Distribution of forces and 'hydrodynamic clustering' in a shear thickened colloid

5 Pages·0.13 MB·English

by  John R. Melrose

Distributions of forces and ’hydrodynamic clustering’ in a shear thickened colloid.
John R. Melrose
Polymers & Colloids group, Cavendish Laboratory, Madingley Road, Cambridge, UK. CB3 OHE
Sheared concentrated colloids with short range polymer coats are examined via simulations. Dis-
tributionsofforcearefoundtobesumsofexponentials. The’hydrodynamicclustering’underlying
the shear thickening effect is shown, in this system, to be a network of percolating coat contacts,
0 with coats both in compression and extension. The geometry and kinetics of this network are ex-
0 plored along with its relation to the bulk stress tensor. Particles experience strong fluctuations in
0 force over epochs circa 10% strain. Density fluctuations up to glassy volume fractions give rise to
2 scattering peaks at low Q in the thickening regime. There is some commonality in the physics of
n this colloid particulate system and other granular media.
a
J Pacs Numbers: 83.50.-v, 47.20.Ft, 83.50.Qm, 83.70.Hq

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1
What is the physics of concentrated particles in flow The algorithm for simulating Eq. (1) with pair terms
]
t ? Detailed observation is hard, however, simulations of isgivenin[13]. Thelubricationforceopposestherelative
f
o powders/granular media [1], [2] have provided much in- motionoftheparticles. Atthehighconcentrationsofin-
s sight. Theyhaveshownexponentialdistributionsofforce terestitisargued[13]thatthesetermswilldominatethe
.
t and force-chains. The origin of the distribution of force long rangedmobility matrix in shearflow. In the case of
a
has motivated much theory [3-6], although in the main ahardsphere modelthe dragcoefficientα(h) isgivenby
m
for static systems. the wellknownReynolds formulawhichdivergesas1/h.
-
d Thisletterexaminesforcetransmissioninsteadystates Theparticlessimulatedherehavecrudemodelsforshort
n of sheared colloid systems. It concerns shear thickening, range polymer coats with both a conservative repulsive
o a dramatic effect in which, at high shear rates, concen- force and a lubrication interactionmodified from that of
c
trated colloids develop high, and sometimes diverging, hard spheres. The coats define a particle contact which
[
viscosities [7]. Understanding thickening is of consider- is crucial to the physics discussed below.

1 able interest [8-12]. The concept of ’hydrodynamic clus- Experiment [8],and simulation [12], [14] establish that
v
tering’hasbeenintroducedtoexplainthickening,butits thickening is dominated by contributions from the lubri-
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precise meaning has not been explored. The main aim cation forces. However, Reynolds lubrication is insuffi-
9
1 ofthis letter is to clarify ’hydrodynamicclustering’, but cienttodrivestrongthickening. Crudemodelsofparticle
1 on route to this commonalities will be found with dry pairsshearingpasteachother[15],leadtotheconclusion
0 powders. thatanydivergenceofbulkpropertiescanonlybe asthe
0
This work considers colloids of rigid particles sus- logarithmoftheinversegapofclosestapproachh . It
0 min
pended in a fluid. Particle motion is overdamped. The was suggested[15]that N-body effects may be necessary
/
t equation of motion is to explain the stronger divergences of experiment.
a
m However,simulationsofsphereswithReynoldslubrica-
R·V+F +F =0, (1)
C B tion [12] [14] do show only logarithmic divergence,much
-
d where V,F ,F are 6N velocity/angular-velocity and weaker than experiment [11]. The correct interpretation
n C B
colloid and Brownian forces/torque vectors and R is a oftheargumentsinref.[15]isnotthatN-bodyeffectsare
o
necessary,butthatlubricationinteractionsstrongerthan
c 6N×6N dragmatrix ofdissipative hydrodynamicinter-
: actions. Reynolds are necessary. Lubrication interaction can be
v
i The results below are, in the main, for a model which stronglyenhancedbypolymercoatsontheparticles[16].
X On compression of the coats the lubrication divergence
includes justthe squeezemodeofhydrodynamiclubrica-
r tion forces between near neighbours. For spheres i and switches to a higher power of (1/h) than Reynolds lu-
a
j with velocities v and v this determines a force f on brication. The strength of its lubrication coefficient, de-
i j ij
pendsonitsporosity(the smallerthepores,thestronger
particle i
the lubrication coefficient; see [12] for a plot of this in-
fij =−α(h)(vi−vj)·nijnij, (2) teraction). The coat also has an elastic modulus. It was
shown previously [12] that this model fits experimental
where n is the unit vector from centre i to j, α(h) the
ij data [11].
drag coefficient andh is the gap between the sphere sur-
Simple pair arguments [17] explain the onset of thick-
faces. Rismadeupofsuchpairterms. Intheabsenceof
ening. Thickeningoccursoncethebulkstressissufficient
dissipations for shear modes, the particles are ’friction-
to compressthe coats againsttheir conservative(spring)
less’. Some simulations have also been performed that
forcestoapointatwhichtherelaxationtimeofthecoats,
include the shear mode.
1
τ is longer than the inverse shear rate 1/γ˙. This is con- time step over a decade of variation. On the linear-log
c
firmed by the simulations. plot the distributions at high force are clearly seen to be
Althoughwecanunderstandtheonsetofthickeningat exponential.
the pair level,there isalsoa dependence onvolumefrac-
tion,withstrongthickeningoccurringonlyforφ >0.50.
v
200
Simulations do reveal many body effects. Simulations
0
[18] motivated a model of thickening by hydrodynamic
clustering: agglomerates of particles bound together by sity 150
lubrication forces. Assuming such clusters are rigid, ex- o −20
c
s
plains enhanced stress. However lubrication forces are e vi 100
onlyactiveunderrelativemotion,sorigidityandagglom- v
eration under lubrication are not obviously compatible. elati −40101 103
R 50
Indeedthe’clusters’arereportedtobetransient[18]and
observedto rapidly disappearafter cessationof flow [9]-
0
they exist in response to the stress. 101 102 103 104 105
In the simulations below, the thickening regime when Shear rate (Peclet)
γ˙τc > 1 is identified with a network of coat contacts of FIG. 1. Viscosity vs dimensionless shear rate (Peclet).
mean coordination in the range 5-6. It is this network Simulation at φv = 0.54 of colloid spheres with model
that constitutes ’hydrodynamic clustering’. The geome- polymer coats of thickness 0.01 diameters and porisity
tryandkineticsofthisnetworkwillbeexploredindetail. dp = 0.0005diam.. Open symbols squeeze lubrication only,
The simulated model has a coat thickness set at 0.01d closed symbols with shear terms. Circles N = 50 particles,
squaresN =200andtrianglesN =2000. Insetnormalstress
and other parameters choosen to fit experiments on per-
differencesN1/Pe(opencircles)andN2/Pe(opendiamonds).
spex particles [11]. The shear rate is defined as the di-
mensionless Peclet number. Figure 1 shows simulation
data at volume fraction φ = 0.54. Below Pe = 200, In the ordered phase at Pe = 100 the distribution
v
the viscosity is close to flat and the particles flow in an is a single exponential for the bonds in extension and
ordered array of strings aligned along the flow direction. dominated by a single exponentialfor the bonds in com-
The system jumps from ordered flow to disordered flow pression (there is negligible 2nd exponential in the tail).
fromPe=200to Pe=500whereit continues to be dis- These are not observable on the scale of figure 2.
ordered and thicken at higher rates. This order-disorder In the thickening regime at Pe = 500,1000,3000 the
transition during the initial stages of thickening is com- distributions over the full range are fit by the sum of
mon for mono-disperse spheres, however it is not suffi- twoexponentialdecays,one particular example is shown
cientnornecessaryforthickeningwhichisafeatureofthe in the inset. The system is characterised by two well
disorderedflow. We report elsewhere [19] systems where separated characteristic forces. Bulk averages and the
the onset of thickening is from regimes of disorder and thickeningaredominatedbythegrowthofthehighforce
charged systems at φ <0.50 which show order disorder distributions. Examination shows that the distribution
v
changes with minimal thickening. In any case, this let- with the decay at higher force (henceforth the contact
ter is a study of the steady state thickening effect within network) is just that of the bonds with coats in con-
the disordered regime. (The steady state should be con- tact; 45-50% of the bonds are in the contact network;
trasted with jamming in hard spheres [20]. A jamming the mean number of contacts per particles is 5.2,5.7,5.9
at a critical shear stress can also occur for models with forPe=500,1000,3000respectively. Itisnotedthatthe
coat interactions if the springs have a maximal force) coordinationnumber isapproaching,frombelow,thatof
Thickening is accompanied by large normal stress dif- an isostatic network of frictionless particles [6]. In the
ferences. In agreement with experiment, N1 is negative orderedregimeapercolatingcontactnetworkalsoexists,
[21]andthecoatlubricationforcesarethedominantcon- but is of lower coordination circa 3.
tribution to the stress tensor. At Pe=200,γ˙τ >1 and The finding of an exponential distribution of forces is
c
remains so up to circa Pe = 104 where the spring force common to recent simulations [1] [2]and experiment [22]
of the highly compressedcoatsgrowsrapidly andthe di- ondrygranularmedia. Inshakenpowdersabimodaldis-
vergence of the lubrication coefficient weakens, bringing tribution was found [2]. In sheared powders large fluc-
γ˙τ close to unity and lowering the thickening. tuations in normal stress were reported [23]. Note, we
c
Figure 2 shows, for several shear rates, the probabil- have also investigated force distributions when aggrega-
itydistributionforthemagnitudeofthelubricationforce tion forces are present [24] and for hard spheres with
on individual bonds. Bonds are defined as Voronoi near Reynolds lubrication and short repulsive springs. Both
neighbours. Data is shown at four different shear rates. these show exponential distributions. The distribution
It was confirmed that the effects reported are indepen- is not the result of a particular force law. Physical ar-
dent of systems size above N=500 and independent of gument for this distribution have been given [3]; it is
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geometric in origin. ofshearrate. Forthebondsinextensionitsprincipalaxis
lies above the extension direction, thus enhancing their
100 contributionto N1, it tends more to the extension direc-
tionthe higherthe shearrate. Thelubricationforcesare
10−4 symmetricfromextensiontocompression,butthespring
forces are not. It is the latter force that is responsible
10−8
y 10−4 fortheasymmetrybetweenthefabricofthecompression
y densit 10−12 100 102 104 106 108 and extension wings.
bilit Peclet Com./Ext. N1/Pe visco x (flow) y (grad.)
a
ob Pe=100 Com. 0.7 0.4 -0.43 0.9
Pr 10−8
Pe=100 Ext. -0.2 0.06 0.27 0.95
Pe=500 Com. -4 19 -0.75 0.65
Pe=500 Ext. -6 9 0.58 0.81
Pe=1000Com. -5 34 -0.74 0.66
10−12 Pe=1000Ext. -12 19 0.62 0.78
−1e+07 −5e+06 0 5e+06 1e+07
(compression) Sq. Lub. Force (extension) Pe=3000Com. -11 55 -0.74 0.68
FIG. 2. Probability density of forces (units kT/d) on Pe=3000Ext. -22 37 0.67 0.74
bonds. From the most inner pair to the most outer pair
Pe = 500,1000 and 3000 . Inset, a log-log plot of the dis- Table 1 Contents are described in the text.
tribution of force magnitudes for the compressive bonds at
Pe=1000 (e.g. the circles in the main plot 2nd in from the 1000
left). Also shown in the inset is the fit of the whole distribu-
tion too a sum of two exponentials (solid line) and the fit of r
o
just thehigh force part (dashedline) toa single exponential. ns
e
s t
s 0
e
The geometry of the network was studied. Overall, r
st
the network is branched. By thresholding on force or al
c
interparticle gap, clusters of M particles were defined o
and their radius of gyration tensor examined. At high of l
force above the high characteristic forces, these had one ue −1000
al
eigenvalue scaling as M and others scaling roughly root v
n
M. Although linear at high force, they are not straight e
g
’force-chains’[5]. Bysummingjustoverthecontactspar- ei
tial stress tensors, σ, and a partialfabric tensor, F, were
−2000
studied. 0 50 100 150 200
F =Xnijnij. (3) strain %
FIG. 3. Short history of the most negative (compression)
ij
andmostpositive(extension)eigenvaluesofthestresstensor
For the systems at Pe=500,1000 and 3000, table 1 summedjustforthebondssurroundingoneparticularparticle
gives data from the high force distributions. For the in the simulations at Pe=3000. The epoch shown is from
Pe=100 system it shows data for bonds with |f | > strain 50 to 52.
bond
2 × 103. The flow geometry has y(x) the shear gra-
dient(flow) direction with (x = −y) the compression
and (x = y) the extension directions. The table shows The kinetics of the networks was examined. It is
the viscosity σ /Pe and first normal stress difference straightforward to define an effective deformation ten-
xy
N1 = (σxx −σyy), and the x and y components of the sor for a cluster. ’Rigid’ clusters of particles defined on
normalisedprincipaleigenvectorofthefabrictensor. The force or gap thresholds would have very low extension
orderedflow at Pe=100has contacts in compressionand and compression (relative to the applied shear) and just
extension just close to the gradient direction. For the rotate in the flow. No clear evidence of these was found,
thickened states contacts both in compression and ex- although some large clusters of contacts did have eigen-
tensioncontributetothe shearstress. However,contacts valuesforthesymmetricpartoftheirdeformationtensors
in extension/compression contribute roughly 2/3 to 1/3 as low as 1/4of that for affine shear. Figure 3 shows the
of the normal stress difference. For contacts under com- largefluctuationsinstressonaparticleduringflow;they
pression,thefabrictensorhasitsprincipalaxisjustbelow build up and decay over epochs of 10% or so strain, at
the compression direction and is relatively independent theappliedshearratethisismuchfasterthanthenatural
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relaxation of the coat. They are driven by the flow that pressionandextensionareinvolved. Theoriginofnormal
is changing the geometrically determined stress concen- stress differences is due to the fabric of the contact net-
trationinthe system. (Inthe frameco-rotatingwiththe work. LowQdensityfluctuationsobservedinexperiment
flow the systems experience a changing biaxial stress.) arealsoassociatedwiththecontactnetwork. Thegrowth
Largechangesinthe distributionofforcesundera’small ofthehighforcedistributionandthepercolatingcontact
change’inthedirectionofappliedstresshasbeentermed network could be taken as an ’order-parameter’ for the
’fragility’ [5]. It is unclear if the 10% strain represents a thickening transition.
’small change’ and whether the changes in the network I acknowledge the support of the Unilever col-
are an example of ’fragility’. loid physics program, EPSRC soft-solids Grant No
A key question is whether the contact networks have GR/L21747 and funds from IFPRI . I have benefited
an inherent length scale. The data does suggest some from many useful discussions with R. Ball, R. Farr, L.
correlation with density fluctuations. The structure fac- Silbert, G. Soga, A. Catherall, D. Grinev, S. Edwards
torinthethickeningregimehaspeaksorientedalongthe and M. Cates.
flow direction inside the near neighbour ring. Similar
peaks were observed in Neutron scattering from thick-
enedcolloids[8]. Selecting particleswith highforcesand
computing a partial structure factor also resulted in low
Q-peaks oriented along the flow direction. Typically a
particle must have candidate bonds for high forces close
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provide this. The distribution of a local volume fraction 5397 (1999).
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336 (1996).
thisispeakedat0.54. Ifthedistributioniscomputedjust
[6] D Grinev and R.C. Ball, cond-matt/9810124 (1999).
overparticlesselectedonahighthresholdofforcemagni-
[7] H. A.Barnes, J. Rheol. 33, 329 (1989).
tude,thepeakisshiftedtoaroundlocalvolumefractions
[8] J.W.BenderandN.J.Wagner,J.Rheol.40,899(1996).
0.58-0.6, circa the hard sphere glass transition. Only to
[9] R. J. Butera, M. S. Wolf, J. Bender and N. J. Wagner,
this degree can the network be considered ’clustering’. Phys. Rev lett 77, 2117 (1996).
However, some particles at low local volume fractions [10] M. K.Chow and C. F. Zukoski,J. Rheol. 39, 15 (1995).
can also carry high forces. [11] W. J. Frith, P. D’Haene, R. Buscall and J. Mewis, J.
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[14] T. H. Phung, J. F, Brady and G. Bossis, J. Fluid Mech.
the generic importance of contact geometry in particle
313, 181 (1996).
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[15] G. Marrucci and M. M. Denn, Rheol. Acta 24, 317
results. It is noted that the force network here does not
(1985).
conform to the simple notion of ’force chains’ prevalent [16] G.H.FredricksonandP.Pincus,Langmuir7,786(1991).
in much of the literature, it is branched andcoordinated
[17] R. C. Ball and J. R. Melrose, Proc. of the 8th Tohwa
closetotheiso-staticlimit,butveryhighforcepartswere Symposium, pg 49, (1998).
like short segments of random walks. The existence and [18] G. Bossis and J. F. Brady, J. Chem Phys. 80, 5141
relevance of exponential distributions of force have been (1984), ibid91, 1866 (1989).
questioned, but figure 2 shows that, at least in flow, this [19] A. Catherall and J. R.Melrose, unpublished(1999).
is a significant feature. [20] J.R.MelroseandR.C.Ball,EurophysLett(1995);R.S.
For the colloid community, the results have given a Farr, J. R. Melrose and R. C. Ball, Phys. Rev. E 55(6),
7203 (1997).
much needed clarification and detailing of the nature of
[21] H. M. Laun, J. Non-Newtonian Fluid Mech. 54, 87
’hydrodynamic clustering’. There are no rigid clusters
(1994).
as some have assumed, but a network of contacts whose
[22] D.M.Meuth,H.M.JaegerandS.R.Nagel,Phys.Rev.E
natural relaxation is slower than the shear time. This 17, 3164 (1998).
networkflows with largefluctuations in lubricationforce
[23] B. Miller, C. O’Hern, R. P. Behringer, Phys. Rev. Lett.
drivenbythechanginggeometryofcontactsinthe shear 77, 3110 (1996).
flow. It had not been realised that bonds both in com-
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[24] L. E. Silbert, R. S. Farr, J. R. Melrose, and R. C. Ball.
J. Chem. Phys. 111, 1 (1999).
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