text: 18 repulsive potential attractive potential ∼ (cid:28) φ b/b when b b if (cid:15)ˆ = (cid:15)/(21/γb ) < 1 − ∼ (cid:28) φ −(cid:15)π/2 ∼0−b/(cid:15) when b 0(cid:28) (cid:15) if (cid:15)ˆ = (cid:15)/(21/γb0) > 1 φ(cid:15) π/2 b/(cid:15) when b (cid:15) (cid:15) 0 TABLE II: Summary of the expansions of the angle φ with a Plummer softening in the potential for hard collisions (cid:15) 10-2 100 10-3 10-4 10-1 10-5 AAAAAAAA AAAAAAAA BBBBBBBB BBBBBBBB CCCCCCCC CCCCCCCC 10-6 10-2 10-1 100 101 10-2 10-1 100 101 X X 100 10-1 AAAAAAAA BBBBBBBB CCCCCCCC 10-1 10-2 10-2 10-3 10-3 AAAAAAAA BBBBBBBB CCCCCCCC 10-4 10-4 10-1 100 101 102 10-1 100 101 102 X X FIG. 15: Numerical computations for repulsive FIG. 16: Numerical computations for attractive potentials with Plummer softening. Top: Graph of φ potentials with Plummer softening (hard scattering). (cid:15) normalized to the angle without softening Top: Graphs of φ (continuous line) and the theoretical (cid:15) φ (continuous line) and of the leading order term prediction Eq. (??) (dotted lines) as a function of b/b 0 0 (dotted line) given in Eq. (??) as a function of b/b for for diﬀerent values of γ and (cid:15)/b = 1/10. Bottom: same 0 0 diﬀerent values of γ and (cid:15)/b = 1/10. Bottom: Graph of quantity for (cid:15)/b = 10 and the theoretical prediction 0 0 φ for (cid:15)/b = 10 and the leading order expansion given Eq. (??) (dotted lines). The dotted blue lines (cid:15) 0 in Eq. (??). The dotted blue lines corresponds to φ . corresponds to φ . 0 0 tween the exact integration Eq. (??) with the asymptotic predictions Eqs. (??) and (??). We have also studied, for duce a value of the softening larger than a critical value γ > 2, for which value of the softening, there is no for- (which we have calculated explicitly) to regularize com- mation of pairs for any value of b. We have seen that pletely the problem. In Fig. ?? we illustrate this behav- introducing a softening (cid:15) > 0 automatically regularizes ior. The continuous red curve corresponds to the case in the angle φ for any value b, except one, for which there is which (cid:15) > (cid:15)∗(b , γ). In this case, φ is a regular function 0 (cid:15) orbiting, except for some particular softenings, in which of b, as it can be seen in the inset. The dashed green the divergences also disappear. It is necessary to intro- curve corresponds to the case in which (cid:15) < (cid:15)∗(b , γ), for 0 Y Y Y Y ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: repulsive

label: table

text: potential

text: attractive

text: potential

text: φ

text: b/b

text: when

text: b

text: b

text: if

text: φ

text: when

text: b

text: if

text: when

text: b

text: TABLE

text: II:

text: Summary

label: caption

text: of

text: the

text: expansions

text: of

text: the

text: angle

text: φ

text: with

text: a

text: Plummer

text: softening

text: in

text: the

text: potential

text: for

text: hard

text: collisions

font: default

text: AAAAAAAA

text: AAAAAAAA

text: BBBBBBBB

text: BBBBBBBB

text: CCCCCCCC

text: CCCCCCCC

text: X

text: X

text: AAAAAAAA

text: BBBBBBBB

text: CCCCCCCC

text: AAAAAAAA

text: BBBBBBBB

text: CCCCCCCC

text: X

text: X

text: FIG.

text: Numerical

text: computations

text: for

text: repulsive

text: FIG.

text: Numerical

text: computations

text: for

text: attractive

text: potentials

text: with

text: Plummer

text: softening.

text: Top:

text: Graph

text: of

text: φ

text: potentials

text: with

text: Plummer

text: softening

text: (hard

text: scattering).

text: normalized

text: to

text: the

text: angle

text: without

text: softening

text: Top:

text: Graphs

text: of

text: φ

text: (continuous

text: line)

text: and

text: the

text: theoretical

text: φ

text: (continuous

text: line)

text: and

text: of

text: the

text: leading

text: order

text: term

text: prediction

text: Eq.

text: (dotted

text: lines)

text: as

text: a

text: function

text: of

text: b/b

text: (dotted

text: line)

text: given

text: in

text: Eq.

text: as

text: a

text: function

text: of

text: b/b

text: for

text: for

text: diﬀerent

text: values

text: of

text: γ

text: and

text: Bottom:

text: same

text: diﬀerent

text: values

text: of

text: γ

text: and

text: Bottom:

text: Graph

text: of

text: quantity

text: for

text: and

text: the

text: theoretical

text: prediction

text: φ

text: for

text: and

text: the

text: leading

text: order

text: expansion

text: given

text: Eq.

text: (dotted

text: lines).

text: The

text: dotted

text: blue

text: lines

text: in

text: Eq.

text: The

text: dotted

text: blue

text: lines

text: corresponds

text: to

text: φ

text: corresponds

text: to

text: φ

text: tween

text: the

text: exact

text: integration

text: Eq.

text: with

text: the

text: asymptotic

text: predictions

text: Eqs.

text: and

text: We

text: have

text: also

text: studied,

text: for

text: duce

text: a

text: value

text: of

text: the

text: softening

text: larger

text: than

text: a

text: critical

text: value

text: γ

text: for

text: which

text: value

text: of

text: the

text: softening,

text: there

text: is

text: no

text: for-

text: (which

text: we

text: have

text: calculated

text: explicitly)

text: to

text: regularize

text: com-

text: mation

text: of

text: pairs

text: for

text: any

text: value

text: of

text: b.

text: We

text: have

text: seen

text: that

text: pletely

text: the

text: problem.

text: In

text: Fig.

text: we

text: illustrate

text: this

text: behav-

text: introducing

text: a

text: softening

text: automatically

text: regularizes

text: ior.

text: The

text: continuous

text: red

text: curve

text: corresponds

text: to

text: the

text: case

text: in

text: the

text: angle

text: φ

text: for

text: any

text: value

text: b,

text: except

text: one,

text: for

text: which

text: there

text: is

text: which

text: γ).

text: In

text: this

text: case,

text: φ

text: is

text: a

text: regular

text: function

text: orbiting,

text: except

text: for

text: some

text: particular

text: softenings,

text: in

text: which

text: of

text: b,

text: as

text: it

text: can

text: be

text: seen

text: in

text: the

text: inset.

text: The

text: dashed

text: green

text: the

text: divergences

text: also

text: disappear.

text: It

text: is

text: necessary

text: to

text: intro-

text: curve

text: corresponds

text: to

text: the

text: case

text: in

text: which

text: γ),

text: for

text: Y

text: Y

text: Y

text: Y

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: 19 implies that φ < π/2. If the interaction is attractive, 20 AAAAAA it will be deﬂected to the bottom left quadrant, which 18 BBBBBB implies φ > π/2 (see Fig. 1). 11 16 In the regime of hard collisions, the situation is very 10 14 9 diﬀerent between the repulsive and attractive case: (i) 12 8 for repulsive∼interactions, the angle of closest appr→oach 7 scales as φ b/b . This is what one expects for b 0: 0 10 6 for vanishing impact factor, particle bounce one on each 8 5 other, coming back in their original directions and op- 4 6 1.55 1.6 1.65 posite sign of the velocity; (ii) for attractive interactions with γ < 2, the leading contribution is 4 ∼ π 2 φ − . (64) 2 γ 0 → 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 We see therefore, that for b/b 0 the angle of deﬂection 0 X depends on the exponent of the interaction potential γ. Of course, the deﬂection angle is the same for Coulomb FIG. 17: Plot of φ as a function of b for γ = 5/2 and (cid:15) (repulsive) and gravitational interaction (γ = 1) two diﬀerent values of the softening. The red Equation (??) implies that, when γ approaches the continuous curve corresponds to a value of (cid:15) slightly value of 2, the angle φ increases, diverging in the limit larger than (cid:15)∗(b , γ) and the dashed green one to a → 0 γ 2. This is due to the eﬀective potential created by value of (cid:15) slightly smaller than (cid:15)∗(b , γ). 0 the angular momentum, which scales with the distance as 1/r2. When the exponent γ of the attractive potential is larger than 2, the angular momentum term cannot, in general, prevent the system to collapse and the particles which φ diverges for b = b ((cid:15)). (cid:15) (cid:93) crash. Studying the distance of closest approach r we min VII. CONCLUSIONS have found two diﬀerent behaviors whether γ is smaller or larger than 2: • → − In this paper we have studied the scattering of t∼wo par- If γ < 2, in the limit γ 2 (for any b smaller than ticles interacting with a central potential v(r) 1/rγ. some critical value which we have calculated explic- This is a generalization of the Rutherford formula of the itly), the value of r tends to 0. The trajectories min scattering of two particles interacting via a Coulomb or in this limit is a succession of smaller and smaller gravitational force. Unlike the original case, it is not loops embedded one in the other. An example of possible to compute in general the de(cid:54) ﬂection angle of the such trajectories was given in Fig. ??. particles analytically for general γ = 1. We have then • If γ > 2, the particles do not crash if the impact calculated the asymptotics of the angle of deﬂection for factor is larger than some critical value, which we the two limiting cases in which we are interested in: the have calculated. For impact factor slightly larger weak collisions regime, in which the particles trajectories than this critical value, we have trajectories with are weakly perturbed, and the strong collision regime, in ∼ r b . The particles then orbite with distance which they are strongly perturbed. Combining the ana- min 0 r forming a binary, which will be destroyed in a lytical expressions and the numerical integration of the min ﬁnite time. We gave an example of such trajectories equation of motion, we have derived the phenomenology in Fig. ??. we detail as follows. In the regime of soft collisions, attractive and repulsive We have also studied the eﬀect of introducing a reg- interactions give a very similar re(cid:18)sult:(cid:19)the angle of closest ularization at small scales in the potential. The conclu- approach scales as sions are detailed in Subsect. ??. A practical application appears naturally in the con- ∼ ∓ b γ text of astrophysics or plasma physics, when we are inter- φ π/2 A(γ) , (63) b ested in calculating the average change of velocity due to 0 the collisions. It is usual (see e.g. [? ]) to decompose the where A(γ) > 0, b is the impact factor, and b a char- relative velocity of the particles before the collisions V as 0 acteristic scale which depends on the reduced mass, the the sum of its component along the direction of the initial coupling constant and the relative asymptotic velocity of relative velocity e(cid:107) and the component perpendicular to the particles. The minus sign corresponds to the repul- it e⊥, i.e., sive interaction and the positive sign to attractive inter- V = V⊥e⊥ + V(cid:107)e(cid:107). (65) actions. This is what it is expected: with no interac- tion, the reduced particle suﬀers no deﬂection, and then It is possible to compute the average change of veloc- φ = π/2. If the interaction is repulsive, the reduced ity ∆V⊥ and ∆V(cid:107) after a collision has been completed particle will be deﬂected in the top left quadrant, which integrating over all the impact factors b: Y ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: implies

text: that

text: φ

text: If

text: the

text: interaction

text: is

text: attractive,

text: AAAAAA

text: it

text: will

text: be

text: deﬂected

text: to

text: the

text: bottom

text: left

text: quadrant,

text: which

text: BBBBBB

text: implies

text: φ

text: (see

text: Fig.

text: In

text: the

text: regime

text: of

text: hard

text: collisions,

text: the

text: situation

text: is

text: very

text: diﬀerent

text: between

text: the

text: repulsive

text: and

text: attractive

text: case:

text: (i)

text: for

text: repulsive∼interactions,

text: the

text: angle

text: of

text: closest

text: appr→oach

text: scales

text: as

text: φ

text: b/b

text: This

text: is

text: what

text: one

text: expects

text: for

text: b

text: for

text: vanishing

text: impact

text: factor,

text: particle

text: bounce

text: one

text: on

text: each

text: other,

text: coming

text: back

text: in

text: their

text: original

text: directions

text: and

text: op-

text: posite

text: sign

text: of

text: the

text: velocity;

text: (ii)

text: for

text: attractive

text: interactions

text: with

text: γ

text: the

text: leading

text: contribution

text: is

text: π

text: φ

text: γ

text: We

text: see

text: therefore,

text: that

text: for

text: b/b

text: the

text: angle

text: of

text: deﬂection

text: X

text: depends

text: on

text: the

text: exponent

text: of

text: the

text: interaction

text: potential

text: γ.

text: Of

text: course,

text: the

text: deﬂection

text: angle

text: is

text: the

text: same

text: for

text: Coulomb

text: FIG.

text: Plot

text: of

text: φ

text: as

text: a

text: function

text: of

text: b

text: for

text: γ

text: and

text: (repulsive)

text: and

text: gravitational

text: interaction

text: (γ

text: two

text: diﬀerent

text: values

text: of

text: the

text: softening.

text: The

text: red

text: Equation

text: implies

text: that,

text: when

text: γ

text: approaches

text: the

text: continuous

text: curve

text: corresponds

text: to

text: a

text: value

text: of

text: slightly

text: value

text: of

text: the

text: angle

text: φ

text: increases,

text: diverging

text: in

text: the

text: limit

text: larger

text: than

text: γ)

text: and

text: the

text: dashed

text: green

text: one

text: to

text: a

text: γ

text: This

text: is

text: due

text: to

text: the

text: eﬀective

text: potential

text: created

text: by

text: value

text: of

text: slightly

text: smaller

text: than

text: γ).

text: the

text: angular

text: momentum,

text: which

text: scales

text: with

text: the

text: distance

text: as

text: When

text: the

text: exponent

text: γ

text: of

text: the

text: attractive

text: potential

text: is

text: larger

text: than

text: the

text: angular

text: momentum

text: term

text: cannot,

text: in

text: general,

text: prevent

text: the

text: system

text: to

text: collapse

text: and

text: the

text: particles

text: which

text: φ

text: diverges

text: for

text: b

text: b

text: crash.

text: Studying

text: the

text: distance

text: of

text: closest

text: approach

text: r

text: we

text: min

text: VII.

text: CONCLUSIONS

text: have

text: found

text: two

text: diﬀerent

text: behaviors

text: whether

text: γ

text: is

text: smaller

text: or

text: larger

text: than

text: In

text: this

text: paper

text: we

text: have

text: studied

text: the

text: scattering

text: of

text: t∼wo

text: par-

text: If

text: γ

text: in

text: the

text: limit

text: γ

text: (for

text: any

text: b

text: smaller

text: than

text: ticles

text: interacting

text: with

text: a

text: central

text: potential

text: v(r)

text: some

text: critical

text: value

text: which

text: we

text: have

text: calculated

text: explic-

text: This

text: is

text: a

text: generalization

text: of

text: the

text: Rutherford

text: formula

text: of

text: the

text: itly),

text: the

text: value

text: of

text: r

text: tends

text: to

text: The

text: trajectories

text: min

text: scattering

text: of

text: two

text: particles

text: interacting

text: via

text: a

text: Coulomb

text: or

text: in

text: this

text: limit

text: is

text: a

text: succession

text: of

text: smaller

text: and

text: smaller

text: gravitational

text: force.

text: Unlike

text: the

text: original

text: case,

text: it

text: is

text: not

text: loops

text: embedded

text: one

text: in

text: the

text: other.

text: An

text: example

text: of

text: possible

text: to

text: compute

text: in

text: general

text: the

text: ﬂection

text: angle

text: of

text: the

text: such

text: trajectories

text: was

text: given

text: in

text: Fig.

text: particles

text: analytically

text: for

text: general

text: γ

text: We

text: have

text: then

text: If

text: γ

text: the

text: particles

text: do

text: not

text: crash

text: if

text: the

text: impact

text: calculated

text: the

text: asymptotics

text: of

text: the

text: angle

text: of

text: deﬂection

text: for

text: factor

text: is

text: larger

text: than

text: some

text: critical

text: value,

text: which

text: we

text: the

text: two

text: limiting

text: cases

text: in

text: which

text: we

text: are

text: interested

text: in:

text: the

text: have

text: calculated.

text: For

text: impact

text: factor

text: slightly

text: larger

text: weak

text: collisions

text: regime,

text: in

text: which

text: the

text: particles

text: trajectories

text: than

text: this

text: critical

text: value,

text: we

text: have

text: trajectories

text: with

text: are

text: weakly

text: perturbed,

text: and

text: the

text: strong

text: collision

text: regime,

text: in

text: r

text: b

text: The

text: particles

text: then

text: orbite

text: with

text: distance

text: which

text: they

text: are

text: strongly

text: perturbed.

text: Combining

text: the

text: ana-

text: min

text: r

text: forming

text: a

text: binary,

text: which

text: will

text: be

text: destroyed

text: in

text: a

text: lytical

text: expressions

text: and

text: the

text: numerical

text: integration

text: of

text: the

text: min

text: ﬁnite

text: time.

text: We

text: gave

text: an

text: example

text: of

text: such

text: trajectories

text: equation

text: of

text: motion,

text: we

text: have

text: derived

text: the

text: phenomenology

text: in

text: Fig.

text: we

text: detail

text: as

text: follows.

text: In

text: the

text: regime

text: of

text: soft

text: collisions,

text: attractive

text: and

text: repulsive

text: We

text: have

text: also

text: studied

text: the

text: eﬀect

text: of

text: introducing

text: a

text: reg-

text: interactions

text: give

text: a

text: very

text: similar

text: angle

text: of

text: closest

text: ularization

text: at

text: small

text: scales

text: in

text: the

text: potential.

text: The

text: conclu-

text: approach

text: scales

text: as

text: sions

text: are

text: detailed

text: in

text: Subsect.

text: A

text: practical

text: application

text: appears

text: naturally

text: in

text: the

text: con-

text: b

text: γ

text: text

text: of

text: astrophysics

text: or

text: plasma

text: physics,

text: when

text: we

text: are

text: inter-

text: φ

text: A(γ)

text: b

text: ested

text: in

text: calculating

text: the

text: average

text: change

text: of

text: velocity

text: due

text: to

text: the

text: collisions.

text: It

text: is

text: usual

text: (see

text: e.g.

text: to

text: decompose

text: the

text: where

text: A(γ)

text: b

text: is

text: the

text: impact

text: factor,

text: and

text: b

text: a

text: char-

text: relative

text: velocity

text: of

text: the

text: particles

text: before

text: the

text: collisions

text: V

text: as

text: acteristic

text: scale

text: which

text: depends

text: on

text: the

text: reduced

text: mass,

text: the

text: the

text: sum

text: of

text: its

text: component

text: along

text: the

text: direction

text: of

text: the

text: initial

text: coupling

text: constant

text: and

text: the

text: relative

text: asymptotic

text: velocity

text: of

text: relative

text: velocity

text: and

text: the

text: component

text: perpendicular

text: to

text: the

text: particles.

text: The

text: minus

text: sign

text: corresponds

text: to

text: the

text: repul-

text: it

text: e⊥,

text: i.e.,

text: sive

text: interaction

text: and

text: the

text: positive

text: sign

text: to

text: attractive

text: inter-

text: V

text: V⊥e⊥

text: actions.

text: This

text: is

text: what

text: it

text: is

text: expected:

text: with

text: no

text: interac-

text: tion,

text: the

text: reduced

text: particle

text: suﬀers

text: no

text: deﬂection,

text: and

text: then

text: It

text: is

text: possible

text: to

text: compute

text: the

text: average

text: change

text: of

text: veloc-

text: φ

text: If

text: the

text: interaction

text: is

text: repulsive,

text: the

text: reduced

text: ity

text: ∆V⊥

text: and

text: after

text: a

text: collision

text: has

text: been

text: completed

text: particle

text: will

text: be

text: deﬂected

text: in

text: the

text: top

text: left

text: quadrant,

text: which

text: integrating

text: over

text: all

text: the

text: impact

text: factors

text: b:

text: Y

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 (cid:40) solids are far less sensitive to contact changes than are the individual trajectories. Namely, under ensemble av- eraging the slope of the stress-strain curve is the same be- f(cid:126)el = −k(δij) δij rˆij , δij > 0 (2) fore and after the ﬁrst contact change [30], and changes ij (cid:126)0, δ < 0. ij in the density of states are negligible [32]. These re- sults show that linear elastic constitutive relations remain The prefactor k is the contact stiﬀness, which generally useful near jamming, but they say nothing about when depends on the overlap Hooke’s law eventually does break down. Recent experiments [13, 21] and simulations [14, 24, 33] − provide evidence for a two stage yielding process, where k = k0 δα 2 . (3) packings ﬁrst soften and only later establish steady shear ﬂow. Yet it remains unclear precisely how rate depen- Here k is a constant and α is an exponent parameter- dence, nonlinearity, and contact changes contribute to 0 izing the interaction. In the following we consider har- the breakdown of linear elasticity. In order to unravel monic interactions (α = 2), which provide a reasonable these eﬀects, it is necessary to vary strain, strain rate, model for bubbles and droplets that resist deformation pressure, and system size simultaneously and systemati- due to surface tension; we also treat Hertzian interac- cally – as we do here for the ﬁrst time. Using simulations tions (α = 5/2), which correspond to elastic spheres. of viscous soft spheres, we ﬁnd that Hooke’s law is valid We perform simulations using two separate numerical within a surprisingly narrow window bounded by viscous methods. The ﬁrst is a molecular dynamics (MD) algo- eﬀects at small strain and nonlinear softening at large rithm that integrates Newton’s laws using the velocity- strain. The size of the linear elastic window displays Verlet scheme. Each disk is assigned a uniform mass power law scaling with pressure and correlates with the m = πR2 proportional to its volume. Energy is dissi- accumulation of not one, but an extensive number of con- i i pated by viscous forces that are proportional to the rel- tact changes. ative velocity ∆(cid:126)v c of neighboring particles evaluated at The basic scenario we identify is illustrated in Fig. 1, ij the contact, which presents ensemble-averaged stresses versus strain. Shear is applied via a constant strain rate γ˙ at ﬁxed 0 volume. We identify three characteristic scales, each of f(cid:126)visc = −τ k(δ ) ∆(cid:126)v c , (4) ij 0 ij ij which depend on the initial pressure p: (i) For strains ∗ ∗ ∗ below γ ≡ γ˙ τ , where τ is a time scale, viscous 0 stresses are signiﬁcant and Eq. (1) underestimates the where τ is a microscopic relaxation time. Viscous 0 stress needed to deform the material. This crossover forces can apply torques, hence particles are allowed to strain vanishes under quasista†tic shear (γ˙0 → 0, ﬁlled rotate as well as translate. squares). (ii) Above a strain γ the material softens and In addition to MD, we also perform simulations using Hooke’s law overestimates the stress. This crossover is a nonlinear conjugate gradient (CG) routine [34], which rate-independent, consistent with plastic eﬀects. (iii) keeps the system at a local minimum of the potential en- † For strain rates above a scale γ˙ (triangles), Eq. (1) is ergy landscape, which itself changes as the system under- never accurate and there is no strain interval where the goes shearing. The dynamics are therefore quasistatic, material responds as a linear elastic solid. i.e. the particle trajectories correspond to the limit of vanishing strain rate. Bubble packings consist of N = 128 to 2048 disks in a 50:50 bidisperse mixture with a 1.4:1 diameter ratio. I. SOFT SPHERES: MODEL AND Shear is implemented via Lees-Edwards “sliding brick” BACKGROUND boundary conditio(cid:88)ns. The stress ten(cid:88)sor is given by We ﬁrst introduce the soft sphere model and summa- 1 1 rize prior results regarding linear elasticity near jamming. σ = f r − m v v , (5) αβ ij,α ij,β i i,α i,β 2V V A. Model ij i We perform numerical simulations of the Durian bub- where V is the volume (area in two dimensions) of ble model [4], a mesoscopic model for wet foams and the packing, f(cid:126) is the sum of elastic and viscous contact ij emulsions. The model treats bubbles/droplets as non- forces acting on particle i due to particle j, and (cid:126)v is the i Brownian disks that interact via elastic and viscous forces velocity of particle i. Greek indices label components when they overlap. Elastic forces are expressed in terms along the Cartesian coordinates x and y. The conﬁn- of the overlap δ = 1 − r /(R + R ), where R and R ing pressure is p = −(1/D)(σ + σ ), where D = 2 is ij ij i j i j xx yy denote radii and (cid:126)r points from the center of particle i the spatial dimension, while the shear stress is σ = σ . ij xy to the center of j. The force is repulsive and acts along The second term on the righthand side of Eq. (5) is a ki- the unit vector rˆ = (cid:126)r /r : netic stress, which is always negligible in the parameter ij ij ij ##LTLine## ##LTLine##

label: equation

text: solids

text: are

text: far

text: less

text: sensitive

text: to

text: contact

text: changes

text: than

text: are

text: the

text: individual

text: trajectories.

text: Namely,

text: under

text: ensemble

text: av-

text: eraging

text: the

text: slope

text: of

text: the

text: stress-strain

text: curve

text: is

text: the

text: same

text: be-

text: −k(δij)

text: δij

text: rˆij

text: δij

text: fore

text: and

text: after

text: the

text: ﬁrst

text: contact

text: change

text: and

text: changes

text: ij

text: δ

text: ij

text: in

text: the

text: density

text: of

text: states

text: are

text: negligible

text: These

text: re-

text: sults

text: show

text: that

text: linear

text: elastic

text: constitutive

text: relations

text: remain

text: The

text: prefactor

text: k

text: is

text: the

text: contact

text: stiﬀness,

text: which

text: generally

text: useful

text: near

text: jamming,

text: but

text: they

text: say

text: nothing

text: about

text: when

text: depends

text: on

text: the

text: overlap

text: Hooke’s

text: law

text: eventually

text: does

text: break

text: down.

text: Recent

text: experiments

text: and

text: simulations

text: provide

text: evidence

text: for

text: a

text: two

text: stage

text: yielding

text: process,

text: where

text: k

text: δα

text: packings

text: ﬁrst

text: soften

text: and

text: only

text: later

text: establish

text: steady

text: shear

text: ﬂow.

text: Yet

text: it

text: remains

text: unclear

text: precisely

text: how

text: rate

text: depen-

text: Here

text: k

text: is

text: a

text: constant

text: and

text: α

text: is

text: an

text: exponent

text: parameter-

text: dence,

text: nonlinearity,

text: and

text: contact

text: changes

text: contribute

text: to

text: izing

text: the

text: interaction.

text: In

text: the

text: following

text: we

text: consider

text: har-

text: the

text: breakdown

text: of

text: linear

text: elasticity.

text: In

text: order

text: to

text: unravel

text: monic

text: interactions

text: (α

text: which

text: provide

text: a

text: reasonable

text: these

text: eﬀects,

text: it

text: is

text: necessary

text: to

text: vary

text: strain,

text: strain

text: rate,

text: model

text: for

text: bubbles

text: and

text: droplets

text: that

text: resist

text: deformation

text: pressure,

text: and

text: system

text: size

text: simultaneously

text: and

text: systemati-

text: due

text: to

text: surface

text: tension;

text: we

text: also

text: treat

text: Hertzian

text: interac-

text: cally

text: as

text: we

text: do

text: here

text: for

text: the

text: ﬁrst

text: time.

text: Using

text: simulations

text: tions

text: (α

text: which

text: correspond

text: to

text: elastic

text: spheres.

text: of

text: viscous

text: soft

text: spheres,

text: we

text: ﬁnd

text: that

text: Hooke’s

text: law

text: is

text: valid

text: We

text: perform

text: simulations

text: using

text: two

text: separate

text: numerical

text: within

text: a

text: surprisingly

text: narrow

text: window

text: bounded

text: by

text: viscous

text: methods.

text: The

text: ﬁrst

text: is

text: a

text: molecular

text: dynamics

text: (MD)

text: algo-

text: eﬀects

text: at

text: small

text: strain

text: and

text: nonlinear

text: softening

text: at

text: large

text: rithm

text: that

text: integrates

text: Newton’s

text: laws

text: using

text: the

text: velocity-

text: strain.

text: The

text: size

text: of

text: the

text: linear

text: elastic

text: window

text: displays

text: Verlet

text: scheme.

text: Each

text: disk

text: is

text: assigned

text: a

text: uniform

text: mass

text: power

text: law

text: scaling

text: with

text: pressure

text: and

text: correlates

text: with

text: the

text: m

text: proportional

text: to

text: its

text: volume.

text: Energy

text: is

text: dissi-

text: accumulation

text: of

text: not

text: one,

text: but

text: an

text: extensive

text: number

text: of

text: con-

text: i

text: i

text: pated

text: by

text: viscous

text: forces

text: that

text: are

text: proportional

text: to

text: the

text: rel-

text: tact

text: changes.

text: ative

text: velocity

text: c

text: of

text: neighboring

text: particles

text: evaluated

text: at

text: The

text: basic

text: scenario

text: we

text: identify

text: is

text: illustrated

text: in

text: Fig.

text: ij

text: the

text: contact,

text: which

text: presents

text: ensemble-averaged

text: stresses

text: versus

text: strain.

text: Shear

text: is

text: applied

text: via

text: a

text: constant

text: strain

text: rate

text: γ˙

text: at

text: ﬁxed

text: volume.

text: We

text: identify

text: three

text: characteristic

text: scales,

text: each

text: of

text: −τ

text: k(δ

text: c

text: ij

text: ij

text: ij

text: which

text: depend

text: on

text: the

text: initial

text: pressure

text: p:

text: (i)

text: For

text: strains

text: below

text: γ

text: γ˙

text: τ

text: where

text: τ

text: is

text: a

text: time

text: scale,

text: viscous

text: stresses

text: are

text: signiﬁcant

text: and

text: Eq.

text: underestimates

text: the

text: where

text: τ

text: is

text: a

text: microscopic

text: relaxation

text: time.

text: Viscous

text: stress

text: needed

text: to

text: deform

text: the

text: material.

text: This

text: crossover

text: forces

text: can

text: apply

text: torques,

text: hence

text: particles

text: are

text: allowed

text: to

text: strain

text: vanishes

text: under

text: quasista†tic

text: shear

text: (γ˙0

text: ﬁlled

text: rotate

text: as

text: well

text: as

text: translate.

text: squares).

text: (ii)

text: Above

text: a

text: strain

text: γ

text: the

text: material

text: softens

text: and

text: In

text: addition

text: to

text: MD,

text: we

text: also

text: perform

text: simulations

text: using

text: Hooke’s

text: law

text: overestimates

text: the

text: stress.

text: This

text: crossover

text: is

text: a

text: nonlinear

text: conjugate

text: gradient

text: (CG)

text: routine

text: which

text: rate-independent,

text: consistent

text: with

text: plastic

text: eﬀects.

text: (iii)

text: keeps

text: the

text: system

text: at

text: a

text: local

text: minimum

text: of

text: the

text: potential

text: en-

text: For

text: strain

text: rates

text: above

text: a

text: scale

text: γ˙

text: (triangles),

text: Eq.

text: is

text: ergy

text: landscape,

text: which

text: itself

text: changes

text: as

text: the

text: system

text: under-

text: never

text: accurate

text: and

text: there

text: is

text: no

text: strain

text: interval

text: where

text: the

text: goes

text: shearing.

text: The

text: dynamics

text: are

text: therefore

text: quasistatic,

text: material

text: responds

text: as

text: a

text: linear

text: elastic

text: solid.

text: i.e.

text: the

text: particle

text: trajectories

text: correspond

text: to

text: the

text: limit

text: of

text: vanishing

text: strain

text: rate.

text: Bubble

text: packings

text: consist

text: of

text: N

text: to

text: disks

text: in

text: a

text: bidisperse

text: mixture

text: with

text: a

text: diameter

text: ratio.

text: I.

text: SOFT

text: SPHERES:

text: MODEL

text: AND

text: Shear

text: is

text: implemented

text: via

text: Lees-Edwards

text: “sliding

text: brick”

text: BACKGROUND

text: boundary

text: The

text: stress

text: is

text: given

text: by

text: We

text: ﬁrst

text: introduce

text: the

text: soft

text: sphere

text: model

text: and

text: summa-

text: rize

text: prior

text: results

text: regarding

text: linear

text: elasticity

text: near

text: jamming.

text: σ

text: f

text: r

text: m

text: v

text: v

text: αβ

text: ij,α

text: ij,β

text: i

text: i,α

text: i,β

text: V

text: A.

text: Model

text: ij

text: i

text: We

text: perform

text: numerical

text: simulations

text: of

text: the

text: Durian

text: bub-

text: where

text: V

text: is

text: the

text: volume

text: (area

text: in

text: two

text: dimensions)

text: of

text: ble

text: model

text: a

text: mesoscopic

text: model

text: for

text: wet

text: foams

text: and

text: the

text: packing,

text: is

text: the

text: sum

text: of

text: elastic

text: and

text: viscous

text: contact

text: ij

text: emulsions.

text: The

text: model

text: treats

text: bubbles/droplets

text: as

text: non-

text: forces

text: acting

text: on

text: particle

text: i

text: due

text: to

text: particle

text: j,

text: and

text: is

text: the

text: i

text: Brownian

text: disks

text: that

text: interact

text: via

text: elastic

text: and

text: viscous

text: forces

text: velocity

text: of

text: particle

text: i.

text: Greek

text: indices

text: label

text: components

text: when

text: they

text: overlap.

text: Elastic

text: forces

text: are

text: expressed

text: in

text: terms

text: along

text: the

text: Cartesian

text: coordinates

text: x

text: and

text: y.

text: The

text: conﬁn-

text: of

text: the

text: overlap

text: δ

text: r

text: /(R

text: R

text: where

text: R

text: and

text: R

text: ing

text: pressure

text: is

text: p

text: σ

text: where

text: D

text: is

text: ij

text: ij

text: i

text: j

text: i

text: j

text: xx

text: yy

text: denote

text: radii

text: and

text: points

text: from

text: the

text: center

text: of

text: particle

text: i

text: the

text: spatial

text: dimension,

text: while

text: the

text: shear

text: stress

text: is

text: σ

text: σ

text: ij

text: xy

text: to

text: the

text: center

text: of

text: j.

text: The

text: force

text: is

text: repulsive

text: and

text: acts

text: along

text: The

text: second

text: term

text: on

text: the

text: righthand

text: side

text: of

text: Eq.

text: is

text: a

text: ki-

text: the

text: unit

text: vector

text: rˆ

text: /r

text: netic

text: stress,

text: which

text: is

text: always

text: negligible

text: in

text: the

text: parameter

text: ij

text: ij

text: ij

text: ##LTLine##

text: ##LTLine##


text: 7 100 100 101 p = 10−7.0 p = 10−5.0 a) p = 10−4 10−1 b) p = 10−6.5 p = 10−4.5 10−1 10−2 p pp == 1100−−65..05 p = 10−4.0 N = 1024 10−2 10−3 pp==1100−−54.5 100 10−3 NN ==122586 1100−−54 ppp===111000−−−433.5 N =512 N = 1024 p=10−2.5 1100−−21 10−140−610−510−410−3N10=−1022140−1 10−160−610−510−410−31p=01−0−2210−1 10−3 10−4 γ γ 10−1 10−5 10−6 p 103 10−7 ∆=0.6 ∆=0.3 c) 10−8 ∆∆==00..54 slope=2/3 102 10−9 10−8 10−7 10−6 10−5 10−4 10−3 p 101 10−2 10−3 10−2 10−1 100 101 102 γ/p2/3 100 10−1 FIG. 6. (main panel) The dimensionless shear modulus of τ = 0.5 quasistatically sheared Hertzian packings plotted versus the 10−2 p = 10−5 ... 10−2 rescaled strain γ/p2/3. (inset) Pressure-dependence of the N = 128 ... 1024 crossover strain γ†. 10−3 10−4 10−4 10−3 10−2 10−1 100 101 102 103 104 isostatic contact number zc. In so doing we will answer γ/p the question ﬁrst posed by Schreck and co-workers [25], who asked how many contact changes a packing can ac- FIG. 7. The contact change density shown for (a) varying cumulate while still displaying linear elastic response. system size and (b) varying pressure. (c) Data collapse for We begin by investigating the ensemble-averaged con- pressures p = 10−2 . . . 10−5 in half decade steps and system tact change density ncc(γ) ≡ [Nmake(γ) + Nbreak(γ)]/N, sizes N = 128 . . . 1024 in multiples of 2. Dashed lines indicate where Nmake and Nbreak are the number of made and bro- slopes of 1 and 1/2. ken contacts, respectively, accumulated during a strain γ. Contact changes are identiﬁed by comparing the contact (cid:32) (cid:33) network at strain γ to the network at zero strain. In Fig. 7a we plot n for packings of harmonic particles − cc at pressure p = 10 4 and varying system size. The data 1 γ n (γ) (cid:39) as γ → 0 . (15) collapse to a single curve, indicating that ncc is indeed cc N γ(1) cc an intensive quantity. The eﬀect of varying pressure is shown in Fig. 7b. There are two qualitatively distinct From Fig. 7 it is apparent that n remains linear in regimes in ncc, with a crossover governed by pressure. † cc (1) γ up to the crossover strain γ . We conclude that γ To better understand these features, we seek to col- cc describes the strain between successive contact changes lapse the ncc data to a master curve. By plotting N ≡ † n /pτ versus y ≡ γ/p, we obtain excellent collapse for over the entire interval 0 ≤ γ < γ . In the softening cc regime the strain between contact changes increases; it τ = 1/2, as shown in Fig. 7b for the same pressures as in scales as n ∼ γ1/2 (see Fig. 7c). Fig. 7a and system sizes N = 128 . . . 1024. The scaling cc function N ∼ y for small y, while N ∼ yτ for y1. The †Let us now re-interpret the softening crossover strain γ ∼ ∆z2 (c.f. Eq. (6)) in terms of the coordination of rescaled strain y provides further evidence for a crossover † the contact network. We recall that ∆z = z − z is the scale γ ∼ p/k, now apparent at the microscale. More- c diﬀerence between the initial contact number z and the over, the fact that data for varying system sizes all col- isostatic value z , which corresponds to the minimum lapse to the same master curve is an important indicator c † number of contacts per particle needed for rigidity. The that γ is an intensive strain scale that remains ﬁnite in excess coordination ∆z is therefore an important char- the large system size limit. acterization of the contact network. The contact change The scaling collapse in Fig. 7c generalizes the results † density at the softening crossover, n , can be related to of Van Deen et al. [30], who determined the strain cc ∆z via Eq. (15), while making use of Eq. (6), scale γ(1) ∼ (p/k)1/2/N associated with the ﬁrst con- cc tact change. To see this, note that the inverse slope † † (dγ/dncc)/N represents the average strain interval be- n ≡ n (γ ) ∼ ∆z . (16) cc cc tween contact changes at a given strain. Hence the initial (1) slope of n is ﬁxed by γ : cc cc γ0G/σ †γ ccn τp/ccn n cc ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: 4 G. F. Wong et al. Table 1. Table showing the diﬀerent sources observed, the date observed, the amount of time taken to observe, map size in arcmin, central position of the map and reference position. Region Date Time Map Size Map Size Central map position Reference position × Observed Duration (R.A. Dec) (Pixels) R.A. (J2000) Dec (J2000) R.A. (J2000) Dec (J2000) (cid:48)× (cid:48) × OriA 2013-09-24 2.4 h 18.9 17.5 42 36 05:35:19.2 –05:18:47.7 05:40:30.0 –05:16:16.0 (cid:48)× (cid:48) × SgrA 2013-09-23 2.8 h 20.4 16.0 42 42 17:45:58.9 –28:58:22.1 17:55:00.0 –28:00:00.0 (cid:48)× (cid:48) × 2013-09-24 1.3 h 20.4 10.5 42 21 17:45:58.9 –28:43:28.3 17:55:00.0 –28:00:00.0 90 64 -5°15'00" -28°40'00" 75 56 18'00" 60 45 45'00" 48 21'00" 30 40 24'00" 15 50'00" 0 32 27'00" 45s 30s 15s 35m00s 5h34m45s Right Ascension (J2000) 24 55'00" Figure 3. Integrated H92α emission map towards OriA; con- 16 tours at 20, 40, 60...140 Kkms1. 8 -29°00'00" days; data from each day were processed separately before 0 merging the datasets during the imaging stage. H92α inte- 30s 15s 46m00s 45s 30s 17h45m15s Right Ascension (J2000) grated and channel maps towards Sgr A are shown in Figs. 4 and 5 respectively. While the data cube has a channel width − of 1.13 km s 1, the channel map in Fig. 5 uses a channel − Figure 4. Integrated H92α emission map with incomplete width of 15 km s 1. Because of the complex emission struc- continuum subtraction towards SgrA; contours at 20, 30, − ture of the region, an attempt was made to select a line-free 40...70 Kkms 1. channel range to create a polynomial to remove continuum emission from the spectra. Examining the attempted con- tinuum subtraction, we see the Northern region with con- (Fig. 6 and Table 2) may be compared to a previous ob- tinuum emission succ−essfu◦lly(cid:48)removed. However, around the servation of the same transition by Cesarsky (1973), which Southern peak (δ < 28 55 ) broadband emission can still used the DSN Goldstone antenna, then 64 m in aperture, (cid:48) be seen. Continuum emission is still present within Figs. 4 with beamwidth of 2.5 . No pointing coordinates are given, and 5. but the published line parameters (T = 4.65 K, V = ± − ± − A LSR 4 DISCUSSION –2.7 2 km s 1, ∆ν = 26 2 km s 1), agree with our results (Table 2) within the error intervals, except for line ampli- 4.1 Orion A tude; but even here, the discrepancy of 0.42 K is less than Our data constitute the ﬁrst fully-sampled single-dish 10%. RRL map of Ori A, though sparsely-sampled maps made Fig. 7 is replotted from Sorochenko (1965), with obser- up of selected beam positions have previously been pub- vational data points added by taking the line brightnesses lished by Jaﬀe & Pankonin (1978) and by Peimbert et al. from Table 3 divided by the emission measure of the Hii re- × − (1988); the RRL structure (Fig. 3) is similar to that seen in gion (EM of 4.0 106 cm 6 pc taken from McGee & Newton (cid:48)(cid:48) a 23 GHz continuum map (resolution 42 , Wilson & Pauls 1981), and shows the variation of RRL brightness with elec- 1984). Many hydrogen RRL transitions have been observed tron density. All the lines denoted by pentagons are observed with single-dish telescopes towards Ori A, mostly towards at similar positions, and the ﬁlled pentagon is the value similar positions, so that the beams overlap. Table 3 lists taken from our map. Triangles denote observations with no H(n)α RRL transitions observed towards positions close to position given. The model curves show that we should expect the peak brightness that we observe in Ori A. The spec- signiﬁcant changes in the relative line brightnesses around − trum corresponding to the peak brightness in our H92α map 10 GHz as the electron density increases from 100 cm 3 MNRAS 000, 1–8 (2015) )0002J( noitanilceD 1−s mK.K )0002J( noitanilceD 1−s mK.K ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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