Computational materials discovery: the case of the W–B system

By means of variable-compositional evolutionary algorithms, in combination with first-principles calculations, the compositions, structures and mechanical properties of the W–B system have been theoretically investigated. As well as confirming the experimental observations (including their crystal structures) for the four known compounds W₂B, WB, WB₂ and WB₃, the new stable compound W₈B₇ and two nearly stable compounds, W₂B₃ and WB₄, have also been predicted in the ground state. The elastic properties and estimated Vickers hardnesses of all these borides have been systematically derived. The results show that, among these borides, hP6-WB₂ exhibits the largest ultra-incompressibility along the c axis, with the highest C₃₃ value (953 GPa, comparable with that of the most incompressible diamond). hP16-WB₃ exhibits the highest hardness of 36.9 GPa, in good agreement with the experimentally measured data from 28.1 to 43.3 GPa, close to the superhard threshold, and oC8-WB shows the highest bulk modulus of about 350 GPa. The new stable compound W₈B₇ crystallizes in the monoclinic mP15 phase, with infinite zigzag B chains running parallel to the W-atom layers, resulting in a relatively high estimated hardness of 19.6 GPa. The anisotropic Young's modulus E and torsion shear modulus G_t have been derived for both oC8-WB and hP16-WB₃. The current state of research and the historic inconsistency of the W–B system are briefly summarized, in particular clarifying the fact that the previous experimentally attributed hP20-WB₄ is in fact the defect-containing hP16-WB₃.

Computing details

Figures

(W2B_I4mcm_tI12)

Crystal data

Tetragonal, I4/MCM	c = 4.7710 Å
a = 5.5706 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.16981	0.66981	0.50000	0.01267*
B1	−0.00000	0.00000	0.75000	0.01267*

Bond lengths (Å)

B1—B1^i

2.385

B1—B1^ii

2.385

Symmetry codes: (i) −x, −y, −z+1; (ii) −x, −y, −z+2.

(W2B_I4m_tI12)

Crystal data

Tetragonal, I4/M	c = 4.7531 Å
a = 5.5822 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.17687	0.66376	0.00000	0.01267*
B1	−0.00000	0.00000	0.21976	0.01267*

Bond lengths (Å)

B1—B1^i

2.089

Symmetry code: (i) −x, −y, −z.

(W8B7_Pm_mP15_relaxed)

Crystal data

Monoclinic, PM	c = 4.5313 Å
a = 6.2915 Å	β = 110.2861°
b = 6.1409 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.16274	0.25286	0.93547	0.01267*
W3	0.65319	0.25350	0.93531	0.01267*
B1	0.80599	0.25010	0.52642	0.01267*
B3	0.30488	0.25470	0.52600	0.01267*
W5	0.01122	−0.00000	0.34846	0.01267*
W6	0.51184	−0.00000	0.35276	0.01267*
B5	0.86768	−0.00000	0.77451	0.01267*
B6	0.36532	−0.00000	0.76458	0.01267*
W7	0.01544	0.50000	0.35643	0.01267*
W8	0.51064	0.50000	0.35689	0.01267*
B7	0.86744	0.50000	0.76846	0.01267*

Bond lengths (Å)

B1—B5	1.863	B5—B1^i	1.863
B1—B7	1.848	B6—B3^i	1.864
B3—B6	1.864	B7—B1^ii	1.848

Symmetry codes: (i) x, −y, z; (ii) x, −y+1, z.

(WB_Cmcm_oC8_relaxed)

Crystal data

Orthorhombic, CMCM	b = 8.4849 Å
a = 3.1776 Å	c = 3.1015 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	0.35518	0.75000	0.01267*
B1	−0.00000	0.06114	0.75000	0.01267*

(WB_I41amd_tI16_relaxed)

Crystal data

Tetragonal, I41/AMD	c = 16.9587 Å
a = 3.1395 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.50000	0.50000	0.30300	0.01267*
B1	0.50000	0.50000	0.15613	0.01267*

Bond lengths (Å)

B1—B1^i

1.892

B1—B1^ii

1.892

Symmetry codes: (i) x, −y+1/2, −z+1/4; (ii) x, −y+3/2, −z+1/4.

(WB2_ReB2_P63mmc_hP6)

Crystal data

Hexagonal, P63/MMC	c = 7.7481 Å
a = 2.9279 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.33333	0.66667	0.25000	0.01267*
B1	0.33333	0.66667	0.54080	0.01267*

(WB2_RuB2_Pmmn_oP6_relaxed)

Crystal data

Orthorhombic, PMMN	b = 4.6558 Å
a = 2.9188 Å	c = 4.2306 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	0.50000	0.37995	0.01267*
B1	0.00000	0.69277	0.88454	0.01267*

Bond lengths (Å)

B1—B1^i	1.795	B1—B1^iii	1.835
B1—B1^ii	1.835

Symmetry codes: (i) −x, −y+1, z; (ii) x+1/2, −y+3/2, −z+2; (iii) x−1/2, −y+3/2, −z+2.

(WB2_R-3m_hR9)

Crystal data

Trigonal, R3M	c = 11.6042 Å
a = 2.9316 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	−0.00000	0.00000	0.01267*
B1	0.66667	0.33333	0.13880	0.01267*

(WB2_WB2_P63mmc_hP12)

Crystal data

Hexagonal, P63/MMC	c = 14.0334 Å
a = 3.0178 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.33333	0.66667	0.63524	0.01267*
B1	0.33333	0.66667	0.02318	0.01267*
B5	0.33333	0.66667	0.25000	0.01267*
B7	−0.00000	0.00000	0.25000	0.01267*

(WB2_R-3m_hR18_relaxed)

Crystal data

Trigonal, R3M	c = 21.0521 Å
a = 3.0166 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	−0.00000	0.57632	0.01267*
B1	0.00000	−0.00000	0.16795	0.01267*
B7	0.00000	−0.00000	0.68185	0.01267*

Bond lengths (Å)

B1—B1^i	1.742	B7—B7^iv	1.855
B1—B1^ii	1.742	B7—B7^v	1.855
B1—B1^iii	1.742	B7—B7^vi	1.855

Symmetry codes: (i) y−1/3, x+1/3, −z+1/3; (ii) y−1/3, x−2/3, −z+1/3; (iii) y+2/3, x+1/3, −z+1/3; (iv) y−1/3, x+1/3, −z+4/3; (v) y−1/3, x−2/3, −z+4/3; (vi) y+2/3, x+1/3, −z+4/3.

(WB2_AlB2_P6mmm_hP3)

Crystal data

Hexagonal, P6/MMM	c = 3.3637 Å
a = 3.0189 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	0.00000	0.00000	0.01267*
B1	0.33333	0.66667	0.50000	0.01267*

(W2B3_Imm2_0GPa_relaxed)

Crystal data

Orthorhombic, IMM2	b = 6.0988 Å
a = 3.0439 Å	c = 5.2743 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	−0.00000	0.25112	0.60700	0.01267*
B1	0.50000	−0.00000	0.78268	0.01267*
B3	0.50000	−0.00000	0.44167	0.01267*
B5	0.00000	−0.00000	0.26431	0.01267*

Bond lengths (Å)

B1—B3	1.799	B3—B5^i	1.786
B3—B5	1.786	B5—B3^ii	1.786

Symmetry codes: (i) x+1, y, z; (ii) x−1, y, z.

(WB3_0Gpa_R-3m_relaxed)

Crystal data

Trigonal, R3M	c = 9.4083 Å
a = 5.2136 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	−0.00000	−0.00000	0.16627	0.01267*
B1	0.66502	−0.00000	0.00000	0.01267*

(WB3_0Gpa_P63mmc_relaxed)

Crystal data

Hexagonal, P63/MMC	c = 6.3130 Å
a = 5.2003 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	0.00000	0.75000	0.01267*
W3	0.66667	0.33333	0.25000	0.01267*
B1	0.00000	0.33470	0.00000	0.01267*

(WB3_0Gpa_P-3m1_relaxed)

Crystal data

Trigonal, P3M1	c = 9.3657 Å
a = 2.9453 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.66667	0.33333	0.29377	0.01267*
B1	0.66667	0.33333	0.05425	0.01267*
B5	0.66667	0.33333	0.53373	0.01267*
B3	0.33333	0.66667	0.13036	0.01267*

(WB3_0Gpa_P-6m2_relaxed)

Crystal data

Hexagonal, P6M2	c = 4.7176 Å
a = 2.9341 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	−0.00000	−0.00000	0.01267*
B1	0.33333	0.66667	0.67338	0.01267*
B3	0.00000	−0.00000	0.50000	0.01267*

(WB4_0GPa_R-3m_relaxed)

Crystal data

Trigonal, R3M	c = 16.5775 Å
a = 2.9488 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.66667	0.33333	0.83333	0.01267*
B1	0.66667	0.33333	0.97042	0.01267*
B7	0.33333	0.66667	0.92615	0.01267*

Bond lengths (Å)

B1—B7^i	1.854	B7—B1^iv	1.854
B1—B7^ii	1.854	B7—B1^v	1.854
B1—B7^iii	1.715	B7—B1^iii	1.715

Symmetry codes: (i) x, y−1, z; (ii) x+1, y, z; (iii) y, x, −z+2; (iv) x, y+1, z; (v) x−1, y, z.

(exp_WB4_0GPa_relaxed)

Crystal data

Hexagonal, P63/MMC	c = 6.4530 Å
a = 5.3606 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.33333	0.66667	0.25000	0.01267*
W3	0.00000	0.00000	0.25000	0.01267*
B1	0.33698	0.00000	−0.00000	0.01267*
B13	0.33333	0.66667	0.61854	0.01267*

Bond lengths (Å)

B1—B1^i	1.748	B13—B1^vii	1.935
B1—B1^ii	1.806	B13—B1^viii	1.935
B1—B1^iii	1.806	B13—B1^ix	1.935
B1—B13^iv	1.935	B13—B1^x	1.935
B1—B13^v	1.935	B13—B1^xi	1.935
B13—B1^vi	1.935	B13—B13^xii	1.697

Symmetry codes: (i) −x+1, −y, −z; (ii) y, −x+y, −z; (iii) x−y, x, −z; (iv) −x+1, −y+1, z−1/2; (v) −y+1, −x, −z+1/2; (vi) y, −x+y+1, z+1/2; (vii) x−y, x, z+1/2; (viii) −x+1, −y+1, z+1/2; (ix) x, y+1, −z+1/2; (x) −x+y+1, −x+1, −z+1/2; (xi) −y, x−y, −z+1/2; (xii) −y+1, −x+1, −z+3/2.

(WB4_new)

Crystal data

Hexagonal, P63/MMC	c = 10.9959 Å
a = 2.9550 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.66667	0.33333	0.75000	0.01267*
B1	0.66667	0.33333	0.11106	0.01267*
B5	0.66667	0.33333	0.54578	0.01267*

Bond lengths (Å)

B1—B5^i	1.725	B5—B1^vi	1.851
B1—B5^ii	1.851	B5—B1^vii	1.851
B1—B5^iii	1.851	B5—B5^viii	1.981
B1—B5^iv	1.851	B5—B5^ix	1.981
B5—B1^i	1.725	B5—B5^x	1.981
B5—B1^v	1.851

Symmetry codes: (i) −y+1, −x+1, −z+1/2; (ii) −x+2, −y+1, z−1/2; (iii) −x+1, −y+1, z−1/2; (iv) −x+1, −y, z−1/2; (v) −x+1, −y+1, z+1/2; (vi) −x+1, −y, z+1/2; (vii) −x+2, −y+1, z+1/2; (viii) y+1, x, −z+1; (ix) y, x, −z+1; (x) y, x−1, −z+1.

(W2B5_old_hP14_relaxed)

Crystal data

Hexagonal, P63/MMC	c = 14.2420 Å
a = 3.0964 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.33333	0.66667	0.13546	0.01267*
B7	0.33333	0.66667	0.97341	0.01267*
B1	0.00000	−0.00000	0.00000	0.01267*
B3	0.00000	−0.00000	0.25000	0.01267*
B5	0.33333	0.66667	0.75000	0.01267*

(W2B5_stable_hP14_relaxed)

Crystal data

Hexagonal, P63/MMC	c = 15.6962 Å
a = 3.0185 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	−0.00000	0.39715	0.01267*
B1	0.00000	−0.00000	0.25000	0.01267*
B3	0.33333	0.66667	0.30270	0.01267*
B7	0.33333	0.66667	0.49728	0.01267*

(W2B5_R-3m_hR21_relaxed)

Crystal data

Trigonal, R3M	c = 21.3755 Å
a = 3.0948 Å

Data collection

Refinement

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
W1	0.00000	0.00000	0.42342	0.01267*
B1	0.00000	0.00000	0.16883	0.01267*
B7	0.00000	0.00000	0.31486	0.01267*
B13	0.00000	0.00000	−0.00000	0.01267*