Table Of ContentStructural anomalies offluids:
Originsinsecond andhigher coordinationshells
William P. Krekelberg∗
Department of Chemical Engineering, University of Texas at Austin, Austin, TX 78712.
Jeetain Mittal†
LaboratoryofChemicalPhysics, NIDDK,National InstitutesofHealth, Bethesda, MD20892-0520.
Venkat Ganesan‡ and Thomas M. Truskett§
DepartmentofChemicalEngineeringandInstituteforTheoreticalChemistry,UniversityofTexasatAustin,Austin,TX78712.
8
0
Compressing or cooling a fluid typically enhances its static interparticle correlations. However, there are
0
2 notableexceptions. Isothermalcompressioncanreducethetranslationalorderoffluidsthatexhibitanomalous
waterlike trends in their thermodynamic and transport properties, while isochoric cooling (or strengthening
n of attractiveinteractions) can have asimilareffect onfluidsof particleswithshort-range attractions. Recent
a
simulationstudiesbyYanetal.[Phys.Rev.E76,051201(2007)]ontheformertypeofsystemandKrekelberg
J
etal. [J.Chem. Phys. 127, 044502 (2007)] onthelatterprovide examples wheresuch structural anomalies
1 can be related to specific changes in second and more distant coordination shells of the radial distribution
1
function.Here,weconfirmthegeneralityofthismicroscopicpicturethroughanalysis,viamolecularsimulation
and integral equation theory, of coordination shell contributions to the two-body excess entropy for several
]
t relatedmodelfluidswhichincorporatedifferentlevelsofmolecularresolution.Theresultssuggestthatintegral
f equation theory can be an effective and computationally inexpensive first-pass tool for assessing, based on
o
thepairpotentialalone, whethernewmodelsystemsaregoodcandidatesforexhibitingstructural(andhence
s
. thermodynamicandtransport)anomalies.
t
a
m
I. INTRODUCTION ofaqueoussolutionproperties.5,6,7,8,9,10,11,12,13,14,15
-
d Although the second shell of the RDF has received com-
n parativelylessattention,thereisevidencethatitalsocontains
o Abulkequilibriumfluidistranslationallyinvariant;i.e.,its structural information relevant for understanding nontrivial
c one-particle density, ρ(1)(r) = ρ, is constant. Nonetheless, behaviorsofliquids.Onenotablefeatureistheshoulder16that
[ assuming spherically-symmetric interactions, the local den-
itdevelopsnearthefreezingtransition,whichinturnbecomes
1 sity ρg(r) surrounding a reference particle is a function of a pronouncedsplit peak17,18,19,20,21,22,23,24,25,26 in supercooled
v distancerfromitscenter,whereg(r)istheradialdistribution liquid and glassy states. Analysis of the configurations that
6 function(RDF) of the fluid.1 Althoughthe RDF dependson
giverisetothisstructuralmotifindicatethatitreflectsfrustra-
6 boththeformoftheinterparticleinteractionsandthethermo- tionoficosahedral23andemergingcrystalline16,26orderinthe
8 dynamic state, some features of its shape are fairly general.
fluid. Understandinghowthesethese typesofstructuralfea-
1
For example, the RDF vanishes for r less than the effective
. turesconnecttorelaxationprocessesofsupercooledliquidsis
1 exclusiondiameteroftheparticles. Forlargerr, itshowsan
anactiveareaofresearch(see,e.g.,Refs.27,28,29,30).
0 oscillatorydecaytowardunitywithpeakslooselycorrespond-
In this work, however, we focus on the second and
8
ingtocoordination“shells”. Awayfromthecriticalpoint,the
0 higher coordination shells of the RDF for a different rea-
structure of the RDF typically persists for distances compa-
: son: to understand their role in the structural anomalies
v rabletoafewparticlediameters,reflectingtheshortrangeof
of fluids. Interparticle correlations of most fluids are en-
i theinterparticlecorrelations.
X hanced upon (i) compression or (ii) cooling (alternatively,
r Studiesoftheliquidstatehaveprimarilyfocusedonthepar- strengtheningofinterparticleattractions). Nonetheless,there
a
ticlesinthefirstcoordinationshell. Thisisdueinparttothe are a few systems of scientific interest that exhibit no-
importantrole that nearestneighborsare expectedto play in tably different behaviors. For example, compression in-
determiningmanyphysicochemicalproperties. Forexample, duced disordering occurs in water and other fluids with
boththenon-idealcontributiontotheequationofstateofthe anomalouswaterliketrendsintheirthermodynamicandtrans-
hard-spherefluid(see,e.g.,Ref.2)andthecollisionfrequency portproperties.31,32,33,34,35,36,37,38,39,40,41,42,43,44 Cooling(orat-
in Enskog theories for transport processes3 scale with the traction) induced disordering, on the other hand, can oc-
“contact”densityρg(σ),whereσistheparticlediameter.The cur in fluids of particles with short-range attractive (SRA)
hard-sphereequationofstateisthestandardreferencesystem interactions,37,45suchasconcentratedsuspensionsofcolloids.
forperturbationtheories.1 Italsoaccuratelypredictshowthe These anomalies do not appear to be first-shell effects.
thermodynamicsof“hard-sphere”colloidalsuspensionsrelate Rather,theyreflecthowstructuringinsecondandmoredistant
to their structure, as has recently been experimentally veri- coordinationshellsrespondstochangesinthermodynamicor
fied by confocalmicroscopy.4 Furthermore, analysisof first- systemparameters.Forexample,Yanetal.43recentlydemon-
shellcontributionstohydrationstructureandthermodynamics stratedinaninsightfulpaperhowthestructuralanomalyofthe
helpstounderstandandmakepredictionsaboutawidevariety five-sitetransferableinteractionpotential(TIP5P)model46for
2
water is quantitativelyrelated to compression inducedtrans- tropy,whichmeasuresstructuralorder,61 andquasi-universal
lational disorderingof molecules in the second coordination excess entropy scalings62,63,64 for the transport coefficients.
shell. Similarly,Krekelbergetal.45haveshownthatthecool- All of this suggests that investigations like the present one,
ing(orattraction)inducedstructuralanomalyofasquare-well which probe the physics of structural anomalies, might also
SRAfluidisduetoweakeningofsecond-andhigher-shellpair provideinsightsintodynamicandthermodynamicanomalies
correlations. aswell.
The goalhere is to study the generality of the abovefind-
ings. It is known that a number of models, with vary-
ing levels of molecular resolution, can qualitatively pre- II. METHODS
dict the structural anomalies of the aforementioned sys-
tems.31,32,33,34,35,36,37,38,39,40,41,42,43,44,45 But do the anomalies
Weusedmoleculardynamicssimulationandintegralequa-
exhibited by lower resolution models have the same micro-
tiontheorytoexaminevariousmodelsfromtwoclassesofflu-
scopic origins as those of more detailed models? Moreover,
ids knownto exhibitstructuralanomalies: those with water-
canthebehaviorofthelowerresolutionmodelsbepredicted,
likepropertiesandthosecomprisingparticleswithSRAinter-
at least qualitatively, by integral equation theory? If so, it
actions. Fortheintegralequationtheoryanalysis,wenumer-
would suggest that integralequation theorymight serve as a ically solved the Ornstein-Zernike equation65 together with
valuablefirst-passtoolinassessing,basedonthepairpotential
an approximate closure relation using the method of Labik
alone,whethernewmodelsystemsmightbegoodcandidates etal.66Inthediscussionofthemodelsbelow,wementionthe
forexhibitingstructuralanomalies.
specific closures employed and provide further details about
Furthermore, although the structurally anomalous trends themolecularsimulations.
analyzed here are interesting in their own right, there is a
We did not perform a systematic study here to determine
more compelling reason to try to understand their origins. which of manypossible closure relations1 providesthe most
In short, they appear to be closely linked to other distinc-
quantitatively accurate description for each model. Rather,
tivedynamicandthermodynamicbehaviors. Forexample,in
our focus was to explore whether integral equation theory
addition to being “structurally anomalous”, cold liquid wa-
solvedwithstandardclosurerelations,suchasPercus-Yevick
terisalso“dynamicallyanomalous”inthatitsself-diffusivity (PY)67orhypernetted-chain(HNC)68,caninfactqualitatively
increases upon isothermal compression and “thermodynam-
predict both the structural anomalies and their microscopic
ically anomalous” in that its volume increases upon iso-
originsin the RDF. Molecularsimulationsof the modelsys-
baric cooling. Errington and Debenedetti31 first noticed that
temsprovidethedatanecessarytomakethatbasicdetermina-
theseparticularanomaliesformacascadeinthetemperature-
tion.
density plane for the extended simple point charge (SPC/E)
model47ofwater.31Specifically,thethermodynamicanomaly
occurs only for state points that also exhibit the dynamic
A. Waterlikefluidmodels
anomaly. The dynamic anomaly, in turn, is only present for
statesthatalsoexhibitthestructuralanomaly.Strongcorrela-
tionsbetweenthesethreebasictypesofanomalieshavesince We investigated two waterlike models: (1) the SPC/E47
been documented for a wide variety of model systems with modeland(2)a lowerresolution“core-softened”36,69 model.
waterlikeproperties.32,33,36,39,40,43,44,48,49,50,51,52 We chose the SPC/E model because it represents one of the
mostcommonlystudiedeffectivepairpotentialsforwater,and
A similar connection between structural and dynamic
it is known to qualitatively reproduce many of water’s dis-
anomalies has now also been identified for model SRA
fluids.37,45 Inthose systems, the mostcommonlystudied dy- tinctivethermodynamic,dynamic,andstructuralproperties.31
As a result, it provides a reasonable baseline against which
namicanomalyisanincreaseinself-diffusivityuponcooling
to comparesimpler, lower resolution models. Details of our
(or strengthening of interparticle attractions), which can oc-
curatsufficientlyhighparticleconcentrations.53,54,55,56,57,58,59 moleculardynamicssimulationsof the SPC/E modelare the
Krekelberg et al.45 discovered that the self-diffusivity sameasreportedinRef.37.
The core-softened model36,69 that we studied is more
anomaly for a square-well SRA fluid occurs only for state
schematic. ItisdefinedbytheeffectivepairpotentialU (r)
points that also exhibit the cooling (or attraction) induced CS
[seeFigure1(a)],
structural anomaly discussed above. In other words, it ap-
pearsthatSRAfluidscanalsodisplayacascadeofanomalies
qualitativelysimilartothoseofwaterlikefluids. σ 12 σ 6 r 2
U (r)=4ǫ − +5ǫexp − −0.7 ,
CS
Although structural and dynamic properties of these sys- "(cid:18)r(cid:19) (cid:18)r(cid:19) # " (cid:18)hσi (cid:19) #
tems show unusual dependencies on quantities like temper- (1)
ature or density, the correlations between structure and dy- whereǫisthecharacteristicenergyscale. Themainideabe-
namics are often similar to those found in simpler liquids hind this potentialis that it has two differentkindsof repul-
(e.g.,thehard-spherefluid).37,38,39,45,60Infact,itwasrecently sions that act at different length scales. The harsh (σ/r)12
demonstrated60 that the cascade of anomalies of one water- repulsion defines the effective hard-core diameter (σ), while
likemodelsystemcanbesemi-quantitativelypredictedbased the softer Gaussian repulsion extends to considerably larger
only on knowledge of the state dependencies of excess en- distances. The end result is that the average interparticle
3
separation, and hence the density, of this fluid can depend discuss some of its main features below. The colloids are
sensitively on both temperature and pressure. The model sphericalandtheireffectiveinteractionsconsistofthreeparts.
is similar to cold water in that it favors locally open (low- The first is a steeply repulsive, essentially hard-sphere(HS),
density)structuresatmoderatepressureandlowtemperature, contribution U (r) = k T(2a/r)36, where 2a is the col-
HS B
but can collapse to denser structures when compressed or loiddiameter,k isBoltzmann’sconstant,andT istempera-
B
heatedenoughto overcomethe softGaussian repulsion. Al- ture.Thesecondtermrepresentstheaforementionedpolymer-
thoughthislowresolutionmodeldoesnotprovideanaccurate induced depletion attraction U (r), approximated by the
AO
molecular-leveldescriptionof water, it doesqualitativelyre- Asakura-Oosawa73 potential. The strength of this attraction
producemanyofits peculiarthermodynamic,structural, and isproportionaltothevolumefractionofpolymersinsolution
kineticbehaviors.36,38,40,69 φ , whiletherangeiscontrolledbytheradiusofgyrationof
p
To compute the properties of the core-softened model, the polymers R , set in this case to a/5. A soft repulsion
g
we performedmoleculardynamicssimulationsin the micro- U is also added to the effective interparticle potential71 to
R
canonical ensemble using N = 1000 identical particles of preventfluid-fluidphaseseparation. Figure1(b)displaysthe
mass m. We used the velocity-Verlet technique for inte- totalcolloidalpotentialU = U +U +U forthree
SRA HS AO R
grating the equations of motion with a time step of ∆t = differentpolymerconcentrationsφ .Thedetailsofthemolec-
p
0.002σ m/ǫ. For the integralequationtheoryanalysis, we ulardynamicssimulationsthatweperformedforthisfluidare
employed the HNC closure. We chose the HNC approxi- thesameasthosereportedinRef.74,withoneexception. In
p
mation because of its ability to describe the structure of an- theoriginalstudy,aweaklypolydispersesystemwasinvesti-
otherfluid witha softGaussianrepulsion,theGaussian-core gated.Here,allparticlesconsideredhadidenticalradiusaand
model.70 We investigated both the SPC/E and core-softened mass m. The advantageof focusingon a monodispersesys-
modelsovera widerangeof densityandtemperature,where temisthatthepaircorrelationsareunambiguouslydescribed
theyareknowntoexhibitstructuralanomalies.31,36,38,40,69 by a single RDF, which facilitates the analysis discussed in
thenextsection. FortheintegralequationtheoryofthisSRA
8 fluid, we employed the PY closure. The PY approximation
(a) isanaturalchoicehereduetoitssimplicityanditsabilityto
6 describethestructureofliquidswithharshlyrepulsive,short-
ε
rangepotentials,1(inparticular,otherSRAfluids57).
/
S4 We also considered a simpler model SRA fluid: a system
C
U ofidenticalsquare-wellparticleswithattractivewelldepth-ǫ
2
andwidth0.03σ,whereσ representsthehard-corediameter.
This modelis similar to othersknown to exhibitstructural45
0
1 2 3 anddynamic45,58anomalies.WealsousethePYclosureinour
r / σ integralequationtheory analysis of this fluid for the reasons
mentionedabove.
9
φ =0.0
(b) p
T 6 0.2
B
k 0.4
/ 3 C. Quantificationofstructuralorder
A
R 0
S
U Foreachofthemodelfluids,wecalculatedthestatedepen-
-3 denciesof−s /k ,
2 B
2 3 4
r/a − s2 =2πρ ∞r2{g(r)lng(r)−[g(r)−1]}, (2)
k
B Z0
FIG. 1: (a) Pair potential of the core-softened model UCS(r/σ)/ǫ
[see Eq. (1)]. (b) Pair potential of the model SRA fluid wheres2 isthetranslationalpair-correlationcontribution75,76
USRA(r/a)/kBT discussedinthetextforvariousvaluesofpolymer to the excess entropy and ρ is the number density. We used
concentrationφp.FurtherdetailsonthisSRAmodelareprovidedin the orientationally averaged oxygen-oxygen RDF in Eq. 2
Refs.71and72. for the analysis of SPC/E water. It has been shown that
−s /k not only quantifies the translational order exhibited
2 B
by a fluid (the tendencyof pairs of particles to adapt prefer-
ential separations),61 but it also strongly correlates with the
B. SRAfluidmodels transportcoefficients(see,e.g.,Refs. 37,45,64). Othertrans-
lationalorder parametershave also been introducedto study
ThefirstSRAfluidmodelthatweconsideredqualitatively thestructureofmolecularandcolloidalfluids,31,61,77butthese
describes a solution of (explicit) colloidal particles attracted measuresareknowntocorrelatestronglywiths ,31,61andthus
2
to one another by depletion interactions due to the presence weexclusivelyusethelatterinouranalysis.
of(implicit)non-adsorbingpolymers. Thedetailsofthecol- To understand how the various coordination shells of the
loidal pair potentialare providedin Refs. 71 and 72, but we RDFcontributeto−s /k ,wealsoinvestigatedthecumula-
2 B
4
tiveorderintegralIs2(r),definedas45 3 ¶ · ¸ T=220
r T 240
Is2(r)=2πρZ0 r′2{g(r′)lng(r′)−[g(r′)−1]}dr′. (3) / k2B2.52 223680000
s
NotethatI (r)→−s /k asr →∞. - (a)
s2 2 B 1.5
Finally,weadoptedthefollowingcriteriatoidentifystruc-
turallyanomalousbehavior. 0.8 1 ρ [g/cm3] 1.2
¶ · ¸
∂[−s ]
2 <0, ρ−structuralanomalies (4a)
∂ρ 4
(cid:18) (cid:19)T
)
∂∂[k[−Ts2/]ǫ] >0, T−structuralanomalies (4b) g( r 2 (b) (d) (f)
(cid:18) B (cid:19)ρ
As indicated in the Introduction, waterlike fluids ex- 0
hibit ρ−structural anomalies37,60 and SRA fluids display
T−structuralanomalies.45 2
r ) (c) (e) (g)
(Is21 ρ=0.85 ρ1=.005.9 ρ1=.215.15
III. STRUCTURALANOMALIES 0.9 1.15 1.30
0
0.3 0.6 0.3 0.6 0.3 0.6
A. Waterlikefluids r [nm]
1. SPC/Ewater FIG. 2: Structural data for the SPC/E water model ob-
tained from molecular dynamics simulations. (a) Structural or-
der parameter −s2/kB as a function of density ρ at T =
First, we discuss the simulation results for how −s /k
2 B 220K,240K,260K,280K and 300K. Vertical dotted lines are at
(i.e., translational order) of the SPC/E water model changes ρ = 0.9g/cm3 and ρ = 1.15g/cm3, the approximate bound-
with density ρ. As can be seen in Figure 2(a), SPC/E water ariesfortheregionofanomalousstructuralbehavior. (Lowerpanel)
displaystheρ−structuralanomaliesofEq.(4a)overtheden- Orientationallyaveragedoxygen-oxygenradialdistributionfunction
sity range 0.9g/cm3 ≤ ρ ≤ 1.15g/cm3 and T < 280K. g(r) and cumulative order integral Is2(r) along the T = 220K
To gain insights into the origins of this behavior, we ex- isotherm[blackcircles,dashedcurvein(a)]forthreedifferentden-
amine the orientationallyaveragedoxygen-oxygenRDF and sityregions: (b,c)ρ≤0.9g/cm3 [uptomaximumin−s2(ρ)/kB],
I as a function of ρ along the T = 220K isotherm for (d,e)0.9g/cm3 ≤ ρ ≤ 1.15g/cm3 [betweenmaximumandmin-
thsr2ee different density regions: (1) the initial increase of imumin−s2(ρ)/kB], (f,g)ρ ≥ 1.15g/cm3 [beyond minimumin
−s (ρ)/k atlowdensities[ρ≤0.9g/cm3,Figs.2(b,c)],(2) −s2(ρ)/kB].Theregionsareindicatedbycirclednumbersalongtop
2 B of(a)andlowerpanel. Inthelowerpanel,arrowsindicatedirection
theanomalousdecreaseof−s (ρ)/k atintermediatedensi-
2 B ofincreasingdensity;dashedverticallineisatr=0.31nmanddot-
ties[0.9g/cm3 ≤ ρ ≤ 1.15g/cm3,Figs.2(d,e)],and(3)the
tedverticallineisatr =0.57nm,theapproximatelocationsofthe
increase of −s2(ρ)/kB at high densities [ρ > 1.15g/cm3, firstandsecondminimaing(r),respectively.
Figs.2(f,g)].
.
Compressingthefluidinthelower-densityregion(1)(ρ ≤
0.9g/cm3)hasrelativelylittleeffectontheRDF[Fig.2(b)],
butit doeslead to a small netincreasein translationalorder. 0.9g/cm3 ≤ ρ ≤ 1.15g/cm3] result in a pronounced de-
As can be seen from the behavior of Is2(r) in Fig. 2(c), the crease in −s2(ρ)/kB, i.e. the ρ-structural anomaly. As can
changescomeprimarilyfromthesecondshell. Thereasonis be seen in Fig. 2(d), the main implications of compression
thatthecoordinationnumberofwater(approximatelyfour,re- for the interparticle correlations are a dramatic flattening of
flectinglocaltetrahedralhydrogen-bondingto nearestneigh- the second coordination shell and an associated shifting in-
bors)isinsensitivetochangesindensityoverthisrange.78As wardofthesemoleculesintotheinterstitialspacebetweenthe
aresult,theincreaseofρiscompensatedbyaslightdecrease first and second shells. These structuralchanges are consis-
inthefirstpeakoftheRDF, andthusthefirst-shellcontribu- tent with the earlier simulation observation78 that high local
tiontothestructuralorderremainslargelyunchanged. Inthe densitycan force a fifth moleculefromthe secondshell into
second shell, however, the change in density does not affect theperipheryoftheotherwisefour-coordinatedfirstshell. In-
theRDF(i.e.,thestrengthofthecorrelationswiththecentral spection of I (r) [Fig. 2(e)] confirms that the decrease in
s2
molecule). This means that compression induced hydrogen- structuralorderingis almost entirely due to reducedcorrela-
bond bending has allowed more total water molecules into tionsbetweenthecentralandsecond-shellmolecules.Infact,
the second shell, which in turn leads to an overall increase Yanetal. convincinglydemonstratedthatasimilarstructural
intranslationalorder. anomalyinTIP5Pwatercanalsobeattributedtocompression
Ontheotherhand,furtherincreasesindensity[region(2), inducedtranslationaldisorderingofthesecondshell.43
5
Can these structural changes explain water’s self-diffusity ¶ · ¸
3 T*=0.2
anomaly?Sciortinoetal.78argued,basedonmolecularsimu-
0.3
lationresults,thatthepresenceofafifthmoleculeinthefirst B2 T* 0.4
k 0.5
claotoiorndainlaatniodnroshtaetlilosniaglnmifioctaionntlsyolfowtheerscetnhterablawrraieterrsmfoorletrcaunlse-. s / 21 0.6
-
This suggests that second-shell waters play a central role in (a)
water’s increasedmobilityundercompression. Interestingly,
0
sincetheself-diffusivityofSPC/Ewaterisstronglycorrelated 0 0.2 ρ* 0.4 0.6
tos2overtheseconditions,37onecanindependentlydrawthe 2 ¶ · ¸
sameconclusionfromthedatainFig.2.
Finally, we observe that, at sufficiently high densities [re-
gion(3),ρ>1.15g/cm3],translationalorderagainincreases r ) 1
upon compression. This is “normal” behavior for dense liq- g(
uids, and it simply reflects the fact that smaller volumes (b) (d) (f)
force particles to adopt locally ordered (i.e., efficient) pack-
0
ingstructures.31,61
1 (g)
) (e)
r (c)
(
2. Core-softenedmodel Is20.5 0.05 0.08 0.175
0.06 0.11 0.25
0.08 0.175 0.35
Inthissection,weinvestigatehowthetranslationalorderof 0
0 2 4 0 2 4 0 2 4
the lower resolution, core-softenedmodel of Eq. 1 responds r/σ
tochangesindensity. First, weconsidertheresultsfromthe
moleculardynamicssimulations. One striking featureof the
FIG. 3: Structural data obtained from molecular dynamics sim-
datais thatthebehaviorof−s /k asa functionofreduced
2 B ulations of the core-softened potential discussed in the text. (a)
densityρ∗ =ρσ3,displayedinFig.3(a),isqualitativelysim- Structural order parameter −s2/kB as a function of reduced den-
ilar to that of SPC/E water [see Fig. 2(a)]. Specifically, the sityρ∗ = ρσ3 atT∗ = kBT/ǫ = 0.2,0.3,0.4,0.5and0.6,where
core-softenedmodelalsodisplaysρ-structuralanomaliesover σ is the particle diameter, and ǫ is the energy scale of the poten-
thedensityrange0.08 ≤ ρ∗ ≤ 0.175thatbecomemorepro- tial (see Eq. (1)). Arrow indicates direction of increasing T∗, and
nouncedatlowertemperature. vertical dotted lines are at ρ∗ = 0.08 and ρ∗ = 0.175, the ap-
Clearly,thecore-softenedmodelisverydifferentfromthe proximate boundaries of theregion of anomalous structural behav-
ior. (Lower panel) Radial distribution function g(r) and cumula-
SPC/E model in that the former does not provide a molec-
ular description of water, and thus it does not favor the for- tiveorderintegralIs2(r)alongtheT∗ =0.3isotherm[redsquares,
dashedcurvein(a)]forthreedensityregions:(b,c)ρ∗ ≤0.08[upto
mationoftetrahedrallycoordinatedhydrogen-bondnetworks,
−s2(ρ∗)/kB maximum],(d,e)0.08 ≤ ρ∗ ≤ 0.175[betweenmaxi-
etc. Nonetheless, as we explain below, the main “micro- mumandminimumin−s2(ρ∗)/kB],(f,g)ρ∗ ≥0.175[beyondmin-
scopic” origins of its density-dependent trends in structural imumin−s2(ρ∗)/kB].Theregionsareindicatedbycirclednumbers
orderarebasicallythesameasthosefortheSPC/Emodel. alongtopof(a)andlowerpanel.Inlowerpanels,arrowsindicatedi-
In order to appreciate the similarity between these two rectionofincreasingdensity;numbersinlegendsindicatevaluesof
models, it is helpful to first notice one difference. In the ρ∗;verticaldashedlineisatr = 1.5σ andverticaldottedlineisat
SPC/E model, the attractive “hydrogen-bond” interactions r = 3.5σ,theapproximatelocationsofthefirstandsecondminima
promote the formation of a first coordination shell, even at ing(r),respectively.
relatively low density. In contrast, since there are no attrac-
tionsinthecore-softenedmodel,theGaussianrepulsionpre-
ventsthe “first” coordinationshell(nearthe hard-corediam- molecules(5,6,etc.)penetratingthefirstshellofanotherwise
eter, 1.0 ≤ r/σ ≤ 1.5) from forminguntil sufficiently high four-coordinatedcentral water molecule. Once this physical
density(ρ∗ &0.1).Ontheotherhand,the“second”coordina- relationshipbetweenthetwomodelsisrecognized,thesimi-
tionshell(againsttheGaussianrepulsion,1.5 ≤ r/σ ≤ 3.5) larities between their structuralpropertiesare easy to under-
ispresentevenatlowdensity. stand.Toillustratethis,wecarriedoutastructuralanalysisof
From a qualitative perspective, one might consider each thecore-softenedmodelidenticaltothatpresentedabovefor
core-softenedparticle as effectivelyrepresentinga cluster of theSPC/Emodel.
water molecules (e.g., a central water molecule and its four In particular, we examined the behavior of the RDF and
nearest neighbors).79 In fact, Yan et al. have recently pre- I [see Fig. 3(b-g)] for the core-softened model as a func-
s2
sented evidence that a mapping of this sort has quantitative tion of density along the T∗ = k T/ǫ = 0.3 isotherm
B
meritwhenonecompares,inappropriatelyreducedform,the for three differentdensity regions: (1) the initial increase of
behaviors of a core-softened ramp model to TIP5P water.79 −s (ρ∗)/k atlowdensities[ρ∗ ≤0.08,Figs.3(b,c)],(2)the
2 B
Whenviewedfromthisperspective,theformationofthefirst anomalousdecreaseof−s (ρ∗)/k atintermediatedensities
2 B
shellin thecore-softenedmodelathighdensityqualitatively [0.08 ≤ ρ∗ ≤ 0.175, Figs. 3(d,e)], and (3) the increase of
corresponds, in the molecular picture, to additional water −s (ρ∗)/k athighdensities[ρ∗ >0.175,Figs.3(f,g)].
2 B
6
Asdiscussedabove,the“first”shelloftheRDFisnotpop- ¶ · ¸
2 T*=0.2
ulatedinthismodelatlowdensitybecausethecore-softened
0.3
particlesthemselvesloosely representa centralwater and its B T* 0.4
k 0.5
foofutrhenecaorrees-tsonfetiegnhebdorflsu.idIn[rtehgisiovnie(w1),,tρh∗e i≤nit0ia.0l8c]omhapsreasnsieofn- s / 21 0.6
fectthatissimilar tothatseenforSPC/E water. Themodest - (a)
increasein−s /k thatisobservedisdueto theincreasein
2 B 0
densityandaminorenhancementofstructuringinthesecond 0 0.2 ρ* 0.4
shell(1.5≤r/σ ≤3.5). 2 ¶ · ¸
Further compression of the core-softened model [region
(2), 0.08 ≤ ρ∗ ≤ 0.175] leads to an anomalousdecrease in
structural order [Fig. 3(a)]. Figs. 3(d) and (e) indicate that r ) 1
thedisorderingisagainduetoaflatteningandshiftinginward g(
of the second shell. Moreover, the “first” shell of the core- (b) (d) (f)
softenedparticlesbeginstoemerge,whichschematicallyrep-
0
resents, in the approximatemolecularview discussed above,
1 (c) (e) (g)
thatadditionalwatermoleculesareeffectivelypenetratinginto
)
thefour-coordinatedfirstshell. r
Similar to SPC/E water, it is known that there is a strong (Is20.5 0.05 0.075 0.165
correlationbetweenexcessentropyandself-diffusivityforthe 0.06 0.125 0.250
core-softenedmodel.38Thisinformation,togetherwiththere- 0.075 0.165 0.350
0
sults shown here, support the view that the self-diffusivity 0 2 4 0 2 4 0 2 4
anomaly of the core-softened model is also linked to its r/σ
density-dependentsecond-shellstructure.
As expected, at higher density [region (3), ρ∗ ≥ 0.175], FIG.4: Structuraldatafor thecore-softened waterlikemodel from
compression leads to an increase in structural order due to integral equation theory. (a) Structural order parameter −s2/kB
as a function of reduced density ρ∗ = ρσ3 at the same values of
simple-liquid-like structuring of particles in the first coordi-
T∗ = kBT/ǫ as in Fig. 3(a), where σ is the particle diameter,
nationshell[Figs.3(f,g)].Inshort,thequalitativeresponseto
and ǫ is the energy scale of the potential (see Eq. (1)). Arrow in-
changesindensityofthestructuralorderanditscoordination- dicates direction of increasing T∗, and vertical dotted lines are at
shellcontributionsforthecore-softenedmodelareverysim- ρ∗ = 0.075 and ρ∗ = 0.165, the approximate boundaries of the
ilar to those of the more detailed SPC/E water model. This regionofanomalousstructuralbehavior. (Lowerpanel)Radialdis-
finding is consistent with the recent demonstration that one tribution function g(r) and cumulative order integral Is2(r) along
can approximately map the anomalies of TIP5P water onto theT∗ = 0.3isotherm[reddashedcurvein(a)]forthreedifferent
thoseofasimilartwo-scaleramppotential.79 densityregions: (b,c)ρ∗ ≤ 0.075 [upto−s2(ρ∗)/kB maximum],
Asafinalpoint,weshowinFig.4thattheintegralequation (d,e) 0.075 ≤ ρ∗ ≤ 0.165 [between maximum and minimum in
theoryofthecore-softenedmodelcanqualitativelypredictall −s2(ρ∗)/kB],(f,g)ρ∗ ≥0.165[beyondminimumin−s2(ρ∗)/kB].
The regions are indicated by circled numbers along top of (a) and
of the trends shown in Fig. 3. The ability of this approach
lowerpanel. Inlowerpanels,arrowsindicatedirectionofincreasing
to reproduce the structural features seen in simulations, to- density; numbers in legends indicate values of ρ∗; vertical dashed
getherwiththe quasi-universalconnectionbetweenstructure
lineisatr = 1.5σ andverticaldottedlineisatr = 3.5σ, theap-
andtransportcoefficientsofliquids,62,63,64 suggeststhatinte- proximatelocationsofthefirstandsecondminimaing(r),respec-
gralequationtheorymightserveasavaluablefirst-passtoolin tively.
assessingwhetherothermodelsystemsrepresentgoodcandi-
datesforexhibitingstaticanddynamicanomalies. However,
iftheintentionistoultimatelyuseitasaquantitativelyaccu- that, at sufficiently high valuesof φ , the opposite trend can
c
rate predictive tool, then more comprehensiveinvestigations be observed;37,45 i.e., attractions counterintuitively decrease
of alternative closure relationships, in the spirit of Ref. 36, theamountofstructuralorder.
willbenecessary. In this section, we briefly discuss how we used molecular
simulation and integral equation theory to gain insights into
this trend. We accomplished this by exploring the various
B. SRAfluids coordination-shellcontributionsto−s2/kBforthetwomodel
SRAfluidsdiscussedinSectionIIB.
Oneofthekeyaspectsofshort-rangeattractive(SRA)flu-
ids is that their structurally anomalous behavior occurs as a
functionofthereducedinterparticleattractivestrengthǫ/k T 1. Colloid-polymermixture
B
atconstantparticlepackingfractionφ ,where−ǫrepresents
c
the well depth of the interparticle attraction. In most typi- We begin by investigating the behavior of the model
calatomicormolecularfluids,onefindsthatstructuralorder colloid-polymersystem71,72bymolecularsimulation. Theef-
(−s /k )increaseswithǫ/k T.SRAfluidsareanomalousin fective colloid-colloid pair potential for this model was pre-
2 B B
7
6 ¶ · terparticle attractions considerablyincreases but narrowsthe
firstpeakoftheRDF[Fig.5(b)].Thesetwoeffectsessentially
φ=0.4
kB5 c cancelsothatthefirst-shellcontributionstoIs2(r)areinsensi-
-s / 24 (a) atilvseotsoliφgphtolyvesrhtihfitsthraenhgieg[hseerecFoiogr.d5i(nca)t]i.oHnoswheelvlsero,fattthraectRioDnFs
inwardanddiminishtheiroverallcorrelationwiththecentral
particle.Theselattermodificationstothestructureofthesec-
3
0 0.1 φ 0.2 0.3 ondandhighercoordinationshellsgiverisetotheanomalous
¶ p · decreaseinthestructuralorderofthissystem. Theyarealso
4
consistentwithbehaviorobservedintherecentKrekelberget
4
(b) (d) al.45simulationsofthesquare-wellSRAfluiddiscussedinthe
)
r 2 Introduction.Amicroscopicinterpretationofthistrendisthat
g( 2 SRAinteractionsdriveweakparticleclusteringatlowφp(ex-
plainingthesharpeningandnarrowingofthefirstpeak). This
clustering, in turn, opens up channels of free volume in the
0 0
(c) (e) fluidanddisruptstheuniformhard-sphere-likepackingorder
4 inthesecondandhighercoordinationshells.59,74,80,81
) 3
r Under conditions where the aforementioned structural
(Is2 φ0p.0=50.0 2 φ0p.2=0.1 oanfotmhealflyuiodc.c45urVs,eriyncsriemasileasritnoφthpeawlsaoteirnlickreeaflsueidthsedmiscoubsislietdy
2 0.1 0.25 above,itisknownthats andself-diffusivityarestronglycor-
2
2 4 6 2 4 6 related for the model colloid-particle mixture.37 As a result,
r/a the self-diffusvity anomaly appears to also derive from sub-
tle structuring effects in the second and higher coordination
FIG.5: Structural data obtained frommolecular dynamics simula- shells.
tionsofthemodelcolloid-polymerSRAfluiddiscussedinthetext. Asonewouldexpect,however,increasingφ ultimatelyin-
p
(a)Structuralorderparameter−s2/kBasafunctionofpolymervol- creasesstructuralorder,iftheinteractionsaresufficientlyat-
umefractionφp(i.e.,strengthofcolloidattractions)atcolloidpack- tractive [region (2), φ ≥ 0.1]. The attractions lead to the
ingfractionφc =0.4. Verticaldottedlineatφp =0.1,thelocation formationofstronglybpondedparticleclusters,59,74,80,81which
oftheminimumin−s2(φp)/kB. (Lowerpanel)Radialdistribution
isreflectedbytheincreasedheightofthefirstpeakoftheRDF
functiong(r)andcumulativeorderintegralIs2(r)alongtheisochore
φc =0.4[blackcirclesin(a)]fortwopolymerconcentrationranges: [Fig.5(d)]andtheassociatedriseinthefirst-shellcontribution
(b,c)φp ≤ 0.1[belowminimumin−s2(φp)/kB],(d,e)φp ≥ 0.1 toIs2(r)[Fig.5(e)].
[above−s2(φp)/kBminimum].Theregionsareindicatedbycircled Inclosing, we test in Figs. 6 and 7 whetherintegralequa-
numbersalongtopof(a)andlowerpanel.Inlowerpanels,arrowsin- tiontheoryisabletoqualitativelycapturetheseattractionin-
dicatedirectionofincreasingφp;theparameteraindicatescolloidal duced structural changes for both model systems introduced
particleradius;verticaldashedlineisatr = 3aandverticaldotted inSectionIIB: thecolloid-polymerfluidandthesquare-well
lineisatr = 5a,theapproximatelocationsofthefirstandsecond
fluid,respectively.ComparisonofFig.6withFig.5andFig.7
minimaing(r),respectively.
with Figure 4 of Ref. 45 demonstrate that this is indeed the
case. The success of these predictions strengthens the case
thatintegralequationtheorywillbeausefultoolinassessing
sentedearlierinFig.1(b)forseveralvaluesofpolymerpack- whetherfuturemodelsystemsofinterestmightdisplaystruc-
ingfractionφ . Sincethereducedwell-depthofthispotential, turalanomalies.
p
ǫ/k T, scales as φ , we analyze structuralorder below as a
B p
functionofthelatter.
In particular, Fig. 5(a) illustrates how −s /k varies as a C. Conclusions
2 B
functionofφ ataparticlepackingfractionofφ = 0.4. As
p c
expectedforSRAfluids,−s2(φp)/kB exhibitsaminimumat Althoughthestructuralorderofafluidisusuallyenhanced
φp ≈ 0.1. In other words, this fluid displays the structural byisothermalcompressionorisochoriccooling,afewnotable
anomalyofEq.(4b)forφp ≤ 0.1. Tounderstandtheorigins systems show the opposite behaviors. Specifically, increas-
ofthistrend,westudytheRDFandthecumulativeorderin- ingdensitycandisruptthestructureofwaterlikefluids,while
tegralIs2(r)asafunctionofφp intwoqualitativelydifferent lowering temperature (or strengthening of attractive interac-
regions: (1) the anomalous decrease in −s2(φp)/kB at low tions) can weaken the correlationsof fluids with short-range
polymerconcentrations(lowinterparticleattractions),and(2) attractions.Thetwo-bodytranslationalcontributiontotheex-
the“normal”increaseof−s2(φp)/kB athigherpolymercon- cessentropyprovidesaquantitativemeasureofthesechanges
centrations(higherinterparticleattractions). in structural order. It is a particularly insightful quantity to
What specific changes to coordination shell structure ex- studybecause(i)itscontributionsfromthevariouscoordina-
plain the attraction induced disordering that occurs at small tion shells of the radial distribution function can be readily
φ [region(1), φ ≤ 0.1]? First, note thatstrengtheningin- determined,and(ii)itcorrelatesstronglywithself-diffusivity,
p p
8
¶ ·
12
¶ ·
B 9 10
k
-s / 2 6 φ (a) / k2B 68
c s
-
30 0.1 φ 0.2 4 φc (a)
p
4 ¶ · 0 1 ε / k T 2 3
B
¶ ·
4
(b) (d) (c)
) (b)
g( r2 2 )4 4
r
(
g2 2
0 0
(c) (e)
0 0
4 (c) (d)
) 4 9
r
( Is22 φ00p..01=30.0 2 φ00p..22=50.1 ( r )Is26 ε/kBT=0.0 6 ε/kBT=0.9
0.3 2.0
3
2 4 6 2 4 6 0.9 3.0
r/a 4
1 2 3 4 1 2 3 4
r/σ
FIG. 6: Structural data for the model colloid-polymer SRA
fluid discussed in the text from integral equation theory. (a)
FIG.7:Structuraldataforthesquare-wellfluiddiscussedinthetext
Structural order parameter −s2/kB as a function of poly- obtainedfromintegralequationtheory. (a)Structuralorder param-
mer volume fraction φp at colloid packing fractions φc = eter −s2/kB asa function of reduced attractivestrength ǫ/kBT at
0.3,0.325,0.35,0.375,0.4,0.425,0.45,0.475 and 0.5. Arrow in-
particlepackingfractionsφc =0.4,0.45,0.5,0.525,0.55,0.56and
dicates direction of increasing φp, and vertical dotted line is at 0.57. Arrow indicates direction of increasing φc, and vertical dot-
φp = 0.1, the approximate boundary of the region of anomalous tedlineisatǫ/kBT = 0.9,theapproximateboundaryoftheregion
structuralbehavior. (Lowerpanel)Radialdistributionfunctiong(r)
ofanomalousstructuralbehavior. (Lowerpanel)Radialdistribution
a[dnadshcuedmuvliaotlievtecourrdveeriinnte(ag)r]alfoIrs2t(wro)aploolnygmtehreciosoncchenotrreatφiocn=ra0n.g4e7s5: function g(r) and cumulative order integral Is2(r) along the iso-
(b,c)φp ≤ 0.1[belowminimumin−s2(φp)/kB],(d,e)φp ≥ 0.1 crahnogreesφ:c(b=,c)0.ǫ5/5k[BdTash≤ed0b.9lu[ebceulorvweminin(aim)]ufomrtiwno−astt2r(aǫc/tikvBeTst)r/enkgBt]h,
[above−s2(φp)/kBminimum].Theregionsareindicatedbycircled (d,e) ǫ/kBT ≥ 0.9 [above −s2(ǫ/kBT)/kB minimum]. The re-
numbersalongtopof(a)andlowerpanel.Theparameteraindicates
gions are indicated by circled numbers along top of (a) and lower
colloidradius. Inlowerpanels,arrowsindicatedirectionofincreas-
panel. Theparameterσindicatescolloiddiameter. Inlowerpanels,
ingφp,verticaldashedlineisatr=3a,andverticaldottedlineisat arrowsindicatedirectionofincreasingǫ/kBT;verticaldashedlineis
r =5a,theapproximatelocationsofthefirstandsecondminimain
atr =1.4σandverticaldottedlineisatr =2.3σ,theapproximate
g(r),respectively.
locationsofthefirstandsecondminimaing(r),respectively.
whichallowsittoprovideinsightsintothedynamicanomalies
ofthesefluids.
Acknowledgments
Here, we have presented a comprehensive study, by both
molecularsimulationandintegralequationtheory,oftheco-
ordinationshellcontributionstothetwo-bodyexcessentropy WethankAnatolMalijevskyforprovidinganefficientcode
forseveralmodelsystems.Thesemodelsincorporatedifferent for numerical solution of the integral equation theories ana-
levels of molecular resolution, but all exhibit the aforemen- lyzed in this work. W.P.K. acknowledges financial support
tionedstructuralanomalies. Theresultsofthisstudysupport of the NationalScience Foundationfor a GraduateResearch
theemergingviewthatthestructuralanomaliesofthesefluids Fellowship.T.M.T.acknowledgesfinancialsupportoftheNa-
cangenerallybeattributedtoquantifiablechangesinthesec- tionalScienceFoundation(CTS0448721),theDavidandLu-
ond and higher coordination shells of the radial distribution cilePackardFoundation,andtheAlfredP.SloanFoundation.
function. Theyalsodemonstratethatintegralequationtheory V.G.acknowledgesfinancialsupportoftheNationalScience
canserveasaneffectivefirst-passtoolforassessing,basedon Foundation(CTS-0347381)andtheRobertA.WelchFounda-
thepairpotentialalone,whethernewmodelsystemsaregood tion. Computersimulationsforthisstudywereperformedat
candidatesforexhibitingstaticanddynamicanomalies. theTexasAdvancedComputingCenter(TACC).
9
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