Distribution of Diffusion Constants and Stokes-Einstein Violation in supercooled liquids

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by  Shiladitya Sengupta

Distribution of Diffusion Constants and Stokes-Einstein Violation in supercooled
liquids
Shiladitya Sengupta1,2 and Smarajit Karmakar2
1Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced
Scientific Research, Jakkur Campus, Bangalore 560 064, India.
2TIFR Centre for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi,Hyderabad 500075, India.
ItiswidelybelievedthatthebreakdownoftheStokes-Einsteinrelationbetweenthetranslational
3 diffusivity and the shear viscosity in supercooled liquids is due to the development of dynamic
1 heterogeneity i.e. the presence of both slow and fast moving particles in the system. In this
0 studywedirectly calculatethedistributionofthediffusionconstantforamodelsystemfordifferent
2 temperaturesinthesupercooledregime. Wefindthatwithdecreasingtemperature,thedistribution
n evolvesfromGaussian tobimodalindicatingthatonthetimescaleoftheαrelaxationtime,mobile
a (liquidlike)andlessmobile(solidlike)particlesinthesystemcanbeunambiguously identified. We
J alsoshowthatlessmobileparticlesobeytheStokes-Einsteinrelationeveninthesupercooledregime
and it is the mobile particles which show strong violation of the Stokes-Einstein relation. Finally,
7
we show that the degree of violation of the Stokes-Einstein relation can be tuned by introducing
] randomly pinned particles in the system.
h
c
e INTRODUCTION Therearehowever,indirect evidences, e.g. the univer-
m
sal exponential tail in the Van Hove functions [24] seen
- for many supercooled liquids, which supports the exis-
t In normal liquids, the shear viscosity (η) is related to
a tence of a distribution of diffusivity (relaxation times).
the translational diffusion constant (D) of a particle dif-
t
s fusing through it via the Stokes-Einstein (SE) relation Consider an extreme case where the system has regions
.
t [1–3]asD = cT,where c is aconstantwhichdepends on with two diffusivity - one for “solid like” (Ds ) and the
a η
other for “liquid like” regions(D ). Hence a distribution
m the details of the particle and boundary conditions and l
ofdiffusivity canbe writtenasp(D)=Aδ(D )+Bδ(D )
T is the temperature. Although, originally derived for s l
-
d a macroscopicprobe particle in the hydrodynamic limit, where A and B are fixed by the normalizationcondition
n andthe amountofsolidlike andliquidlikeregions. Now
the SE relation also holds for the self diffusion of liq-
o if we calculate the van Hove correlation function as
uid particles at high temperatures [4]. However, it has
c
[ been shown extensively [5–20] that when a liquid is su-
percooledthe SErelationbreaksdowni.e. themeasured Gs(x,t)=Z dDp(D)g(x|D), (1)
1
self diffusivity becomes much larger than the value pre-
1v dicted by the SE relation. Often the shear viscosity is where g(x|D) = √4π1Dtexp(cid:16)−4xD2t(cid:17) then one may show
8 substituted by the relaxation time (τ) which is cheaper that the van Hove function will have a long tail and de-
1 to compute and more commonly treated in theories. pending on the distribution of the p(D), the tail of the
1
Phenomenological arguments e.g. by Stillinger and distribution can be either exponential or Gaussian [25].
.
1 Hodgdon[4]andbyTarjusandKivelson[11]haveshown In general the exponential tail has been reported [24]
0
that by considering supercooledliquids to consistof mo- which, as mentioned in [25], might be due to the small
3
bile “fluid-like” and less mobile “solid-like” regions, the range of the data.
1
: decoupling between the translational diffusion (D) and The main aim of this study is to calculate directly the
v
the relaxation time (τ) is explained naturally because distribution of the diffusivity from the simulation data.
i
X the average diffusion constant is predominantly deter- From the previous discussion, one may expect that in
r mined by the “fluid like” regions whereas the average deeply supercooled liquids the distribution of diffusivity
a
relaxation time is dominated by the “solid-like” regions. (p(D))willbebimodalingeneral. Unlikepreviousworks
The existence of clusters (with finite lifetime) of mobile [26–28] where the presence oftwo types of particles with
andlessmobileparticleshasbeendirectlyshowninmany distinct mobilities are inferred indirectly from the dis-
different studies [21–23] and is known as the dynamical tribution of displacements, our method directly and un-
heterogeneity (DH). However,the above-mentionedphe- ambiguouslyshowsthatthedistributionofthediffusivity
nomenological theory of decoupling is based on the ex- becomesbimodalbelowsometemperature. Notethatthe
istence of a distribution of diffusivity (relaxation times) bimodal nature of the distribution only says that there
whereas previous studies on DH have typically analyzed are two types of particles in the system, but it does not
the distribution of particle displacements. Hence not prove that these particles are clustered together to form
much direct information about the existence and nature “solid like” and “fluid-like” regions. Hence bimodal dis-
of the diffusivity (relaxation time) distribution is avail- tributionsofdiffusivity aloneisnotenoughtojustify the
able. picture of supercooled liquids being a sparse mixture of
2
“fluid like” and “solid like” regions. Using random pin- where p(D)dD = P(Dτ)d(Dτ). The choice of Dτ as
ninggeometrywefurthershowthatindeedonehas“fluid our variable is due to the fact that D changes by several
like” and “solid like” regions in the system up to some orders of magnitude in the studied temperature range
time scale of the order of α relaxation time, τ . whereas Dτ changes relatively modestly with decreasing
α
We end this section by noting that the phenomeno- temperature and it will be easier to compare the distri-
logical explanation of decoupling does not tell anything bution obtained for different temperatures.
about the nature and origin of the heterogeneity. Mi-
croscopictheoriestounderstanddecouplingfromthedy- 35 100
namic heterogeneity e.g. mode coupling theory (MCT)
[29], random first-order transition (RFOT) theory [30– 30 τ)10−3
x,
32],sheartransformationzone(STZ)theory[33],dynam- 25 G(s10−6
icalfacilitation[34]andtheobstructionmodel[35]donot
mutually agree onthe originand the nature of heteroge- Dτ)20 10−9
neous dynamics. Besides, the decoupling of D and τ P(15 −5 −2 x0 2 5
may also be explained in terms of a growinglength scale
10
[27, 36] which does not directly require a distribution of
diffusivity (relaxation times). 5
0
10−3 10−2 10−1 100
DISTRIBUTION OF DIFFUSIVITY Dτ
Below we explain briefly the method used to extract FIG.1: Thesolidlineshowstheresultsoftheiterativescheme
used to calculate the distribution of diffusivity ( see text for
the distribution of diffusivity directly from the van Hove
details ) along with the original distribution shown as sym-
correlation function G (x,τ) using the iterative algo-
s bols. The agreement is really encouraging and inset shows
rithm suggested in [37] and recently used in [25] for the
the corresponding comparison for the van Hove correlation
diffusion processes in biological systems. We start with functions.
the definition of van Hove correlationfunction
We tested on a toy model whether the above men-
G (x,τ)= δ[x (x (τ) x (0))] , (2)
s h − i − i i tionediterativeschemeconvergestothecorrectsolution.
WestartedwithadistributionP(Dτ)andcalculatedthe
where the < . > implies averaging over the time origin.
van Hove correlation function G (x,τ) using Eq.3, and
Now ifwe assumethat particledisplacements arecaused s
then used this to recalculate the probability distribution
by diffusion processes and there is a distribution of local
P(Dτ) using Eq.6. In Fig.1, we have compared the ob-
diffusivity p(D), then we can express G (x,τ) in terms
s
tained distribution P(Dτ) with the input distribution.
of p(D) as
The agreement is really amazing with moderate number
D0 of iterations. In the inset we have compared the van
Gs(x,τ)=Z p(D).g(x|D).dD, (3) Hove function obtained from the converged distribution
0
ofdiffusivitywiththeexactone. Herealsotheagreement
whereg(xD)=1/√4πDτexp x2/4Dτ andD0 isthe is near perfect. Note that the iterative scheme does not
| −
upper limit of diffusion consta(cid:0)nt and w(cid:1)ill be equal to depend at all on the initial guess distribution.
diffusivity for a free diffusion. Now given the G (x,τ) After establishing the rapid convergence of the itera-
s
wecalculatethedistributionofdiffusivityp(D)following tive scheme to the correct solution we tried to calculate
[37] as the distribution of diffusivity for a model glass forming
liquid, the Kob-Andersen Model [38]. To compare the
pn+1(D)=pn(D) ∞ Gs(x,τ)g(xD)dx, (4) van Hove functions for different temperatures we cal-
Z Gn(x,τ) | culated it for the time τ when the mean square dis-
s m
−∞
placement becomes half the inter particle diameter i.e.
where pn(D) is the estimate of p(D) in the nth iteration <∆r2(τ )>=0.50asshowninFig.2. The temperature
m
with p0(D)=(1/D )exp( D/D ) and
avg avg dependence of τ is very different from the α-relaxation
− m
time τ calculated from the normalized two point corre-
D0 α
Gns(x,τ)=Z pn(D).g(x|D).dD. (5) lationfunctionas<Q(τα)>=1/e. IntheinsetofFig.2,
0 we have shown τ and τ as a function of temperature.
m α
ThisisanothermanifestationoftheStokes-Einstein(SE)
Similarly
violation in this model.
Pn+1(Dτ)=Pn(Dτ) ∞ Gs(x,τ)g(xD)dx, (6) To extract the distribution function of the diffusivity
Z Gn(x,τ) | oneneedstosupplysomewhatsmoothlyaverageddataof
s
−∞
3
T = 0.450 thediffusivityP(Dτ )asfunctionoftemperatureandin
m
T = 0.470
T = 0.500 Fig.4, we showed the van Hove correlationfunctions cal-
T = 0.520 culated from these distributions of diffusivity along with
T = 0.550
100 T = 0.600 the simulationdata. Theagreementbetweenthe simula-
T = 0.700
T = 0.800 tiondataandthecalculatedonesisfantastic. Animpor-
T = 0.900
tantpointtomentionhereisthattailsofthesevanHove
T = 1.000
> correlationfunctions cannot be completely describedby
2∆r10−2 104 <<∆Q(rτ2)(τ>)=>=1/e0.5 0 asingleexponentialfunctionoverthewholerangeatleast
< for the low temperature data ( T 0.50). Rather they
≤
τ102 can be better fitted by two exponential functions.
10−4 100 T = 0.450
0.4 0.6 T 0.8 1 102 TT == 00..550500

10−2 10−1 100 101 102 103 104 T = 0.700
t T = 1.000
)m100
FIG.2: MeanSquareddisplacement(MSD)fordifferenttem- x,τ
perature and the horizontal line indicates the time where G(s
mean squared displacement reaches 0.50 in the units of par-
10−2
ticles diameter for different temperature. The corresponding
timeisdenotedasτm here(seetextfordetails). Inset: Com-
parison of the α-relaxation time τα calculated from the nor-
malized two point correlation function as < Q(τα) >= 1/e 10−4
with τm. −4 −3 −2 −1 x0 1 2 3 4
FIG. 4: The van Hove correlation functions (solid line) as
50 obtainedfromtheiterativescheme(seetextfordetails)along
T = 0.450
with the simulation data (symbols). The curves are shifted
T = 0.470
upwardforclarityandtopointoutthatthelinegoesthough
40 T = 0.500
thedata points overthe whole range of thedata.
T = 0.520
) T = 0.550
m 30
τ T = 0.600
D
T = 0.700
(
P 20 T = 0.800 10−2 Ds
T = 1.000 Dl
10 10−3 100
100 −3 10−2 10−1 100 D10−4 θ(T)10−1
Dτm
10−5
FIG.3: Calculated distributionofthediffusivityfordifferent 10−2
0.4 0.6 0.8 1
temperatures. Notice the appearance of the bimodality in 10−6 T
the distribution just below the onset temperature T = 1.00. 0.4 0.5 0.6 0.7 0.8 0.9 1
T
Clearbimodaldistributioninthesupercooledregimeconfirms
thattherearetwodifferenttypesofparticlesintermsoftheir
FIG. 5: The temperature dependence of the diffusivity asso-
mobility up to time scale of typical relaxation time. The
appearance of more peaks in the distribution at still further ciated with the solid like (Ds) and liquid like (Dl) particles.
These values are calculated from the peak positions of the
lower temperature is really interesting, indicating possibility
distributions in Fig.3. Inset : The Stokes-Einstein violation
ofextremelyslowtomoderately slowtoveryfast particlesin
parameter θ(T) = Dτα for the two types of particles. One
very deep supercooled state. T
sees clearly that solid like particles obey Stokes-Einstein re-
lation to a reasonable accuracy over the whole temperature
range and it is the liquid like particles which show strong
the van Hove correlationfunction Gs(x,τm) for the iter- Stokes-Einstein violation.
ativeschemeto convergefast. Inthis studywefittedthe
extreme tail of the calculated G (x,τ ) using exponen- Now looking at Fig.3, we can see that just below the
s m
tial function as this part is in general noisy and difficult onset temperature (T = 1.00), the distribution starts to
to average. In Fig.3, we have shown the distributions of becomebimodalandtwopeaksclearlyemergeattemper-
4
0
10 ρimp=0.005 cles in the system, there will always be instances where
ρimp=0.010 thesefrozenparticlesarepartofthese“solidlike”regions.
ρimp=0.020 In that case, due to the frozen particles the relaxation
ρimp=0.040 of these regions will be hindered further and the relax-
T −1 ρimp=0.080 ationtimeofthewholesystemwillincreasedramatically
/α10 ρimp=0.160 with increasing density of these frozen particles. How-
τ
D ever, these frozen particles will have very small effect on
diffusivity which is mainly governed by the “fluid like”
particles, so the diffusivity will not change that dramat-
−2 ically. In this scenario one expects to see an enhance-
10
ment of the Stokes-Einstein breakdown if one increases

0.5 1 1.5 2 2.5 3 thenumberdensityρ oftherandomlyfrozenparticles.
T imp
Onthecontraryiftheclustersof“solid-like”particlesare
FIG. 6: Stokes-Einstein violation parameter has been shown not present in the system then we expect that changing
asafunctionoftemperaturefordifferentdensityofthefrozen the density of randomly frozen particles will have little
particles ρimp. One can clearly see that the deviation from effect on the relaxation dynamics of these regions i.e.
theStokes-Einsteinrelationbecomesstrongerwithincreasing no dependence of the SE violation parameter with ρ .
imp
density of the frozen particles.
In Fig.6, we have shown the temperature dependence of
the Stokes-Einstein violation parameter θ(T) = Dτα for
T
different pinning densities ρ . The deviation of θ(T)
imp
ature around T = 0.60. At further lower temperatures from being a constant as a function of temperature with
the distribution seems to show existence of shoulder or increasing ρ is very dramatic.
imp
another peak but the peak at large diffusivity remains
ihnatvaectclaelatrhloyugdhemwointhstrdaetcerdeatshinatg tpheearke haerieghtwt.oTtyhpuesswoef 10−1 ρρiimmpp==00..000150,,ωω==00.. 115893
particlesinthesupercooledliquid. Oneismoresolidlike ρimp=0.020,ω=0.207
(less mobile) than the other in the time scale of the or- 10−2 ρimp=0.040,ω=0.225
der of α relaxation time, τ . It is worth noticing that ρimp=0.080,ω=0.259
α
the width of the distribution increases with decreasing −3 ρimp=0.160,ω=0.295
D10 ρimp=0.200,ω=0.301
temperature which indicates increase of DH leading to
stronger SE breakdown[17].
−4
10
In the top panel of Fig.5, we have shown the temper-
ature dependence of the diffusivity associated with the
−5
10
solidlike(D )andthe liquidlike(D )particles. Herewe
s l
took the peak position as an estimator for the diffusion 100 101 102τα 103 104 105
constants for solid and liquid like particles. At the low-
est three temperatures shoulders seem to appear in the FIG. 7: Diffusivity (D) plotted as a function of relaxation
distributions, but we calculated the peak position to be time (τα) for different density of the frozen particles ρimp
the position of the dominant peak. In the lower panel of in log-log to show the power law relationship between these
Fig.5,wecalculatedtheStokes-Einsteinviolationparam- quantities expected from the fractional Stokes-Einstein Re-
eter θ(T)= Dτα for the two set of particles. One clearly lation D ∝ τα−1+ω, with ω ≥ 0. Notice that the exponent
sees that solidTlike particles obey the Stokes-Einsteinre- ω increase with increasing pinning density ρimp indicating
a stronger breakdown of the Stokes-Einstein Relation. The
lation over the whole temperature range, whereas the
solid line has a slope equal to -1.0.
liquid like particles show strong SE violation leading to
overallviolation of the SE relationin the liquid. A simi- In literature people often use a fractional Stokes-
lar conclusion is reached in [26]. Einstein relation [9, 14, 35] to fit the data where the
After establishing the fact that two types of particles original relation breaks down. We tried similar fits to
can be identified in the system on a time scale of α re- the low temperature data using the form D τ 1+ω,
∝ α−
laxation time, we now turn to the question of whether with ω 0. In Fig.7, we have shown that our data can
≥
the “solid-like”particlesgrouptogetherto formclusters. bewellrepresentedbythefractionalStokes-Einsteinrela-
Toanswerthisquestionweperformedanothersetofsim- tioninthelowtemperaturerangewithω increasingwith
ulation experiments where we randomly freeze (random increasing ρ . Now it is important to mention that
imp
pinning geometry) some of the particles and then ask in the past many people have tried to tune the Stokes-
what is the effect of this on the dynamics of the system. Einstein breakdown exponent to better understand the
Ifthe “solidlike”particlesareclusteredtogethertoform physics behind it. In those studies either different inter-
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5
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