Fig. 1 Atomistic structures of six STGBs rotating around [0001] axis of Mg: a $({12\bar{3}0})[0001]$/21.8°-A STGB, b $( {12\bar{3}0})[0001]$/21.8°-T STGB, c $(13\bar{4}0)[0001]$/32.3° STGB, d $(23\bar{5}0)[0001]$/13.2°-A STGB, e $(23\bar{5}0)[0001]$/13.2°-T STGB, f $( {15\bar{6}0})\left[ {0001} \right]$/42.1° STGB

Fig. 2 Schematic illustrations of six STGBs in hcp Mg after relaxation: a $( {12\bar{3}0})[0001]$/21.8°-A STGB, b $( {12\bar{3}0} )[0001]$/21.8°-T STGB, c $( {15\bar{6}0} )\left[ {0001} \right]$/42.1° STGB, d $(13\bar{4}0)[0001]$/32.3° STGB, e $(23\bar{5}0)[0001]$/13.2°-A STGB, f $(23\bar{5}0)[0001]$/13.2°-T STGB

Fig. 3 Segregation energy profiles of solutes Zn, Al, Ag, Ca, and Gd segregated at various sites of: a $( {12\bar{3}0})[0001]$/21.8°-A STGB, b $( {12\bar{3}0})[0001]$/21.8°-T STGB, c $(13\bar{4}0)[0001]$/32.3° STGB, d $(23\bar{5}0)[0001]$/13.2°-A STGB, e $(23\bar{5}0)[0001]$/13.2°-T STGB, f $( {15\bar{6}0} )\left[ {0001} \right]$/42.1° STGB

Fig. 4 Schematic illustrations for the solute-induced structural transformation behavior of $( {13\bar{4}0})[0001]$/32.3° STGB. a, c Initial structure with solute Zn occupying site 9 and site 8; b, d relaxed structure with Zn atom substitution; e, g initial structure with solute Ca occupying site 8 and site 6; f, h relaxed structure with Ca atom substitution. The gray and purple balls represent Zn and Ca atoms, respectively

Fig. 5 Correlation between solute segregation energy and the V of GB sites: a Zn; b Al; c Ag; d Ca; e Gd

Fig. 6 Correlation between solute segregation energy and the V × SBL of GB sites: a Zn; b Al; c Ag; d Ca; e Gd

Fig. 7 a Schematic illustration of $( {25\bar{7}0})[0001]$/27.8° STGB and GB features: b V and c V × SBL distribution of this STGB. Segregation energy profiles for solutes: d Zn and e Ca segregated at 20 candidate sites of $( {25\bar{7}0} )[0001]$/27.8° STGB

Fig. 8 Grain boundary features: a V and b V × SBL distribution of the six studied STGBs. The number of potential segregation sites per unit area characterized by: c large V and d small V × SBL of the six STGBs

Fig. 9 a, b Segregation energy, c, d strengthening energy, e, f grain boundary energy of group IA, IIA, IIIA, IVA, VA solutes, 3d TMs (Cu and Zn), and all 4d TMs at $(13\bar{4}0)[0001]$/32.3° STGB. Solutes are placed at their most energetically favorable segregation sites

Fig. 10 GB segregation energy of 26 solutes at the $(13\bar{4}0)[0001]$/32.3° STGB and its mechanical (pink columns) and chemical (green columns) contributions

Fig. 11 GB strengthening energy of 26 solutes at the $(13\bar{4}0)[0001]$/32.3° STGB and its mechanical (pink columns) and chemical (green columns) contributions

Fig. 12 Relationships between strengthening energy and a, b -ICOHP values and c, d Δ-ICOHPF values for: a, c solutes occupying site 2, b, d solutes occupying site 3

Fig. 13 Strengthening maps consisting of the GB segregation energy and strengthening energy for the studied 26 solutes. The solutes are classified into three groups according to their effect on GB cohesion: cohesion enhancer (blue triangle), cohesion embrittler (red circle), and no effect (purple pentagon)