UNIVERSAL NUCLEATION BEHAVIOUR OF SHEARED SYSTEMS
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U NIVERSAL N UCLEATION B EHAVIOUR OF S HEARED S YSTEMS
A P REPRINT
Amrita Goswami Indranil Saha Dalal∗
Department of Chemical Engineering Department of Chemical Engineering
arXiv:2011.01020v1 [cond-mat.soft] 2 Nov 2020
Indian Institute of Technology Kanpur Indian Institute of Technology Kanpur
amritag@iitk.ac.in indrasd@iitk.ac.in
Jayant K. Singh∗
Department of Chemical Engineering
Indian Institute of Technology Kanpur
jayantks@iitk.ac.in
November 3, 2020
A BSTRACT
The study of shear-affected nucleation is a burgeoning field, with far-reaching implications for
industry and several branches of science. In this work, we combine a modified CNT approach with
equilibrium and non-equilibrium simulations to describe the shear-dependent nucleation behaviour
of the TIP4P/2005, TIP4P/Ice, mW water models, and a Lennard-Jones system. We are able to
generate extensive results for low to moderate supercoolings and a wide range of shear rates, for
which brute-force molecular dynamics is impractical. Our analyses reveal that a maximum in the
nucleation rate with shear, at a particular supercooling, is universal. Furthermore, we demonstrate
how simple parabolic fits can accurately describe the nucleation behaviour of every system analyzed.
The temperature dependence of this nucleation behaviour is also examined systematically, unearthing
the possibility of anomalous properties of water.
Keywords rare-event, nucleation, shear, seeding, Classical Nucleation Theory
1 Introduction
Nucleation events are at the heart of several complex natural phenomena and processes of industrial importance [1–6].
Each stage of crystallization, from the onset to the kinetics, is of prime importance in diverse fields ranging from
climatology [7–9], cryopreservation [10], microbiology [11] to materials science [12] and geology [13]. Despite a
large body of work, many aspects of nucleation remain notoriously impervious to theory, simulations, and experiments
[14–17].
Much of the existing literature is concerned with the nucleation of quiescent systems at molecular scales [18–30].
Practically, however, fluids can rarely be approximated as static systems and are in a continuous state of flux. Flow
dynamics and shear effects on nucleation are an active area of scientific discourse, often generating disparate predictions
and conclusions [31–36]. Some studies indicate that the presence of shear inhibits the nucleation rate [37, 38], while
others assert that the nucleation rate is enhanced by shear [39–43]. Recent simulation studies uniformly conclude that
there is a non-monotonic dependence of the nucleation rate on shear [44–48].
In the past, the homogeneous nucleation of sheared hard-sphere (HS) colloids, glassy systems [44–46, 49], and more
recently mW water under shear [47, 48], has been studied using theory and simulations. Water is a highly anomalous
liquid exhibiting several anomalies in the supercooled regime [50], but efforts have not been made to distinguish the
shear-dependent nucleation behaviour of water from other systems. On the other hand, there is evidence that the
non-monotonic behaviour of the nucleation rate with shear is a trait shown by HS colloids and mW water both [45–48].
∗
Corresponding Author
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In this work, we investigate the effects of shear on the rigid water models TIP4P/2005 [51], TIP4P/Ice [52], the
coarse-grained mW water model [53], and the Lennard-Jones system [54]. For the first time, we systematically examine
the dual influences of temperature and shear on the nucleation rates for different systems. We probe the underlying
common nucleation behaviour, while exploring the possibility of anomalies in the properties of water.
2 Theory and Methods
2.1 The Seeding Technique and CNT for Sheared Systems
In this work, we utilize the formalism of Goswami et. al. [48], in which shear is explicitly incorporated in the Classical
Nucleation Theory (CNT) equations. The approach couples the extended CNT equations with the seeding method for
calculating equilibrium properties in the absence of shear.
The free energy of a crystal nucleus in a bulk homogeneous nucleating system, under the effect of a simple volume-
preserving shear γ̇, is given by [46]
7 η 2 γ̇ 2 1 η 2 γ̇ 2 4 3
4 |∆µ0 |
F (R) = − πR3 0
+ 4πR 2
σ 0 1 + 2
+ πR , (1)
3 v 24 G 2 G 3
where F (R) is the free energy of formation of a cluster of radius R, |∆µ0 | is the chemical potential difference between
the thermodynamically stable crystal phase and the metastable liquid phase when no shear is applied, σ0 is the surface
tension or the interfacial free energy of the nucleus at zero shear, v 0 is the volume of one molecule in the crystal phase,
η is the fluid viscosity, and G is the shear modulus of the nucleus.
2 2
7 η γ̇
The 24 G2 term in Eq.(1) is a corrective shape factor accounting for the deformation of the nucleus into an ellipsoid.
In the quintessential seeding technique [19, 55], a solid crystalline cluster of known size is enclosed within an
equilibrated supercooled liquid. The method directly yields an estimate for the interfacial free energy σ0 , at the
temperature T for which the inserted seed is critical. At the maximum of the free energy in Eq.(1), we obtain the critical
nucleus size N ∗ :
3
πσ03 v 02 24G2 + 7η 2 γ̇ 2
N∗ = . (2)
162G3 2G|∆µ0 | − η 2 γ̇ 2 v 0
The height of the free energy barrier for nucleation, corresponding to the critical nucleus size N ∗ , is obtained from
πσ03 v 02 (24G2 + 7η 2 γ̇ 2 )3
F (N ∗ ) = . (3)
648G4 (2G|∆µ0 | − η 2 γ̇ 2 v 0 )2
We note that both Eq.(3) and Eq.(2) predict that the free energy barrier and critical nucleus size N ∗ will rise for
increasing shear rates.
The steady-state nucleation rate, J, can be estimated using the following CNT-based expression, derived using an
approach similar to that of Auer et. al. [18, 56]
F (N ∗ )
−
J = ρl Zf + e kB T
, (4)
where the nucleation rate J is the current or flux across the free energy barrier, in the cluster-size space and is in units
of the number of nucleation events per unit volume per unit time, f + is the rate of attachment of particles to the critical
cluster, ρl is the number density of the supercooled liquid, and Z is the Zeldovich factor.
The Zeldovich factor Z is related to the curvature at the top of the free energy barrier. Z captures the probability of
multiple re-crossings of the energy barrier. Z thus accounts for the possibility that postcritical clusters could still shrink
without growing due to such a barrier re-crossing [57]. Z can be calculated using [48]:
s 31 s
|F 00 (N ∗ )| v0 σ0 (24G2 + 7η 2 γ̇ 2 )
Z= = . (5)
2πkB T 36πN ∗2 6G2 kB T
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The expression for the shear rate-dependent attachment rate f + is given by [48]:
7η 2 γ̇ 2
+ 24Dl ∗ 2
f = (N ) 3 1+ . (6)
λ2 24G2
2 2
The shape factor 7η γ̇
24G2 is zero for perfectly spherical nuclei. Even if the shape factor is negligible, the net effect of
shear is that of the enhancement of f + due to the concomitant increase in both Dl and N ∗ .
Thus the height of the free energy barrier F (N ∗ ), and the steady-state nucleation rate J can be recast in terms of the
shear rate and equilibrium properties |∆µ0 | and σ0 , calculated when the shear rate is zero. The other quantities required
for the calculation of J are the quantities f + , v 0 , η and G.
For shear rates low enough that the fluid behaves as a Newtonian liquid, the viscosity η at a particular temperature is
constant. For the water and Lennard-Jones systems studied, we observe that the non-monotonic behaviour with respect
to shear occurs for shear rates before the onset of shear-thinning. Thus, we assume that the shear viscosity is constant at
a particular temperature for the shear rates considered in this work.
We assume that the shear modulus G of the nuclei is isotropic, although this is not strictly true for ice. However, the
variation in G for both hexagonal ice (Ih) and amorphous ice is typically in the range of ≈ 3 − 4.5 GP a [58–60] and
does not significantly impact the calculated nucleation rates [48].
2.1.1 Calculation of the Shear Viscosity
In this work, we have used non-equilibrium molecular-dynamics (NEMD) simulations to estimate the shear viscosities
for various water models and the Lennard-Jones system at moderate and high supercooling.
Figure 1: The shear viscosity estimated from NEMD simulations for different water models at 298 K and 1 atm.
The black dotted line represents the experimentally obtained viscosity of water. The viscosities for the TIP4P/Ice,
TIP4P/2005 and mW water models are denoted by filled square symbols in blue, red and green, respectively. The
standard errors were calculated from 7 − 8 independent trajectories.
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Figure 1 shows the viscosity results obtained from NEMD simulations. The viscosity plateaus in the regime of
Newtonian fluid behaviour, which is used to estimate the shear viscosity at zero shear. The viscosity decreases rapidly
for higher shear rates, at which significant shear-thinning is observed.
Table 1: Shear viscosities (in mPas) at 298 K and 1 atm for the water
models studied in this work.
TIP4P/Ice TIP4P/2005 mW Experiment [61]
1.799 ± 0.054 0.876 ± 0.008 0.317 ± 0.014 0.896
The viscosity at 298 K for the TIP4P/2005 obtained from Green-Kubo calculations is 0.855 cP [62]. This agrees well
with our estimate of the shear viscosity in Table 1.
From Figure 1 and Table 1, we can surmise that the TIP4P/2005 yields the viscosity closest to the experimental data.
Our results show that TIP4P/Ice overestimates and mW underestimates the viscosity significantly.
2.1.2 Calculation of the Diffusion Coefficient
The diffusion coefficient Dl has been reported to depend linearly on the shear rate for the mW water model [48].
The diffusion coefficient is also influenced by the temperature [63]. We expect that a linear expression for the two-
dimensional diffusion coefficient (Dl ), at each temperature, can be determined for both rigid water models and the
Lennard-Jones system.
In this work, the shear rate is imposed in the x dimension. For a particular temperature and shear rate, we can determine
Dl , in the yz plane. We exclude the x dimension to rule out the motion due to convection. The two-dimensional Dl can
be estimated using the following relation [64]
1
l = lim h[y(t) − y(0)]2 + [z(t) − z(0)]2 i, (7)
t→∞ 4t
where Dl is the two-dimensional self-diffusion coefficient, t is the time duration measured from a reference time origin,
and h[y(t) − y(0)]2 + [z(t) − z(0)]2 i is the ensemble average over all particles and time origins.
3 Simulation Details
Equilibrium and non-equilibrium molecular dynamics (NEMD) trajectories were obtained using LAMMPS (Large-scale
Atomic/Molecular Massively Parallel Simulator) [65]. The SLLOD algorithm [66] and the Lees-Edwards boundary
conditions [67, 68] were used to generate simple shear flows in the x dimension. The Nose-Hoover thermostat was used
to control the temperature [69].
We have analyzed four different systems in this work: the rigid water models TIP4P/2005 [51] and TIP4P/Ice [52], the
coarse-grained monoatomic water (mW) model [53], and the truncated and shifted Lennard-Jones (LJ) potential [54]
for Argon.
We simulated the mW model at 235 K, 240 K, 255 K, and 260 K. The melting point temperature for the mW model
is 274.6 K, at 1 bar [53]. We used a time-step of 5 f s for the Velocity-Verlet integration scheme. The system size was
4096 water molecules.
Temperatures of 237.5 K, 232.5 K and 232.5 K were simulated for the TIP4P/2005 model, corresponding to super-
coolings of 14.5 K, 19.5 K and 29.5 K, respectively. For TIP4P/Ice, simulations were performed for supercoolings
of 14.5 K, 19.5 K and 34.5 K, at temperatures of 257.5 K, 252.5 K, and 237.5 K, respectively. The long-range
electrostatics were treated with the particle-particle particle-mesh (PPPM) algorithm [70]. The rigid water molecules
were constrained by the SHAKE algorithm [71]. Systems were composed of 4029 water molecules for both TIP4P/2005
and TIP4P/Ice.
Reduced units, in terms of the depth of the interaction potential, , and the distance at which the potential vanishes, σ,
have been used throughout for the LJ potential. The LJ interactions were truncated at 8.5 Å. The number of atoms in
the simulations was 13500, and the density at each temperature was taken to be that at a pressure of −0.02/σ 3 .
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For all systems, the two-dimensional diffusion coefficients in the absence and presence of shear were calculated using
the VMD Diffusion Coefficient Tool [72]. Long simulation times of at least 200 ns for mW and LJ, and ≈ 100 − 200 ns
for the rigid water models were maintained to ensure that the linear diffusive regime was attained.
The viscosity at each temperature for every model was estimated by a nonequilibrium method with periodic shear flow
[73]. Errors were estimated from independent NEMD simulations for each data point.
4 Results and Discussion
4.1 Viscosity of Supercooled Liquids
Figure S1 depicts the NEMD calculations of the shear viscosity, at various supercoolings, for the TIP4P/2005, TIP4P/Ice
and mW water models. Table 2 presents the results of these viscosity calculations for the water models, and compares
the values with experimental data at the nearest temperatures.
Table 2: Shear viscosities at various temperatures and 1 atm for the water
models studied in this work.
Model Temperature (K) Viscosity (mPas) Experiment (mPas)
TIP4P/2005 237.5 12.44 ± 2.59 15.3 ± 3 [74]
TIP4P/2005 232.5 24.79 ± 2.87 -
TIP4P/2005 222.5 117.6 ± 12.2 -
TIP4P/Ice 257.5 15.4 ± 0.29 3.558 ± 0.156 [75]
TIP4P/Ice 252.5 23.17 ± 0.76 4.738 ± 0.21 [75]
TIP4P/Ice 237.5 175.3 ± 10 15.3 ± 3 [74]
mW 260 0.487 ± 0.0057 3.059 ± 0.134 [75]
mW 255 0.519 ± 0.0062 3.967 ± 0.174 [75]
mW 240 0.683 ± 0.006 12.68 ± 0.555 [75]
mW 235 0.683 ± 0.006 25.88 ± 1.13 [75]
For moderate and low supercooling, we observe that TIP4P/2005 approaches experimental results the most closely.
The shear viscosities calculated for LJ are listed in Table S1.
4.2 Effect of Shear on Diffusion for Supercooled Liquids
The applied shear rate has the effect of enhancing the diffusion coefficient. It has been shown that a linear relationship
with shear can be established for the diffusion coefficients at a constant temperature, for the mW model [48]:
Dl = D0 + cγ̇, (9)
where Dl is the two-dimensional diffusion coefficient for a particular shear rate γ̇, at a temperature T , D0 is the
diffusion coefficient when the shear rate is zero, and c is a fitting parameter with units of squared length.
The linear fits for the diffusion coefficients at various temperatures are shown in Figure S2 and Figure S3 for TIP4P/2005
and TIP4P/Ice, respectively. Table 3 contains the values of D0 and c calculated from the linear fits to the data.
Table 3: Values of D0 and c for the TIP4P/2005, TIP4P/Ice and mW
models. The data for the mW model were obtained from Goswami et. al.
[48]
Model Temperature (K) D0 (×10−11 m2 /s) c (×10−20 m2 )
TIP4P/2005 237.5 15.08 1.18
TIP4P/2005 232.5 7.73 1.25
TIP4P/2005 222.5 2.92 1.20
TIP4P/Ice 257.5 15.16 1.19
TIP4P/Ice 252.5 9.47 1.17
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Model Temperature (K) D0 (×10−11 m2 /s) c (×10−20 m2 )
TIP4P/Ice 237.5 1.72 1.22
mW 260 371.5 0.485
mW 255 334.5 0.533
mW 240 249.6 0.617
mW 235 214.5 0.663
We observe that similar behaviour is exhibited by the LJ system (Figure S4). Table S2 contains the values of D0 and c
for LJ.
4.3 Temperature Dependence of Transport Properties
Figure 2: Power law fits to the shear viscosities for the (a) TIP4P/2005 model, (b) TIP4P/Ice model, and (c) mW model.
The fitted expression A0 ( TTs − 1)−γ has been shown by solid lines in red, blue and green for the TIP4P/2005, TIP4P/Ice
and mW water models. The black filled square symbols denote the viscosity values obtained from NEMD simulations.
The filled blue triangles show the shear viscosities calculated by Lee et. al. [76], for the TIP4P/2005 model.
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Figure 3: Power law fits to the diffusion coefficients at zero shear, D0 , for the (a) TIP4P/2005 model, (b) TIP4P/Ice
model, and (c) mW model. Solid lines in red, blue and green depict the fitted power law expression for the TIP4P/2005,
TIP4P/Ice and mW water models, respectively. The black filled square symbols represent the values of D0 .
The viscosity of water at different temperatures can be fitted to power law expressions to obtain the temperature
dependence of viscosity [75, 77]. We have fitted the viscosities, estimated from NEMD simulations, using a power law
of the following form:
−β
T
η(T ) = A0 −1 , (8)
Ts
where A0 , Ts and γ are fitting parameters. A0 has units of dynamic viscosity, Ts has units of temperature and β is a
constant.
Figure 2 shows the power law fits to the viscosity data for TIP4P/2005, TIP4P/Ice and mW.
The transport properties of water, including the diffusion coefficient at zero shear, can also be fitted to power law
expressions of the form in Eq. (8) [78]. In the case of fits to the diffusion coefficients, A0 has units of diffusivity. In the
power law expression of Eq.(8), Ts corresponds to the mode-coupling temperature in the mode-coupling theory picture
[79–81], such that Ts is greater than the glass transition temperature Tg . Ts can also be interpreted as a ‘common
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singularity temperature’ [78]. The theories predict that a power law fit to diffusion coefficient data would yield a
common Ts value for all dynamic properties, i.e., a common singularity temperature.
This prediction has been shown to be consistent with certain experimental data and simulation results, using the SPC/E
water model [77]. On the other hand, conflicting experimental results have also violated these theories, showing that Ts
obtained from power law fits to diffusion coefficient data is lower than Ts obtained from power law fits to viscosity
values [75].
In order to test the agreement of our simulation results with mode-coupling theory predictions, we also fit D0 for the
TIP4P/2005, TIP4P/Ice and mW models.
Figure 3 depicts the power law fits to the diffusion coefficients in the absence of shear for TIP4P/2005, TIP4P/Ice and
mW.
Table 4: Comparison of Ts obtained from power law fits to the shear
viscosities and diffusion coefficients D0 for the TIP4P/2005, TIP4P/Ice
and mW water models.
Model Ts from fit to η (K) Ts from fit to D0 (K) Tg
TIP4P/2005 206 ± 5 209 ± 16 193, 199, 197 [82]
TIP4P/Ice 220 ± 1.7 205 ± 9.5 201, 209, 193 [82]
mW 207 ± 5.3 205 ± 12 150 [83]
Table 4 lists the Ts values calculated from the power law fits shown in Figure 2 and Figure 3. For all the water models,
Ts is greater than the glass-transition temperature Tg . The Ts values for TIP4P/2005 and mW agree within the limit of
errors; however, the Ts diverges for TIP4P/Ice.
4.4 Universal Nucleation Behaviour for Supercooled Liquids
Simulations have shown that the nucleation rate exhibits non-monotonicity with respect to the applied shear rate for
colloidal systems, at a constant temperature [44, 46]. This non-monotonicity has also been observed for the mW water
model[47, 48]. Here, we show that the non-monotonic behaviour of the shear-dependent nucleation rates is exhibited by
the rigid water models TIP4P/2005 and TIP4P/Ice, as well as the LJ system. The non-monotonicity is thought to arise
from the conflicting effects of shear on the free energy barrier and the kinetic pre-factor [46]. We recall that Eq. (3)
predicts that the free energy barrier grows with increasing shear rates, thus retarding nucleation. Concomitantly, the
shear rate also enhances the kinetic pre-factor, which is favourable for nucleation.
Table 5: Values of the interfacial energy σ0 , chemical potential µ0 , volume
of a single ice molecule v 0 , and liquid density rhol , required for the
calculation of the nucleation rate for the water models studied in this
work. The attachment length λ has been taken to be 3.5 Å for all cases.
σ0 and µ0 were obtained from Espinosa et. al. [25].
Temperature
Model (K) σ0 (mN/m) µ0 (kcal/mol) v 0 (×10−29 m3 ) ρl (g/cm3 )
mW 260 29.5 0.0669 3.0534 1.0022
mW 255 29.0 0.0895 3.0503 1.0024
mW 240 28.9 0.1553 3.0441 1.0031
mW 235 28.55 0.1746 3.044 1.0031
TIP4P/2005 237.5 25.9 0.0612 3.242 0.970
TIP4P/2005 232.5 25.0 0.0801 3.238 0.967
TIP4P/2005 222.5 20.4 0.1137 3.235 0.925
TIP4P/Ice 257.5 26.3 0.0629 3.555 0.952
TIP4P/Ice 252.5 25.4 0.0826 3.292 0.947
TIP4P/Ice 237.5 23.7 0.1335 3.285 0.932
Table 5 and Table S3 list the values of the parameters required to calculate the nucleation rates.
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Figure 4: (a) Family of parabolas fitted to the normalized relative nucleation rate varying with shear for the rigid water
models TIP4P/2005 and TIP4P/Ice. ∆T represents the supercooling for each model and temperature. (b) Parabolic fits
to the normalized relative nucleation rate for the coarse-grained mW model. (c) Parabolas describing the trend in the
normalized relative nucleation rate with the shear rate for the LJ system. Each parabola describes the behaviour of the
nucleation rate with shear rate, for a particular temperature and system. T ∗ refers to the reduced temperature in units of
/kB .
We introduce a normalized relative nucleation rate, defined with respect to J0 , the nucleation rate at zero shear, and
Jmax , the highest nucleation rate observed at a particular temperature [48].
J−J0
Figure S5 shows the variation of the dimensionless relative nucleation rates Jmax −J0 with the shear rate for the
TIP4P/2005, TIP4P/Ice, mW and LJ models, at different supercoolings. For every model and for all temperatures,
J−J0
non-monotonic behaviour is observed: Jmax −J0 increases up to a maximum and falls beyond an optimal shear rate. We
J−J0
define the optimal shear rate, γ̇opt , as the shear rate for which Jmax −J0 is maximized.
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We have found that, for all the systems studied in this work, parabolic fits approximate the nucleation rate behaviour
with very high accuracy (Figure S6). A family of parabolas with the following form can describe the nucleation
behaviour at a particular temperature T and supercooling ∆T :
J − J0 2
= 1 − a (γ̇ − γ̇opt ) , (11)
Jmax − J0
where γ̇opt is the vertex of the parabola corresponding to the maximum of the nucleation rate curve, a is a temperature
and system dependent parameter determining the width of the parabola.
J−J0
Figure 4 shows the family of parabolas describing the variation of Jmax −J0 with shear, for the water models and LJ
system. In fact, normalizing the shear rates by dividing by the optimal shear rates yields a parabola with a = 1 and a
vertex at γ̇/γ̇opt = 1 in all cases.
Figure 5: Variation of the normalized relative nucleation rate with the normalized shear rate, γ̇/γ̇opt , plotted alongside
the corresponding parabolic fit for every system and temperature considered in this work. The parabolic equation
2
1 − (γ̇/γ̇opt − 1) has been denoted by a dotted cyan line.
J−J0
Figure 5 visually presents the excellent agreement of the parabolic fit with Jmax −J0 , plotted against the shear rates
normalized by the optimal shear rates, for every system.
The most prominent inference that can be drawn from Eq. (11) and Figure 5 is that there exists a single maximum
nucleation rate, at any given temperature for every system. For shear rates higher than γ̇opt , the nucleation rate decreases.
Despite the complex interactions of shear-dependent terms in Eq. (4), the simple functional form of Eq. (11) works
well for each system analyzed. The results suggest that this behaviour is a fundamental property of liquids that obey the
formalism outlined in Section 2.1.
4.5 Dependence of the Optimal Shear Rates on Temperature
Previous work on the mW model suggests that the shear-dependent nucleation rates have a non-linear dependence on
the temperature [48]. The nucleation rate curves for TIP4P/2005 and TIP4P/Ice, in Figure S5 and Figure 4, also reveal
the possibility of a non-linear dependence on the temperature.
This non-linear temperature dependence of the nucleation rate curves could arise from the inclusion of several
temperature-dependent parameters in the expression for the nucleation rate (Eq.(4)). The transcendental nature of the
nucleation rate expression prevents us from directly solving an analytical expression for the optimal shear rate.
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In order to approximate the nucleation rate curves and obtain optimal shear rates for intermediate temperatures, we use
various assumptions and employ statistical inference, detailed in the Supporting Information.
Figure 6: Variation of the predicted optimal shear rates, at different supercoolings, for the (a) TIP4P/2005 model,
(b) TIP4P/Ice model, and (c) the mW model. The optimal shear rates at each temperature were calculated using
approximations described in the Supporting Information, and are denoted by filled blue squares. The optimal shear
rates obtained using the more precise values listed in Table 5 are depicted as open red circles.
Figure 6 depicts the variation of the predicted optimal shear rates with supercooling for the TIP4P/2005, TIP4P/Ice
and mW water models. The more exact calculations for each supercooling and water model are shown alongside the
predicted values. For all the water models, the predicted optimal shear rates increase continuously up to a maximum.
Figures S7-S9 graphically present the agreement of the predicted nucleation rate curves and the nucleation rate curves
obtained in the preceding section using Table 5. It is also evident from Figure 6(a) and Figure 6(b) that the rigid water
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models seem to follow a similar trend. The maximum in the optimal shear rates for TIP4P/2005 and TIP4P/Ice is
observed at a supercooling of ≈ 22 K and ≈ 24 K, respectively. On the other hand, the maximum in the optimal shear
rates is observed at a supercooling of ≈ 55.6 K for the mW water model, which is significantly higher compared to
that of the rigid water models.
Figure 7: Temperature dependence of the optimal shear rates, at different temperatures, for the LJ system. The predicted
optimal shear rates at each temperature were calculated using approximations described in the Supporting Information,
and are plotted as filled blue squares. The open red circles denote the optimal shear rates obtained using the values
described in Table S3.
Figure 7 presents the variation in the predicted optimal shear rates with temperature for the LJ system. Clearly,
monotonic behaviour is observed in the temperature range 0.52 ≤ T ∗ ≤ 0.59. This trend is distinct from that of both
the rigid and coarse-grained water models. The distinctive behaviour, manifested by the rigid water models and the
coarse-grained mW model, hints that this may be an anomalous property of water.
In addition, although the predicted data offer valuable insight into the temperature dependence of the optimal shear rates
and nucleation curves, there are shortcomings associated with the assumptions used. Evidently, the approximations are
more reliable for the coarse-grained mW model and LJ model, compared to the rigid water models. Deciphering the
intricate composite effects of temperature on the nucleation rate is non-trivial and requires further investigation.
5 Conclusions
In summary, we have reported the effects of shear on the nucleation rates at different temperatures, for the TIP4P/2005,
TIP4P/Ice, mW water models and the LJ system. Nucleation events at low and moderate supercoolings are notoriously
difficult to simulate, and such extensive calculations are virtually inaccessible to brute-force molecular dynamics. By
employing a formalism based on modified CNT equations, we were able to obtain nucleation rate curves for a wide
range of shear rates at moderate and low supercoolings.
The transport properties and their relationship with the shear rate were analyzed for the water models and LJ system.
We found that, in all cases, the two-dimensional diffusion coefficient increase linearly with the applied shear rate, at a
particular temperature. We also noted that, of the TIP4P/2005, TIP4P/Ice, and mW water models, TIP4P/2005 shows
the closest match with experimental viscosities.
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In accordance with previous simulation results for colloids, glassy systems and mW water [45–48], we confirmed that
the nucleation rate curves exhibit non-monotonic behaviour with shear, at a particular supercooling. We devised an
J−J0
expression for a family of parabolas , describing the normalized relative nucleation rate Jmax −J0 with the shear rate
γ̇. Despite the complicated dependence of shear in the transcendental equation for the nucleation rate, the parabolic
fits yield excellent agreement to the nucleation rate curves and are valid for every system considered in this work. We
infer that the existence of a maximum in the nucleation rate with shear is a universal property of systems that obey the
extended CNT equations.
We systematically examined the temperature dependence of the nucleation rate curves for all the systems considered in
this work. We defined optimal shear rates,γ̇opt , as the shear rates corresponding to the maximum in the nucleation
rate, for a temperature T . We discovered that, for every water model, the optimal shear rates have a non-monotonic
dependence on the temperature, increasing up to a maximum for a particular supercooling. The trend of the optimal
shear rates with supercooling is similar for the rigid water models TIP4P/2005 and TIP4P/Ice. Although a maximum is
observed in the optimal shear rates for mW as well, the behaviour is markedly different from that of the rigid water
models. This difference could be attributed to the fact that mW is a coarse-grained water model, devoid of hydrogens.
On the other hand, the LJ system shows monotonic dependence on the temperature, in the range 0.52 ≤ T ∗ ≤ 0.59. We
speculate that the non-monotonic behaviour of the optimal shear rates, and consequently of the nucleation rate curves,
with temperature could be an anomalous property of water.
Thus, we have uncovered both commonalities in the nucleation behaviour and also the potentially anomalous behaviour
of water. Our results provide insight into the previously unexplored, intriguingly complex interplay of temperature and
shear, affecting the nucleation rate.
6 Acknowledgements
This work was supported by the Science and Engineering Research Board (sanction number STR/2019/000090 and
CRG/2019/001325). Computational resources were provided by the HPC cluster of the Computer Center (CC), Indian
Institute of Technology Kanpur.
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