Anisotropic Local Stress and Particle Hopping in a Deeply Supercooled Liquid

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by  Sarika Bhattacharyya

Anisotropic Local Stress and Particle Hopping in a Deeply Supercooled Liquid
∗
Sarika Bhattacharyya and Biman Bagchi
Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India
The origin of the microscopic motions that lead to stress relaxation in deeply supercooled liquid
remainsunclear. Weshowthatinsuchaliquidthestressrelaxationislocallyanisotropicwhichcan
serve as the driving force for the hopping of the system on its free energy surface. However, not
5
all hopping are equally effective in relaxing the local stress, suggesting that diffusion can decouple
0
from viscosityevenatlocal level. Ontheotherhand,orientational relaxation isfoundtobealways
0
coupled to stress relaxation.
2

n
a Dynamicsofsupercooledliquidshowmanyfascinating fectsofthesehopping. Therefore,onecannotexplorethe
J
properties, namely the non-exponentiality in various re- relationship between hopping and the total stress relax-

1 laxationfunctions ofdynamicalvariables,like the stress, ation.
3
density, composition, polarization and orientation[1, 2,

3, 4, 5]. One often observes a very slow power law de- The experiments on the other hand are successful in
]
h cay in the intermediate to long times[6]. Although these showing the decay of the SAF. However, except a re-
p have drawn attention of experimentalists and theoreti- cent work using single molecule spectroscopy [9], these
- cians and have been extensively studied, many aspects experiments are macroscopic and do not provide enough
m
still remain ill-understood. It is believed that the nature information of the microscopic motions in the system.
e
of the microscopic motion undergoes a drastic change
h
at a temperature T (a temperature substantially be- Majumdar[10]hadearlierdiscussedtheimportanceof
c B
s. low the melting temperature, Tm). At the temperature localrelaxationmodes(ofwavelengthlessthantheshort
c TB, the continuous hydrodynamic type motion, which is range order) in giving rise to non-exponentiality in the
i prevalent at higher temperature changes to discontinu- stressrelaxationfunction. Thisworkdiscussedrelaxation
s
y ous motion. It is also believed that TB is close to the in terms of relaxation within small regions, surfaces and
h temperature where effects of the underlying free energy alsovolumes,withprogressivelengtheningoftimescales.
p landscape on the dynamics are felt for the first time[1]. However, in that analysis the basic mechanism of relax-
[ T isalsofoundtobeclosetothemodecouplingtemper- ation was still assumed to be continuous.
B
1 ature, Tc. This change in the nature of the microscopic
v motion is believed to be the originof the experimentally In this Letter we demonstrate for the first time that
6 observed α-β bifurcation [7] and also the cross-over be- in the deeply supercooled liquid (where hopping is the
5 tween the rotational and translational relaxation times only surviving large amplitude motion), there is a close
1 [8]. relationshipbetweenthelocalstressandtheorientational
1 and translational hopping. The local SAF is anisotropic
0
Due to the complexity of the problem, computer sim- and is found to change drastically during the hopping,
5
ulationshaveplayedakeyroleinaugmentingourunder- thus showing that the local stress and the hopping of a
0
standinginthisarea. Inparticular,simulationsallowone particle are intimately connected. The anisotropy in the
/
s todirectlylookatthemicroscopicevents. Thecomputer localstresscouldbethedrivingforceforhopping. Asthe
c
i simulation studies of the stress auto correlationfunction free energy of the system can be expressed in terms of
s inthesupercooledliquidcouldsuccessfullyreproducethe the position dependent stress in a generalizedGinzburg-
y
powerlawbehaviorofthestressauto-correlationfunction Landau formulation [11], the change of the anisotropic
h
p (SAF) [3, 4]. However, in the deeply supercooled liquid, stress due to hopping should be regarded as the driving
: onefindsthatwithinthesimulationtime,therelaxation, force for the transitions of the system between different
v
after an initial decay (typically 10-20%) becomes fully minima of the free energysurface. However,not all hop-
i
X arrested. The microscopic origin of the subsequent de- pings are effective in relaxing the stress.
r cay is unclear. The computer simulation studies further
a show that the orientationaland translationalhopping of Our solvent is represented by binary Lennard-Jones
particles are the only mode present and hence the stress mixture, which has been extensively studied [1, 12, 13,
relaxation can happen only via hopping. However, since 14] and is known to be a good glass former, and our
the relaxation time is much much longer (could be of soluteprobesareprolateellipsoids. Pressureiskeptcon-
the order of ms or sec), the computer simulation results, stantby Andersen’s pistonmethod [15] while inthe case
(which can explore mostly upto nano second regime and of temperature, a damped oscillator method has been
sometimes micro second regime), cannot include the ef- adopted which keeps temperature constant at each and
every time step [16]. The piston mass involved here is
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0.0027(m /σ ) which is regarded as optimum [16, 17].
A A
Theinteractionbetweentwoellipsoidswitharbitraryori-
∗Forcorrespondence: [email protected] entationsis assumedto be givenbythe Gay-Berne(GB)
2
potential [18],
12
σ0
UGB = 4ǫ(rˆ,uˆ1,uˆ2)"(cid:18)r−σ(rˆ,uˆ1,uˆ2)+σ0(cid:19) (1)
6
− σ0
(cid:18)r−σ(rˆ,uˆ1,uˆ2)+σ0(cid:19) #
whereuˆ1uˆ2aretheaxialvectorsoftheellipsoids1and2.
rˆisthevectoralongtheintermolecularvectorr=r2−r1,
where r1 and r2 denote the centers of mass of ellipsoids
1 and 2 respectively. σ(rˆ,uˆ1,uˆ2) and ǫ(rˆ,uˆ1,uˆ2) are the
orientation-dependentrangeandstrengthparametersre-
spectively. σ and ǫ depend on the aspect ratio κ. The
minor axis of the ellipsoid is equal to the diameter of
the larger solvent and the major axis is 3 times that of
the minoraxis. Finally,the interactionbetweenasphere
and an ellipsoid is accounted for by a modified GB-LJ
potential given below
12 6
σ(θ) σ(θ)
U =4ǫ Ei − Ei (2)
Ei Ei
r r
"(cid:18) (cid:19) (cid:18) (cid:19) #
where ’E’denotes the ellipsoids and ’i’ can be ’A’ or ’B’.
The expression for σ(θ) is available [19].
Ei
The ellipsoidin binarymixture systemwith the above
mentioned potential is a well behaved system and it
can also exhibit the experimentally observed anoma-
lous viscosity dependence of the orientational correla-
tion time [19]. Four ellipsoids were placed far from each
other in a binary mixture of 500 particles with num-
ber of ’A’ particles, NA = 400 and number of ’B’ type FIG.1: (a)Showsthedisplacementofthe1sttaggedellipsoid
particles N = 100. The reduced temperature is ex- over 1000 τ. There is a hopping of the ellipsoid around 300
B
pressedas,T∗(=k T/ǫ ),thereducedpressureas,P∗(= τ. In the inset we plot the orientational correlation function
B A
Pσ3/ǫ ). and the reduced density as ρ∗(=ρσ3). The obtained at different intervals. (b) Same as figure 1 (a) but
A AA A ∗
time is scaled by τ = (mAσA2/ǫAA). The time step of fTo∗r=t0h.e8.2nd tagged ellipsoid. The plots are at P =10 and
the simulation is .002 τ and the system is equilibrated
for 1.5 × 105 steps anpd the data collection step is 5 ×
6 ∗
10 . ThestudieshavebeenperformedatT =0.8andthe
P∗=6 and 10. are associated with orientational hopping. In the inset
At P∗ =6, both hopping and continuous motions ex- we also plot the orientational time correlation functions
ist in the system, thus the stress relaxationcould not be (OCF), before, during and after the hopping. Figure
directly correlated with the hopping. At P∗=10, only 1(a)showsthetrajectoryofthefirsttaggedellipsoidand
inset shows its orientational time correlation function,
microscopic motion that survives is hopping. In a recent
study, we have reported observation of correlated trans- < P2(uˆi(0)uˆi(t)) > /<P2(uˆi(0)uˆi(0))>, were P2 is the
second order Legendre polynomial. The hopping takes
lational and orientational hopping [20] at this pressure.
After extensive simulations we could find only two dif- place in 20τ and the displacement is 0.5σ. Here the el-
lipsoid hopping is accompanied by hopping of 5-6 of its
ferent kinds of motions. The translational hopping was
either associated with correlated hopping of 5-6 nearest nearest neighbors. The OCF decays only during the pe-
riod of hopping. In figure 1 (b) the trajectory of the
neighbors or it exhibited a motion in a ring like tunnel.
While it is possible that other types of motion like iso- second tagged ellipsoid and in the inset its orientational
time correlation functions are shown. Note that in this
lated hopping can exist, we could not find them. The
hopping rate was found to be 2 × 107 where both the case the displacement of the particle is large (1.1σ) and
it alsotakesplace overa verylong periodof time (50 τ).
type of motions occurred with almost equal frequency.
In the following we focus onthe stress relaxationand its Here we find that the taggedparticle is moving in a ring
∗ ∗ like tunnel. Although the orientational correlation func-
relation vis-a-vis hopping at P =10 and T =0.8. The
reduced density of the system is 1.41. tion decays during the hopping, it’s decay is less when
compared to that of the 1st tagged particle.
In figure 1 we show two different kinds of spatial hop-
ping observed in our simulations. Both the hoppings The motions in a string like and ring like tunnel in a
3
8000 zx 10000
zy
yx
6000 xz 8000
yz
xy
> >
σ(t) 4000 σ(t) 6000
σ<(0) 2000 σ(0)< 4000 zzyxxyxz
yz
0 2000 xy
−2000 0
0 0.5 1 1.5 2 0 2 4 6 8
time (t) time (t)
6000 8000
zx
zx zy
zy yx
4000 yx 6000 xz
xz yz
σ(t) > yxzy σ(t)> xy
0) 2000 0) 4000
σ( σ(
< <
0 2000
−2000 0
0 0.5 1 1.5 2 0 2 4 6 8
time (t) time (t)
FIG.2: (a)Thedifferentcomponentsofthelocalstressauto- FIG.3: (a)Thedifferentcomponentsofthelocalstressauto-
correlation (obtained from the sum of the stress of the 1st correlation (obtained from the sum of the stress of the 2nd
ellipsoid and its nearest neighbors) function obtained before ellipsoid and its nearest neighbors) function obtained before
hopping, between 320-330 τ. (b)is the same as 2 (a) but the hopping, between 1590-1630 τ. (b) is the same as 3 (a) but
components of the SAF are obtained after hopping between the components of the SAF are obtained after the hopping,
(340-350τ). Theplots are at P∗=10 and T∗=0.8. between 1670-1710τ. The plots are at P∗=10 and T∗=0.8.
deeply supercooled liquid have been reported earlier by respectively. Before the hopping there is an anisotropy
other authors, although they did not discuss this large of the stress. The yz and zy components of the stress
displacements over a long time for the ring like motion are much larger than others, and also their correlations
[21]. ceaseto decay. This leads to the hopping of the ellipsoid,
In the following part we discuss in details the local mostly in the x direction. During and after the hopping
stressrelaxationassociatedwiththesetwodifferentkinds there is an exchange of the stress. After the hopping
of hoppings. The local stress around the ellipsoid is althoughthe yz component ofthe stressrelaxesandalso
obtained by summing over the stress on the ellipsoid the t=0 value of all the components reduces, the SAF
and its nearest neighbors. The stress has six compo- in the xz and zx components cease to decay. Thus, this
nents and the stress auto-correlation function is given kind of motion in a ring like tunnel does not lead to the
by < σ (0)σ (t) >, where i,j=x,y,z. In case of only relaxation of all the components of the stress.
ij ij
Lennard-Jonesfluid, σ =σ ,butforparticlesinteract- Theorientationalcorrelationfunctionofthe1sttagged
ij ji
ing via GB and GB-LJ potential this is not so. particle relaxesmore ( inset of figure 1 (a)) comparedto
Figures 2 (a) and (b) show the SAF around the 1st thatofthe2ndtaggedparticle(insetoffigure1(b))when
taggedellipsoid,beforeandaftertheperioditishopping, computedinthe respectiveintervalswheretheyarehop-
respectively. Before the hopping there is an anisotropy ping. From figure 2 and 3 we found that the local stress
ofthestress. Thexyandyxcomponentsofthestressare relaxation takes place when the 1st ellipsoid is hopping
much larger than the others and also their correlations where as when the second ellipsoid is hopping although
cease to decay. This anisotropy leads to a hopping of the there is an exchange of stress between it’s components,
ellipsoid, mostly in the z direction. During the hopping thelocalSAFdoesnotcompletelyrelax. Thereisadirect
thereisarelaxationoftheSAFinxyandyxcomponents connectionbetweenthelocalstressandthe orientational
and after hopping all the components relax. Note that relaxationfunctions,implyingthatrotationandviscosity
the total stress in all the components are also lower. are coupled even in a localized region.
Figures 3 (a) and (b) show the SAF around the 2nd Inordertounderstandwhathappenstothe surround-
taggedellipsoid,beforeandaftertheperioditishopping, ingofthelocalregionwhenthereisarelaxationofstress
4
due to hopping we have studied the stress auto correla- lipsoid(Fig.1b)althoughittranslatesmore,thestressre-
tion function of a bigger region of 2 σ around the 1st laxationduring its hopping is less. Thus suggesting that
ellipsoid.Wefoundthereareabout62-67particlesinthis translationalmotion and viscosity are decoupled even in
region. Although there is an anisotropy of the compo- a localized region. On the other hand, the orientational
nents of the stress in this larger region, this anisotropy relaxation and also the stress relaxation is more for the
cannotbecorrelatedwiththedirectionofhopping. Some firstellipsoid. Thussuggestingthattheorientationalmo-
of the components of SAF build up immediately after tionalwaysremainscoupledtoviscosity. Thisisinagree-
hopping and in a later time it relaxes. Similar analysis mentwiththeexperimentalresultsandinfactprovidesa
when done in a even bigger region shows that it takes microscopic explanation of the results which are known
longer for the stress of that region to relax and also the for a long time. There can be an apparent connection
effect of the hopping is less. between the stress tensor and the momentum circula-
In conclusion we demonstrated that the direction of tion. Thus it is possible that the non-decaying SAF im-
the hopping of the tagged particle is determined by the pliesthatmomentumcirculationexistsinadeeplysuper-
anisotropy in the stress. Anisotropic stress relaxation is cooled liquid Since the anisotropic stress contributes to
differentwhenthereisamany-particlehoppingandthere the free energyofthe system,a changein the anisotropy
is a motion in a ring like tunnel. Although there is an drives the system from one free energy minimum to the
exchange of stress between the components due to the other. When the anisotropy in the stress disappears and
particle motion, the stress relaxationis less in a ring like all the SAF relaxes then the system definitely moves to
motion. Interestingly, the effect of hopping is found to a lower free energy minimum.
spreadovertheadjoiningregionlikerippleswiththeam- ThisworkwassupportedbyagrantfromCSIR,India.
plitude decreasing with increasing distance from the el- We thank Srikanth Sastry Arnab Mukherjee and Rajesh
lipsoid. We notethatinthe caseofthesecondtaggedel- Murarka for helpful discussions.
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