Anomalous structural and mechanical properties of solids confined in quasi-one-dimensional strips

arX

iv:0

705.

3135

v2 [

cond

-mat

.sof

t] 1

Apr

200

8

Anomalous structural and mechanical properties of solids confined in quasi one

dimensional strips

Debasish Chaudhuri1 and Surajit Sengupta2

1 Max Planck Institute for the Physics of Complex Systems,

Nothnitzer Strasse 38, 01187 Dresden, Germany ∗

2 S. N. Bose National Center for Basic Sciences,

JD Block, Sector 3, Salt Lake, Kolkata - 700098, India †

(Dated: November 20, 2013)

We show using computer simulations and mean field theory that a system of particles in twodimensions, when confined laterally by a pair of parallel hard walls within a quasi one dimensionalchannel, possesses several anomalous structural and mechanical properties not observed in the bulk.Depending on the density ρ and the distance between the walls Ly , the system shows structuralcharacteristics analogous to a weakly modulated liquid, a strongly modulated smectic, a triangularsolid or a buckled phase. At fixed ρ, a change in Ly leads to many reentrant discontinuous transitionsinvolving changes in the number of layers parallel to the confining walls depending crucially on thecommensurability of inter-layer spacing with Ly. The solid shows resistance to elongation butnot to shear. When strained beyond the elastic limit it fails undergoing plastic deformation butsurprisingly, as the strain is reversed, the material recovers completely and returns to its originalundeformed state. We obtain the phase diagram from mean field theory and finite size simulationsand discuss the effect of fluctuations.

PACS numbers: 62.25.+g, 64.60.Cn, 64.70.Rh, 61.30.v

I. INTRODUCTION

Recent studies on various confined systems [1, 2, 3, 4,5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] have shownthe possibility of obtaining many different structures andphases depending on the range of interactions and com-mensurability of a microscopic length scale of the sys-tem with the length scale of confinement. Since internalstructures often play a crucial role in determining localdynamical properties like asymmetric diffusion, viscosityetc. and phase behavior [5], such studies are of consider-able practical interest. Charged particles interacting viaa screened Coulomb potential when confined in a one di-mensional parabolic well showed many zero temperaturelayering transitions [3]. At high temperatures this clas-sical Wigner crystal melts and the melting temperatureshows oscillations as a function of density [3]. Such os-cillations are characteristic of confined systems, arisingout of commensurability. In Ref. [6, 7] a similar layer-ing transition is found in which the number of layers ina confined liquid with smectic modulations changes dis-cretely as the wall-to-wall separation is increased. Theforce between the walls oscillates as a function of thisseparation depending crucially on the ratio of the sepa-ration to the thickness of the smectic layers. Confinedcrystals always align one of the lattice planes along thedirection of confinement [3, 4] and confining walls gen-erate elongational asymmetry in the local density pro-file along the walls even for the slightest incommensu-

∗Electronic address: [email protected]†Electronic address: [email protected]

ration [8]. For long ranged interactions, extreme local-ization of wall particles has been observed [4, 10]. Hardspheres confined within a two dimensional slit boundedby two parallel hard plates show a rich phase behav-ior [1, 2, 19, 20, 21, 22]. For separations of one to fivehard sphere diameters a phase diagram consisting of adazzling array of up to 26 distinct crystal structures hasbeen obtained [2, 19]. Similarly confined systems withmore general interactions also show a similar array ofphases[16]. In an earlier experiment on confined steelballs in quasi one dimensions (Q1D), i.e. a two dimen-sional system of particles confined in a narrow channelby parallel walls (lines), vibrated to simulate the effect oftemperature, layering transitions, phase coexistence andmelting was observed [15]. A recent study on trappedQ1D solid in contact of its own liquid showed layeringtransitions with increase in trapping potential[17]. Lay-ering fluctuations in such a solid has crucial impact onthe sound and heat transport across the system[23, 24].

In this paper, we show that a Q1D solid strip of lengthLx confined within parallel, hard, one dimensional wallsseparated by a distance Ly has rather anomalous proper-ties. These are quite different from bulk systems in one,two or three dimensions as well as from Q1D solid stripswith periodic boundary conditions (PBCs). We list be-low the main characteristics of a Q1D confined strip thatwe demonstrate and discuss in this paper:

1. Re-entrant layer transitions: The nature of the sys-tem in Q1D depends crucially on the density ρ andwidth of the channel Ly. The number of layersof particles depends on the ratio of the interlayerspacing (fixed mainly by ρ) and Ly. With increasein channel width Ly at a fixed ρ we find many re-entrant discontinuous transitions involving changes

http://arxiv.org/abs/0705.3135v2

mailto:[email protected]

mailto:[email protected]

2

in the number of layers parallel to the confining di-rection. The phase diagram in the ρ − Ly plane iscalculated from Monte Carlo simulations of systemswith finite size as well as mean field theory (MFT).While all the phases show density modulations inthe direction perpendicular to the wall, we identifydistinct analouges of the bulk phases viz. modu-lated liquid, smectic, triangular solid and buckledsolid.

2. Anomalous elastic moduli: A solid characterizedby a periodic arrangement of particles offers resis-tance to elongation as well as shear. The Q1D con-fined solid is shown to have a large Young’s modu-lus which offers resistance to tensile deformations.On the other hand the shear modulus of the sys-tem is vanishingly small so that layers of the solidparallel to the confinining wall may slide past eachother without resistance.

3. Reversible failure: Under externally imposed ten-sile strain the deviatoric stress shows an initial lin-ear rise up to a limiting value which depends on Ly.On further extension the stress rapidly falls to zeroaccompanied by a reduction in the number of solidlayers parallel to the hard walls by one. However,this failure is reversible and the system completelyrecovers the initial structure once the strain is re-

duced quite unlike a similar solid strip in presence ofPBCs in both the directions. The critical strain forfailure by this mechanism decreases with increas-ing Ly so that thinner strips are more resistant tofailure. We show that this reversibility is relatedto the anomalies mentioned above. Namely, theconfined solid, though possessing local crystallineorder, retains the ability to flow and regenerate it-self. In this manner portions of the Q1D confinedsolid behave like coalescing liquid droplets. A pre-liminary study of this reversible failure mechanismwas reported in Ref.[8].

4. Displacement fluctuation and solid order: Longwavelength displacement fluctuations in Q1D areexpected to destabilize crystalline order beyond acertain length scale. While we always observe thispredicted growth of fuctuations, in the case of con-fined system, the amplitude depends crucially onthe wall separation. If Ly is incommensurate withthe interlayer spacing, then local crystalline orderis destabilized. Otherwise, fluctuations are kinet-ically suppressed in the confined system at highdensities. Finite size effects also tend to saturatethe growth of fluctuations. Solid strips of finitelength therefore exhibit apparent crystalline orderat high densities both in simulations as well as inexperiments[3].

We have used an idealized model solid to illustratethese phenomena. Our model solid has particles (disks)which interact among themselves only through excluded

volume or “hard” repulsion. We have reasons to believe,however, that for the questions dealt with in this pa-per, the detailed nature of the inter particle interactionsare relatively irrelevant and system behavior is largelydetermined by the nature of confinement and the con-straints. Our results may be directly verified in experi-ments on sterically stabilized “hard sphere” colloids[25]confined in glass channels and may also be relevantfor similarly confined atomic systems interacting withmore complex potentials. Our results should hold, atleast qualitatively, for systems with fairly steep repulsiveinteractions[24, 36, 37].

This paper is organized as follows. In the next sec-tion, we introduce the model confined solid and discussthe geometry and basic definitions of various structuraland thermodynamic parameters. We then introduce thevarious possible structures with their basic characteris-tics in section III. In section IV, this will be followed bythe results of computer simulations, in the constant NAT(number, area, temperature) ensemble, exploring the de-formation and failure properties of this system and therelation of the various structures described in section IIIto one another. In section V, we provide a finite sizephase diagram obtained from simulations and compareit with an MFT calculation. In section VI we discussour results with emphasis on the role of long wave lengthfluctuations in the destruction of crystalline order in lowdimensions and conclude giving some outlook for futurework.

II. THE MODEL AND METHOD

The bulk system of hard disks where particles i and j,in two dimensions, interact with the potential Vij = 0 for|rij | > d and Vij = ∞ for |rij | ≤ d, with d the hard diskdiameter and rij = rj − ri the relative position vectorof the particles, has been extensively[28, 29, 30, 31, 32]studied. Apart from being easily accessible to theo-retical treatment[33], experimental systems with nearly“hard” interactions[25] are available. The hard disk freeenergy is entirely entropic in origin and the only ther-modynamically relevant variable is the number densityρ = N/A or the packing fraction η = (π/4)ρd2. Accu-rate computer simulations[31] of hard disks show that forη > ηf = 0.719 the system exists as a triangular latticewhich melts below ηm = 0.706. The melting occurs pos-sibly through a two step continuous transition from solidto liquid via an intervening hexatic phase[31, 32]. Elas-tic constants of bulk hard disks have been calculated insimulations[32, 34]. The surface free energy of the harddisk system in contact with a hard wall has also beenobtained[35] taking care that the dimensions of the sys-tem are compatible with a strain-free triangular lattice.

The wall- particle interaction potential Vwall(y) = 0 ford/2 < y < Ly −d/2 and = ∞ otherwise. Here, evidently,Ly is the width of the channel. The length of the channelis Lx with Lx ≫ Ly. Periodic boundary conditions are

3

G1

G2

G3

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

a

a

G

G2

1

x

y

FIG. 1: The confined solid is shown along with the centeredrectangular (CR) unit cell. For an unstrained triangular lat-tice ax = a0 and ay =

√3a0. G1, G2 and G3 denote the

directions of the three reciprocal lattice vectors (RLV). Notethat the third reciprocal lattice direction G3 is equivalent toG2, even in the presence of the walls.

assumed in the x direction(Fig.1).Before we go on to describe the various phases we ob-

serve in this model system, it is instructive to considerhow a triangular lattice (the ground state configuration)may be accomodated within a pair of straight hard walls.For the channel to accommodate nl layers of a homoge-neous, triangular lattice with lattice parameter a0 of harddisks of diameter d, (Fig.6) it is required that,

Ly =

√3

2(nl − 1)a0 + d . (1)

For a system of constant number of particles and Ly,a0 is a function of packing fraction η alone. We defineχ(η, Ly) = 1 + 2(Ly − d)/

√3a0, so that the above con-

dition reads χ = integer = nl (the commensurate con-figuration) and violation of Eq.(1) implies a rectangularstrain away from the reference triangular lattice of nl

layers. The lattice parameters of a centered rectangular(CR) unit cell are ax and ay (Fig. 1). In general, fora CR lattice with a given Ly, ay = 2(Ly − d)/(nl − 1)and ax = 2/ρay, ignoring vacancies. There are two dis-tinct classes of close packed planes in the CR lattice.Due to the presence of confinement, even for a triangularlattice, the set of planes with reciprocal lattice vector(RLV) G1 = y 4π

ayperpendicular to the walls are dis-

tinct from the equivalent set of planes with the RLV’sG2 = x4π

aycos(π

6) + y 4π

aysin(π

6) and G3 = x 4π

aycos(π

6) −

y 4πay

sin(π6) (Fig.1).

Anticipating some of the discussion in section IV, wepoint out two different but equivalent “pictures” forstudying the deformation behavior of confined narrowcrystalline strips. In the first picture, the stress is re-garded as a function of the “external” strain. Using theinitial triangular solid (packing fraction η0) as reference,the external strain associated with changing Lx at a con-stant N and Ly is ǫ = (Lx − L0

x)/L0x. In this case we

12

FIG. 2: (Colour online) Solid: Left panel shows a picture of103 uncorrelated, superimposed configurations of a high den-sity (η = 0.85) solid phase. The wall to wall separation iscommensurate with a ten layered solid at the given density.The colour code is such that red (light) means high local den-sity and blue (dark) means low density. The right panel showsthe corresponding structure factor which shows a pattern typ-ical for a two dimensional triangular solid.

obtain the stress as an oscillating non-monotonic func-tion of ǫ. On the other hand, internally, the solid is, freeto adjust nl to decrease its energy (strain). Therefore,one may, equivalently, calculate strains with respect to areference, distortion-free, triangular lattice at η. Usingthe definition εd = εxx − εyy = (ax − a0)/a0 − (ay −√

3a0)/√

3a0 = ax/a0 − ay/√

3a0 and the expressions for

ax, ay and a0 = 2(Ly − d)/√

3(χ − 1) we obtain,

εd =nl − 1

χ − 1− χ − 1

nl − 1, (2)

where the number of layers nl is the nearest integer to χso that εd has a discontinuity at half -integral values ofχ. For large Ly this discontinuity and εd itself vanishesas 1/Ly for all η. This “internal” strain εd is relatednon-linearly to ǫ and may remain small even if ǫ is large.The stress is always a monotonic function of εd. Notethat a pair of variables η and Ly (or alternately ǫ and χ)uniquely fixes the state of the system.

We have carried out extensive Monte Carlo (MC) sim-ulations in the constant NAT ensemble using standardMetropolis updates for hard disks (i.e. moves are re-jected if they lead to overlaps). The initial conditionfor our simulations is the perfectly commensurate trian-gular lattice with a fixed number of layers nl and at afixed packing fraction η. After equilibration for typicallyabout 106 Monte Carlo steps (MCS), a strain ǫ is im-posed by rescaling Lx. Since the walls are kept fixed,this strain reduces η. The sequence of phases and thenature of the transitions among them induced by thisstrain is discussed below.

III. STRUCTURES AND PHASES

In Q1D long wavelength fluctuations[37] are expectedto destroy all possible order except for those imposed

4

FIG. 3: (Colour online) Buckled phase: A small incommen-suration is introduced by increasing the wall to wall sepa-ration from the value commensurate to a high density tenlayered triangular solid at η = 0.89. This reduces the pack-ing fraction to η = 0.85 and produces this phase. The up-per panel shows a picture of 103 superimposed configurations.The colour code for the local densities is the same as Fig.2.Note that different portions of triangular solid are displacedalong y- direction to span the extra space introduced betweenthe walls. Lines are drawn to identify this shift in triangularregions. The lower panel shows the corresponding structurefactor where the peak in G1 direction is diminished. Someextra weak peaks corresponding to super-lattice reflectionsappear at lower values of the wave-number.

by explicit (as opposed to spontaneous) breaking of sym-metry. The confining potential, in the case of a Q1Dsystem in a hard wall channel, explicitly breaks thecontinuous rotational symmetry down to a Z2 symme-try (rotations by angle ±π). This immediately leadsto all 2n-atic bond orientational orders, like nematic(n = 1), square (n = 2), hexatic (n = 3) etc. whichremain nonzero throughout the phase diagram. This sit-uation is therefore similar to a system in a periodic laserpotential[38, 39, 40]. Apart from orientational symme-try, the confining potential explicitly breaks translationalsymmetry perpendicular to the walls leading to a den-sity modulation in that direction. Any fluctuation whichleads to global changes in the direction of layering orin the layer spacing for fixed wall to wall separation isstrongly suppressed.

For finite systems (and over finite - but very long -times) of confined Q1D strips one obtains long-lived,metastable ordered phases which are observable even inexperiments[15]. It is these ‘phases’ that we describe inthis section. For convenience in nomenclature, we con-tinue to use the terms solid, modulated liquid and smec-tic to denote these phases, though keeping in mind thatthe distinctions are largely quantitative and not quali-tative. For example, a weak solid-like local hexagonalmodulation is present on top of the smectic layering or-

der in what we call the smectic phase. The smectic de-velops continuously and smoothly from the modulatedliquid. We also denote the sharp changes in observables(eg. strength of diffraction peaks) as “phase transitions”though they can never be classified as true phase tran-sitions in the sense of equilibrium thermodynamics. Weshow, however, that these phase transitions may be de-scribed within MFTs (section V) although they mightloose meaning when fluctuations are considered in thelimit of infinite observation time.

In our description of the phases we make use of thestructure factor

SG =

⟨

1

N2

N∑

j,k=1

exp(−iG.rjk)

⟩

,

where rij = rj − ri with ri the position vector of par-ticle i. We shall use particularly the values of SG forthe reciprocal lattice vectors G = ±G1(η) and ±G2(η).Notice that ±G2(η) and ±G3(η) as shown in Fig.1 areequivalent directions. A plot of SG for G = (Gx, Gy) inthe two dimensional plane gives the diffraction patternobservable in scattering experiments. For every phasediscussed below, the diffraction pattern always shows atleast two peaks corresponding to ±G1 since they repre-sent the density modulation imposed by the walls. Therelative strengths of these peaks, of course, depend onthe structure of the phase concerned.

If the separation between the hard walls is kept com-mensurate such that χ = nl, an integer, at high den-sity we obtain a perfect two dimensional triangular solid

(Fig.2). The solid shows a diffraction pattern which istypical for a two dimensional triangular crystal. We showlater that appearances can be deceptive, however. Thistriangular “solid” is shown to have zero shear moduluswhich would mean that it can flow without resistancealong the length of the channel like a liquid. Stretch-ing the solid strip lengthwise, on the other hand, costsenergy and is resisted. The strength of the diffractionpeaks decreases rapidly with the order of the diffraction.In strictly two dimensions this is governed by a non-universal exponent dependent on the elastic constants[41]. In Q1D this decay should be faster. However, largersystem sizes and averaging over a large number of con-figurations would be required to observe this decay, sinceconstraints placed by the hard walls make the systemslow to equilibrate at high densities. For a general valueof χ the lattice is strained which shows up in the rel-ative intensities of the peaks in the diffraction patterncorresponding to G2 and G3.

A little extra space introduced between the walls start-ing from a high density solid phase gives rise to bucklinginstability in y- direction and the system breaks into sev-eral triangular solid regions along the x- direction (Fig.3).Each of these regions slide along close packed planes (cor-responding to the reciprocal directions G2 or G3) withrespect to one another. This produces a buckling wavewith displacement in y- direction travelling along the

5

0 5 10y/d

0

1

2

3

4

η(y)

FIG. 4: (Colour online) Smectic: A confined triangular solidat η = 0.85 is strained in x-direction to a packing fraction η =0.73. The upper panel shows a picture of 103 superimposedconfigurations. The colour code for local density is same asFig.2. The lower left panel shows the density modulation iny- direction. The lower right panel shows the correspondingstructure factor where the only remaining strong peaks arein G1 direction identifying a smectic order (solid-like in y-direction and liquid-like in other directions). This smecticphase, however, possesses hexagonal modulations producingweak triangular order leading to faint secondary spots in thestructure factor.

length of the solid. In conformity with the quasi two di-mensional analog [1, 21, 22] we call this the buckled solid

and it interpolates continuously from χ = nl to nl ± 1layers. This phase can also occur due to the introductionof a compressional strain in x-direction keeping Ly fixed.The diffraction pattern shows a considerable weakeningof the spots corresponding to planes parallel to the wallsSG1

together with generation of extra spots at smallerwave-number corresponding to the buckled super-lattice.The diffraction pattern is therefore almost complemen-tary to that of the smectic phase to be discussed below.We do not observe the buckled solid at low densities closeto the freezing transition. Large fluctuations at such den-sities lead to creation of bands of the smectic phase withina solid eventually causing the solid to melt.

At low enough densities or high enough in-commensuration (χ half integral) the elongated densityprofiles in the lattice planes parallel to the walls can over-lap to give rise to what we denote as the smectic phase(Fig.4) in which local density peaks are smeared out in x-direction but are clearly separated in y-direction givingrise to a layered structure. The diffraction pattern showsonly two strong spots (SG1

) which is typical correspond-ing to the symmetry of a smectic phase. We use this fact

0 5 10x/d

0

0.5

1

1.5

2

2.5

η(y)

FIG. 5: (Colour online) Modulated liquid: A confined trian-gular solid at η = 0.85 is strained in x-directon reducing thedensity to η = 0.6. The upper panel shows a picture of 103 su-perimposed configurations. The colour code of local densitiesis the same as Fig.2. The lower left panel shows the densitymodulation in y- direction which is like the smectic phase butthe modulation dies out at the center. The structure factorin the lower right panel shows a ring like pattern which is atypical signature of liquid. Superimposed on this are strongsmectic-like peaks in G1 direction.

as the defining principle for this phase. Note that unlikeusual smectics there is no orientational ordering of direc-tor vectors of the individual particles since hard disks areisotropic.

At further lower densities the relative displacementfluctuations between neighbors diverges and the diffrac-tion pattern shows a ring- like feature typical of a liquidwhich appears together with the smectic like peaks in thedirection perpendicular to the walls. We monitor this us-ing the relative Lindemann parameter[55], given by,

l = 〈(uxi − ux

j)2〉/a2

x + 〈(uyi − uy

j)2〉/a2

y (3)

where the angular brackets denote averages over config-urations, i and j are nearest neighbors and uα

i is the α-th component of the displacement of particle i from it’smean position. This phase is a modulated liquid (Fig.5).The density modulation decays away from the walls andfor large Ly, the density profile in the middle of the chan-nel becomes uniform. A clear structural difference be-tween smectic and modulated liquid is the presence ofthe ring pattern in the structure factor of the latter, acharacteristic of liquid phase (compare Fig.4 and 5). Wemust emphasize here again that the distinction betweena modulated liquid and a smectic is mainly a questionof degree of the layering modulation. The two structuresmerge continuously into one another as the density is in-creased. Also when the modulated liquid co-exists with

6

Gi

ρ

0

0.5

1

1.5

2

2.5

3

0.69 0.73 0.77 0.81 0.85

3

3

3

3

3

3

3

3

3

3

3

3

3

333

33

33

3

3

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

22

2222

22

2

2

2

2

2

2

2

2

2

2

η

l,

FIG. 6: Results of NAT ensemble MC simulations of N =nx × ny = 65 × 10 hard disks confined between two parallelhard walls separated by a distance Ly = 9 d. For each η, thesystem was allowed to run for 106 MCS and data averagedover a further 106 MCS. At η = 0.85 we have a strain freetriangular lattice. Plots show the structure factors SGi

, i =1(+), 2(⋄) for RLVs Gi(η), averaged over symmetry relateddirections, as a function of η. Also plotted in the same graphis the Lindemann parameter l(✷). The lines in the figure area guide to the eye.

the solid, the layering in the former is particularly strongdue to proximity effects.

IV. MECHANICAL PROPERTIES AND

FAILURE

In this section we present the results for the mechan-ical behavior and failure of the Q1D confined solid un-der tension. We know that fracture in bulk solids oc-curs by the nucleation and growth of cracks[42, 43, 44,45]. The interaction of dislocations or zones of plasticdeformation[45, 46] with the growing crack tip deter-mines the failure mechanism viz. either ductile or brittlefracture. Studies of the fracture of single-walled carbonnanotubes[47, 48] show failure driven by bond-breakingwhich produces nano cracks which run along the tube cir-cumference leading to brittle fracture. Thin nano-wiresof Ni are known[49, 50] to show ductile failure with ex-tensive plastic flow and amorphization. We show thatthe Q1D confined solid behaves anomalously, quite un-like any of the mechanisms mentioned above. It showsreversible plastic deformation and failure in the constantextension ensemble. The failure occurs by the nucleationand growth of smectic regions which occur as distinctbands spanning the width of the solid strip.

We study the effects of strain on the hard disk trian-gular solid for a Ly large enough to accommodate onlya small number of layers nl ∼ 9 − 25. The Lindemannparameter l diverges at the melting transition [51]. Wealso compute the quantities SG=G1

and SG=G2.

In Fig.6 we show how SG2, SG1

and l vary as a func-tion of externally imposed elongational strain that re-

duces η. Throughout, SG2< SG1

6= 0. This is a conse-quence of the hard wall constraint[35] which induces anoblate anisotropy in the local density peaks of the solid offfrom commensuration (nonintegral χ). As η is decreasedboth SG1

and SG2show a sharp drop at η = ηc1

whereχ = χ∗ ≈ nl − 1/2 [Fig. 7 (inset) ]. For η < ηc1

we getSG2

= 0 with SG16= 0 signifying a transition from crys-

talline to smectic like order. The Lindemann parameterl remains zero and diverges only below η = ηc3

(≈ ηm) in-dicating a finite-size- broadened “melting” of the smecticto a modulated liquid phase.

To understand the mechanical response of the con-fined strips, we compute the deviatoric stress built upin the system as a function of applied strain. The stresstensor of a bulk system interacting via two body cen-tral potential has two parts: (i) A kinetic part σK

λνleading to an isotropic pressure and (ii) A virial termdue to inter-particle interaction σint

λν . The free-particle-like kinetic component of the stress βσK

λν = −ρδλν andthe component due to inter-particle interaction σint

λν =−〈∑<ij> rλ

ijfνij〉/S with fν

ij the ν-th component of inter-particle force, S = LxLy area of the strip. The expres-sion for the stress tensor for the bulk system of hard diskstranslates to[52]

βσλνd2 = −d2

S

∑

<ij>

⟨

δ(rij − d)rλijr

νij

rij

⟩

+ Nδλν

.(4)

The presence of walls gives rise to a potential whichvaries only in the y-direction perpendicular to the walls.Therefore, strains ǫxx and ǫxy do not lead to any changein the wall induced potential. As a consequence theconjugate stresses for the confined system σC

xx = σxx

and σCxy = σxy. However, a strain ǫyy does lead to a

change in potential due to the walls and therefore a newterm in the expression for conjugate stress appears[53],σC

yy = σyy + σw with σw = −〈∑<iw> fwi yiw〉/S where w

denotes the two confining walls. This expression can beeasily understood by regarding the two walls as two addi-tional particles of infinite mass[54]. Thus, to obtain thecomponent of the total stress normal to the walls fromMC simulations we use

βσCyyd2 = βσyyd2 − d2

S

[⟨

∑

i

yiδ(yi − d/2)

⟩

+

⟨

∑

i

(Ly − yi)δ(Ly − yi − d/2)

⟩]

. (5)

The deviatoric stress, σC = σCxx − σC

yy, versus strain,ǫ = ǫxx − ǫyy (ǫyy = 0) curve for confined hard disks isshown in Fig. 7. For η = η0 (ǫ = 0) the stress due to theinter-particle interaction is purely hydrostatic with σxx =σyy as expected; however, due to the excess pressure fromthe walls the solid is actually always under compressionalong the y direction, thus σC

xx > σCyy. At this point

the system is perfectly commensurate with channel widthand the local density profiles are circularly symmetric.

7

0 0.1 0.2 0.3 0.4ε

-5

0

5

10

15

20β

σC d

2

IncreaseDecrease

2

2

2

2

2

2

Cr Sm+Cr

Sm

χ

n l

*

FIG. 7: (Colour online) A plot of the deviatoric stress σC

versus external strain ǫ obtained from our MC simulations of65 × 10 hard disks initially at η = 0.85 and Ly = 9d. Datais obtained by holding at each strain value for 2 × 104 MCSand averaging over a further 3 × 104 MCS. The entire cycleof increasing ǫ(◦) and decreasing to zero (✷) using typicalparameters appropriate for an atomic system, corresponds toa real frequency of ω ≈ 25K Hz. Results do not essentiallychange for ω = 1K Hz − 1M Hz. The line is a guide to theeye. The vertical lines mark the limit of stability of the crystal(Cr), the two phase region (Cr+Sm) and the onset of thesmectic phase (Sm). Inset shows the variation of the criticalχ∗ with nl, points: simulation data; line: χ∗ = nl − 1/2.

Initially, the stress increases linearly, flattening out at

the onset of plastic behavior at η<∼ ηc1

. At ηc1, with the

nucleation of smectic bands, σC decreases and eventu-ally becomes negative. At ηc2

the smectic phase spansthe entire system and σC is minimum. On further de-crease in η towards ηc3

, σC approaches 0 from below(Fig. 7) thus forming a loop. Eventually it shows a smallovershoot, which ultimately goes to zero, from above, asthe smectic smoothly goes over to a more symmetric liq-uid like phase – thus recovering the Pascal’s law at lowenough densities. If the strain is reversed by increasingη back to η0 the entire stress-strain curve is traced backwith no remnant stress at η = η0 showing that the plas-tic region is reversible. For the system shown in Fig.1 weobtained ηc1

≈ 0.77, ηc2≈ 0.74 and ηc3

≈ 0.7. As Ly is

increased, ηc1merges with ηc3

for nl>∼ 25. If instead, Lx

and Ly are both rescaled to keep χ = nl fixed or PBCsare imposed in both x and y directions, the transitionsin the various quantities occur approximately simultane-ously as expected in the bulk system. Varying nx in therange 50 − 5000 produces no qualitative change in mostof the results.

Reversible plasticity in the confined narrow strip is instark contrast with the mechanical response of a simi-lar strip in the absence of confinement. In order to showthis, we study a similar narrow strip of 65×10 hard disksbut now we use PBCs in both the directions. At pack-

0 0.05 0.1 0.15 0.2ε

-20

-10

0

10

20

30

β σ

d2

IncreaseDecrease

FIG. 8: (Colour online) A plot of the deviatoric stress σversus external strain ǫ obtained from our MC simulations of65×10 hard disks, in presence of PBCs in both the directions,initially a triangular lattice at η = 0.85. Data is obtained byholding at each strain value for 2 × 104 MCS and averagingover a further 3×104 MCS. The entire cycle of increasing ǫ(◦)and decreasing to zero (✷) using typical parameters appropri-ate for an atomic system, corresponds to a real frequency ofω ≈ 100K Hz. The line through the points is guide to the eye.

ing fraction η = 0.85 the initial geometry (L0x = 65ax,

Ly = 10ay with ay =√

3ax/2) contains a perfect tri-angular lattice. We impose strain in a fashion similarto that described in Fig.7. The resulting stress-straincurve is shown in Fig.8. With increase in strain ǫ thesystem first shows a linear (Hookean) response in the de-viatoric stress σ = σxx − σyy, flattening out at the onsetof plastic deformation below ǫ ∼ 0.1. Near ǫ = 0.1 witha solid-liquid transition (solid order parameter drops tozero with divergence of Lindeman parameter at the samestrain value) the deviatoric stress σ decreases sharply tozero in the liquid phase obeying the Pascal’s law. Unlikein the confined strip, with further increase in strain σdoes not become negative and fluctuates around zero inabsence of wall induced density modulations. With de-crease in strain, starting from the liquid phase, the stressσ does not trace back its path. Instead first a large nega-tive stress is built up in the system as we decrease strainup to ǫ ∼ 0.04. With further decrease in ǫ, the stressσ starts to increase and at ǫ = 0 the system is under ahuge residual stress βσd2 = 25. The configuration of thestrip at this point shows a solid with lattice planes ro-tated with respect to the initial stress-free lattice. Thissolid contains defects. Note that in the presence of hardconfining walls, global rotation of the lattice planes costlarge amounts of energy and would be completely sup-pressed. The generation of defects is also difficult in aconfined system unless they have certain special charac-teristics which we describe later.

For a confined Q1D strip in the density range ηc2<

η < ηc1we observe that the smectic order appears within

narrow bands (Fig. 9). Inside these bands the number of

8

FIG. 9: (Colour online) Plot of χ(x, t) as a function of thechannel length x/d at η = 0.76 after time t = (a)5 × 105 and(b)2×106 MCS for N = 103×10 with Ly = 9d. Note that χ =10 in the solid and χ = 9 in the smectic regions. Arrows showthe coalescence of two smectic bands as a function of time.Calculated diffraction patterns for the (c) 10-layered solid and(d) 9-layered smectic regions. (e) Close up view of a crystal-smectic interface from 103 superimposed configurations at η =0.77. The colour code of the local density is the same asFig.2. Note the misfit dislocation in the inter-facial region.A Burger’s circuit is shown to identify the Burger’s vector

corresponding to this dislocation as ~b = yay/2 + xax/2.

layers is less by one and the system in this range of η isin a mixed phase. A plot (Fig.9 (a) and (b)) of χ(x, t),where we treat χ as a space and time (MCS) depen-dent “order parameter” (configuration averaged numberof layers over a window in x and t), shows bands in whichχ is less by one compared to the crystalline regions. Afternucleation, narrow bands coalesce to form wider bandsover very large time scales. The total size of such bandsgrow as η is decreased. Calculated diffraction patterns(Fig. 9 (c) and (d)) show that, locally, within a smecticband SG1

≫ SG2in contrast to the solid region where

SG1≈ SG2

6= 0.Noting that χ = χ∗ = nl − 1/2 when the solid fails

(Fig. 7 inset), it follows from Eq. 2, the critical strainε∗d = (4nl − 5)/(2nl − 3)(2nl − 2) ∼ 1/nl. This is sup-ported by our simulation data over the range 9 < nl < 14.This shows that thinner strips (smaller nl) are more re-sistant to failure. At these strains the solid generatesbands consisting of regions with one less particle layer.Within these bands adjacent local density peaks of theparticles overlap in the x direction producing a smec-tic. Within a simple density functional argument[59]

it can be shown that the spread of local density profilealong x-axis αx ∼ 1/

√

SG2and that along y-direction is

αy ∼ 1/√

2SG1+ SG2

[61]. In the limit SG2→ 0 (melt-

ing of solid) αx diverges though αy remains finite as SG1

remains positive definite. Thus the resulting structurehas a smectic symmetry.

A superposition of many particle positions near a solid-smectic interface [see Fig. 9(e)] shows that: (1) Thewidth of the interface is large, spanning about 10 − 15particle spacings. (2) The interface between nl layeredcrystal and nl − 1 layered smectic contains a dislocationwith Burger’s vector in the y- direction which makes upfor the difference in the number of layers. Each bandof width s is therefore held in place by a dislocation-anti-dislocation pair (Fig. 9). In analogy with classicalnucleation theory[55, 56], the free energy Fb of a singleband can be written as

Fb = −δFs + Ec +1

8πb2K∆ log

s

a0

(6)

where K∆ is an elastic constant, ~b = yay/2 + xax/2 isthe Burger’s vector, δF the free energy difference be-tween the crystal and the smectic per unit length and Ec

the core energy for a dislocation pair. Bands form whendislocation pairs separated by s > 1

8π b2K∆/δF arise dueto random fluctuations. To produce a dislocation paira large energy barrier of core energy Ec has to be over-come. Though even for very small strains εd the elasticfree energy becomes unstable the random fluctuationscan not overcome this large energy barrier within finitetime scales thereby suppressing the production of nl − 1layered smectic bands up to the point of ε∗d. In principle,if one could wait for truly infinite times the fluctuationscan produce such dislocation pairs for any non-zero εd

though the probability for such productions exp(−βEc)[β = 1/kBT , inverse temperature] are indeed very low.Using a procedure similar to that used in Ref.[32, 40, 53],we have monitored the dislocation probability as a func-tion of η (Fig.10). For confined hard disks, there areessentially three kinds of dislocations with Burger’s vec-tors parallel to the three possible bonds in a triangu-lar solid. Only dislocations with Burger’s vectors havingcomponents perpendicular to the walls, cause a change innl and are therefore relevant. The dislocation formationprobability is obtained by performing a simulation wherethe local connectivity of bonds in the solid is not allowedto change while an audit is performed of the number ofmoves which tend to do so. Since each possible distortionof the unit cell (see Fig.10 - inset) can be obtained by twospecific sets of dislocations, the dislocation probabilitiesmay be easily obtained from the measured probability ofbond breaking moves. Not surprisingly, the probabilityof obtaining dislocation pairs with the relevant Burger’svector increases dramatically as η → ηc1

(Fig.10) and ar-tificially removing configurations with such dislocationssuppresses the transition completely. Band coalescenceoccurs by diffusion aided dislocation “climb” which athigh density implies slow kinetics. The amount of im-

9

Pd

0

0.002

0.004

0.006

0.008

0.01

0 0.05 0.1 0.15 0.2 0.25

ε

FIG. 10: Dislocation probabilities of a 65 × 10 system areplotted as a function of strain starting from a triangular solidat η = 0.85 and Ly = 9d. The corresponding bond-breakingmoves are depicted in the inset panel. Arrows show the direc-tions of the bond-breaking moves. + show dislocation prob-abilities for those Burger’s vectors which have componentsperpendicular to the walls (topmost figure in the inset). Onthe other hand probabilities corresponding to the other twotypes of moves (denoted by ∗) remain zero.

posed strain fixes the total amount of solid and smecticregions within the strip. The coarsening of the smec-tic bands within the solid background in presence of thisconservation, leads to an even slower dynamics than non-conserving diffusion. Therefore the size of smectic bandL(t) scales as L ∼ t1/3[57]. Throughout the two-phaseregion, the crystal is in compression and the smectic intension along the y direction so that σ is completely de-termined by the amount of the co-existing phases. Alsothe walls ensure that orientation relationships betweenthe two phases are preserved throughout. As the amountof solid or smectic in the system is entirely governed bythe strain value ǫ the amount of stress σC is completelydetermined by the value of strain at all times regardlessof deformation history. This explains the reversible[58]plastic deformation in Fig. 7.

V. THE MEAN FIELD PHASE DIAGRAM

In this section we obtain the phase diagram fromMonte Carlo simulations of 65 × nl hard disks, nl be-ing the number of layers contained within the channel.We compare this with a MFT calculation. At a givenLy we start with the largest number of possible layers ofparticles and squeeze Lx up to the limit of overlap. Thenwe run the system over 5 × 104 MCS, collect data overfurther 5 × 104 MCS and increase Lx in steps such thatit reduces the density by δη = 0.001 in each step. At agiven Ly and nl the value of Lx that supports a perfect

triangular lattice has a density

η△ =

√3π

8

nl(nl − 1)

Ly(Ly − 1). (7)

In this channel geometry, at phase coexistence, the stresscomponent σxx of the coexisting phases becomes equal.The other component σyy is always balanced by the pres-sure from the confining walls. Thus we focus on the quan-tity px = −σxx as Lx is varied:

px = − 1

Ly

∂F

∂Lx= ρ2

∂fN

∂ρ,

∂fN

∂ρ=

px

ρ2. (8)

The free energy per unit volume fV = (N/V )(F/N) =ρfN . One can obtain the free energy at any given densityρ by integrating the above differential equation startingfrom a known free energy at some density ρ0,

fV (ρ) = ρf0

N(ρ0) + ρ

∫ ρ

ρ0

px

ρ2dρ . (9)

We discuss below how f0N (ρ0) for solid and modu-

lated liquid phases are obtained. The free energy ofthe solid phase may be obtained within a simple ana-lytical treatment viz. fixed neighbor free volume theory(FNFVT), which we outline below. More detailed treat-ment is available in Ref.[27]. The free volume vf (η, χ)may be obtained using straight forward, though rathertedious, geometrical considerations and the free energyf0

N(η, χ) = −ρ ln vf (η, χ). The free volume available to aparticle is computed by considering a single disk movingin a fixed cage formed by its nearest neighbors which areassumed to be held in their average positions. The freevolume available to this central particle (in units of d2)is given entirely by the lattice separations b = a0(1+ǫxx)

and h =√

3a0(1 + ǫyy)/2 where a0 is lattice parameterof a triangular lattice at any given packing fraction ηand ǫxx = (nl − 1)/(χ − 1), ǫyy = 1/ǫxx. As stated inSec.II, χ is obtained from channel width Ly and packingfraction η. vf is the area available to the central testparticle. Note that the effect of the confining geome-try is incorporated in the lattice separations b, h. TheFNFVT free energy has minima at all χ = nl. For halfintegral values of χ the homogeneous crystal is locally un-stable. Although FNFVT fails also at these points, thisis irrelevant as the system goes through a phase transi-tion before such strains are realized. In the high densitytriangular solid phase, we know f0

N (ρ0 = ρ△), exactly,from the fixed neighbor free volume theory (FNFVT). Itis interesting to note that, apart from at ρ△, this FN-FVT can estimate the solid free energy quite accuratelyat small strains ǫxx < 4% around ρ△. To obtain thesolid branch of free energy fS

V (ρ) from simulations, weintegrate the px − ρ curve starting from the FNFVT es-timate at ρ = ρ0 ≡ ρ△.

For the confined fluid phase we use a phenomenologi-cal free energy [62] of hard disk fluid added with a sim-ple surface tension term due to the scaled particle the-ory [63]. We outline this in the following. For the bulk

10

(unconfined) liquid phase one can obtain f0N (ρ = ρl) up

to a fairly large volume fraction ηl ∼ 0.81 where theliquid reaches its metastable limit in bulk, using the phe-nomenological expression[62],

βf0

N (ρl) =(2ηc − 1) ln(1 − 2ηc−1

ηcηl) − ln(1 − ηl

ηc)

2(1 − ηc)

+ ln ρl − 1 (10)

where ηc is the close packed volume and depends on Ly.We have checked from the simulations that this equa-tion of state shows good agreement with the simulatedpx − η diagram in a confined channel up to a densityof η = 0.65. Above this density, px falls below and py

goes above this equation of state estimate. Integratingpx from η0 = ηl ≡ 0.6 using the above mentioned form off0

N (ρl) we find the bulk part of the liquid branch of freeenergy fL

V (ρ). In order to incorporate the effect of con-finement approximately, we add a scaled particle theory(SPT) [63] surface energy contribution to the bulk freeenergy. Within SPT, the interfacial tension (interfacialenergy per unit length) of a hard disk fluid in contactwith a hard wall can be calculated as γ = γ + psptd/2[64] where γ = (ρd/2β)[1/(1 − η) − 1/(1 − η)2] and theSPT equation of state is βpspt/ρ = 1/(1− η2) [63]. Thusthe SPT surface energy per unit volume ST = γ/Ly =ρd/2β(1−η)Ly is added with the bulk contribution fL

V (ρ)to account for the confinement.

At thermodynamic equilibrium, the chemical potentialµ = −(∂fV /∂ρ)V,T and pressure for the coexisting phasesmust be same. Thus from a plot of free energy vs. densityobtaining common tangent to solid and liquid branchesof free energies means finding coexisting densities of solidand liquid having the same chemical potential. Once thetwo branches of free energy are obtained in the abovementioned way, we fit each branch using polynomials upto quadratic order in density. The common tangent mustshare the same slope and intercept. These two conditionsuniquely determine the coexisting densities. The onset ofbuckling instability can be found from the slope of thepx − η curve at high densities – a negative slope beingthe signature of the transition. The estimates of the co-existing densities and onset of the buckling instabilities inour simulations are thus obtained and shown using filledsymbols in the phase diagram (Fig.11).

To obtain the phase diagram from MFT (lines inFig.11) we use the above mentioned phenomenologicalfree energy of hard disk fluid[62] together with the SPTsurface tension term. For the solid we use the FNFVTfree energy for all densities. We take care to minimizethe free energy of the solid with respect to choices of nl

first. Then we find out the minimum of the solid andliquid free energies at all densities. The coexisting den-sities are obtained from common tangent constructionsfor this minimum free energy. Our MFT phase diagramso obtained is shown along with the simulation results inthe η − Ly plane (Fig.11).

In Fig.11 points are obtained from simulations and the

Ly

nl =9

nl =10

nl=11

8

8.5

9

9.5

10

0.65 0.7 0.75 0.8 0.85 0.9

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

η

FIG. 11: (Colour online) Phase diagram. The continuous linesdenote the phase boundaries obtained from mean field theory.The white, vertically shaded and horizontally shaded regionsindicate single phase, two-phase coexistence and forbidden re-gions, respectively. The single phase regions of low densitiesare liquid and high densities are solid phases. On the green(dashed) lines nl = 9, 10, 11 the system is commensurate withthe channel width such that exactly integral number of layersare contained. Points denoted by filled ◦, filled △, and filled✷ are obtained from simulations. Regions with densities lessthan the filled ◦ are modulated liquid and above filled △ aresolid. In the intervening region, we have a solid-smectic co-existence. The filled ✷ mark the onset of buckling instability.Smectic order increases smoothly and continuously as densityincreases. The liquid within the coexistence region is highlymodulated.

continuous curves from theory. The regions with den-sities lower than the points denoted by filled circles aresingle phase modulated liquid. Whereas the regions withdensities larger than the points denoted by filled trian-gles indicate the single phase solid. The modulations inthe liquid increase with increasing density and at highenough ρ the structure factor shows only two peaks typ-ical of the smectic phase. This transition is a smoothcrossover. All the regions in between the filled circlesand filled triangles correspond to solid-smectic coexis-tence. The filled squares denote the onset of buckling athigh densities. The high density regions shaded by hor-izontal lines are physically inaccessible due to the harddisk overlap. The MFT prediction for the solid-liquidcoexistence is shown by the regions between the continu-ous (red) lines (shaded vertically). The unshaded whiteregions bounded by red lines and labeled by the numberof layers denote the solid phase. All other unshaded re-gions denote liquid phase. For a constant channel width,a discontinuous transition from liquid to solid via phasecoexistence occurs with increasing density. However, theMFT predicts that the solid remelts at further higherdensities. Notice that the simulated points for the on-set of buckling lie very close to this remelting curve.Since the MFT, as an input, has only two possibilities

11

of solid and fluid phases, the high density remelting lineas obtained from the MFT may be interpreted as theonset of instability (buckling) in the high density solid.The MFT prediction of the solid-fluid coexistence regionshows qualitative agreement with simulation results forsolid-smectic coexistence. The area of phase digram cor-responding to the solid phase as obtained from simulationis smaller than that is predicted by the MFT calculation.This may be due to the inability of MFT to capture theeffect of fluctuations. From the simulated phase diagramit is clear that if one fixes the density and channel widthin a solid phase and then increases the channel widthkeeping the density fixed, one finds a series of phase tran-sitions from a solid to a smectic to another solid having alarger number of layers. These re-entrant transitions aredue to the oscillatory commensurability ratio χ that oneencounters on changing the channel width. This is purelya finite size effect due to the confinement. It is importantto note that for a bulk hard disk system, solid phase isstable at η > 0.719, whereas, even for a commensurateconfined strip of hard disks, the solid phase is stable onlyabove a density η = 0.75. With increase in incommen-suration the density of the onset of solid phase increasesfurther. This means that confinement of the Q1D stripby planar walls has an overall destabilizing effect on thesolid phase.

The phase diagram calculated in this section is a MFTphase diagram where the effects of long wavelength (andlong time scale) fluctuations are ignored. For long enoughQ1D strips fluctuations in the displacement variableshould increase linearly destroying all possible sponta-neous order and leading to a single disordered fluid phasewhich is layered in response to the externally imposedhard wall potential. However, even in that limit the lay-ering transitions from lower to higher number of layersas a function of increasing channel width, survives[6].

Our simple theory therefore predicts a discontinuoussolid-fluid transition via a coexistence region with changein density or channel width. However, details like thedensity modulation, effects of asymmetry in density pro-file, vanishing displacement modes at the walls and mostimportantly nucleation and dynamics of misfit disloca-tions crucial to generate the smectic band mediated fail-ure observed in simulations are beyond the scope of thistheory. Also, the effect of kinetic constraints which stabi-lize the solid phase well inside the two phase coexistenceregion is not captured in this approach. We believe, nev-ertheless, that this equilibrium calculation may be usedas a basis for computations of reaction rates for address-ing dynamical questions in the future.

Before we end this section, it is useful to compare ourresults with similar calculations in higher dimensions vizhard spheres confined within hard parallel plates forminga quasi two dimensional film[1, 2, 22]. Commensurationeffects produce non-monotonic convergence of the phaseboundaries to that of the bulk system. The appearenceof novel phases e.g. the buckled solid not observed inthe bulk is also a feature showed by all these systems;

-0.02 -0.01 0 0.01 0.02ε

xy

-8

-4

0

4

8

βd2

σ xy

PBCConfined

FIG. 12: (Colour online) Shear stress vs. shear strain atη = 0.85. A system of 40 × 10 hard disks simulated withperiodic boundary conditions and Ly commensurate with tenlayered triangular solid gives a shear modulus µ = 398±4. Onthe other hand when the same system of 40×10 hard disks isconfined within a commensurate channel, that fits 10-layers oflattice planes, the shear modulus drops drastically to µ = 0.

on the other hand these quasi two dimensional systemsshow two kinds of solids – triangular and square – and no‘smectic like’ phase. The effect of fluctuations also shouldbe much less in higher dimensions and one expects theobserved phases to have less anomalous properties.

VI. DISCUSSION AND CONCLUSION

One of the key definitions of a solid states that a solid,as opposed to a liquid, is a substance which can retain itsshape due to its nonzero shear modulus. Going by thisdefinition, a Q1D solid strip even of finite length confinedwithin planar, structureless walls is not a solid despite itsrather striking triangular crystalline order. Indeed, theshear modulus of the confined solid at η = 0.85 is zero,though the corresponding system with PBC show finitenonzero shear modulus (See Fig.12). This is a curiousresult and is generally true for all values of 4 < nl < 25and η investigated by us. Confinement induces stronglayering which effectively decouples the solid layers closeto the walls allowing them to slide past each other ratherlike reentrant laser induced melting[38, 39, 40]. This im-mediately shows that the only thermodynamically stablephase in confined Q1D channel is a modulated liquid, thedensity modulation coming from an explicit breaking ofthe translational symmetry by the confinement.

To understand the nature and amount of fluctu-ations in the confined Q1D solid we calculate theauto-correlation of the displacement along the channel,B(x) = 〈(ux(x) − ux(0))2〉 for a layer of particles neara boundary. The nature of the equilibrium displacementcorrelations ultimately determines the decay of the peakamplitudes of the structure factor and the value of the

12

10 100 1000x/d

0.001

0.002

0.003

0.004

0.005

<[u

x(x)-

u x(0)]

2 ]>/d

2

PBCConfined

FIG. 13: (Colour online) Flucuation < (ux(x)−ux(0))2 > /d2

on a lattice plane (line) for 5000× 10 commensurate triangu-lar solid (η = 0.85) with PBC in both directions is comparedwith that in presence of hard channel confinement. Averagesare done over 100 configurations. ◦ denotes the data obtainedin presence of PBC in both directions and ✷ denotes the dataobtained for confined system. In both cases the initial log-arithmic growth (tracked by dashed and solid straight linesrespectively) crosses over to a faster linear increase beforesaturating due to finite size effects. Confinement apparentlyreduces the amount of fluctuations without afecting the gen-eral behavior.

equilibrium elastic moduli [55]. It is known from har-monic theories, in one dimension B(x) ∼ x and in two di-mensions B(x) ∼ ln(x). In the Q1D system it is expectedthat for small distances, displacement fluctuations willgrow logarithmically with distance which will crossoverto a linear growth at large distances with a crossoverlength xc ∼ Ly lnLy [36, 37]. We calculate this correla-tion for a system of 5000 × 10 particles averaged over10, 30, 50, 100 configurations each separated by 103

MCS and equilibrated over 105 MCS at η = 0.85. Wecompare the results obtained for a strip with confine-ment and a strip with PBC’s in both directions, takingcare that in each case the channel width is commensuratewith the inter-lattice plane separation. With an increasein the number of configurations over which the averagesare done, fluctuations reduce and converge to a smallnumber. This happens for both the cases we study. It isinteresting to notice a logarithmic to linear cross-over ofthe fluctuations near x = 10d for both the cases. Sincethe harmonic theory[37] ignores effects due to bound-aries, one can conclude that this general behavior is quiterobust. At large distances, displacement correlations areexpected to saturate due to finite size effects. The mag-nitude of fluctuation at saturation for the confined sys-tem (∼ 0.0035) is noticably lower than that in presenceof PBC (∼ 0.0045) (Fig.13). Thus the introduction ofcommensurate confinement can have a stabilizing effectof reducing the overall amount of fluctuations present ina Q1D strip with PBC.

We have separately calculated the elastic modulus of a

40 × 10 confined hard disk solid at η = 0.85 under elon-gation (Young’s modulus) in the x and the y directions.For an isotropic triangular crystal in 2D these should beidentical. We, however, obtain the values 1361 and 1503(in units of kBT/d2 and within an error of 3%) respec-tively for the two cases. The Young modulus for thelongitudinal elongation is smaller than that in the direc-tion transverse to the confinement and both these valuesare larger than the Young modulus of the system underPBC (Y = 1350). This corroborates the fact that thenon-hydrodynamic component of the stress σxx − σyy isnon-zero even for vanishingly small strains as shown inFig.7. Therefore even if we choose to regard this Q1Dsolid as a liquid, it is quite anomalous since it triviallyviolates Pascal’s law which states that the stress tensor ina liquid is always proportional to the identity. This is be-cause of the explicit breaking of translational symmetryby the confinement. Lastly, commensurability seems toaffect strongly the nature and magnitude of the displace-ment fluctuations which increase dramatically as the sys-tem is made incommensurate.

As Lx ≈ Ly → ∞ all such anomalous behavior isexpected to be localized in a region close to the wallssuch that in the central region a bulk 2d solid is recov-ered. This crossover is continuous, though oscillatorywith commensurabilities playing an important role, andextremely slow (∼ 1/Ly). It is therefore difficult to ob-serve in simulations.

What impact, if any, do our results have for realis-tic systems? With recent advances in the field of nanoscience and technology[65, 66] new possibilities of build-ing machines made of small assemblage of atoms andmolecules are emerging. This requires a knowledge ofthe structures and mechanical properties of systems upto atomic scales. A priori, at such small scales, thereis no reason for macroscopic continuum elasticity theory,to be valid[67]. Our results corroborate such expecta-tions. We have shown that small systems often showentirely new behavior if constraints are imposed lead-ing to confinement in one or more directions. We havealso speculated on applications of reversible failure as ac-curate strain transducers or strain induced electrical orthermal switching devices [26, 27]. We believe that manyof our results may have applications in tribology[16] inlow dimensional systems. The effect of corrugations ofthe walls on the properties of the confined system is aninteresting direction of future study. The destruction oflong ranged solid like order should be observable in nanowires and tubes and may lead to fluctuations in transportquantities [68].

VII. ACKNOWLEDGMENT

The authors thank M. Rao, V. B. Shenoy, A. Datta, A.Dhar, A. Chaudhuri and A. Ricci for useful discussions.Support from SFB-TR6 program on “Colloids in externalfields” and the Unit for Nano Science and Technology,

13

SNBNCBS is gratefully acknowledged.

[1] M. Schmidt and H. Lowen, Phys. Rev. Lett. 76, 4552(1996).

[2] A. Fortini and M. Dijkstra, J. Phys.: Condens. Matter18, L371 (2006), cond-mat/0604169.

[3] G. Piacente, I. V. Schweigert, J. J. Betouras, and F. M.Peeters, Phys. Rev. B 69, 045324 (2004).

[4] R. Haghgooie and P. S. Doyle, Phys. Rev. E 70, 061408(2004).

[5] R. Haghgooie and P. S. Doyle, Phys. Rev. E 72, 011405(2005).

[6] P. G. de Gennes, Langmuir 6, 1448 (1990).[7] J. Gao, W. D. Luedtke, and U. Landman, Phys. Rev.

Lett. 79, 705 (1997).[8] D. Chaudhuri and S. Sengupta, Phys. Rev. Lett. 93,

115702 (2004).[9] L.-W. Teng, P.-S. Tu, and L. I, Phys. Rev. Lett. 90,

245004 (2003).[10] V. M. Bedanov and F. M. Peeters, Phys. Rev. B 49, 2667

(1994).[11] R. Bubeck, C. Bechinger, S. Neser, and P. Leiderer, Phys.

Rev. Lett. 82, 3364 (1999).[12] Y.-J. Lai and L. I, Phys. Rev. E 64, 015601(R) (2001).[13] R. Bubeck, P. Leiderer, and C. Bechinger, Europhys.

Lett. 60, 474 (2002).[14] M. Kong, B. Partoens, and F. M. Peeters, Phys. Rev. E

67, 021608 (2003).[15] P. Pieranski, J. Malecki, and K. Wojciechowski, Molecu-

lar Physics 40, 225 (1980).[16] C. Ghatak and K. G. Ayappa, Phys. Rev. E 64, 051507

(2001); K. G. Ayappa and C. Ghatak, J. Chem. Phys.117, 5373 (2002); H. Bock, K. E. Gubbins, K. G. Ayappa,J. Chem. Phys. 122, 094709 (2005).

[17] A. Chaudhuri, S. Sengupta, and M. Rao, Phys. Rev. Lett.95, 266103 (2005).

[18] M. Koppl, P. Henseler, A. Erbe, P. Nielaba, and P. Lei-derer, Phys. Rev. Lett. 97, 208302 (2006).

[19] P. Pieranski, L. Strzelecki, and B. Pansu, Phys. Rev.Lett. 50, 900 (1983).

[20] S. Neser, C. Bechinger, P. Leiderer, and T. Palberg, Phys.Rev. Lett. 79, 2348 (1997).

[21] T. Chou and D. R. Nelson, Phys. Rev. E 48, 4611 (1993).[22] M. Schmidt and H. Lowen, Phys. Rev. E 55, 7228 (1997).[23] D. Chaudhuri, A. Chaudhuri, and S. Sengupta, J. Phys.:

Condens. Matter 19, 152201 (2007).[24] A. Chaudhuri, D. Chaudhuri, and S. Sengupta, Phys.

Rev. E 76, 021603 (2007).[25] I. W. Hamley, Introduction to Soft Matter: polymer, col-

loids, amphiphiles and liquid crystals (Wiley, Cluchester,2000).

[26] S. Datta, D. Chaudhuri, T. Saha-Dasgupta, and S. Sen-gupta, Europhys. Lett. 73, 765 (2006).

[27] D. Chaudhuri and A. Dhar, Phys. Rev. E 74, 016114(2006).

[28] B. Alder and T. Wainwright, Phys. Rev. 127, 359 (1962).[29] J. A. Zollweg, G. V. Chester, and P. W. Leung, Phys.

Rev. B 39, 9518 (1989).[30] H. Weber and D. Marx, Europhys. Lett. 27, 593 (1994).[31] A. Jaster, Physica A 277, 106 (2000).

[32] S. Sengupta, P. Nielaba, and K. Binder, Phys. Rev. E61, 6294 (2000).

[33] J. P. Hansen and I. R. MacDonald, Theory of simple

liquids (Wiley, Cluchester, 1989).[34] K. W. Wojciechowski and A. C. Branka, Phys. Lett. A

134, 314 (1988).[35] M. Heni and H. Lowen, Phys. Rev. E 60, 7057 (1999).[36] A. Ricci, P. Nielaba, S. Sengupta, and K. Binder, Phys.

Rev. E 74, 010404(R) (2006).[37] A. Ricci, P. Nielaba, S. Sengupta, and K. Binder, Phys.

Rev. E 75, 011405 (2007).[38] E. Frey, D. R. Nelson, and L. Radzihovsky, Phys. Rev.

Lett. 83, 2977 (1999).[39] W. Strepp, S. Sengupta, and P. Nielaba, Phys. Rev. E

63, 046106 (2001).[40] D. Chaudhuri and S. Sengupta, Europhys. Lett. 67, 814

(2004).[41] D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457

(1979).[42] A. A. Griffith, Philos. Trans. Roy. Soc. A 221, 163

(1920).[43] J. A. Hauch, D. Holland, M. P. Marder, and H. L. Swin-

ney, Phys. Rev. Lett. 82, 3823 (1999).[44] D. Holland and M. Marder, Phys. Rev. Lett. 80, 746

(1998).[45] J. S. Langer, Phys. Rev. E 62, 1351 (2000).[46] R. Lofstedt, Phys. Rev. E 55, 6726 (1997).[47] T. Belytschko, S. P. Xiao, G. C. Schatz, and R. S. Ruoff,

Phys. Rev. B 65, 235430 (2002).[48] M. F. Yu et al., Science 287, 637 (2000).[49] H. Ikeda et al., Phys. Rev. Lett. 82, 2900 (1999).[50] P. S. Branicio and J.-P. Rino, Phys. Rev. B 62, 16950

(2000).[51] K. Zahn, R. Lenke, and G. Maret, Phys. Rev. Lett. 82,

2721 (1999).[52] O. Farago and Y. Kantor, Phys. Rev. E 61, 2478 (2000).[53] D. Chaudhuri and S. Sengupta, Phys. Rev. E 73, 011507

(2006).[54] F. Varnik, J. Baschnagel, and K. Binder, Phys. Rev. E

65, 021507 (2002).[55] P. M. Chaikin and T. C. Lubensky, Principles of Con-

densed Matter Physics (Cambridge University Press,Cambridge, 1995).

[56] J. S. Langer, in Solids far from equilibrium, edited byC. Godreche (Cambridge University Press, Cambridge,1992), Vol. III, p. 169.

[57] A. Bray, Advances in Physics 51, 481 (2002).[58] T. Roopa and G. V. Shivashankar, Appl. Phys. Lett. 82,

1631 (2003).[59] T. V. Ramakrishnan and M. Yussouff, Phys. Rev. B 19,

2775 (1979).[60] J. Wu and Z. Li, Ann. Rev. of Phys. Chem. 58, 85 (2007).[61] D. Chaudhuri and S. Sengupta, Indian Journal of Physics

79, 941 (2005).[62] A. Santos, M. L. de Haro, and S. B. Yuste, J. Chem.

Phys. 103, 4622 (1995).[63] E. Helfand, H. L. Frisch, and J. L. Lebowitz, J. Chem.

Phys. 34, 1037 (1961).

http://arxiv.org/abs/cond-mat/0604169

14

[64] M. Heni and H. Lowen, Phys. Rev. E 60, 7057 (1999).[65] R. Gao, Z. Wang, Z. Bai, W. deHeer, L. Dai, and M.

Gao, Phys. Rev. Lett. 85, 622 (2000).[66] A. N. Cleland and M. L. Roukes, Appl. Phys. Lett. 69,

2653 (1996).

[67] I. Goldhirsh and C. Goldenberg, European Physical Jour-nal E 9, 245 (2002).

[68] A. Bid, A. Bora, and A. K. Raychaudhuri, Phys. Rev. B72, 113415 (2005).

Anomalous structural and mechanical properties of solids confined in quasi-one-dimensional strips

Documents

Transcript of Anomalous structural and mechanical properties of solids confined in quasi-one-dimensional strips