Classical nucleation theory for the crystallization kinetics in sheared liquids

While statistical mechanics provides a comprehensive framework for the understanding of equilibrium phase behavior, predicting the kinetics of phase transformations remains a challenge. Classical nucleation theory (CNT) provides a thermodynamic framework to relate the nucleation rate to thermodynamic quantities such as pressure difference and interfacial tension through the nucleation work necessary to spawn critical nuclei. However, it remains unclear whether such an approach can be extended to the crystallization of driven melts that are subjected to mechanical stresses and flows. Here, we demonstrate numerically for hard spheres that the impact of simple shear on the crystallization rate can be rationalized within the CNT framework by an additional elastic work proportional to the droplet volume. We extract the local stress and strain inside solid droplets, which yield size-dependent values for the shear modulus that are about half of the bulk value. Finally, we show that for a complete description one also has to take into account the change of interfacial work between the strained droplet and the sheared liquid. From scaling reasons, we expect this extra contribution to dominate the work formation of small nuclei but become negligible compared to the elastic work for droplets composed of a few hundreds of particles.

Figure 1

Solid-liquid coexistence under shear. Sketch of the formation of a solid droplet within the sheared melt. We distinguish in orange “solidlike” particles that have a high bond-order symmetry from liquid particles. For the thermodynamic modeling, we employ Gibbs concept of a sharp dividing surface separating the homogeneous droplet (orange area) from the surrounding liquid. The inset shows the simple-shear flow geometry with strain rate $\dot{\gamma }$ used in our simulations.

Figure 2

Shear viscosity and shear modulus. (a) Shear viscosity $\eta$ as a function of the liquid density ${\rho }_l$ for $\dot{\gamma }=0.0,0.036$, and 0.084. The solid line for $\dot{\gamma }=0$ is a Vogel-Fulcher-Tammann fit [50]. Solid lines for $\dot{\gamma }>0$ are guides to the eye. (b) Bulk shear modulus $G_{\infty }$ as a function of the solid density ${\rho }_s$ for the WCA solid (empty black circles) and for true hard spheres (filled purple circles) taken from Ref. [51]. The solid line is a linear regression of $ln{G}_{\infty }$.

Figure 3

Extracting kinetics and free-energy barriers. (a) Splitting probability $P_B$ as a function of the nucleus size $n$ for various imposed shear rates $\dot{\gamma }$. Lines are fits to Eq. (14). (b) Time evolution of the largest nucleus $n\left(t\right)$ showing multiple unsuccessful nucleation events for $\dot{\gamma }\simeq 0.06$. The red horizontal line indicates the quiescent critical nucleus size $n_c(\dot{\gamma }=0)\simeq 40$. (c) Mean first passage time as a function of $n$. Solid lines are the model functions from (a) but scaled by the nucleation time ${\tau }_x=1/j$. The inset shows a zoom for data at small strain rates. (d) Free-energy reconstruction using $P_B$ (cf. main text). Simulations in this figure are performed at $\varphi \simeq 0.542$.

Figure 4

Extracting response coefficients. (a) Nucleation rate $k$, (b) nucleation barrier $W$, (c) critical nucleus size $n_c$, and (d) Zeldovich factor $z_c$ as functions of the squared shear stress ${\sigma }_l^2$ in the ambient liquid. Response coefficients $\{a_k,a_W,a_n,a_z\}$ are extracted from the lines, which are linear regressions only taking into account data with ${\sigma }_l^2<0.15$. Packing fractions are ranging from 0.539 (dark red) to 0.553 (light orange).

Figure 5

Testing CNT predictions. Relative test of response coefficients $\{-a_k/\mathrm{\Delta }{F}_0,a_W,2a_n/3,-a_z\}$ as a function of the packing fraction $\varphi$. The inset shows $a_W$ compared with the bulk prediction $a_W^{\infty }=1/\left(G_{\infty }\mathrm{\Delta }{P}_{\infty }\right), G_{\infty }$ and $\mathrm{\Delta }{P}_{\infty }$ being the bulk solid shear modulus and the bulk pressure difference, respectively. The green line is an exponential decay.

Figure 6

Seeding of droplets under shear. (a) Evolution of the droplet size for ten independent fleeting trajectories starting from a seed with $n\simeq 300$ for $\varphi \simeq 0.518$ and $\dot{\gamma }\simeq 0.036$. (b) Evolution of total shear stress in the system. The red line corresponds to our parametrization ${\sigma }_l(\dot{\gamma },{\rho }_l)=\eta (\dot{\gamma },{\rho }_l)\dot{\gamma }$ [cf. Fig. 2]. (c) Critical droplet sizes $n_c$ as a function of ${\sigma }_l^2$. Lines are linear regressions. (d) Response coefficients $a_W$ and $2a_n/3$ as a function of the packing fraction $\varphi$. Empty and filled symbols are from seeded and direct simulations, respectively.

Figure 7

Density and stress profiles. (a) Average radial density distribution $\rho \left(r\right)$ with respect to the center of mass of droplets for $\varphi \simeq 0.542, \dot{\gamma }\simeq 0.036$ and nucleus sizes $n=80,250,500,$ and 1000 from black to light orange. (b) Average stress profile $\sigma \left(r\right)$. The horizontal dashed line indicates our parametrization of the liquid shear stress [cf. Fig. 2].

Figure 8

Solid strain reconstruction. (a) Snapshot of a configuration with a strained droplet generated by the seeding method with $n=1000$ at $\varphi \simeq 0.542$ and $\dot{\gamma }\simeq 0.036$. (b) Evolution of the average solid and liquid stress after switching off the shear flow. (c) Probability distribution of two times the off-diagonal of the Green tensor $2E_{xy}$ for $t=5{\tau }_B$. The solid line is a normal distribution. (d) Evolution of the average solid and liquid strain after switching off the imposed flow.

Figure 9

Shear modulus. (a) Solid shear stress and (b) strain as a function of the droplet radius $R_0$ for strain rates $\dot{\gamma }=0.036,0.06$, and 0.084. (c) Shear modulus $G={\sigma }_s/{\gamma }_s$ (filled symbols) of small droplets compared with $G_{\infty }\left({\rho }_s\right)$ (empty symbols) employing the actual droplet density ${\rho }_s$. The red horizontal line indicates the bulk estimate $G_{\infty }\left({\rho }_s^{\infty }\right)\simeq 99$. All simulations in this figure are performed at $\varphi \simeq 0.542$. (d) Corrected response coefficient $a_W$ (empty disk, obtained in analogy with Fig. 4 but plotting vs ${\sigma }_s^2$) and $a_W^{\text{CNT}}$ (cross, employing $G\simeq 65$ and $\mathrm{\Delta }P\simeq 0.60$).

Figure 10

Elastic work. Empty symbols are the excess “free energy” $\mathrm{\Delta }F$ [Eq. (19)] and filled symbols corresponds to the direct evaluation [Eq. (20)] of the elastic work $W_e$ as a function of nucleus size $n$ at $\varphi \simeq 0.542$ for strain rates $\dot{\gamma }\simeq 0.036,0.06,0.084$ (bottom to top). Dashed lines model the elastic work as $W_e\left(n\right)\sim n^{1+1/3.}$. The inset shows the difference $\mathrm{\Delta }W(n;\dot{\gamma })=\mathrm{\Delta }F(n;\dot{\gamma })-W_e(n;\dot{\gamma })$, where $W_e(n;\dot{\gamma })$ is taken from an extrapolation of $W_e$ to smaller sizes $n$.

Figure 11

Brownian dynamics vs molecular dynamics. Comparison between the nucleation rates $k$ extracted from BD and MD simulations as a function of the square of the strain rate ${\dot{\gamma }}^2$.

Figure 12

The fast growth method. (a) Excess chemical potential ${\mu }_l-{\mu }_l^{\text{id}}$ as a function of the liquid density ${\rho }_l$. Shown are results from the fast growth method (purple symbols), Widom insertion method (black symbols), and thermodynamic integration (black solid line). (b) Insertion work $w_{\text{ex}}$ as a function of the strain rate $\dot{\gamma }$ for various packing fractions $\varphi$. Lines are quadratic fits.