As mentioned above, Ti₃C₂ can be prepared from the Ti₃AlC₂ by means of etching. The as-synthesized Ti₃C₂ has the fairly complex surface morphology with the mixed terminal groups (F, O, and OH) occurring.^[37] The hydroxylated derivative is not included in our computations because the H atoms are relatively easy to replace.^[17,38,39] The surface functionalizations of oxygen and fluoride are focused. The surface morphology of Ti₃C₂T₂ (T = F or O) has previously shown that MXene with complete and symmetrical surface terminations is more thermodynamically stable than those with partial and asymmetric surface terminations.^[17,25,40] Therefore, three possible configurations of Ti₃C₂T₂ monolayer are constructed by saturating the dangling bond of surface Ti(1) atoms. In more detail, the T-terminations are equally and homogeneously distributed on both surfaces of monolayers, corresponding to the top position on Ti(1) atoms (1-top site), the hollow position on C atoms (2-hollow site), and the hollow position on Ti(2) atoms (3-hollow site) as shown in Fig. 1.

We start with considering structural optimizations to characterize the crystal structures of monolayers by using two methods, including the cases with and without van der Waals dispersion corrections. The structural information and calculated E_form are listed in Table 1. The calculated value of E_form is sensitive to the van der Waals dispersion correction. For the same Ti₃C₂T₂ configuration, the calculated E_form by the DFT+D2 method is 1 eV higher than that by the DFT method. However, it is found that the ground state configuration with van der Waals dispersion correction is similar to that without van der Waals dispersion correction, the T-terminations preferentially bind to the 3-hollow sites, which is also in agreement with the previous theoretical result.^[23,25,28] This is because structural characteristics of Ti₃C₂T₂ single layer determine the sites that the terminations prefer to occupy. The Ti atoms tend to form six-fold coordination in this structure, thus the Ti-centered distorted octahedral structure of surfaces is constructed through T-terminations occupying the 3-sites.^[41] The trends of energy variation after surface functionalization obtained by the two methods are also consistent with each other. The calculated E_form of Ti₃C₂ monolayer is obviously larger than that of the functionalized derivative. The positive value of E_form indicates that the Ti₃C₂ monolayer is difficult to form spontaneously through the selective etching of Al layers. On the contrary, the negative value of E_form shows that the Ti₃C₂T₂ monolayers obtained by the selective etching of the Ti₃AlC₂ phase are stable. Actually, the Ti₃C₂ monolayer without surface terminations has not yet been synthesized, and the presence of surface terminations conduces to the formation of monolayer.

Table 1.

Table 1.

Table 1. Optimized structural data (in unit Å) and calculated Eform values (in unit eV) of Ti3C2, Ti3C2F2, and Ti3C2O2 monolayers. . System t Bond lengths Eform DFT/DFT+D2 DFT/DFT+D2 DFT/DFT+D2 Ti(1)–C Ti(2)–C Ti3C2 4.62/4.72 2.04/2.05 2.19/2.20 2.87/3.86 Ti3C2F2-1 site 4.67/4.72 2.06/2.06 2.20/2.20 −6.23/−5.22 Ti3C2F2-2 site 4.79/4.85 2.11/2.12 2.17/2.18 −7.15/−6.14 Ti3C2F2-3 site 4.66/4.70 2.06/2.07 2.18/2.19 −8.03/−6.93 Ti3C2O2-1 site 4.94/4.96 2.13/2.14 2.19/2.19 −1.84/−1.19 Ti3C2O2-2 site 5.10/5.13 2.24/2.24 2.13/2.13 −5.68/−4.89 Ti3C2O2-3 site 5.14/5.17 2.21/2.21 2.17/2.18 −7.28/−6.38

Table 1. Optimized structural data (in unit Å) and calculated Eform values (in unit eV) of Ti3C2, Ti3C2F2, and Ti3C2O2 monolayers. .

The structural properties of ground state of Ti₃C₂ and Ti₃C₂T₂ monolayer are investigated. The t is defined as the vertical distance between the top Ti(1) atom and bottom Ti(1) atom. Based on our structural models, the difference between structure properties obtained by using DFT and DFT+D2 methods is limited to 0.5%. Therefore, the optimized structure of monolayer is not very sensitive to the van der Waals dispersion correction. Compared with the Ti₃C₂ monolayer, the T-terminated Ti₃C₂ monolayer has larger calculated t and the surface Ti(1)–C bond length, implying that surface terminations strongly interact with the original Ti₃C₂ structure, which is consistent with previous theoretical result.^[23,25]

After a thorough understanding of the structure properties of monolayers, we subsequently extend our computations to their adsorption behaviors of the single M (M = Li and Na) atom. Those adsorption configurations of one M atom on three possible adsorption sites of surfaces are considered (1-, 2-, and 3-sites as shown in Fig. 1), corresponding to a chemical stoichiometric ratio for each of Ti₃C₂M_0.25 and Ti₃C₂T₂M_0.25. To quantitatively evaluate the adsorption behaviors, the calculated E_ads1, and optimal adsorption height (h) and Hirshfeld charge of the adsorbed atom are calculated by using the DFT and DFT+D2 methods. The h is defined as the vertical distance from the M atom to the surface layer of Ti(1) atom. The Hirshfeld charge is defined as being relative to the deformation density, this is the difference between atomic charge densities after and before relaxation. After full geometry relaxation, the relative stabilities of adsorption configurations can be determined through calculating the value of E_ads1.

The most stable adsorption configurations with their calculated E_ads1, h, and Hirshfeld charge are shown in Fig. 2. For the Ti₃C₂M_0.25 configurations, the M atom prefers to sit at the hollow site due to repulsive interaction between the positively charged M and Ti atoms. When the functional groups exist, the M atom migrates to the nearest hollow site on the terminations that occupy the 1-top site, except the Ti₃C₂O₂Na_0.25 configuration obtained by using the DFT method. In previous analysis of the ground state configuration of Ti₃C₂T₂ monolayers, the T-terminations preferentially bind to the 3-hollow sites. Therefore, the M atom tends to stay away from the site that the terminations most prefer to occupy within the monolayer. The relative structural stability of adsorption configuration depends mainly on the possible charge transfer between adsorbed atom and terminal atom. If adsorbed metal atoms can provide sufficient electrons for the terminal atoms, then the T-terminations occupying the 1-top sites become the most stable configuration of Ti₃C₂T₂M_0.25. Here it is clearly observed that the calculated E_ads1 of Ti₃C₂T₂ monolayer with or without van der Waals dispersion corrections, is lower than that of Ti₃C₂ monolayer. Further, the calculated value of Hirshfeld charge for Ti₃C₂T₂ monolayer is higher than that of Ti₃C₂ monolayer. The low adsorption energy and the large charge transfer suggest that the M atom prefers to adsorb on the T-terminated Ti₃C₂ surface. This may relate to the strong interaction between functional group and metal atom.

Fig. 2.

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	Fig. 2. The most stable adsorption configurations (side and top views) of (a) Ti3C2Li0.25 and Ti3C2T2Li0.25 systems, (b) Ti3C2Na0.25 and Ti3C2T2Na0.25 systems obtained by using DFT method, and (c) Ti3C2Li0.25 and Ti3C2T2Li0.25 systems, (d) Ti3C2Na0.25 and Ti3C2T2Na0.25 systems obtained by using DFT+D2 method with their calculated values of Eads1, h, and Hirshfeld charge.

After a thorough understanding of the adsorption behaviors of one M atom on the surfaces, we check the variation of adatom concentration as a function of lattice parameters c. In this subsection, the multilayer incorporation of metal atom into nanosheets is focused by performing the DFT method. To check the possible multilayer incorporation, the quasi-symmetrically adsorption configurations are used, corresponding to the chemical stoichiometric ratio of Ti₃C₂M_2y and Ti₃C₂T₂M_2y. Here, three different incorporation atom concentrations (y = 1–3) are considered. Since M atoms’ orderings can be different for the same incorporation concentration, the geometry optimization is performed for all possible structures, here more than 100 structures are considered. To investigate the structural stabilities, the calculated E_ads2 as a function of lattice constant c is analyzed. The energetically favorable structures with the same lattice constant are determined by comparing their relative E_ads2. The calculated E_ads2 values of the most energetically favorable structures of different intercalation layers as a function of lattice constant c are shown in Fig. 3.

Fig. 3.

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	Fig. 3. Calculated Eads2 values of most energetically favorable structures of different intercalation layers as a function of lattice constant c.

In Fig. 3, the calculated E_ads2 of intercalating the same number of atomic layers into the interlayer spacing of nanosheet decreases gradually with the increase of lattice constant c. The critical value of lattice constant c of intercalating the same number of atomic layers is determined by obtaining the inflection points between the positive and negative values of adsorption energy, thus, the minimum lattice volume and the maximum volumetric capacity can be predicted in the bulk-stacking Ti₃C₂ and Ti₃C₂T₂ nanosheet with the determined number of M atomic layers. When each additional layer of M atoms is intercalated, the calculated E_ads2 has a process of change from high energy to low energy with the increase of lattice constant c. For example, under the same lattice constant c (c = 37.04 Å), the calculated E_ads2 value of Ti₃C₂F₂Li₄ nanosheet and that of Ti₃C₂F₂Li₆ nanosheet are greatly different from each other, which phenomenon is attributed to the increase of the Coulomb repulsive between the M atoms with the number of intercalated atoms increasing. With the increase of lattice constant c, the calculated E_ads2 of Ti₃C₂F₂Li₆ nanosheet decreases gradually. Once calculated E_ads2 is lower than zero, multilayer adsorption structure is stable, indicating that the interlayer spacing variation should affect the number of intercalating atomic layers, and the intercalation layer number increases with lattice constant c increasing. In the multilayers studied above, the distance from the bottom of one Ti₃C₂T_x sheet to the top of the next sheet is around 5.0 Å,^[20] indicating that the sheets are bound mainly by van der Waals interaction, and at most two layers’ M atoms can be incorporated into the interlayer spacing between two layered hexagonal structures. As the lattice expands along the c-axis direction in a wide range, more metal atoms can be incorporated into the interlayer spacing. When the chemical stoichiometric ratio of M atom to Ti atom are larger than 2y/3 (y = 1 or 2), some M atoms are driven to penetrate into the sub-adjacent region of the surface, resulting in the multilayer adsorption in the space region. A similar multilayer adsorption behavior has also been found in previous work.^[26] Overall, the prepared Ti₃C₂T_x monolayers and the expansion of the interlayer spacing of Ti₃C₂T_x nanosheets will be helpful in increasing the number of incorporation atom layers. Actually, the interlayer-expanded Ti₃C₂ has been synthesized experimentally, and its Li storage capacity was improved.^[31,42,43]

After the thorough calculation for E_ads2 of all the configurations, we subsequently extend our computations to their theoretical capacity, and the optimal ratio of the capacity to structure is obtained in the stacking structure. The optimum lattice constant c and the calculated capacities corresponding to different incorporation layers are listed in the following Table 2. The value of Li capacity can be increased from 251.16 mAh·g⁻¹ (1280.75 mAh·cm⁻³) to 768.68 mAh·g⁻¹ (1767.61 mAh·cm⁻³), when the c-axis value is expanded from 17.04 Å to 43.04 Å. The value of calculated C_MNa can be increased from 218.32 mAh·g⁻¹ to 413.13 mAh·g⁻¹ when the c-axis value is expanded from 19.04 Å to 43.04 Å, and the value of calculated C_VNa can reach 1146.18 mAh·cm⁻³ when the value of the c axis varies in the same range. The maximum value of the calculated C_MNa accords well with the previously reported value of 413.0 mAh·g⁻¹.^[26] The difference in calculated C_M between Li and Na atoms with the same valence state can be attributed to their different atomic weights in the case of the same incorporation layers, thus, the heavier ions have lower gravimetric capacity. The difference in calculated C_V is partially due to their difference in optimal structure in the case of the same incorporation layers, thus, the large lattice volume has lower volumetric capacity. The theoretical value of the calculated capacity varies with the length of the lattice constant c, however, this change is not the process of linear increase. For example, the increased percentage of the theoretical C_VLi of Ti₃C₂ nanosheet slows down with each additional intercalation layer, and the theoretical C_VNa of Ti₃C₂O₂ nanosheet decreases with the lattice constant c increasing. Therefore, the proper mass load and interlayer spacing of the experimental sample are essential to obtain optimal performance of electrode material.

Table 2.

Table 2.

Table 2. Calculated optimized Vcell values (in unit Å3), and calculated CVM values (in units mAh·cm−3) and CMM values (in mAh.·g−1) of Ti3C2M2y and Ti3C2T2M2y systems. . System y Vcell CVLi CMLi Vcell CVNa CMNa Ti3C2M2y 1 171.74 1037.26 295.42 204.39 871.57 251.03 2 253.37 1406.16 548.86 351.32 1014.16 413.13 3 334.99 1595.33 768.68 – – – Ti3C2F2M2y 1 155.42 1146.18 244.28 204.39 871.57 213.12 2 269.69 1321.07 459.50 351.32 1014.16 360.37 3 351.32 1521.17 650.55 – – – Ti3C2O2M2y 1 139.09 1280.75 251.16 155.42 1146.18 218.32 2 253.37 1406.16 471.63 318.67 1118.02 367.79 3 302.34 1767.61 666.73 – – –

Table 2. Calculated optimized Vcell values (in unit Å3), and calculated CVM values (in units mAh·cm−3) and CMM values (in mAh.·g−1) of Ti3C2M2y and Ti3C2T2M2y systems. .

Further structural analysis can find that the interaction between the different Ti₃C₂T₂ layers is ignored, when the c-axis value is expanded to more than 43.04 Å, because the layer spacing of surface terminations between two layered hexagonal structures exceeds the distance of the van der Waals interaction. In fact, the Ti₃C₂T_x monolayer is difficult to prepare into the experimental samples, and sample generally belongs to a mixture of few-layer, multilayer and even bulk structure, their lattice constant c is in a range between 19.50 Å and 35.04 Å.^[20,44] It is determined that two to four layers of Li atoms can be incorporated into the interlayer space of experimental samples, and one to two layers of Na atoms can be incorporated into the same interlayer space. The optimal values of theoretical capacity corresponding to experimental samples are 1406.16 mAh·cm⁻³ for calculated C_VLi, 548.86 mAh·g⁻¹ for calculated C_MLi, 1146.18 mAh·cm⁻³ for calculated C_VNa, and 251.03 mAh·g⁻¹ for calculated C_MNa, respectively. Although, the theoretical capacity of Na atom is much lower than that of Li atom, which is in agreement with the experimentally observed Na capacity.^[32,45,46] However, the theoretical C_Na value is comparable to that in the case of other electrode material, such as 146 mAh·g⁻¹ of MoS₂.^[9] The Ti₃C₂T_x nanosheets are a good candidate for being used as an electrode material in NIBs.

In experiments, bare Ti₃C₂ without any surface termination has never been achieved, thus understanding the effect of surface terminations on storage performance is particularly important.^[47,48] Here, taking the Ti₃C₂Li_2y and Ti₃C₂T₂Li_2y as the representative cases, the influence of T-terminations on calculated value of C_V is analyzed. The optimum lattice constant c for incorporation one atomic layer on each surface of single layer is calculated to be 21.04 Å for Ti₃C₂, 19.04 Å for Ti₃C₂F₂, and 17.04 Å for Ti₃C₂O₂. The optimum lattice constant c for incorporation two atomic layers is calculated to be 31.04 Å for Ti₃C₂ and Ti₃C₂O₂, and 33.04 Å for Ti₃C₂F₂. The optimal lattice constant c for incorporation three atomic layers is calculated to be 41.04 Å for Ti₃C₂, 43.04 Å for Ti₃C₂F₂, and 37.04 Å for Ti₃C₂O₂. The optimized lattice parameter of Ti₃C₂O₂Li_2y nanosheet is smaller than that of the Ti₃C₂Li_2y nanosheet and Ti₃C₂F₂Li_2y nanosheet. On the basis of the small lattice parameter, the Ti₃C₂O₂Li_2y system generally possesses a small molar volume. Thus, the O-terminated Ti₃C₂ nanosheet can hold more incorporation atoms than any other nanosheets when their volumes are equal. The O functional group as an excellent chemical modifier of surfaces can greatly improve the volumetric capacity of Ti₃C₂ system. However, a process of surface functionalization with fluorine group is not responsible for increasing the volumetric capacity. The higher volumetric capacity performance can be obtained by regulating the main surface termination of sample in experimental preparation due to the effect of different surface functional groups on theoretical capacity. The above analysis of the effect of functionalized surface on theoretical capacity is consistent with previous experimental and theoretical results.^[12,17,25]

It is noticed that the number of intercalation Na atom layers on each surface is only two, when the value of c-axis lattice constant varies in a range from 19.04 Å to 43.04 Å. Although, the Li/Na atoms intercalated into nanosheets have a similar operating mechanism which may be the formation of Ti₃C₂M_2y and Ti₃C₂T₂M_2y after intercalating M atoms into the space vacated by the Al atoms.^[24] However, the intercalation layers’ number of Na atom is far less than that of Li atom. We seek to understand the reason why the theoretical capacity of Na atoms is lower than that of Li atoms. Taking the twelve stable configurations of Ti₃C₂M_2y system and Ti₃C₂T₂M_2y system with the c-axis lattice constant of 25.04 Å (y = 1) and 35.04 Å (y = 2) for example, the formation process of M-ions’ intercalation can be divided into two parts. First, the Ti₃C₂ substrate is strained to the configuration identical to that of the Li/Na-intercalated surfaces. Second, the Li/Na-ions are intercalated into the nanosheets.

In the first process, there are two kinds of energy losses that can be obtained for the substrate after deformation. First, the calculated total energy of Ti₃C₂ substrates of Ti₃C₂ nanosheet and Ti₃C₂T₂ nanosheet after crystal deformation has had an energy cost (E_ec1 = |(E_system(deformation Ti₃C₂ substrates) − E_system(perfect Ti₃C₂ substrates)|), second, the calculated total energy of Ti₃C₂T₂ nanosheet after crystal deformation also has an energy cost (E_ec2 = |(E_system(deformation Ti₃C₂T₂ nanosheets) − E_system (perfect Ti₃C₂T₂ nanosheets)_|)). The calculated values of E_ec1 and E_ec2 reveal the degree of substrates’ deformation. Once the calculated value is greater than zero, the substrate produces a larger lattice deformation. The calculated E_ec1 value of the Na atom incorporation is similar to that of the Li atom incorporation into corresponding nanosheet as shown in Fig. 4. The energy loss of Ti₃C₂ substrate is almost negligible in the Li/Na-ion insertion processes. The Ti₃C₂ substrate only shows slight structural distortion in the insertion processes, which is consistent with previous experimental result.^[32] This excellent structural stability of substrate is a necessary condition for the mechanical and electrochemical stability of electrode materials. Furthermore, the calculated E_ec2 value of the Na atom incorporation is smaller than that of the Li atom incorporation into corresponding nanosheet. Although Li-ion has a smaller ion radius than Na-ion, the functionalized substrate with Li incorporation needs to produce a larger shift of surface termination, which can adjust the substrate to adapt the ion intercalated into nanosheets. Obviously, the E_ec1 and E_ec2 do not embody the origin of the low Na capacity.

Fig. 4.

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	Fig. 4. Calculated Eec1, Eec2, and Eec3 values of most favorable configuration of Ti3C2M2y and Ti3C2T2M2y nanosheet for c-axis lattice constant of 25.04 Å (y = 1) and 35.04 Å (y = 2), where y represents the number of intercalation layers on one side of single layer.

In the second process, the energy cost (E_ec3 = E_ads/h_totave) is determined after the metal atoms have adsorbed on the surfaces, here, E_ads = Eadatom@system − Esystem − 8yE_atom, with E_{adatom@system} and E_system being the total energy of adsorbed system and unadsorbed system, E_atom being the energy of per metal atom from the bulk metal, y representing the number of adsorption layers on every surface of single layer, h_totave denoting the average vertical height from the outermost layer of metal atom to the surface layer of Ti(1) atom. The coefficient 8×y represents the total number of adsorption atoms on the both sides of single layer. The calculated E_ec3 value reveals the binding degree of substrate to metal atoms, which mainly includes the interaction of polar surfaces with cation and the electrostatic exclusion between adsorbed cation.^[49] Once the calculated value is less than zero, the substrate produces the binding energy to the metal atoms. The calculated E_ec3 values are also shown in Fig. 4, and corresponding configurations are displayed in Fig. 5. The calculated E_ads value of Na atom is higher than that of Li atom, and the calculated h_totave value of Na atom becomes larger that of Li atom, thus, the calculated E_ec3 value exhibits a maximum value at Na-ion intercalation. Clearly, the calculated E_ec3 values reveal that the substrate has ability to more weakly bind Na atom than Li atom. This suggests the reason why low Na capacity is related directly to the coupling strength of the Na-ion with substrate and the electro-static exclusion between adsorbed Na-ions.

Fig. 5.

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Fig. 5. Twelve stable configurations of (a) Ti3C2Li2 and Ti3C2T2Li2, (b) Ti3C2Na2 and Ti3C2T2Na2 systems with the c-axis lattice constant of 25.04 Å, and (c) Ti3C2Li4 and Ti3C2T2Li4, (d) Ti3C2Na4 and Ti3C2T2Na4 systems with the c-axis lattice constant of 35.04 Å. Note: htotave = have(top) +have(bottom), where have(top) and have(bottom) are the average vertical height from the upper or lower layers of Li/Na atoms to the corresponding surface layer of Ti(1) atoms.

In addition, the dispersive electron cloud plays a key role in analyzing the interaction of the ions with substrate and the electrostatic exclusion between ions. The electron density difference of the (110) atomic plane among the twelve stable configurations is examined as shown in Fig. 6. For the Ti₃C₂T₂ substrate, the local electron transfer between the Li/Na and substrate surface is more apparent than for the Ti₃C₂ substrate. The presence of F- or O-terminations conduces to the electron transfer from the Li/Na atom to the surface. It is noticed that the interaction between the Na-ions and the substrate is weaker than that between the Li-ions and substrate. This weaker interaction can further increase the calculated value of E_ec3. Furthermore, the common charge quantity between the Na-ion layers is significantly less than that between the Li-ion layers. This characteristic indicates that the electrostatic exclusion between the Na-ions layers is greater than that between the Liion layers. Based on the above analysis, the reason for the low Na capacity is attributed to the weak interaction between the Na atoms and the substrate and the strong electrostatic exclusion between adsorbed Na atoms. Therefore, the energy cost at zero temperature and the electron distribution can explain why the theoretical capacity of Na atoms is lower than that of Li atoms on Ti₃C₂T_x nanosheet.

Fig. 6.

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Fig. 6. Electron density difference of the (110) atomic plane among (a) Ti3C2Li2, (b) Ti3C2F2Li2, (c) Ti3C2O2Li2, (d) Ti3C2Na2, (e) Ti3C2F2Na2, (f) Ti3C2O2Na2 nanosheets for c-axis lattice constant of 25.04 Å, and among (g) Ti3C2Li4, (h) Ti3C2F2Li4, (i) Ti3C2O2Li4, (j) Ti3C2Na4, (k) Ti3C2F2Na4, (m) Ti3C2O2Na4 nanosheets for c-axis lattice constant of 35.04 Å. Blue color indicates the loss of electron density and red color indicates the enrichment of electron density compared with the sphere electron density in atoms. The color scale is in units of e.·Å.−3.