Special Issue on "Theoretical and Computational Chemistry"
J. Indian Chem. Soc.,
Vol. 96, July 2019, pp. 741-751

Importance of translational jump in diffusion of hydrophobic solute in supercooled water:
Solute size dependence
Vikas Dubey, Shivam Dueby †, Shakkira Erimban† and Snehasis Daschakraborty*
Department of Chemistry, Indian Institute of Technology Patna, Patna-801 106, Bihar, India
E-mail: snehasis@iitp.ac.in
Manuscript received online 02 April 2019, revised and accepted 27 April 2019
It has been reported in many experimental and simulation studies that small solutes – dissolved in supercooled water (orsupercooled liquid in general) – violates the Stokes-Einstein equation (SE), a hydrodynamic relation connecting the self-diffusion
coefficient of the solute, the viscosity of the solvent, the radius of the solute, and the temperature of the system. However,with the increase in the solute’s size, the validity of the SE equation is gradually attained. It is generally believed that thepresence of translational jump occurrence of the small solute in supercooled water is responsible for the SE breakdown.
However, a quantitative estimation of the jump contribution for the self-diffusion of a solute is still lacking. Here, we have
performed molecular dynamics (MD) simulations for four different nonpolar solutes (with increasing solute size), separately
dissolved in water. By adopting a recent technique for direct identification of translational jump occurrence of the solute
molecule, we have successfully calculated the jump-diffusion coefficient of the solute and its contribution to the total selfdiffusion of the solute molecule in liquid water both at room temperature and supercooled regime. As the solute size decreases,
the contribution of jump-diffusion increases at the supercooled regime. We have also separated the jump-diffusion from thetotal diffusion of the solute to obtain the residual diffusion, which better follows the SE equation. This is direct evidence forthe increased importance of translational jump-diffusion for the observed breakdown of the SE relation for small solutes atthe supercooled regime. These new findings can assist in comprehending many experimental results where the breakdownof the SE equation is observed.
Keywords: Supercooled water, molecular dynamics, hydrophobic solute, Jump-diffusion.

1. Introduction
Supercooled water (water below its freezing point) is afascinating liquid with a large number of anomalous properties. The increasing violation of the Stokes-Einstein (SE)equation, D = kBT/6r, connecting the translational diffusion coefficient D and medium viscosity , with lowering temperature is one of them1–4. The above breakdown starts
showing up below T ~2.1Tg for water, while for molecularglass forming liquid the SE breakdown is prominent only
below T ~1.2Tg4. Similar to the self-diffusion of the liquid molecules, the translational diffusion of a tracer in liquid also
violates the SE equation, connecting D with the viscosity of the liquid medium5–14.
Many possible ways are available to quantify the extent
of violation of the SE equation. Experimental and simulation

works generally study the coupling/decoupling of D of thetagged particle from  of the medium instead of direct verification of the SE equation4. Sometimes, due to unavailabilityof accurate viscosity data, structural relaxation time (average translational relaxation time) acts as a proxy based on
the fact that the structural relaxation time is generally proportional to the viscosity of the medium2.
There are many explanations for the above decouplingof the translational diffusion of the tagged particle from theviscosity of the medium at the supercooled regime. The
gradual increase of the microscopic spatiotemporal heterogeneity with cooling down the liquid is one of the key reasons7,8,15–30. The decoupling is also explained from the second liquid-liquid critical point and the position of the Widom
line. Jump translation of a tagged particle in a supercooled† Both the authors contributed equally.

JICS-3741

J. Indian Chem. Soc., Vol. 96, July 2019liquid is a crucial factor for the observed decoupling between
the translational diffusion and the viscosity28. Goldstein put
forward a theory on the temperature dependent mechanismof translational diffusion of liquid molecules18,24,31–38. The
theory suggests that while at the low temperature a tagged
particle in the liquid diffuse via large step jump from one
saddle point to another in the rugged free energy landscape,the particle translates via small step Brownian motion at ahigher temperature. Several theoretical and computer simulation studies confirmed the characteristics of this picture18,24,31–38.
A recent study by two of us has successfully calculated –for the first time using the radius of gyration method – thejump-diffusion coefficient DJ of the water molecules at a wide
range of temperatures including the deeply supercooled regime39. The above study has addressed the increasing importance of translational jump diffusion of the water moleculesin the diffusion-viscosity decoupling with decreasing temperature. By separating DJ from the overall diffusion coefficient
D the residual diffusion coefficient DRes is obtained. DRes
remains strongly coupled with the viscosity at the studied
temperature range.
The present work focusses on the effect of solute size on
jump-diffusion coefficient of the solute molecule in supercooled water. It was observed earlier that the solute’s diffusion gets increasingly decoupled from the medium viscosity
as the solute/solvent molecular size ratio decreases40. For
the diffusion of Lennard-Jones (LJ) solute in LJ solvent, threedifferent regimes were identified based on the diffusivity ofthe solute with variable sizes. The validity of the SE equation
is evidenced when the solute/solvent size ratio is higher than
~0.5. The “small solute” regime was seen for solute/size ratio less than ~0.17. In the intermediate regime, where thediffusivity of the solute is very high, it is termed as the “levitation effect”.
Even though the above studies have suggested the importance of translational jump-diffusion of a small solute forthe breakdown of the SE relation in supercooled water (liquid), more direct and quantitative estimation of DJ is still
missing. The present work is an MD simulation attempt forthe direct evaluation of DJ of four different hydrophobic solutes in water and prediction of the percentage contribution
of DJ to the overall diffusion of the water molecules D.
742This study provides insight into the role of the translational jump-diffusion of the solute molecules for the increasing breakdown of the SE relation with decreasing the solute
size at the supercooled regime. Moreover, the exploration of
the jump-diffusion of the hydrophobic solute in supercooled
water is also important in cold denaturation of the protein.
Enhanced fluctuation of some protein domains, which has
been witnessed at low temperature, plays a vital role for cold
denaturation of the protein41. It is interesting to note that highfluctuating domains are generally localized within the protein core regions. It is well known that the watersoluble proteins have a hydrophobic core, in which side
chains are buried from water, which stabilizes the folded
state42,43.
The organization of the rest of the paper is as follows. In
Section 2 we describe the simulation protocol and the forcefield parameters of the molecules. Results and discussions
are detailed in Section 3. Section 4 summarizes the results
and offer concluding remarks.

Scheme 1. Molecular structures of the four hydrophobic solutes arranged in order of increasing size from left to right.

2. Simulation details
We have used GROMACS package44 for performing the
MD simulation. The system includes 2000 water molecules
and a solute molecule. We have considered four differenthydrophobic solutes: methane, neopentane, tetra-tertbutylmethane (TTB), and fullerene. The structures of themolecules are presented in Scheme 1. The initial system isbuilt using Packmol software45. The above choice of solutes
provides a range of solute radius from ~2.0 Å to ~6.0 Å. The
solutes are modeled by OPLS/AA force field46 while watermolecules are modeled byTIP4P/2005 force field47. It wasobserved that the OPLS/AA force fields for methane 48,
neopentane49, and fullerene50 works reasonably well. For

Dubey et al.: Importance of translational jump in diffusion of hydrophobic solute in supercooled water
TTB, we have not found any literature work on MD simulation study. TIP4P/2005 water model has proven efficacy inpredicting various experimental structural and dynamical
properties of water at a wide range of temperatures including deeply supercooled regime.
Simulations are carried out at two different temperatures,
240 K and 300 K. Since the freezing temperature of TIP4P/
2005 water is 252 K at 1 atm pressure, 240 K temperature
represents the supercooled regime. Therefore, this study
focuses on the dynamics of the solutes in water at room temperature and in the supercooled regime. We have maintained
the pressure at 1 bar for both the temperatures throughout
the simulations. First, we use the steepest-descent algorithm
for classical energy minimization of the systems. The classically energy minimized configuration is then equilibrated for
50 ns time using NPT ensemble, during which the temperature is kept constant at the respective desired temperaturesusing the Nosé-Hoover thermostat51,52 and maintained pressure using the Parrinello-Rahman barostat53. The final box
length is presented in Table 1. We have chosen 0.5 ps and 1
ps for the pressure and temperature coupling constant respectively. Further, we have performed production runs of 1
s for 240 K and 500 ns for 300 K for each of the solutes
using NPT ensemble. These simulations sums to a total of 6
s long simulation.

3. Results and discussion
3.1. Hydration structure
We first study the hydration structure (both the spatial
and orientational) around the different hydrophobic solutes.
We calculate the radial distribution functions (g(r)) between
the center-of-mass (COM) of the solute and solvent water
OW. Fig. 1 exhibits the g(r) plots for four different solutes at
two temperatures. As expected, the g(r) peaks shift towards
higher r values with increasing the solute size. More interestingly, for the alkanes, the first peak intensity decreases
with increasing the hydrophobic solute size. This is as perearlier simulations, where a similar reduction of first peakheight was observed on increasing the size of the methane
cluster55. The above reduction is supposedly due to a gradualdewetting transition. However, for fullerene, the intensity of
the first peak of g(r) is higher than those for the other alkanes. The current simulated g(r) is in good agreement withearlier simulated g(r)56. The greater intensity of the fullerene-

We have used the Leapfrog-Verlet algorithm for solving
the equations of motions with a time step of 2 fs. The production trajectory is saved at a regular interval of 100 fs. A
cut-off distance of around half of the box length is applied for
LJ interactions. Particle Mesh Ewald summation technique
is used to handle the long-range Coulomb interactions. During the simulations, all the bonds are constrained with LINCS
algorithm54.
Table 1. The average box lengths for the systems at two different
temperatures
SystemBox length (Å)
T = 240 K

T = 300 K

Methane

39.32

39.17

Neopentane
TTB

39.33
39.37

39.19
39.24

Fullerene

39.42

39.29

Fig. 1. Radial distribution functions between the COM of the four different solute molecules and water Ow atoms at the temperatures: 300 K (a) and 240 K (b).

743

J. Indian Chem. Soc., Vol. 96, July 2019
water g(r) peak compared to the alkane-water g(r) peak
may be due to the difference of shape factor. While C60 isperfectly spherical, the alkanes are having tetrahedral geometry. Therefore, the hydration structure around C60 is more
spherical and regular compared to that around the alkanes.
This is also suggested by the larger width of the alkane-water g(r) compared to that of the C60-water g(r).
The g(r) peaks at long distance r are clearly visible beyond the second peak for T = 240 K, while they are not visible at 300 K temperature. These long distance peaks areprobably emanating from the collective structural ordering(lowering of entropy) of supercooled water molecules. A recent study57 has already focussed on the temperature-dependence of water-water g(r) at higher r. This study has indicated the possible HDL-LDL transition as a possible origin of
the g(r) peaks at longer r.
We now present the angular structure of the solvent water molecules around the hydrophobic solutes. Fig. 2 presents P(r,), which is a two-dimensional probability density
distribution of the angle  – between water OW…solute COM
vector and the water OW-HW bond vector – and the distancer between water OW atom and the solute’s COM at T = 240 K
for four different solutes (Similar pictures are obtained for T= 300 K but not shown here). It is evident from Fig. 2 that theFig. 2. Two-dimensional probability density distribution, P(r,), of theangle  – between water OW…solute COM vector and thewater OW-HW bond vector – and the distance r between water
OW atom and the solute’s COM at T = 240 K for four different
solutes.

744

water molecules at the interface of the hydrophobic solute
are oriented in a particular fashion. Either the water molecules are oriented tangential with the solute surface (tangential OH with  ~ 65º), or they are aligned towards the bulk
water (bulk OH with  ~ 180º). No dangling OH bonds arefound near the solute (dangling OH with  ~0º), which confirms the absence of H-bonds between the solute and thesolvent water molecules. The peaks – situated at the second
hydration shell positions – indicate that the majority of thesecond shell solvent water OH bonds are aligned towards
the solute such that these water molecules form H-bonds
with the interfacial water solvent molecules. The above orientational order is almost invariant to the solute size andtemperature, except slight changes of the peak intensity. The
above hydration structures of the four solutes confirm thehydrophobicity of the solutes in TIP4P/2005 water solvent58–69.
3.2. Diffusion of the solutes: Size dependence
We calculate self-diffusion coefficient D of the solute molecules dissolved in water at two different temperatures using the mean square displacement (MSD) route. MSD is calculated using the equation, |r(t)|2 = |ri (t) – ri (0)|2. Here,r(t) and r(0) are the positions of the solute molecule at time t
and time t = 0 respectively. Fig. 3 presents MSDs against
time for four different solutes, each at two temperatures. In
all cases, MSDs show three distinct regions: (i) the ballistic
part (|r(t)|2  t2) at short time, (ii) the diffusive part((|r(t)|2t)) at the longer time, and (iii) the sub-diffusive
part at intermediate time (|r(t)|2t  ; 0 <  < 1 ). This
sub-diffusive part of MSD appears from the rattling motion of
the solute molecule, which is momentarily trapped inside thesolvent’s cage24,38,39,70–77. At room temperature, the subdiffusive regime is more prominent at an intermediate time
for the methane solute. The above region decreases with
the increase of the solute size. For fullerene, this regime isalmost invisible. However, at T = 240 K, the sub-diffusive
part is much more intense for all the solutes. This indicatesthat the trapping of a hydrophobic solute in supercooled water
solvent cage is significant irrespective of the solute’s size,
while at room temperature the trapping is mainly importantfor small hydrophobic solutes24,38,39,70–77. D is calculated
by fitting the MSDs of the solutes at diffusive limit using thefollowing equation, |r(t)|2 = 6Dt.
Table 2 lists the D values of the solutes at two temperatures. Clearly, at both the temperatures, the D value de-Dubey et al.: Importance of translational jump in diffusion of hydrophobic solute in supercooled water
as the solute’s D value increases by ~5 times as the solute’s
size is decreased by ~2.5 times. The stronger breakdown of
the SE equation at 240 K temperature is very similar to thedecoupling of diffusion of supercooled liquid from viscosity.
We will see in the next section, the importance of the jumptranslation for the self-diffusion of the small solute in supercooled water.

Fig. 3. Mean square displacement of the solute molecules as functions of time at the two temperatures, 300 K (a) and 240 K (b).

creases with increasing solute’s size. The solutes diameters
have been determined using the following equation, S =2(r1S–W – r1W–W), where r1S–W and r1W–W are the 1st peak position of the g(r) between the solute and water OW and the 1st
peak position of the water OW-water OW g(r). S values ofthe solutes are listed in Table 2. Table 2 clearly shows thatwhile the solute’s size is decreased by ~2.5 times on replacing the solute fullerene by methane, solute’s D value increases by ~10 times at 240 K temperature. This strongly
contradicts to the behavior predicted from the SE equation,
where D is just inversely proportional to the solute’s diameter (that is the increase of D value at the same proportion
with the decrease of solute’s size). At room temperature,
however, the violation of the SE equation is not so intense
Table 2. The calculated diameters of the solutes (S = 2(r1S–W –r1W–W)), and the total self-diffusion coefficient D of the solutes at two
temperatures
SystemS (Å)

Methane

4.46

0.124

1.744

Neopentane
TTB

6.78
9.38

0.046
0.034

1.150
0.451

Fullerene

10.66

0.015

0.308

D/10–5 (cm2/s)
T = 240 K
T = 300 K

3.3.Calculation of the jump-diffusion coefficient
Now, we turn our focus on estimating the translational
jump contribution to the total self-diffusion coefficient of thesolutes at two temperatures. Here, the key is to identify areal translational jump occurrence of the solute correctly. Multiple ways are possible for identifying a jump event. The simplest of all is via following over time the displacement of thesolute molecule from its position at t = 0 and finding the point
when a rapid change of displacement happens24,38,39,78–80.
This method was only successful in tracing the presence ofjump event in the trajectory of the solute molecule and understand the mechanistic pathway of a translational jump
event. Unfortunately, being based on manual identificationof the jump events, this approach fails in a quantitative analysis of jump-diffusion coefficient.
Another method, based on oscillator model theory81, wasused to calculate the jump-diffusion coefficient (neck diffusion) of a small solute particle in a viscous liquid. This method
captured the increased relevance of jump-diffusion of a small
solute molecule through solvent cages82. However, it is notclear whether the same theory is applicable for estimating
the jump-diffusion coefficient of bigger solute and solvent
molecules.
We have used here a different method, previously developed by Raptis et al.83,84 and later modified by Araque etal.85. This method has been recently used by three of us forunderstanding the importance of translation jump of supercooled water molecules for increasing decoupling of the selfdiffusion coefficient of water from viscosity with decreasing
temperature39. In this method, we first split the long trajectory of a solute molecule and calculate the radius of gyration
Rg for each of the trajectory segments. Rg, for a trajectory
segment of length t (n time space), is calculated using thefollowing equation39,83–85,
Rg (t, t) =1 n [ri (t ;t )  r CM (t ;t )] 2
n i 1(1)745

J. Indian Chem. Soc., Vol. 96, July 2019
where ri (t; t) and rCM (t; t) are the position of the solute ati-th time frame and the center of mass position of the trajectory segment of length t. rCM (t; t) is calculated by thefollowing equation,rCM (t; t) =1 n
 r i (t ;t )n i 1(2)

Consideration of the value of t in eqs. (1) and (2) must be
done judiciously. It is not recommended to use the same tvalue for different solutes and temperature. Because the diffusion of a solute changes significantly with its size and temperature, the consideration of the same t for all conditions
may not produce correct results. For example, a trajectory
segment of t =100 ps duration at 240 K temperature explores much less phase space than that at 300 K. Similar
situation arises when trajectory segments of the same lengthis used for two different solutes. All the above problems canbe solved if we consider t as the time t*, at which the nonGaussian parameter 2(t) goes maximum. We calculate 2(t)using the following equation,
2(t) = 3r4 (t)/5r2 (t)2 – 1(3)

where, r2 (t) is the MSD and r 4 (t)= |ri (t) – ri (0)|. Fig. 4

presents 2(t) as a function of time for four different solutes,
each at two temperatures. The peak intensity of 2(t) increases with decreasing solute size. This indicates an increase of temporal heterogeneity with decreasing solute size.
But, t* (listed in Table 3) shifts towards higher value as thesolute size increases. Now, comparison of 2(t) plots between T = 240 K and 300 K temperatures reveals that thepeak heights for all the solutes at 240 K temperature arehigher than those at 300 K temperature. Therefore, t time

Fig. 4. Non-Gaussian parameter 2(t) as a function of time for all thesolute molecules at the two different temperatures, 300 K (a)and 240 K (b). The time t* for the maximum value of 2(t) arelisted in Table 3.

duration increases with increasing temperature and decreasing the solute size.
As we split the trajectory into segments, each with theduration of t time, and we calculate Rg for all these trajectory segments, our next goal is to calculate the distance traversed by the molecule in a trajectory segment (t). Thiscan be calculated using the equation, (t) = 2Rg83–85. Fig. 5

Table 3. Solute size dependence of the time scale t* (the time when the non-Gaussian parameter 2(t) is the maximum), the two crossing
distances (r1 and r2) between simulated and theoretical self-part of the van Hove correlation functions, the number of successful jumpoccurrences nJ, the jump frequency J, and the average jump length <J > at two different temperatures
SystemMethaneT (K)240

t* (ps)9.0

r1 (nm) Cage
0.11

r2 (nm) Jump
0.28

nJ
2461

Jump frequency J (ns–1)
2.50<J > (Å)
3.27

Neopentane

300
240

1.2
19

0.15
0.08

0.41
0.18

92
740

0.18
0.74

4.36
2.00

300

1.2

0.10

0.26

55

0.11

2.72

TTB

240
300

17
1.8

0.08
0.08

0.17
0.20

19
18

0.02
0.04

1.81
2.09

Fullerene

240

28

0.06

0.14

12

0.01

1.47

300

2.8

0.08

0.19

10

0.02

1.94

746

Dubey et al.: Importance of translational jump in diffusion of hydrophobic solute in supercooled water

Fig. 5. The plot of (t) as a function of time for four different solutes at T = 240 K temperature. The horizontal lines indicate the cutoff distancer2, listed in Table 3. The circles, which are above the cutoff distance r2, represent the jump trajectory segments. The jump trajectory
segments for the two (t) values (indicated by vertical red arrows in panel (a)) are shown by three-dimensional positional phase space in
Fig. 7.

plots (t) as a function of time for all four solutes at T = 240
K. Evidently, we have to identify the big successful jumps,
which significantly contribute to the jump-diffusion of the solute. For that, we use the self-part of the van Hove correlation
function GSsimu (r, t ), which can be calculated from the following equation86,87.
1

N

GSsimu (r, t ) = N  i   r  |r i (t 0 )  r i (t )| t(4)0

It is well known that GSsimu (r, t ) deviates maximum from thetheoretical Gaussian behavior GStheo (r, t ) = [3/2r2 (t)]3/2
exp [–3r2/2r2 (t)]P at t* time, when 2(t) is maximum (seeFig. 4). We have plotted both the GSsimu (r, t ) and GStheo (r, t )in Fig. 6 for four solutes and at two temperatures.
Fig. 6 shows that GSsimu (r, t ) deviate from the Gaussianity
at the most for methane solute, but as the solute size increases GSsimu (r, t ) becomes more and more Gaussian innature. For fullerene, we see practically no difference between GSsimu (r, t ) and GStheo (r, t ). This indicates that the
dynamical heterogeneity in solute’s translation diffusion decreases with increasing the size of the solute. This is consis-

tent with the decrease in the peak height of 2(t) with increasing solute size. Also, as expected, the deviation of GSsimu(r, t ) from the Gaussian behavior is less at 300 K temperature than that at 240 K temperature. In all cases, GSsimu (r, t )crosses GStheo (r, t ) at two r values: r1 and r2. At the smaller rlimit (r < r1) the actual displacement of the solute is less thanthat predicted from the Gaussian GStheo (r, t ). Therefore, thetrajectory segment with (t) less than r1 is a cage trajectory.
On the other hand, beyond r > r2 distance the actual displacement of the solute molecule is larger than the displacement, predicted from GStheo (r, t )85. Hence, the trajectory segments, with (t)> r2, can be termed as jump trajectory. Table
3 lists the numerical values of r1 and r2 for different solutes
at two temperatures. Fig. 7 presents several examples ofjump trajectory segments. Clearly, the present method for
identifying the jump event is working as expected.
As the methodology of identifying a jump trajectory is described above, the next step is to count the jump events bycounting the peaks of (t), which cross the cutoff distancer2. Table 3 summarizes the number of jump events nJ, thejump frequency J (number of jump events per 1 ns), and747

J. Indian Chem. Soc., Vol. 96, July 2019average jump length <J > (averged over nJ). We see thatnJ and J are maximum for methane solute and both thevalues decrease with increasing solute’s size and temperature. This essentially indicates that the importance of jumptranslation in solute’s diffusion increases with decreasing thesolute’s size and the temperature.
Using J and <J>, listed in Table 3, we now calculate
the jump-diffusion coefficient DJ using the following equation39,
1
DJ = —
J2J 
6Fig. 6. The self-part of the van Hove correlation functions (solid line)
and the corresponding ideal Gaussian distribution (GStheo (r, t ))(dashed line) at time t* when 2(t) is maximum. The left handpanels are for T = 240 K, while results for room temperature
are presented in right hand panels.

(5)

The above equation has been written based on the approximation that the translation jump events stochastically contribute to the jump-diffusion of the solute. The DJ values arelisted in Table 4. The percentage contribution of DJ , J =(DJ /D)×100, to the overall diffusion coefficient D has also
been listed in Table 4. For methane at T = 240 K, J ~35,
which indicates that the jump diffusion of methane contributes approximately 35% to the overall diffusion of methanein supercooled water at 240 K temperatures. For neopentane,
the value of J decreases to ~11.J further decreases to 0.3
for the larger solute molecule. At the room temperature J isalways less than 1. These observations are the most important results in this study since it provides a quantitative support to the general understanding that the enhancement of
the diffusion coefficient on decreasing solute size in the supercooled is due to the gradual dominance of large translational jump.
It was observed earlier that a residual diffusion coeffi-

Fig. 7. The jump trajectory segments of methane for the two (t) values (indicated by vertical red arrows in Fig. 5a) in three-dimensional
positional phase space at T = 240 K temperature. The time evolution is color coded.

748

Dubey et al.: Importance of translational jump in diffusion of hydrophobic solute in supercooled water
cient DRes can be calculated by subtracting DJ from the overall
diffusion coefficient D using the following equation39,
DRes = Dt – DJump(6)

From the definition, DRes is that component of the overall
diffusion, which completely excludes molecular jump translation. Therefore, DRes is primarily the small step diffusion
process as in the standard Einstein’s Brownian motion. We
have calculated DRes for all the solutes and the temperatures and listed the numerical values in Table 4. Table 4 clearlyshows that while the D value increases by ~10 times onchanging the solute from fullerene to methane (the solute
size decreases by ~2.5 times) at 240 K temperature, DRes
increases by ~5 times. However, for room temperature, both
Table 4. Different components of diffusion coefficients: total selfdiffusion coefficient (D), diffusion coefficient due to jump (DJ), andthe residual diffusion coefficient (DRes) of different solute molecules
at two different temperatures. Percentage contribution of DJ to D isalso listed in the last column
SystemT (K)

D/10–6

D/10–8

D/10–6

J =(cm2/s)(cm2/s)(cm2/s)(DJ /D)×100

Methane

240
300

1.24
17.4

43.0
5.80

0.81
17.4

34.7
0.34

Neopentane

240

0.46

4.90

0.41

10.7

TTB

300
240

11.5
0.34

1.40
0.10

11.5
0.34

0.13
0.30

Fullerene

300
240

4.51
0.15

0.30
0.04

4.51
0.15

0.07
0.27

300

3.08

0.13

3.08

0.04

the D and DRes values increase by ~5 times on the samedecrease of the solute size. Therefore, DRes, at 240 K, behaves with solute size very similarly to that at 300 K temperature. This strongly indicates that the excess enhancement of diffusion of a small solute, sometimes termed as‘levitation effect40, has an overwhelming contribution fromthe jump-diffusion of the small solute.
4. Conclusion
In conclusion, we have presented a detailed MD simulation study for understanding the hydrophobic solute’s size
dependence on the self-diffusion coefficient of the solute forboth room-temperature and supercooled water. We havereported an increasing diffusion coefficient with decreasing
solute’s size at both the temperatures. However, the increaseof the diffusion is almost double in supercooled water than in
JICS-4

water at room temperature. This apparent breakdown of theSE equation for methane and neopentane has been explainedby the higher jump-diffusion of the smaller solutes in supercooled water. We have quantitatively estimated the role oftranslational jump-diffusion in the enhanced diffusion ofsmaller solutes. By careful consideration of the translational
jump trajectories of the solute, we have calculated the jumponly diffusion coefficient DJ. As the solute size is increased,
the contribution of DJ to the overall diffusion increases. By
subtracting DJ from D, we obtain the residual diffusion coefficient DRes. The effect of enhanced diffusion for the methane solute at 240 K compared to that for fullerene is reduced
while we consider DRes of the solute instead of D. This is aclear proof of increasing importance of translational jumpdiffusion of the small solute molecule for the observed violation of the SE equation. These new findings can help in elucidating many experimental studies featuring molecular transport properties in more complex chemical and biological environments, where strong diffusion-viscosity decoupling isprevalent.
Acknowledgement
We acknowledge IIT Patna for financial assistance andcomputational resources. Vikas and Shivam acknowledge
IIT Patna for research fellowships. Shakkira acknowledges
DST INSPIRE Fellowship (INSPIRE No. IF180090) for funding. We acknowledge SERB Early Career Research Award(File No. ECR/2017/002335) for funding.
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