Glass-Transition Properties from Hard Spheres to Charged Point Particles

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by  Anoosheh Yazdi

Glass-Transition Properties from Hard Spheres to Charged Point Particles
Anoosheh Yazdi,1 Alexei Ivlev,2 Sergei Khrapak,2 Hubertus Thomas,2
Gregor E. Morfill,2 Hartmut Lo¨wen,3 Adam Wysocki,3 and Matthias Sperl1
1Institut fu¨r Materialphysik im Weltraum, Deutsches Zentrum fu¨r Luft- und Raumfahrt, 51170 K¨oln, Germany
2Max-Planck-Institut fu¨r extraterrestrische Physik, 85741 Garching, Germany
3Institut fu¨r Theoretische Physik II: Weiche Materie,
Heinrich-Heine-Universit¨at Du¨sseldorf, Universita¨tsstrasse 1, 40225 Du¨sseldorf, Germany
4 (Dated: January 29, 2014)
1
The glass transition is investigated in three dimensions for single and double Yukawa potentials
0
for the full range of control parameters. For vanishing screening parameter, the limit of the one-
2
component plasma is obtained; for large screening parameters and high coupling strengths, the

n glass-transition properties crossover to the hard-sphere system. Between the two limits, the entire
a transition diagram can be described by analytical functions. Different from other potentials, the
J
glass-transition and melting lines for Yukawa potentials are found to follow shifted but otherwise

8 identical curvesin control-parameter space.
2
PACSnumbers: 64.70.Q-,66.30.jj,64.70.ph,64.70.pe

]
h
c
I. INTRODUCTION screening length λ. The second (longer-ranged) Yukawa
e
m potential is specified by a relative strength ǫ, and a rela-
tive inverse screening length α < 1. In the limit of van-
- The physics of ordered (crystals) and disordered
t (glasses)solidstatesandtheirinterrelationhasbeensub- ishingscreening,onerecoverstheone-componentplasma
a
(OCP), the simplest model that exhibits characteristics
t ject to investigations in various model systems [1, 2].
s ofchargedsystems[9]. Motivatedbythesuccessofmode-
. Such model systems capture typical features of more
t coupling theoryfor idealglasstransitions (MCT) for the
a complex matter and often allow for the variation of the
m interparticleinteractionstoexplorephysicalregimesoth- hard-sphere system (HSS), cf. [10], in the following,
the glass-transitionshall be calculatedwithin MCT [11].
erwise not accessible. A qualitatively strong variation
-
d concerns the distinction between hardandsoft repulsion Sincefortime-reversibleevolutionoperators,i.e.,Newto-
n as in the Yukawa potential which describes the range nianandBrowniandynamics,theglassydynamicswithin
o MCTareidentical[12],thecalculationsareapplicableto
from excluded-volume to charge-basedinteractions.
c both complex plasma and charged colloids.
[ Yukawa potentials are realized in both colloidal sus-
pensions [3–5] and complex plasmas [2], and since in
1
complex plasmas the damping can be tuned, this of-
v II. METHODS
fers a way for the comparison of Brownian and New-
9
tonian dynamics with the same particle-particle inter-
6
2 action in experimental systems [6]. While in sterically We consider a system of N point-like particles in a
7 stabilized colloidal suspensions, the interaction can typ- volumeV ofdensityρ=N/V interactingviathepairwise
. ically be well-approximated by the hard-sphere interac- repulsive potential in Eq. (1). We investigate the glass
1
0 tion[1],forchargedparticlesinsuspensions,hard-sphere transitions in two cases: the single Yukawa (ǫ = 0) and
4 plusYukawainteractionismoreappropriate. Incomplex the double Yukawa potential (ǫ > 0). Within MCT, the
1 plasmas, the average interparticle distance compared to glass transition is defined as a singularity of the form
: theparticles’diametersistypicallylargeenoughtoallow factorfq =limt→∞φq(t)thatisthelong-timelimitofthe
v
i for an approximation of point-like particles and hence density autocorrelation function. In the liquid state, fq
X a screened Coulomb potential for point particles is ap- iszero,whileintheglassstate,fq >0. Atthetransition,
ar pprleoxprpialatsem. aIsnaaldsodiatiosnectoondthreepsucrlseievneinlegnlgetnhgtshc,alienacroismes- tthhee sfoorlumtiofanctoofr[s1a3d]opt their critical values fqc ≥ 0. fq is
from the non-equilibrium ionization-recombination bal- f
q
ance [7, 8] which gives rise to a double Yukawa potential = q[fk], (2)
1 f F
q
at interparticle distances r as −
which is the long time limit of the full MCT equations
U(r) Γ ofmotion. f isdistinguishedfromothersolutionsofthe
= [exp( κr)+ǫexp( ακr)] . (1) q
k T r − − Eq. (2) by its maximum property, thus it can be calcu-
B
lated using the iteration f(n+1)/(1 f(n+1)) = [f(n)]
Distance r is giveninunits ofthe meaninterparticledis- q − q Fq k
[14] with f(0) =1 and the memory kernel given by
tance1/√3ρwiththedensityρ=N/V forN particlesina k
volumeV. ThecouplingparameterisΓ=Q2√3ρ/(kBT), [f ]= 1 d3kSqSkSp[q kc +q pc ]2f f , (3)
withthechargeQ,andκ=1/(λ√3ρ)istheinverseofthe Fq k 16π3 q4 · k · p k p
Z
2
where p = q k; all wave vectors are expressed in nor- n
o
malized units−. Note that the number density does not nsiti
laepnTpgehtaherosecnxalpylelsiicniipntluyutsninittosthtohefe1k/Eer√q3n.ρe(.l3F)qa,resitnhceeswtaetiecxsptrreuscstuthree 4 glass tra m elting
factors S . The Fourier transformed direct correlation
q Γ
functions cq, are relatedto structure factors throughthe g10
Ornstein-Zernike (OZ) relation o
l
3
c2
γq = q , (4) fluid
1 c
q
−
where the spatial Fourier transform of γ is γ(r) =
q
2
h(r) c(r)andh(r)isthetotalcorrelationfunctionwhich 0 2 4 6 8 10
− κ
is related to structure factor through S = 1+h . We
q q
close the equations by the hypernetted-chain (HNC) ap-
FIG.1. Glass-transitiondiagramforthesingleYukawapoten-
proximation,
tial(filledcircles). Transition pointsareshowntogetherwith
the full curve exhibiting Eq. (7). For comparison, a similar
c(r)=exp[ U(r)/(k T)+γ(r)] γ(r) 1, (5)
− B − − curveis shown for the melting of thecrystal.
whereU(r)istheinteractionpotential. Itwasfoundear-
lier that HNC captures well various structural features
for κ > 0, the glass-transition line moves to higher crit-
for repulsive potentials, especially also for the OCP [15].
ical coupling strengths Γc(κ). Figure 1 shows for ref-
For the HSS, the quality of HNC is known to be inferior
erence the melting curve for weakly screened Yukawa
tothePercus-Yevick(PY)approximationincertainther-
systems, described by Γ(κ) = 106eκ/ 1+κ+κ2/2
modynamicaspects[9],soweexpectHNCtovaryinper-
[17, 18]. This expression has been suggested originally
formance for different parameter regions of the Yukawa (cid:0) (cid:1)
on the basis of the Lindemann-type arguments, cf. [19].
potentials in Eq. (1).
The Lindemann criterion states that the liquid-crystal
We solve Eq. (4) and Eq. (5) by iteration and use the
phase transition occurs when in the crystal the root-
usual mixing method in order to ensure convergence [9]. mean-square displacement δr2 of particles from their
We iterate n times from an initial guess, c(0)(r), until a h i
equilibrium positions reaches a certain fraction of the
self-consistent result is achieved, i.e.,
mean interparticle distance. Within the simplest one-
dimensionalharmonicapproximationthisyieldsthescal-
1/2
R ing U′′(r =1) δr2 /T const., where primes denote the
c(n+1)(r) c(n)(r)2dr <δ, (6) h i ≃
"Z0 | − | # sYeuckoanwdadienrtiveraatcivteiownitthhirselsepaedcstttootdhiestmanecltei.ngApcuprliveedatbootvhee,
where the value of the constant is determined from the
with δ =10−5, where R is the cut-off length of c(r). We
conditionΓ 106atmelting ofthe OCPsystem(κ=0)
employR=47.1239andameshofsizeM =2396points. ≃
[20]. Thisexpressionforthemeltingcurveiswidelyused
Consequently, the resolution in real and Fourier space is
due to its particular simplicity and reasonable accuracy:
∆r = R/M = 0.0197 and ∆q = π/R = 0.0667, respec-
DeviationsfromnumericalsimulationdataofRef.[21]do
tively. We use an orthogonality-preservingalgorithm for
not exceed several percent, as long as κ . 8. Moreover,
the numericalcalculation of Fourier transforms [16]. For
similar arguments can be used to reasonably describe
a particular κ we begin the computation of c(r) at a
freezingofothersimplesystems,e.g. Lennard-Jones-type
small coupling parameter Γ, successively increase Γ, and
fluids [22]. Remarkably, when comparing the predicted
use the outcome as an initial guess for the subsequent
glass-transitionwiththemeltingcurve,oneobservesthat
calculation.
bothtransitionlines runinparallel. The glass-transition
line is described by the function
III. SINGLE-YUKAWA POTENTIAL Γc(κ)=Γc eκ 1+κ+κ2/2 −1 , (7)
OCP
A. Glass-Transition Diagram whichis shownas solid line(cid:0)in Fig.1, i.e.,(cid:1)the glasstran-
sition is found at 3.45 of the coupling strength of the
TheMCTresultsforthesingleYukawacaseareshown melting curve.
in Fig. 1. The filled circles for different Γ and κ indi- The fit quality given by Eq. (7) is remarkable for two
cate the glass transition points calculated by Eq. (2). distinct reasons: First, the potential changes quite dras-
For κ 0, the glass transition for the OCP limit is tically along the line from a long-ranged interaction at
found a→t Γc = 366. When screening is introduced low κ to the paradigmatic hard-sphere system at very
OCP
3
large κ to be detailed below. Such simplicity along d as the unit of length. For HNC, the transition point
control-parameterdependent glass-transitionlines is not is found at a packing fraction of ϕc = 0.525. This
HSS
to be expected and not observedfor other potentials, cf. value as well as the behavior of f in Fig. 2 is very close
q
the square-well system [23, 24]. Second, the non-trivial in HNC and the PY approximation where ϕc =0.516
HSS
changes along the transition lines are apparently quite [14]. Itisseenthatthedistributionoff isdominatedby
q
similar for the transition into ordered and disordered a peak at interparticle distances which indicate the cage
solids alike, and Eq. (7) applies to both. For the men- effect [11, 14]; oscillations for higher wave vectors follow
tioned square-wellsystem, orderedand disordered solids this length scale in a way similar to the static structure
have no such correlation [24]. factor. For both PY and HNC, the peak positions for f
q
Since both MCT and the structural input involve ap- coincide,fortheprincipalpeakeventhepeakheightsare
proximations,typicallytheglasstransitionsarefoundfor almostidentical. ForHNC,thef aretypicallyabovethe
q
higher couplings than predicted, the deviation is around PY solutions resulting in a 10% larger half-width of the
10% in the densities for the HSS [11]. While one can ex- distribution of the f for the HNC. The predicted devia-
q
pectthatabsolutevaluesfortransitionpointsneedtobe tionsbetweenHNC andPYaremostlyindistinguishable
shiftedtomatchexperimentalvalues[10],thequalitative when comparing to experiments except for the small-q
evolution of glass-transition lines with control parame- limit where experimental results favor the PY-MCT cal-
ters is usually quite accurate and even counterintuitive culation, cf. [11, 25].
phenomena like melting by cooling have been predicted Forthe Yukawapotential,overallthe criticalformfac-
successfully [23]. Hence, we assume the description of torsexhibitsimilarfeaturesasfortheHSS.Differentfrom
the liquid-glass transitions in the single Yukawa system the HSS, in the OCP limit the form factors vanish for
to be qualitatively correct. the limit q 0. This anomaly for charged systems cor-
→
responds to the small wave-vector behavior in the static
structure S q2 for q 0 [9]. Since in the OCP, mass
q
B. Glass-Form Factors and charge fl∝uctuations→are proportional to each other,
the conservation of momentum implies the conservation
of the microscopic electric current, and hence no damp-
21 ing of charge fluctuations in the long wave-length limit.
0.8 18 Considering Eq. (3) we shall demonstrate, that fq q2
∝
15 for small wave vectors.
HSS log10Γ12 Denoting θ as the angle between q and k we can ex-
0.6 9 pand the direct correlationfunctions as:
6
f 3 1 1
q0.4 0 10 20κ30 40 50 c|q−k| =ck−c′kq cosθ+ 2q2cos2θc′k′− 6q3cos3θc′k′′ (8)
where the primes represent the respective first, second
0.2 andthirdderivativesofc withrespecttok. Substituting
k
Eq. (8) into Eq. (3) leads to
OCP
0 [f ] S α+q2S β, (9a)
0 5 10 15 20 25 Fq k → q q
q
where [26]
FIG. 2. Critical glass-form factors fq for the glass transition
inthesingleYukawasystem. Forincreasingscreeningparam- α= 1 ∞dkk2S2[c2 + 2kc c′ + 1k2c′2]f2, (9b)
eterκ,theinsetshowsthelocationoftherespectivetransition 4π2 k k 3 k k 5 k k
Z0
points on the MCT-transition line, cf. Fig. 1, with the same
symbols as in themain panel. The full curveshows thesolu- and
tion for the HSS within the HNC approximation. The result
for HSS within thePY approximation [14] is shown dashed. β = 1 ∞dkk2S2[1c′2+ 1 k2c′′2+ 2kc′c′′
4π2 k 3 k 28 k 5 k k
Z0 (9c)
The different points on the glass-transition lines shall 1 1 1
+ c′c′′+ kc c′′′+ k2c′c′′]f2.
be discussed in detail in the following. For the well- 3 k k 15 k k 21 k k k
knowncaseoftheglasstransitionintheHSS,thecritical
formfactorsareshownbyafullcurveinFig.2. Different The term linear in q in Eq. (9a) vanishes. Similarly, the
from earlier results calculated for S within the PY ap- small-q expansion of the static structure factor in the
q
proximation [14], here we also show the HSS within the OCP reads [27]
HNC approximation to be consistent with the Yukawa
results. The control parameter for the HSS is the pack- q2 q4
S(q)= + [cR(0) 1]+ (q6) (10)
ing fraction ϕ = ρ(π/6)d3 with the hard-core diameter k2 k4 − O
D D
4
where k2 = 4πΓ represents the inverse Debye length, 2
D
and cR(q) = c(q) cS(q) is the regular term of the di- 2.5
−
rectcorrelationfunction,assumingthatatlargedistances
2.0
particles can only be weakly coupled, which creates the 1.6 dc
singular term cS(q) = U(q)/k T. From Eq. (9a) and eff1.5
B
Eq. (10) we get − ϕc
ϕc1.2 eff 1.0
0 20 40 60 80 100
α β α κ
=q2 +q4[ + (cR(0) 1)]+ (q6). (11)
Fq k2 k2 k4 − O
D D D
0.8
FromEq.(2)onecanconcludethatf hasthesamelimit
q
HSS
asF , hence wehaveshownthatf q2 for vanishingq.
q q
∝
Fornon-vanishingscreening,κ>0,thesmall-q behav- 0.4
ioroftheformfactorsischaracterizedbyfiniteintercepts 0 20 40 60 80 100
κ
at q =0. This regular behavior is ensured by the q 0
limit of cS = 4πΓ/(q2+κ2). For larger wave vect→ors,
the f firsqt dec−rease in comparisonto OCP – cf. κ=5.7 FIG. 3. Effective packing fraction ϕceff for Yukawapotentials
q alongthetransitionlineinFig.1. Thehorizontaldashedline
(×)and14.0(H)inFig.2–beforeincreasingbeyondthe shows the HSS-HNClimit of ϕcHSS =0.525. The inset shows
OCPresultforκ&30. Forverylargescreening,theform the effective hard-sphere diameter, dceff = lnΓc/κ equivalent
factors of the Yukawa potential apparently approachthe
to the effective densities. In both plots, the dotted curves
HSS case. display the small-κ asymptotes derived from Eq. (7).
5
C. HSS Limit ε = 0 ε = 0.01
By setting U(d )/k T 1 for ǫ = 0 in Eq. (1) one
eff B
∼
can define an effective diameter that becomes a well-
4
defined hard-core diameter for κ . Along the glass- ε = 0.2
→∞
transition line Γc(κ) the effective packing fraction and Γ
0
diameter are given (with logarithmic accuracy) by g1
o
l
π lnΓc 3 3
ϕc = , dc =lnΓc/κ, (12)
eff 6 κ eff
(cid:18) (cid:19)
whereonlythedefinitionofthepackingfractionhasbeen
used. Figure 3 displays the effective packing fractions 2
along the single-Yukawa transition line up to κ 100. 0 5 10 15 20 25
≈ κ
For small κ, the large effective diameter yields consid-
erable overlaps among the particles and hence a pack-
FIG. 4. Glass transition diagram for double Yukawa poten-
ing fraction beyond unity. The effective hard-sphere di-
tials with α=0.125, ǫ= 0.2 (diamonds) and 0.01 (squares).
ameter can be seen in the inset of Fig. 3. For κ & 40
ThesingleYukawadata(filledcircles)isshowntogetherwith
the Yukawa potentials’ effective diameter dc reaches its
eff the analytical description by Eq. (7) (solid curve labeled
asymptotic value. Together with the findings on the fq ǫ = 0). The single Yukawa points are scaled according to
thisestablishesthecrossoveroftheglass-transitionprop- Eq. (13) for ǫ=0.2, and shown by open circles. Dotted and
ertiesoftheYukawasystemtothehard-spherelimit. The dashedcurvesrepresentscaledversionsofEq.(7)forǫ=0.01
relation in Eq. (7) fits effective diameters and densities and ǫ = 0.2, respectively. The solid curves labeled ǫ = 0.01
wellforsmallerκ.10andunderestimatesthecalculated and ǫ=0.2, respectively,show thesolution of Eq. (14).
values for larger κ, as expected.
cases, for small κ the transition lines start at OCP and
IV. DOUBLE-YUKAWA POTENTIAL follow the single-Yukawa line. After a crossover regime,
for κ & 15 for ǫ = 0.01 and κ & 10 for ǫ = 0.2, the
transitions are described well by rescaling the original
A. Glass-Transition Diagrams
single-Yukawa results according to
Progressingtowardsthe double Yukawapotentials, we Γ′ =Γ/ǫ, κ′ =κ/α. (13)
show in Fig. 4 the results of MCT calculations for the
same relative screening α=0.125 and a weak (ǫ=0.01) In Fig. 4, scaling by Eq. (13) is demonstrated by trans-
as well as a strong (ǫ = 0.2) second repulsion. In both forming the MCT results for ǫ = 0 (full circles) into a
5
rescaledversion(opencircles)forǫ=0.2whichcompares rc
s ε = 0
well to the full MCT calculation for the double Yukawa
0.080
potential(diamonds). Similarly, formula(7) canbe used ε = 0.2
to describe all double Yukawa results for small screen-
ing lengths, and the results for large screening lengths 0.076
by scalingEq.(7) with Eq.(13). The dotted anddashed
curvesinFig.4exhibitthescaledcurvesforǫ=0.01and
0.072
0.2, respectively. The linear combination of the analyti-
cal descriptions for both length scales reads
0.068
Γc(κ)/Γc = e−κ(1+κ+κ2/2)
OCP (14)
+ǫe−κα(1+κα+κ2α2/2) −1 ,
(cid:2) 0.064
(cid:3)
and is demonstrated by the solid line for ǫ = 0.01 in
0 20 40 60 80 100
Fig. 4. It is seen that Eq. (14) describes the MCT re- κ
sultsfordoubleYukawapotentialsfortheentirerangeof
control parameters including crossover regions. In con- FIG. 5. Localization length for single Yukawa (full circles)
clusion, the MCT predictions for both single and double anddoubleYukawa(diamonds)potentialwithα=0.125and
Yukawapotentialscanberationalizedbyasingleanalyt- ǫ=0.2. Theopencirclesshowthesingle-Yukawadatascaled
according to Eq. (13). The horizontal dashed line shows the
icalformula(7)whichtraces the melting curve,captures
HSSlimit for rc.
the interplay between large and small repulsive length s
scales,andextendsforallparametersfromOCPtoHSS.
sults for the localization length are always close to the
values usually assumed for the Lindemann criterion.
B. Localization Lengths
Another length scale resulting from the dynamical V. CONCLUSION
MCT calculations is given by the localization length.
It is defined from the long-time limit of the mean- In summary, we have demonstrated above the full
squared displacement δr2(t) = r(t) r(0)2 as rsc = glass-transition diagram for the single and double
h| − | i
limt→∞δr2(t)/6. For the glass transition in the HSS, Yukawa systems. While some parallel running lines for
MCTpredictsalocalizationlengthwithinHNCofrc/d= limitedparameterrangeshavebeenshownearlierforlog-
p s
0.0634. This scale is quite close to the classical result of arithmiccorepotentialplusYukawatail[28,29],herewe
a Lindemann length [19]. describe the transition diagrams by analytical formulae.
For the single and double Yukawa potential, the evo- InparticularitcouldbeshownhowtheHSSlimitcontin-
lution of the localization length with κ is demonstrated uously evolves into the OCP limit. We have shown that
in Fig. 5. From a value of rc =0.070for OCP, the local- the glass-transition lines resulting from the combination
s
ization lengths increase for the single Yukawa potential, of HNC and MCT – two rather complex nonlinear func-
reach a maximum around κ 10 and decrease to the tionals – can be described analytically over their entire
≈
values for HSS for large κ. The maximum can be in- rangefromtheOCPlimitforsmallκtotheHSSlimitfor
terpreted as follows: The widths of the distributions in largeκ. Qualitatively,the behaviorof the transitionline
f seen in Fig. 2 correspond to an inverse length scale can be estimated by the Lindemann criterion for melt-
q
equivalent to r c, and the smaller width of the f mean ing [19], while quantitatively, glass transition and crys-
s q
an increase of r c. For large κ, the localization length tal melting are following remarkably similar trends for
s
needs to approachthe HSS value, hence the r c decrease stronger coupling.
s
again. Both trends together yield a maximum. It is important to note that the present calculations
ThelocalizationlengthsforthedoubleYukawasystem were performed for point particles with various degrees
follows the single-Yukawa results for small κ . 5 as ob- ofchargingandscreening. The limit ofthe HSS emerges
served in Fig. 4 and hence increases; for κ&5, the dou- fromthatcalculationswithoutactualexcludedvolumein
ble Yukawa system approaches the scaled single-Yukawa the potentials. With the important difference of a finite
results shown by the circles. For larger κ, the evolution hard-sphere radius being present, the possibility that in
followsthescaledsingle-Yukawaresultsandwhiledeviat- additionto a Coulombcrystala dilute systemofcharges
ingforκ&50fromthescaledresults,ascaledmaximum may also form a Coulomb glass was explored in the re-
is reached around κ 80. stricted primitive model for a mixture of charged hard
≈
Altogether, the variation of the localization lengths spheres [30] and the hard-sphere jellium model [31] as
is around 10% which is small compared to other glass- well as for a system of charged hard spheres to describe
transition diagrams [24]. Hence we conclude that for charge-stabilized colloidal suspensions [32]. In conclu-
both single- and double-Yukawa potentials the MCT re- sion, the present calculations offer exhaustive analyti-
6
cal descriptions for glass transitions over a wide range firm or challenge the unified picture presented above.
of quite different interaction potentials. The predictions Financial support within the ERC (Advanced grant
should motivate data collapse from computer simulation INTERCOCOS,projectnumber267499)isgratefullyac-
anddifferentexperimentalmodelsystemsinordertocon- knowledged.
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