Local structure controls shear and bulk moduli in disordered solids

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by  M. Schlegel

Local structure controls shear and bulk moduli in disordered solids
M. Schlegel1, J. Brujic2, E. M. Terentjev3 and A. Zaccone4
1Department of Engineering, University of Cambridge, Trumpington Street, CB2 1PZ Cambridge, U.K.
2Physics Department, New York University, New York, NY 10003, USA
3Cavendish Laboratory, University of Cambridge,
JJ Thomson Avenue, CB3 0HE Cambridge, U.K. and
4Department of Chemical Engineering and Biotechnology, University of Cambridge,
New Museums Site, Pembroke Street, CB2 3RA Cambridge, U.K.
Paradigmaticmodelsystems,whichareusedtostudythemechanicalresponseofmatter,arerandom
networks of point-atoms, random sphere packings, or simple crystal lattices; all of these models as-
sumecentral-forceinteractionsbetweenparticles/atoms. Eachofthesemodelsdiffersinthespatial
6 arrangement and the correlations among particles. In turn, this is reflected in the widely different
1 behaviours of the shear (G) and compression (K) elastic moduli. The relation between the macro-
0 scopic elasticity as encoded in G, K and their ratio, and the microscopic lattice structure/order,
2 is not understood. We provide a quantitative analytical connection between the local orientational
order and the elasticity in model amorphous solids with different internal microstructure, focusing
n
onthetwooppositelimitsofpackings(strongexcluded-volume)andnetworks(noexcluded-volume).
a
J The theory predicts that, in packings, the local orientational order due to excluded-volume causes
less nonaffinity (less softness or larger stiffness) under compression than under shear. This leads
3
to lower values of G/K, a well-documented phenomenon which was lacking a microscopic explana-
1
tion. Thetheoryalsoprovidesanexcellentone-parameterdescriptionoftheelasticityofcompressed

] emulsions in comparison with experimental data over a broad range of packing fractions.
t
f
o PACSnumbers:
s
.
t
a Oneoftheoverarchinggoalsofsolidstatephysicsisto vanishes. This state of affairs has been revealed in simu-
m find a universal relationships between the lattice struc- lationstudies2,4,atleastsincethe1970’s5. Furthermore,
- ture of matter in the solid state and its mechanical re- the same phenomenon is well documented also in disor-
d sponse. From this point of view, it is important to sim- dered atomic solids6 and non-centrosymmetric crystals
n plify the details of the interactions between the building (e.g. piezoelectrics) 7.
o
blocks (atoms, particles) in order to single out the rele- However,thereisnomechanisticunderstandingofthis
c
[ vant physics and general laws. The framework of lattice phenomenon, nor analytical theories able to describe
dynamics successfully provided the link between atomic- it, beyond the somewhat obvious observation that the
1
levelstructureandmacroscopicpropertiesofsimplecrys- internal structure of packings is different from that
v
tal lattices1. Our understanding is instead much more of random networks, due to the self-organization and
2
1 limitedwhenstructuraldisorderplaysanimportantrole, mutual excluded-volume of particles in the packing,
3 such as in glasses, liquids and other disordered states of which are absent in random isotropic networks. Below
3 matter2,3. we provide a quantitative connection between structure
0 With the advent of computer simulations, it became and elasticity based on nonaffine lattice dynamics which
.
1 clear that disordered solids, which are of paramount im- shows that the local self-organization of the particles
0 portance in many areas of technology and life sciences, with excluded-volume leads to a higher degree of bond-
6 cannot be described simply as perturbations about the orientational order8,9 in randomly packed structures
1 crystalline order. In this context, an unsolved problem compared to isotropic random networks. In turn, this
:
v is the striking difference in the elastic deformation be- leads to a significantly higher bulk modulus and a lower
i haviour of random networks and random packings. For nonaffinity under compression.
X
networks, the shear modulus G and the compression
r
a modulus K display the same dependence on the coor- Results
dination number z which represents the average number Nonaffine lattice dynamics. Our main tool is the
of elastic springs per node of the network. Therefore, Born-Huang free energy expansion10, suitably modified
G ∝ K ∝ (z−z ), and both moduli vanish at the same to account for the structural disorder in terms of the
c
critical coordination z which is dictated by isostaticity. nonaffinityofthedisplacements(asexplainedbelow). In
c
It is different for random packings where only the shear order to make analytical calculations, we neglect the ef-
modulusscaleslinearly asG∝(z−z ),whereasthebulk fectofthermalfluctuations(i.e. weoperateintheather-
c
modulusvanishesonlyatacoordinationmuchlowerthan mallimit,whichisapplicabletogranularsolidsandnon-
z . Thismeansthatpackingshaveacomparativelylarger Brownian emulsions), and we focus on harmonic central-
c
bulk modulus, with respect to random networks, and re- forceinteractionsbetweentheparticles. Thusweneglect
main well stable against compression also near, at, and boththebendingresistancewhentheparticlesslidepast
even below the critical coordination where shear rigidity eachother,aswellastheeffectofstressedbonds. Itisim-
2
portant to emphasize that both these effects can provide Theory of elastic moduli. Upon carrying out the
rigidity to certain lattices, which are otherwise floppy formaltreatmentwiththestandarddynamical(Hessian)
or unstable when only central forces between atoms are matrix18 H andtheexpressionforthedisorder-induced
ij
active. This fact is well known e.g. in the context of force (defined for the example of shear deformation γ in
inorganic network glasses10,11. the{xy}planeasf =Ξxyγ), thenonaffinecontribution
i i
The key to understanding the elasticity of amorphous to the free energy of deformation can be evaluated as
lattices is nonaffinity10. In a nutshell: the applied ex- shown in several places in the recent literature13,14. It
ternal deformation induces a deformation at the micro- has been shown that the elastic constants are given by
scopic level of interatomic bonds. If the interatomic dis- C =CA −CNA with the nonaffine correction due
ιξκχ ιξκχ ιξκχ
placementsaresimplyproportionaltotheappliedoverall to disorder given as
deformation field, then the deformation is called affine,
andonecanexpandthefreeenergyinpowersofsmallin- (cid:88) (cid:16) (cid:17)−1
CNA = Ξιξ H Ξκχ. (1)
teratomicdisplacementsandtakethecontinuumlimitof ιξκχ i ij j
ij
themicroscopicdeformationforeithersheardeformation
or compression1. In other words, the microscopic inter-
The affine part of the elastic constants is pro-
particle displacements are directly proportional to, and
vided by the affine Born-Huang lattice dynamics,
uniquely determined by the applied macroscopic strain.
which is exact for centrosymmetric lattices: CA =
Differentiating the free energy twice with respect to the ιξκχ
1 R2κ(cid:80) nι nκnξ nχ. Here κ is the effective spring
macroscopic strain yields the shear modulus G and the 2V 0 ij ij ij ij ij
bulk modulus K, depending on the geometry of the ap- constant of the interatomic (interparticle) interaction,
plied deformation (shear or hydrostatic compression, re- which is harmonic near the equilibrium, V is the total
spectively). volume of the system, and R0 is the equilibrium sepa-
As was first realized by Lord Kelvin12, and more re- ration length between nearest neighbours spheres of di-
cently emphasized by Alexander10 and Lemaitre and ameter σ. nιij is the ι = x,y,z Cartesian coordinate of
Maloney13,theaffineapproximationisstrictlyvalidonly the unit vector which defines the orientation of the bond
betweentwobondedneighboursiandj. Inthenonaffine
forcentrosymmetriccrystallattices. Thereasonbecomes
relaxationterm, theforceperunitstrainactingonevery
evident if one considers the forces which are transmitted
atom is given analytically, for the case of shear deforma-
to a test atom in the lattice upon deforming the solid.
tion, by13 Ξxy = −R κ(cid:80) n nxny. It is easy to check
Every neighbour transmits a force which is cancelled by i 0 j ij ij ij
thelocal inversionsymmetryinthecentrosymmetricBra- thatΞxiy =0foracentrosymmetriclattices. Asshownin
vais cell (see Fig.1a below). As a result, there is no local Ref.14,undertheassumptionofcentral-forceinteraction,
netforceactingontheatomsofthelatticeintheiraffine andforarandomnetworkofequalharmonicspringswith
positions, and the old affine free energy expansion1 suf- numberdensityofnodesN/V,theshearmoduluscanbe
ficestocorrectlydescribetheelasticdeformation. Witha evaluated analytically as
disordered or non-centrosymmetric lattice, the situation
1 N
is different. The forces that every atom receives from G=G −G = κR2(z−z ). (2)
its neighbours no longer cancel, because the local inver- A NA 30V 0 c
sion symmetry is violated. The net force acting on every
Theproportionalitytoz iscontributedbytheaffineterm
atom has to be relaxed via additional atomic displace-
CA above,wherethesum(cid:80) nι nκnξ nχ canbeeval-
ments, called nonaffine displacements13. These motions, xyxy ij ij ij ij ij
under the action of the disorder-induced local forces, are uated in mean-field averaging, (1/2)(cid:80)ijnxijnyijnxijnyij (cid:39)
associated with a total work, which is an internal work (zN/2)(cid:104)nxnynxny(cid:105), where the quantity (zN/2) repre-
ij ij ij ij
done by the system (hence negative, by thermodynamic sents the total number of bonds in the system. The fac-
(cid:80)
convention). tor1/2infrontofthe ...isrequiredbecausethesum
ij
Theworkdonebynonaffinedisplacementsrepresentsa counts the bonds twice. Further, (cid:104)nxnynxny(cid:105) = 1/15
ij ij ij ij
quoteofinternallatticeenergywhichcannotbeemployed for a random isotropic distribution of bond orientations.
to react to the applied deformation. Therefore, the free The nonaffinity of the amorphous solid is encoded in
energy of deformation can be written as F =F −F , thequantityCNA ∝−z ,whichdefinesthecriticalnum-
A NA ιξκχ c
to distinguish the affine contribution F from the non- ber z = 2d = 6 of bonds at which the shear modulus
A c
affine contribution due to disorder14,15, −F . The vanishesbyvirtueofthenon-affinesofteningmechanism.
NA
fact that non-centrosymmetric lattices (e.g. piezoelec- This result is valid for random networks where bonds
tric crystals) are affected by nonaffine distortions of the haverandomlydistributedorientationsinthesolidangle.
primitive cell16,17, however, does not necessarily mean Inthatmodel,anybond-orientationalorderparameteris
that they are unstable or soft. These materials are, of identically zero and the average rotational symmetry is
course, fully rigid and do exhibit a large value of shear isotropic. For the more general case where correlations
modulus, provided that they have a sufficient atomic co- between bond-orientation vectors of nearest-neighbours
ordination, well above the isostatic limit, and a fairly are important, it can be shown (see the Supplementary
large value of spring constants. Information) that the nonaffine correction term reduces
3
d e
a b c
FIG. 1: Geometry of particles, bonds and forces: (a) In a centrosymmetric lattice the forces acting on every particle
cancel by symmetryand leavethe particle force-free. Henceno additionaldisplacementsare requiredto keeplocalmechanical
equilibrium on top of the affine displacements dictated by the applied strain. (b) In a jammed packing, there is a remarkable
degree of local orientational order: due to excluded-volume correlations it can still happen that two particles make an angle
equal to 180o across the common neighbour at the center of the frame, leading to cancellation of local forces. This effect is
significant under compression, thanks to isotropy, but negligible under shear. (c) In a random network, the probability of
having this cancellation of forces is much smaller. In this case, nonaffine displacements are required on all particles (nodes)
to keep local equilibrium under the non-vanishing sum of nearest-neighbour forces. This limit has the strongest nonaffinity
and the lowest values of elastic moduli. (d) The excluded volume cone: a bond, for example along the z-axis, leads to an
excluded-conewherenothirdparticlecanexist. R istheequilibriumbonddistance,σrepresentsthediameteroftheparticles.
0
(e)Theframe-rotationtricktoevaluatethecontributionsoflocalexcluded-volumecorrelationstothenonaffineelasticmoduli.
Here, for simplicity, only the special case of φ =φ =0, i.e. both ij and iq lying in the plane xz, has been illustrated.
ij iq
tothefollowingform,afterreplacingthesumoverbonds negative correction to the elastic constants, thus soften-
by the average: ing the material. The random network is thus the oppo-
siteextremetotheperfectcentrosymmetrichardcrystal.
N (cid:88)
CNA =κR23 (A +B ), (3) In the random network model, which served for
ιξκχ 0 V α,ιξκχ α,ιξκχ
long time as a structural model for many inorganic
α=x,y,z
glasses11,19,thenodesarejustpoint-atomswithzerovol-
whereA ≤0andB ≥0aredefinedasfollows: ume,σ =0. Thisisaveryimportantfeaturebecausethe
α,ιξκχ α,ιξκχ
absence of any excluded-volume hindrance between such
atoms allows them to be placed at random positions in
(cid:68) (cid:69)
A = nαnι nξ nαnκnχ (4a) space. Suchamodelisclearlyapplicableonlytosystems
α,ιξκχ ij ij ij iq iq iq
wherethebondlengthismuchlargerthantheatomicdi-
(cid:68) (cid:69)
Bα,ιξκχ = nαijnιijnξijnαijnκijnχij . (4b) ameterσ (whichisthecasefornetworkglassesandsome
amorphous semiconductors). The limit σ/R → 0 thus
0
Here (cid:104)...(cid:105) represents an angular average, in the solid an- correspondstotherandomnetworkmodel. Theopposite
gle, over the orientations of bonds ij and iq as explained limit, σ/R0 = 1, corresponds to the jammed packing,
in15. It is important to note that in A , we average wheresphericalparticlesarebarelytouchingtheirneigh-
α,ιξκχ
overallpossibleorientationsoftwobondstotheatomsj bours. In this limit, the excluded-volume repulsion be-
andq,respectively,measuredfromacommonatomi. For tweenspheresinclosecontactplaysaveryimportantrole
the average in B , one only needs to consider bonds in the self-organization and in the local structure of the
α,ιξκχ
between the particles i and j as discussed in14. Hence, packing. Inparticular,duetoexcluded-volume,thereare
it is evident that A is non-zero only if the orienta- restrictions on the available portion of solid angle where
α,ιξκχ
tions of the two bonds ij and iq are correlated (that is, a nearest-neighbour can sit. It is therefore significantly
theorientationofij doesdependontheorientationofiq, morelikely,incomparisonwiththerandomnetworkcase,
and vice versa). If there is no correlation, meaning that that a particle j makes an angle of 180o with a particle
given a certain orientation of iq in the solid angle, ij can q directly across a third particle i placed at the center of
have any random orientation in the solid angle with the the frame (Fig.1b), due to the existence of sectors in the
same probability, then A = 0. This is so because solid angle (as measured from the central particle) that
α,ιξκχ
the average can be factored out into the product of two areforbidden. Hence,thelocalorientationalorderinthe
averagesoftripletseachofthetype(cid:104)nαnι nξ (cid:105),andeach jammedpacking, welldocumentedinpreviousstructural
ij ij ij studies8,9, isimportantalsointhedeterminationofelas-
angular average vanishes separately, as one can verify by
tic moduli. In the following we are going to focus our
insertion.
detailed calculations on the jammed packing limit with
The limit where any two bonds ij and iq are uncor-
R =σ.
related, and A = 0, defines the geometry of the 0
α,ιξκχ
random network4 (Fig.1c). The random network limit Weimplementedaminimalmodel,inspiredbythegra-
represents the case where nonaffinity makes the largest nocentricmodelofgranularpackings20,fortheexcluded-
4
volume correlations which allows an explicit evaluation is:
of the two-bond angular-correlation terms A for
α,ιξκχ  
−sin(φ )
jammed packings. If the bond iq has a given orienta- e ×n iq
tion in the solid angle, parameterised by the pair of an- t= z iq = cos(φiq) . (6)
|e ×n |
gles {ϕ ,θ } then, clearly, the bond ij can have any z iq 0
iq iq
orientation in the solid angle apart from those orienta-
The rotation matrix R is defined by the Rodrigues’
tions delimited by the excluded cone depicted in Fig.1d.
The angular average for the orientation of ij is thus re- formula25
stricted to the total solid angle Ω minus the excluded (cid:2) (cid:3)
R=cos(θ )1+sin(θ ) t +(1−cos(θ ))t⊗t (7)
cone, which gives the allowed solid angle as Ω−Ωcone, iq iq × iq
with Ω = π(σ/R )2. The probability density distri-
cone 0
where 1 represents the identity matrix. Further, we de-
bution ρ of bond orientations is taken to be isotropic
fined
for iq, that is ρ = 1/4π. For ij, instead, the proba-
iq
bility that it takes a certain orientation is a conditional  
0 −t t
z y
one, because it depends on the orientation of iq. Hence, (cid:2) (cid:3)
t = tz 0 −tx (8)
the conditional probability for the orientation of ij is ×
−t t 0
y x
ρ (Ω | Ω ) = 1/(4π − Ω ), for Ω ∈ Ω−Ω ,
ij ij iq cone ij cone  
and ρij(Ωij | Ωiq) = 0 for Ωij ∈ Ωcone. In the sec- 0 0 cos(φiq)
tion below we use these considerations to evaluate the = 0 0 sin(φiq).
excluded-volume correction to the nonaffine moduli en- −cos(φiq) −sin(φiq) 0
coded in A .
α,ιξκχ
Next, we look at the integral I defined as:
Evaluation of the excluded-volume correlations αιξ
term in the moduli. The excluded-volume correlation (cid:90)
term contributing to the elastic moduli is given by Iαιξ = nαijnιijnξijsin(θij)dθijdφij. (9)
Ω−Ωcone
(cid:68) (cid:69)
Aα,ιξκχ = nαijnιijnξijnαiqnκiqnχiq (5a) This integral occurs in the expression for Aα,ιξκχ, and
considering that ρ (Ω | Ω ) = const in the allowed
(cid:90) (cid:90) ij ij iq
= ρij(Ωij |Ωiq)ρiq(Ωiq)nαiqnκiqnχiq solid angle Ω−Ωcone for ij, we have factored ρij(Ωij |
Ω Ω−Ωcone Ωiq) = const out of the ij integral leaving a product
×nαnι nξ dΩ dΩ . between Iαιξ and ρij(Ωij | Ωiq) inside the integral of
ij ij ij ij iq
Eq.5(b),
(5b)
(cid:90)
A = I ρ (Ω |Ω )ρ (Ω )nαnκnχdΩ .
Toevaluatetheaboveintegralitisnecessarytofirstiden- α,ιξκχ αιξ ij ij iq iq iq iq iq iq iq
Ω
tify the correlation between ij and iq and then devise a
(10)
strategy to evaluate the integral in the above equation.
As is shown in the SI, in the new rotated frame, one
A solution can be found by exploiting the symmetry obtains:
of the problem, and, in particular, the rotational invari-
ance. ThelocalCartesianframecenteredontheparticlei I =(cid:90) π (cid:90) 2π nαnι nξ sin(cid:16)θ˜ (cid:17)dθ˜ dφ˜ . (11)
isrotatedsuchthatthez-axis(fromwhichtheazimuthal αιξ θ˜ij=θmin φ˜ij=0 ij ij ij ij ij ij
angles θ and θ are measured) is brought to coincide
ij iq
with the unit vector niq defining the orientation of the θmin is determined by the excluded volume cone as
bond iq (see Fig.1e for illustration of the special case θmin =2ψ =2·arcsin(σ/2R0).
where iq and ij lie in the xz plane). This trick reduces We recall that nα is defined as the α Cartesian co-
ij
the number of variables in the problem: instead of deal- ordinate of the bond unit vector n and is related to
ij
ing with two sets of angles, {ϕ ,θ } and {ϕ ,θ }, we the bond unit vector of the rotated frame n via
ij ij iq iq ij,rot
need to consider only one set {ϕ˜ij,θ˜ij}, which gives the nij = R · nij,rot, with R given by Eq.(7). The bond
orientation of the bond ij in the rotated frame. Upon unit vector in the rotate frame nij,rot is defined by the
suitably defining the rotation matrix, the above integral pairofanglesθ˜ ,φ˜ whichrepresenttheintegrationvari-
ij ij
is much simplified. ables in Eq.(11). Therefore, we can now use Eq.(11) to-
Therotationisdefinedaroundanaxist(paralleltoe gether with Eq.(10) to arrive at the following expression
y
in the special case of φij =φiq =0 illustrated in Fig.1e), for Aα,ιξκχ:
and perpendicular to both e and n , with an angle of
z iq (cid:90) π (cid:90) 2π (cid:90) π (cid:90) 2π
θiq (usual convention of rotation: counter clockwise if A = ρ ρ nαnκnχ
axis vector points in the direction of the viewer). Here, α,ιξκχ θiq=0 φiq=0 θ˜ij=2ψ φ˜ij=0 ij iq iq iq iq
e and e denote the unit vectors along the y and z axis, (cid:16) (cid:17)
y z ×nαnι nξ sin θ˜ sin(θ )dθ˜ dφ˜ dθ dφ .
respectively, of the Cartesian frame centered on particle ij ij ij ij iq ij ij iq iq
i. Therefore, the unit vector t defining the rotation axis (12)
5
With the last Eq.(12), we have reduced the original in- The latter is strongly anisotropic and causes the forces
tegral for A to a much simpler integral with well- transmittedbyneighbourstobemisalignedsuchthatthe
α,ιξκχ
definedintegrationlimitsinthesolidangle. Theintegral cancellationofnearest-neighbourforceswithsameorien-
canbeeasilyevaluatedusingρ =1/4π,whichaccounts tationandoppositedirectionisnotaseffective. Ourthe-
iq
for the fact that the orientation of iq can be freely cho- oretical predictions match the known effect of vanishing
sen, whereas ρ (Ω |Ω )=1/(4π−Ω )=1/3π due oftheratioG/K attherigiditytransition4 z =2d=6.
ij ij iq cone iso
to the restriction imposed by excluded-volume. Theanalysisforthecentrosymmetriccrystalbasedonthe
From the evaluation of the integral we obtain the fol- affine assumption can be found in Born’s work and gives
lowing numerical values of the coefficients, the constant ratio1 G/K = 0.6, independent of z. The
same ratio is also found in the simulations of Ref.2. This
α x y z
limit is captured by our general framework of disordered
Aα,xxxx −0.0304 −0.00357 −0.00357 (13) lattice dynamics, as both sets of coefficients Aα,ιξκχ and
Aα,xyxy −0.00357 −0.00357 −0.000149 Bα,ιξκχ areidenticallyzeroforcentrosymmetriccrystals,
Aα,xxyy −0.00982 −0.00982 −0.00327 giving GNA =0 and KNA =0.
We have seen above that the shear modulus does not
We also recall that B = 1/7, B = 1/35,
x,xxxx y,xxxx
completelyvanishattheisostatictransition,butremains
B = 1/35 B = B = 1/35, B =
z,xxxx x,xyxy y,xyxy x,xxyy
small and equal to G = 0.0218, and that the ratio
B = 1/35, B = B 1/105 as obtained in corr
y,xxyy z,xyxy z,xxyy
G /K isabout0.26. Hence,ourtheorygivesanor-
Ref.14. Using these values of coefficients in Eq.(3), for corr corr
der of magnitude O(10−1), instead of O(0), as many nu-
shearinthexy planewefind: G=(1/30)κR2(N/V)(z−
0 merical simulations seem to suggest upon extrapolation
z )+G ,wherez =2d=6andthecorrectionterm
iso corr iso
to z =z . On the other hand, however, our theory is the
duetoexcluded-volumecorrelationsisG =0.0218,in c
corr
onlyanalyticalapproachwhichpredictsasubstantialdif-
units of κR2(N/V). The anisotropy of the shear field
0 ference, close to one order of magnitude, between K and
leavesasmallprojectionoftheinterparticleforcesinthe
G. In many amorphous and other non-centrosymmetric
directionoftheopposingbonds, whichleavesnonaffinity
materials, the difference between shear and bulk modu-
nearly intact under shear.
lus is about a factor 4, like for example in crystalline ice
Discussion
and quartz7,24, which is very consistent with our result.
The non-zero, though small, G predicted by the an-
corr
alytical theory might be due to model approximations Our theoretical predictions are presented in Fig.2a,b
whichareintrinsicallydifferentfromapproximationsand for the shear and the bulk moduli, respectively. It is
assumptions done in numerical simulations. For exam- evident that the random network is the overall softest
ple, we always overestimate the excluded-volume cone systembecauseeveniftheshearmodulusisbasicallythe
by not considering the deformability of the soft parti- same as for the jammed packing (apart from the rela-
cles in jammed packings. If this was properly taken tivelysmalltermGcorr =0.0218inthepackingmodulus
into account, it would lead to a smaller excluded-volume which we neglected in the plot), its bulk modulus is sig-
coneandweakercorrelations,hencetoahighernonaffin- nificantlysmaller. Thereasonisthatthebulkmodulusof
ity than predicted in this approximation. In turn, that thepackingbehavesclosertotheaffinedeformationlimit
would yield an even smaller, practically negligible, value due to the reduction of nonaffinity caused by excluded-
of G . Another, though related, source of inaccuracy volumecorrelations,asexplainedabove. Intriguingly,the
corr
is the neglect of deviations from the average nearest- same behaviour (soft shear modulus, quasi-affine bulk
neighbour distance R . These deviations are possible if modulus) is well known to occur in atomic amorphous
0
the particles are allowed to deform slightly at contact. materials, such as amorphous Gallium6. In the random
Therearealsootherdifferencesintermsofboundarycon- network, instead, the nonaffinity is strongest because no
ditionsandthestructureofthepackingcannotobviously cancellationofforcesduetolocalparticlecorrelationscan
beexactlythesamefortheoryandsimulations. Further, occur. This microscopic mechanism thus explains what
we do not take into account local chemistry-related ef- observed in recent numerical simulations where this dif-
fects at the interface between grains/drops (which may ference between packings and networks was investigated
controlhowthecreationofexcesscontactsz−z depends numerically4. What was interpreted as an ”anomalous”
c
uponφunderdifferentphysico-chemicalconditions21–23). behaviour, can be explained mechanistically based on
This is so because we want to focus on the more general nonaffinity.
many-body physics which controls the mechanical defor- Finally,ourmicroscopictheoryprovidesaquantitative
mation behaviour (i.e. how G and K vary with z). predictionofmoduliandofthediscontinuousjumpofthe
In a similar way, for the bulk modulus we obtain bulk modulus at the jamming transition, quantified by
K = (1/18)κσ2(N/V)(z − z ) + K . In this case K . Weintroducetheshorthandβ =(1/30)κσ2(N/V)
iso corr corr
K = 0.087, always in units of κR2(N/V), is signif- and α = (1/18)κσ2(N/V) for the prefactors of G and
corr 0
icantly larger. The reason why K ≈ 4G lies in K, respectively, for convenience of notation. Recalling
corr corr
thefactthattheforcestransmittedbyneighboursareon that κ has units of N/m, σ is a length and N/V is in
average cancelling each other effectively under isotropic units of m−3, it is clear that α and β are measured
compression, though not to the same extent in shear. in units of Pa, although here we discuss their calcu-
6

0.5 0.7
fcc 2.5 Data from Jorjadze et al.
fcc 0.6 bcc Fit with Kcorr/=1.5
0.4 bcc 2.0
2(N/V)G/R000..23 crystal 2(N/V)K/R0000...345 packing crystal P-P [kPa]C11..05
packing
0.2
0.5
0.1
random network a 0.1 random network b 0.0 c
0.0 0.0
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
z-zc z-zc z-zc
FIG. 2: Theoretical predictions in different limits across the disorder spectrum: (a) Theoretical predictions for
the shear modulus G near the isostatic limit z ≥ z , for crystals, jammed packings and random networks. The small term
iso
G = 0.0218 which contributes to the packing shear modulus has been neglected in line with the considerations presented
corr
in the text. (b) Similar predictions for the bulk modulus K for crystals, jammed packings and random networks, where now
K ismakinganimportantcontributiontothepackingbulkmodulus. (c)FitofexperimentaldataofRef.26 oncompressed
corr
emulsion, using our Eq.(14) with the only fitting parameter given by α≈0.17 kPa.
lated values in units of κσ2(N/V). Calculating the slope equation to get
G ≈ β(z −z ), we find β ≈ 0.60 for the shear mod-
iso
ulus, in good agreement with the value β ≈ 0.75 found δP =P −P = Kcorr(δz)2+ 2α(δz)3. (14)
in the simulations of Goodrich et al.2. For the jump c z2 3z2
0 0
in the bulk modulus at jamming, using the short-hand
K ≈α(z−z )+K ,ourtheorygivesK /α=1.50, Theone-parameterfitcomparisonbetweentheanalyt-
iso corr corr
whichisoftherightorderofmagnitudebutsmallerthan ical theory, given by Eq.(14) and the experimental data
thevalueK /α=4.50givenbyGoodrichetal.2. This of Ref.26 is shown in Fig.2c. The only fitting parameter
corr
discrepancy might be due to the obviously different ap- is α∝κ/R0 which is directly proportional to the spring
proximationsandassumptionsdoneinnumericalsimula- constantofthedrop-dropinteraction,hencecontainsthe
tion protocols, which were discussed at the beginning of dependence on the particular chemistry of the emulsion,
this section. andinverselyproportionaltothedropdiameter. Ourfit-
ting accounts for both creation of excess contacts with
pressure, and nonaffine particle rearrangements, and is
Comparison with compressed emulsions abletoprovideaone-parameterfitofthedata. InRef.26
We also compared our prediction for the jump of com- the same data were modelled by accounting for the cre-
pressibilitywithrecentexperimentsoncompressedemul- ation of excess contacts only, and neglecting rearrange-
sions26. In the experiment, different values of pressure ments,whichrequirestwoadjustableparameters. Hence,
appliedtothepackingwererecorded,andthevaluesofz amorequantitativedescriptionofexperimentaldatacan
corresponding to the different pressure values were mea- be achieved using the new framework proposed here.
suredusingafluorescentdyeintheinterparticlecontacts Conclusions
between emulsion droplets. The output of this measure- We showed that the mechanical response of solids is
mentisacurverelatingδP =P−P toδz =z−z ,where strongly affected by the degree of local orientational
c c
we have to interpret z as the limit of isostaticity. The order of the lattice, whether fully enforced (as in centro-
c
bulk modulus is defined in terms of pressure and coordi- symmetric crystals), low (as in random networks), or
nation z via K = −V(dP/dV) = −V(dP/dδz)dδz/dV. intermediate due to excluded-volume constraints in
There is a one-to-one mapping between the volume frac- jammed packings). In particular, intermediate degrees
tion occupied by the drops, φ, and the contact number, of orientational order are very relevant for amorphous
z, in compressed emulsions, which was determined em- solids as documented by numerical simulations and
√
pirically in Ref.26 to be δz =z δφ, with z =10.6, for experiments (see e.g. Refs.8,9). Our theory shows that
0 0
their system. Using this relation, and the definition of the lower the local orientational order, the stronger is
volume fraction φ = V /V, one obtains: dδz/dV = the role of internal nonaffine deformations which always
√ drops
−z V /2 δφV2 = −z2φ/2δzV. Upon replacing in soften the mechanical response. With excluded-volume
0 drops 0
the formula for K, we finally have a relationship be- correlations, as in packings, there is significant local
tween K, δz, and δP, given by K = φz2/2δz(dP/dδz). orientational order9 and two bonds can have the same
0
We can thus replace our theoretical expression for K = orientationacrossacommonneighbour,duetoexcluded-
αφδz+K where α is the only fitting parameter con- volume correlations. The forces transmitted by these
corr
tainingthespringconstant,andintegratethedifferential nearest-neighbours cancel each other completely under
7
compression, thus considerably reducing nonaffinity and inversion symmetry can lead to a better understanding
softening for the compression mode. For lattices with of the role of phonon lattice instabilities on the critical
strong excluded-volume like random packings (but also temperature of superconductors27.
atomic materials like amorphous Gallium), our theory
predictsthatthebulkmoduluscanbeafactorof4larger
than the shear modulus, which is in semi-quantitative or
at least qualitative agreement with both simulations2,4
and experiments on atomic6 and molecular materi-
Acknowledgments
als7. Furthermore, our theory provides an excellent
quantitative description of the dependence of the bulk
modulus of compressed emulsions on the microscopic This work was supported by the Theoretical Con-
coordination number, with just one fitting parameter in densed Matter programme grant from EPSRC. M.S.
thecomparisonwithexperiments26. Wealsoexpectthat thanks the Konrad-Adenauer-Stiftung for their financial
our lattice dynamics framework for materials that lack support.
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14 Zaccone,A.&Scossa-Romano,E.,Approximateanalytical Competing financial interests
description of the nonaffine response of amorphous solids. The authors declare no competing financial interests.