Department of Physics
The dynamical susceptibility
in the Lorentz model
Candidate:
Fabrizio Camerin
Supervisor:
Emanuela Zaccarelli
This thesis project has been carried out within the Erasmus Mundus Joint Master Degree AtoSiM
It is well known that glass-forming systems display an increase in the maximum of the dynamical susceptibility, which is a feature associated to dynamical heterogeneities, i.e. atoms (or colloids) that move differently in different environments. This work assesses the dynamical susceptibility for a model that has served for decades as prototype for the description of heterogeneous transport, the Lorentz model. Its non-uniform dynamics is entailed by the presence of point-like intruders that meander a hierarchy of isolated pockets (or clusters) and a percolating fixed matrix.
By performing extensive, large-scale Molecular Dynamics simulations of the Lorentz model, it’s revealed, unexpectedly, that the dynamical susceptibility does not exceed its typical long-time limit at all wave-vectors. No maximum and no signs of dynami-cal heterogeneities are observed. To understand these findings, a specifidynami-cally designed cluster-resolved theory has been designed and compared with cluster-resolved simula-tions. This approach has proved to be crucial to demonstrate that the dynamical suscep-tibility needs to be appropriately treated in order to be a faithful indicator to quantify dynamical heterogeneities.
This thesis project represents the conclusion of a two-year Master program in which I met great people I’d like to thank.
First of all, I would like to thank Emanuela, my supervisor, for her competence, her pa-tience but most of all for her humanity and generosity. Without her strong commitment and her resolute guide, this work wouldn’t have been possible.
This project is also the outcome of an international collaboration with prof. Thomas Franosch of the University of Innsbruck (Austria), who actively participated in the designing of the research especially regarding its theoretical implications. I am grateful for his support and I believe this synergy will continue in the future.
A special thank to the AtoSiM guys for having been my second family, and to all the AtoSiM Team that allowed me to participate in such a prestigious European study-program.
Thanks for the long and short talks, visits, messages, to my parents, to my always-present old friends Filippo, Sara, Martina, Davide, Martina, Monica, Chiara, Michele, Giulia, Martina, and to my roman friend Lorenzo. I am also thankful to Jose and Javi for the intriguing discussions we had in these months of internship.
Finally, thanks to Rome, Amsterdam and Lyon for being the wonderful cities that host me throughout this great experience, and thanks to all the friends that I left around whose moments together I’ll never forget.
Abstract v
Acknowledgements vi
1 Dynamical heterogeneities 1
1.1 Introduction . . . 1
1.2 The Lorentz model . . . 3
1.2.1 Fractal behavior . . . 5
1.3 Evidences of heterogeneous dynamics . . . 7
1.3.1 Introduction to the cluster-resolved scaling theory . . . 7
1.3.2 The Mean-Square Displacement . . . 11
1.3.3 The Self-Intermediate Scattering Function . . . 12
2 A cluster-resolved approach 15 2.1 The Dynamical Susceptibility χ4 . . . 15
2.1.1 Cluster-resolved Dynamical Susceptibility - Scaling Theory . . . . 16
2.2 Intermezzo: the Dynamical Susceptibility in a model of permanent gels and in glasses . . . 20
2.3 All-cluster χ4 - Numerical Analysis . . . 21
2.4 Cluster-resolved χ4 - Numerical Analysis . . . 24
2.4.1 Cluster analysis and wave-vector dependence . . . 25
2.4.2 Size-scaling of the χ4 . . . 29
2.4.3 Cluster-cluster interactions . . . 30
2.4.4 Infinite cluster . . . 31
3 The reliability of the χ4 as indicator of dynamical heterogeneities in the Lorentz model 35 3.1 Why is there a discrepancy between the all-cluster and the cluster-resolved calculations? . . . 35
3.1.1 The true Lorentz model: one tracer at the time . . . 35
3.1.2 N -particle analysis . . . 37
3.2 All-cluster χ4: is it an accessible quantity for the Lorentz model? . . . 38
4 Conclusions 41
Bibliography 43
Dynamical heterogeneities
1.1
Introduction
Diffusion in heterogeneous and disordered media has been, in the last decades, a rele-vant field of study that permeates the material and biological sciences. In particular, the slowing down of the dynamics has been observed in porous materials [1] and in crowded environments, such as cells and biological molecules [2,3]. Dynamical arrest is also observed in largely investigated systems like glasses, that are amorphous solids that microscopically resemble the typical disorder of the liquid state. The glass transition, that can be reached by lowering the temperature of the system or by compressing it, still remains nowadays poorly understood. In fact, densely packed or supercooled liquids become too viscous to flow on short timescales, making their experimental realization challenging. Moreover, they do not not exibit the typical large-scale density fluctuations observed at an ordinary critical point, thus nullifying any attempt to treat it in a similar way [4].
A step towards the construction of a unified theory of the glassy state is constituted by the concept of dynamical heterogeneities, which refers to the existence, in amorphous materials, of spatial fluctuations in the local dynamics due to presence of atoms moving in different ways in a disordered structure. Consequently, a qualitative distinction be-tween materials close to the glass transition and ordinary fluids can be attempted based on the presence or not of domains where the amplitude of single-particle displacements are correlated.
A fascinating interplay between the typical glass transition singularities and the slow-down of the dynamics in amorphous structure has been studied in porous media in which dense liquids are confined. An example is directly given by the sodium-silicate
melts, where the silicon-oxygen component displays diffusion coefficients that are orders of magnitude smaller than the ones of the alkali metal. As it would happen in a porous environment, the sodium ions meander through an arrested structure and percolates the fixed matrix [5]. A simplified model that describes this case might be the one studied theoretically by Krakoviack [6] and numerically by Kim et al. [7], that is the so-called “quenched-annealed” binary mixture, in which a fluid of hard spheres is confined in a hard sphere mold. If the concentration of the fluid in the porous structure becomes very small, the dynamics of the system is uniquely governed by the excluded volume of the matrix, as the interaction among the fluid particles becomes negligible.
This ideal representation corresponds to the so-called Lorentz model, originally employed to explain the resistance of metals for electron transport. In this case, the matrix is con-stituted by frozen spheres distributed in a random way in space and the percolating fluid is depicted as point-like intruders; the formation of aggregates of spheres and their overlap is permitted due to the independent placement of the obstacles.
At low densities of excluded volume, the clusters are dispersed in space and transport is not affected, while, at higher densities, the void space available to the tracer is re-duced causing a decrease of the value of the diffusion coefficient. At a certain critical density, the system undergoes a so-called percolation transition of the void space and im-plies subdiffusion for the non-interacting tracers. In other words, the motion is globally slowed down and, in contrast to a normal diffusion process, it occurs with a non-linear relationship to time. In case the density of the matrix exceeds the percolation thresh-old, at which the diffusion of the tracer is fully anomalous, the motion shifts to localized.
As a matter of fact, the Lorentz model is too idealized to be considered as a refer-ence prototype for glassy systems. In fact, it does not take into account their complex slow-rearranging dynamics, as the only interactions that appear are the ones between the tracer particles and the fixed matrix. Despite this fact, it’s recognized as a valuable minimal model to predict macroscopic properties that concern transport phenomena in random and porous structure, which is of primary importance for possible applications in the field of material sciences, nanochemistry and biology. Moreover, due to the presence of a hierarchy of isolated pockets in which the intruders are able to diffuse, it encodes all the peculiarities needed to study a non-uniform dynamics, implicitly helping to shed light on the unascertained glass transition.
The quantification of the dynamical heterogeneities of the Lorentz model constitutes the primary goal of this thesis project. In order to do so, I will calculate the so-called dy-namical susceptibility χ4, i.e. the fluctuations of the self-intermediate scattering (ISF)
function. This indicator has been largely employed in the last two decades and applied to a variety of models for glasses and gels [8–10], but not yet to the Lorentz model.
The reason why this indicator might be helpful in the quantification of heterogeneous dynamics resides precisely in the microscopic description given above for glasses. As the origin of the non-uniform dynamics relates to atoms that diffuse differently in different regions of the systems, one has to resolve the dynamics both in space and time, and quantify the deviation from the average behavior. This is conventionally done with the self-intermediate scattering function whose decay in time signals the faster or slower decorrelation of the atoms. As a consequence, an increase in its fluctuations, i.e. an increase in the χ4, is representative for heterogeneous dynamics.
The calculation of the dynamical susceptibility has been approached from a theoret-ical and numertheoret-ical point of view.
The following sections are dedicated to a general overview of the Lorentz model and to a treatment of the evidences of heterogeneous dynamics. Furthermore, I’m going to introduce the so-called cluster-resolved scaling theory applied to the mean-square dis-placement of the tracers to fully appreciate these proofs.
The dynamical susceptibility has been calculated thanks to extensive, large-scale Molec-ular Dynamics simulations of the Lorentz model. Our research includes also an entirely new cluster-resolved approach that, together with the theoretical bases, will be treated in full in Chapter 2.
Chapter 3 is instead dedicated to the presentation and discussion of our final results, whereas I will draw the conclusions of our work underlining its future perspectives in Chapter 4.
1.2
The Lorentz model
The Lorentz model is constituted by an array of frozen obstacles of a certain density. The region of void space, strictly speaking, is explored by a single structureless tracer; for the moment, we assume the equivalence to a gas of non-interacting point-particles. In part of the literature, the Lorentz model is also named the “Swiss-cheese model” [11] by analogy to the Swiss cheese. In fact, one may describe this model through uniformly sized spherical holes randomly placed in a transport medium, see Figure 1.1.
The properties of the model are characterized by a unique relevant parameter, that is the dimensionless obstacle density n∗ = nσd, where σ is the “exclusion” radius of the spheres and d the dimension of the system. Alternatively, one may employ the accessible volume fraction, the so-called porosity of the system. The probability that N obstacles do not occupy a point in a large volume V , for the three-dimensional case, is (1 − 4πσ3/3V )N. Taking the limit for V → ∞ and N → ∞, the volume fraction accessible to the tracer
Figure 1.1: Typical trajectories of an ant in 2D Lorentz model slightly below the critical percolating density. Obstacles are omit-ted for clarity and different colors encode different initial conditions [17].
is [12,13]
Φ = e−43πnσ 3
. (1.1)
For a moderate value of n∗ , the void space consists of many isolated regions that coex-ists with a connected one that spans through the entire system. The latter corresponds to the infinite percolating cluster. Above a critical density the system undergoes a local-ization transition, that means that the particle is trapped by the obstacles. Long-range transport occurs only on the infinite cluster below the critical density.
The present model can be treated as a continuum percolation problem where, close to the percolation transition, the transport is limited and dominated by some narrow bot-tlenecks that, in three dimensions, are bounded by three interpenetrating holes [14,15]. The formation of these bottlenecks, that make the connections from one side to the other one of the void cluster very weak, is due to purely geometrical reasons, and it is reasonable to expect that the strength of the transport in that region has a power-law distribution1, as all the physical properties at criticality [16]. Thus, the critical behavior of the transport properties, described by the critical exponents, is different from the one
1
Supposing that the transport medium is an electrical conductor, the Lorentz model may be mapped to a so-called random resistor network. This model may be described by an arbitrary network of nodes connected by links, each being a resistor. In such a system, the resistance (or the conductance) plays the same role of the diffusion coefficient in the Lorentz model.
of a standard discrete model, where the spacing among the obstacles is fixed and uni-form. Despite this, it’s worth to note that the exponents that describe the geometrical properties of the system do not have a different counterpart in a discrete model but, instead, they are universal (see below); the reason is that geometrical exponents depend on how the necks are connected and not on their width [14].
Specifically, the universal properties of a percolating cluster are the strength of the in-finite network and the correlation or connectivity length [16]. The former is defined as the probability that an arbitrarily selected site belongs to the infinite cluster and, as the critical density is approached from below2, it nullifies following a power-law
Φ∞∝ (−)β, (1.2)
where = n∗−n∗c n∗
c is the so-called separation parameter and n ∗
c the critical density at
which the infinite percolating cluster is formed. The correlation or connectivity length, defined as the typical size of the largest non-percolating or finite cluster, is expected to diverge, for small , as
ξ ∝ ||−ν. (1.3)
1.2.1 Fractal behavior
At the critical density, the infinite cluster becomes a self-similar structure with a certain fractal dimension.
In fact, it is not possible to identify a characteristic length, considering that void struc-tures can be recognized on many length scales. For this reason, it is also difficult to define univocally, for the entire system, the “mass” (or volume) of the percolating clus-ter. As it is more intuitive, let us look at a classical percolation on a lattice, where the mass M can simply be established as the number of occupied cells within a certain area Ld. For a uniformly occupied system, one expect that the mass is linearly proportional to the area: the bigger the linear length, the heavier the percolating cluster. It appears evident, however, that in case the system is not uniform, as it happens at the critical density, the former proportionality is lost and a new dimension, smaller than the one of the system, comes into play, that is the Hausdorff-Besicovitch or the fractal dimension. Thus,
M ∝ Ldf. (1.4)
In other words, the volume of the infinite cluster is sub-extensive, and its volume frac-tion in an infinite system tends to 0. For the Lorentz model, since the largest obstacles
2It is worth to note that, for the Lorentz model, the strength of the infinite network behaves in the
opposite way with respect to the classical percolation in a lattice. In the latter case in fact, below the critical threshold it is null whereas above it tends to increase with the power law as in Eqn.1.2.
bundles are typically of size ξ for n∗ → n∗c, the percolating cluster remains homogeneous on length-scales larger than the correlation length and ramified, with holes on all smaller scales. It follows that the correlation length can be interpreted as being a typical dis-tance above which the infinite cluster is statistically self-similar, i.e. fractal [13,16,18]. It is worth to note that, at the critical density, not only the infinite cluster becomes self-similar but also the total void space. This implies that all finite clusters resemble the infinite one on scales bigger than the typical microscopic scale σ, but smaller compared to their size, such that the “finite horizon” of the cluster has not been explored yet. In particular, the large clusters with linear dimension Rs σ are fractals whose volume
scales as Rdsf. The linear dimension of a certain cluster is also known as the radius
of gyration3, matematically defined as the root mean square distance of the cluster’s ramifications from its center of mass.
A hyperscaling relation can be introduced to relate the fractal dimension of the infi-nite cluster to the exponents of the geometrical properties through the real dimension of the system. In fact, in presence of the infinite cluster, one can infer that M = Φ∞Ld
or Φ∞ = MLd. In addition to this fact, using Eqn. (1.3), Φ∞ ∼ L −β
ν. Equating the two
expression one gets M ∼ L−βνLd that, compared with Eqn.(1.4), gives the relation
df = d −
β
ν. (1.5)
For a three-dimensional Lorentz model df = 2.53 (β = 0.41, ν = 0.88), confirming the
subextensivity of the total weight of the self-similar structure [13].
The dynamics of structureless particles on fractal objects is also self-similar, and allows the introduction of new critical exponents that are independent of the ones introduced above. The self-similarity of a random walk has been demonstrated by Havlin and Ben-Avraham in 1987 [18]. In their review paper, they take a series of n consecutive steps of a random walk as a single big step; for large n, their distribution tends to a Gaussian. Nonetheless, when one consider two random walks with n1 and n2 steps, there are two
differences: the time needed for a step, given that in the first case t1 ∝ n1 and in the
second case t2 ∝ n2, and the average length of a step respectively equal to n1/21 and n 1/2 2 .
Consequently, scaling the time by a factor of λ and the length by a factor of λ1/2, the two walks will be equivalent. In other words, upon dilation of the space by a factor of λ1/2, the number of step increases by a factor of λ showing that the fractal dimension
3This quantity has been largely used in the study of polymers, being objects typically more
compli-cated than simple geometrical figures. For a two dimensional case, if one turns a certain cluster around an axes that passes perpendicularly to its center of mass, the kinetic energy and angular momentum of this rotation are the same as if all sites were on a ring of radius Rs. The s subscript indicates an s-sized
of the random walk, appropriately defined, is dw=
log λ
log λ1/2 = 2. (1.6)
In general, the mean square displacement can be expressed as
δr2∼ t2/dw. (1.7)
The fractal dimension of a random walk plays a central role for the anomalous diffusion. The values that it may assume in different regimes are discussed in the following.
1.3
Evidences of heterogeneous dynamics
The research on the Lorentz model, in the last ten years, has been mainly focused on pro-ducing the proofs of heterogeneous dynamics by studying the mean square displacement of the tracers and the self-intermediate scattering function. Both of them have been treated numerically and theoretically by applying the so-called cluster-resolved scaling theory. Even though this first part of the theory has been already derived by Kammerer et al. in 2008 [19,20], I present it here more detailed and especially more functional for the further developments presented in Chapter 2.
1.3.1 Introduction to the cluster-resolved scaling theory
The first fact to realize, when it comes to study the scaling relations of a percolating system, is that one can refer to two possible ensembles: the restricted and the all-cluster ones. The former refers to intruder whose motion is confined to one all-cluster only, while the latter considers the motion on cluster of all sizes, making it required to average. Therefore, it is necessary to introduce a quantity that defines the cluster size distribution. This is usually achieved employing the so-called cluster number ns that,
multiplied by the number of sites per cluster s, gives the probability that a site belongs to an s-sites cluster. This quantity is expected to decay, at the percolation threshold, as a power law like [16]
ns∼ s−τ(s → ∞), (1.8)
where τ is the Fisher exponent that can be related to the fractal dimension of the system via
τ = 1 + d df
Now, the largest finite cluster is of the order of the correlation length and, consequently, the following scaling law is expected for the cluster-size distribution
ns(s, ) = R −d−df
s n±(Rs/ξ). (1.10)
The prefactor, that encodes the scaling behavior at the fixed point, is derived taking into account the proportionality between the number of sites and the dimensions of the system, and Eqns. (1.8) - (1.9). The scaling function n±describes how the critical point repels trajectories (in the sense of renormalization group) that are close but not on the critical manifold. Moreover, we expect that the cluster-size distribution is self-similar for clusters small compared to the correlation length implying n±(Rs/ξ 1) = const.,
while clusters larger than the correlation length are suppressed, n±(Rs/ξ 1) → 0.
Under these premises, the scaling relations for the dynamical quantities can be elab-orated starting from the Van Hove self-correlation function, which represents the proba-bility that the tracer particle has been displaced by ~r in time t. Its definition it’s given for N structureless particles with time-dependent positions ~Ri(t), i = 1, . . . , N , distributed
uniformly in the void space enclosed in a box of volume Ld. The thermodynamic limit N → ∞, Ld→ ∞ with fixed particle density N/Ld is anticipated. Consequently,
G(~r, t) = 1 N X i î hδ(~r − [ ~Ri(t) − ~Ri])i ó av, (1.11)
where h·i denotes a thermal average and [·]av an average over different realizations of
the host fixed matrix (“average over disorder”). In the thermodynamic limit G(~r, t) becomes independent of the system size and is isotropic. We also define a fluctuating van-Hove self-correlation function
Γ(~r, t) = 1 N
X
i
δ(~r − [ ~Ri(t) − ~Ri]), (1.12)
such that its thermal and disorder averages gives back Eqn. (1.11) as
G(~r, t) = [hΓ(~r, t)i]av. (1.13)
To take into account that the void space is the disjoint union of clusters Cs, we may
partition the sum over all particles by collecting firstly by clusters and, secondly, by sorting according to their size s. Sums over all particles can therefore be performed by replacing P
i 7→ PCs
P
i∈Cs where
P
particles that are confined to cluster Cs. We introduce the fluctuating van-Hove
self-correlation function on cluster Cs as
Γ(~r, t, Cs) = 1 Ns X i∈Cs δ(~r − [ ~Ri(t) − ~Ri]). (1.14)
Note that each term in the sum is the same, since an expectation value has to be taken. Therefore it makes sense to normalize by the number of particles in that cluster Ns = N (s/ΦLd), the last relation follows by assuming that all clusters carry the same
density of particles. Here Φ = Φ() is the porosity of the system, defined in (1.1); this behaves regularly in the vicinity of the critical point, thus one may safely replace Φ 7→ Φc.
As a consequence of these rearrangements, the van-Hove self-correlation function follows as a sum over all clusters and a subsequent thermal and disorder averages
G(~r, t) = X Cs Ns N hΓ(~r, t, Cs)i av . (1.15)
Close to the critical point, all large clusters look alike provided distances and times are measured in terms of their natural scales. The scaling hypothesis implies
hΓ(~r, t, , Cs)i = R−ds G ±
F(~r/Rs, tR −dw
s , Rs/ξ). (1.16)
Here, distances ~r are compared to the cluster size Rs whereas times t are compared to
the time it takes to explore a cluster t ∼ Rdw
s . At the critical point Rs/ξ → 0 for any
finite cluster and the last argument may be ignored. The scaling function G+F refers to a system above the critical point, > 0, while G−F is for lower densities. Finally, the prefactor R−ds in the scaling law, Eq. (1.16), is chosen to guarantee normalization
Z
ddr Γ(~r, t, , Cs) = 1, (1.17)
after substituting by dimensionless argument ~r 7→ ~r/Rs.
A similar scaling prediction for the infinite cluster can be made. In fact, by rewriting the scaling hypothesis for the finite clusters in an equivalent form that allows to take the limit of an infinite cluster, one gets
hΓ(~r, t, , Cs)i = ξ−dG±∞(~r/ξ, tξ−dw, Rs/ξ), (1.18)
where the reference scale for the infinite cluster has been taken to be the largest linear size, i.e. the correlation length; the last argument may be ignored as it always infinite.
Taking this into account, one arrives at the scaling hypothesis
hΓ(~r, t, C∞)i = ξ−dG∞(~r/ξ, tξ−dw), (1.19)
with G∞ is the corresponding scaling function.
In order to obtain the all-cluster averaged van-Hove self-correlation function one has to sum over all cluster-resolved van-Hove functions, each one weighted by its volume. At first, let us consider only the contributions of the finite clusters.
After averaging, the sum over all finite clusters is replaced by [P
Cs<∞. . .]av7→ ΦL dR
dsn(s, ) . . ., where the factor ΦLd accounts for the fact that the number of clusters grows with the
accessible volume. Since the fraction of particles Ns/N located on cluster Csis precisely
s/ΦLd, one arrives at the scaling prediction, Eqn (1.16) at
X Cs<∞ Ns N hΓ(~r, t, , Cs)i av = Z ∞ 0 ds sn(s, )Rs−dG±F(~r/Rs, tR−ds w, Rs/ξ). (1.20)
The scaling behavior of the integral can be extracted by substituting s ∼ Rdf s, i.e.
ds ∼ Rdf−1
s dRs, and employing the scaling prediction of the cluster-size distribution
n(s, ), Eqn. (1.8), . . . ∼ Z ∞ 0 Rdf−1 s dRsRsdfRs−d−dfn(Rˆ s/ξ)R−ds G±F(~r/Rs, tR−ds w, Rs/ξ) ∼ Z dRsRdsf−2d−1ˆn(Rs/ξ)G±F(~r/Rs, tR−ds w, Rs/ξ) = ξdf−2dGˆ± F(~r/ξ, tξ −dw), (1.21)
with ˆG±F another general scaling function.
On the other hand, the infinite cluster contributes with its volume fraction Φ∞() ∼
(−)β ∼ ξ−β/ν, such that
Φ∞()hΓ(~r, t, C∞)i ∼ ξ−d−β/νG∞(~r/ξ, tξ−dw) (1.22)
∼ ξdf−2dG
∞(~r/ξ, tξ−dw). (1.23)
where Eqn. (1.5) has been used in the last step. Hence, we find that both the contribu-tions of the hierarchy of finite clusters and the infinite cluster scale in the same way.
To sum up, the all-cluster-averaged van Hove self-correlation function satisfies
1.3.2 The Mean-Square Displacement
The second moment of the Van-Hove self-correlation function defines the mean-square displacement. Using a similar approach to the one presented above, one can define the cluster-resolved mean-square displacement4 as
δr2(t, , Cs) = t2/dwδrF2(tR −dw
s , R1/νs ), (1.25)
δr2(t, , C∞) = t2/dwδr2∞(tξ−dw), (1.26)
where the scaling behavior is the one expected from Eqn. (1.7).
The all-cluster average is obtained by employing these two relations, each one weighed by the respective probability that a site belong to a cluster of a certain dimension. In other words, δr2(t, ) =X s snsδr2(t, , Cs) + Φ∞δr2(t, , C∞) ∼ t2/zδr±2(ˆt), (1.27)
where δr±2 is the scaling function that describes the approach to the percolation thresh-old from the localized and from the diffusive regimes respectively, ˆt = t/tx with tx that
defines the crossover timescale up to which the diffusion and localization regimes occur, and z := 2dw
2+df−d [12, 13] is the critical exponent that encodes for the scaling behavior
at the critical point.
H¨ofling et al. [17] studied the mean-square displacement for an all-cluster ensemble in a three-dimensional Lorentz model. As one can observe from Figure 1.2a, at least three different behaviors are reveled depending on the density.
Looking at the upper lines of the graph, one can observe a quite sharp transition from a ballistic regime, where the displacement of the tracer depends on the square of the time, to a diffusive one, thus confirming that at low densities of the fixed matrix, the movement of the tracers is not influenced at all. At greater excluded volume, it is clear the appearance of an intermediate time dependence that becomes even larger at densi-ties approaching the critical one: the increasing in the disorder in the system acts as a friction that slows down the diffusion process but not necessary prevents it.
The subdiffusive regime is obeyed over more than five decades at n∗c = 0.839, and it corresponds to a value of the critical exponent z = 6.25 for three-dimensional percola-tion. This regime, as underlined by the theory, is the result of the average MSD among tracers located in finite and infinite clusters. For densities higher than the critical one,
4
As most of the clusters have linear dimension much smaller than the correlation length, i.e. R1/νs
(a) All-cluster MSD [17] (b) Infinite cluster MSD [13] Figure 1.2: Mean-square displacement (MSD) for the two typical ensembles for
differ-ent densities in a 3D Lordiffer-entz model. Black lines locate the critical density.
the dynamics becomes localized: the tracers are only confined in closed bundles and are not free to diffuse everywhere. In this regime, the tracer can explore, at maximum, a certain area and consequently the mean-square displacement tends to a plateau at long times.
Figure 1.2b shows the mean-square displacement for the tracers located on the infinite cluster only. In here, as expected, the walk dimension of the tracers at criticality is lower and equal to 4.81 as the particles experience less spacial restriction. Obviously, no localized regime is reported as the infinite cluster ceases to exist just above the critical density.
1.3.3 The Self-Intermediate Scattering Function
Another relevant quantity that confirms the Lorentz model to be strongly heterogeneous in the dynamics is the self-intermediate scattering function, which constitutes a precur-sor of the dynamical susceptibility.
Physically, the self-ISF contains information on the decorrelation among particles: the faster its decay, the shorter the average relaxation time. Mathematically, the intermediate scattering function is merely the Fourier transform of the Van-Hove self-correlation function defined in Section 1.3.1, even though, especially in simulations, a more convenient and effective approach is the calculation via the single-particle tra-jectories. For how it is defined, its maximum value is 1 and the long-time limit is 0, which corresponds to the total decorrelation of the tracers. In order to study the scaling behavior, the first definition is more favorable
F (q, t, ) = 1 N X i î hexpÄi~q · ~Ri(t) äó av= Z ddrei~q·~rG(~r, t, ). (1.28)
with ~q as a wave-vector. As for the Van-Hove self-correlation function, we shall introduce the fluctuating self-intermediate scattering function
Φ(~q, t) = 1 N X i expÄi~q · ~Ri(t) ä = Z ddrei~q·~rΓ(~r, t), (1.29)
where Γ(~r, t) is defined in Eqn. (1.12). Similarly, for the cluster-resolved quantities, Φ(~q, t, Cs) = 1 Ns X i∈Cs expÄi~q · ~Ri(t) ä = Z ddrei~q·~rΓ(~r, t, Cs), (1.30) such that [hΦ(~q, t)i]av= F (q, t). (1.31)
However, it’s important to realize that, in this form, the oscillating factor in the Fourier transform kills effectively all the contributions of distances larger than |~r| & 1/q, that are the ones for which a scaling prediction is defined. Thus, we rewrite in the following form
1 − F (q, t, ) = F (0, t, ) − F (q, t, ) =
Z
ddrî1 − ei~q·~róG(~r, t, ), (1.32)
such that now large distances have a more relevant contribution to the integral. The scaling prediction then becomes5
1 − F (q, t, ) = ξdf−2d Z ddrî1 − ei~q·~róG(~r/ξ, tξ−dw) = ξdf−d Z dd(r/ξ)î1 − ei~q·~róG(~r/ξ, tξ−dw) = ξdf−dF (qξ, tξ−dw). (1.33)
It’s also interesting to look at the so-called nonergodicity parameter f (q, ) defined as limt→∞F (q, t, ). A non-vanishing f (q, ) indicates that dynamic correlations are
per-sistent, implying a non-ergodic dynamics; roughly speaking, it measures the fraction of particles that are trapped on a length scale of 2π/q [22]. The scaling prediction is the same of the self-intermediate scattering function
1 − f (q, ) = ξdf−df (qξ).ˆ (1.34)
For qξ 1 all finite clusters are explored and contribute unity to the nonergodicity parameter, while the contribution of the infinite cluster is always zero: in the long-time limit the particles just run away to infinity and thus have an ergodic behavior. The scaling prediction is independent of the wavenumber and states
1 − f (q, ) ∼ ξdf−d ∼ Φ
∞(), (1.35)
(a) Density-dependence of the non-ergodic pa-rameter [22]
(b) All-cluster ISF at n∗c [22]
Figure 1.3: The self-intermediate scattering function and the non-ergodic parameter shows that the dynamics at low wave-vectors is never ergodic due to the presence of a hierarchy of finite clusters.
which is the weight of the infinite cluster. Since above the percolation threshold, n∗ > n∗c, the infinite cluster ceases to exist, the non-ergodic parameter tends, for low wave-vectors, to 1 (Figure 1.3a). The extrapolation to the null wave-number exibits a sharp transition at the critical density.
Naturally, the slow-down of the dynamics is also signaled by the self-ISF at low wave-vectors. In fact, the cluster size distribution at the percolation threshold is wide and the shape of the self-ISF is due to the contribution of many different diffusive processes that occur on different clusters: their superposition produces the highly non-Gaussian curves reported in Figure 1.3b. As a comparison, the Gaussian approximation of the self-ISF and its first-order correction are reported in the same Figure as broken and dotted lines, respectively.
Brief summary
In Chapter 1, after a brief contextualization of the issue of the dynamical heterogeneities in various systems, I presented the Lorentz model and its peculiarities, with special refer-ence to the its fractal behavior and its relation with a percolating system. Subsequently, I focused the attention to the main evidences the scientific community has produced to demonstrate the non-uniformity in the dynamics of the model. For this reason, I presented our tailored re-adaptation of the cluster-resolved scaling theory and the cor-responding numerical results published in the literature. In the next Chapter, I’m going to introduce the dynamical susceptibility and our first related numerical evidences.
A cluster-resolved approach
2.1
The Dynamical Susceptibility χ
4In order to obtain the mathematical expression for the dynamic four-point susceptibility, or simply dynamical susceptibility, let us first define the connected pair self-van-Hove correlation function as Gc4(~r0, ~r00, t) = 1 N X ij î hδ(~r0− [ ~Ri(t) − ~Ri])δ(~r00− [ ~Rj(t) − ~Rj])i ó av − 1 N X ij î hδ(~r0− [ ~Ri(t) − ~Ri])ihδ(~r00− [ ~Rj(t) − ~Rj])i ó av, (2.1)
where h· · · i denotes a thermal average and [· · · ]avan average over realizations of the host
matrix as in Chapter 1. In terms of the fluctuating van-Hove self-correlation function, Eqn. (1.12), this can also be expressed as1
Gc4(~r0, ~r00, t) = N
hΓ(~r0, t)Γ(~r00, t)i − hΓ(~r0, t)ihΓ(~r00, t)i
av. (2.2)
The application of the Fourier transform to the Equation above yields
Z ddr0ddr00ei~q(~r0−~r00)Gc4(~r0, ~r00, t) = 1 N X ij h hei~q[ ~Ri(t)− ~Ri]e−i~q·[ ~Rj(t)− ~Rj]ii av − 1 N X ij h hei~q[ ~Ri(t)− ~Ri]ihe−i~q·[ ~Rj(t)− ~Rj]ii av, (2.3)
1The factor N in Eqn. (2.2) accounts for the expectation that fluctuations decrease as 1/N with the
system size, i.e. Gc4should be an intensive quantity in the thermodynamic limit.
where one can recognize the fluctuating self-intermediate scattering function, Eqn. (1.29), such that the final result of the Fourier transform is
· · · = Nîh|Φ(~q, t)|2ió
av− N
î
|hΦ(~q, t)i|2ó
av=: χ4(q, t), (2.4)
with χ4(q, t) the dynamic four-point susceptibility. The two natural variables are
respec-tively the wave-number q and the time t. In practice, this formula permits an efficient measure of the degree of dynamical heterogeneity especially in computer simulations where the dynamics is easily resolved both in space and time.
Being the Lorentz model strongly heterogeneous due to its compartmentalization (tracers can diffuse and relax in different manners), we anticipate χ4(q, t) to display a divergence
as the percolation threshold is approached.
Besides, it can be demonstrated that the long-time limit of the dynamical susceptibility tends to 1 in case the dynamics is ergodic. By substituting the exponentials in Eqn. (2.3) with the Euler expression, one gets
χ4(q, t) = 1 N X ij î
hcos ~q∆ ~Ricos ~q∆ ~Rji + hsin ~q∆ ~Risin ~q∆ ~Rji
ó av − 1 N X ij î
hcos ~q∆ ~Riihcos ~q∆ ~Rji + hsin ~q∆ ~Riihsin ~q∆ ~Rji
ó
av,
(2.5)
where ∆ ~Ri = ~Ri(t) − ~Ri and the imaginary part drops algebraically. In the long-time
limit the second term of the expression goes to zero as well as the terms of the sum for which i 6= j. Consequently, we are left with
· · · = 1 N
X
i
î
hcos ~q∆ ~Ricos ~q∆ ~Rii + hsin ~q∆ ~Risin ~q∆ ~Rii
ó
av= 1. (2.6)
2.1.1 Cluster-resolved Dynamical Susceptibility - Scaling Theory
The strategy to obtain a cluster-resolved dynamical susceptibility is again to decompose the global expression, Eqn. (2.4), taking into account the presence of the clusters. I’m going to rely on the same approach and notation that I have presented before for the self Van-Hove function.
The substitution of the cluster-resolved fluctuating ISF, Φ(~q, t) =P
Cs(Ns/N )Φ(~q, t, Cs), reveals that χ4(q, t) = N X Cs,C0s NsNs0 N2 hΦ(~q, t, Cs)Φ(~q, t, Cs0)∗i − hΦ(~q, t, Cs)ihΦ(~q, t, Cs0)∗i av . (2.7)
To make theoretical progress we assume that contributions arising from different clusters do not contribute. In other words the fluctuations of the self-ISF on distinct clusters are uncorrelated. By assuming this, the global four-point susceptibility becomes a weighted average over all clusters
χ4(q, t, ) = X Cs Ns N χ4(~q, t, , Cs) av , (2.8)
where Ns is again the number of particles in the cluster Cs. Here, the cluster-resolved
four-point susceptibility it’s been implicitly defined as χ4(~q, t, Cs) = NshΦ(~q, t, Cs)Φ(~q, t, Cs0)
∗i − hΦ(~
q, t, Cs)ihΦ(~q, t, Cs0) ∗i
. (2.9)
At this stage, one can formulate a scaling hypothesis for the latter quantity. The basic insight is that, at the critical point, all large but finite clusters look alike and all ob-servables should behave identically provided lengths and times are measured in natural units, i.e. the linear size of the finite cluster Rs and the time t ∼ Rsdw it takes to
ex-plore it. Furthermore, by scale invariance, all large but finite clusters should resemble the infinite one on small length and times scales, such that the ”finite horizon” has not been explored yet. This remains true slightly above or below the critical point on length scales up the correlation length ξ. Finally, as the fluctuations of the ISF on large clusters should grow with the number of particles on this cluster, Eqn. (2.9), it is true that Ns∼ Rsdf by recalling that bigger clusters host more tracers.
Hence, these insights suggest the following scaling law for the cluster-resolved dynamical susceptibility
χ4(~q, t, , Cs) = Rdsfχˆ4(qRs, tR−ds w, Rs/ξ). (2.10)
The strongest fluctuations should emerge at wave-numbers that match the size of the cluster qRs ∼ 1; for higher wave-numbers only fluctuations on a smaller fraction of the
cluster are probed whereas, for lower wave-numbers, the cluster is not resolved anyway. Since the finite clusters look exactly like the infinite one, provided their finite size has not been probed yet, a scaling form for the infinite cluster can be inferred by switching to new variables, similarly to what has been worked out for the van-Hove correlation function. Hence,
χ4(~q, t, , C∞) = ξdfχˆ∞4 (qξ, tξ−dw). (2.11)
In the thermodynamic limit, it’s allowed to replace [P
Cs<∞]av. . . 7→ ΦLd
R
dsn(s) . . .. Since the fraction of particles Ns/N located on cluster Csis precisely s/ΦLd, one arrives,
for the finite clusters, at X Cs<∞ Ns N χ4(~q, t, , Cs) av = Z ∞ 0 ds sn(s, )χ4(~q, t, , Cs). (2.12)
The scaling behavior of the integral can be extracted by substituting s ∼ Rdf
s , i.e ds ∼
Rdf−1
s dRs and employing the scaling prediction of the cluster-size distribution n(s, ),
Eqn. (1.10), . . . ∼ Z ∞ 0 Rdf−1 s dRsRsdfRs−d−dfn(Rˆ s/ξ)Rsdfχˆ4(qRs, tR−ds w, Rs/ξ) = ξ2df−dχˇ 4(qξ, tξ−dw). (2.13)
On the other hand, the contribution of the infinite clusters comes with its volume fraction Φ∞() such that
Φ∞()χ4(~q, t, , C∞) = ξ−β/νξdfχˆ∞4 (qξ, tξ−dw) = ξ2df−dχˆ∞4 (qξ, tξ−dw). (2.14)
Thus, both the infinite cluster and the hierarchy of finite clusters scale in the same way giving, as prediction for the total four-point susceptibility,
χ4(q, t, ) = ξ2df−dχˆ4(qξ, tξ−dw). (2.15)
The most general scaling form for the dynamical susceptibility can be obtained by triv-ially adding the box size L as another scaling variable:
χ4(q, t, , L) = ξ2df−dχˆ4(qξ, tξ−dw, L/ξ), (2.16)
Other scaling predictions can be made by specializing to the critical point. For = 0, the correlation length diverges ξ → ∞, so that an appropriate form of the scaling law is required to perform the limit. Trading off ξ in terms of q−1, one arrives at
χ4(q, t, = 0) = qd−2dfχ(tq˜ dw). (2.17)
For times t q−dw the system has not approached the cutoff length 1/q, and the
formula, consequently, should be independent of the wave-number. This happens only if ˜χ(τ 1) ∼ τ(2df−d)/dw, where τ = tqdw. Hence,
On the other hand, for large times, the fluctuations should saturate, i.e. become time-independent, which implies ˜χ(τ 1) = const.. Then,
χ4(q, t, = 0) ∼ qd−2df = q−(2−η), (2.19)
where the exponent relations γ = ν(2df− d) and γ/ν = 2 − η (Fischer relation) have
been used.
In simulations, by looking at the smallest wave-number that can be reached, qmin, Eqn.
(2.15) becomes
χ4(qmin, t, ) = ξ2df−dχˆ4(qminξ, tξ−dw), (2.20)
and, in order to see a dependence on , i.e. to verify how the critical point is approached in terms of the dynamical susceptibility, we restrict everything to the regime ξ L ∼ 1/qmin, that is qminξ 1. Then, the first argument is always small, and we are allowed
to replace it by zero. This yields
χ4(qmin, t, ) = ξ2df−dχˆˆ4(tξ−dw). (2.21)
At large times, the hierarchy ξ t1/dw L holds, i.e the tracer had enough time to
explore the correlation volume and the fluctuations should not depend on time anymore. Thus, ˆχˆ4(τ 1) = const. which implies
χ4(qmin, t, ) ∼ ξ2df−d. (2.22)
Moreover, by working out Eqn. (2.22), one can recognize the scaling form of the mean cluster size defined, in general percolation theory, as S =R
dss2n(s) ∼ ||−γ, with the integral that extends to the largest clusters of volume ξdf. Let us demonstrate it by
directly employing this last expression:
S = Z dss2n(s) ∼ Z dRs ds dRs R2df s R −d−df s ∼ Z ξ dRsRdsf−1R2ds fR −d−df s ∼ ξ2df −d∼ ||−ν(2df−d) = ||−γ. (2.23)
On the contrary, if one looks at small times, the hierarchy t1/dw ξ L holds, and it
can be argued that the system hasn’t reached the finite correlation length yet. Therefore to make the dependence on ξ cancel, we impose ˆχˆ4(τ 1) ∼ τ(2df−d)/dw
χ4(qmin, t, ) ∼ t(2df−d)/dw. (2.24)
This theoretical derivation shows how a consistent cluster-resolved approach may be used in order to describe the all-cluster dynamical susceptibility. Moreover, it has been
demonstrated that, by studying directly the fluctuations of the self-intermediate scatter-ing function, one can get information regardscatter-ing the percolation transition via the critical exponents.
2.2
Intermezzo: the Dynamical Susceptibility in a model
of permanent gels and in glasses
If one had to compare the Lorentz model with other systems in order to foresee the time dependence of the dynamical susceptibility, most similarities would be found for permanent gels. In this model, clusters of solute are permanently kept together while suspended in a solvent; aggregates of particles can diffuse, but the nature of the hetero-geneity remains static.
(a) (b)
Figure 2.1: The time and the wave-vector dependence of the dynamical susceptibility are reported in left and right panel, respectively, for increasing densities (from bottom to top). The black line in (b) shows that the relation between the long-time χ4and the mean cluster size si verified, χ∞4 ∼ kη−2 [24].
Figure 2.1a shows the dynamical susceptibility at the lowest wave-vector for different densities approaching the critical one (from bottom to upper lines).
In particular, one can notice an increase of the dynamical susceptibility before it reaches a plateau at long-times; the stabilization value is always higher as the critical density is approached. Thus, the dynamical susceptibility will never reaches the long-time limit we would observe if the dynamics is ergodic: correlation among particles that resides on the very same cluster will always be present. The right panel, Figure 2.1b, reveals that the long-time limit of the χ4 just analyzed is the highest one it can be registered, since
smaller values are observed at smaller length scales (or at higher wave-vectors, in these graphs indicated with k). In addition, close to the critical density (highest curve), the expected relation with the mean cluster size is verified. Despite these affinities, in the case of permanent gels, one treats a finite number of particles of which the system is constituted whereas, in the Lorentz model, one deals with tracers exploring a portion of empty volume.
A different behavior is instead observed for glasses. This is a typical case where the dynamical heterogeneities have a transient nature: the χ4(t) has a peak on timescales
of the order of the typical relaxation time of the fluid. For glasses, it coincides with the average time requested by the particles to exit the “cage” in which they typically are encapsulated. At longer times, the dynamical susceptibility decreases and reaches the typical long-time limit of 1. Figure 2.2 shows what just reported, for different temper-atures. Here, the shape of the χ4 is coherent with the slow-rearranging dynamics that
characterizes glass-forming liquids.
Figure 2.2: Time dependence of the dynamical susceptibility. For each temperature, χ4has a maximum which shifts to larger times
and has a larger value when T is decreased, revealing the increasing length scale of dynamic heterogeneity in liquids that approach the glass transition [4].
2.3
All-cluster χ
4- Numerical Analysis
To study numerically the dynamical susceptibility, we run extensive, large scale Molec-ular Dynamics simulations of the Lorentz model.
In particular, we employ a square simulation box of size L = 100σ (see below) where periodic boundary conditions are applied. The initial configuration of the system is prepared by inserting in the box 839000 spheres to obtain a reduced obstacle density of n∗ = n∗c = 0.839; the spheres’ position is fixed. Afterwards, other 1000 point-particles have been placed in the empty volume serving as intruders that explore the finite and
Figure 2.3: The upper lines show the average MSD of the tracers for a low density of obstacles: the theoretical curve and the numerical calculation are superimposed even if the simula-tion box is small. This is not true at the critical density where perfect matching is observed only if the MSD is cal-culated for a box of L = 100. The inset shows a detail of the Figure: smaller simulation boxes imply a more pronounced divergence at long times.
the infinite clusters; the tracers are inserted in a completely random way, no previous information regarding the location or the linear size of the clusters is known.
The number of tracers was chosen to be 1000 for the simulation to be feasible in the most reasonable amount of time and for the results to have a good enough statistics: for a system prepared in this way the necessary computing time was approximately 25 days on the supercomputer to study 7 decades in time.
The final size of the box is set in agreement to previous shorter simulations where the average mean-square displacement of the tracers have been compared to the theoretical expectations. This is reported in Figure 2.3 for n∗ = 0.50 and for the critical density. One can notice that, in the first case, the simulation result is perfectly over-imposed to the theoretical curve even for a small box (L = 50σ). On the contrary, when the diffu-sion is fully anomalous, a deviation appears: only if the box has a linear size of 100σ (green line), the theoretical prediction is reproduced and no finite-size effect emerge. Concerning the units in which the quantities are reported hereafter, the mass is ex-pressed in units of the tracers’ mass m = 1, the length in units of σ (the radius of the spheres), and the wave-numbers have a mesh of 100σ2π . Finally, the time is expressed in units of t0 =
q
mσ2
Figure 2.4: Dynamical susceptibility for 1000 tracers randomly dis-placed in the empty volume of a box of size 100σ. No dy-namical heterogeneities evidence is observed.
here kBT is fixed to 1.
Figure 2.4 shows the results for χ4 in the above-mentioned simulation conditions. The
dynamical susceptibility have been calculated by using Eqn. (2.4). In order to improve the quality and the reliability of the outcomes, each wave-vector q employed in the cal-culation is constituted by a subset of vectors whose components have been combined such that they have the same modulus.
Quite astonishingly, however, we register no evidence of dynamical heterogeneities for all the wave-vectors analyzed since the value of the χ4 always stands below 1; Figure
2.4 does not even display a transient maximum and contrasts with the results shown for the permanent gels.
Two are the possible explanations for what’s been found in these calculations. Firstly, the number of tracers we have inserted in the system might have been too small to effectively detect the fluctuations of the self-intermediate scattering function; secondly, the contribution of the different clusters to the total χ4 might have badly averaged out,
such that the system all together does not turn to be heterogeneous even though its separate constituents (the single clusters) might have been.
In the following section, I’m going to treat and analyze the single clusters by recall-ing the cluster-resolved approach described previously.
2.4
Cluster-resolved χ
4- Numerical Analysis
The numerical study of the cluster-resolved dynamical susceptibility goes through the identification of the clusters in the system.
The first attempt has been based on a grid method: the distribution of the obsta-cle spheres in the simulation box is mapped onto a three-dimensional grid which allows to determine the exact location of the connected regions, their distribution in size and their gyration radii. After the recognition of the grid boxes “occupied” by the spheres, the three-dimensional simulation box is split in two-dimensional layers in which the con-nected clusters are labeled. By joining the layers one-by-one and by applying periodic boundary conditions, it is possible to reconstruct all the clusters of the system.
This method has been tested by checking the critical exponents of the cluster size distri-bution nsand by verifying the relation between the radius of gyration of the clusters and
their size, see Figure 2.5 for a test case. The result shows that the expected relation is not well verified. It’s likely that the size of the cubes constituting the three-dimensional grid are not small enough to detect properly the narrow channels, so that the clusters identified do not resemble the true distribution. Although an increase in the “resolution”
Figure 2.5: Radius of gyration of the clusters as a function of their size. The size is measured in terms of the number of cubes com-posing the three-dimensional grid. In this test case the side of each cube has size < 0.1σ. The red line indicates the expected relation Rs= s1/df.
of the grid would represent a possible solution, the method would not be computation-ally feasible anymore due to limitation in the physical memory of the supercomputer2.
Consequently, the clusters have been “reconstructed” through the single points of the trajectories of the particles. This might produce an underestimate value of the di-mension of the clusters especially if (a) the host matrix is rich in narrow channels (that happens for sure at the critical point), (b) the simulations are not long enough and (c) the statistics is poor. Moreover, one has to keep in mind (d) that the cluster-resolved dynamical susceptibility (and, similarly, the all-cluster quantity) should depend on the number of particles diffusing in the cluster under investigation, following Eqn. (2.9). For this reason, each cluster has to be explored by a congruous number of tracers in order to properly probe the fluctuations of the self-ISF.
Thus, in practice, making use of the output of the very first simulations, we reconstructed the trajectories of all the 1000 tracers obtaining 1000 estimates of gyration radii. From those, we chose several representative values and, for each one, at least three different tracers that according to the “first estimate” were diffusing on similar distances. We then restarted an equal amount of simulations by forcing 1000 tracers to be on the same cluster. All this procedure has been repeated for two different realizations of the obstacle matrix.
This scheme, built by taking into account all the possible weaknesses outlined above, is fundamental to give a full description of the cluster-resolved dynamical susceptibility, which is the goal at this stage3. We now proceed by verifying numerically the theoretical assumptions.
2.4.1 Cluster analysis and wave-vector dependence
Let us analyze, at first, the computed gyration radii for a series of clusters with respect to the mean-square displacement (MSD) of the particles diffusing in the same volume. The MSD constitutes a good approximation of the so-called “hydrodynamic radius” Rh,
which typically specifies the largest extension of soft colloids measured, for instance, by Dynamic Light Scattering experiments. In this case, it represents the radius that includes all the ramifications of the cluster, being calculated from all the points in the trajectory. Curiously, as it can be noticed in Figure 2.6a, the established relation be-tween the hydrodynamic and the gyration radius in polymers [26] is valid also here, since
2
Another method, always based in the partition of the system, would require both the Voronoi and the Delaunay tesselation, as described in [25]. However, for reasons related to the time available and for the complexity of the algorithm, this has not been implemented so far.
3With this procedure, it’s not possible to reconstruct the full cluster size distribution and the
(a)
(b)
Figure 2.6: (a) The Figure reports the values of Rh and Rs for the analyzed clusters.
The slope of the line is 1.34. The radii are reported in units of σ. (b) “Reconstruction” of a cluster whose Rh = 23.8 and Rs = 17.8 using the
trajectories of two particles specifically inserted in that pocket. The sphere of radius Rh includes all the point of the trajectories, whereas the other
one is smaller and its center correspond to the center of mass of the cluster. The axes represents the simulation box.
the former is always 1/3 bigger than the second one. Figure 2.6b, depicts an example of a cluster in the simulation box reconstructed via multiple trajectories.
Figure 2.7: Cluster-resolved dynamical susceptibility as a function of time for a small cluster whose Rs∼ 1.
Figure 2.8: Cluster-resolved dynamical susceptibility as a function of time for a medium-sized cluster whose Rs∼ 7.
different sizes and then averaged among similar ones. As one can notice from the exam-ples reported in Figures 2.7, 2.8, 2.9, the outcomes are considerably different compared to the all-cluster evidence. First of all, the value of the dynamical susceptibility exceeds 1 for certain wave-vectors and, after an initial transient, it stabilizes at a specific value, similarly to what has been observed for permanent gels. The fact that no drop in its value is observed is consistent with the static nature of the heterogeneities; in fact, no
Figure 2.9: Cluster-resolved dynamical susceptibility as a function of time for a big cluster whose Rs∼ 17.
rearrangement can occur at any time. The development of temporary maxima/oscilla-tions, that occurs on medium-big clusters, might be related to oscillations revealed also in the mean-square displacement. These may originate from interim deflections of the particle’s trajectory into the fractal ramifications of the clusters. It’s likely, however, that by averaging among many more similar clusters and realizations, the fluctuations disappear.
Figure 2.10: On average, for all the clusters investigated, q∗R s ∼ 1.
This suggests that χ4 has to be studied at a lenght-scale
Another important issue relates to the wave-vector at which the highest fluctuations of the self-ISF occur, q∗. As predicted theoretically, the highest dynamical susceptibility is registered at big wave-vectors for small clusters and viceversa for greater pockets. This trend has been found for all the cluster investigated, as shown in Figure 2.10, where the relation q∗Rs∼ 1 is valid. Thus, this confirms that, if the dynamics in the cluster is not
studied through an appropriate length-scale, the fluctuations are not properly depicted. For instance, let us consider the case of a medium/big-sized cluster where, for large wave-vectors, the long-time limit of 1 is observed. In this case, one can imagine that on small length scale the relaxation times of the tracers does not change in any appreciable way. Consequently, their self-intermediate scattering functions look alike among each other and the fluctuations are not relevant.
2.4.2 Size-scaling of the χ4
Having checked that different-sized clusters exhibit the maximum value of the dynamical susceptibility at different wave-vectors, let us move to analyze the χ4 for wave-vectors
not necessarily representative of the dimension of a certain cluster.
Figure 2.11a reports the trend of the long-time dynamical susceptibility χ∞4 as a function of the wave-vectors for cluster with four different gyration radii. Except to verify once more that q∗Rs ∼ 1, one can notice the similarity in the shape of the different curves,
that is highlighted even more after proper rescaling (Figure 2.11b). These relevant findings confirm the validity of the scaling function in Eqn. (2.10).
On the other hand, is there a relation between the values of the dynamical susceptibility at q∗ among distinct clusters? In principle, by looking at Eqn. (2.10), a clear scaling in terms of the size should be noticeable. Figure 2.12 is obtained by grouping together,
(a) (b)
Figure 2.11: Long-time dynamical susceptibility χ∞
4 as a function of q for different
clusters. Bigger clusters have their maximum dynamical susceptibility at lower q. Upon proper rescaling, the curves are almost superimposable.
Figure 2.12: The dynamical susceptibility doesn’t change if the number of inserted particles is the same in each cluster. This is verified by averaging χ4(q∗) according to three different
non-uniform meshes. The wide error bars suggest that the statistic has to be improved.
according to three different meshes, the values of χ∗4≡ χ4(q∗) by size-similarity and by taking their average. The values of dynamical susceptibility always oscillates around the same value (∼ 40) for all the three groupings. This is actually coherent with the procedure we adopted. In fact, we inserted in every cluster the same amount of particles and, if we assume it is true that the fluctuations of the self-ISF depend on the number of particle on each cluster, everything is consistent. Instead, a remark has to be done on the error bars: their amplitude indicates that the deviations from their average values are not negligible. For this reason, more realizations and more clusters should be considered in order to improve the statistics.
2.4.3 Cluster-cluster interactions
The scaling theory for the cluster-resolved dynamical susceptibility is built on the as-sumption that contributions coming from different cluster are completely negligible. This, however, has to be verified numerically.
For this reason, we expressly inserted particles into two separate clusters and we calcu-lated the total dynamical susceptibility by not making any distinction regarding their actual position; we then compared these outcomes with the results of simulations per-formed separately in the two above-mentioned clusters. Figure 2.13 shows, for two
Figure 2.13: Correlations on different clusters are not relevant. The χ4
calculated for all the 1000 tracers placed on two different clusters (full lines) is recovered when the two are treated separately (dotted lines). This analysis has been performed for several wave-vectors, two of which are here reported.
wave-vectors, that the average of the latter results is perfectly superimposed to the total dynamical susceptibility, thus confirming the consistency of the theoretical assumption.
2.4.4 Infinite cluster
The calculation of the dynamical susceptibility has been carried out also in the infinite cluster. In Figure 2.14, in the following page, I report some preliminary results (not yet averaged over different realizations and clusters) of the dynamical susceptibility for various densities approaching the critical point, namely n∗ = 0.75, 0.80, 0.82, 0.83, and for the critical density n∗c = 0.839.
By looking at the lowest wave-vector, Figure 2.14a, one can get information regard-ing the global dynamics, that is verified to be increasregard-ingly heterogeneous as the critical density n∗c is approached.
At progressively lower length-scales, a temporary maximum appears. This is possibly due to limited and momentary deviation of the particles in certain ramifications of the infinite cluster that are probed only at smaller scales. As the dynamics is ergodic, the long-time limit of 1 is observed. A similar behavior is found for glass-forming liquids whose time dependence of the dynamical susceptibility has been described in Section 2.2.
(a) qσ = 0.06 (b) qσ = 0.13
(c) qσ = 0.25 (d) qσ = 0.38
(e) qσ = 0.63 (f) qσ = 1.26
Figure 2.14: Time dependence of the dynamical susceptibility for the infinite cluster at different densities. Each subfigure refers to a different wave-vector. At the lowest lenght-scale one can get information regarding the global dynamics and notice that the dynamical heterogenities of the system increase as the critical density is approached. At high wavevectors, a similar behavior to the one found in glasses is observed.
These preliminary data are currently being extended and will be used to verify the theoretical predictions of the scaling theory via Eqn. (2.19).
Brief summary
The calculations of the dynamical susceptibility, based on 1000 tracers randomly dis-tributed in the entire system, showed no evidence of dynamical heterogeneities. We then employed a cluster-resolved approach through which we characterized and analyzed the single clusters of the system both theoretically and numerically. Instead, the final section has been dedicated to study of the infinite cluster at different densities and wave-vectors. The discrepancy found between the all-cluster and the restricted ensembles is investi-gated in the next Chapter.
The reliability of the χ
4
as
indicator of dynamical
heterogeneities in the Lorentz
model
The cluster-resolved study turned out to be crucial for the understanding of the all-cluster behavior. In fact, we verified the increase of the dynamical susceptibility for the single clusters of the Lorentz model. This would mean that, by applying the cluster-resolved theory, Eqn. (2.8), the global dynamical susceptibility increases over its typical long-time limit, disavowing the evidences we obtained for the all-cluster χ4 (Section 2.3)
but confirming the similarity of our results to the ones of the permanent gels.
As a consequence, it is evident that an innovative approach to the all-cluster calculations has to be applied in order to clarify whether or not (or in which conditions) the predicted increase of the χ4 is effectively detectable. I am going to discuss this issue and the final
implications in the following Sections.
3.1
Why is there a discrepancy between the all-cluster and
the cluster-resolved calculations?
3.1.1 The true Lorentz model: one tracer at the time
Since the reason why the total χ4 didn’t capture correctly the heterogeneities it is
amenable to (a) a sampling issue, (b) interactions wrongly taken into account or (c) 35
wrong averages among separate clusters, we have decided to verify what would be the outcome if only one tracer was involved in the calculation of the dynamical suscepti-bility. Up to this step, in fact, we have treated the Lorentz model as an ensemble of non-interacting tracers assuming the equivalence to the “true” Lorentz model in which
(a) Single-cluster calculation. A decrease in the number of tracers implies a reduction of the χ4. However, a non-direct proportionality is unveiled.
(b) All-cluster calculation. The single and the 1000 tracers χ4 are perfectly
superimposable. No proportionality between the χ4 and tracers appears.
Figure 3.1: One-tracer and 1000-tracers dynamical susceptibility in a single cluster and in the all-cluster ensemble. Two distinct wave-vectors are studied in both cases.
distinct replicas of a single diffusing tracer are analyzed. By applying this protocol to a single cluster and to the results that gave rise to the all-cluster evidence, it’s conceivable to find out where the complication resides.
Figures 3.1a - 3.1b show the calculation of the dynamical susceptibility for 1 and 1000 tracers for a single-cluster and in the all-cluster approach, respectively; two wave-vectors are analyzed. The one-particle χ4has been averaged among several distinct single-tracer
calculations to improve the statistics.
By looking at these Figures, it is evident that in the all-cluster case the 1 and 1000 tracers dynamical susceptibility are superimposed whereas, in the other situation, only the 1000 tracers calculation shows the presence of heterogeneities in the dynamics. We have already established that the dynamical susceptibility should be proportional to the intruders, but this does not seem to be the case, at least in the all-cluster simulation. Instead, a non-direct proportionality appears in the single cluster calculation, meaning that there can be a non-trivial phenomenon behind it. For this reason, we now move to analyze the N -particle dynamical susceptibility.
3.1.2 N -particle analysis
Figure 3.2 displays the calculation of the dynamical susceptibility for different numbers of tracers in a single finite cluster. As for the 1-tracer χ4, the N -particle susceptibility
shown here has been averaged among an ever-increasing number of sets, the smaller the amount of particles in each one. It’s worth to note that a decrease in the number of the particles corresponds to a decrease in value of the dynamical susceptibility. However, this reduction is proportional to number of tracers treated up to 100, while below no significant drop is revealed and, consequently, this proportionality is lost! The inset in
(a) N -particle χ4 for qσ = 0.50 (b) N -particle χ4for qσ = 0.50 rescaled by N
Figure 3.2: If the number of tracers in the clusters does not exceed 100, the correlation effect between the tracers and the matrix is not reveled. The inset shows the direct proportionality between tracers and χ4 above n.tracers = 100.
