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The valence electron structures and thermal and electric properties of Na||Sb-Pb-Sn liquid metal battery are systematically studies with solid and molecular empirical electron theory (EET). The theoretical studies show that the thermal and electric properties are strongly related to the valence electron structure of electrode. The cathodic alloys Na1–xIAx (IA = K, Rb, Cs) are designed by doping IA group alkali metals (K, Rb, Cs) into Na electrode since the melting points of IA group metals (K, Rb, Cs) are all lower than that of sodium. The theoretical bond lengths and cohesive energy of cathodic alloys Na1–xIAx match the experimental ones well. The theoretical studies show the decreasing tendency of melting point, cohesive energy and electric potential with increasing doping content x in Na1–xIAx alloys, which is due to the modulation of valence electron structure of IA group dopants. According to the analyses of valence structures, the number of lattice electrons decreases with the increasing of the doping content x for the cathodic alloy and causes the melting point, electric potential and cohesive energy to decline. It reveals that the IA group dopant modulates the valence electron structure of cathodic alloy, and induces the electron transformation from lattice electron to covalent electron in s orbital. The anode products such as NaSb3, NaSn, Na15Sn4 and NaPb are formed by transporting Na ions into the anode alloy Sb-Sn-Pb. The calculated bond-lengths and melting points fit the observed ones well for these anode products. Owing to their complex structures with various atomic occupations in unit cell, the thermal property or electric property is not only relative to lattice electron, but also depends on the covalent electron. The sublattice plays an important role in the forming of the four anode products. The lattice electrons are supplied by Na at 4f sites in Na3Sb, Na at 16e and Sn at 32g sites in NaSn, Sn at 16c and Na at 48e sites in Na15Sn4, and Na at 16f and Pb at 32g sites in NaPb, respectively. The open-gate voltage is closely related to the lattice electrons and inversely proportional to the average number of lattice electrons per atom. The open-gate voltage of NaSb3 is the largest among the anode products, however, its averaged number of lattice electron per atom is the least. Since the lattice electron number of NaSn is the largest among the anode products, the open-gate voltage of NaSn is the least. It implies that the lattice electron plays a very important role in Na||Sb-Pb-Sn liquid metal battery, which can modulate the valence electron structures and thermal and electric properties. -
Keywords:
- solid and molecular empirical electron theory /
- liquid metal battery /
- valence electron structure /
- open gate voltage
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图 1 阴极Na1–xIAx合金的n电子数与熔点的关联图
Figure 1. Correlations between various electron numbers and melting points of Na1–xIAx cathode alloys.
图 2 阴极Na1–xIAx合金的各种电子数与结合能的关联图
Figure 2. Correlations between various electrons and cohesive energy of Na1–xIAx cathode alloys.
图 3 阴极Na1–xIAx合金的各种电子数与电势的关联图
Figure 3. Correlations of various electrons and electric potentials for Na1–xIAx anode alloys.
表 1 Na1–xIAx合金键距计算
Table 1. Calculation of bond distances of Na1–xIAx alloy.
Na1–xIAx $ {I}_{\alpha } $ $ {D}_{\mathrm{uv}}\left({n}_{\alpha }\right) $/Å ${\bar{D}}_{\mathrm{uv}}\left({n}_{\alpha }\right) $/Å $ {n}_{\rm{A}}$ $ { I}_{\alpha } $ $ {D}_{\mathrm{uv}}\left({n}_{\alpha }\right) $/${ \text{Å} }$ ${\bar{D} }_{\mathrm{uv} }\left({n}_{\alpha }\right)/{ \text{Å} }$ $ {n}_{\alpha } $ |∆D|/${\text{Å} }$ Na 8 3.7296 3.7502 0.05160 6 4.3004 4.3210 0.00810 0.0206 Na0.99K0.01 8 3.7381 3.7538 0.05220 6 4.3103 4.3260 0.00820 0.0157 Na0.99Rb0.01 8 3.7430 3.7572 0.05220 6 4.3157 4.3300 0.00820 0.0143 Na0.99Cs0.01 8 3.7454 3.7628 0.05220 6 4.3187 4.3361 0.00810 0.0175 Na0.98K0.02 8 3.7465 3.7574 0.05290 6 4.3202 4.3310 0.00823 0.0108 Na0.98Rb0.02 8 3.7564 3.7643 0.05290 6 4.3310 4.3389 0.00820 0.0079 Na0.98Cs0.02 8 3.7611 3.7755 0.05290 6 4.3370 4.3513 0.00817 0.0143 Na0.97K0.03 8 3.7550 3.7610 0.05348 6 4.3301 4.3361 0.00829 0.0060 Na0.97Rb0.03 8 3.7697 3.7713 0.05351 6 4.3463 4.3479 0.00825 0.0016 Na0.97Cs0.03 8 3.7769 3.7881 0.05354 6 4.3552 4.3665 0.00821 0.0113 Na0.96K0.04 8 3.7635 3.7647 0.05411 6 4.3399 4.3411 0.00834 0.0012 Na0.96Rb0.04 8 3.7831 3.7785 0.05415 6 4.3616 4.3570 0.00829 0.0047 Na0.96Cs0.04 8 3.7926 3.8009 0.05419 6 4.3735 4.3818 0.00824 0.0082 Na0.95K0.05 8 3.7719 3.7684 0.05474 6 4.3498 4.3463 0.00840 0.0036 Na0.95Rb0.05 8 3.7965 3.7856 0.05479 6 4.3769 4.3661 0.00834 0.0109 Na0.95Cs0.05 8 3.8084 3.8136 0.05484 6 4.3918 4.3971 0.00827 0.0052 
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表 2 Na阴极合金的价电子结构
Table 2. Valence electron structures of cathode Na based alloy.
Na1–xIAx nc ns np nl R(1) Na 0.4614 0.4606 0.0008 0.5386 1.4181 Na0.99K0.01 0.4668 0.4660 0.0008 0.5332 1.4217 Na0.98K0.02 0.4722 0.4713 0.0008 0.5278 1.4254 Na0.97K0.03 0.4776 0.4767 0.0009 0.5224 1.4290 Na0.96K0.04 0.4830 0.4821 0.0009 0.5170 1.4327 Na0.95K0.05 0.4884 0.4875 0.0009 0.5116 1.4363 Na0.99Rb0.01 0.4668 0.4660 0.0008 0.5332 1.4235 Na0.98Rb0.02 0.4722 0.4713 0.0008 0.5278 1.4289 Na0.97Rb0.03 0.4776 0.4767 0.0009 0.5224 1.4343 Na0.96Rb0.04 0.4830 0.4821 0.0009 0.5170 1.4397 Na0.95Rb0.05 0.4884 0.4875 0.0009 0.5116 1.4451 Na0.99Cs0.01 0.4668 0.4660 0.0008 0.5332 1.4263 Na0.98Cs0.02 0.4722 0.4713 0.0008 0.5278 1.4345 Na0.97Cs0.03 0.4776 0.4767 0.0009 0.5224 1.4428 Na0.96Cs0.04 0.4830 0.4821 0.0009 0.5170 1.4510 Na0.95Cs0.05 0.4884 0.4875 0.0009 0.5116 1.4592
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表 3 阴极Na1–xIAx合金的熔点、结合能与电势
Table 3. Melting point, cohesive energy, and electric potentials of cathode Na1–xIAx alloy.
掺杂量x 原子 杂阶 掺杂 杂阶 $ \bar{T}_{\rm{m}} $/K $ {E}_{\mathrm{c}} $/(eV·atom–1) $ {\bar{E}}_{\mathrm{c}} $/(eV·atom–1) $\left| { {\Delta E}_{\mathrm{c} } }/{ {E}_{\mathrm{c} } }\right|/{\%}$ 电势/V 0 Na 3 — — 336.76 1.113 1.165 4.67 0.1482 0.01 Na 2 K 4 336.64 1.111 1.164 4.77 0.1481 0.01 Na 2 Rb 4 336.45 1.110 1.163 4.77 0.1480 0.01 Na 2 Cs 4 336.02 1.103 1.161 5.26 0.1478 0.02 Na 2 K 4 336.64 1.109 1.163 4.39 0.1481 0.02 Na 2 Rb 4 336.14 1.108 1.161 4.78 0.1478 0.02 Na 2 Cs 4 335.29 1.110 1.158 4.32 0.1475 0.03 Na 2 K 4 336.60 1.103 1.162 5.35 0.1480 0.03 Na 2 Rb 4 335.85 1.109 1.159 4.51 0.1476 0.03 Na 2 Cs 4 334.58 1.108 1.154 4.15 0.1471 0.04 Na 2 K 4 336.57 1.107 1.162 4.97 0.1479 0.04 Na 2 Rb 4 335.57 1.103 1.157 4.90 0.1474 0.04 Na 2 Cs 4 333.88 1.109 1.151 3.79 0.1467 0.05 Na 2 K 4 336.55 1.108 1.161 4.78 0.1478 0.05 Na 2 Rb 4 335.30 1.107 1.156 4.43 0.1472 0.05 Na 2 Cs 4 333.20 1.108 1.147 3.52 0.1463
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表 4 正极合金的晶体结构
Table 4. Crystal structures of anode alloys.
合金 空间群 a/$\text{Å}$ b/$\text{Å}$ c/$\text{Å}$ 原子 占位 x y z Sb 2c 0.3333 0.6666 0.2500 Na3Sb P63mmc (194) 5.355 5.355 9.496 Na1 2b 0 0 0.2500 Na2 4f 0.3333 0.6666 0.5830 NaSn I41/acd (142) 10.460 10.460 17.390 Sn 32g 0.0696 0.1260 0.9362 Na1 16f 0.6258 0.8758 0.1250 Na2 16e 0.8724 0 0.2500 Sn 16c 0.2083 0.2083 0.2083 Na15Sn4 I43d (220) 13.140 13.140 13.140 Na1 12a 0.3750 0 0.2500 Na2 48e 0.1270 0.1548 0.9670 Pb 32g 0.0696 0.1186 0.9383 NaPb I41/acd (142) 10.580 10.580 17.746 Na1 16e 0.2500 0.1250 0.5000 Na2 16f 0.1250 0.3750 0.6250
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表 5 阳极合金的键距
Table 5. Bond distances of the anode alloy.
合金 键序 成键原子 $ {I}_{\alpha } $ $ {D}_{\mathrm{uv}}\left({n}_{\alpha }\right) $/$\text{Å}$ $ {\bar{D}}_{\mathrm{uv}}\left({n}_{\alpha }\right)/$$\text{Å}$ $ {n}_{\alpha } $ |ΔD|/$\text{Å}$ Na3Sb 1 Sb-Na2 6 3.0975 3.0910 0.39055 0.0065 2 Sb-Na1 4 3.1685 3.1620 0.19416 0.0065 3 Na1-Na2 4 3.1780 3.1715 0.18030 0.0065 4 Na1-Na1 6 3.4769 3.4704 0.03738 0.0065 5 Sb-Na1 12 3.4813 3.4748 0.05846 0.0065 6 Na2-Na1 12 3.4813 3.4748 0.05630 0.0065 7 Na2-Na1 12 4.4310 4.4245 0.00147 0.0065 8 Na2-Na2 2 4.7575 4.7510 0.00064 0.0065 NaSn 1 Sn-Sn 2 2.9748 3.0201 0.42650 0.0453 2 Sn-Sn 4 2.9925 3.0378 0.39849 0.0453 3 Na1-Sn 4 3.3355 3.3808 0.07506 0.0453 4 Na1-Sn 4 3.3592 3.4045 0.06854 0.0453 5 Na2-Sn 4 3.3974 3.4427 0.13064 0.0453 6 Na2-Sn 4 3.4231 3.4684 0.11837 0.0453 7 Na1-Sn 4 3.4870 3.5323 0.04197 0.0453 8 Na2-Sn 2 3.5225 3.5678 0.08083 0.0453 9 Na2-Sn 4 3.5482 3.5935 0.07324 0.0453 10 Na1-Na2 4 3.6148 3.6601 0.03985 0.0453 11 Na1-Na2 4 3.6658 3.7111 0.03277 0.0453 12 Na1-Na1 1 3.7218 3.7671 0.01197 0.0453 13 Sn-Sn 2 3.7406 3.7859 0.02257 0.0453 14 Sn-Sn 2 4.3780 4.4233 0.00196 0.0453 15 Na1-Na2 4 4.4919 4.5372 0.00138 0.0453 16 Na1-Sn 4 4.6674 4.7127 0.00045 0.0453 17 Na2-Na2 1 4.7095 4.7548 0.00132 0.0453 Na15Sn4 1 Sn-Na2 24 3.2378 3.2854 0.20850 0.0476 2 Na2-Na2 24 3.2624 3.3100 0.19499 0.0476 3 Na1-Na2 24 3.3425 3.3901 0.11017 0.0476 4 Na2-Na2 12 3.3468 3.3944 0.14830 0.0476 5 Sn-Na2 24 3.4049 3.4525 0.12127 0.0476 6 Sn-Na2 24 3.4189 3.4665 0.11589 0.0476 7 Na1-Na2 24 3.5026 3.5502 0.06555 0.0476 8 Sn-Na1 24 3.5482 3.5958 0.05582 0.0476 9 Na2-Na2 24 3.8138 3.8614 0.03261 0.047 10 Na2-Na2 24 3.9794 4.0270 0.01906 0.0476 11 Na2-Na2 12 4.1712 4.2188 0.01023 0.0476 NaPb 1 Pb-Pb 2 3.1464 3.1452 0.33477 0.0013 2 Pb-Pb 4 3.1618 3.1606 0.31556 0.0013 3 Pb-Na2 4 3.3653 3.3641 0.19895 0.0013 4 Pb-Na1 4 3.3888 3.3876 0.08237 0.0013 5 Pb-Na2 4 3.4215 3.4203 0.16035 0.0013 6 Pb-Na2 4 3.4847 3.4835 0.12582 0.0013 7 Pb-Na1 4 3.4929 3.4917 0.05524 0.0013 8 Pb-Na1 4 3.5549 3.5537 0.04354 0.0013 9 Pb-Na1 4 3.6172 3.6160 0.03428 0.0013 10 Pb-Pb 2 3.6418 3.6406 0.05001 0.0013 11 Na1-Na2 8 3.6967 3.6955 0.03479 0.0013 12 Na2-Na2 1 3.7406 3.7394 0.06488 0.0013 13 Pb-Pb 2 4.4008 4.3996 0.00272 0.0013 14 Na1-Na2 4 4.5455 4.5443 0.00134 0.0013 15 Pb-Na2 4 4.7513 4.7501 0.00097 0.0013
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表 6 阳极产物的价电子结构
Table 6. Valence electron structures of anode products
合金 原子 杂阶 nc ns np nl R(1) Na3Sb Sb 2 3.0000 0.5694 2.4306 0 1.4279 Na1 4 1.0000 0.9982 0.0018 0 1.3070 Na2 2 0.4614 0.4606 0.0008 0.5386 1.4181 NaSn Sn 1 2.0000 0 2.0000 2.0000 1.3990 Na1 1 1.0000 0.9982 0.0018 0 1.3070 Na2 4 0 0 0 1.0000 1.5133 Na15Sn4 Sn 4 3.6638 0.8319 2.8319 0.3362 1.3990 Na1 4 1.0000 0.9982 0.0018 0 1.3070 Na2 3 0.5350 0.5340 0.0010 0.4650 1.4029 NaPb Pb 2 2.0962 0.0481 2.0481 1.9038 1.4300 Na1 4 1.0000 0.9982 0.0018 0 1.3070 Na2 1 0 0 0 1.0000 1.5133
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表 7 正极合金的熔点、结合能与电势
Table 7. Melting point, cohesive energy, and electric potentials of anode alloy.
合金 Tm/K [35] $ \bar{T}_{\rm{m}} $/K |${\Delta {T}_{\mathrm{m} } }/{ {T}_{\mathrm{m} } }$|/% 电势/V n β Ec/(eV·atom–1) Na3Sb 1129 1142.96 1.2 1.1520 4 0.60 1.766 NaSn 851 813.16 4.4 0.7343 5 0.60 2.103 Na15Sn4 681 746.16 9.6 0.9074 3 0.71 1.318 NaPb 645 630.68 2.2 0.8263 6 0.60 1.559
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表 8 电池的开路电压
Table 8. Open gate voltages of the battery.
Na1–xIAx 开路电压/V Na3Sb NaSn Na15Sn4 NaPb Na 1.0038 0.5861 0.7592 0.6781 Na0.09K0.01 1.0039 0.5862 0.7593 0.6782 Na0.98K0.02 1.0039 0.5862 0.7593 0.6782 Na0.97K0.03 1.0040 0.5863 0.7594 0.6783 Na0.96K0.04 1.0041 0.5864 0.7595 0.6784 Na0.95K0.05 1.0042 0.5865 0.7596 0.6785 Na0.99Rb0.01 1.0040 0.5863 0.7594 0.6783 Na0.98Rb0.02 1.0042 0.5865 0.7596 0.6785 Na0.97Rb0.03 1.0044 0.5867 0.7598 0.6787 Na0.96Rb0.04 1.0046 0.5869 0.7600 0.6789 Na0.95Rb0.05 1.0048 0.5871 0.7602 0.6791 Na0.99Cs0.01 1.0042 0.5865 0.7596 0.6785 Na0.98Cs0.02 1.0045 0.5868 0.7599 0.6788 Na0.97Cs0.03 1.0049 0.5872 0.7603 0.6792 Na0.96Cs0.04 1.0053 0.5876 0.7607 0.6796 Na0.95Cs0.05 1.0057 0.5880 0.7611 0.6800 nl/atom 0.2693 1.2500 0.3645 1.0682
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表 A1 IA族元素的乙种杂化表
Table A1. B type hybrid table of IA group
σ 1 2 3 4 Chσ 1 0.5386 0.4650 0 Ctσ 0 0.4616 0.5350 1 nTσ 1 1 1 1 nlσ 1 0.5386 0.4650 0 ncσ 0 0.4616 0.5350 1 Rσ(1) H 0.3708 0.3289 0.3222 0.2800 Li 1.3260 1.2089 1.1440 0.9860 Na 1.5133 1.4551 1.4308 1.3070 K 1.9628 1.8794 1.8601 1.7820 Rb 2.0870 2.0270 2.0175 1.9570 Cs 2.2140 2.2260 2.2279 2.2400 注: $ l, \; m, \;n, \; \tau $: 1 0 0 0
$l{'}, \; m{'}, \;n{'}, \; \tau {'}$: 0.9982 0.0018 0 0
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表 A2 VA族元素的甲种杂化表
Table A2. A type hybrid table of VA group
σ 1 2 3 4 Chσ 1 0.5694 0.1983 0 Ctσ 0 0.4306 0.8017 1 nTσ 3 or 5 3 or 5 3 or 5 3 or 5 nlσ 0 0 0 0 ncσ 3 or 5 3 or 5 3 or 5 3 or 5 Rσ (1) N 0.7000 0.7517 0.7973 0.8200 P 1.0980 1.1173 1.1343 1.1428 As 1.1800 1.2390 1.2911 1.3170 Sb 1.3560 1.4279 1.4919 1.5230 Bi 1.3990 1.4455 1.5044 1.5290 注: $ l, \; m, \; n, \; \tau $: 1 2 0 1; $ l{'}, \; m{'}, \; n{'}, \; \tau {'} $: 0 3 0 1
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表 A3 IVA族元素的甲种杂化表
Table A3. A type hybrid table of IVA group
σ 1 2 3 4 5 6 Chσ 1 0.9502 0.8320 0.1681 0.0481 0 Ctσ 0 0.0498 0.1680 0.8319 0.9519 1 nTσ 4 4 4 4 4 4 nlσ 2 1.9040 1.6640 0.3360 0.0960 0 ncσ 2 2.0960 2.3360 3.6640 3.9040 4 Rσ(1) C 0.7630 0.7630 0.7630 0.7630 0.7630 0.7630 Si 1.1700 1.1700 1.1700 1.1700 1.1700 1.1700 Ge 1.2230 1.2230 1.2230 1.2230 1.2230 1.2230 Sn 1.3990 1.3990 1.3990 1.3990 1.3990 1.3990 Pb 1.4300 1.4300 1.4300 1.4300 1.4300 1.4300 注: $ l, \; m, \; n, \; \tau $; 2 2 0 0; $ l{'}, \; m{'}, \; n{'}, \; \tau {'}; $ 1 3 0 1
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