Finding the ‘Goldilocks Zone’: cationic size and tilting of carbodiimide and cyanamide anions

Juan Medina-Jurado; Alex J. Corkett; Richard Dronskowski

doi:10.1515/zkri-2023-0049

Published by De Gruyter (O) December 25, 2023

Finding the ‘Goldilocks Zone’: cationic size and tilting of carbodiimide and cyanamide anions

Juan Medina-Jurado , Alex J. Corkett and Richard Dronskowski

From the journal Zeitschrift für Kristallographie - Crystalline Materials

https://doi.org/10.1515/zkri-2023-0049

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Abstract

Solid-state carbodiimides and cyanamides are a group of compounds that generally shows motifs of layered structures, with an alternating sequence of metal cations and NCN²⁻ anions. A study of the relationship between the ionic sizes in the cation layer and the crystal structures found is presented using the geometric characteristics of the NCN²⁻ coordination environment. The results of this analysis reveal a ‘Goldilocks zone’ of cationic sizes, from Al³⁺ to Tl⁺, that are capable of forming stable “layered” metal carbodiimide structures. Furthermore, by employing a vectorial approach a correlation between the size difference of the cations and the degree of tilting of the NCN²⁻ framework has been found, capable of predicting the likelihood of new phases crystallizing in such structures.

Keywords: cyanamides; carbodiimides; crystal chemistry; size; tilting

Corresponding author: Richard Dronskowski, Chair of Solid-State and Quantum Chemistry, Institute of Inorganic Chemistry, RWTH Aachen University, 52056 Aachen, Germany, E-mail: drons@HAL9000.ac.rwth-aachen.de

Appendix

1 Mathematical background of size estimation.

From Figure A, given a cation A with size r_A, the relation of determining the size of B is given as follows:

Figure A:

Arrangement of the ions A and B in the trigonal prism coordinating to the top N atom, as well as the geometric relation between the bond lengths, d_AN and d_BN, and the distance between the ions, l.

If d_AN = r_A + r_N and d_BN = r_B + r_N, then d AN ´ = d AN 2 − d 2 and d BN ´ = d BN 2 − d 2

Therefore:

(1) l 3 2 = d AN ′ + d BN ´ 2 − ( l 2 ) 2

The size of the B cation therefore depends on r_A, l, and d. For a given A cation, the minimum and maximum B cationic size would be given by:

(2) f ( r A , l , d ) min < r B = f ( r A , l , d ) < f ( r A , l , d ) max

To find suitable minimum and maximum values for l and d, let us assume A and B to be the same cation. Then, from eq. (1) we obtain

(3) l 3 2 = d AN ´ + d AN ´ 2 − ( l 2 ) 2

which manifests that the l and d values are strongly related to each other, and a priori it is impossible to find a unique solution for eq. (3) given some cation A. As already mentioned, however, the overall effect of the balance of l (related to the size of the base formed for by cations) and d (related to the distance from N to the base formed by the cations) is the tilting of the NCN²⁻ unit, so instead of restricting (l, d) it is better to delimit θ. Then, eq. (2) can be rewritten as:

(4) f ( r A , l , d ) θ min < r B = f ( r A , l , d ) < f ( r A , l , d ) θ max

Thus, we may solve eq. (1) for each cation A, using (l, d)_min from the compound with the lowest tilting (NiNCN, θ = 0°, l = 3.153 Å, d = 1.085 Å) and (l, d)_max from the compound with the highest tilting (Hf(NCN)₂, θ = 27.72°, l = 3.476 Å, d = 1.310 Å). Hence, the minimum and maximum allowed size for a cation A with ionic radii r_A are given by:

r min = ( l min 3 2 − ( r A + r N ) 2 − d min 2 ) 2 + d min 2 + ( l min 2 ) 2 − r N

r max = ( l max 3 2 − ( r A + r N ) 2 − d max 2 ) 2 + d max 2 + ( l max 2 ) 2 − r N

For example, starting with Fe²⁺ and r_A = 0.780 Å, the minimum and maximum sizes would be given by:

r min = ( 3.153 3 2 − ( 0.780 + 1.46 ) 2 − 1.085 2 ) 2 + 1.085 2 + ( 3.153 2 ) 2 − 1.46

r min = 0.603 Å

r max = ( 3.476 3 2 − ( 0.780 + 1.46 ) 2 − 1.310 2 ) 2 + 1.310 2 + ( 3.476 2 ) 2 − 1.46

r max = 1.022 Å

With a similar analysis equation (1) for ions from Al³⁺ to Tl ⁺ may be solved. The results are presented in the next table:

ion	r_A (Å)	r_Bmin (Å)	r_Bmax (Å)	ion	r_A (Å)	r_Bmin (Å)	r_Bmax (Å)
Al³⁺	0.535	0.726	1.186	Ho³⁺	0.901	0.556	0.954
Cr³⁺	0.615	0.682	1.128	Dy³⁺	0.912	0.552	0.949
Ni²⁺	0.690	0.644	1.078	Tb³⁺	0.923	0.549	0.943
Sn⁴⁺	0.690	0.644	1.078	Gd³⁺	0.938	0.543	0.936
Hf⁴⁺	0.710	0.635	1.065	Eu³⁺	0.947	0.541	0.931
Mg²⁺	0.720	0.630	1.058	Cd²⁺	0.950	0.540	0.930
Zr⁴⁺	0.720	0.630	1.058	Sm³⁺	0.958	0.537	0.926
Cu²⁺	0.730	0.626	1.052	Nd³⁺	0.983	0.529	0.914
Zn²⁺	0.740	0.621	1.046	Pr³⁺	0.990	0.527	0.910
Sc³⁺	0.745	0.619	1.043	Ca²⁺	1.000	0.524	0.906
Co²⁺	0.745	0.619	1.043	Ce³⁺	1.010	0.521	0.901
Li⁺	0.760	0.612	1.034	Na⁺	1.020	0.518	0.897
Fe²⁺	0.780	0.603	1.022	Hg²⁺	1.020	0.518	0.897
In³⁺	0.800	0.595	1.010	La³⁺	1.032	0.515	0.891
Mn²⁺	0.830	0.583	0.993	Ag⁺	1.150	0.487	0.843
Lu³⁺	0.861	0.571	0.976	Sr²⁺	1.180	0.481	0.832
Yb³⁺	0.868	0.568	0.972	Pb²⁺	1.190	0.479	0.828
Tm³⁺	0.880	0.564	0.966	Ba²⁺	1.350	0.459	0.778
Er³⁺	0.890	0.560	0.960	K⁺	1.380	0.457	0.771
Y³⁺	0.900	0.557	0.955	Tl⁺	1.500	0.454	0.745

2 Mathematical background of anionic tilting

Figure 5 suggests a relation between the tilting angle and the distortion | 2 δ ‾ | according to sin θ = | 2 δ ‾ | d NCN . That is to say that a large distortion in the N position as regards the base cations translates into a large tilting angle. Figure 5b allows to derive an expression for the distortion, | 2 δ ‾ | , according to:

| 2 δ ‾ | = d AN ´ − d BN ´ = d AN . cos α A − d BN . cos α B , where sin α A = d d AN < sin α B = d d BN

Then, since cos α_A > cos α_B we may write:

cosα_A = cosα_B·(1 + k), k > 0

Therefore, | 2 δ ‾ | = cos α B · ( d AN · ( 1 + k ) − d BN ) or

(5) | 2 δ ‾ | = cos α B · ( r A − r B + k · d AN )

For the case of compounds with empty sites, inspecting the coordination environment of NCN²⁻ lets one deduce that the behavior of the empty sites is similar to that of a large cation. Then, from equation (5) assuming that the site of cation A is empty we obtain:

(6) | 2 ε ‾ | = cos α B . ( r ⊡ − r B + k · d ⊡ N )

Here r_⊡ is the size of the empty site and d_⊡N is the virtual distance between the empty site and the N atom. If we consider that the size of the empty site is proportional to the size of the ion B, we may write: r_⊡ = m·r_B, m > 1. Therefore, eq. (6) becomes | 2 ε ‾ | = cos α B · ( r B · ( m − 1 ) + k · d ⊡ N ) , so one realizes that the distortion in the N atoms and, also, the tilting angle, is proportional to the size of the B cation, r_B.

The results of calculating the total distortion for 21 binary, ternary, and quaternary compounds are presented in the next table:

Compound	Space group	Environment from Figure 6	δ ‾ total (Å)	θ_total (°)
FeNCN	P6₃/mmc	(c)	0.0	0.0
CoNCN	P6₃/mmc	(c)	0.0	0.0
NiNCN	P6₃/mmc	(c)	0.0	0.0
Sm₂(NCN)₃	C2/m	(f)	≈r_B = 0.958	10.67
Hf(NCN)₂	Pbcn	(h)	≈2 × 0.71 = 1.420	27.72
LiAl(NCN)₂	Pbcn	(a)	≈2 × (0.76 − 0.535) = 0.450	9.81
LiIn(NCN)₂	Pbcn	(a)	≈2 × (0.80 − 0.76) = 0.080	6.74
LiYb(NCN)₂	Pbcn	(a)	≈2 × (0.868 − 0.76) = 0.216	1.39
LiY(NCN)₂	Pbcn	(a)	≈2 × (0.90 − 0.76) = 0.280	3.07
NaSc(NCN)₂	Pbcn	(a)	≈2 × (1.02 − 0.745) = 0.550	10.27
NaIn(NCN)₂	Cmcm	(a)	≈2 × (1.02 − 0.80) = 0.440	10.61
Li₂Hf(NCN)₃	R 3 ‾ c	(d)	≈ 3 × (0.76 − 0.71) = 0.087	4.23
Li₂Zr(NCN)₃	R 3 ‾ c	(d)	≈ 3 × (0.76 − 0.72) = 0.069	4.69
Li₂Sn(NCN)₃	Pnna	(d)	≈ 3 × (0.76 − 0.69) = 0.121	3.45
Li₂Sn(NCN)₃	Pnna	(e)	≈(0.76 − 0.69) = 0.070	4.46
Na₂Sn(NCN)₃	Pnna	(d)	≈ 3 × (1.02 − 0.69) = 0.572	13.15
Na₂Sn(NCN)₃	Pnna	(e)	≈(1.02 − 0.69) = 0.330	5.76
Li₂MgSn₂(NCN)₆	P 3 ‾ 1m	(i)	≈(0.76 − 0.69) + 0.72 = 0.790	6.08
Li₂MnSn₂(NCN)₆	P 3 ‾ 1m	(i)	≈(0.76 − 0.69) + 0.83 = 0.900	11.51
Na₂MnSn₂(NCN)₆	P 3 ‾ 1m	(i)	≈(1.02 − 0.69) + 0.83 = 1.160	13.66
Na₂FeSn₂(NCN)₆	P 3 ‾ 1m	(i)	≈(1.02 − 0.69) + 0.78 = 1.110	16.72
Na₂CoSn₂(NCN)₆	P 3 ‾ 1m	(i)	≈(1.02 − 0.69) + 0.745 = 1.075	16.28
Na₂NiSn₂(NCN)₆	P 3 ‾ 1m	(i)	≈(1.02 − 0.69) + 0.69 = 1.020	16.14

Research ethics: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved the submission.
Competing interests: The authors declare no conflict of interest regarding this article.
Research funding: Juan Medina-Jurado is grateful for the financial support from Deutscher Akademischer Austauschdienst (DAAD). Alex Corkett is indebted to the Deutsche Forschungsgemeinschaft (DFG) for funding (project number 441856704).
Data availability: All data are available within the manuscript.

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Received: 2023-10-25

Accepted: 2023-12-05

Published Online: 2023-12-25

Published in Print: 2024-01-29

Finding the ‘Goldilocks Zone’: cationic size and tilting of carbodiimide and cyanamide anions

Abstract

1 Mathematical background of size estimation.

2 Mathematical background of anionic tilting

References

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