Article
Cite This: J. Phys. Chem. B 2018, 122, 7569−7583

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Importance of Solvents’ Translational−Rotational Coupling forTranslational Jump of a Small Hydrophobic Solute in Supercooled
Water
Vikas Dubey, Nitesh Kumar, and Snehasis Daschakraborty*
Department of Chemistry, Indian Institute of Technology Patna, Patna 801103, Bihar, India

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S Supporting Information*

ABSTRACT: Despite clear evidence of sudden translational jumpoccurrence of a solute in supercooled water, a detailed mechanism of thisjump is still lacking. A previous work [Indra, S.; Daschakraborty, S. Chem.
Phys. Lett. 2017, 685, 322−327] put forward a mechanism of this jumpfrom an initial solvent cage to a ﬁnal one. The proposed mechanism is
astoundingly similar to that of the electron/proton transfer reaction inaqueous solution. The above study identiﬁed the spatial prearrangement(rearrangement before the jump occurrence) of cage forming water solvent
molecules as the actual reaction coordinate. However, the study completely
ignored the contribution of the orientational prearrangement of solvent
water molecules. In this study, we have monitored both the spatial and theorientational prearrangements of water solvent molecules at subzero
temperatures during the jump occurrence of the solute. We have foundoverwhelming contributions of both the spatial and orientational
prearrangements of water, which symmetrize the hydration structure at the initial and ﬁnal cage positions to facilitate thejump event. Through a systematic temperature dependence study (from T = 240 to 270 K), we have found clear evidence that astrong synchronization between translational and rotational prearrangements of the solvent water molecules is crucial for thesolute’s jump from one solvent cage to another in supercooled water (below T = 252 K). The above translation−rotationsynchronization is probably due to the cooperative movement of solvent water molecules forming clusters in the supercooled
region. Since these cooperative dynamics are the consequence of the spatiotemporal heterogeneity in the medium, we infer thatthe large-amplitude translational jump of the nonpolar solute probably stems from the spatiotemporal heterogeneity ofsupercooled water. At temperatures above the melting point, this cooperativity is partly lost since the translational andorientational prearrangements become somewhat independent of each other.

1. INTRODUCTION
Decoupling between the self-translation and rotational
diﬀusions of supercooled water (water below its freezing
point) has been witnessed in recent experimental studies.1−4
An increasing order of violation of the Stokes−Einstein (SE)relationconnecting the self-translation diﬀusion coeﬃcient(DT) of the solvent through the radius (r) of the same
molecule with the viscosity of the medium (η) by the equation
DT = kBT/6πηris witnessed with gradual cooling of thesystem from room temperature. On the other hand, theStokes−Einstein−Debye (SED) relationconnecting the selfrotational diﬀusion coeﬃcient (DR) through the volume (V) of
the solvent molecule with the viscosity of the medium (η) by
the equation DR = kBT/6ηVremains almost valid.
Similar decoupling is observed in other glass-forming liquids
and solution of a tracer. Here, the translation diﬀusion of thesolvent5−16 or the tracer17−29 is decoupled from the rotational
diﬀusion of the same entity. We note that the above violationof the SE relation is very similar for both the tracer and thesolvent because the dynamics of the dissolved solute isnaturally inﬂuenced by the solvent’s motion. In most of the© 2018 American Chemical Society

cases, larger violation of the SE relation compared to the SED
relation was found. However, the opposite picture also does
exist in the literature. A molecular dynamics (MD) simulation
predicted lager violation of the SED relation than the SE
relation in supercooled water.17,23 This is exactly opposite ofwhat has been observed in the recent experiment.1 A more
recent MD simulation,16 however, supported the experimentand showed that the violation of the SE relation is larger thanthat of the SED relation when the viscosity is incorporated in
the calculation.
Multiple factors can cause the above decoupling between thetranslational and rotational diﬀusion in supercooled water.
Microscopic spatiotemporal heterogeneity can be one ofthem.30−49 It is also possible that the violation of SE relationin supercooled water can be due to the crossing of the Widom
line, arising from the liquid−liquid critical point.3,4 Solute size
is another factor for the above behaviors.20,24,28 It is
Received: April 4, 2018
Revised: July 11, 2018
Published: July 12, 2018
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In this article, we explore more detailed pathway of the jumpmechanism of a very simple nonpolar solute, xenon (Xe), inwater at various temperatures, ranging from T = 240 to 270 K.
We study the role of both the spatial and orientational
rearrangements of water solvent molecules around the Xe
solute on the translation jump. Xe is indisputably a
prototypical hydrophobic solute, and its hydration structure(both the spatial and orientational) resembles other popularly
known hydrophobic solutes like methane, ethane, etc.,
immersed in liquid water.82−89 Our primary objective is tomonitor the translational and rotational rearrangements ofnearby solvent water molecules with time during the solute’s
jump from one cage to another in supercooled water and water
above freezing temperature. For the present study, we use MD
simulation technique, which we detail below.

worthwhile to mention that a recent MD simulation study hassuggested that a strong coupling between the translation androtational motions of a polyatomic ion in water may incite aviolation of the SE relation.21 A hypothesis was put forward byGoldstein on the eﬀect of temperature on the mechanism oftranslation of liquid molecules.50 The theory states that at low
temperature, the liquid molecules translate in the rugged freeenergy landscape and therefore hop from one free-energy
minima to another through saddle points. On the other hand,
the liquid molecules translate via free Brownian motion at hightemperature as the thermal energy becomes comparable to thebarrier heights or higher. Several theoretical and computer
simulation studies conﬁrmed characteristics of this picture.51−57 However, the detailed molecular mechanism forthe solute’s hopping from one free-energy minima to another isstill elusive. Furthermore, the exploration of the detailed
pathway of a translational jump of a hydrophobic solute insupercooled water is also important in cold denaturation ofprotein. A simulation analysis evidenced increase of ﬂuctuationof some protein domains with decreasing temperature of thesystem, which plays an important role in cold denaturation ofprotein.58 Also, the unexpectedly high ﬂuctuating domains are
generally localized within the protein core regions. It is wellknown that the water-soluble proteins have a hydrophobic
core, in which side chains are buried from water, which
stabilizes the folded state.59,60
A recent study by one of us has focused on the pathway of
the translational jump of a model nonpolar solute from one
solvent water cage to another using classical molecular
dynamics (MD) simulation.61 The solute rattles inside aclathrate-like cage of solvent water.62 After some waiting time,
the solute escapes (partly through large-amplitude translationaljump) from the cage and then gets trapped in a new cage. Thisrattling−jump−rattling mechanism is consistent with Goldstein’s picture of molecular translation in liquid at low
temperature.50−57 This study identiﬁed the solvent coordinate
as the key reaction coordinate for the jump process. Thissolvent activated process is very similar to other solution-phasereactions like the Ando−Hynes picture (Grotthuss mechanism) of proton transfer (PT),63−66 the Marcus picture ofelectron-transfer reaction,67 and the Laage−Hynes rotational
jump mechanism of water.68,69 Presolvation of water moleculesaround the solute’s future solvent cage position starts evenwhen the solute continues rattling inside its initial cage. The
solute jumps to the ﬁnal cage only when the latter becomes
structurally and energetically comparable to the initial cage,
accommodating the solute. Therefore, the reaction coordinate
for the solute’s jump is the collective solvent coordinate insteadof the solute’s rattling motion inside the cage.70
Notwithstanding the fact that the above study provided agreat detail of the solute’s jump mechanism from one water
solvent cage to another and explored the role of spatial
prearrangement of solvent water molecules, the role of therotational prearrangement was completely ignored. Since there
is a unique orientational structure of water solvent around ahydrophobic solute,61,71−82 the jump of the solute to a newcage will be highly disfavored until the desired orientational
hydration structure is attained via orientational prearrangement of the solvent water molecules around the new cage
position. Therefore, the orientational prearrangement ofsolvent water is inevitable around the initial and ﬁnal cage
positions before a successful jump of the hydrophobic solutefrom the initial to the ﬁnal solvent water cage.

2. METHOD AND MODEL2.1. Simulation Details. We perform MD simulation
using GROMACS package.90 One Xe solute molecule isdissolved in 500 water molecules in a cubic simulation box
having periodic boundary conditions on all sides. TIP4P/2005
model91 of water, being one of the most successful models ofwater at subzero temperatures, is chosen here. The Xe solute ismodeled as 12-6 Lennard-Jones (LJ) particle with force-ﬁeld
parameters described elsewhere.84,85 The solute−solvent crossparameters are deduced from the Lorentz−Berthelot mixing
rules, which was proven to be correct for an aqueous solutionof Xe.92 We have further validated this combination of theforce ﬁelds of the Xe solute and water solvent in Section S1 of
the Supporting Information (SI). We have compared thesimulated diﬀusion coeﬃcient of the solute with anexperimental diﬀusion coeﬃcient of the same solute at T =298 K. The unavailability of experimental results at the current
temperature range (T = 240−270 K) has enforced us to checkthe validity of this combination of the force ﬁeld at room
temperature only.
We have simulated four diﬀerent temperatures, 240, 250,
260, and 270 K, of which only the ﬁrst two temperatures
represent the supercooled state of water since TIP4P/2005
water freezes at 252 K.91 Equilibrations of all of these systems
are performed for 10 ns. During the equilibration, thetemperature is kept constant at the respective desired
temperatures using the Nosé−Hoover thermostat,93,94 andthe pressure at 1 atm using the Parrinello−Rahman barostat.95
We have set 0.5 ps for both the pressure and temperature
coupling constant. We then extend the simulation for another50 ns for obtaining the production trajectory. The Leapfrog−
Verlet algorithm is used to solve the equations of motions
every 1 fs time. The production trajectory is saved at a regularinterval of 50 fs. A cutoﬀ distance around half of the box lengthis applied for LJ interactions. The particle mesh Ewald
summation technique and LINCS algorithm96 are used tohandle the long-range Coulomb interactions and to constrain
the water O−H bonds, respectively. A 50 ns long production
trajectory is utilized for analyses.
The validity of the above simulation protocol is ensured by
the excellent agreements of the simulated density of the systemwith the experimental density97,98 (see Table S1 in the SI) andthe simulated diﬀusion coeﬃcients of the solvent watermolecules (calculated from the mean square displacement(MSD) plot as a function of time) with the experimentallymeasured diﬀusion coeﬃcients99 (see Figure S2 in the SI) at allof the four temperatures. We note that while the above
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experimental data are for pure water, the current simulation
data are for the aqueous solutions of one hydrophobic solute.
A quite good agreement between the experiment and thecurrent simulation suggests that the small hydrophobic solute
does not inﬂuence much the overall density and the diﬀusion
coeﬃcient of the solvent water, despite the fact that theinterfacial water solvent molecules (next to the hydrophobic
solute) have appreciably slower dynamics compared to thebulk water.100−109 To elucidate further, we have carried out
additional four simulations of pure water (500 TIP4P/2005
water molecules) at the same four diﬀerent temperatures (T =240, 250, 260, and 270 K) using the same simulation protocol
described in the previous paragraph. The ﬁnal simulated
densities are listed in Table S1 in the SI. These densities match
well with the available experimental data. MSDs and thediﬀusion coeﬃcients of the pure water at the four temperatures
are reported in Figure S3 and Table S2 in the SI. The
simulated diﬀusion coeﬃcients of pure water are slightly closer
to the experimental diﬀusion coeﬃcients of pure water thanthat from our original simulations of the aqueous solutions of
the Xe solute. This indicates that the solute reduces the overall
diﬀusivity of water solvent only very slightly.

3. RESULTS AND DISCUSSION
3.1. Trapping of the Solute inside Solvent Water
Cage. Trapping of the solute inside solvent water cage can beevidenced from the mean square displacement (MSD) of thesolute.55,56,110−118 MSD is calculated using the equation⟨|Δr(t)|2⟩ = ⟨|r(t) − r(0)|2⟩, where r(t) is the time-dependent
position vector of the solute and r(0) is the solute’s position at
t = 0. Figure 1a exhibits MSD as a function of time for four
diﬀerent temperatures. The qualitative nature of the MSD
plots match that of other supercooled liquids. The plateau
regionin between the regular ballistic regime and thediﬀusive regime at the longer timeis subdiﬀusive withfractional time dependence ⟨|Δr(t)|2⟩ ∝ tα. This subdiﬀusive
regime stems from the rattling motion of the solute inside thesolvent water cage. We note that power law ﬁtting of the MSD
plots are carried out in between two stipulated time limits.
While the starting time of the ﬁtting is the moment when theballistic regime completes, which is ∼400 fs, the upper limit is
the time when the non-Gaussian parameter α2(t) is maximum(i.e., the time when the solute’s dynamics deviates at the mostfrom the Gaussian behavior). The method of calculating α2(t)is detailed elsewhere.11 Figure 1b presents α2(t) of the solute
as functions of time at four diﬀerent temperatures. The maxima
of α2(t) plots are located at t = 7.3, 5.8, 3.3, and 2.2 ps at T =240, 250, 260, and 270 K, respectively. The power law ﬁttingsof the MSDs in the above time slices give the value of the
exponents α as ∼0.23, ∼0.25, ∼0.27, and ∼0.35 for T = 240,
250, 260, and 270 K, respectively. Therefore, with decreasing
temperature, the rattling part of the solute is further reinforced
and therefore the solute gets trapped inside the water solvent
cage for a longer time. The enhanced caging eﬀect, withlowering the temperature, directly indicates the increased cage
lifetime,37,55,56,110−118 which may be due to the strengtheningof water···water tangential H-bonds in the solvent cage withdecreasing the temperature. The stronger H-bonding of cagewater molecules at a lower temperature must increase the Hbond lifetime and thereby slows down the solvent cage
opening process.118,119 Along with water’s structural transformations (stronger H-bonds etc), one may also expect theincrease of the cage lifetime and decrease of the diﬀusion

Figure 1. (a) Mean square displacement of the solute and (b) nonGaussian parameter α2(t) as functions of time at four diﬀerent
temperatures, T = 240, 250, 260, and 270 K.

coeﬃcient to be simply related with the decrease of kinetic
energy of Xe in solution at a lower temperature.
We now study the structure of the solvent water cage. The
spatial and orientational structures of the water moleculesaround the solute are shown in Figure 2. Figure 2a presents
two-dimensional probability density distribution (P(r,θ)) of
the angles θbetween water OW···Xe vector and the water
OW−HW bond vectorand the distances r between water OW
atom and the Xe solute at T = 250 K. Similar ﬁgures at theother three temperatures (T = 240, 260, and 270 K) arepresented in Figure S4 in the SI. Figure 2a shows that the ﬁrsthydration-shell water molecules (interfacial water molecules)(r ∼ 5.5 Å) are arranged in such a way that the majority water
OH bonds are oriented either nearly tangential with the solute(tangential OH with θ ∼ 65°) or toward the bulk water (bulk
OH with θ ∼ 180°). No water OH bonds are found aiming
toward the solute (dangling OH with θ ∼ 0°), persuading no
H-bonds between the solute and the solvent water molecules.
The above hydration structure of Xe solute guarantees that thissolute acts perfectly like a small hydrophobic solute in TIP4P/
2005 water solvent.71−81 We note that the peak situated at r ∼6.5 Å and θ ∼ 0° indicates that the majority of the second-shellsolvent water OH bonds are aligned toward the solute suchthat these water molecules form H-bonds with the interfacial
water solvent molecules.
Figure 2b presents radial distribution function (g(r))
between the solute and water OW at four diﬀerent temperatures. The ﬁrst peak (intensity ∼ 2.3) is located at ∼4 Å andranges up to ∼5.5 Å distance. The slight increase in the ﬁrst
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oriented at 0° < θ < 65°, while the other half is aligned at 65° <θ < 180°. With decreasing the temperature, we ﬁnd a slightincrease of the orientational order of water since both thetangential and bulk peak heights increases to some extent bydecreasing the temperature. Figure 2c also indicates that theprobability of ﬁnding the dangling OHs is nil.
3.2. Translational Jump of the Solute: Time-Dependent Solute Coordinate. We now turn our focus on thetranslational jump of the solute from one solvent water cage toanother. We ﬁrst calculate the displacement of the solute along
the trajectory with respect to its position at the starting pointof the simulation trajectory. Then, we search for thetranslational jumps of the solute. Here, we deﬁne a jump by
the large displacement of the solute at a short time interval.
This method of identifying jumps suﬃces the purpose ofunderstanding the jump mechanism of the solute.
Figure 3a presents an example of a such jump occurrence
when the solute moves by ∼4 Å in ∼1 ps. This motion is much

Figure 2. Spatial and orientational structure of water moleculesaround the hydrophobic solute. (a) Radial distribution function
between solute and water O atom (red solid line: values given by theright-hand-side ordinate axis) and two-dimensional probability
density distribution of the angle θ between water O−H bond andthe water O atom···Xe vectors in the y axis (values given by the lefthand-side ordinate axis) and the distance (between the Xe solute andsolvent water O atom) in the x axis. (b) Radial distribution functions
between Xe and water O atoms at the four diﬀerent temperatures. (c)Angular distribution of θ for the ﬁrst-hydration-shell (r < 5.5 Å) watermolecules at four diﬀerent temperatures.

Figure 3. (a) Small portion of the solute’s displacement from 50 ns
long trajectory at T = 250 K to show a representative jump. The largeamplitude jump of the solute from one cage to another is highlighted
by the arrow; t = 0 ps is set at the time when the solute particle jumps
from the initial to ﬁnal cage. (b) Comparison of the probability
distributions between the 41 jump distances, presented by a dashedline, and the total displacement of the solute (solid line), calculated at
1 ps interval. Note that the time required for completion of the jumpis ∼1 ps. Similar plots at other three temperatures are presented in
Figures S5 and S6 in the SI.

peak height indicates slightly more ordering of interfacial
hydration structure around the solute by decreasing thetemperature. This is consistent with earlier studies.61,82 The
normalized probability distribution (p(θ)) of the angles θbetween the Xe···water OW vector and the water OH bondvectorsis presented in Figure 2c for the interfacial solvent
water molecules. Visibly, the majority of the interfacial water
OH orientations are either tangential or bulk. The probabilityof the bulk OH orientation is slightly higher than that of thetangential OH orientation. However, the integration of P(θ) in
the region 0° < θ < 65° is almost equal to that in the region
65° < θ < 180°. So, nearly half of the cage water OH bonds are

faster than the normal diﬀusion of the solute with a diﬀusion
coeﬃcient value of 1.75 × 10−5 cm2/s, which suggests the
movement of ∼0.4 Å in ∼1 ps. We have considered a set ofsuch 41 large-amplitude jumps of the solute throughout the full
50 ns trajectory at T = 250 K. The number of translational
jumps of 23, 49, and 71 are considered at T = 240, 260, and
270 K, respectively. In this section, we show the result only for
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and ﬁnal cages, respectively. This is essentially based on theassumption that the solute moves stochastically inside a solvent
cage and covers the entire spherical space such that its average
position is identical to the COM positions of the cage. We
note that Ri and Rf mimic the positions of the acid and base,respectively, in an acid−base PT complex.122,123 Equation 1 isnormalized in the sense that QJ = −1 when the solute is inside
the initial cage and QJ = 1 when the solute is inside the ﬁnal
solvent cage. Figure 4a presents trajectory-averaged ⟨QJ⟩ as a

T = 250 K as the representative temperature. (The results forother three temperatures are shown in the SI.) The jumpoccurrence times and their amplitudes for all of the four
temperatures are listed in Table S3 in the SI. Figure 3b
presents the normalized distribution of displacements of thesolute in 1 ps time duration (1 ps time is required forcompletion of a jump occurrence) for the full trajectory at T =250 K. The above distribution is obtained from samples of thestart-to-end vector distances of 1 ps segments of the full 50 ns
trajectory by progressively shifting the origin by 100 fs. The
distribution of the 41 translational jump amplitudes of thesolute is also plotted in the same ﬁgure. (The distributions at
all of the four temperatures are presented in Figure S5 in theSI.) Since the displacement distributionhaving a peak at 0.65
Åhas a long distance tail, it provides a clear signature oflarge-amplitude displacement of the solute. The peak of the 41
jump amplitude distribution is located around the tail of thedisplacement distribution. This conﬁrms the validity of our
choices of the translational jump occurrences of the solute.
Before calculating the solute jump coordinate, we investigate
the contribution of these 41 translational jumps in the overall
diﬀusion of the solute at T = 250 K. The detailed method for
calculating jump-only diﬀusion coeﬃcient Djumpsdiﬀusion
coeﬃcient solely due to the translational jumps of the solute
is described elsewhere.120,121 Djumps is estimated as Djumps = (1/6)νλ2, where ν is the jump frequency and λ is the average jumplength; ν is calculated by dividing the total number of jumps,
considered in the calculation, by the total simulation trajectory
length and λ is actually the linear distance between the initial
and ﬁnal cage positions of the solute. In the present case at T =250 K, ν = 0.82 ns−1, and λ = 3.5 Å. With both the numbers,
we obtain Djumps = 1.92 × 10−7 cm2/s, which is ∼10% of theoverall diﬀusion coeﬃcient of the solute at T = 250 K (1.93 ×
10−6 cm2/s), obtained from the MSD plot (see Figure 1a). We
note that the present method of identifying a translational
jump is completely based on the displacement of the solutewith respect to the position at the starting time of the
production trajectory. Despite the fact that this method suﬃces
the objective of the present work, which is to comprehend thetranslational jump mechanism of the solute in supercooled
water, the translational jump frequency is undermined in thismethod. A better way of jump characterization, such as amethod based on the radius of gyration of solutes
trajectory,120,121 can increase the contribution of the jumps
in the overall diﬀusion of the solute. However, the applicabilityof the latter method in understanding the explicit mechanismof the jump from one solvent cage to another, especially theidentiﬁcation of reaction coordinate (as we have done here), isa matter of concern.
Now, we calculate the solute jump coordinate, QJ, which isactually the position of the solute as a function of time during
the jump from one water solvent cage to another. We obtain
the QJ parameter using the following equation61
QJ =[r(t ) − R i] + [r(t ) − R f ][r(t ) − R i] − [r(t ) − R f ]Figure 4. (a) Solute jump coordinate as a function of time. Timeevolution of the 41 trajectory-averaged jump coordinate ⟨QJ⟩ (blacksolid line) with t = 0 ﬁxed at the jump occurrence time for T = 250 K.
While the vertical dashed line represents the jump occurrence time (t= 0 ps), ⟨QJ⟩ = 0 is indicated by horizontal dashed line. The inset
displays ⟨QJ⟩ during the jump occurrence from the initial cage to theﬁnal cage. (b) Speed of the Xe solute as a function of time. The speed
shows quick increase at the jump occurrence time (t = 0 ps).

function of time for T = 250 K. (⟨QJ⟩ for all of thetemperatures are shown in Figure S6 in the SI.) The trajectory
averaging has been performed by ﬁrst setting the jumpoccurrence time at t = 0 ps and then averaging QJ for all of the
41 jumps (for T = 250 K). Figure 4a shows that ⟨QJ⟩ remains
almost constant at ⟨QJ⟩ = −1 before the jump occurrence timeand quickly shifts to ⟨QJ⟩ = +1 after the jump occurrence via⟨QJ⟩ = 0, when the solute is in the middle of the two water
solvent cages. It is evident, from the inset of Figure 4a, that thejump occurrence completes within ∼1 ps time interval. The
near constancy of the solute jump coordinate at ⟨QJ⟩ = −1
(before the jump occurrence) and ⟨QJ⟩ = 1 (after the jumpoccurrence) indicates that while the solute rattles inside thesolvent cage, the overall motion of the solvent cage is almost
negligible. However, the solvent water molecules forming theinitial and ﬁnal cages around the solute diﬀuse individually. We
will see in the next section the time-dependent solvent
reorganization to facilitate the solute’s jump occurrence.

(1)

We note that the above equation is motivated from thenormalized proton transfer (PT) coordinate, a parameter tomonitor the position of a transferring proton from a proton
donor acid to a proton acceptor base during PT reaction.122,123
Here, r(t) is the time-dependent position of the solute and Ri
and Rf are the average positions of the solute inside its initial
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along the solute’s jump trajectory at T = 250 K. (The resultsfor other three temperatures are presented in Figures S7−S9 in
the SI). These seven diﬀerent instants connote seven diﬀerent
time intervals between t = −10 and 10 ps, where t = 0 ps refers
to the jump occurrence time. These seven time intervals are:(i) t = −10 to −7 ps, (ii) t = −7 to −4 ps, (iii) t = −4 to −1 ps,(iv) t = −1 to 1 ps, (v) t = 1−4 ps, (vi) t = 4−7 ps, and (vii) t= 7−10 ps.
Figure 5a,g shows that the distribution near the Ri position is
distinctly diﬀerent from the one near the Rf position. Figure 5ashows that before the jump occurrence time, the distribution
around Ri position is due to the hydrophobic hydration. This isconsistent with the fact that the solute is situated inside theinitial cage before the jump occurrence time. On the other
hand, the distribution around the Rf position is merely
parabolic with the distance from the Rf position. This parabolic
nature signiﬁes the increasing volume eﬀect of the sphericalshells with distance. Figure 5g shows that after the jumpoccurrencewhen the solute reaches the ﬁnal solvent cageacomplete reversal of the above hydration structure occurs near
the two cage positions. While the distribution around the Rf
position resembles the hydrophobic hydration structure, thedistribution around the Ri position is parabolic with the
distance. The similar nature of Figure 5a,gincluding acomplete switch of the hydration structures around the twocage positionsguarantees the successful jump of the solutefrom the initial to the ﬁnal water solvent cage.
Comparative analyses of Figure 5a−c show that the
distributions around the R i position tend toward anequivalence with the distribution around the Rf positions as
the time elapses toward the jump occurrence time t = 0 ps.
Finally, at the jump occurrence time, the spatial distributions
around the Ri and Rf positions become almost identical (seeFigure 5d). This indicates complete symmetry of the hydration
structures around the initial and ﬁnal cage positions. The
above symmetrization of the spatial hydration structure, bymeans of solvent water prearrangement around the two cages,
is the key for the successful jump of the solute from one cage
to another. The above spatial prearrangement was alreadyobserved in the preceding paper.61 The similar picture emerges
while proton or electron hops from one site to another inaqueous solution.63−67 Comparative analyses of Figure 5e−gshow that after the jump occurrence time, the distributions
gradually diﬀer from each other with time and the solute getsstabilized inside the ﬁnal solvent cage position. The timedependent behavior of the solvents’ spatial rearrangement will
be discussed in Section 3.3.2.
Now, we turn our focus on how the orientational structuresof the solvent water molecules near the Ri and Rf positions
change with time during the jump of the solute. The
orientation of solvent water molecules, which is actually the
angular orientation of water OH bond vector, is characterized
by an angle θ between the following two vectors: (i) the vector
between the solvent water OW atom and Ri (or Rf) positionand (ii) water OW···HW bond vector. Figure 6 plots the
distribution of the angle θ (corrected over solid angle) at sevendiﬀerent time intervals for the water near the Ri and Rf
positions (r < 5.5 Å) at T = 250 K.71−81 (The distributions
for the other three temperatures are exhibited in Figures S10−S12 in the SI.) Figure 6a,g, the orientational structure ofsolvent water molecules before and after the jump occurrence,shows that the angular distribution near the Ri position is
distinctly diﬀerent from the one near the Rf position both

To elucidate further, we have plotted in Figure 4b, thetrajectory-averaged speed of the solute, SXe, against time.
Notably, SXe shows a quick rise at t = 0 ps. Therefore, thesolute moves faster at the jump occurrence time than before
and after the jump occurrence.
3.3. Role of Solvent Coordinate. 3.3.1. Equilibrium
Structure. We now turn our focus to the role of solvent water
for the jump occurrence of the solute. First, we examine how
the spatial structure of the water solventforming the initial
and the ﬁnal cageschanges with time during the jumpoccurrence of the solute. The objective of this calculation is tounderstand how the ﬁnal cage builds up during the jumpoccurrence at the cost of the dissolution of the initial cage.
We have plotted, in Figure 5, the spatial distribution of watermolecules around the initial solvent cage position Ri and theﬁnal solvent cage position Rf at seven diﬀerent instants of time

Figure 5. Spatial distribution of the water solvent molecules aroundthe Ri (blue line) and Rf (red line) positions at seven diﬀerent timeintervals during the solute’s jump at T = 250 K.
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the angular distribution around the Xe solute (see Figure 2c).
On the other hand, the angular distribution around the Ri
position becomes almost uniform. The above role reversal
around the two cage positions implies the successful
completion of the complete process of the solute’s jumpfrom one cage to another.
A closer inspection of Figure 6a−c reveals that theorientational distributions around the Ri and Rf positions
gradually symmetrize as the time approaches the jumpoccurrence time t = 0 ps. This is evident in Figure 6d, wherethese two distributions become almost identical. Thissymmetrization of orientational and spatial structures61 is
crucial for the successful jump of the solute from one cage toanother. Figure 6e−g shows that the two orientational
distributions again start diﬀering from each other with timeafter the jump occurrence time as the solute gets trapped inside
the ﬁnal solvent cage position.
Figures 5 and 6 indicate that the transformations of thespatial and orientational structures of the cage-forming solvent
water molecules are not one-step processes. Instead, theyrequire continuous rearrangement of water solvent molecules
near the two cages such that the hydration structures near the
initial cage are akin to that near the ﬁnal cage position. Thesetime-dependent spatial and orientational rearrangements ofcage solvent water molecules will be discussed in the following.
3.3.2. Time-Dependent Rearrangement. In the preceding
section, we have shown the spatial and orientational
distributions of solvent water around the two reference
positions, Ri and Rf, at seven diﬀerent times along the jump
trajectory. We now follow the solvent water rearrangement
process with time.
Spatial rearrangements of water solvent molecules aremonitored separately in two key regions (R1 and R2). The
regions R1 and R2 are identiﬁed from Figure 5 by the following
distances criteria. The region R1 is identiﬁed by the spaceinside the distance r = 3.5 Å from either of the two reference
positions. On the other hand, the R2 region is limited by the
distance criteria 3.5 Å < r < 5.0 Å. The following is therationale of the separate analysis at the two diﬀerent regions.
Comparative analyses of Figure 5a−g show that while thespatial distribution of the water molecules inside the R1 region
continuously increases (decreases), the spatial distribution
inside the R2 region decreases (increases) gradually around theRi (Rf) position with time. Therefore, the nature of changes ofthe spatial distribution depends on the location of the water
solvent molecules around the reference positions (whether R1
or R2).
Figure 7a plots the number of solvent water molecules (NW)
inside the R1 region around the Ri and Rf positions against timefor T = 250 K. These are 41 jump trajectory-averaged plots.
Figure 7a shows that while the number of water molecules NW
around the Rf position decreases from ∼4 to ∼2, NW near theRi position increases from ∼2 to ∼4. At the jump occurrencetime (t = 0 ps), NW ∼3 near both the reference positions, Ri
and Rf. Therefore, the symmetrization of spatial hydration
structures near the Ri and Rf positions inside the R1 region
occurs due to the above prearrangement of ∼1 water molecule
before the jump occurrence time. This is important for solute’s
jump occurrence from the initial to the ﬁnal solvent cage.
Figure 7b presents the time-dependent number of water
solvent molecules, NW, located in the R2 region around the Ri
and Rf positions. NW increases with time near Rf position froma value of ∼9 to ∼13. Therefore, the inclusion of ∼4 water

Figure 6. Probability distribution of angles θ between water O−Hbond vector and the vector between solvent water O···Ri (Rf) position
for the cage solvent water molecules at T = 250 K. The blue and redlines indicate the distributions around the Ri and Rf positions,
respectively.

before and after the jump occurrence times. Before the jumpoccurrence time (see Figure 6a), the angular distribution
around Ri is similar to that around Xe solute (see Figure 2c).
On the contrary, the distribution around Rf position is almost
uniform, indicating an approximately random orientation ofwater molecules around the ﬁnal cage. After the solute jumps
to the ﬁnal cage (see Figure 6g), the above picture simply ﬂips.
Now, the angular distribution around the Rf position resembles
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orientation is liable for H-bond formation with the solute.
Therefore, D type is also called as dangling. The second type ofwater OH bond (TI) is almost tangential to the solute surface(slightly inward). We call this group of OH bonds innertangential ones. The third group (TO) is in between thetangential and the bulk orientation. We term this group as
outer tangential. The last group of water OHs (B) are oriented
toward the bulk environment, and therefore, we call this groupas bulk OH.
Figure 8a−d exhibits time-dependent arrangement of theabove four diﬀerently oriented water solvent molecules near

Figure 7. Number of solvent water molecules in two regions: (a) R1and (b) R2 around the Ri (blue line) and the Rf (red line) positions as
a function of time for T = 250 K. (Similar ﬁgures for T = 240, 260,
and 270 K are presented in Figures S13, S14, and S15 in the SI,
respectively.) The two regions R1 and R2 are speciﬁed from Figure 5.
The region R1 denotes the space inside the sphere of r < 3.5 Å fromthe initial or the ﬁnal reference position of the solute, while the region
R2 denotes the spherical shell of width 3.5 Å < r < 5.0 Å.

molecules occurs in the R2 region of the Rf position. On theother hand, near the Ri position, NW decreases from a value of∼13 to ∼9 after the jump occurrence of the solute. Therefore,
the inclusion (exclusion) of ∼4 water molecules in (from) theR2 (R1) region is the key rearrangement of solvent wateraround the ﬁnal (initial) cage positions.
Presumably, about two of the total four newly added watermolecules in the R2 region of the Rf position come from the R1
region of the same position. On the other hand, about two of
the total four departed water molecules in the R2 region of theRi position go to the R1 region at the same position. The
remaining (two) water molecules may diﬀuse from the R2
region of the Ri position to the same region of the Rf position.
This spatial rearrangement of solvent water near the initial andﬁnal cages of the solute is crucial. Importantly, half of the total
rearrangement occurs before the jump occurrence time t = 0
ps. This prearrangement is the key solvent activation for thisjump. (The time-dependent spatial rearrangement of solvent
water at the three other temperatures, T = 240, 260, and 270
K, are shown in Figures S13, S14, and S15 in the SI,
respectively.)
We now focus on the time-dependent orientational
rearrangement of solvent water molecules near the initial andﬁnal cage positions, Ri and Rf. Here, we follow with time therearrangement of water solvent molecules separately for four
diﬀerent orientations, characterized by the angle between
solvent water OW···Xe solute vector and solvent water OW−HWbond vector (θ). The four diﬀerent orientations are: (i) D (0°< r < 45°), (ii) TI (45° < r < 85°), (iii) TO (85° < r < 145°),and (iv) B (145° < r < 180°). We note that the D-type

Figure 8. Number of solvent water OH bonds as a function of timenear the Ri (blue line) and Rf (red line) positions of the solute. The
four panels present results separately for four diﬀerently orientatedsolvent water molecules. They are: (a) D (0° < r < 45°), (b) TI (45°< r < 85°), (c) TO (85° < r < 145°), and (d) B (145° < r < 180°).
These four regions are identiﬁed from Figure 6. These results are for
T = 250 K. (Similar ﬁgures for T = 240, 260, and 270 K are presented
in Figures S16, S17, and S18 in the SI, respectively.)

the Ri and Rf positions. Figure 8a shows that the number ofdangling water OH decreases from ∼6 to ∼3 near the Rf
position. Increment by the same number occurs near the Ri
position. The angular prearrangement of water OH bonds
starts near the Ri and Rf positions at least 5 ps before the jumpoccurrence time and continues to happen until a ﬁnal solvent
cage is formed around the solute after the jump occurrence
time. We note that the symmetrical hydration structures areformed around the two solvent water cage positions at thejump occurrence time (t = 0 ps). Figure 8b shows that the
number of TI water OHs near the Rf position increases from avalue of ∼13 to ∼19, while the same number of water OH(∼6) decreases near the Ri position due to the jump of thesolute. Similar to the prearrangement of the dangling OHs, theprearrangement of the TI water OHs occur by ∼50%. It is clear
in Figure 8c that the number of TO OHs near the Rf position
decreases from ∼19 to ∼16, while the same number of waterOH (∼3) increases near the Ri position due to the jump. Also,
half of the total solvent orientational rearrangement occurs by
the jump occurrence time (t = 0 ps). This is crucial for the
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jump occurrence of the solute from the initial to the ﬁnal cage.
Finally, Figure 8d shows that the number of B-type water OH
bonds increases from ∼4 to ∼6 near the Rf position, while thesame number of water OH (∼2) decreases near the Ri position
due to the jump. Similar to the prearrangement of D OHs, theprearrangement of the TI OHs occurs by ∼50% at the jumpoccurrence time. (The time-dependent orientational rearrangements of solvent water at the three other temperatures, T =240, 260, and 270 K, are shown in Figures S16, S17, and S18 in
the SI respectively.)
The above prearrangement symmetrizes the hydration
structures around the two solvent cage positions, as seen in
Figures 5 and 6. This symmetrization facilitates the solute’s
jump from one cage to another. The similar prearrangement isalso essential in proton-transfer reaction63−66 in water androtational jump motion68,69 of liquid water molecules.
3.4. Synchronization of Spatial and Orientational
Coordinates of Solvent Water. In this section, we compare
the solvent spatial and orientational coordinates during thejump along with the solute jump coordinate. We note that thesolute coordinate is actually the jump coordinate QJ presented
in Section 3.2. The solvent spatial and orientational
coordinates are extracted from Figures 7 and 8, respectively.
The methodology is discussed below.
First, we identify four diﬀerent segments of each of the four
rearrangement lines in Figure 7. This segmentation is based on
the rate of change of NW. For instance, we have shown in
Figure 9a the four diﬀerent steps of the spatial rearrangementof the solvent water molecules in the R1 region near the Ri and
Rf positions at T = 250 K. To quantify the above, we have ﬁteach segment with linear equations. The slopes of the straightlines, signifying the rate of the rearrangement processes, arelisted in Table 1. These four steps are described below. First,

Table 1. Rate of the Four Diﬀerent Steps of the Spatial
Rearrangement of Solvent Water in Two Regions (R1 andR2) of the Initial (Ri) and the Final Solvent Water Cage
Position (Rf)rate of rearrangement (ps−1)
regions

initial/ﬁnal cage

I

II

III

IV

R1

Ri
Rf
Ri
Rf

0.077
−0.052−0.2050.096

0.748−1.159−0.8833.313

0.206−0.164−0.162
0.397

0.017
0.042
−0.096
0.022

R2

the rearrangement process starts slowly with a rate less than 0.2
ps−1. Next, the process speeds up right before the jumpoccurrence time with a rate of ∼0.8−3 ps−1. The process thenslows down to a rate of ∼0.2−0.4 ps−1 immediately after thejump occurrence time and then almost levels oﬀ ∼5 ps after
the jump occurrence time. Interestingly, the above rearrangement rates are comparable near the Ri and Rf positions at thesame time interval. Therefore, the spatial rearrangement of thesolvent water molecules near the Ri position occurs almost
parallel to the solvent rearrangement near the Rf position. This
motivates one to obtain an average picture of the spatial
rearrangement of water solvent molecules. We will come backto the above averaged picture later in this section.
A similar picture emerges for time-dependent reorientational
rearrangement of solvent water molecules for diﬀerently
orientated water solvent molecules (D, TI, TO, B) at four
diﬀerent temperatures. Unlike translation, we identify ﬁve stepsof the orientational rearrangement of the solvent watermolecules near the two cage positions in Figure 8. Figure 9b
represents these ﬁve steps of the orientational rearrangementof the TI solvent water OH bonds near the two cages at T =250 K. To get a more quantitative picture, we have ﬁt the ﬁvesteps of the rearrangement process with linear equations, theslopes of which are listed in Table 2. We note that the slopes
Table 2. Rate of Five Diﬀerent Steps of the Orientational
Rearrangement of Four Diﬀerently Orientated Solvent
Water (D, TI, TO, and B) around the Initial (Ri) and Final
Solvent Water Cage Positions (Rf)rate of rearrangement (ps−1)
regions
D
TI
TO
B

initial/ﬁnal
cage

I

II

III

IV

V

Ri
Rf
Ri
Rf
Ri
Rf
Ri
Rf

0.031
−0.0210.017
0.027−0.018−0.008−0.016−0.0030.103−0.089−0.2690.435
0.330−0.327−0.1220.150

0.817−1.276−1.8621.603
0.655−1.574−0.8590.548

0.192−0.101−0.2620.442
0.132−0.383−0.070
0.144

0.028
0.028
−0.074
0.029
0.166
0.065
−0.0370.027

determine the rates of the rotational rearrangements. The
above ﬁve diﬀerent steps of the orientational rearrangement
are: (i) insigniﬁcant rearrangement with negligible rate, (ii)slow rearrangement with a rate of 0.1−0.4 ps−1, (iii) fasterrearrangement with a rate of 0.6−2.0 ps−1, (iv) slower
rearrangement with a rate of 0.1−0.4 ps−1, and (v) almost
no rearrangement. Similar to the spatial arrangement, the ratesof the rotational rearrangement processes near the Ri and Rf
positions are very close to each other at the same time interval.

Figure 9. (a) Four diﬀerent steps of the spatial rearrangement of thesolvent water molecules in the R1 region near the Ri and positions at
T = 250 K and their ﬁttings with linear equations. (b) Five steps ofthe orientational rearrangement of the TI solvent water OH bonds
near the two cages at T = 250 K and the linear ﬁts.
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Figure 10. Solvent water translational coordinate ⟨QST⟩ (blue lines), orientational coordinate ⟨QSO⟩ (red lines), and the solute jump coordinate⟨QJ⟩ (black lines) as functions of time at four diﬀerent temperatures: (a) T = 240 K, (b) T = 250 K, (c) T = 260 K, and (d) T = 270 K.

Therefore, the orientational rearrangement of the solvent watermolecules near Ri occurs almost parallel to that near the Rf
position. This is the basis for calculating the average water
solvent orientational coordinate.
Now, we calculate the normalized translational and rotational solvent coordinates using the following equations
Q ST =[NW (t ) − NW (0)] + [NW (t ) − NW (∞)][NW (t ) − NW (0)] − [NW (t ) − NW (∞)]

Q SO =[NOH(t ) − NOH(0)] + [NOH(t ) − NOH(∞)][NOH(t ) − NOH(0)] − [NOH(t ) − NOH(∞)]⟨Q SO⟩ =D
TI
TO
B+ (Q SO+ Q SO+ Q SO+ Q SO)R f ](2)

In eq 2, NW(t) is the number of water molecules in either the
R1 or the R2 region around the Ri and Rf positions of the soluteat time t. We note that NW(t) is plotted as a function of time in
Figure 7. NW(0) and NW(∞) are the numbers of watermolecules in either the R1 or the R2 region around the Ri or Rf
position at t = −10 and 10 ps, respectively. In eq 3, NOH(t) is
the time-dependent number of water OH aligned in either ofthe four possible orientations (D, TI, TO, and B) near the Ri or
Rf position. NOH(0) and NOH(∞) are the values of NOH(t) at t
= −10 and 10 ps, respectively. Equations 2 and 3 are thenormalized versions of the water solvent spatial and orientational coordinates in the sense that both the values of QST andQSO are ca. −1, while the solute resides in the initial solvent
cage before the jump occurrence and ∼1 when the solute stays
inside the ﬁnal solvent cage after the jump occurrence.
Next, we obtain the average spatial and orientational solvent
coordinates ⟨QST⟩ and ⟨QSO⟩ using the following equations
1
R1
R2
R1
R2
[(Q ST+ Q ST)R i + (Q ST+ Q ST)R f ]
4(5)

In eq 4, the component QXST is the solvent spatial coordinate in
the X region. The quantity QYSO, in eq 5, is the solvent
orientational coordinate for Y-type-oriented solvent watermolecules.
⟨QST⟩ and ⟨QSO⟩ are plotted in Figure 10 against time atfour diﬀerent temperatures. The solute jump coordinate ⟨QJ⟩ is
also plotted in the same ﬁgure. Apparently, both ⟨QST⟩ and⟨QSO⟩ start increasing well before the solute actually jumps
from the initial to the ﬁnal water solvent cage at the jumpoccurrence time (t = 0). This indicates decoupling of thesolute jump coordinate from the solvent coordinate. We notethat this decoupling, actually originated from jump translationof the solute, may give rise to serious violation of SE relation,
which connects the solute’s translation with the viscosity of thesolvent medium. The jump rotation (instead of small angle
rotation) of water, 68,69 ionic liquid, 124 deep eutectic
mixture,125,126 and some polyatomic ion8 also induce serious
breakdown of SED relation. Both the ⟨QST⟩ and ⟨QSO⟩ reachthe value 0, indicating half prearrangement of the solvent watermolecules to symmetrize the spatial and orientational
structures of the initial and ﬁnal cages, when the solute
jumps to the new cage. Therefore, half of the full solvent
rearrangement is the key for the jump process. The rest of therearrangement of the solvent water occurs after the jumpoccurrence time. Closer inspections of Figure 10a,b further
reveal that at the supercooled condition (T = 240 and 250 K),⟨QST⟩ and ⟨QSO⟩ almost overlap at almost all of the times fromt = −10 to 10 ps. This indicates that the spatial andorientational rearrangement of solvent water molecules aroundthe Ri and Rf positions occur almost at the same time. Thissynchronization of translational and rotational motion can also
be termed as translation−rotation coupling. We note that the(3)⟨Q ST ⟩ =

1
D
TI
TO
B
[(Q SO+ Q SO+ Q SO+ Q SO
)R i
8

(4)
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Figure 11. Plot of the solvent water translational coordinate ⟨QST⟩ against the solvent water orientational coordinate ⟨QSO⟩ at four diﬀerent
temperatures: (a) T = 240 K, (b) T = 250 K, (c) T = 260 K, and (d) T = 270 K. The blue straight lines are drawn through the data for visual guide.

jump occurrence time. As we decrease the temperature to T =260 K, we see, in Figure 11, that ⟨QST⟩ increases almost to 0
while ⟨QSO⟩ increases to −0.5. Then, ⟨QSO⟩ increases to 0,
while ⟨QST⟩ increases only slightly to reach the value of exactly
0. After the jump occurrence of the solute, both the ⟨QST⟩ and⟨QSO⟩ increase together at the same rate, indicating rotationdriven translation of the solvent water during postjump
rearrangement. This is similar to that at T = 270 K. As wefurther decrease the temperature to T = 250 K (2 K below thefreezing temperature, T = 252 K), we see much stronger
coupling between ⟨QST⟩ and ⟨QSO⟩ at all times during thejump event. Both translation and rotation occur almost at thesame rate. The similar coupling is also seen at T = 240 K. Thisfurther strengthens our proposition that there is an absolutenecessity of synchronized translational and rotational movement of solvent water for successful translation jump of a
nonpolar solute from one cage to another in supercooled
water. However, the jump event does not require similar
synchronization above the melting temperature of water. The
above observation derives an important conclusion. The
rotation-driven translational motion of solvent water in the
supercooled state can be a crucial factor for the breakdown of
the SE relation, which is observed in supercooled water. This
picture is consistent with a recent simulation study,8 suggesting
the role of rotation-assisted translation of a polyatomic ion in
violation of the SE relation.

above coupling of the two motions is very diﬀerent from theconventional coupling between translational and rotational
diﬀusion of solvent. While the rotational coupled translationalmotion purely refers to the concerted movement of translational and rotation (in the analogy of proton-coupled electrontransfer reaction), the coupling between translational androtational diﬀusions refers to the validity of the twohydrodynamic relations, SE and SED relations. The synchronization of translational and rotational motions of supercooled
water is the consequence of water solvent molecules forming
cluster and moving cooperatively.1−49,57,127−137 This cooperative dynamics in supercooled liquid is usually related to thepresence of spatiotemporal heterogeneities. This is the mostimportant result, which infers that the spatially heterogeneous
dynamics can result in jump translation of a nonpolar solute insupercooled water. With increasing the temperature of themedium above the freezing temperature, 252 K, the above
overlap between the translation and rotational solvent
coordinate decreases, indicating weak water solvent translation−rotation coupling for solute’s jump motion from one
solvent cage to another. This is evident from Figure 10c,d for T= 260 and 270 K, respectively.
To elucidate further, we have plotted ⟨QST⟩ against ⟨QSO⟩ in
Figure 11 at four temperatures. Figure 11d shows that at thehighest temperature, T = 270 K, ⟨QST⟩ increases from ca. −1
to ∼0 without much change of ⟨QSO⟩. This indicates theoccurrence of translational prearrangement almost solely
without signiﬁcant rotational prearrangement of solvent
water. Next, the translation takes the lead and the rotation isalmost switched oﬀ. Therefore, the two processes occur almost
independently from each other and give rise to the suitablestructure of the cages for enabling the jump of the solute.
Finally, after the jump occurrence, both the ⟨QST⟩ and ⟨QSO⟩
increase together at the same rate, indicating rotation driventranslational rearrangement of the solvent after the solute’s

4. CONCLUSIONSIn this paper, we have put forward a mechanism of thetranslational jump of a nonpolar solute in TIP4P/2005 water
both below and above its freezing temperature, T = 252 K. We
have identiﬁed the water solvent’s translational and rotational
rearrangement as the reaction coordinates for the translational
jump of the solute from one solvent cage to another. The
solute jumps from the initial to the ﬁnal cage only when the
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spatial and orientational prearrangement of the solvent watermolecules symmetrize the hydration structure around thesolvent cage positions. Half of the total solvent rearrangement
completes by the solute’s jump occurrence time. Interestingly,
a similar symmetrization of hydration structure aroundelectron/proton donor and acceptor drives the electron/
proton transfer reaction (which are the solvent activatedreactions) in the forward direction.63−67
The systematic temperature dependence study has shownthat the above picture remains valid at all of the temperatures,
we have studied here. However, the comparative analysis
between the spatial and orientational prearrangements ofsolvent water at diﬀerent temperatures dictates a very diﬀerent
story. In the supercooled region, both the translational androtational prearrangements occur at the same time to facilitate
the jump process. The above synchronization between thetranslational and the rotational motion of solvent water stems
from the strong translation−rotation coupling in the supercooled medium. At this point, we must emphasize that theabove translational−rotational coupling refers to the translational and rotational movements of water in a concerted way.
This is diﬀerent from the conventional deﬁnition of coupling oftranslational and rotational diﬀusion, which is purely hydrodynamic in nature because the latter coupling is characterized
by the validity of the SE and SED relations. The above
concerted translational and rotational movements are the resultof cooperativity of the solvent water molecules in supercooled
condition. The cooperativity actually originated from thespatiotemporal heterogeneity of the medium.1−49,57,127−137
Therefore, this study infers that the spatially heterogeneous
dynamics in supercooled water can cause a solute molecule todiﬀuse via large-amplitude jumps from one solvent cage toanother. As the temperature is increased above the freezing
point, the above coupling decreases and the solvent’s spatial
prearrangement becomes somewhat independent of theirrotational prearrangement near the solvent cage positions.
Finally, the present work explores the detailed pathway ofjump translation of a very simple hydrophobic solute Xe insupercooled water. Due to that fact that the Xe solute closely
resembles another relatively simple hydrophobic solute, likemethane, ethane, etc., the current mechanism should beequally applicable to the translational jump of other small,
purely hydrophobic solutes. However, the translational jumpmechanism of slightly more complex solutes, like hydrophilic
and amphiphilic solutes, where iceberg formation is not seen to
occur,107−109 can be very diﬀerent. More studies need to be
conducted in this context.

■■

at the four diﬀerent temperatures, (vi) spatial andangular distribution functions around the Ri and Rf
positions at seven diﬀerent time intervals during thesolute’s jump occurrence, and (vii) time-dependent
spatial and angular rearrangement of solvent watermolecules near the Ri and Rf positions during the jumpoccurrence of the solute (PDF)

AUTHOR INFORMATION

Corresponding Author*E-mail: snehasis@iitp.ac.in.
ORCID

Snehasis Daschakraborty: 0000-0002-2694-7142
Notes

The authors declare no competing ﬁnancial interest.

■

ACKNOWLEDGMENTS
V.D. acknowledges IIT Patna for a research fellowship. The
authors acknowledge IIT Patna for ﬁnancial and computational
support. S.D. thanks Dr. Sandipa Indra and Prof. Ranjit Biswas
for fruitful discussion.

■

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ASSOCIATED CONTENT

S Supporting Information*

The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/acs.jpcb.8b03177.
(i) Checking the validity of the combination of thesolute’s and water solvent molecule’s force ﬁeld, (ii)checking the validity of the simulation methodology andparameters by calculating the simulated densities of thesystem and the diﬀusion coeﬃcients of water at the four
temperatures, (iii) spatial and orientational hydration
structure of the Xe solute at three diﬀerent temperatures,(iv) table listing the jump occurrence times and theiramplitudes at four diﬀerent temperatures, (v) displacement distribution and the normalized jump coordinates
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DOI: 10.1021/acs.jpcb.8b03177
J. Phys. Chem. B 2018, 122, 7569−7583