Lamellar order, microphase structures and glassy phase in a field theoretic model for charged colloids

0.36 MB·English

by  Marco Tarzia

Lamellar order, microphase structures and glassy phase in a field theoretic model for
charged colloids
Marco Tarziaa and Antonio Coniglioa,b
a Dipartimento di Scienze Fisiche and INFN sezione di Napoli, Universit`a degli Studi di Napoli “Federico II”,
Complesso Universitario di Monte Sant’Angelo, via Cinthia, 80126 Napoli, Italy and
b Coherentia CNR-INFM
(Dated: February 6, 2008)
7
0 In this paper we present a detailed analytical study of the phase diagram and of the structural
0 properties of a field theoretic model with a short-range attraction and a competing long-range
2 screened repulsion. We provide a full derivation and expanded discussion and digression on results
n previouslyreported brieflyin M.Tarzia and A.Coniglio, Phys. Rev. Lett. 96, 075702 (2006). The
a modelcontainstheessentialfeaturesoftheeffectiveinteractionpotentialamongchargedcolloids in
J polymeric solutions. We employ the self-consistent Hartree approximation and a replica approach,
3 and we show that varying the parameters of the repulsive potential and the temperature yields a
2 phasecoexistence,alamellarandaglassyphase. Ourresultssuggestthattheclusterphaseobserved
in charged colloids might be the signature of an underlying equilibrium lamellar phase, hidden on

] experimental time scales, and emphasize that the formation of microphase structures may play a
ft prominent role in theprocess of colloidal gelation.
o
s PACSnumbers: 64.60.Cn,64.70.pf,82.70.Dd
.
t
a
m
I. INTRODUCTION σ,andthegyrationradiusofthepolymers,R (typically
g
- one has 0.05<R /σ <0.2). On the other hand, the ad-
g
d
ditionofsaltincreasesthe numberofionsinthe solution
n Colloidal suspensions are solutions of solid (or liquid)
responsible for the screening of the electrostatic repul-
o
mesoscopicparticlesimmersedintoanothersubstance[1].
c sion, thereby reducing the amplitude and increasing the
These systems, like blood, proteins in water, milk, inks,
[ screening length of the Yukawa potential.
or paints, are ubiquitous in our everyday life and are

1 extremely important in biology and industry. Due to Charged colloids in polymeric solutions have recently
v
their potential applications for designing new materi- raisedalotofinterest,bothfromanexperimental[4,5,7]
1
als with a wide range of viscoelastic properties, in the and a theoretical [8, 9, 10, 11, 12, 13, 14] point of view.
6
last few years there has been much interest in the role Inthese systems the competitionbetweenattractiveand
5
1 of the inter-particle potential on controlling the struc- repulsive interactionsondifferent length scales stabilizes
0 tural and dynamical properties of colloidal systems. By the formationofaggregatesofanoptimalsize andshape
7 appropriately varying some control parameters (such as (cluster phase), characterized by a peak of the struc-
0 the composition of the solvent, the coating of the parti- ture factor around a typical wave vector, k . Experi-
m
/
t cles, the concentration of the polymers into the solvent, mentally, such cluster phase, made up by approximately
a
...) the effective interaction potential between colloids monodisperse equilibrium aggregates, can be clearly ob-
m
can be suitably tuned in the experiments. It is possi- served using confocal microscopy at low volume fraction
-
ble to realize a hard-sphere system [2]; by adding non and low temperature (high attraction strength) [4, 5, 7].
d
n adsorbing polymers the hard-sphere interaction can be By appropriately tuning the controlparameters (i.e., in-
o complemented by a short-range attraction, induced by creasing the volume fraction or decreasing the temper-
c depletion forces [3]. Recent experimental works outlined ature) the system progressively evolves toward a gel-
v: that in some cases a residual net charge on the surface like non-ergodic disordered state [5, 7] (colloidal gela-
i of colloidal particles may be present [4, 5], thereby in- tion),wherestructuralarrestoccurs. Althoughintensely
X
ducing a long-range electrostatic repulsion screened by studied both experimentally and numerically, a theoret-
r the presence of ions in the solution. The resulting ef- ical understanding these phenomena is still lacking: the
a
fective interaction is therefore givenby a hard-coreterm mechanismsinducingcolloidalgelationarestillunknown
accounting for the excluded volume,a depletion-induced and many gaps remain in our present knowledge of the
narrow attractive shell and a long-rangerepulsive shoul- equilibriumphasediagramofthesesystems. InRef.[8],it
der. This kind of potential is well approximated by the hasbeenproposedthatcolloidalgelationisrelatedtothe
DLVO (after Derjaguin, Landau, Verwey and Overbeek) formation of a Wigner glass, whose blocks are compact
potential[6]. Interestinglyenough,theparametersofthe and thermodynamically stable clusters, with a residual
potential can be suitably tuned in the experiments. The long-range repulsion between them. In Ref. [10] it was
depth of the attractive shell is controlled by the concen- instead highlighted the percolative nature of the struc-
tration of the polymers, φ , which therefore plays the tural arrest in charged colloids, suggesting that perco-
p
role of an inverse temperature, whereas its range is pro- lation of a spanning network of clusters with long living
portional to the ratio between the radius of the colloids, bondsplaysacrucialrole. Morerecently,ithasbeensug-
2
gestedthattheformationofpartiallyorderedanisotropic in two [11, 12] dimensions.
domains and microphase structures may be responsible The paper is organized as follows: in Sec. II we de-
for physical gelation in these systems [11, 13, 14]. scribe the model; in Sec. III we study the model within
In a recent letter [13], we have introduced a φ4 model the self-consistent Hartree approximation; in Sec. IV we
with competition between a short-range attraction, de- analyzethedynamicsofthesysteminthesphericallimit;
scribed by the Ginzburg-Landau Hamiltonian, and a inSec.Vwestudytheglasstransition;inSec.VIwecom-
long-range screened repulsion, described by a Yukawa pute the pair correlation function of the system in the
potential [15]. Albeit schematically, the model contains real space, showing the emergence of competing length
theessentialfeaturesoftheeffectiveinteractionpotential scales and emphasize the prominent role of microphase
among chargedcolloids in polymeric solutions and sheds structures; in Sec. VII we summarize the results found:
new light on the structural properties of these systems. we discuss the resulting phase diagram and stress the
Herewepresentadetailedandextensiveanalyticalstudy connections with charged colloids. Some details of the
ofthismodel,andafullderivationandexplanationofthe calculations are reported in Apps. A-D.
results previously reported briefly in [13].
We show that depending on the control parameters,
II. THE MODEL
there is a region of the phase diagram where usual
phase separation between a colloid-rich and a colloid-
poor phase takes place. Conversely, as the screening We consider the standard three dimensional φ4 field-
length and/or the strength of the repulsion exceeds a theorywiththeadditionofarepulsivelong-rangepoten-
threshold value, phase separation is prevented. In this tial [13]:
case, at moderately high temperature the competition
1
between attraction and repulsion has the effect to pro- [φ]= d3x f(φ)+ ( φ(x))2 + [φ], (1)
LR
duce microphase structures, i.e., partially ordered mod- H 2 ∇ H
Z (cid:20) (cid:21)
ulated domains which in the terminology of particle sys-
where φ(x) is the scalar order parameter field, related
tems, correspond to the cluster phase. These modulated
to the concentration of colloidal particles. The local free
structures are the precursors of a first order transition
energy has the usual Ginzburg-Landau form:
to an equilibrium lamellar phase found at lower temper-
atures. By using a replica approachfor systems without r g
f(φ)= 0 (φ(x))2+ (φ(x))4. (2)
quenched disorder [16, 17], and by employing the Self-
2 4
ConsistentScreeningApproximation(SCSA)[18,19],we
alsoshowthepresenceofaglasstransitionlineinthelow The coefficient r0 is a temperature dependent mass pro-
temperature region,once the firstordertransitionto the portionalto the deviation r0 ∝T −TcMF from the mean
lamellar phase is avoided. The mechanism responsible field transition temperature in absence of frustration.
for the glass transition in this case turns out to be com- The long-range repulsive interaction is described by a
pletely different from that of molecular glass-formers: as Yukawa potential:
a matter of fact, we find that glassiness is not due to
the presence of a hard-core interaction, which is totally [φ]= W d3xd3x′ e−|x−x′|/λ φ(x)φ(x′). (3)
absent in our model; it is instead due to the formation HLR 2 x x′
ZZ | − |
of the microphase structures which order up to the size
The local term favours the formation of a uniform con-
of correlation length. The geometric frustration, arising
densed phase, while the long-range term energetically
fromarrangingsuchmodulatedstructuresinadisordered
frustrates this condensation. In the momentum space
fashion, leads to a complex free energy landscape and,
the Hamiltonian of Eq. (1) reads:
consequently, to a dynamical slowing down.
Our results suggest that the cluster phase observed in
V d3k 4πW
colloidalsuspensionsshouldbefollowed,upondecreasing [φ]= r +k2+ φ φ (4)
H 2 (2π)3 0 λ−2+k2 k −k
the temperature (or increasing the volume fraction), by Z (cid:20) (cid:21)
an equilibrium periodic phase (a columnar or a lamellar gV d3k d3k d3k
1 2 3
+ φ φ φ φ .
phase, depending on the volume fraction). If, instead, 4 (2π)3 (2π)3 (2π)3 k1 k2 k3 −k1−k2−k3
this ordered phase is avoided, a structural arrest, corre- Z
sponding to the gel phase observed in the experiments The model has been also studied in [20] in the context
and in numerical simulations, should eventually occur. of microemulsion. The parameters W and λ are, respec-
The existence of ordered phases in colloidal suspensions tively, the strength and the range of the repulsive po-
and the fact that the transition to the gel phase could tential. For W = 0 we obtain the canonical short-range
occurinametastableliquid,arenovelpredictions,which ferromagnet. Interestingly, for λ we recover the
→ ∞
have never been considered before. Recently, the pres- case of Coulomb interaction. The latter model was first
ence of such ordered phases in atomistic model systems introduced in Ref. [21], in the context of cuprate sys-
of charged colloids interacting via the DLVO potential, tems. Then, it has been further studied in many other
hasbeenunambiguouslyshownbothinthree[11,14]and papers (see, for instance, Refs. [22, 23, 24, 25, 26, 27])
3
consistently [33]. For a one component order parameter
1 W there are six ways of choosing the two factor of φ to be
paired in φ2 among the four factor of φ in the quar-
0.08 P tic term ghφ4/i4. Hence, the self-consistent Hartree ap-
0.8 proximationconsistsinsubstitutingthetermgφ4/4with
0.04 F 3g φ2 φ2/2. Withthissubstitution, theHamiltonianbe-
h i
comes quadratic:
0.6 P
L
T 00 1 2 3 λ Z = φke−β2V (d2π3k)3 r0+3ghφ2i+k2+λ−4π2W+k2 φkφ−k.
0.4 F Z D R (cid:2) (cid:3)
(5)
It is then possible to compute Z and evaluate the corre-
0.2 lation function:
L
G(k)= φ φ φ φ . (6)
k −k k −k
h i−h ih i
0
0 Wc 0.1 4πW 0.2 In the paramagnetic phase, hφki=0, one obtains:
T
G(k) =
r +3g φ2 +k2+ 4πW
0 h i λ−2+k2
FIG. 1: (color online) Main frame: Phase diagram of the T
modelobtainedwithintheself-consistentHartreeapproxima- = , (7)
r+k2+ 4πW
tion, in the frustration (4πW)-temperature (T) plane, for a λ−2+k2
fixed range of the repulsive potential, λ = 2 (and g = 1,
where the renormalized mass term, r, has been defined
r0 = −1). The temperature is rescaled with respect to the
as:
critical temperatureof theunfrustrated model, Tc(W =0)=
2π2/3Λ. The (red) continuous curve, Tc(W,λ), corresponds r r +3g φ2 . (8)
to the second order phase transition from the paramagnetic ≡ 0 h i
totheferromagneticphase(withtheusualHartreecriticalex-
We now recall that:
ponents, ν = 1 and γ = 2), whereas the (blue) dashed curve
corresponds to the fluctuation-induced first order transition d3k
φ2 =G(x,x)= G(k), (9)
from the paramagnetic to the lamellar phase, TL(W,λ). Wc h i (2π)3
is the critical threshold of the repulsion strength, Eq. (11). Z|k|<Λ
5are set in such a way that the elementary length of a lat-
where Λ is the ultraviolet cutoff. As a result, Eqs. (8)
tice spacing, is Inset: System phase diagram in the range
and (9) yield:
(λ)-frustration (W) plane for a fixed temperature T = 0.1,
showing therelative position of the different phases. d3k T
r =r +3g , (10)
0 (2π)3 r+k2+ 4πW
Z|k|<Λ λ−2+k2
where it has been used to describe the phenomenology
which provides a self-consistent equation for the renor-
of a wide variety of systems, where competing interac-
malized mass r. It can be shown that, apart from the
tions on different length scales stabilize pattern forma-
factor 3, this approximation becomes exact for an N-
tion and the creation of spatial inhomogeneities (for a
component order parameter in the limit N [33].
reviewsee[28]). Thesesystemsincludemagneticsystems →∞
Inorderto solvethe self-consistentequation,Eq.(10),
and dipolar fluids characterized by long-range Coulom-
it is convenient to define:
bic interactions [29], mixtures of block copolymers [30],
water-oil-surfactant mixtures [31] and doped Mott insu- 1
W , (11)
lator, including high T superconductors [32]. As a con- c ≡ 4πλ4
c
sequence, our model allows to describe and to interpret
and distinguish between two cases: W < W and W >
inanunifiedfashionthephenomenologyofabroadrange c
W .
of different physical systems. c
A. W <Wc
III. PARAMAGNETIC, FERROMAGNETIC
AND LAMELLAR PHASES WITHIN THE
HARTREE APPROXIMATION ForW <Wcthecorrelationfunction,G(k),behavesas
in the standard unfrustrated case: it is a monotonically
decreasing function of k, with a maximum in k = 0,
In this section we solve the model, Eq. (1), within the | |
where it equals:
self-consistent Hartree approximation. This approxima-
tion amounts in replacing one factor φ2 in the quartic T
term by its average, namely φ2 , to be determined self- G(k=0)= r+4πWλ2. (12)
h i
4
Therefore, as r 4πWλ2, the susceptibility, χ = slowing down due to colloidal gelation from that due to
→ −
G(k = 0)/T, diverges. This corresponds to a usual kinetic of phase separation.
second order phase transition towards a ferromagnetic Nevertheless, the system still undergoes a first order
phase. FromEq.(10), one cancompute the criticaltem- transition to a lamellar phase when the temperature is
perature, T (W,λ), given by the following relation: loweredbelowT (W,λ)(bluedashedlineinFig.1). Such
c L
firstorderphasetransitionisinducedbythefluctuations,
3gTc Λdk 1+k2 = r 4πWλ2. (13) as first discussed by Brazovskii for a related model [34]
2π2 Z0 1+ WWc +k2 − 0− TanhdeatrsaanlssiotiofonutnedmipnertahteurCeoTluLo(mWb,iλc)ccaasne (bλe→det∞erm) i[n27ed].
(cid:16) (cid:17) onequatingthefreeenergyoftheparamagneticphaseto
It is possible to show that the usual Hartree critical ex-
that of the lamellar phase [27], as explained in App. B.
ponents are found, i.e., ν = 1 and γ = 2 (see App. A).
The lamellar phase is characterizedby a periodic vari-
In the terminology of colloidal suspensions, this means
ation of the order parameter with wave-length l =
thatforT <T thesystemundergoesaphaseseparation m
c 2πk−1:
between a colloid-richand a colloid-poor phase. The de- m
pendence of the critical temperature, T (W,λ), upon W
c φ =m(δ(k k )+δ(k+k )), (16)
(for λ = 2 and r0 = 1) is plotted in the main frame of h ki − m m
−
Fig. 1 (red continuous line in Fig. 1), showing that the
wherek isgiveninEq.(14). NotethatasW approaches
only effect of the repulsive interaction for W < W is to m
c W fromabove,accordingtoEq.(14),k vanishes. This
decreasethenumericalvalueofT : forW =0werecover c m
c signals the fact that the size of the stripes diverges on
the usual critical temperature of the unfrustrated sys-
theboundarybetweenthelamellarandtheferromagnetic
tem in the Hartree approximation, Tc(W =0)=−23πg2Λr0 phase.
(notice that r0 <0), whereas Tc vanishes as W Wc. Within the fluctuation approach described in App. B,
→
it is also possible to study the stability of other kinds of
orderedphases,suchascolumnarorperiodicallyordered
B. W >Wc clusterphases,whichoccurin numericalsimulations[12,
14]. In fact, it turns out that these phases can be stable
Conversely, if W >Wc, the correlation function has a besidesthelamellarphase,atlowervolumefractions,i.e.,
maximum around a finite value of the wave vector, km: for φk=0 <0. However, in this paper we only focus on
thehcase iφ = 0, (i.e., d3x φ(x) = 0) where the
k=0
1/2 1/2 h i h i
k = √4πW λ−2 =(4π)1/4 √W W . onlyperiodicallyorderedstablephaseisthelamellarone.
m − − c R
(cid:16) (cid:17) (cid:16) p (cid:17)(14)
As we shall discuss later on, the peak in the correlator
C. Phase diagram within the Hartree
indicates that the system establishes microphase struc-
approximation
tures with inverse domain size given by k , which char-
m
acterize the incipient periodic order. Such modulated
structures are the analog of the cluster phase observed Theresultsfoundwithintheself-consistentHartreeap-
in colloidal systems characterized,also in this case, by a proximationare summarizedin Fig. 1, where the system
peakinthestructurefactoraroundacharacteristicwave phasediagraminthefrustration(4πW)-temperature(T)
vector corresponding to the inverse of the typical size of plane at fixed screening length λ = 2 (main frame) and
the clusters [5, 7, 8, 9, 10]. inthescreeninglength(λ)-frustration(W)planeatfixed
In k =k , the correlation function is given by temperatureT =0.1(inset),isplotted,showingtherela-
m
tivepositionsoftheparamagnetic(P),theferromagnetic
T (F) and the lamellar (L) phases.
G(k =k )= . (15)
| | m r+2√4πW λ−2 AccordingtoEq.(11),bychangingλ,Wc changesand,
− hence, the phase diagram modifies. For instance, in the
Ttiohnerse,fχo(rek, G=(|kk|)=, dkivme)rgaenfdo,rcronserquentλly−,2the2fl√u4cπtuWa-. Ccoonusleoqmubenlciem,itth,eλfe→rrom∞a,gwneetihcapvheatsheartedWucces→to0t.heAasxias
m sp
| | → ≡ −
As a result, also the integral of the right hand side of W = 0, for T < T (W = 0). Conversely, for λ 0 we
c
→
Eq. (10) diverges, and the self-consistent Hartree rela- have that W and the lamellar phase disappears.
c
→ ∞
tion can be satisfied only at T = 0. This implies that It is interesting to observe that the dependence of the
for W > W the paramagnetic phase is stable at all fi- threshold W on the screening length λ is exactly the
c c
nite temperatures, its spinodal line, where the suscepti- same of that found in numerical simulations of a model
bility for k = k diverges, χ(k ) , is located at system for charged colloids interacting via a DLVO po-
m m
| | → ∞
T =0. ThereforeaboveW there isno phaseseparation. tential [9]: the authors find that for W < W the sys-
c c
This result is quite important for designing new materi- tem tends to phase separate, whereas for W > W the
c
als as well as in the experimental and numerical study preferred structures are elongated (and eventually quasi
ofcolloidalsystems,where itis crucialto distinguishthe one-dimensional) finite clusters. Physically, such thresh-
5
tooldnivaanl,uEe,qW. (c4,),caunpbtoetinhteelropwreetsetdoredxepranindikn2g: the Hamil- +W d3x′e−|x−xx′|/λφxα′(x′,t) . (19)
#
Z | − |
V d3k
[φ] r +4πWλ2 (17) In the limit N we recover the spherical model
H ≃ 2 Z (2π)3h(cid:0) 0 (cid:1) replacing N1 Nβ=→1φ2β∞(x,t) by hφ2α(x,t)i. Here h·i rep-
+ 1 4πWλ4 k2 φ φ resents an ensemble average over the initial configura-
− k −k tions. By asPsuming the translational invariance of the
gV d3k1 d3k2 d(cid:0)3k3 (cid:1) i pair correlation function G(x,x′,t) = φ(x,t)φ(x′,t) =
+ 4 (2π)3 (2π)3 (2π)3 φk1φk2φk3φ−k1−k2−k3. G(x x′ ,t), the quantity φ2(x,t) h= G(0,t) Si(t)
Z | − | h i ≡
becomes independent on the position x. Thus, dropping
At W = Wc the coefficient of the k2 term vanishes, im- the index α, taking the Fourier transformoverthe space
plying that there is no energetic cost associated to the of Eq. (19) and multiplying both sides by φ ( k,t) we
α
creation of interfaces between high density and low den- obtain the equation of motion for the time d−ependent
sity regions,leadingto patternformationandto the cre- structure factor, G(k,t)= φ (k,t)φ ( k,t) :
α α
ation of spatial inhomogeneities. h − i
∂G(k,t) 4πW
= Mk2 r+gS(t)+k2+ G(k,t).
∂t − λ−2+k2
IV. DYNAMICS IN THE SPHERICAL LIMIT (cid:20) (cid:21) (20)
By replacing 1 N φ2(x,t) by S(t) we have, in fact,
N β=1 β
In this section we study the dynamics of the model in
enabled a linearization of Eq. (19), “preaveraging” the
thecaseofanN-componentorderparameterinthelimit P
nonlinear term. Now Eq. (20) can be readily integrated
N , after an instantaneous quench from high tem-
to obtain:
→ ∞
perature. The spherical approximation is equivalent to
theHartreeapproximationdescribedintheprevioussec- G(k,t)=G(k,0)e−Mk2 Q(t)+k2t+λ−4π2W+kt2 , (21)
tion (apart from the factor 3 in Eq. (10)), and allows to
(cid:2) (cid:3)
study analytically the dynamics. We solve the time de-
where Q(t) is defined by
pendent Cahan-Hilliard equation [35], following the ap-
proachtakeninRefs.[36]. Weobtainanexactanalytical t
solution for the evolution of the time dependent struc- Q(t)= dt′[gS(t′)+r]. (22)
ture factor of the system, G(k,t), and for the peak po- Z0
sition, k (t). In agreement with the results presented
max To complete the solution, S(t) must be computed self-
in the previous section, we find that as W < W , the
c consistently. As shown in App. C, one can obtain
usual spinodal decomposition occurs, and the maximum
an expression for the time dependent structure factor,
of the structure factor approacheszero as t−1/4 for large
Eq. (C7), and an equation for the evolution of the peak
times. This corresponds to phase separation between a
position:
high density colloid-rich phase and low density colloid-
poor one. On the other hand, for W > Wc, provided exp Mt 1 4πW k4 (t) (23)
that we are on the spinodal line of the lamellar phase − (λ−2+k2 (t))2 max
(T =0), the peak position approaches a non-zero value, (cid:20) (cid:18) max (cid:19) (cid:21)
k ,giveninEq.(14),andthestructurefactorasymptot- 4πWλ−2 1/2 √Mt
m = P 1 ,
icallyapproachaδ-functionpeakedaroundk ,signaling × − (λ−2+k2 (t))3 k (t)
m (cid:20) max (cid:21) max
alamellarorder. However,sincethe paramagneticphase
isstableatallfinitetemepratures,forT >0wefindthat where the function P is given in Eq. (C5). Taking the
the lamellar phase starts to form locally, but then fades logarithmofbothsidesofEq.(23)anddividingbyMtwe
awayafterlongtime andthe systemstayshomogeneous: obtain an asymptotic expression for kmax(t), Eq. (C8),
k (t) approaches a limiting value where the structure which is readily analyzed in the limit t . Careful
max → ∞
factor vanishes exponentially. analysis show that depending on the values of W and λ,
Let us consider the Cahan-Hilliard equation [35]: different solutions of this equation are found. In partic-
ular, one needs again to distinguish between the cases
∂φ(x,t) δ [φ(x,t)] W <W and W >W .
=M 2 H , (18) c c
∂t ∇ δφ(x,t)
where M is related to the mobility, and H is given in A. W <Wc
Eq.(1). ForanN-componentorderparametertheabove
equation reads:
ForW <W thesolutionofEqs.(23)and(C8)isgiven
c
by:
∂φ (x,t) g N
α = M 2 r+ φ2(x,t) 2 φ (x,t)
∂t ∇ " N β −∇ ! α k∞ lim kmax(t)=0, (24)
βX=1 ≡t→∞
6
corresponding to the usual spinodal decomposition. Us-
ingEq.(C8)we findthatthattheusualN →∞domain Σ(T) 2e-04 Σ(Td)
growth exponent of the canonical short-range ferromag-
net with conserved order parameter is found:
k (t) t−1/4. (25) 1e-04
max
∼
2e-04
Obviously,thissolutioncanbeobtainedprovidedthatwe
are below the critical point, r < 4πWλ2 [T < T (W)],
− 0 c 0
otherwisenospinodaldecompositiontakesplaceandthe 0 W 0.2 0.4
system remains in the paramagnetic phase. c 4πW
1e-04
B. W >Wc
ForW >Wc thesolutionofEqs.(23)and(C8)isgiven 0
1 1.05 1.1 1.15 1.2
by:
T/T
k2 =√4πW λ−2 =k2 , (26) K
∞ − m
where the wave vector k coincides with that defined in
∞
Eq.(14). Inthiscase,thepeakpositionreachesanonzero FIG. 2: (color online) Main frame: Complexity, Σ, as a
steadystatevalueforlargetimesandthestructurefactor function of T/TK for 4πW = 0.5 (red dashed curve) and
4πW =0.08(blackcontinuouscurve),forafixedvalueofthe
asymptotically approachesa δ-function peaked arounda
range of the repulsive potential, λ = 2, and for g = 1 and
finite value of the wave vector k . This situation cor-
m r0 =−1. At high temperature (T >Td) the system is in the
responds to the lamellar phase and k is the inverse
m paramagnetic phase and the complexity is zero. At T = Td
domain size. In particular, for earlier times (such that thecomplexitydiscontinuouslyjumpstoafinitevalue,signal-
4πW k4 (t)), using Eq. (C8), we have that: ing the emergence of a complex free energy landscape. The
≪ max
complexitydecreasesasthetemperatureisdecreasedandvan-
k (t) 3lnt 1/4 t−1/4. (27) ishes at TK where the thermodynamic transition to a 1RSB
max ∼ 4 glassy phase takes place. Inset: Complexity at the dynami-
(cid:18) (cid:19)
caltransitiontemperature,Σ(T ),plottedasafunctionofthe
d
Thus, apart from the logarithmic factor, in this regime
strength of the repulsive potential, 4πW. As W decreases,
kmax(t) behaves as the standard solution of the unfrus- Σ(Td) decreases. At W =Wc, the glassy phase disappears.
trated short-range ferromagnet in the large N limit in
the case of usual spinodal decomposition. However, at
later times, kmax(t) saturates at km as a power law: whereWsp =(λ−2 r)2/16π. Consequently,P,Eq.(C5),
−
asymptoticallyvanishesinanexponentialfashion,signal-
k2 (t) k2 +Ct−1/2. (28)
max ∼ m ing that the periodic pattern fades. At the spinodal line
According to Eq. (C7), the structure factor G(k,t) can we have that W =W , which gives back Eq. (28).
sp
be approximated by a Gaussian centred about km and In conclusion, the analysis of the dynamics in the
with width sphericallimitindicatesthatforW <W thesystemun-
c
(4πW)1/4 dergoes a phase separation between a colloid-rich phase
∆k , (29) and a colloid-poor one. On the other hand, for W >W
∼ √4Mt(√4πW λ−2) and T > 0 the system starts to form microphase strucc-
−
h i tures and periodically ordered modulated domains, as
whichinthelimitt approachesaδ-functioncentred
→∞ signaled by a peak in the structure factor which grows
in k , as stated before.
m in time andwhose positionseems to approachk . How-
m
However, this situation can occur only provided that
ever, since the paramagnetic phase is stable at all finite
weareonthespinodallineofthelamellarphase,located
temperatures, the microphase structures have a finite
at T = 0. As discussed in the previous section, the ex-
lifetime. The lamellar order is fully established only at
pression of the spinodal line reads (both in the Hartree
T =0.
approximation and in the large N limit):
r (W,λ)= 2√4πW +λ−2. (30)
sp
−
V. THE GLASS TRANSITION
Indeed, as r >r (W,λ) (i.e., T > 0), from Eq. (C8) we
sp
have that k (t) approaches exponentially a limiting
max
value, k2 = (r+λ−2)/2: Sincetheparamagneticphaseisstableatallfinitetem-
∞ − peratures,the firstordertransitionto thelamellarphase
e−M(W−Wsp)t can be kinetically avoided. It is then possible to super-
k2 (t)=k2 +C , (31)
max ∞ × √Mt coolthe system in the (meta-stable) homogeneousphase
7
below the first order transition line, T (W,λ), as much In order to get an explicit expressionfor Σ we introduce
L
as one can obtainsupercooledliquids by performing fast replicas. The replicated free energy reads:
enoughcoolingsbelow the melting temperature avoiding
crystallization. Experimental [37] and numerical[38, 39] F(m)= lim T ln ψ Z˜[ψ] m, (36)
resultsshowthat,indeed,thesesystemsexhibitlong-time −u→0+ m D
relaxationssimilartothatobservedinglasses. Recentre- Z (cid:16) (cid:17)
sultshaveclearlyshownthepresenceofaglasstransition from which, evidently, F˜ can be obtained as
in the Coulomb case (λ ) [25, 26, 27, 40].
→∞ ∂mF(m)
F˜ = , (37)
∂m
(cid:12)m=1
A. General formalism (cid:12)
(cid:12)
and hence, according to Eq. (35)(cid:12)
In orderto study the glasstransitioninour model, we
1 ∂F(m)
use a replica approach introduced in Ref. [16] and then Σ= . (38)
T ∂m
developed in Ref. [17], formulated to deal with system (cid:12)m=1
(cid:12)
without quenched disorder,which allowsto compute the (cid:12)
Using Eq. (32) and integrating o(cid:12)ver ψ, we get:
complexity, Σ.
Our aim is to derive the free energy landscape of the
m m
model. The equilibrium free energy, defined as F = Z(m) = lim φa exp β [φa]
TlnZ, is relevant only if the system is able to explore u→0+ D − H
− Z aY=1 Xa=1
the entire phase space. This is not the case in the glassy
m
u
phase, where the system is frozen in metastable states. + d3xφa(x)φb(x) . (39)
Inordertoscanthelocallystablefieldconfigurations,we 2m !
a,b=1Z
introduce an appropriate symmetry breaking field ψ(x), X
and compute the following partition function [16]: The partition function of Eq. (39) is formally equivalent
to that of a system in a quenched random field analyzed
Z˜[ψ]= φ exp β [φ] u d3x[ψ(x) φ(x)]2 , by means of the replica trick, such as the random field
D − H − 2 − Ising model (RFIM). The only difference is that in this
Z (cid:18) Z (cid:19)
(32) casethelimitm 1mustbetaken. Wecanthususethe
→
where u denotes the strength of the coupling. Since u> techniquesdevelopedtodealwithsuchsystems[19]. The
0, the introduction of the pinning field ψ(x) forces the matrix correlation function of the problem of Eq. (39)
order parameter φ(x) to assume configurations close to obeys the following Dyson equation:
ψ(x). Therefore, the free energy
u
G−1(k)=G−1(k)δ +Σ (k) , (40)
f˜[ψ]= T lnZ˜[ψ] (33) ab 0 ab ab − m
−
where G−1(k) is the propagator of the free unreplicated
will be low if ψ(x) equals to configurationswhich locally 0
theory, Eq. (7), and Σ (k) is the self-energy in the
ab
minimize [φ]. Thus, in order to scan all metastable
H replica space. If we find that, due to the ergodicity-
states, we have to sample all configurations of the field
breaking coupling constant u, Σ (k) has finite off-
ψ, weighted with exp( βf˜[ψ]): ab
− diagonals elements, we can conclude that there must be
an energy landscape sensitive to that infinitesimal per-
ψf˜[ψ] exp βf˜[ψ] turbation, leading to a glassy dynamics. On the other
F˜ = lim D − . (34) hand, if the self-energy is diagonal, the dynamics is er-
u→0+ R ψ exp (cid:16)βf˜[ψ] (cid:17)
godic and the system is in the liquid phase.
D −
R (cid:16) (cid:17)
Therefore, F˜ is a weighted average of the free energy
in the various metastable states. If there are only a B. The Self-Consistent Screening Approximation
few local minima (i.e., a non-extensive number of local
minima) of free energy f˜ equal to F, we have that F˜ In the following we use the self-consistent screening
equals the true free energy F of the system. However,in approximation(SCSA) [18, 19, 25, 26], which consists in
case of the emergence of an exponentially large number introducing a N-component version of the model with
of metastable states with large barriers between them, φ = (φ ,...,φ ) and coupling constant g/N and sum-
1 N
a nontrivial contribution arises from the above integral ming self-consistently all the diagrams of order 1/N (see
even in the limit u 0+ and F˜ differs from F. This Fig.7). Thisapproximationisexactuptoorder1/N. At
→
allows to identify the configurationalentropy Σ [16, 17]: the end of calculations we will consider the scalar case,
N = 1 [41]. Since the attractive coupling between repli-
F =F˜ TΣ. (35) cas is symmetric with respect to the replica index, we
−
8
ity for different values of the strength of the repulsion,
1
f W, and of the temperature T. The model undergoes
G(k) k a glass transition of the same nature of that found in
discontinuous spin glasses [13, 25, 26]. In Fig. 2 the be-
200 0.5 haviour of the configurational entropy is plotted for two
different values of W as a function of the temperature.
For a given value of the strength of the repulsion W, at
high temperature, (k) vanishes, leading to a vanishing
ξ-1 00 k k 1 complexity, correspFonding to the fact that the system
m
is in the liquid phase. At T = T (W) the configura-
d
100
tional entropy jumps discontinuously to a finite value,
signaling the emergence of an exponentially large num-
ber of metastable states. At this temperature a glassy
dynamics sets in, corresponding to a non-zero value of
the longtime correlationfunction (k). The complexity
F
0 decreases as the temperature is decreased and vanishes
0 k k 1 at T (W). At this temperature, called the Kauzmann
m K
temperature,anidealthermodynamicglasstransitionto
a 1RSB glassy phase takes place. Note that, in agree-
FIG.3: Main frame: Momentum dependenceof thecorre- ment with the results presented in the previous section,
lation function, G(k), for 4πW = 0.2 and λ = 2 at T = Td, theglasstransitiondisappearsforN ,whenthecor-
showingthatitispeakedaroundthetypicalmodulationwave rections due to the diagrams of order→1/∞N are not taken
vectorkm withwidthξ−1,givenbytheinverseofthecorrela- into account.
tion length, signaling the fact that microphase structures es-
In order to characterize the nature of the glass transi-
tablish over a finite range ξ. Inset: Momentum dependence
tion, we examine the properties of the correlation func-
of the non ergodicity parameter, fk, for the same values of
W,λ and T. tions [13]. In the main frame of Fig. 3 the diagonal part
of the correlator, (k), is plotted as a function of the
G
wavevector atthe dynamicaltransitiontemperatureT ,
d
assume the following structure of the matrix correlation showing that the correlation function is clearly peaked
function: around a maximum located in k , whose value is given
m
in Eq. (14). As shown in the next section, the width
G (k)=[ (k) (k)]δ + (k), (41)
ab G −F ab F of such maximum, ξ−1, allows to identify the inverse of
i.e., with diagonal elements (k) and off-diagonal ele- the correlationlength. The shape of (k) indicates that,
ments (k). It can be shoGwn that for system with although no periodic order occurs ( φG = 0), a lamel-
F h kmi
quencheddisorder(suchas the RFIM), this ansatz turns lar structure of wave length l = 2πk−1 over a finite
m m
out to be equivalent to the one-step replica symmetry range ξ is formed. These microphase structures are the
breaking ansatz (1RSB). While the diagonal correlator analogofthe clusterphaseobservedincolloidalsystems.
can be interpreted as the usual (one time) correlation Thesystemformsamosaicofsuchmodulatedstructures,
function, i.e., periodically ordered over length scales smaller than the
correlation length ξ, and randomly assembled in a dis-
T (k)= φ φ , (42)
k −k
G h i ordered fashion over larger length scales. The defects
theoff-diagonaltermcanbeinterpretedasmeasuringthe andthe imperfections in the perfect stripe arrangements
long-time correlations: give rise to the tails in the correlation function. The
lower is the temperature, the more pronounced is the
T (k)= lim φ (t)φ (0) . (43)
F t→∞h k −k i peak of (k) around the maximum in km, signaling the
G
fact thatξ increasesas the temperature is decreased. At
Hence, (k) vanishesinthe paramagneticphasewhile is
F T = T , where ξ > 2l , these microphase structures es-
finite in the glassy one. d m
tablishoveraleng∼thlargerthantheir modulationlength
As shown in Ref. [26] and in App. D, within the
andbecomes frozen. The glasstransitionarisesfromthe
SCSA the expressionsof the correlatorscan be found by
fact that there are many possible configurations to ar-
solving numerically a set of coupled integral equations,
rangesuchmodulatedstructuresinadisorderedfashion,
Eqs. (D8)-(D16). After that, according to Eqs. (D19)
leading to the emergence of metastable states.
and (D20) we are able to compute the complexity, Σ.
The characteristic wave length, k−1, dominates also
m
thedynamicsasindicatedbythemomentumdependence
C. Results of the non ergodicity parameter
We have fixed the range of the repulsive potential to φk(t)φ−k(0) (k)
f lim h i = F , (44)
λ=2andwehavestudiedthebehaviourofthecomplex- k ≡t→∞ φk(t)φ−k(t) (k)
h i G
9
r-r
T sp
s
e l
0.004 l
a m
c
s

h ξ
0.002 t
r (W) g
d n
r (W) e
0.1 K l
c
00 W 0.2 0.4 ti
c 4πW is
r
e
t l
c
GLASS a D
r ξ
a
h
c
0
W 0.1 0.2 4πW T* T
c
FIG. 4: (color online) Main frame: Dynamical (blue curve FIG. 5: (color online) Temperature dependence of the char-
and triangles) and thermodynamical (red curve and dia- acteristic length scales. At high temperatures, T > T⋆, the
monds)transitiontemperaturetotheglassyphase,asafunc- systembehavesasafluidofchargesoflinearsizeξ(redcurve)
tionofthestrengthoftherepulsivepotential,4πW,forλ=2. andscreeninglengthlD (blackcurve). AtT⋆ thisdescription
Inset: Values of the renormalized mass, rd and rK, corre- breaksdown and thecompetition between attraction and re-
sponding, respectively, to the dynamical and the thermody- pulsion produces microphase structures with interstripe dis-
namical glass transition temperatures, as a function of 4πW. tance lm (blue curve), established over a finite range ξ. The
Inthefigurethedifferencebetweenrd(W)(redcurveandtri- modulation length, lm, decreases as the temperature is de-
angles) andrK(W)(bluecurveanddiamonds) andthemini- creased and at low temperature equals 2πkm−1. Conversely,
mumvalueoftherenormalizedmass,rsp(W),isplotted. The the correlation length, ξ, increases as the temperature is de-
figure shows that as W → Wc, both rd and rK approach creased and diverges at T → 0 at the spinodal line of the
rsp(W),signaling that Td,K(Wc)=0. homogeneous phase. Glassiness emerges when ξ >∼ 2lm. On
the contrary, the modulated structures progressively fade as
ξ<∼lm.
plotted in the inset of Fig. 3 at T = T . Basically, f
d k
represents the height of the plateau reached by the cor-
relation function of the density fluctuations of wave vec-
tor k after infinite time. FromEq. (44) one immediately
observes that as (k)=0 (T >T ), also the non ergod-
d
F
icityparameter,f ,vanishes: thesystemisinanergodic
k
phase, since all the correlation functions relax to zero.
On the other hand, at T = T the ergodicity is broken.
d
The presence of a maximum of fk located in |k| = km rsp(W) is the minimum value of the renormalized mass
signalsthefactthatstructuralarrestismorepronounced (corresponding to T = 0), whose expression is given in
over length scales of order lm. The width of the maxi- Eq. (30). The figure clearly shows that the glassy phase
mum of the non ergodicity parameter, Υ−1, can be tied disappears as W W , where we find that r (W )
c d,K c
to the emergence of a new length scales. Indeed, the r (W ), i.e., T (W→) = T (W ) = 0. (Nevertheless, w→e
sp c d c K c
hreepigrhesteonftsththeeptlyapteicaauldleencgatyhssacsa|l|eko|v−erkwmh|i>chΥd−ef1e,catsndanΥd hreasvuelttshcaatnlibmeWin→teWrpc+rTetde,Kd(iWfo)ne6=obTsde,Krv(eWsct)ha=t,0a)c.coTrdhiensge
imperfections in the periodically modulated pattern are toEq.(14),thesizeofthemodulatedstructuresincreases
allowedtowanderandto diffuseoverlongtimes [25,26]. as W is decreased and diverges as W W . In fact, for
c
→
InFig.4the dynamicaltransitiontemperatureT (W) purely geometrical reasons, the larger is the modulation
d
and the ideal one T (W) are plotted as a function of length,thesmalleristhenumberofmetastableconfigura-
K
the frustration 4πW, for a fixed value of the range of tionstoarrangethemicrophasestructuresinadisordered
the repulsive potential, λ = 2. It is also interesting to fashion. As a consequence, the number of metastable
studythebehaviourofr (W)andr (W),i.e.,thevalues states (i.e., the complexity) vanishes as W W and
d K c
→
of the renormalized mass r corresponding, respectively, the glassy phase disappears. This is clearly outlined by
to the dynamical and to the thermodynamical transi- theinsetofFig.2,whichshowsthatthecomplexitycom-
tion temperature. More precisely, in the inset of Fig. 4, puted at the dynamical transition temperature tends to
r (W) r (W)andr (W) r (W)areplotted,where zero as W W .
d sp K sp c
− − →
10
VI. MODULATED STRUCTURES AND the range of the repulsive potential. Therefore, the sys-
CHARACTERISTIC LENGTH SCALES tem at high temperatures, T > T⋆, behaves as a fluid
of charges of linear size ξ and screening length l . The
D
In this section we analyze more accurately the nature temperature dependence of ξ and lD is shown in Fig. 5.
of the microphase structures which are formed due to
the competitionbetweenattractionandrepulsionondif-
ferent length scales. This investigation provides many
B. Low temperature: r<r⋆
insights on the nature of the cluster phase observed in
colloidalsuspensionsandonthephysicalmechanismsin-
ducing the structural arrest in these systems. Conversely, for r < r⋆ (i.e., T < T⋆) the argument
Let us consider the case W > W , where the correla- of the square root of Eq. (46) becomes negative. Conse-
c
tion function, G(k), is peaked around the typical mod- quently, ζ2 and κ2 assume complex values. The analytic
ulation wave vector, k . In the following we compute continuation of Eq. (48) to the low temperature region
m
the correlation function in real space (within the self- reads:
consistentHartreeapproximation,or,equivalently,inthe
T
spherical limit) and we show that different length scales G(x) = e−κ1|x| 2κ κ λ−2 cos(κ x)
emergeinthe system[23, 42]. Thepaircorrelatorinreal 16π xκ1κ2 1 2− 2| |
space is determined by computing the Fourier transform | +| κ2 κ2 hλ(cid:0)−2 sin(κ x)(cid:1), (51)
of G(k), given in Eq. (7): 2− 1− 2| |
(cid:0) (cid:1) i
G(x) = 1 d3kG(k)eik·x (45) where κ = κ1 +iκ2. This expression implies that, al-
(2π)3 Z though no periodic order occurs (hφkmi= 0), a lamellar
T k(λ−2+k2)sin(kx) structureofwavelengthlmoverafiniterangeξisformed.
= dk , The modulation length is given by:
2π2 x (k2+ζ2)(k2+κ2)
| |Z
where −1/2
l =2π κ −1 =4π 2 rλ−2+4πW r λ−2 ,
m 2
| | − −
(r+λ−2) (r λ−2)2 16πW
ζ2,κ2 = ± − − . (46) h p i (52)
2 whereas, according to Eq. (51), the correlation length
p
The above equation marks a crossover temperature T⋆, reads:
correspondingtothetemperatureatwhichtherenormal-
−1/2
ized mass equals: ξ = κ −1 =2 r+λ−2 +2 rλ−2+4πW ,
1
| |
r⋆ =λ−2+2√4πW. (47) h(cid:0) (cid:1) p i (53)
andgivestherangeoverwhichthemicrophasestructures
are formed. The temperature dependence of the charac-
A. High temperature: r>r⋆ teristic length scales present in the system is shown in
Fig. 5. The correlation length increases as the tempera-
ture is decreased and diverges as T goes to zero, where
For high enough temperatures, r > r⋆ (i.e., T > T⋆),
the spinodal line of the paramagnetic phase is located.
the argumentofthe squarerootofthe righthandsideof
At low temperatures we have l = 2πk−1, where k is
Eq.(46)ispositive. Consequently,ζ2 andκ2 assumereal m m m
the characteristic wave vector defined in Eq. (14). Con-
values andthe integral,Eq.(45), canbe easily evaluated
versely, as T T⋆−, the modulation length l diverges
byapplyingtheresiduetheoremtothepoleslyingonthe → m
as (r⋆ r)−1/2. Interestingly enough, it turns out that
imaginary axis at k = iζ, iκ, leading to −
± ± the glass transition emerges at temperatures at which
T (ζ2 λ−2)e−ζ|x|+(λ−2 κ2)e−κ|x| ξ 2lm, as conjectured in Ref. [26]: the microphase
G(x)= − − . (48) str≥uctures form over a length larger than their modu-
4π x(ζ2 κ2)
(cid:2) | | − (cid:3) lation length and become frozen. The glass transition
The above expression clearly indicates the emergence of arises from the fact that there are many possible config-
two characteristic length scales, ξ = ζ −1 and l = urations to arrange such modulated structures in a dis-
D
κ −1. The former plays the role|ℜo{f t}h|e correlation orderedfashion,leadingtoagreatnumberofmetastable
|ℜ{ }|
length of the canonical short-range ferromagnet and at states. Asthetemperatureisincreased,theclusterphase
high temperature behaves as: continuously fades approximately at the temperature at
which ξ < l : the microphase structures are ordered
ξ ∼r−1/2. (49) over a len∼gthmscale smaller than their own modulation
length and the system becomes homogeneous. Such a
The otherlengthl ,instead,is arenormalizedscreening
D
crossovertemperature at which approximately the mod-
length, and for high temperature tends to
ulated structures fades and the system is homogeneous
l λ, (50) corresponds to the black dotted line of Fig. 6.
D
∼