A novel particle tracking method with individual particle size measurement and its application to ordering in glassy hard sphere colloids

6.2 MB·English

by  Mathieu Leocmach

A novel particle tracking method with individual particle size measurement and its
application to ordering in glassy hard sphere colloids
Mathieu Leocmach∗ and Hajime Tanaka†
Institute of Industrial Science, University of Tokyo,
4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
Particle tracking is a key to single-particle-level confocal microscopy observation of colloidal sus-
pensions, emulsions, and granular matter. The conventional tracking method has not been able to
provide accurate information on the size of individual particle. Here we propose a novel method to
localisesphericalparticlesofarbitraryrelativesizesfromeither2Dor3D(confocal)imageseitherin
diluteorcrowdedenvironment. Moreoverthismethodallowsustoestimatethesizeofeachparticle
reliably. We use this method to analyse local bond orientational ordering in a supercooled poly-
3 disperse colloidal suspension as well as the heterogeneous crystallisation induced by a substrate.
1 For the former, we reveal non-trivial couplings of crystal-like bond orientational order and local
0 icosahedral order with the spatial distribution of particle sizes: Crystal-like order tends to form in
2 regions where very small particles are depleted and the slightly smaller size of the central particle
stabilizes icosahedral order. For the latter, on the other hand, we found that very small particles
n
areexpelledfromcrystalsandaccumulatedonthegrowthfrontofcrystals. Weemphasizethatsuch
a
information has not been accessible by conventional tracking methods.
J

0
3
I. INTRODUCTION width σ to yield a soft peak per particle. Local intensity

maxima within this blurred image give the coordinates
]
t of the particles with pixel resolution. Sub-pixel resolu-
f Experimental physicists often need to recognize ob-
o tion (0.1 0.3 pixels error) can be achieved by taking
jects, count them, follow them and characterise them.
s the centre∼of mass of a neighbourhood around the lo-
. Perrin [1] had to count colloids by hand to establish
t cal maxima. The extension of this algorithm to localize
a the sedimentation-diffusion equilibrium. Nowadays com-
m puter vision algorithms are used routinely in the lab particles in three-dimensional (3D) confocal microscopy
images has been done in two ways: either tracking parti-
to track hundreds of thousands of objects as diverse as
-
d stars in a galaxy [2], tracers in a microfluidic device [3], clesineachconfocalplaneandreconstructingtheresults
n pattern formation in polymer systems [4, 5], dust in a (2D-flavour) [22, 23], or full image analysis on three di-
o mensional pictures (3D-flavour) [24].
plasma, bacteria [6, 7] or viruses in a living cell [8]. In
c
all these cases, tracking is possible if the particles are ei- The choice of the width σ of the blurring kernel is
[
ther point-like and far apart, or several pixel wide and critical: if it is too small, then the intensity profile is flat

1 almostmonodisperseinsize(atleastthesmallestdimen- near the centre of a particle, leading to multiple and ill-
v sion for anisotropic objects [6, 7]). To our knowledge, localized maxima per particle; if it is too large, then the
7
algorithmthatallowtrackingofpolydisperseparticlesin peaks of nearby particles overlap, leading to shifts in the
3
2 crowded environments have not reached the soft matter detectedpositions[25,26], orevenfusionoftheparticles
7 community. (onlyoneparticledetectedinsteadoftwo). Ifthecolloids
. Overall particle size distribution in colloidal suspen- arefairlymonodisperseonecanargue(atleastinthe3D-
1
0 sionsandemulsionsinfluencescrystallisation[9–13],glass flavour)thatthereexistsarangeofpossiblewidthwhere
3 forming ability [9, 10, 12–14], sedimentation [15, 16] and the choice of σ has almost no effect on the number of
1 emulsionstability[15,17]amongotherphysicalphenom- particles detected. Choosing σ within this range gives
: ena. It can be characterized by various methods that confidence in the localisation results.
v
i rely on measurements done in well-controlled environ- However,wefoundthatnosuch“goodblurwidth”ex-
X ments [18–20]. However the local size distribution is not ists in a sample of moderate (6 7%) polydispersity (see
−
r accessible experimentally in situ and has thus not been Fig. 1c-h). The detection of smaller particles with small
a
studied so far. blurring is incompatible with the detection of the larger
Particle-level microscopy experiments usually access particles,andconversely. Thisunacceptablefailureofthe
the coordinates of the particles via the algorithm pro- Crocker and Grier [21] algorithm, as well as the want of
posed by Crocker and Grier [21]. The original noisy im- theparticles’radiidata,triggeredourdesignofanovello-
age is blurred by convolution with a Gaussian kernel of calisation algorithm that would be robust even for a sys-
temofanyfinitepolydispersity,whichisunavoidableand
sometime desired in real experiments. Recently Kurita
and Weeks [27, 28] have designed a sizing method using
∗ [email protected];Presentaddress: Labora- particlecoordinatesfromconfocalexperiments. However
toire de Physique, CNRS UMR 5672, Ecole Normale Suprieure their method do not work at the image processing level
deLyon,46alled’Italie69364Lyoncedex07,France. and relies on coordinates extracted via the Crocker and
† [email protected] Grier [21] algorithm. If these coordinates are wrong or
2
c d e
a b f g h
FIG.1. Visualisationoftheresultsofvarioustrackingmethodsforthesameportionofimage. (a)Multiscale3Dtracking. (b)
Reconstruction from multiscale 2D tracking. (c-h) Crocker and Grier method in 3D with blurring radius increasing from 2px
to 4.5px by steps of 0.5px. The circles on each picture are the result of 2D multiscale tracking of each XY slice of the 3D
pictures. Sphere are displayed with radii determined by the tracking methods in (a) and (b), and equal to the blurring radius
for (c)-(h).
if some particles are missed (as shown in Fig. 1), the conclude in Section V.
output of their method could not be exact.
The key notion to detect objects of unknown and pos-
sibly diverse sizes in an image is the scale space [29]. A II. LOCALISATION AND SIZING METHOD
popularimplementationforisotropicobjects(or“blobs”)
is the Scale Invariant Feature Transform (sift) of Lowe
Inthissection,wewillstartbyrecallingtheprincipleof
[30]. It is often used to match between different images sift,thenwewillexplainourmethodintheidealcaseof
from complex objects consisting of many rigidly linked
anisolatedbinaryball,toaddsuccessivelyfinitedilution
blobs (e.g., to create a large scale image from overlap-
and difference in brightness between the particles.
ping pictures) [30–32]. To our knowledge, this method
has never been used for the quantitative localisation and
sizing of independent single-blob objects like spherical
A. Scale invariant feature transform
colloids, droplets in an emulsion, or crystal nuclei.
Here we apply this new particle tracking method to
ThesiftconsistsinconvolvingtheoriginalimageI by
studyhowthesizesofparticlesaffectlocalstructuralor-
Gaussiankernelsg oflogarithmicallyincreasingwidths
deringinasupercooledcolloidalsuspensionandthepro- σs
σ to obtain a series of blurred images G
cess of heterogeneous nucleation from a substrate. We s s
reveal non-trivial local couplings between such orderings
s>0, G =I(cid:63)g , (1)
and the spatial distribution of particle sizes, which may ∀ s σs
providecrucialinformationforourunderstandingonhow
the polydispersity influences liquid dynamics and order- where (cid:63) is the convolution operator and
ing phenomena.
The organization of our paper is as follows. In Sec- s>0, σ =2s/nσ , (2)
s 0
∀
tion II, we will describe our localisation method and its
results on synthetic but more and more realistic data. with n a fixed integer. Following Ref. [30] we use
In Section III of the paper, we will focus on the specific σ = 1.6 and n = 3. Bright objects in the original im-
0
case of 3D confocal data. In Section IV we will apply age appear as bright blobs in the blurred images, and
our method to a crystallizing glass of polydisperse hard theblobsfusetogetherasthekernelwidthincreases(see
spheres observed by confocal microscopy and discuss the Fig. 1c-h). This can be seen as a series of low-pass fil-
influenceofsizedistributionontheorderingprocess. We ters in the frequency domain. If we take the difference
3
sions (e.g., confocal microscopy images) in two ways: ei-
200µm 50µm
ther by extracting 2D blobs independently on each slice
andthenreconstructing3Dobjects(Fig.1b)orbywork-
ing directly in three dimensions [31, 32] (Fig. 1a). We
foundthattheformermethodispronetoerrors, missing
aboutatenthoftheparticlesinourbestimplementation
(Fig. 1b). The 3D results presented in this paper are
obtained solely by the later method.
Westressthatavolumetricimplementationof siftim-
plies a large amount of data (typically more than 1Gb
for a (256px)3 picture) and thus requires careful mem-
FIG. 2. Application of sift in 2D. (a) Fluorescent droplets ory management. Our best implementation (C++) on a
in a microfluidic device. The wall of the device is visible 4coresi7computertakeslessthat10stoextractthepo-
in the top-left corner. (b) Nucleation under phase contrast sitions and scales of 104 particles in such a volumetric
∼
microscope. The blue circles are the result of our algorithm. picture. A much slower implementation (Python+Scipy,
For any tracking algorithm some detection failure near the single core) takes 1s to deal with (1600px)2 2D images
edge of the image are unavoidable and indeed visible here like Fig. 2b. We would expect real-time processing for
since we show the whole pictures used for tracking.
2D images on GPU-enabled implementations.
between consecutive blurred images we obtain a comb of
B. Sub-pixel and sub-scale resolution
band pass filtered versions of I:
Here we assume perfect noiseless, distortion-free, im-
s>1, DoG =G G (3)
s s s−1
∀ − ages. Theobjectstolocalisearethus(pixellised)ballsof
ThedifferenceofGaussians(DoG)responsefunctionde- uniform intensity. To mimic the low resolution of exper-
fined in this way depends on the position in space(cid:126)r and imental images in our test images (see Fig. 3), we draw
on the scale s. Bright objects in the original image are uniformly white balls on a 4 to 16 times larger image
detected as local minima in DoG. With this procedure andthenwereduceaccordinglytheresolutionusingarea
any feature with a radius as small as 2px and as large resampling.
asitsdistancetotheedgesoftheimagecanbedetected. The DoG constructed above is defined on a d+1 di-
Furthermore,theintensityoftheresponseisoptimalata mensional grid, with d the spatial dimensionality. One
σthatcanberelatedtothesizeoftheobject(seebelow). caninterpolateitasacontinuousfunctionoftheposition
Thus a local minima in both space and scale in the re- (cid:126)randthescalesandthuslocalisetheminimawithapre-
sponsefunctionDoGindicatesbothlocalisationandsize cisionbelowthegridsize,whichcorrespondstosub-pixel
of an object, without any assumption on the target size. resolution on the position and sub-scale sizing [30]. We
Because of the inherent polydispersity of many soft found that second order estimate of the spatial deriva-
matter systems, the possible applications of sift-based tives and first order estimates of the scale derivatives
localisation are countless. For example, droplets of an gave the best precision.
emulsion can be followed through a microfluidic device. The object-by-object optimal scale determination al-
Fig. 2a shows the result of our version of the sift on a lows us to perform the spatial sub-pixel resolution step
very polydispered and dense emulsion observed by wide- for each object on an image that is blurred just enough
fieldfluorescencemicroscopy[33]. Thesizesextractedare to have neither a flat intensity profile nor an influence of
obviouslycorrectexceptwhenthesystemcannotbecon- the overlap with a nearby object’s blob (an effect that
sideredas2D. Anotherpossible applicationisnucleation plagues the Crocker and Grier [21] algorithm [25, 26]).
rate measurement. When a phase transition proceeds We found that if for a given object the DoG is minimum
via nucleation growth, the sizes of the nuclei are very di- atσs,thebestimagetouseisGs−1. Thisleadstoaspa-
verse, makingtheirautomaticcountingdifficultbyother tial resolution below 0.1 pixels in the worst case (when
methods. As shown in Fig. 2b the sift allows to count particles are at hard-core contact), which is the same
and to measure the size of nuclei during a liquid-liquid as the average precision claimed by Crocker and Grier
transition in a water-glycerol mixture [34]. [21]. Moreover, when particle’s surfaces are further than
However, to the extent of our knowledge, the sift has 1 pixel, the error on the positions is less than 0.02 pixels
notbeenusedinaphysicscontexttoobtainreliablemea- (0.3% of the diameter) (see Fig. 3c).
surements of particles numbers and sizes from well cal-
ibrated images. We will address this reliability in the
following. C. Sizing at infinite dilution
Wewillmainlycopewiththethree-dimensionalexten-
sion of the sift. As in the case of the Crocker and Grier The analytical response of a binary ball to a Gaussian
[21] algorithm, the sift can be extended in three dimen- blur, at the centre of the ball is a function of dimension-
4
15 Therefore, in each of our discrete DoG images the value
)
%
at the centre of the particle can be expressed as
s ( 2
u r
radi 10 erro DoG(R,σs,α)=DoG(xs,α)=G(xs/α)−G(xs), (4)
put ve 0
Out 5 elati with α = 21/n. Given sub-scale refinement, this can be
r written as a continuous function of σ:
R
0
0 5 10 15 0 5 10 15
DoG(R,σ,α)=DoG(x,α)=G(x/α) G(x). (5)
a Inputradius b Inputradius(px) −
HereitisclearthatminimizingDoG(x,α)withrespect
x) 0 to x yields a value x∗ that depends only on α. Exact
p
1 calculation yields (see Appendix):
0
.0
error( −5 4px rij 4px x∗ =R/√2σ∗ =(cid:114)1dlnαα−2, (6)
rij 10 −
− where d is the spatial dimensionality. Practically one
8 9 10 11 12 13 14 15 16 17 obtains σ∗, the value of σ that minimises DoG(R,σ,α),
c rij(px) byapolynomialfitfordiscretedataσj. Eq.(6)allowsto
4.02 translate the n-dependent σ∗ to the parameter-free real
radiusoftheparticle,R. TheerroronRdoesdependon
4
the number of subdivisions. We found that with n = 3
px) 3.98 the radius of an isolated pixelated ball can be indeed
(
R measured within 0.3% relative error with this method
3.96 Nocorrection
(see Fig. 3b).
Oneiteration
3.94 Thescales=0.5corresponds(viaEq.(2)andEq.(6))
tothesmallerdetectableradiusR 3.5px(σ =1.6,
8 9 10 11 12 13 14 15 16 17 min 0
≈
n = 3). In order to detect small objects, Lowe [30] rec-
d rij(px) ommends to double the size of the input image using
linear interpolation prior to building the first level of the
Nocorrection
n 1,000
o pyramid. This method fares relatively well in noiseless
uti Oneiteration images (see Fig. 3b) despite larger errors, but we found
distrib 500 Twoiterations timhaatgeits hdausettoobneoiuseseadndwidthecocanrveoliunticoonnfaorctaeflamctisc.roIsncoapdy-
e edges
Siz dition doubling an image size implies a 8-fold increase
in memory consumption (reaching 60GB in the case of
0
4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 a (512px)3 original image). Our implementation allows
e R(px) this on 64bit computers by relying on memory mapped
files, nevertheless it is mostly impractical. All following
testsresultsareobtainedwithoutrelyingondoubledim-
FIG. 3. Results from perfect images. (a-b) Sizing of an iso-
ages.
lated sphere. Left of the vertical line our algorithm uses a
doubled image. (c) Localisation error and (d) sizing error
function of the distance between two particles. Oscillations
are due to off-lattice centre position. (e) Size distribution D. Edge and overlap removal
extracted from digitized configuration of 4000 monodisperse
hard spheres at 0.50 volume fraction (the vertical line indi- The difference-of-Gaussian function has a strong re-
cates the input radius). The tail to the right is due to par-
sponse not only at the centre of bright objects but also
ticles on the edges of the image who have fewer neighbours
alongtheiredges. Toeliminatethesespuriousdetections,
andthusaremore‘dilute’. (d)and(e)alsoshowtheeffectof
Lowe [30] suggests to construct the local Hessian matrix
finite dilution correction up to convergence.
around each minimum of the DoG and then compare its
eigenvalues to identify elongated objects.
We found that this was often not enough, especially
less ratio x = R/√2σ: G(x) (see Appendix for an exact in crowded environments where an isolated void induces
expression). TheDoGresponsefunctionisthedifference local minima of the DoG response with rather isotropic
between the two such functions. However, the choice of signaturesontheedgeofnearbyparticles. Anothercase
the width of the two functions is not arbitrary: we make not covered by the Hessian technique is the physically
the difference between two consecutive blurred images, hierarchical structure of many soft matter systems, e.g.,
the image blurred by σ and the image blurred by σ . particlesformingclusters. Inthissituationouralgorithm
s+1 s
5
detects a blob for each particle and a much larger blob where the matrix and the right hand side are computed
for the cluster. In both cases the DoG response of the given ( A , r ,σ∗) and iteratively R (k), with the
{ j} { ij} i { j}
spuriousfeatureissmaller(lessnegative)thantheoneof upper parenthesised index indicating the iteration rank.
the valid particles covering the same portion of space. The results of Eq. (6) are good starting values for the
Our method to remove the spurious feature is as fol- radii.
low: we looked for pairs of particles closer to each other Using Eq. (7), the elements of the matrix simplify to
thanthelargeroftheirradii(assuminginfinitedilution).
This means the centre of one of the particles is situated
∂2DoG ∂2DoG
inside the other. In the physical systems we study, this i =A (r =r ,R=R(k),σ =σ∗). (10)
cannot be correct due to excluded volume effects, thus ∂Rj∂σ j ∂R∂σ ij j i
we remove the feature of lesser DoG response. We name
thismethod‘halfoverlapremoval’. Onemaybetempted
In principle, Eq. (9) is a N N system of equa-
to implement a full overlap removal (no pair of parti- ×
tions. However the DoG functions and its derivatives
cle closer than the sum of their radii) to eliminate even
are rapidly decaying functions, thus the matrices are ac-
more spurious detection. We found that the gain was
tuallyverysparse(aboutasmanynon-zerocoefficientsas
extremely limited (0.01% of the detected features sup-
particles in the first coordination shell), alleviating dra-
pressed) and that imprecision in positioning or in sizing
matically the computational burden when using sparse
caused valid particles to be discarded.
system solvers. Fig. 3d shows the result of such correc-
Wenotethattheorientationofnon-sphericalparticles
tion for two identical particles, where Eq. (9) converges
or the deformation of soft objects can be monitored in
in a single iteration. In a many body case (see Fig. 3e)
principle by using the local Hessian matrix.
the convergence is reached in two iterations; however,
the extremely low error of the dilute case is not totally
recovered ( 3% relative error rather than 0.3%).
E. Finite dilution ≈
The Gaussian (and DoG) response of a particle de-
cays rapidly away from its surface (see Eq. (21). We
F. Brightnesses
found that even at contact the position shift induced by
a nearby particle was less than a tenth of a pixel (see
Fig. 3c). However when two particles are closer than To solve Eq. (9), one needs knowledge of the bright-
a few times the blurring width, their influence on each nesses A . In a first approximation, they can be as-
i
{ }
other cannot be neglected: the minima of the DoG is ef- sumedequaltoaconstant,whichallowstosimplifythem
fectivelyshiftedtowardsmallerscales, leadingtosmaller out. This is often a sensible approximation. Neverthe-
radii if one uses Eq. (6) out of the dilute context (see less, the particles in an experimental image are not uni-
Fig. 3d). formly bright due to synthesis imperfection (quantity of
The response of N particles of radii R can be su- dye fixed by each particle) and photo bleaching. If one
i
{ }
perimposed at any scale σ and at any point of space, in does not take into account the relative brightness of the
particular we define DoG as the response at the centre particles, less bright particles will appear smaller.
i
of particle i. A better approximation is to measure during the sift
(cid:88) processthevalueoftheDoGresponseatthepositionand
DoGi( Aj , rij , Rj ,σ)= AjDoG(r =rij,R=Rj,σ)s,cale of each particle, i.e. DoG ( A , r , R ,σ∗).
{ } { } { } j Given the (iterative) values of thei {Rj}(k{),ijo}ne{cajn}solive
(7) { j}
Eq.(7)togetaniterativevalueof A (k). Withrespect
wherethefunctionDoG(r,R,σ)istheresponseofapar- { i}
to the brightnesses, Eq. (7) is a linear system of N equa-
ticle of radius R at distance r from its centre; r is the
ij
tions with N unknowns, thus directly solvable. It is also
distances between particles i and j and A are the re-
i
spective brightness of the particles. The{sif}t algorithm as sparse as Eq. (9).
sypieelcdtst{oσσi∗}atsoσ∗t,htahtufsordiafflelrie,nDtiaotGinigisEmq.in(7im)wumithwrietshpercet- aloTnog wsuimth tuhpe,rtahdeiicinoeaffijcoiiennttsite{rAati}ivecapnrocbeessc:omputed
i
to σ: R (0) Eq.(6) σ∗
{ i} ←−−−−{ i}
∂DoG repeat
i, i( A , r , R ,σ∗)=0. (8)
∀ ∂σ { j} { ij} { j} i A (k+1) Eq.(7) DoG , R (k), σ∗ , r
{ i} ←−−−−{ i} { i} { i} { ij}
The system defined by Eq. (8) is non-linear with re- R (k+1) Eq.(9) R (k), A (k+1), σ∗ , r
spect to Rj but can be solved iteratively by Newton’s unt{il ci}onverge←n−c−e−−{ i} { i} { i} { ij}
{ }
method:
In our tests, we found that for both cases with and
(cid:20)∂2DoGi(cid:21) (cid:16) R (k+1) R (k)(cid:17)= ∂DoGi, (9) withoutthebrightnessdeterminationthisalgorithmcon-
∂R ∂σ × { j} −{ j} − ∂σ verges quickly in one or two iterations.
j
6
III. APPLICATION TO 3D CONFOCAL Using Eq. (11) we obtain
MICROSCOPY IMAGES
(cid:68) (cid:69) (cid:68) (cid:69)
2 2
[y] [y]
X Z
|F | |F |
A. Effect of a point spread function = . (15)
2 2
[h] [h]
X Z
|F | |F |
Real images suffer from optical limitations, e.g., the
Here indicatestheFouriertransformalongaxisX. In
point-spread function (psf) of the microscope. In par- pointFsXcanning confocal imaging, the psf has negligible
ticular, spherical particles observed by confocal mi-
lobes along X and Y (X only for line scanning), thus we
croscopy appear elongated along Z. The influence of
have
suchanisotropicdistortionontheGaussianandDoGre-
(cid:68) (cid:69)
sponses is not trivial and most of the equations given in 2
[y]
Z
Appendixdonothaveanalyticalequivalents. Inparticu- Z[h]2 = (cid:68)|F | (cid:69). (16)
lar, the minimum of the DoG response is found at larger |F | [y]2
X
|F |
valuesofR/σ,thusthenaiveuseofthemethodsdetailed
intheprevioussectionleadtooverestimateparticlesizes.
For example an Hermitian kernel (real valued spectrum)
In addition, the overlap of neighbouring particle im-
is
agesisnolongernegligiblewhentheparticlesarealigned
along Z. This leads to large imprecision in particle (cid:118)(cid:117)(cid:68) 2(cid:69)
pmoesnittio(ne.sg,.,esipsoelcaitaeldlypianirthoef pcaarsteicolefsa,ninistoetrrfoapceic, ceonlvloiriodna-l FZ[h]=(cid:117)(cid:117)(cid:116)(cid:68)|FZ[y]|2(cid:69). (17)
[y]
X
gel). In Fig. 4d the radial distribution function of al- |F |
most monodisperse colloids with short range attraction
Fig.4b-cshowstheparticleslocalizedfromtheoriginal
displaysaspuriousshoulderbeforeitsfirstpeak,indicat-
image and from the deconvolved image. Deconvolution
ing that particles centres are often found closer than the
mends both size overestimation and imprecision in z co-
sum of their hard core radii, as illustrated in Fig. 4b.
ordinate. This also translates in the radial distribution
We found that these issues can be better addressed by
function (Fig. 4d), where the spurious shoulder before
pre-processingtheimages(asdetailedbelow)ratherthan
post processing the sift output via analytical methods. the first peak disappears.
B. Deconvolution C. From optical to hard-core radius
Theimageyacquiredbyamicroscopecanbeexpressed The method exposed above relies on the uniformity of
as thebrightnesswithineachparticletoextractopticalradii
that is a good approximation to the physical (hard core)
y =x(cid:63)h+(cid:15), (11)
radii. However a colloidal particle often shows a smooth
where x is the perfect image, h is the psf of the micro- intensityprofileinitselfunderthemicroscope,lessbright
at the edge than at the centre of the particle. This can
scope and (cid:15) is the noise independent of both x and h.
be due to inhomogeneous fixation of the dye during the
The process of estimating x from y and some theoreti-
colloid synthesis, but the bottom line is the in-plane psf
cal or measured expression of h is called deconvolution.
not corrected for in our above deconvolution method.
Deconvolutioninthepresenceofnoiseisadifficultprob-
Under such conditions, the particles are detected
lem [35]. Hopefully here we do not need to reconstruct
smaller than their expected sizes. The real to optical
the original image, but only our first Gaussian blurred
versionofit, i.e., G =x(cid:63)g startingfromy =y(cid:63)g . size ratio has to be set using our knowledge of the sam-
0 σ0 0 σ0 ple. For example one may use the position of the first
Indeed, after a reasonable amount of blur in three di-
peak of the g(r) to measure the average hard-core diam-
mensions,thenoisecanbeneglectedandwethusobtain:
eter. In general the real to optical size ratio may depend
on the size of the particles and may evolve in time (i.e.,
y0 G0(cid:63)h, (12) becauseofphotobleaching). Wepresentbelowadetailed
≈
analysis of these issues for a case system.
or in Fourier space
We also note that darker edges lead to smaller influ-
enceofaparticleontheDoGresponseofitsneighbours.
[y ]= [G ] [h]. (13)
0 0
F F ×F Asmalleropticalradiusmakestheparticlesopticallyfur-
Once [h] is known the deconvolution reduces to a ther from each other. Accordingly, we found that finite
F
simple division in Fourier space. Let us measure [h] in dilution corrections were less important in real images
F
an isotropic system where we can write than in our synthetic test images. For the same reason,
(cid:68) (cid:69) (cid:68) (cid:69) thescalingfactormustbemeasuredandappliedafter—
2 2
X[x] = Z[x] . (14) not before — finite dilution corrections.
|F | |F |
7
2
1.5
)
r
(g 1
0.5
0
0 1 2 3 4
a b c d r[µm]
FIG. 4. Deconvolution. Detail of the same YZ slice of (a) original confocal image, (b) previous blurred by σ = 1.6, (c)
0
previousdeconvolvedbymeasuredkernel. Circlesindicatethetrackedparticlespositionandsizewhenusingwhether(b)or(c)
as first Gaussian layer G . All three centres are in the slice ±0.5px. (d) Radial distribution function of almost monodisperse
0
sticky spheres localised (blue) without and (red dash) with deconvolution.
IV. REVEALING POLYDISPERSITY EFFECTS vent. At the top of the sample (close to the objective
ON STRUCTURAL ORDERINGS IN A GLASSY lens and thus allowing much clear imaging) the volume
COLLOIDAL HARD SPHERE LIQUID fraction drops and all crystallites melt. Finally, we set
back the thermostat to the density-matching tempera-
Hereweapplyourtrackingmethodtoaccesscouplings ture, allowing the top of the sample to slowly return to
between the spatial particle size distribution and local its supercooled state. We could then observe the hetero-
structural ordering in a supercooled colloidal liquid with geneous nucleation from the beginning. We tracked the
size polydispersity. sample from top to bottom and we are thus sure that
the crystallite actually form at the wall and do not come
from the rest of the sample. Namely, crystallisation in
this sample is caused only by heterogeneous nucleation
A. Experimental
on the substrates, presumably due to the wall-induced
enhancement of crystal-like bond orientational ordering
We used pmma (poly(methyl methacrylate)) colloids [37].
stericallystabilizedwithmethacryloxypropylterminated
pdms (poly(dimethyl siloxane)) and fluorescently la-
belledwithrhodamineisothiocyanatechemicallybonded
to the pmma. The colloids were suspended in a sol- B. Global size distribution
vent mixture of cis-decalin and cyclohexyl-bromide for
both optical index and density matching. To screen We localise the particles using our method with a pre-
any (weak) electrostatic interactions, we dissolved tetra- blur of only σ = 1.0 (see Section IIA) to be able to
0
butylammonium bromide salt, to a concentration of detect the smaller particles (R 2px) without ex-
min
≈
300nmol/L [36]. The estimated Debye screening length pensive oversampling. We checked that choosing higher
is 13nm, well below the length scale of the colloids σ truncates the size distribution but has no other effect
0
that can be considered as hard spheres. The 8bit on its shape. After removing half-overlapping features,
graylevel data was collected on a Leica SP5 confocal mi- we applied a single iteration of finite dilution corrections
croscope, using 532nm laser excitation and voxel size of for both intensities and radiito obtainthe opticalradius
(283nm)2 293nm. of each particle.
×
To realize a more precise density matching, the tem- We then estimate the hard-core diameter function of
peraturewascontrolledonbothstageandobjectivelens. theopticalradiusRbylocatingthefirstpeakofthepar-
This setup allows us to alter the buoyancy simply by a tialradialdistributionfunctiong (r)oftheparticleshav-
R
temperature change, thus to set the effective gravity up- ingR closetoR. AsshowninFig.5a,therealtooptical
i
ward,downwardoralmostnull. Aftercarefulshearmelt- size ratio is around 1.5 and is rather constant respective
ing,thesamplewasfilledintoa100µm 1mmcapillary to R, thus a single overall scaling factor is enough. It
×
(Vitrocom) and set on the microscope stage. We then can be determined by a single partial g (r) with R near
R
spent a few days to find the temperature corresponding the peak of the size distribution, or by locating the peak
to exact density matching (within 0.1K). This waiting of g(rˆ), with rˆ = r /(R +R ). We found that the
ij ij i j
timewasenoughforcrystallitestoformonbothtopand real-to-optical size ratio was increasing with time due to
and bottom walls. Then, we heat up our sample by a photobleaching (see Fig. 5b). We fit this increase by a
few degrees compared to the density-matching tempera- linear relation and applied the resulting time-dependent
ture in order to make the colloids heavier than the sol- ratio to obtain the real size of each particles.
8
The consistency of our method can be checked by
constructing the radial distribution function g(r). In
monodispersehardspheres, theg(r)shouldhaveasharp 20
first peak at r =2R corresponding to hard core contact. firstpx]
[ 15
Polydispersityimplieshardcorecontactsatvariousrand the()r
thusbroadensthepeak. Onecanrecoverasharppeakby ofgR 10
constructing g(rˆ) (this time rˆis scaled by the real radii, onof
nottheopticalones). InFig.5dwesuccessfullyusedthe sitiak 5
oe
sizes measured by our method to rectify the first peak. Pp
0
Fig. 5c shows the size distribution only 140 dry col- 2 2.5 3 3.5 4 4.5 5 5.5 6
loids measured on high resolution scanning electron mi- a
Detectedradius[px]
croscopy (sem) images. We also show the size distribu-
tion obtained by our in situ measurements ( 1.7 106 1.6
∼ × al
ipnasrteasnwtaenlleowuitshsitzhiengfo).rmTehreonmcaeinaspoelavkenotfstwheellilnatgeorfc2o5m%- opticatio 1.55
ipnlinragdoiufstihsetaSkEeMn inmteoaasuccroemunetn.tsHocowmevpelrettehlye smmiassllessatmhe- eal-to-sizer 1.5
R
tailtowardsmallsizesfeaturingtwolowpeaksthatprob- 1.45
ablycorrespondtosecondaryandternarynucleationdur-
0 10 20 30 40 50 60
ing the synthesis of the colloids [38, 39].
b
Time[h]
From our data we compute an overall volume fraction
of 0.60, almost constant during the experiment. The n 4 g(r)
polydispersity estimated from sem data is 6.2%. This is utio g(rˆ)
coherent with a fit of the main peak of our data (6.9%), rib )r
however the polydispersity of the whole distribution in- dist (g 2
cluding the tail toward small sizes is 14.8%. We will see
e
below how such a complex size distribution affects the Siz
physical behaviour of the system. 0
11 22 33 44 55 0 0.5 1 1.5
c Diameters[µm] d r/σpeak,rˆ
C. Polydispersity and structural heterogeneity in a
supercooled colloidal liquid FIG.5. Insitu sizingofcolloidsinaglass. (a)Positionofthe
firstpeakofthepartialg (r)functionoftheopticalradiusR.
R
Hard sphere supercooled liquids and glasses contain Solidlinecorrespondstoareal-to-opticalsizeratioof1.5. (b)
Timedependenceofthereal-to-opticalsizeratio. Solidlineis
both medium ranged crystal-like bond orientational or-
dering (mrco) and local icosahedral ordering [40]. The a linear fit. (c) Size distribution estimated by our algorithm
(dashed line). Comparison with the estimation from sem of
later acts as a frustration against the expansion of the
only140dryparticles(steps)ispossibleonce25%ofswelling
former and thus against crystallisation. Both kind of
is taken into account (full line). (d) First peak of the radial
structurescanbedetectedusingbondorientationalorder distributionfunctionwith(fullline)andwithout(dashed)the
(boo) introduced by Steinhardt et al. [41] (see Fig. 6). individual sizes data.
As we have described elsewhere [40], crystal-like bond
ordering is well described by the scalar bond order pa-
rameter Q , and the icosahedral bond ordering by w . ratio near 0.8 seems to stabilize icosahedral order. This
6 6
fcc or hcp crystals under thermal vibrations typically is consistent with a recent simulation study by Shimono
have Q > 0.4 [42]. We distinguish mrco from liquid and Onodera [43].
6
structures by a threshold at Q6 = 0.25. The perfect 13 Wenotethatthetailofsmallparticlescharacteristicof
particles icosahedron is the minimum of w6, at negative our overall size distribution is less pronounced in mrco
values. We consider that a neighbourhood is icosahedral and almost disappears in well-formed crystals (Fig. 7a).
when w6 < 0.023. The width of the main peak of the distribution does
−
In Fig. 7, we compare the overall size distribution to not change significantly upon crystal-like ordering. This
the size distribution within each remarkable structure. meansthatasmallamountofpolydispersityisacceptable
The roles of the particle at the centre of an icosahedron inside mrco, coherently with the results for the (defec-
and of the particles surrounding it are asymmetric. The tive) crystalline lattice by Fasolo and Sollich [11]. How-
size distribution of the particles at the surface of icosa- ever mrco cannot appear where markedly smaller parti-
hedra is almost identical to the overall size distribution, cles are present. We also checked the difference between
however the size distribution of the particles at the cen- fcc (w < 0) and hcp (w > 0) to find that they have
4 4
ter of icosahedra features a second peak at radii about bothexactlythesamesizedistribution(notshown). This
80%smallerthanthemainpeak. Acentretosurfacesize suggeststhatthesizedistributioncontrolstheformation
9
1
0.8 t=0 t=40h
z 0.6
ρ
0.4
0.2
0
0.1
ρ
0
5 10 15 20 25 5 10 15 20 25
FIG. 6. Structure visualisation at t = 0 (left) and t = 40h z/(2R ) z/(2R )
peak peak
(right). Small particles (R < 1.5µm) are shown in green,
crystal-like ordered particles (Q >0.25) in red and icosahe-
6 FIG. 8. Instantaneous density profiles at t = 0 (left) and
dralparticlesandtheirneighboursinpurple. Otherparticles
t = 40h (right). Top: large particles (R > 1.5µm) with-
are not shown.
out smoothing (black line) and with a Gaussian smooth-
ing of width 2R (gray area). Bottom: small particles
peak
(R < 1.5µm) with the same color code. For all figures, the
all
on 10 reddashedcurveisthesmootheddensityoforderedparticles
uti MRCO (Q >0.25) irrespective of their size.
b crystalline 6
distri 5
e
z fractionationappendslaterinthecrystallisationprocess.
Si
0
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
a D. Polydispersity and heterogeneous crystallisation
R[µm]
in a supercooled colloidal liquid
all
n 10
o
uti ico. centre As explained above, we triggered heterogeneous crys-
distrib 5 ico. surf. ttaiml enusctelepatwioencaonnctohnesttroupctwaallz-odfeopuerndceonnttadiennesri.tyAptreoaficlhe
e
z foranyspeciesofinterest. InFig.8weshowattwotime
Si
stepsthedensityprofilesoflargeparticles(R>1.5µm),
0
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 small particles (R<1.5µm) and particles with medium
b R[µm] to high crystalline order (Q6 > 0.25). Icosahedral par-
ticles (not shown) are an order of magnitude fewer than
small particles (see Fig. 6) and are basically irrelevant.
FIG. 7. Size distribution within local structures in a super-
Nearthewalls,thedensityprofilesshowoscillationschar-
cooled hard sphere colloidal liquid. (a) With increasing local
acteristic of layering (see, e.g., Refs. [37, 44]), whose
crystallinebondorientationalorder: allparticles,MRCOpar-
wavelength corresponds to the peak of the size distri-
ticles (0.4>Q >0.25) and crystalline particles (Q >0.4).
6 6
Crystallineorderedregionsarelesspolydisperse. (b)Particles bution (the ‘majority species’). We note that the small
at the centre or on the surface of icosahedra (w <−0.023). particles also shows layering on the same wavelength, al-
6
A small central particle with large surface particles tend to though with a smaller amplitude.
stabilize icosahedra. Once the short oscillations removed via a Gaussian
smoothing of width 2R , one can see that the den-
peak
sity of large particles is constant from top to bottom of
of mrco but has no influence on its symmetry (proba- the sample (the slight dip on the edges is due to a small
bly,thecrystalpolymorph). Westressthatthiscoupling misalignment between our z axis and the normal of the
of mrco and icosahedral ordering with the spatial dis- wall). It confirms the accuracy of our density matching
tribution of particle sizes does not imply fractionation, but also stresses that crystallisation takes place without
but rather suggests that a rather homogeneous size dis- an increase in density. The smoothed density profile of
tribution tends to help or stabilize mrco and a slightly smallparticlesstartsalmostflat. Howeverthesmallpar-
smaller size of a particle also stabilizes the formation of ticles are expelled from the forming crystal, and as the
icosahedral order around it. We will see below that the crystalline front propagates it pushes in front of itself a
10
Q V. CONCLUSION
6
0.5
We have presented a new method to extract the coor-
dinates of particles of very diverse sizes even when they
0.4
areveryclose(atcontact)toeachother. Thisopensnew
experimental possibilities in soft matter systems where
0.3
the particles are polydisperse (e.g., colloids, emulsions,
andgranularparticles)orevenchangingsize(e.g., phase
0.2
separationdynamics). Ourmethodalsoreliablyextracts
thesizeofindividualparticle,whichallowsainsitu anal-
0.1
ysis of the consequences of naturally occurring size dis-
tributions(beyondmonodisperse,bidisperseorGaussian
0
distribution).
We showed that in the case of colloidal hard sphere, a
FIG. 9. Heterogeneous nucleation. Successive computer re-
sizedistributionwithamainpeakandatailtowardsmall
constructionofthefirstthreelayersfromthewall,every12h,
sizes induces different behaviours than what would be
seen from the fluid side. Particles sizes are according to our
method. Small (R < 1.5µm) particles are shown in green. expected from a Gaussian size distribution. Our method
Other particles are coloured according to their crystal-like allowed us to measure this distribution properly and to
order. Note how the small particles are expelled from the lookintotheinteractionsbetweensizedistribution, local
forming crystallites to concentrate in the grain boundaries. ordering and heterogeneous crystal nucleation in a glass.
We found that the presence of a long tail toward small
sizes in the size distribution, was leading to highly poly-
crystallinematerialsthatmaybeofinterestforengineer-
ing purposes. We hope that this novel particle tracking
growingridgeofsmallparticles, veryapparentafter40h method with a capability of particle-size determination
in both Fig. 8 and Fig. 6. will be applied to a wide field in soft matter physics.
We confirmed this view by looking at the behaviour of
small particles and crystalline order in a constant plain
ACKNOWLEDGEMENTS
(Fig. 9 and Supplementary Movie 1). Before crystalli-
sation, small particles are scattered randomly and mrco
WearegratefultoKoshiHasatani,AnthonyGenotand
developwherefewverysmallparticlesarepresent. Then,
crystallisation proceeds from the mrco, pushing away Yannick Rondelez for providing the image of Fig. 2a; to
Ken-ichiro Murata for providing the image of Fig. 2b;
the small particles that accumulate in the grain bound-
to John Russo for providing us with simulation data of
aries. Asaconsequence,heterogeneouscrystallisationdo
dense hard spheres, which are used in Fig. 3e; to Hideyo
not proceed via a spatially homogeneous layer-by-layer
Tsurusawa for the synthesis of the colloid and the con-
growth, as observed at low polydispersity [45], but by
focal data of Fig. 4; and to Rebecca Rice and C. Paddy
many nucleation-growth events reflecting the initial fluc-
tuations of the mrco. This dynamics is reminiscent of Royall for sem measurements of the size distribution of
dried particles. This work was partially supported by
the homogeneous nucleation case [46–48] but enhanced
a grant-in-aid from the Ministry of Education, Culture,
by the wetting of the wall by crystalline bond orienta-
Sports, Science and Technology, Japan and also by Ai-
tional order [37].
hara Project, the FIRST program from JSPS, initiated
by CSTP.
Tosumup, thesmallparticlesaffectthegrowthofthe
crystal by coupling ordering to diffusion. Without the
small particles, moving grain boundaries implies mov-
ing particles only locally and thus propagation is easy. APPENDIX: GAUSSIAN RESPONSE OF A
With very small particles, moving grain boundaries im- BINARY BALL
plies moving them by the same amount by diffusion. In
the same way, the crystallisation front has to push the
A binary ball of radius R is convolved by a Gaus-
R
small particles in front of it, effectively increasing the B
sian kernel of width σ. The response at a distance r
polydispersity in the melt and thus slowing its propa-
(along z-axis without loss of generality) is
gation. Indeed, weeks after preparation our samples was
stillonlypartiallycrystallisedandhighlypolycrystalline.
Wofetnhoetesptehcaitesth‘issmcaolulpplianrgtiicsledsu’.etNoothne-ccoonnsseerrvveeddsnpaetcuieres G(r,R,σ)= 1 (cid:90) e−(z−r)22σ+2y2+x2dxdydz
(2π)3/2σ3
defined by local structural ordering such as icosahedra BR
(18)
cannot produce such a coupling.