Introduction

Liquid-phase exfoliation (LPE)1,2,3,4,5,6,7 is a highly promising approach to large-scale production and dispersion of a wide variety of layered materials (LMs). It affords facile processing of individual nanosheets, which can be deposited on surfaces or combined into free-standing films6, and vertical or horizontal stacks. LPE has been used to create novel multi-ferroic materials for photoconducting cells7,8,9, p-n junctions, field-effect transistors10,11, and memory devices12 using stacks of layered perovskites. Integration of LMs using LPE has potential applications in large-area electronics13 and inkjet printing14.

A wide variety of transition metal dichalcogenides (TMDCs), metal oxides, and perovskites have been exfoliated into 2D layers by electrochemical, sonication and shear methods1,6. Here, we will focus on the sonication approach in which a bulk solid is suspended in a suitable solvent and exfoliated into atomically thin LMs by cavitation phenomenon. Since LMs are characterized by strong in-plane covalent bonds and weak out-of-plane interactions, it is possible to exfoliate LMs by weakening the interlayer van der Waals interaction using ultrasonic cavitation2. It is desirable to keep ultrasonic intensity low in order to avoid sonolysis and defects in 2D materials and to prevent radicals in solvents which may affect dispersion of LMs. Solvents play a critical role in efficient production of 2D materials by LPE. Experimental measurements15,16 of interfacial energy, Hildebrand solubility and Hansen parameter are commonly used to guide the selection of solvents for exfoliation of LMs. Solvents with weak volatility (N,N-dimethylformamide and N-methyl-2-pyrrolidone) and low boiling points (propanol, chloroform) have been successfully used to exfoliate LMs1,15.

Despite a great deal of experimental work, there is very little understanding of atomistic mechanisms underlying LPE. The motivation for the joint experimental and simulation work reported in the paper is to unveil the atomic mechanism of liquid-phase exfoliation and thus facilitate the synthesis of atomically thin layered materials (LMs). Shock exfoliation of LMs by bubble collapse mimics experimental conditions, and experimentalists are trying to optimize the conditions for liquid-phase exfoliation by choosing suitable solvents and shock intensity so that single sheets of defect-free LMs are produced. We have performed molecular dynamics (MD) simulations17,18 in which a MoS2 crystal is suspended in a solvent of water and isopropanol (IPA) containing a cavitation bubble. The system is subjected to a planar shock which initiates a chain of events in the solvent, culminating in the exfoliation of MoS2 into nanosheets. As the shock wave propagates through the solvent, the density of solvent increases to 1.6 g/cc and the number of nearest neighbors of a water molecule increases from 4 to 8, indicating an ice VII like motif. Water in the compressed solvent is not frozen: on the contrary, the self-diffusion coefficient of H2O molecules normal to the direction of shock propagation is increased by 60%. The shock wave impact collapses the cavitation bubble and generates a high-speed nanojet in the solvent. The nanojet impact generates large shear stresses (~10 GPa) on the MoS2 surface and the surface temperature goes up to ~3,000 K. These large shear stresses and elevated temperature initiate exfoliation of MoS2, and shock waves reflected from MoS2 surfaces enhance exfoliation. We have performed LPE experiment with IPA and DI water, which verify the conclusion from the simulations.

Results

Figure 1(a) shows an initial configuration of a simulation in which an MoS2 solid (yellow and pink spheres) is immersed in a 1:1 mixture of H2O and IPA. (For the sake of clarity, only the lower half of the MD box is shown in the figure.) Cavitation is introduced after equilibrating the system under ambient conditions. The ratio of the bubble radius to the shortest distance between the bubble center and MoS2, i.e. the stand-off parameter S, plays a critical role in exfoliation. We performed simulations for several values of S ranging between 1 and 2 and observed exfoliation in the range 1.1 < S < 2.0 with particle velocity Vp = 3.0 km/s. Here we will present results for a bubble of diameter 9.4 nm and S = 1.14. Additional results are presented in the supplementary information.

Figure 1
figure 1

Initial configuration of the shock-induced exfoliation simulation and formation of ice VII motif. (a) Shows the initial setup of the exfoliation simulation. Bulk MoS2 is represented by pink (Mo) and yellow (S) sheets. MoS2 is immersed in a solvent, consisting of water and IPA molecules (1:1 ratio by weight). For clarity, only 2% of the solvent molecules (Oxygen: red, Carbon: cyan, Hydrogen: white) are shown in the lower half of the MD box. The solvent contains a nanobubble of radius R = 4.7 nm. The stand-off parameter, d/R = 1.14, where d is the distance between the bubble center and the closest MoS2 surface. (b) Shows the radial distribution function for oxygen-oxygen in water (red) and oxygen in water and the center of mass of IPA (blue). (c) Shows one water molecule with 8 nearest neighbors. Six of them are H2O and two are IPA molecules.

Full size image

In all simulations, shock is generated by a momentum mirror placed normal to the z direction just outside the MD box. The solvent, MoS2 and bubble are moved towards the momentum mirror with a constant speed Vp at time t = 0. When the solvent molecules cross the mirror, their momenta in the z direction are reversed which creates a planar shock wave in the solvent propagating away from the mirror. Using this approach, we first calculated the Hugoniot (shock-wave velocity Vs as a function of Vp) of the solvent without MoS2 and found that it was similar to the Hugoniot of pure water; see Figure S2 in the Supplementary Information.

Shock simulations are performed at several particle velocities in the range of 0.5–4.0 km/s. Here we present results for the shock velocity Vs = 7.4 km/s corresponding to the particle velocity Vp = 3.0 km/s. A movie of exfoliation is in the Supplementary Material. Under these conditions, the pressure in the solvent rises to 10.5 GPa and the density of water increases from 0.95 g/cc to 1.59 g/cc. This high compression has a dramatic effect on the structure of water. Figure 1(b) shows the radial distribution function go-o(r) for oxygen atoms of water molecules. Here the first peak is located at 2.68 Å and the first minimum at 3.75 Å, whereas in pure water under ambient conditions the first peak and first minimum are at 2.76 Å and 3.34 Å, respectively. The average number of nearest neighbors of water molecules, calculated from the area under the first peak in Fig. 1(b) with a cutoff distance of 3.75 Å, is 8 as opposed to 4 in pure water under ambient conditions. The O-O-O bond-angle distribution for water molecules in the high-density solvent (HDS) peaks around 56◦ and the O-O-IPA distribution peaks around 65◦. These results indicate that the structure of water in the HDS is similar to that of ice VII in that both of them have the same density (1.6 g/cc) and number of nearest neighbors (8), see Fig. 1(c). However, the differences in bond-angle distributions reflect disorder in the first solvation shell of water in the HDS compared to the first solvation shell of ice VII19,20.

The MD results for the structure of water in the HDS are in good agreement with Dolan et. al.’s shock-wave experiment20 on pure water. They observed rapid freezing of water for Vp between 0.5 km/s and 2.0 km/s under isentropic and ramp-wave compression. Freezing occurred within a few nanoseconds above a critical value of pressure (7 GPa) irrespective of the peak pressure generated by shock. Their data clearly show that ultrafast homogenous nucleation of ice VII is feasible under shock compression above 7 GPA.

In the MD simulation, we do not observe complete freezing of H2O into ice VII because the timescale of the applied shock (~20 ps) is much shorter than the timescale (a few ns) in the experiment and also because of the presence of IPA molecules. However, we do observe a significant change in the dynamics of water molecules in the HDS: the self-diffusion coefficient of water, calculated from mean-square displacements normal to the direction of shock-wave propagation, is larger (3.7 × 10−5 cm2/s) than that of pure water under ambient conditions (2.4 × 10−5 cm2/s). The IPA molecules in the HDS diffuse much more rapidly than under ambient conditions: the self-diffusion coefficient for the center-of-mass motion of IPAs in the 50%wt mixture is 2.3 × 10−5 cm2/s in the HDS and 1.4 × 10−6 cm2/s under ambient conditions.

Under the impact of the shock wave, the bubble begins to shrink because the surface tension of the bubble cannot provide enough restoring force to balance the shock-wave compression. Snapshots in Fig. 2 show a time sequence of changes in the shape and size of the nanobubble resulting from the shock impact. As more and more solvent molecules enter the bubble, the proximal side of the bubble changes from spherical to ellipsoidal. The shape of the shock front also changes during bubble shrinkage: the front loses planarity because the solvent molecules entering the bubble have different velocities than the shock-front velocity Vs. The front regains planarity after the bubble disappears. The bubble collapse time – the elapsed time between the onset of bubble shrinkage and complete bubble collapse – is 1.5 ps, see Fig. 2(b). It is in close agreement with the Rayleigh formula for bubble collapse time,

$$\begin{array}{c}\tau =0.45D\sqrt{\frac{\rho }{{\rm{\Delta }}P}}\,,\end{array}$$
(1)

where D is the initial diameter of the bubble, ρ is the fluid mass density (HDS in our case), and ΔP is the pressure difference across the bubble surface. Substituting D = 9.4 nm, ρ = 1.59 g/cc and ΔP = 10 GPa in Eq. (1), we find τ = 1.7 ps. It is remarkable that this estimate agrees so well with the MD result even though the effects of viscosity, surface tension and non-uniformity of the solvent near the bubble surface are ignored in the Rayleigh formula.

Figure 2
figure 2

Snapshots of the collapsing cavitation bubble. (a) Shows the initial configuration of the cavitation bubble and MoS2. The bubble is represented by a shell of liquid molecules on the bubble surface. The standoff parameter is 1.14. At t = 0.2 ps, the shock wave hits the proximal side of the bubble. (b)–(d) Are snapshots showing changes in the shape of the collapsing bubble due to the shock wave at time t = 0.65, 0.85, 1.1 ps. The shock wave is represented by the change in the liquid density (half of the density is shown in the figures). (e) shows changes in the surface area and volume as a function of time while the bubble is collapsing. The surface area and volume are normalized to their respective initial values. The bubble collapses at t = 1.5 ps.

Full size image

At the onset of the bubble collapse, we notice sudden increases in the translational kinetic and rotational energies of solvent molecules at the shock front. These energy jumps are caused by solvent molecules entering the proximal side of the bubble. Velocity streamlines of these molecules are focused towards the bubble center in the form of a high-speed nanojet. Figure 3(a) shows the center-of-mass velocity streamlines of molecules in the nanojet at t = 1.75 ps after the bubble collapse. The nanojet length and width are 6.5 nm and 3 nm, respectively. At the tip of the nanojet, the velocity is 6.1 km/s and the pressure around 20 GPa, which is close to the pressure estimated from the jump condition, P − P0 = ρ VpVs = 21 GPa. The nanojet length increases linearly with the initial diameter of the bubble21. Experiments22,23 show that this linear relationship also holds for micron-to-millimeter size bubbles. The nanojet generates a vortex (see the inset in Fig. 3(b)) whose angular velocity, calculated from the stream velocity \(\overrightarrow{{\rm{\Omega }}}=\nabla \times \overrightarrow{{\rm{v}}}\), ranges between 5 and 15 ps−1. The nanojet hits the MoS2 surface at t = 1.75 ps after the bubble collapse. The impact causes pit formation on the MoS2 surface (indicated by the red region Fig. 3(c)), resulting in an 11% volume reduction at t = 3 ps. The pit is 3 nm wide and 1 nm deep. At t = 3.7 ps, the convex hull volume of MoS2 expands by 20%.

Figure 3
figure 3

Shows cavitation bubble collapse giving rise to a nanojet and a vortex. (a) Snapshot taken at t = 1.75 ps shows velocity streamlines of the nanojet resulting from the bubble collapse. The color represents the magnitude of the stream velocity, which ranges between 3 and 6 km/s. The nanojet length and width are 6.5 nm and 3 nm, respectively, and the pressure at the tip of the nanojet is around 20 GPa. (b) The inset shows a vortex generated by the nanojet. The nanojet impact on MoS2 creates a 3 nm wide and 1 nm deep pit at t = 3 ps. (c) von Mises shear strain distribution in the pit region initiates exfoliation of MoS2.

Full size image

The nanojet impact has a dramatic effect on exfoliation of MoS2. The time evolution of exfoliation is related to pressure, shear stress and temperature distributions in MoS2, see Fig. 4. The kinetic energy imparted by the nanojet significantly increases the pressure in the pit region of the MoS2. At t = 2 ps, the instant pressure in the 4 nm wide and 1 nm deep pit (red region in Fig. 4(a)) varies between 40 and 50 GPa. The pressure in the green and yellow regions around the pit ranges between 10 and 20 GPa, whereas the rest of the MoS2 is still at zero pressure. The pressure begins to drop as the pit region expands to 7 nm in width and 4 nm in depth at t = 2.3 ps. At 3.0 ps the pressure is between 20 and 30 GPa in almost all of MoS2 (see Fig. 4(b)), and after t = 7 ps the pressure drops to 0 GPa. Figure 4(c) shows the pressure distribution after MoS2 exfoliation at t = 40 ps.

Figure 4
figure 4

Pressure, temperature and shear stress distributions in MoS2 after the nanojet impact. (a) At t = 2 ps, the pressure is 40–50 GPa in the red region; 10–20 GPa in the green/yellow region; and around 0 GPa everywhere else. (b) Shows that the pressure is around 10–20 GPa over almost the entire MoS2 at t = 3 ps. (c) Shows that the pressure is released at t = 40 ps. (d) Shows the antisymmetric distribution of shear stress Sxz at t = 2 ps, (−10 to −5 GPa in the blue region, and 5~10 GPa at the center). (e) Shows Sxz spreads to initiate exfoliation at t = 3 ps. (f) Shows that the shear stress is released at t = 40 ps. (g) Temperature in the pit (red) is around 3,000 K, whereas the rest of the sample is around 300 K. (h) shows that the high temperature region becomes 7 nm wide and 1.5 nm deep. Here the temperature in the green and yellow regions is around 1,500 K. (i) Shows that the temperature in MoS2 is uniform and around 1,600 K after exfoliation.

Full size image

Temperature fluctuations closely follow changes in the pressure in MoS2. The local temperature (calculated with peculiar velocity) in the pit ranges between 2,500 and 3,500 K at time at t = 2 ps. Away from the pit, the MoS2 remains at room temperature (indicated by the blue region in Fig. 4(g)). At t = 3 ps, the pit region cools off slightly due to expansion and the temperature distribution ranges between 2,500 and 3,000 K. At t = 4 ps, the temperature in MoS2 drops to 1,000 K except at the surface impacted by the nanojet where the temperature is 1,500 K. The temperature fluctuations persist until the very end of exfoliation, see Fig. 4(i).

The nanojet impact also generates large shear stresses, which initiate exfoliation of MoS2 into nanosheets. Figure 4(d) shows that the shear stress component Sxz in the red (compressive) and blue (tensile) regions fluctuates between 5 and 10 GPa, whereas in the rest of the MoS2 sample the shear stress is negligible. (Note, the shear-stress distribution is antisymmetric along the central plane at x = 100 Å.) A similar pattern is observed in the shear stress component Syz, which is antisymmetric along the plane y = 100 Å (see Figure S2 in Supplementary Information). These large shear stresses initiate exfoliation of MoS2 layers. Figure 4(e) shows that bulk MoS2 has partially exfoliated and the shear stress has dropped to 5 GPa at t = 3 ps. The shear stress is released after exfoliation, see Fig. 4(f).

The exfoliation is significantly enhanced every time the shockwave propagates through MoS2; see Figure S3 in the Supplementary Information. The instantaneous temperature of MoS2 increases to 1,300 K during the first encounter with the shock wave between 1.75 and 3 ps. Subsequently, the convex hull volume increases by 20% as 7,500 H2O and 1,700 IPA molecules flow inside the galleries of MoS2. The temperature of the sample decreases to 1,100 K at t = 12 ps and about 73% of the solvent remains in the MoS2 layers. The shock wave is reflected from the back end of the MD box (z = 0) at t = 8 ps and when the release wave hits MoS2 at t = 12 ps, the temperature of MoS2 increases to 1,400 K and the solvent content in the galleries of MoS2 increases with the addition of 3,000 water and 800 IPA molecules. Eighty percent of the solvent remains between the MoS2 layers after the passage of the release wave. As the shock wave reaches the opposite end of the box (z = 28.7 nm) and reflected again at time t = 18 ps, the temperature of MoS2 increases to 1,650 K at t = 23 ps, and an additional 5,700 H2O and 1,500 IPA molecules enter the galleries of MoS2.

Figure S3 in Supplementary Information shows solvent molecules between MoS2 nanosheets at t = 40 ps. Only 10% of the molecules chosen randomly are shown here. There are 13,500 water and 3,300 IPA molecules inside the sample at t = 40 ps. The volume of the convex hull continues to increase as more nanosheets exfoliate, see Figure S3(b). The swelling parameter of MoS2 plateaus at 2.0. The number of solvent molecules in the galleries of the exfoliated nanosheets increases rapidly when shock waves hit the MoS2 at t = 1.75, 12, and 18 ps; see Figure S3(c). Some of these solvent molecules diffuse out of the galleries. Overall, there is a dynamic balance between the number of solvent molecules entering and leaving the galleries of MoS2 sheets. Figure S3(c) in the Supplementary Information shows that the number of water molecules flowing in and out of the MoS2 sheets is larger than the number of IPA molecules because MoS2 is hydrophobic and water diffuses more rapidly than IPA.

To verify the theoretical claims, experimental exfoliation of MoS2 was performed in IPA and DI water mixture (see Methods section). Figure 5(a) shows the black colored MoS2 powder in IPA + DI Water solvent before exfoliation. After exfoliation, the color of the solvent changes from transparent to light green due to the presence of 2D MoS2 flakes dispersed in the solvent. A slow-motion video was captured during exfoliation to examine different stages of exfoliation. Figure 5(c) shows snapshots of MoS2 powder at different stages of exfoliation. First, a bubble forms at the interface of MoS2 and solvent as a result of sonic wave propagation in the solvent. The bubble expands after 30 s and collapses after 60 s, resulting in shock wave inside the solvent and MoS2. The bubble also carries MoS2 with it, which is indicated by the local change in color of the solvent when the bubble bursts. The process repeats itself several times, resulting in uniform dispersion of 2D MoS2 sheets in the IPA + DI water solvent.

Figure 5
figure 5

Exfoliated MoS2 with solvent between the nanosheets. (a) Commercial MoS2 in IPA + DI water solution (i) before and (ii) after exfoliation. (b) Raman spectrum of the exfoliated MoS2 flakes. (c) Snapshots of MoS2 flakes being exfoliated in IPA + DI water mixture. A small bubble formation takes place at the interface of MoS2 and solvent. The bubble expands after 90 s and collapses at 120 s, taking MoS2 powder along and dispersing it in the solvent. The color change in the liquid is an evidence of this phenomenon.

Full size image

Conclusion

In conclusion, MD simulations reveal atomistic processes underlying liquid-phase exfoliation (LPE) of LMs by sonication. Cavitation phenomenon underlies LPE experiments in which sonication probes are used to generate cavitation bubbles. We find that bubble collapse is a highly energetic process, and this is corroborated by experimental evidence of hot-spot formation in solvents24. Experiments25 indicate that temperatures in hot spots can be as high as 5,000 K. Our simulation shows that shock-induced bubble collapse results in the formation of a high-speed jet whose impact on an MoS2 surface initiates the exfoliation process. The nanojet impact raises the MoS2 surface temperature to 3,000 K and pressure to 20 GPa, and exerts a shear stress of 10 GPa. Exfoliation of MoS2 is initiated by this combination of large shear stress and temperature increase and, finally, exfoliation is significantly enhanced by repeated interactions of MoS2 with release waves resulting from the reflection of shock waves in the system.

The experiment has helped us validate the simulation. Experiments reveal that sonication causes cavitation and the collapse of cavitation bubbles generates shock waves in the solvent. Bubble creation and collapse are captured with high resolution camera and Raman spectra show that bulk MoS2 is exfoliated and flakes of MoS2 are observed. Our simulations are in accord with experimental observations. The simulation results - stress and temperature distributions, size and placement of bubbles, nature of solvent and solvent concentration - can help experimentalists with the optimization and scaling of exfoliation yield.

Method

In the MD simulation, an MoS2 crystal of dimensions (9.8 nm)3 is suspended in an H2O/IPA mixture (1:1 ratio by weight) containing a spherical cavitation bubble of radius 4.7 nm. The system contains 106 atoms and its dimensions are 19.7 nm × 19.7 nm × 28.7 nm in the x, y, and z directions, respectively. Periodic boundary conditions are applied along x and y and fixed boundary condition in the z direction. The simulation is carried out with a combination of force fields: TIP4P/200526 for water, REBO potential27,28 for MoS2, and OPLS-AA force field29 for IPA. (The OPLS-AA is commonly used for organic molecules). The interaction between MoS2 and water is described by a combination of Lennard-Jones (LJ) and electrostatic potentials with force-field parameters taken from Luan et al.30, and we apply the Lorentz-Berthelot combination rule to parameterize LJ interactions between H2O, MoS2 and IPA molecules. Electrostatic forces and energy are calculated with the PPPM method, and slab correction31 is applied to allow for fixed boundaries in the z direction. Force fields are validated by experimental data on contact angles and surface tensions15,32, see the Supplementary Information Section 1. We use the Velocity-Verlet integrator with constraints on O-H bond and H-O-H bond angles of water molecules imposed with the SHAKE algorithm33. All simulations are done with open source package LAMMPS18 and the visualization is done with OVITO and ParaView34,35.

The system is relaxed for 250 ps at 300 K with a time step 0.5 fs and then a spherical nanobubble of radius 4.7 nm is created in the solvent. Exfoliation of MoS2 depends critically on the stand-off parameter, i.e., the ratio of the distance between the bubble center and the nearest MoS2 surface to the bubble diameter. The stand-off parameter was varied between 1 and 2 to determine the optimum value for exfoliation. The system was subjected to planar shock in the z direction using a momentum mirror. The particle velocity was varied between 0.5 and 4.0 km/s and the time step was reduced to 0.1 fs during shock.

To calculate thermal and mechanical properties in the system, the MD box is divided into 20 × 20 × 58 voxels and first the center-of-mass velocity of each voxel, \({\overrightarrow{v}}_{j}={\sum }^{}{m}_{k}{\overrightarrow{u}}_{k}/{\sum }^{}{m}_{k}\) is computed and subtracted from the velocity of each atom inside the voxel to get the thermal velocity \({\overrightarrow{v}}_{k,j}\). The instantaneous “temperature” distribution in the system is determined by calculating the kinetic energy of each voxel:

$$\begin{array}{c}\frac{1}{2}{N}_{f,j}{k}_{B}{T}_{j}=\sum _{k}\frac{1}{2}{m}_{k}{\overrightarrow{v}}_{k,j}^{2},\end{array}$$
(2)

where kB is the Boltzmann’s constant, mkis the mass and \({\overrightarrow{v}}_{k,j}\) is the velocity of the kth atom in the jth voxel, and Nf,j is the number of degrees of freedom in the jth voxel. The summation is over all the atoms in the voxel.

The pressure distribution in the system is calculated from the virial stress tensor for each voxel j:

$${S}_{j}^{\alpha \beta }=\frac{1}{V}\sum _{i\in V}[-{m}_{i}{v}_{ij}^{\alpha }{v}_{ij}^{\beta }+\frac{1}{2}\sum _{k}({r}_{kj}^{\alpha }-{r}_{ij}^{\alpha }){F}_{ik}^{\beta }],$$
(3)

where V is the volume of a voxel,\(\,{r}_{ij}^{\alpha }\) and \({v}_{ij}^{\alpha }\,\)are the Cartesian components of the position and velocity of the ith atom in the jth voxel, respectively, and \({F}_{ik}^{\beta }\) is the force on atom i due to atom k. The outer summation is over the atoms in a voxel, and the inner summation is over the atoms in the neighbor lists of atom i. The pressure in a voxel is given by,

$$\begin{array}{c}{P}_{j}=-\,\frac{1}{3}Tr({S}_{j}^{\alpha \beta }).\end{array}$$
(4)

We have also estimated the exfoliation yield by computing the accessible surface area34,36 of the MoS2 sample. The surface area is calculated with a sphere of radius 4.0 Å. To compute the swelling of the sample, a 3-dimensional convex hull37 is constructed with the largest clusters of MoS2 and the number of IPA and water molecules flowing in MoS2 galleries are counted.

To measure the impact on bulk MoS2, we calculate von Mises local shear strain38 \({\eta }_{k}^{{\rm{Mises}}}\) at each atom k. We choose the initial relaxed configuration of bulk MoS2 and the current configuration to get the local transformation matrix by minimizing

$$\begin{array}{c}\sum _{j}{|{\overrightarrow{r}}_{jk}^{0}{{\boldsymbol{J}}}_{k}-{\overrightarrow{r}}_{jk}^{1}|}^{2}\to {{\boldsymbol{J}}}_{k}={(\sum _{j}{\overrightarrow{r}}_{jk}^{0T}{\overrightarrow{r}}_{jk}^{0})}^{-1}(\sum _{j}{\overrightarrow{r}}_{jk}^{0T}{\overrightarrow{r}}_{jk}^{1}).\end{array}$$
(5)

Here the summation is over the nearest neighbors of atom k, \({\overrightarrow{r}}_{jk}^{0,1}\,\)is the separation of atom j and k at the initial and current configurations. The shear strain of atom k is then computed as

$$\begin{array}{c}{\eta }_{k}^{{\rm{Mises}}}=\sqrt{{\eta }_{xy}^{2}+{\eta }_{yz}^{2}+{\eta }_{zx}^{2}+\frac{{({\eta }_{xx}-{\eta }_{yy})}^{2}+{({\eta }_{yy}-{\eta }_{zz})}^{2}+{({\eta }_{zz}-{\eta }_{xx})}^{2}}{6}},\end{array}$$
(6)

where ηab(a, b = x, y, z) are the six components of the local Lagrangian strain matrix ηk of atom k, and \({{\boldsymbol{\eta }}}_{k}=\frac{1}{2}({{\bf{J}}}_{{\rm{k}}}{{\bf{J}}}_{{\rm{k}}}^{{\rm{T}}}-{\bf{I}})\).

In the experiment, isopropanol (IPA) (99.99%; Sigma Aldrich) and de-ionised (DI) water were mixed in equal proportions. 2 mg of MoS2 powder (99.99%; Sigma Aldrich) was immersed in 50 ml of solvent mixture (IPA + DI water). MoS2 powder was allowed to sonicate for 48 hrs in the solvent. The temperature of the bath was maintained by changing the water bath every hour. Different stages of the exfoliation process were captured by a slow-motion video at 240 fps using a telephoto lens.