $ {10}^{7} \mathrm{V}·\mathrm{m}{·\mathrm{C}}^{-1}{·\mathrm{K}}^{-1} $. 而当外电场E = 10 MV/m时, $ {a}_{0}' $ = $ 3.27\times {10}^{7} \mathrm{V}·\mathrm{m}·{\mathrm{C}}^{-1}{·\mathrm{K}}^{-1} $, 增加了35%. 但是从E = 20 MV/m开始, 唯象系数$ {a}_{0}' $开始减小, 这种现象与下面看到的现象略有不同, 即后面两种材料没有这样明显的先升后降的现象. 这一现象产生的原因在于聚(偏氟乙烯-三氟乙烯) 55 mol%/45 mol%共聚物的相变温度约为70 °C, 其软化温度约为90—100 °C[17]. 这个温度正好是利用介电常数倒数与温度关系的线性拟合得到唯象系数的温度. 由于共聚物的软化, 高电场产生的麦克斯韦应力[18]能对聚合物的形状产生改变, 从而影响其介电性能, 即使得其介电常数升高, 从而使得介电常数倒数与温度曲线的斜率减小, 导致唯象系数$ {a}_{0}' $减小. 这种现象在聚(偏氟乙烯-三氟乙烯-氯氟乙烯)中较弱, 因为其相变温度在室温附近. 图 3 (a)聚(偏氟乙烯-三氟乙烯) 55 mol%/45 mol%共聚物介电常数与温度和电场的关系; (b)介电常数倒数与温度和电场的关系; (c)唯象系数$ {a}_{0}' $与电场的关系, 图中实线是为了观察方便 Figure3. (a) Permittivity of P(VDF-TrFE) 55 mol%/45 mol% as a function of electric field and temperature; (b) reciprocal of permittivity as a function of temperature and electric field; (c) $ {a}_{0}' $ as a function of electric field and linear fitting, and the solid line occurred is guided for observation.
此外, 也研究了聚(偏氟乙烯-三氟乙烯-氯氟乙烯)(P(VDF-TrFE-CFE)) 69.7 mol%/30.3 mol%/6.05 mol%三聚物的唯象系数与电场的函数关系[16]. 图4(a)为介电常数倒数与温度和电场的函数关系. 图4(b)是唯象系数$ {a}_{0}' $与电场E的关系. 从图4(b)可以看出: 当$E = 30$ MV/m时, ${a}_{0}' \approx 4.0\; \times {10}^{7}\;\mathrm{V}·\mathrm{m}·{\mathrm{C}}^{-1}·{\mathrm{K}}^{-1}$; 当$E = 60$ MV/m时, ${a}_{0}' \approx 7.5\; \times {10}^{7}\;\mathrm{V}·\mathrm{m}·{\mathrm{C}}^{-1}{·\mathrm{K}}^{-1}$. 相对于最低点, 增加了约90%. 图 4 (a)聚(偏氟乙烯-三氟乙烯-氯氟乙烯) (P(VDF-TrFE-CFE)) 69.7 mol%/30.3 mol%/6.05 mol%三聚物介电常数倒数与温度和电场的关系; (b)相应的唯象系数$ {a}_{0}' $与电场的关系 Figure4. (a) Reciprocal of permittivity of the (P(VDF-TrFE-CFE)) 69.7 mol%/30.3 mol%/6.05 mol% as a function of temperature and electric field; (b) the corresponding $ {a}_{0}' $ as a function of electric field.
图5(a)显示了Ba0.85Ca0.05Sr0.1TiO3陶瓷介电常数倒数与温度和电场的函数关系[19]. 图5(b)是唯象系数$ {a}_{0}' $与电场E的关系. 在E = 1 MV/m时, ${a}_{0}' \approx 5.0\times {10}^{5}\;\mathrm{V}·\mathrm{m}{·\mathrm{C}}^{-1}·{\mathrm{K}}^{-1}$. 在E = 2 MV/m时, ${a}_{0}' \approx 1.17\times {10}^{6}\;\mathrm{V}·\mathrm{m}{·{\mathrm{C}}^{-1}·\mathrm{K}}^{-1}$. $ {a}_{0}' $增加1倍以上. 图 5 (a) Ba0.85Ca0.05Sr0.1TiO3陶瓷介电常数倒数与温度和电场的函数关系; (b) 相应的唯象系数$ {a}_{0}' $与电场的关系 Figure5. (a) Reciprocal of permittivity of Ba0.85Ca0.05Sr0.1TiO3 ceramics as a function of temperature and electric field; (b) the corresponding $ {a}_{0}' $ as a function of electric field.
图6所示为聚(偏氟乙烯-三氟乙烯)(P(VDF-TrFE) 55 mol%/45 mol%共聚物的电卡强度与温度、电场和频率的关系. 该共聚物在209 MV/m的电场和80 °C的温度下, 具有12 K的电卡绝热温变(dT/dE = 0.5 × 10–7 K·V·m–1[20]). 从图6可知: 1) 计算的电卡强度比用间接法得到的电卡强度大1倍左右; 2) 电卡强度随温度的增加而减小, 原因在于该共聚物的相变温度约为70 °C, 介电常数-温度峰值也在70 °C附近. 由于极化强度随电场的增加而增加, 所以电卡强度也随之增加; 3) 从增加的幅值来看, 在不同温度下电卡强度与电场约为线性关系; 4) 聚合物的电卡强度比锡钛酸锶钡陶瓷的电卡强度(计算值约为 2 × 10–6 K·V·m–1 , 实验值约为0.52 × 10–6 K·V·m–1 [21])小一个数量级. 原因在于虽然聚合物表现出较大的电卡温变(约12 K), 但是其矫顽电场(约50 MV/m)远大于铁电陶瓷的矫顽电场(约2 MV/m). 可见铁电陶瓷比铁电聚合物具有更高的电卡强度. 到目前为止, 实验得到的最大的钛酸钡单晶的电卡强度为2.2 × 10–6 K·V·m–1[22]. 可见, 对于目前研究的电卡制冷材料, 在电卡强度方面还有较大的提升空间. 图 6 聚(偏氟乙烯-三氟乙烯) 55 mol%/45 mol%共聚物的电卡强度(dT/dE)与温度、电场和频率的关系 (a) f = 0.1 kHz; (b) f = 1 kHz; (c) f = 10 kHz; (d) f = 100 kHz Figure6. Electrocaloric strength (dT/dE) of P(VDF-TrFE) 55 mol%/45 mol% copolymer as a function of temperature, electric field and frequency: (a) f = 0.1 kHz; (b) f = 1 kHz; (c) f = 10 kHz; (d) f = 100 kHz
图 1 钛酸钡晶体极化强度(a)和介电常数倒数(b)在外电场作用下与温度的关系
图 2 从图1(a)得到的
图 3 (a)聚(偏氟乙烯-三氟乙烯) 55 mol%/45 mol%共聚物介电常数与温度和电场的关系; (b)介电常数倒数与温度和电场的关系; (c)唯象系数
图 4 (a)聚(偏氟乙烯-三氟乙烯-氯氟乙烯) (P(VDF-TrFE-CFE)) 69.7 mol%/30.3 mol%/6.05 mol%三聚物介电常数倒数与温度和电场的关系; (b)相应的唯象系数
图 5 (a) Ba0.85Ca0.05Sr0.1TiO3陶瓷介电常数倒数与温度和电场的函数关系; (b) 相应的唯象系数
图 6 聚(偏氟乙烯-三氟乙烯) 55 mol%/45 mol%共聚物的电卡强度(dT/dE)与温度、电场和频率的关系 (a) f = 0.1 kHz; (b) f = 1 kHz; (c) f = 10 kHz; (d) f = 100 kHz