Glassy dynamics in confinement: Planar and bulk limit of the mode-coupling theory

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by  Simon Lang

Glassy dynamics in confinement: Planar and bulk limit of the mode-coupling theory
Simon Lang,1 Rolf Schilling,2 and Thomas Franosch1
1Institut fu¨r Theoretische Physik, Leopold-Franzens-Universit¨at Innsbruck,
Technikerstraße 25/2, A-6020 Innsbruck, Austria
2
Institut fu¨r Physik, Johannes Gutenberg-Universit¨at Mainz, Staudinger Weg 7, 55099 Mainz, Germany
(Dated: January 12, 2015)
We demonstrate how the matrix-valued mode-coupling theory of the glass transition and glassy
dynamics in planar confinement converges to the corresponding theory for two-dimensional (2D)
planar and the three-dimensional bulk liquid, provided the wall potential satisfies certain condi-
5 tions. Since the mode-coupling theory relies on the static properties as input, the emergence of
1 a homogeneous limit for the matrix-valued intermediate scattering functions is directly connected
0 to the convergence of the corresponding static quantities to their conventional counterparts. We
2 show that the 2D limit is more subtle than the bulk limit, in particular, the in-planar dynamics
decouples from the motion perpendicular to the walls. We investigate the frozen-in parts of the
n intermediatescatteringfunctionintheglassstateandfindthatthelimitstimet→∞andeffective
a
wall separation L→0 do not commute due to the mutual coupling of the residual transversal and
J
lateral force kernels.
9
PACSnumbers: 64.70.Q-,64.70.pv,05.20.Jj
]
t
f
o
I. INTRODUCTION uids[28–30]. Qualitative[27,29,30]andquantitative[30]
s
. comparisonsoftheMCTresultsforthese2Dliquidswith
t
a Confined liquids have been studied extensively in those from experiments [31] and simulations [30] have
m been reported. The two-dimensional systems are found
physics,inparticulartheir phasebehavior[1, 2], dynam-
tobehavequalitativelysimilartotheirthree-dimensional
- ical properties [3–13], and their structural characteriza-
d counterparts.
tion [14–16]. Furthermore confined liquids are interme-
n
o diatebetweenalow-dimensionalandbulkliquidanddis- Motivated by the numerous experimental and simu-
c play an intriguing interplay of near-range local ordering lational results for the glass transition of confined liq-
[ and the confining length. In case of a slit geometry, i.e., uids [3, 6–13], mostly for a slit geometry, MCT has been

two parallel flat hard walls with effective separation L generalized recently to describe dense liquids in planar
1
v one can elaborate the full range from two-dimensional confinement [32–34]. In contrast to the MCT for the
3 (2D) up to three-dimensional (3D) liquid behavior by glassydynamicsinbulkordisorderedstructures[35,36],
5 varying the wall separation from zero to infinity. For theinhomogeneouspackingintheslitrequirestoconsider
1 instance,forequilibriumphasetransitionsonecanstudy symmetry-adapted matrix-valued intermediate scatter-
2
the crossover [17] from the Kosterlitz-Thouless transi- ing functions to characterize the density fluctuations of
0
tion[18–23]in2Dliquidstotheconventionalphasetran- theconfinedliquid. Inthecaseofhardspheres,thetran-
.
1 sitionsofa3Dliquid. Similarly,thecrossoverbehaviorof sition line, separating the regime of collective frozen-in
0 the glassy dynamics of a quasi-two-dimensionalconfined states from liquid states, has been determined as a func-
5
liquid towards a bulk system provides insight in the na- tion of the slit width. An intriguing multiple reentrant
1
ture of the mechanism of structural arrest. transitionforwallseparationsonthescaleofafewparti-
:
v In bulk systems many features of the slowing down of cle diameters has been predicted and corroborated by
Xi structural relaxation upon cooling or compression have recent molecular-dynamics simulations with controlled
been rationalized in terms of the mode-coupling theory polydispersity [37].
r
a of the glass transition (MCT) developed by G¨otze and While MCT has been successfully tested for planar,
co-workers[24, 25]. The predictions of MCT include the bulk and confined liquids, the natural question arises:
emergenceofanon-triviallong-timelimitoftheinterme- Does MCT for liquids in planar confinement for L → 0
diate scattering functions, called glass-formfactors asso- andL→∞properlyconvergetoMCTfor2Dand3Dliq-
ciated with two-time fractals in the vicinity of the criti- uids, respectively? Particularly, one would like to know
cal point. Particularly, MCT entails a sharp dynamical if the dynamical behavior of a strongly confined liquid
glass transition ,e.g., for a hard-sphere liquid at a criti- isapproximatelydescribedbyatwo-dimensionalsystem.
cal packing fraction. Although in nature this transition To provide an answer is the major goal of the present
is smeared, the various predictions of MCT in three di- work. Inparticular,wewilldemonstratehowthematrix-
mensions have been confirmed in the past two decades valued MCT due to the inhomogeneous structure of the
by experiments and computer simulations [25, 26]. confinedliquidreducestotheMCTofscalarintermediate
Inordertostudythedependenceoftheglasstransition scatteringfunctions for the two-dimensional[27] andthe
on the spatialdimension MCT has also been workedout bulk case [25]. In contrast to the bulk limit, the planar
fortwo-dimensionalsingle-component[27]andbinaryliq- limit L→0 turns out to be rather subtle.
2
The outline of this work is as follows. In the next sec- Throughoutthis paper the microscopicdynamics are as-
tion we introduce the model, the quantities of interest, sumed to be Newtonian generated by the Hamiltonian
and recall the equations of motion for confined liquids H({p~ },{~x }),inparticular,collisionswiththeflathard
n n
withintheMCTapproximation. InSec.IIIwediscussthe walls are elastic thereby conserving momentum parallel
behavioroftheMCTfunctionalforsmallwallseparation to the walls. The trajectory in the N particle phase
in terms of the proper convergence of the static proper- space is denoted by ({~r (t)},{z (t)},{P~ (t)},{Pz(t)})
n n n n
ties towards their two-dimensional counterparts demon- and all calculations are performed in the canonical en-
strated recently [38]. In particular, we show that the semble. The thermodynamic limit N → ∞,A → ∞ is
planar MCT is recovered for all finite times in the limit anticipatedforfixed2Dnumberdensityn =N/A. With
0
of vanishing plate separation. Next we study the fixed- the accessible volume of the particles V = AL, we find
point equation for the glass-form factors for small slit for the 3D number density n=N/V =n /L.
0
widths. In Sec. IV we demonstrate that the MCT equa- Themodulationoftheequilibriumdensityprofilen(z)
tions include the bulk behavior as limiting case as the in the slit is encoded in discrete Fourier components
wall separation becomes infinitely large. Section V pro-
vides a critical assessment of the different convergences n = dzexp(iQ z)n(z), (4)
µ Z µ
and possible implications for the glassy dynamics in ex-
treme confinement. where the mode index µ refers to discrete wave numbers
Q = 2πµ/L, µ ∈ Z and integration is performed over
µ
the accessible slit z ∈[−L/2,L/2].
II. CONFINED LIQUIDS: BASIC QUANTITIES
Thefundamentalvariableofinterestisthemicroscopic
AND MCT
fluctuating density mode
N
Themicroscopicsetup,thederivationsoftheequations
ρ (~q,t)= exp[iQ z (t)]eiq~·~rn(t), (5)
ofmotionoftherelevantdynamicquantitiesofinterestis µ µ n
X
detailedinRef.[33];herewesummarizeitsmainfeatures, n=1
in order to keep the present paper self-contained. where~q =(q ,q )aretheconventionaldiscrete(forfinite
x y
ConsideraliquidofN particlesofmassmbetweentwo A) wave vectors in the x-y-plane. Particle number con-
parallel,planarwallswithcrosssectionAandseparation servationrelatesthetimederivativeofρ (~q,t)tothecur-
µ
H. Then a point in phase space is specified by the setof rentdensitiesparallelandperpendiculartothewalls[33]:
coordinates parallel and perpendicular to the wall ~x =
n
(~rn,zn) and corresponding momenta p~n =(P~n,Pnz), n= jα(~q,t)= 1 N bα(~qˆ·P~ (t),Pz(t))exp[iQ z (t)]eiq~·~rn(t),
1,...,N. Weuseimpenetrablewallswithadditionalwall µ m n n µ n
X
potential of the form U({zn};L)= Nn=1U(zn;L) and n=1 (6)
P with channel index α =k,⊥. Here, the short-hand no-
U (z) for|z|≤L/2,
U(z;L)= w (1) tation for the unit vector ~qˆ = ~q/q and the selector
®∞ for|z|>L/2. bα(x,z) = xδ +zδ has been employed. The emer-
αk α⊥
Here, the effective wallseparationL is introduced as the gence of several decay channels is reminiscent of the
transverselengthaccessibletotheparticles,andtherefore mode-coupling theory of molecules [39, 40], where gen-
we distinguish between point particles andhardspheres, eralized density modes couple to both translational and
rotational currents.
H −σ, hard spheres, The basicquantity ofthe MCT ofliquids inslitgeom-
L= (2)
®H, point particles. etry is the generalization of the intermediate scattering
function
The interaction energy of a particle with either of the
1
walls Uw(z) is assumed to be smooth, with possible di- Sµν(q,t)= hρµ(~q,t)∗ρν(~q,0)i. (7)
N
vergencies at z ≡ ±L/2. In principle, both walls can
interact differently with the liquid, i.e., asymmetric wall We shall make use of a natural matrix notation
potentials U (z) = U (L/2 + z) + U (L/2 − z) with [S(q,t)] = S (q,t), and similarly for other correla-
w − + µν µν
U (x) 6= U (x) are allowed. Periodic boundary condi- tion functions throughout this paper. MCT requires
− +
tions are imposed parallel to the walls in x-y direction. the Fourier coefficients n , the static structure factors
µ
Wesuppresstheparametricdependenceonthewallsepa- S (q) = S (q,t = 0) and the static current-current
µν µν
rationLinthefollowingtoallowforacompactnotation. correlators
The pair interactions V(~x)≡V(~r,z) depend only on the
[J(q)]αβ =Jαβ(q)
mutualdistance|~x|,i.e.,V({~x })= N V(|~x −~x |). µν µν
n n<m n m 1
Then the Hamilton function is speciPfied by = hjα(~q)∗jβ(~q)i
N µ ν
N p~2 n∗
H({p~ },{~x })= n +U(z ) +V({~x }). (3) =δαβv2 µ−ν, (8)
n n 2m n n th n
Xï ò 0
n=1
3
as known input [33] with thermal velocity v = equations which have to be solved self-consistently. The
th
(k T/m)1/2. General properties of the static and dy- investigations performed in the following rely on the
B
namic correlators including symmetry relations have proofs that the mode-coupling equations provide the ex-
been discussed in detail [33]. istenceofuniquesolutions,whichhasbeendemonstrated
Employing the Zwanzig-Mori projection operator for- rigorously for Brownian dynamics in monocomponent
malism [25, 41] exact equations of motion for the collec- simple liquids [25, 43] and mixtures [44] and only re-
tive correlators S (q,t) can be derived [32, 33]. For cently for the MCT with multiple relaxation channels,
µν
later purposes it is more convenient to consider the where the here discussed confined MCT is merely a spe-
Laplace-transformed equations, where the convention cial case [45].
for the transformed matrix correlators is Sˆ (q,z) =
µν
i ∞ dteiztS (q,t), Im[z] > 0 1. The equations in
t=0 µν
thRe Laplace domain for Sˆ (q,z) deal with general- III. CONVERGENCE TO THE PLANAR MCT
µν
ized matrix-valued fraction representations. First, we
express Sˆ (q,z) in terms of current-memory kernels InthissectionwediscusshowtheMCTequationscon-
µν
Kˆ (q,z) [33], verge towards the MCT for 2D liquids, which is one of
µν
the major results of this work. Let us emphasize that
Sˆ(q,z)=− zS−1(q)+S−1(q)Kˆ(q,z)S−1(q) −1, (9) this requires to discuss the theory for small but finite
slit width L > 0. Convergence implies that fluids con-
î ó
fined to small slit widths behave similarly to the truly
which split by the perpendicular and parallel current to
two-dimensionalcase, the errorscan be made arbitrarily
small upon decreasing the slit width.
Kˆ (q,z)= bα(q,Q )Kˆαβ(q,z)bβ(q,Q ). (10)
µν µ µν ν In the limiting regime of small wall separation the in-
X
αβ=k,⊥ plane motion is presumed to be close to the dynamics of
a two-dimensionalsystem. Furthermore,ofallthe struc-
Thekernel[Kˆ(q,z)]αβ =Kˆαβ(q,z)canberepresentedby
µν µν tural properties entering the MCT equations the two-
dimensional structure factor should dominate the equa-
Kˆ(q,z)=− zJ−1(q)+J−1(q)Mˆ (q,z)J−1(q) −1, tions for small slit widths. The MCT equations encode
î ó (11) the confinement by the walls via the structural input in
which involves the force kernel Mˆαβ(q,z). The con- terms of the density modes and the symmetry-adapted
µν
stitutive MCT ansatz expresses the force kernel in the static structure factors both as initial values for the cur-
rentcorrelatorsandfortheintermediate scatteringfunc-
timedomainintermsoftheintermediatescatteringfunc-
tions [33], tions and on the associated direct correlation functions
via the vertices in the MCT functional. One subtlety
1 arises since for small wall separation L > 0 the wave
Mαβ(q,t)= Xα (~q,~q ~q )
µν 2N3 q~X1,q~2µνX11µν22 µ,µ1µ2 1 2 mnuomtiboenrsblQowµ =up2.πµ/L associatedwith the perpendicular
×S (q ,t)S (q ,t)Xβ (~q,~q ~q )∗.
µ1ν1 1 µ2ν2 2 ν,ν1ν2 1 2
(12)
A. Density profile and static correlators
Theverticeshavebeencalculatedrelyingonasystematic
convolution approximation [33], Since the MCT relies on the structural properties of
the fluid as known input, the question of convergence to
Xµα,µ1µ2(~q,~q1,~q2) a planar limit is intimately related to the structure of
=−Nv2 n0δ [bα(~qˆ·~q ,Q ) fluids inextreme confinementandtherapidityofthe ap-
thL2 q~,q~1+q~2 1 µ−µ2 proachtoatwo-dimensionalsystem. Thisissuehasbeen
×c (q )+(1↔2)], (13) addressed in two of our earlier works [38, 46] in terms
µ−µ2,µ1 1
of a cluster and cumulant expansionin case of hard core
andinvolvethedirectcorrelationfunctionsc (q),which and smooth potentials, respectively. There, one can not
µν
are related to the static structure factor S (q) by the onlyestimatetheorderofconvergencebutactuallycalcu-
µν
inhomogeneous Ornstein-Zernike equation [34, 42]. late the leading corrections to both thermodynamic and
The Eqs. (9)-(13) involve the initial conditions structural quantities with respect to a planar reference
S (q,t = 0) = S (q) and Kαβ(q,t = 0) = Jαβ(q) system. Here we shall be interested only in the leading
µν µν µν µν
therebyconstitutingacompletesetofcouplednon-linear terms and quantify the order of the corrections in terms
of Landau symbols O(·) and o(·) as the slit width be-
comes small L→0.
The convergence of the structural quantities has been
1 The frequency z should not be confused with the transversal demonstrated [38] assuming analytic wall potentials ful-
variablez. Thedistinctionisclearfromthecontext. fillingthesmoothnesscriterionU(z =z˜L)−U(0)=O(L),
4
which states that the particle-wall interaction should be the memory kernels. We assume first, that times t
controlledforfixedscaled transversecoordinatesz˜asthe and frequencies z are held fixed independent of the wall
slit width approaches zero. Then the density profile be- separation, while the limit L → 0 is performed. For
comes flat [38] even on the scale of the plate distance this purpose, it is favorable to introduce rescaled modes
n(z) = (n /L)[1+O(L)]. This in turn implies conver- Q˜ = LQ = O(L0), which are independent of the slit
0 µ µ
gence for the Fourier modes of the density, size. Usingtheconvergenceofthedirectcorrelationfunc-
tion, Eq. (17), neglecting terms of O(L2) from Eq. (13),
n = n0 =const. for µ=0, (14) one obtains
µ
®O(L) else.
Xα (~q,~q ,~q )
µ,µ1µ2 1 2
Note,thatthe Fouriermoden isindependentofLfrom
0
the sum rule n0 = dzn(z) ≡ N/A. We assume here =−Nvt2hn0δq~,q~1+q~2 δαk(~qˆ·~q1)c(q1)δµ10δµ2µ
n
the worstcaseofasyRmmetricwalls,whereasfor symmet-
ric walls O(L) can be replaced by the faster convergence +Lδα⊥Q˜µ−µ2c˜µ−µ2,µ1(q1)+(1↔2) . (18)
o
O(L2), see Ref. [38]. Let us repeat a word of warning
here. The existence of the convergence is not guaran- Thereby,wefindthattheleadingcontributionofthever-
teedingeneraland,inparticular,potentialsdivergingat tices in the limit L →0 stems from α =k. Keeping now
the walls, e.g., Lennard-Jones and Coulomb potentials, only the leading order, i.e., we set L = 0, and assuming
do not belong to the class indicated above. Yet, pure a priori that S (q,t) = O(L0) for all (µ,ν), in the 2D
µν
hard walls trivially fulfill the smoothness criterion and thermodynamic limit N → ∞, A → ∞ with n = N/A
0
they constitute the reference case we wish to address. fixed, the nonvanishing memory kernel assumes the fol-
From the convergence properties of the density modes, lowing form
Eq. (14), one can immediately infer the convergence of
the static current correlator,Eq. (8), to Mkk(q,t)=n0v4 d2q1 [~qˆ·~q c(q )]2S (q ,t)S (q ,t)
µν 2 thZ (2π)2 1 1 00 1 µν 2
(cid:8)
v2 for µ=ν,
Jαβ(q)=δαβ th (15) +[~qˆ·~q c(q )][~qˆ·~q c(q )]S (q ,t)S (q ,t)
µν ®O(L) else, 1 1 2 2 0ν 1 µ0 2
+(1↔2) , (19)
which becomes to leading order diagonalwith respect to (cid:9)
the discrete modes. where ~q2 = ~q − ~q1. In particular, for µ = ν = 0 the
Similarly, for the static structure factors it has been memory kernel contains only couplings of S00(q,t)
shown [38] that
d2q
Mkk(q,t)= 1 V(~q,~q ~q )S (q ,t)S (q ,t), (20)
S(q)[1+O(L2)] for µ=ν =0, 00 Z (2π)2 1 2 00 1 00 2
S (q)= (16)
µν
®(1−δµ0)δµν +O(L) else,
and the vertices V(~q,~q ~q ) coincide with the ones of the
1 2
whereS(q)denotes the static structurefactorofthe cor- two-dimensional theory [27]
responding 2D liquid. Thus the structure factor matrix
l[iSm(qit)]µaνnd=inSaµdν(dqi)tiobne,coSme(sqd)ifaogronµal6=as0 wbeecllominesthiede2aDl- V(~q,~q1,~q2)= n20vt4h[(~qˆ·~q1)c(q1)+(1↔2)]2. (21)
µµ
gas-like for L→0.
The remaining memory kernels assuming S (q,t) =
µν
The direct correlation function of the confined liquid O(L0) for all (µ,ν) are of higher order in L, viz.
in the slit converges to
cµν(q)=L2[c(q)δµ0δν0+L2c˜µν(q)+o(L2)], (17) Mkµ⊥ν(q,t)=O(L),
M⊥k(q,t)=O(L),
where c(q) is the corresponding 2D direct correlation µν
function and the correction amplitude c˜ (q) is inde- M⊥⊥(q,t)=O(L2), (22)
µν µν
pendent of the slit width L. The prime observation of
Ref. [38] was that the corrections are O(L2) irrespective as one infers from Eq. (18).
ofthewallpotential,whereasforthestructurefactorsfor The notable property of the MCT functional,
(µ,ν) 6= (0,0) or the density profile the leading correc- Eqs.(19), is thatif the intermediate scatteringfunctions
tions are O(L). This latter property plays an important Sµν(q,t) are diagonal in the mode indices, this property
role for the convergence of MCT for confined liquids. is preservedby Mkk(q,t). Since also the static structure
µν
factorsandcurrentcorrelators,whichserveasinitialcon-
ditions, are diagonal to lowest order in L the equations
B. Mode-coupling theory: t finite, L→0 of motion, Eqs. (9)−(11) do not generate off-diagonal
terms. More formally one can show that all time deriva-
We start the investigation of the convergence of the tives dlS (q,t)/dtl| ,l ∈ N are diagonal, similar to
µν t=0 0
confined MCT towards the planar MCT by discussing Ref. [47]. Since the solutions have been demonstrated
5
to be unique [45], the thus constructed solution remains between both walls. This time scale becomes small for
diagonal for all times t>0. L→0, thus itis separatedfromthe microscopicdynam-
In particular, one finds that the equations of motion icsτ ≪t =1/Ω ofthe planardynamics. Indeed,ifone
0 q
for S (q,t) decouple completely from the remaining di- inspectsEq.(25)onthetimescaleτ,i.e.,zτ =O(1),the
00
agonal ones as we demonstrate below. To simplify no- planar correlator S(q,t) for t/τ = O(1) has not evolved
tation we drop the mode indices and write S(q,t) = for L→0: S(q,t)=S(q). On this rapid scale the MCT
S (q,t) and similarly for the static structure factor kernelsarenegligible,whichisinaccordancewiththeno-
00
S(q)=S (q,t=0). Furthermore we make contactwith tion that they aredesigned to describe the slow dynami-
00
the notation of Ref. [27]: calprocessesleadingtostructuralarrest. Ifoneperforms
the same reasoning as described above neglecting the
Mkk(q,t)=Ω2v2 m(q,t), (23) memory kernels for the MCT equations of motion of the
00 q th
tagged-particle correlator in Ref. [34], then one obtains
with the characteristic frequency Ω2 = q2v2 /S(q). Us- the same equation of motion for the transversal dynam-
q th
ing this result and Eq. (15) it follows from the second ics Sˆ(s)(q,z) = −1/ z−Q˜2L−2v2 /z for µ 6= 0, if
µµ µ th
fraction representation, Eq. (11), that its solutions are
the tagged particle iîs of the same spóecies as the host-
diagonal in (α,β) and in (µ,ν), and that the equations
liquid particles. Thus, the fast transversal dynamics of
for Kˆµkkµ(q,z) and Kˆµ⊥µ⊥(q,z) decouple for all µ. Conse- thecollectiveintermediatescatteringfunctionreducesto
quently, one obtains a closed equation for Kˆkk(q,z) in- the dynamics of the incoherent scattering function, i.e.,
00
volving mˆ(q,z), only. Abbreviating the 2D relaxation Sˆ(s)(q,z) ≡ Sˆ (q,z) for µ 6= 0 and L → 0 indicating
µµ µµ
kernel Kˆ(q,z) = Kˆ (q,z) = q2Kˆkk(q,z) one finds from thattheresidualperpendiculardynamicsismerelyaone-
00 00
Eq. (11) particle dynamics.
We conclude that the equations of motion are capa-
Kˆ(q,z)=− q2vt2h . (24) ble to account for a dynamical decoupling of lateral and
z+Ω2mˆ(q,z) perpendicular degrees of freedom, splitting off the pla-
q
nar glassy dynamics from a fast one-dimensional ideal-
TakingadvantageofthediagonalityofSµν(q)andEq.(9) gas like motion in a finite box. Nevertheless, we expect
one obtains the well-known double-fraction representa- that the true transversal dynamics is not correctly con-
tion tained in the MCT equations, but the fundamental time
scales are still properly reflected.
−S(q)
Sˆ(q,z)= . (25)
z−Ω2/ z+Ω2mˆ(q,z)
q q
(cid:2) (cid:3)
Going back to the temporal domain leads to the gener- C. Mode-coupling theory: L finite, t→∞
alized harmonic oscillator equation
Thepresentcaseallowstostudythe glassformfactors
t
S¨(q,t)+Ω2S(q,t)+Ω2 m(q,t−t′)S˙(q,t′)dt′ =0, (26) as a function of the wall separation L. Beyond a cer-
q qZ
0 taincriticalpoint,theMCTequationsdescribestructural
arrest characterized by nonvanishing glass-form factors,
coinciding with the MCT equation of a 2D liquid [27].
i.e., for confined liquids F (q) := lim S (q,t) 6= 0.
It is interesting to ask what the MCT equations yield µν t→∞ µν
Inexperimentsandsimulations,thesefrozen-inpartsde-
fortheremainingdiagonalcorrelatorsS (q,t)forµ6=0
µµ scribe the plateau values of the intermediate scattering
for small plate separation L. While the equations for
function in the dense or supercooled regime [26, 48, 49].
S (q,t)allowforadirectlimitL=0,the L-dependence
00 We inspect the solutions forthe glass-formfactorsin the
cannottotallybeeliminatedfortheotherquantities. We
limit L → 0. From Eq. (26) one can readily extract
evaluate Eq. (11) again for L = 0, and the contraction
the limit lim lim S (q,t) ≡ F(q), which coin-
with the selectors, Eq. (10), implies for the current ker- t→∞ L→0 00
cides with the glass-formfactors of the planar MCT [27]
nelKˆ (q,z)=−Q˜2L−2v2 /z. The transversalcoherent
µµ µ th in case of structural arrest. Interchanging the limits,
scattering function then becomes
i.e., taking first the limit t → ∞ and then L → 0,
Sˆµµ(q,z)= z−Q˜2−L1−2v2 /z for µ6=0, (27) fliomrmL→fa0cltimorts→d∞iffSerµνq(uqa,lti)tat=ivelliymfLr→om0Fthµνo(sqe)o,btthaeinseedgflraosms-
µ th MCTin2D[27],aswillbedemonstratedinthefollowing.
which is the equation of motion of uncoupled undamped The fixed-point equation for glass-formfactors Fµν(q)
oscillations. Thus for L → 0 and µ 6= 0 the correla- have been derived in Ref. [33] by performing the limit
tors S (q,t) display fast harmonic oscillations with fre- z → 0. Since frozen-in force kernels display a pole at
µµ
quency Ωµ = Q˜µL−1vth := 4πµ/τ(L,T) and are inde- zero frequency, Mαµνβ(q,z) = −(Fαµνβ(q)/z)[1+ o(1)] for
z →0 one obtains
pendent of the planar wave number q. Their basic pe-
riod τ = τ(T,L) = 2L/v is just the time a single par-
th F(q)= S−1(q)+S−1(q)N−1(q)S−1(q) −1, (28)
ticle with thermal velocity v needs for a single bounce
th
(cid:2) (cid:3)
6
with contractions i.e., lim lim S (q,t)6= lim lim S (q,t).
t→∞ L→0 µν L→0 t→∞ µν
This result necessarily implies the existence of an L-
[N−1]µν(q)= bα(q,L−1Q˜µ) dependent diverging time scale τL, on which the lateral
αβX=k,⊥ dynamics couples to the transversal one. We emphasize
×[N−1(q)]αβbβ(q,L−1Q˜ ), (29) that the iteration scheme for the nonergodicity parame-
µν ν ter, see Ref. [33, 45], remains valid for arbitrarily small
and the inverse of the frozen-in part of the MCT func- L>0 and yields always the solutions, where lateral and
tional transversaldegrees of freedom are coupled.
[N−1(q)]αβ :=[J(q)F−1(q)J(q)]αβ. (30)
µν µν
IV. CONVERGENCE TO THE
It has been proventhat these equations exhibit a unique THREE-DIMENSIONAL BULK
maximal solution [33, 45].
The goal here is to find a self-consistent solution for In this subsection, we demonstrate that the MCT for
the glass-form factors F (q) in terms of estimates in confined fluid approaches the standard MCT of the 3D
µν
powers of L. The strategy is to perform a conver- glasstransitionforwallseparationsapproachinfinitypro-
gent iteration suggested in Ref. [33] and to keep track videdthestaticcorrelationsareshort-ranged. Wediscuss
of the respective orders in L. We initialize the iter- first the convergence of the static structure factors and
ation with F (q) = O(L0) for all µ,ν. Then, with the density profile. Then, we use these properties to ex-
µν
the vertices, Eq. (18), one infers for the long-time lim- tract the bulk limit of the MCT for confined liquids.
its lim Mαβ(q,t) = Fαβ(q), that the mode-coupling
t→∞ µν µν
functionals display orders Fkk(q) = O(L0), F⊥k(q) =
µν µν A. Density profile and correlators
O(L), Fk⊥(q) = O(L) and F⊥⊥(q) = O(L2). In con-
µν µν
trast to the frequency-dependent equations of motion,
We assume that the fluid-fluid and fluid-wall interac-
Eq. (11), the limit L = 0 cannot be performed, since
tions are short-ranged and show that the bulk behavior
then the force-kernel matrix F(q) becomes singular and
fortheaveragedensityandthetwo-particlestaticcorrela-
inversion in Eq. (30) is not possible.
tionfunctions is attained asthe wallseparationbecomes
Keepingtheleading-orderestimatesforthekernelsone
large. This limit has to be performed such that the 3D
obtains immediately by Eqs. (28)−(30) the estimates
densityn=n /Lremainsfixed. The spatialdependence
F (q) = O(L0), F (q) = O(L2) (ν 6= 0), F (q) = 0
00 0ν µ0 of the density n(z) arises due to the wall potential U(z)
O(L2) (µ 6= 0) and F (q) = O(L4) (µ,ν 6= 0) as first
µν andthepair-potentialV(|~x−~x′|). Itdisplaysasignificant
iterate. Reinsertingthefirstiterateinthemode-coupling
variation only in the vicinity of the wall, and becomes
functional does not reduce the orders further, as demon-
constant otherwise. Therefore we can write
strated in detail in Appendix. Hence the solutions,
n(z)=n+∆n(z), (32)
F (q)=O(L0), (31)
00
F (q)=O(L2) for µ6=0, where∆n(z)decaysonascaleofawallcorrelationlength
µ0
ξ away from the wall, which can be defined by
F (q)=O(L2) for ν 6=0,
0ν
1
F (q)=O(L4) for µ,ν 6=0, ξ = dz|∆n(z)|. (33)
µν nZ
are the unique solutions characterizing the glass states
Werecallthatnisthebulkdensity. Ingeneralξ depends
in extreme confinement for small but finite L. In fact
on L, but for large wall separation L →∞ it assumes a
the Landausymbols O(·) can be replaced by asymptotic
finite limit depending on temperature and density, only.
proportionality∼(·),i.e.,theorderscannotbeimproved.
Weassumethattheconfinedliquidissufficientlyfaraway
Letusemphasize,thatforallestimateslim F (q),
L→0 µν from a critical point. Then we obtain with Eq. (4) and
the frozen-in parts of the force kernels Fαβ(q) for
µν n0 =nL
α,β =k⊥ are mutually coupled. Thus, the equations for
t → ∞ first do not decouple for small wall separations, n =n [δ +O(ξ/L)], (34)
µ 0 µ0
in striking contrast to the case where the limit L→0 is
since the integral [cf. Eq. (4)] over ∆n(z) is of order ξ.
performed for fixed finite times. In particular, this ob-
Similar considerations apply for the distinct part
servation entails for the glass-formfactor lim F (q),
L→0 00 G(d)(~r,z,z′) of the density-density correlation function
that it does not coincide with the glass-form factors of
G(~r,z,z′) (see Ref. [38] for conventions). We write the
the two-dimensional MCT, Eq. (26); see the discussion
distinctpartasasymptoticexpansionwithrespecttothe
in Appendix below Eq. (A8).
corresponding bulk correlator
In conclusion, we have demonstrated, that the equa-
tions ofmotion displaya delicate dependence in the lim- 1
its L → 0 and t → ∞. Both limits do not commute, G(d)(~r,z,z′)= L G3(dD)(~x)+∆G(d)(~r,z,z′) , (35)
î ó
7
where ~x = (~r,z) and G(d)(~x) denotes the bulk dis- where ~q = ~q−~q ,µ = µ−µ ,ν = ν −ν . The mem-
3D 2 1 2 1 2 1
tinctpartofthedensity-densitycorrelationfunction[50], ory kernel appears to be nondiagonal both in (α,β) and
which displays rotational and translational invariance. (µ,ν). However,asin the two-dimensionallimit we shall
Thecorrections∆G(d)(~r,z,z′)againdecayonawallcor- show that the subspace in which S (q,t) is diagonal,
µν
relation length which we assume to be of same order as i.e.,
ξ.
S (q,t)=S(k,t)δ (41)
The Fourier decomposition 2 of the distinct and the µν µν
corresponding self part [G(s)(~r,z,z′) = n(z)δ(~r)δ(z − remains invariant under the MCT equations.
z′)/n0] follows by Eqs. (34) and (35) Theassumption,Eq.(41),isconsistentwiththediago-
nalityofthe staticstructurefactorS (q) [Eq.(37)], the
µν
Sµ(dν)(q)=S(d)(k)δµν +O(ξ/L), initialcondition for Sµν(q,t). One infers for the memory
S(s)(q)=δ +O(ξ/L). (36) kernel the simplification
µν µν
1
Here we adopt the convention that~k = (~q,Qµ) abbrevi- Mαµνβ(q,t)=2Nn2vt4hX
ates the 3D wave vector, whereas ~q is reserved for two- ~k1
dimensionalvectors,inparticulark =|~k|=(~q2+Q2µ)1/2. × bα ~q·~q1,Q c(k )+(1↔2)
For the structure factor we arrive at q µ1 1
ï Å ã ò
~q·~q
S (q)=S(k)[δ +O(ξ/L)], (37) × bβ 1,Q c(k )+(1↔2)
µν µν q µ1 1
ï Å ã ò
where S(k) is the structure factor of the bulk liquid. ×S(k ,t)S(k ,t)δ (42)
1 2 µν
From the Ornstein-Zernike relation in confined geome-
try [34, 42] one immediately obtains with~k2 =~k−~k1andthecorrespondingwavevectorsread
~k =(~q ,Q ).
i i µi
cµν(q)=L[c(k)δµν +O(ξ/L)], (38) The thermodynamic limit L → ∞,A → ∞,N → ∞
such that N/AL = n restores isotropy in addition to
with the bulk direct correlation function nc(k) = 1 −
homogeneity. Therefore we can choose the direction of
1/S(k)[50]. Thus,thestaticstructurefactorsSµν(q)and the ’external’ wave vector~k =(~q,Q ) in a suitable way.
the direct correlation matrix (c (q)) become diagonal µ
µν Withoutrestrictinggeneralitywechoose~k =(~q =~0,Q ),
for L→∞. µ
which implies for the projections ~qˆ·~q = 0. With k =
i
|~k|=|Q | and Q =~kˆ·~k , the selector simplifies to
µ µi i
B. Mode-coupling theory
bα(~qˆ·~q ,Q )=~kˆ·~k δα⊥. (43)
i µi i
For the bulk limit ofthe MCT equations of motionwe
Thentheonlynonvanishingelementsofthememoryker-
also require the convergence of the static current corre-
lator Jαβ(q). From Eqs. (8) and (34) we obtain nel in the thermodynamic limit become
µν
d3k
Jµανβ(q)=δαβvt2hδµν +O(ξ/L). (39) M⊥µν⊥(q,t)=δµνZ (2π)13V(~k,~k1~k2)S(k1,t)S(k2,t),
(44)
Thus all static correlators become diagonal with respect
with the 3D vertices,
tothediscretemodeindicesµandν inthebulklimitL→
∞. Substituting cµν(q) from Eq. (38) into the memory V(~k,~k ~k )= nv4 (~kˆ·~k )c(k )+(1↔2) 2. (45)
kernel [Eq. (12)] and using Eq. (39) we find 1 2 2 thh 1 1 i
1 Againwemakecontacttoestablishednotationandwrite
Mαµνβ(q,t)=2Nn2vt4hXX M⊥µν⊥(q,t)=δµνΩ2kvt2hm(k,t) with Ω2k =k2vt2h/S(k) and
q~1 µ1ν1 m(k,t) coincides with the MCT kernel for a 3D liq-
× bα ~q·~q1,Q )c(~q ,Q +(1↔2) uid [25].
q µ1 1 µ1 The remaining steps are similar to the 2D case. Re-
ï Å ã ò
× bβ ~q·q~q1,Qν1)c(~q1,Qν1 +(1↔2) ∗ cpolarcreinlagtoinrsEreqd.u(c4e3s)toQµKˆiµbνy(qQ=µ0=,z)k=, tkh2eKˆtµ⊥oν⊥ta(lqc=ur0re,nzt)
ï Å ã ò
[seeEq.(10)]. ByEqs.(39)and(11)itbecomesdiagonal
×S (q ,t)S (q ,t), (40)
µ1ν1 1 µ2ν2 2 Kˆ (q = 0,z) = k2v2 δ in the discrete wave numbers.
µν th µν
We abbreviate Kˆ(k,z) = Kˆ (q = 0,z) = k2Kˆ⊥⊥(q =
µµ µµ
0,z) and arrive at the equation of motion
2 Tdh2erdczondvze′Ant(i~ro,nz,fzo′r)ecxopr(r−eliaQtoµrzs)ienxps(liiQt νgze′o)me−etiq~r·y~r is Aµν(q) = Kˆ(q,z)=−z+Ωk22vmˆt2h(k,z). (46)
R k
8
Diagonalityof S (q) and ofKˆ (q,z) in Eq.(9) implies find that the mode-coupling contributions to the trans-
µν µν
that Sˆ (q = 0,z) = Sˆ(k,z)δ remains diagonal, i.e., verse force fluctuations can be neglected for the descrip-
µν µν
the assumption of Eq. (41) remains consistent. One ob- tionof this rapiddynamics. Itbecomes obviousthat the
tains again a double-fraction representation transversal collective scattering function in this regime
reduces to the incoherent scattering function indicating
Sˆ(k,z)= −S(k) . (47) thatthisdynamicsisamereone-particledynamics. More
z−Ω2/[z+Ω2mˆ(k,z)] generally, we expect that the MCT equations for the in-
k k
coherent dynamics (see Ref. [34]) also converges to its
Finally,theLaplacebacktransfromofEq.(47)yieldsthe
planarandbulkcasedisplayingsimilarfeaturesasfound
3D generalized harmonic oscillator equation
for the collective correlatorsdiscussed here.
t The structural properties and thermodynamic phase
S¨(k,t)+Ω2kS(k,t)+Ω2kZ m(k,t−t′)S˙(k,t′)dt′ =0, behavior of extremely confined fluids can be determined
0
using an effective two-dimensional pair potential, as has
(48)
been shown recently [38, 46]. Hence it would be inter-
whichisthewell-knownMCTequationforglassydynam-
esting to use the two-dimensional MCT with these ef-
ics of a bulk liquid [25]. Since for large wall separations
fective potentials and to compare to the corresponding
we expect S (q,t) = O(ξ/L) for µ 6= ν, we will not
µν
results of the MCT in confinement. Similarly, one could
discuss the MCT equations for these correlators which
comparecomputersimulationsinslitgeometrywithtwo-
describe the dynamics close to the walls. In contrast to
dimensional simulations using the effective pair poten-
the 2Dlimit, the limits t→∞andL→∞docommute,
tial. An approach in the same spirit has been pursued
since the fixed-point equation for the nonergodicity pa-
forcolloid-polymermixtureswheretheeffectiveAsakura-
rameter, Eqs. (28)−(30), converges properly for L →∞
Oosawa depletion interaction has been employed also
to its bulk counterpart [25].
for studying dynamical properties. Similar strategies
have been applied also for asymmetric colloidal mix-
V. SUMMARY AND CONCLUSIONS tures [51, 52] or star polymers [53]. We anticipate that
this works reasonably well also in the present context
although a microscopic justification is lacking.
We have demonstrated that the mode-coupling equa-
tions for confined liquids [32, 33] for finite times con- Let us discuss the limit of small wall separations in
verge to the MCT of the planar and bulk liquid for ef- more detail, since the planar limit displays peculiarities.
fective wall separation L → 0 and L → ∞, respectively. Interestingly, we find that the limits t → ∞ and L → 0
In particular, we have shown how these limiting cases do not commute. If the limit L → 0 is performed first
emerge from the reduction of the matrix-valued theory, for fixed time, then applying the limit t → ∞ leads to
which accounts for the inhomogeneous structure within theglass-formfactorofthecorresponding2Dglassstate.
the slit. In both cases, time is initially finite, while the Takingthereverseorderyieldsaglassstate,whereresid-
limits L → 0 and L → ∞ are performed, respectively. ualmutualcouplingsbetweentheperpendicularandpar-
The recoveryof the correct limits demonstrates the con- allel frozen-in stresses remain. This subtle dependence
sistency and robustness of the MCT ansatz also for con- ontheperformedlimitssuggeststheexistenceofadiver-
fined liquids. gent L-dependent time scale τL, on which the transver-
Several conditions are imposed on the static level to sal ideal-gas-likemotion couples to the lateraldegrees of
guarantee the existence of these limits. For the 3D freedom. This time scale occurs as a consequence of the
limitwehaveassumedshort-rangedparticle-particleand non-commutativity of the limits t → ∞ and L → 0 of
particle-wall interactions and the thermodynamic state theunderlyingMCTequationsinconfinedgeometryand
of the liquid is located sufficiently far away from a crit- provides an interesting prediction for future simulations
ical point. For the 2D limit, we have required that the and experiments.
particle-wallinteractionissuchthatstructuralproperties Let us speculate on the relevance of such a time scale
approach their two-dimensional counterparts, in partic- τ diverging for L → 0 which competes with the struc-
L
ular, the density profile becomes flat sufficiently fast for turalα-relaxationtime. Forinstance,itbecomesconceiv-
L→0. The analyticity of the particle-wallinteractionis able that for sufficiently small L and fine-tuning of the
sufficient for this condition. density, the lateral degrees of freedom are frozen up to
The generalizedmatrix-valuedintermediatescattering timest≪τ andbehaveeffectivelyasatwo-dimensional
L
functionsbecomediagonalinbothlimits. Forfinitetimes glass. At later times the coupling to the transversal de-
in the limit L → 0 we have demonstrated that the 2D greesoffreedomsetsinandyieldseitheradifferentglass
glassy dynamics of the lateral degrees of freedom decou- stateorevenmeltstheglassentirely. Hencethetransver-
ples for L → 0 from the transversal dynamics. While sal fluctuations effectively soften the planar interaction.
the dynamics parallel to the walls coincides with the 2D Such a (partial) melting scenario differs from glass-glass
MCT dynamics, the latter display fast dynamics on a transitions or reentrant transitions, e.g., for attractive
timescaleL/v mimickingideal-gas-likemotioninaone- glasses [52, 54, 55] or binary mixtures [56, 57], since it
th
dimensional box of size L. On this short time scale we is a purely dynamical phenomenon for a single thermo-
9
dynamic state. The emergence of two different regimes in out-of-plane directions yield a generic mechanism to
of the glassy dynamics is based on the assumption that weakly couple to transversaldegrees of freedom.
thetransversaldegreesoffreedomareslowvariableseven
in extreme confinement. Therefore, it would be interest-
ingtotestthispredictionexperimentallyorbycomputer
ACKNOWLEDGMENTS
simulation and verify that two different regimes, sepa-
rated by a divergent time scale τ , indeed behave quali-
L
tatively differently. Potential candidates for experimen- This work has been supported by the Deutsche
tal realizations are superparamagneticcolloidalparticles Forschungsgemeinschaft DFG via the Research Unit
at a liquid surface [31, 58–60] where small fluctuations FOR1394 “Nonlinear Response to Probe Vitrification”.
Appendix A: Glass-form factors in extreme confinement
InthisAppendixweinvestigatetheconvergenceoftheglass-formfactors,i.e.,lim F (q). Wedemonstratehere
L→0 µν
that the estimate F (q)= O(L0), F (q)= O(L2) (ν 6=0), F (q) = O(L2) (µ 6=0) and F (q) = O(L4) (µ,ν 6=0)
00 0ν µ0 µν
yields a consistent solution for the fixed-point equation; see Eqs. (28)−(30). The required ingredient for this analysis
is the asymptotic behavior of the vertices for L→0, Eq. (18). For all α,β, we list here the expanded functionals to
lowest required powers in L [convention ~q =~q−~q ]:
2 1
α=β =k:
v4
Fkk(q)= th (~qˆ·~q )2c(q )2F (q )F (q )
µν 2N Xh 1 1 00 1 µν 2
q~1
+(~qˆ·~q )(~qˆ·~q )c(q )c(q )F (q )F (q )+(1↔2)
1 2 1 2 0ν 1 µ0 2
i
v4
+L2 th (~qˆ·~q )2c(q )c˜∗ (q )F (q )F (q )
2N XXh 1 1 ν−ν2,ν1 1 0ν1 1 µν2 2
q~1 ν1ν2
+(~qˆ·~q )(~qˆ·~q )c(q )c˜∗ (q )F (q )F (q )+(1↔2)
1 2 1 ν−ν1,ν2 2 0ν1 1 µν2 2 i
v4
+L2 th (~qˆ·~q )2c(q )c˜ (q )F (q )F (q )
2N X X h 1 1 µ−µ2,µ1 1 µ10 1 µ2ν 2
q~1 µ1µ2
+(~qˆ·~q )(~qˆ·~q )c(q )c˜ (q )F (q )F (q )+(1↔2) , (A1)
1 2 1 µ−µ1,µ2 2 µ10 1 µ2ν 2
i
α=k,β =⊥:
v4
Fk⊥(q)=L th (~qˆ·~q )c(q )Q˜ c˜∗ (q )F (q )F (q )
µν 2N 1 1 ν−ν2 ν−ν2,ν1 1 0ν1 1 µν2 2
XX
q~1 ν1ν2
v4
+L th (~qˆ·~q )c(q )Q˜ c˜∗ (q )F (q )F (q )
2N 1 1 ν−ν1 ν−ν1,ν2 2 0ν1 1 µν2 2
XX
q~1 ν1ν2
+(1↔2), (A2)
α=⊥,β =k:
v4
F⊥k(q)=L th (~qˆ·~q )c(q )Q˜ c˜ (q )F (q )F (q )
µν 2N 1 1 µ−µ2 µ−µ2,µ1 1 µ10 1 µ2ν 2
X X
q~1 µ1µ2
v4
+L th (~qˆ·~q )c(q )Q˜ c˜ (q )F (q )F (q )
2N 2 2 µ−µ2 µ−µ2,µ1 1 µ1ν 1 µ20 2
X X
q~1 µ1µ2
+(1↔2). (A3)
10
α=β =⊥:
v4
F⊥⊥(q)=L2 th Q˜ Q˜ c˜ (q )c˜∗ (q )F (q )F (q )
µν 2N µ−µ2 ν−ν2 µ−µ2,µ1 1 ν−ν2,ν1 1 µ1ν1 1 µ2ν2 2
Xq~1 µνX11µν22
v4
+L2 th Q˜ Q˜ c˜ (q )c˜∗ (q )F (q )F (q )
2N µ−µ2 ν−ν1 µ−µ2,µ1 1 ν−ν1,ν2 2 µ1ν1 1 µ2ν2 2
Xq~1 µνX11µν22
+(1↔2). (A4)
We insert our estimates for the glass-form factors, Eq. (31), and count the leading orders in L. It turns out that
for estimating the orders, it is sufficient to distinguish between mode index zero (’0’) and a generic non-zero mode
(’¯0’). In this short-hand matrix notation, we find
k0 k¯0 ⊥0 ⊥¯0
k0 O(L0) O(L2) O(L3) O(L1)
Fαβ(q) = k¯0O(L2) O(L4) O(L5) O(L3). (A5)
µν ⊥0 O(L3) O(L5) O(L6) O(L4)
(cid:0) (cid:1) ⊥¯0O(L1) O(L3) O(L4) O(L2)
 
The inversion of this matrix can be done by introducing block matrices
k ⊥
k A B
Fαβ(q) = , (A6)
µν ⊥ C D
(cid:0) (cid:1) ï ò
e.g.,
0 ¯0
0 O(L0) O(L2)
A≡(A )= , (A7)
µν ¯0 O(L2) O(L4)
ï ò
etc. Block-matrix inversion yields
k ⊥
k (A−BD−1C)−1 −A−1B(D−CA−1B)−1
[F−1(q)]αβ = . (A8)
µν ⊥ −D−1C(A−BD−1C)−1 (D−CA−1B)−1
(cid:0) (cid:1) ï ò
An important insight is, that the estimates of each entry, e.g., compare A and BD−1C, are of the same order in
L. The consequence is, that each entry of [F−1(q)]αβ contains contributions from all force kernels Fαβ(q) with the
µν µν
various combinations of α and β. In partic(cid:0)ular, we em(cid:1)phasize that for the entry (α=β =k;µ=ν =0) the scalar
(BD−1C) = Fk⊥[(F⊥⊥)−1] F⊥k =O(L0), (A9)
00 0κ κγ γ0
X
κγ
is ofthe same orderas A =Fkk, the latter correspondingto the arrested2DMCT kernel. This in turn implies that
00 00
A −(BD−1C) does not coincide with the expected 2D result:
00 00
A −(BD−1C) 6=Fkk. (A10)
00 00 00
Explicit inversionleads to
[F−1(q)]αβ (A11)
µν
(cid:0) k(cid:1)0 k¯0 ⊥0 ⊥¯0
k0 O(L0) O(L−2) O(L−3) O(L−1)
k¯0O(L−2) O(L−4) O(L−5) O(L−3)
= .
⊥0O(L−3) O(L−5) O(L−6) O(L−4)
 
⊥¯0O(L−1) O(L−3) O(L−4) O(L−2)
 