# Coagulation-flocculation process on a lattice: Monte Carlo simulations

Viktoria Blavatska1,2111Author to whom any correspondence should be addressed,
Jaroslav Ilnytskyi1,3, and Erkki Lähderanta4,5 1 Institute for Condensed
Matter Physics of the National Academy of Sciences of Ukraine, 79011 Lviv,
Ukraine  2 Dioscuri Centre for Physics and Chemistry of Bacteria, Institute of
Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland  3
Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic
National University, 12 S. Bandera Str., UA-79013 Lviv, Ukraine  4 Department
of Physics, School of Engineering Science, LUT University, Yliopistonkatu 34,
FI-53850 Lappeenranta, Finland  5 Department of Physics, Universitat de les
Illes Balears, Cra Valldemossa, km. 7.5, 07122, Palma, Spain
<EMAIL_ADDRESS>
###### Abstract

Coagulation-flocculation, the physicochemical process widely used for
purification a wastewater, is affected both by chemical details of involved
polymers and by the statistics of their conformations on a large scale. The
latter aspect is covered in this study by employing a coarse-grained modelling
approach based on a combination of two paradigms of statistical mechanics. One
is the self-avoiding walk (SAW) which generates a range of conformations for a
linear polymer of $N_{{\rm SAW}}$ monomers. Another one is a non-trivial
diffusion limited aggregation (DLA) process of $N_{{\rm DLA}}$ impurities
(referred thereafter as “particles”) which describes their coagulation
occurring with the probability $0<p\leq 1$ ($p=1$ recovers a standard DLA).
DLA of diffusive particles is complemented by their irreversible adsorption on
the SAW monomers occurring with the probability equal to one, both processes
resulting in formation of the DLA-SAW agglomerates. The dynamics of formation
of such agglomerates, as well as their fractal dimensions and internal
structure are of practical interest. We consider a range of related
characteristics, such as: (i) absolute $N_{a}$ and relative $n_{a}$ adsorbing
efficiencies of SAW; (ii) effective gyration radius $R_{g{\rm DLA-SAW}}$ of
the DLA-SAW agglomerates; and (iii) the fractal dimension $D_{{\rm DLA-SAW}}$
of these aggregates. These are studied within a wide range for each parameter
from a set $\\{p,N_{{\rm DLA}},N_{{\rm SAW}}\\}$.

###### pacs:

36.20.-r, 36.20.Ey, 64.60.ae

††: Journal of Physics A: Mathematical and Theoretical

##  1 Introduction

Coagulation-flocculation is an important physicochemical process to purify a
wastewater off impurities [1, 2]. A coagulant, typically a short polymer
termed often as a “clarifying agent”, neutralizes the particles’ charge
allowing them to coagulate. A flocculant, added to wastewater, aids further
conglomeration of particles into larger agglomerates, speeding up the process
of sedimentation [3, 4]. High molecular weight flocculants exhibit bridging
flocculation, when a single molecule adsorbs a number of particles resulting
in formation of a necklace-like structure [5]. Polymers commonly used in
applications with inorganic solids such as clays and silts are anionic,
whereas cationic polymers are used to settle organic solids such as animal
waste or vegetation. They can be either synthetic, e.g. alum, lime, ferric
chloride, polyaluminium, or derived from plant parts [6, 7, 8], bacteria [9,
10], and chitosan [11].

In most wastewater purification protocols, coagulation-flocculation is an
intermediate step, prior to the membrane filtration [12]. In this case both
processes take place in bulk. However, there are many situations when the
target particles for aggregation reside on an interface, e.g. the liquid-
liquid, water-air, etc. [13, 14, 15, 16]. Their aggregation is desirable
either for the consequent removal of aggregates, or for the sake of the
interface stabilization and control of its properties [17]. In this case the
problem, obviously, reduces to the case of the two-dimensional one. Therefore,
both the $3D$ and $2D$ cases are equally important from the application point
of view, but in this study we consider the $2D$ case only.

The high molecular weight polymers used in bridging flocculation process are
typically of linear architecture, commonly based on polyacrylamide. In
particular, Gibson et al. examined seven of such polyacrylamide polymers and
they found that the two of them, Drewfloc 2449 and 2468, demonstrate the
highest solid removal efficiency [18]. In general, adsorption mechanism for
such flocculation is considered to be based on hydrogen bonding between the
amide or hydroxyl groups of a polymer and hydroxylated sites on particles’
surfaces [5].

Modelling the coagulation-flocculation faces serious difficulties as it is
intrinsically a multiscale process involving events ranging from an atomistic
to a macroscopic scale. Indeed, interactions between particles and monomers of
a polymer chain, be it a coagulant or flocculant, depends strongly on the
chemical details of both. This aspect can be tackled by ab initio and/or
atomistic molecular dynamics simulations. The aim of the current study is to
cover an opposite, macroscopic, side of the length scales spectrum. Firstly,
the coagulant is considered in an implicit way, where we just assume that due
to its presence the particles are able to coagulate with the tunable
probability $0<p<1$. A linear high molecular weight flocculant is considered
explicitly, assuming that it is able to adsorb particles irreversibly, with
the probability equal to one. Our main focus will be on the macroscopic
properties such as: (i) the efficiency of a flocculant to agglomerate
particles; (ii) effective dimensions of agglomerates; and (iii) their fractal
dimension. All these characteristics define hydrodynamic behavior of
agglomerates during their removal from the wastewater via membrane filtering,
gel chromatography, or other approaches [19, 20]. We should mention that the
modelling of the flocculation as a fractal DLA process have been done before
[21]. Both $2D$ and $3D$ DLA were considered but for the case of a point seed
only.

The description of the mixed solution, containing a polymeric flocculant and
impurity particles, can be performed on required large scale level by
employing the lattice models of polymers [22, 23]. In these terms, a
flocculant is represented via the SAW of the size $N_{{\rm SAW}}$ [24, 25]. On
the other hand, particles aggregation can be described by the DLA process of
the size $N_{{\rm DLA}}$ [26]. The concept of the DLA was initially introduced
[27] as a discrete model of growth of dendritic clusters formed by
irreversible aggregation of small particles. Since then, it became a paradigm
in description of growth phenomena and pattern formation [28, 29], in
particular as a model of colloidal particle aggregation [30, 31], growth of
neuronal trees [32], retinal arterial and venous vessels [33], thin film
nucleation growth process[34], growth of microbial colonies [35] etc.

Experimental studies reveal proportionality between the particles
sedimentation rate and the flocculant molecular weight [36, 37]. An
explanation is given in terms of stronger adhesion, e.g., via hydrogen
bonding, between the particles and the abundant hydroxyl groups in the case of
the high molecular weight glycopolymers [37]. The result, however, might
depend on the chemical composition of a flocculant. It is of interest to
clarify the effect of the flocculant molecular weight, $N_{{\rm SAW}}$,
originated purely from the conformational statistics of long polymers. Other
parameters affecting the flocculation rate that can be pointed out are: the
coagulation probability $p$, and concentration of the particles, provided by
$N_{{\rm DLA}}$.

We should note that, although both the SAW and DLA paradigms are well
established and understood by now, combining them towards description of
particular physical phenomena has not been exploited to its full extent yet.
By doing this, one goes one step further from more simple cases of point-like
seeds or the seed cores with a regular shape. Indeed, the seed core in a form
of a SAW presents a complex structure that can adopt a wide range of
conformations, from stretched to highly coiled ones, affecting the process of
the DLA cluster growth in each case. By altering the probability of
coagulation, $p$, one may shift an emphasis from aggregation of particles to
their diffusion and vice versa, steering the process towards some specific
spatial structure of a cluster.

Therefore, here we link two paradigms of DLA and SAW to model coagulation-
flocculation on a large scale level. In doing this, DLA describes coagulation
of particles, whereas SAW represents a linear high molecular weight flocculant
in a good solvent. We may note that the same model is also applicable to other
related problems, such as: adsorption of colloidal particles from a dilute
aqueous suspension [38, 39]; nanoparticle diffusion and adsorption in polymer
melts [40, 41, 42, 43], etc. Finally, we would like to address the issue of
relative scales of a flocculant and the impurity particles in the lattice type
of modelling. The dimensions of flocculant monomers and of a particle are both
defined via a lattice constant $a$. Since a polymer is modelled as a SAW, $a$
is lower bound by the persistent length $l_{p}$ of a polymeric flocculant,
with the typical values of $l_{p}=0.5-1\textrm{nm}$. The scale $a$, however,
is not upper bound because of the self-similarity feature of a polymer chain
at $a>l_{p}$. Therefore, the monomer can be interpreted as a subchain with
essentially larger dimensions than $l_{p}$, bringing the intrinsic length
scale of a problem into the realms of tens of nanoseconds or further up. More
close comparison of the scales with particular real life polymers is
problematic and, in general, is not intended.

The layout of the paper is as follows. In the next Section 2, we give details
of the computational methods used to construct the SAW and DLA clusters, the
results are presented and discussed in Section 3, followed by Conclusions.

##  2 Models and algorithms

The growth of the SAW is performed sequentially: the $n$th monomer is
generated at the random, but yet unoccupied, lattice site adjacent to the
$(n-1)$th monomer. The process continues until the required number of
monomers, $N_{{\rm SAW}}$, is created. The model is known to capture perfectly
the universal configurational properties of long, flexible polymer chains in a
good solvent [44] and has been studied in detail by both computer simulations
[45, 46, 47] and analytical approaches [48, 49].

The DLA process is typically initiated from a single “seed” particle. The
second particle diffuses from a large distance away from a seed until it
reaches it, in which case two particles merge into a cluster serving as a new
seed for subsequent iterations. The process continues until the cluster of
desired size $N_{{\rm DLA}}$ is built. Since the perimeter sites of a cluster
can be accessed more easily than those in its inner core, DLA cluster is
characterized by a highly branched fractal structure. Its fractal dimension
$D_{{\rm DLA}}$ is found from the relation connecting the cluster size
$N_{{\rm DLA}}$ and its effective linear dimensions given by the gyration
radius ${R_{g}}_{{\rm DLA}}$

${\overline{R_{g}}}_{{\rm DLA}}\sim N_{{\rm DLA}}^{1/D_{{\rm DLA}}},$ (1)

where the averaging ${\overline{(\cdots)}}$ is performed over an ensemble of
different DLA realizations. In general, the fractal dimension gives an
estimate of the space-filling properties of considered structure: the closer
it is to the Euclidean dimension $D$ of embedding space, the more “dense” is
the structure.

Figure 1: (color online) The snapshots of SAW cluster of $N_{{\rm SAW}}=120$
particles (shown with red symbols) with $N_{{\rm DLA}}=1000$ particles
adsorbed during DLA process (shown with blue symbols) with sticking
probabilities $p=1.0$ (a) and $p=0.1$ (b).

Note that the fractal dimension $D_{{\rm DLA}}$, as defined by Eq. (1), can be
referred to as the dynamical one, describing growth of cluster dimensions
$R_{g}$ with the increase of its size $N_{{\rm DLA}}$. In $D=2$, the value
$D_{{\rm DLA}}=1.712(2)$ has been found [50]. Alternatively, $D_{{\rm DLA}}$
can be obtained from the density correlation function for the completely grown
DLA cluster, and smaller value, $D_{{\rm DLA}}\approx 1.66$, is reported in
this case [27, 51]. Apart from a “classic” DLA, the reaction limited cluster
aggregation (RLCA) is also of interest [52, 53, 54, 55]. In this case, the
non-vanishing repulsive forces between particles are taken into account,
resulting in reduction of the aggregation rate, so that the aggregation (in
our case, coagulation) probability is set lower than one. These two regimes,
DLA and RLCA, can be attributed to the rapid and slow colloid aggregation,
respectively, as defined in colloid science [56]. DLA with the non-trivial
coagulation probability $0<p\leq 1$ of diffusive particles in $D=2$ has been
analyzed in Ref. [57]. In the case of rapid aggregation, $p=1$, a highly
porous “classic” DLA cluster is reproduced, see Fig. 1 (a). In a moderately
slow aggregation regime, $p<1$, the process became more diffusion driven
allowing particles more time to explore the voids inside a fractal structure.
As the result, more compact and dense DLA clusters are created that are
characterized by the increase of $D_{{\rm DLA}}$ with the decrease of $p$, see
Fig. 1 (b). In the extremely slow aggregation regime, at $p\to 0$, the limit
of the Eden model is recovered with $D_{{\rm RLCA}}=D$ [53]. Recently, such
modified DLA process has been used to simulate aggregation of wax and
asphaltene particles in a crude oil [58], as well as in the antimicrobial
peptide attack on supported lipid membranes [59].

When DLA is performed on SAW, the seed is not local and some generalizations
of the standard DLA should be discussed. Note also that SAW itself a fractal
object, the exact value for its fractal dimension in $D=2$ is found to be
$D_{{\rm SAW}}=4/3$ [60]. The multi-seed generalization of DLA has been
considered in Ref.[61]. At large separations between individual seeds, the
crossover from a fractal to a uniform aggregate structure was observed at a
certain length scale dependent on the concentration of mobile particles [62,
61]. Another generalization of DLA is obtained for the seed with a non-zero
dimensions. As shown by Wu et al. [63], the fractal dimension of DLA cluster
decreases with an increase of the dimensions of a seed particle. The same
conclusion is derived from the studies of DLA growth on spherical surfaces of
various radius [64, 65]. Besides a seed of a spherical shape, its linear
counterpart has also been considered. In particular, the DLA on a linear seed,
reproducing a fibre, was studied in Refs. [66, 67], where an increase of its
length resulted in gradual decrease of $D_{{\rm DLA}}$ towards $D=1$, the
value of Eucledean dimension of a seed, and the range of singularities becomes
narrower. More complex structure of a seeding core has been also examined
[68]. In this respect, DLA in a SAW as a seed can be interpreted as yet
another extension of the standard DLA process.

To construct SAW on $D=2$ square lattice, we use the pruned-enriched
Rosenbluth Method (PERM) [69], based on the Rosenbluth-Rosenbluth (RR)
algoritm of growing chain [45] and reinforced by enrichment strategies [70].
The first monomer is introduced at a random site of a lattice with the
coordinates $x_{1},y_{1}$. Each following $n$th monomer is added at a randomly
chosen site adjacent to the $(n-1)$th one, such that its coordinates
$\\{x_{n},y_{n}\\}$ satisfy the conditions: $\\{x_{n}=x_{n-1}\pm
1,y_{n}=y_{n-1}\\}$ or $\\{x_{n}=x_{n-1},y_{n}=y_{n-1}\pm 1\\}$ ($n\leq
N_{{\rm SAW}}$, where $N_{{\rm SAW}}$ is the total length of polymer chain).
The weights

$W_{n}\sim\left(\prod_{l{=}2}^{n}m_{l}\right)^{-1}$ (2)

are prescribed to each SAW configuration with $n$ monomers, where $m_{l}$ is
the number of free lattice sites, where $l$th monomer could be potentially
added. When the polymer chain of total length $N_{{\rm SAW}}$ is constructed,
the new one starts from the same starting point, until the desired number of
growing chain configurations are obtained.

Population control in PERM suggests pruning configurations with too small
weights, and enriching the sample with copies of the high-weight
configurations [69]. To this end, two thresholds values $W_{n}^{<}$ and
$W_{n}^{>}$ are chosen depending on the running value of partition sum
$Z_{n}=\sum_{{\rm conf}}W_{n}^{{\rm conf}}$, where summation is performed over
existing configurations of a chain. If the current weight $W_{n}$ of an
$n$-monomer chain is less than $W_{n}^{<}$, the chain is either discarded with
probability $1/2$, or it is kept and its weight is doubled. If $W_{n}$ exceeds
$W_{n}^{>}$, the configuration is doubled and the weight of each copy is taken
as half the original weight. The pruning-enrichment control parameters are
adjusted in such a way that on average 10 chains of total length $N_{{\rm
SAW}}$ are generated per each iteration [71].

At the second stage of our simulation, the ensemble of constructed SAW
clusters are used as seeds for DLA process with $N_{{\rm DLA}}$ particles. We
draw an imaginary circle of radius $R=R_{0}+R_{max}$ [72, 73] (so-called birth
circle, see Fig. 2) around each constructed SAW cluster, where $R_{max}$ is
defined as the distance between the center of DLA-SAW aggregate and the
farthest adsorbed particle with respect to it, and $R_{0}$ is chosen to be
sufficiently large (we used the value of $R_{0}=50$). The model has been
tested for different $R_{0}$ [72], and the increase of the value of this
parameter leads to an increase of average time for diffusive particles to
reach the perimeter of aggregate, but does not influence the quantitative
characteristics of adsorption processes.

Let us define the position of $i$th diffusive particle ($i=1,\ldots,N_{{\rm
DLA}}$) at time $t$ with its coordinates $\\{X_{i}(t),Y_{i}(t)\\}$. Each
particle starts to move from the point randomly chosen on the circle, so that
the coordinates of its initial position satisfy the condition
$X_{i}^{2}(0)+Y_{i}^{2}(0)=R^{2}$. The particle is not allowed to cross the
perimeter of a circle and moves only inside of it, so that at any time step
$t$ we have: $X_{i}^{2}(t)+Y_{i}^{2}(t)\leq R^{2}$. When a particle reaches
the position such that any of its neighboring site contains a monomer of SAW,
i.e. $\forall n$: $\\{X_{i}(t)\pm 1=x_{n},Y_{i}(t)=y_{n}\\}$ or
$\\{X_{i}(t)=x_{n},Y_{i}\pm 1=Y_{n}\\}$, it is adsorbed with probability equal
to one and a new particle starts to move from a new randomly chosen point on a
birth circle. The following particles can be adsorbed not only to a SAW
cluster, but also aggregate with the previously adsorbed particles, with a
chosen probability $p$. The process is stopped when the desired number of
particles $N_{{\rm DLA}}$ are adsorbed.

Figure 2: (color online) The snapshots of constructed DLA-SAW cluster with
$N_{{\rm SAW}}=120$ (shown with red symbols) and $N_{{\rm DLA}}=1000$ (shown
with blue symbols) with sticking probability $p=0.1$, surrounded by birth
circle of the radius $R=R_{0}+R_{max}$.

Note that the double averaging is needed to be performed for any observable of
interest $O$ in the considered problem. First, we perform the averaging on a
fixed SAW cluster over an ensemble of $M$ different DLA realizations

${\overline{O}}=\frac{1}{M}\sum_{i=1}^{M}O^{i}.$ (3)

Also the avaraging over an ensemble of $C$ constructed SAW configurations is
to be performed according to

$\langle{\overline{O}}\rangle=\frac{{\sum_{j=1}^{C}W_{N_{{\rm
SAW}}}^{{j}}{\overline{O}}^{j}}}{Z_{N_{{\rm SAW}}}}$ (4)

with $W_{N_{{\rm SAW}}}^{j}$ being the weight of an $N_{{\rm SAW}}$-monomer
chain in $j$th configuration of SAW as given by (2), $Z_{N_{{\rm
SAW}}}=\sum_{j}^{C}W_{N_{{\rm SAW}}}^{{i}}$ and ${\overline{O}}^{j}$ is the
value obtained after averaging over all constructed DLA realizations on $j$th
SAW configuration.

We applied averaging over $M=10^{3}$ DLA realizations and $C$ up to $10^{4}$
SAW configurations in our analysis below.

##  3 Results

First of all, we revisit the results for the DLA on a single particle seed
constructed on a $D=2$ square lattice with the variable coagulation
probability $p$ [57].

Figure 3: a) Average gyration radius of the DLA clusters obtained from a
single particle seed, as the function of DLA size, $N_{{\rm DLA}}$, at several
values of $p$. b) Estimates for the fractal dimension of the DLA clusters as
the function of $p$.

The gyration radius of DLA cluster is defined as

$R_{g\,DLA}^{2}=\frac{1}{N_{{\rm DLA}}^{2}}\sum_{i=1}^{N_{{\rm
DLA}}}\sum_{j=1}^{DLA}((X_{i}-X_{j})^{2}+(Y_{i}-Y_{j})^{2}).$ (5)

It is subsequently averaged over the ensemble of DLA realizations, according
to Eq. (3), providing an average value ${\overline{R^{2}_{g}}}_{{\rm DLA}}$.
The data obtained are shown in Fig. 3a. Reduction of the $p$ value below $1$
gives more opportunity for particles to diffuse towards the center of a
cluster. This results in formation of a more densely packed structures as
compared with a standard DLA, $p=1$, see Fig. 1. It is clearly seen from Fig.
3a, that the value of ${\overline{R^{2}_{g}}}_{{\rm DLA}}$ decreases with the
decrease of $p$. To evaluate the values of fractal dimensions $D_{{\rm DLA}}$
of generated clusters, the linear least-square fits are performed. To this end
we estimate the lower cutoff for the number of particles $N_{{\rm DLA}}^{min}$
at which the correction to scaling terms become irrelevant. The linear fits
for the average gyration radius is used

$\ln{\overline{R^{2}_{g}}}_{{\rm DLA}}=A+2/D_{{\rm DLA}}\ln N_{{\rm DLA}}.$
(6)

The $\chi^{2}$ value (sum of squares of normalized deviation from the
regression line) divided by the number of degrees of freedom, $DF$ serves as
an estimate for the fit accuracy. An example is given in Table 1. The
estimates for $D_{{\rm DLA}}$ obtained as functions of $p$ are provided in
Fig. 3b.

$N_{{\rm DLA}}^{min}$ | $D_{{\rm DLA}}$ | $A$ | $\chi^{2}/DF$
---|---|---|---
5 | 1.743 (2) | -1.520(3) | 1.7296
10 | 1.742 (2) | -1.523(3) | 1.2942
15 | 1.740 (1) | -1.526(2) | 0.9429
20 | 1.741 (1) | -1.525(2) | 0.7421
Table 1: Results of the least-square fits for evaluation the fractal dimension
$D_{{\rm DLA}}$ when varying the lower cutoff $N_{{\rm DLA}}^{min}$ for the
case $p=0.5$

Figure 4:  Average number of particles $\langle\overline{N_{a}}\rangle$
directly adsorbed on the SAW seed (a) and adsorption efficiency per single
monomer $\langle\overline{n_{a}}\rangle$ (b), as functions of $N_{{\rm SAW}}$
at several values of the coagulation probability $p$.

Now, we turn our attention to the formation of DLA on a seed in a form of SAW,
see Fig. 1. It is intuitively obvious that different perimeter sites of a such
cluster are accessible to diffusive particles with uneven probabilities. The
particles adsorbed on the polymer directly form a solvation shell, which
causes a screening effect for the newly arriving particles. As the result,
when the number of particles $N_{a}$ in a solvation shell reaches certain
saturation value, the new particles are unable to reach the SAW core directly,
but can be adsorbed by the solvation shell particles only.

Let us evaluate the size of the solvation shell of the polymer seed of size
$N_{{\rm SAW}}$, i.e. to estimate the number of particles $N_{a}$ adsorbed
directly by the polymer seed. After performing double averaging, over an
ensembles of SAW and DLA configurations, we obtain the results for
$\langle\overline{N_{a}}\rangle$ presented in Fig. 4a. As expected, with the
decrease of the probability $p$, the number of particles
$\langle\overline{N_{a}}\rangle$ directly adsorbed by a polymer flocculant
increases. This results in formation of a more compact agglomerate, as
illustrated in Fig. 1b. We also introduce the normalized value
$n_{a}=N_{a}/N_{{\rm SAW}}$, which characterizes the efficiency of direct
adsorption of diffusive particles per single monomer of the SAW seed. By
analysing behaviour of $\langle{\overline{n}_{a}}\rangle$ as function of
$N_{{\rm SAW}}$ (Fig. 4b), one notices that the direct adsorption efficiency
is higher for shorter chains, and saturates at large values of $N_{{\rm
SAW}}$.

Let us recall that the underlying SAW seed is itself a not-regular fractal
structure with its effective size (gyration radius $R_{g\,{\rm SAW}}$) taking
on a range of different values in an ensemble of possible SAW configurations.
We analyzed the correlation between values of $R_{g\,{\rm SAW}}$ in given
configuration of SAW and the numbers of particles in solvation shell $N_{a}$,
observed in performing different DLA realizations with this SAW configuration
as a seed. Corresponding results are presented on Fig. 5. The smaller values
of $R_{g\,{\rm SAW}}$ correspond to more compact SAW configurations with
higher fraction of “inner” monomers, screened from incoming diffusive
particles as compared with monomers positioned closer to the SAW outer
perimeter. Indeed, the number of particles $N_{a}$ directly adsorbed to SAW
perimeter is smaller for the case of compact SAW and increase gradually with
an increase of $R_{g\,{\rm SAW}}$. This effect is more pronounced for smaller
$p$ values (Fig. 5b), when direct adsorption of particles on SAW seed is
dominant as compared with particle-particle coagulation, and thus is more
“sensitive” to the subtle peculiarities of the underlying SAW. Note that the
probability distribution for $N_{a}$ as averaged over SAW and DLA realizations
(see Fig. 6) is broader at smaller $p$ due to the same effect.

Figure 5: Correlation between the values of gyration radius $R_{g\,{\rm SAW}}$
of SAW seed with $N_{{\rm SAW}}=40$ monomers and the possible number of
particles in solvation shell $N_{a}$ at $p=1.0$ (a) and $p=0.1$ (b). Figure 6:
Probability distribution of a number of particles in solvation shell $N_{a}$
on a SAW seed with $N_{{\rm SAW}}=40$ monomers with $p=1.0$ and $p=0.1$.

Another parameter of interest is the total number of bonds (contacts) $N_{{\rm
bond}}$ established between SAW and adsorbed diffusive particles. Since each
the monomer of SAW can have contacts with more than one adsorbed particle,
this quantity is not trivial and is expected to be larger than the number of
particles $N_{a}$ in a solvation shell. We prescribe labels $v(n)$ to each
$n$th monomer of the SAW ($n=1,\ldots,N_{{\rm SAW}}$) with the initial values
of $v(n)=0$. When, during the process of diffusion, the $i$th particle became
adjacent to $n$th monomer and gets adsorbed, we increase the label $v(n)$ by
one, so that:

  * •

if $X_{i}(t)=x_{n}\pm 1$ and $Y_{i}(t)=y_{n}$ then $v(n)=v(n)+1$,

  * •

if $X_{i}(t)=x_{n}$ and $Y_{i}(t)=y_{n}\pm 1$ then $v(n)=v(n)+1$

so that at the end $v(n)$ contains the total number of contacts established by
$n$th monomer with adsorbed particles and $N_{{\rm bond}}=\sum_{n}v(n)$.

Figure 7:  The chelation abilities $\langle\overline{n_{bond}}\rangle$ of
polymer chain as functions of $N_{{\rm SAW}}$ at several different values of
the coagulation probability $p$.

We introduce the value $n_{{\rm bond}}=N_{{\rm bond}}/N_{a}$, which
characterizes the chelation efficiency of a polymer chain and is also of
interest from the point of view of wastewater cleaning: the larger is the
average number of contacts per single monomer, the stronger is the ability of
a polymer flocculant to “hold” the particles, which are already adsorbed. At
each fixed value of $p$, $\langle{\overline{n_{{\rm bond}}}}\rangle$ is found
to increase with the length of a polymer chain (see Fig. 7) and gradually
reaches its saturation value. The larger polymer macromolecules are thus found
to be more effective in “keeping” the directly adsorbed impurity particles,
and this efficiency increases with decreasing the parameter $p$.

Figure 8: Average gyration radii of DLA-SAW aggregates, a) At $p=0.1$ and
several fixed values of parameter $N_{{\rm SAW}}$; b) At $N_{{\rm SAW}}=50$
and several fixed values of $p$.

Now we proceed to analyze the structure of agglomerates of $N_{{\rm
SAW}}+N_{{\rm DLA}}$ particles, as given by their fractal dimension. The
scaling of the gyration radii $\langle{\overline{R^{2}_{g}}}\rangle_{{\rm{\rm
DLA-SAW}}}$ in a range of sizes of the SAW seed $N_{{\rm SAW}}$ is analyzed
next. This is related to a real life situations with various degree of
pollution levels and of a pollutant coagulation ability. The data obtained are
presented in both frames of Fig. 8. Here, we observe what we believe is the
crossover between the solvation and DLA regimes, which is especially evident
at small $p$, e.g. $p=0.1$. In this case, particles prefer to deposit
theirselves on a SAW rather than to aggregate. Therefore, at low $N_{{\rm
DLA}}<2N_{{\rm SAW}}$, the process is dominated by solvation (screening) of a
SAW by particles, characterized by very slow increase of effective size
$\langle{\overline{R^{2}_{g}}}\rangle_{{\rm{\rm DLA-SAW}}}$ with $N_{{\rm
DLA}}$. By applying least-square fitting of data in this region to the form
(Eq. (1)), we obtained $1/D=0.35(3)$ (the dashed line in Fig. 9). At $N_{{\rm
DLA}}\sim 2N_{{\rm SAW}}$, a SAW is completely screened, and since then a
normal DLA starts, where the screened SAW plays a role of its seed. At
sufficiently large $N_{DLA}$, the DLA scaling with $D_{{\rm DLA-
SAW}}(p=0.1)=1.868(5)$ is retrieved (the result of least-square fitting of
data in this region to the form (Eq. (1)) are presented with solid line on
Fig. 9), see also Table 2). The crossover is seen for all $N_{SAW}$ being
examined, and it occurs at $N_{{\rm DLA}}\sim 2N_{{\rm SAW}}$ in all cases (as
can be seen from curves of Fig. 8a). At yet smaller $p=0.01$, solvation regime
has a non-linear dependence of $\langle{\overline{R^{2}_{g}}}\rangle_{{\rm{\rm
DLA-SAW}}}$ on $N_{{\rm DLA}}$ due to strongly suppressed DLA in favour of
screening the SAW, whereas at higher $p>0.1$ it gradually disappears, as in
this case the screening of a SAW loses its priority over a DLA, see,
respective curves in Fig. 8b.

Figure 9: Average gyration radii of DLA-SAW aggregates at $p=0.1$ and $N_{{\rm SAW}}=100$. Dashed line represents results of least-square fitting to the form (Eq. (1)) with $1/D=0.35(4)$ (solvation regime), solid line represents results of the least-square fitting to the form (Eq. (1)) with $D_{{\rm{\rm DLA-SAW}}}=1.868(5)$ (DLA regime).  $N^{min}_{{\rm DLA}}$ | $D_{{\rm{\rm DLA-SAW}}}$ | $A$ | $\chi^{2}/DF$
---|---|---|---
50 | 1.92(2) | -1.13(2) | 3.5344
100 | 1.91(1) | -1.18(1) | 2.3247
150 | 1.896(8) | -1.226(7) | 1.5064
200 | 1.896 (6) | -1.261(6) | 1.1872
250 | 1.875 (6) | -1.291(7) | 1.0922
300 | 1.870 (5) | -1.303(6) | 0.9038
350 | 1.868(5) | -1.308(5) | 0.7565
Table 2:  Results of the least-square fitting for evaluation the fractal
dimension $D_{{\rm{\rm DLA-SAW}}}$ with varying the lower cutoff $N_{{\rm
DLA}}^{min}$ at $N_{{\rm SAW}}=50$ and $p=0.1$.

Let $M$ denote the number of realizations of DLA processes (which is 1000 in
our case) on a fixed SAW configuration. For all the monomers
$n=1,\ldots,N_{{\rm SAW}}$ we sum up the number of times $k(n)$, when
diffusive particle was absorbed to this monomer. In such a way, a weight
$w_{n}=k(n)/M$ is prescribed to each monomer. Such distributions of the
hitting points are called the ”harmonic measure” [74], which can be
represented within the multifractal concepts. When studying physical processes
on complex fractal objects, one often encounters the situation of coexistence
of a family of singularities, each associated with a set of different fractal
dimensions [75]; combination of two fractal growth processes, as in our case,
is expected to lead to multifractal features as well [76].

Figure 10: a) Average moments $\langle M(q)\rangle$ (Eq.(8)) at $q=1$ and
several values of $p$ as functions of $R_{g\,SAW}$ in double logarithmic
scale. b) A spectrum of multifractal exponents $d(q)$ as function of $q$ at
several values of $p$.

The different moments of the distribution of observables scale independently,
which is usually referred to as multifractality [77]. The multifractal
spectrum can be used to provide information on the subtle geometrical
properties of a fractal object, which cannot be fully described by its fractal
dimensionality. Indeed, the growth probability distribution in DLA clusters is
a typical example of multifractal phenomena [27, 62, 78].

In our case, the multifractal moments can be defined as

$M(q)=\sum_{n=1}^{N_{{\rm SAW}}}w^{q}(n)$ (7)

When averaged over different configurations of the constructed SAW clusters
and realizations of DLA, they scale with the gyration radius $R_{g}$ of an
underlying SAW core according to:

$\langle\overline{M(q)}\rangle=R_{g\,SAW}^{d(q)}$ (8)

with the exponents $d(q)$ being characterized by a non-linear dependence on
$q$. To estimate the numerical values of $d(q)$ on the basis of data obtained
by us (see Fig. 10), the least square fitting was used. At $q=0$ we just count
the number of sites of the cluster of linear size R, and thus $d(0)$
corresponds to the fractal dimension of the SAW trajectory for each $p$. At
$q>0$, the set of $d(q)$ is found to be non-trivial and dependent on $p$. The
obtained spectrum at several values of $p$ are given on Fig. 10b). At each
$q$, the values of $d(q)$ increase with decreasing the parameter $p$. This
demonstrates an increase of non-uniformity distribution of harmonic measure
(the hitting probabilities of underlying SAW trajectory by incoming diffusive
particles are distributed more with decreasing $p$).

##  4 Conclusions

In this paper we mapped the process of coagulation-flocculation, which is
important from the environmental point of view for cleaning wastewater, onto
the physical model of the DLA of $N_{{\rm DLA}}$ particles that takes place on
a seed represented by a SAW containing $N_{{\rm SAW}}$ monomers. Within this
approach, the DLA particles represent the impurities in suspension, which may
sediment in form of flocs (aggregates) either spontaneously (as a result of
particle-particle coagulation with probability $p$) or due to addition of a
special agent (flocculant). The adsorbing linear polymer chain, represented by
SAW here, serves as a flocculant by establishing multiple bonds with adsorbed
DLA particles.

Despite the fact that both paradigms are well studied over a number of years,
their combination has got less attention, especially in relation to this
particular application. Computer simulations are performed by means of the
pruned-enriched Rosenbluth Monte Carlo algorithm on a $D=2$ square lattice.

Both DLA and SAW processes generate clusters of complex fractal structure. We
found that their combination in the $D=2$ space, at the special case of
$N_{{\rm SAW}}=N_{{\rm DLA}}$, leads to formation of a flocculated agglomerate
with the fractal dimension $D_{{\rm{\rm DLA-SAW}}}=1.618(5)$, which exceeds
the fractal dimension of a pure SAW, but is smaller than $D_{{\rm DLA}}$. The
result is relevant in the respect of the further removal of the agglomerate
containing impurities by means of membrane filtration or gel chromatography.
We also evaluated the properties related to the adsorption efficiency of SAW,
such as: the average size of solvation shell $\langle N_{a}\rangle$, adsorbing
efficiency per monomer $\langle n_{a}\rangle$, and a number of adsorption
bonds between the SAW and the DLA particles $\langle N_{bond}\rangle$
depending on the $N_{{\rm SAW}}$ and the coagulation probability $p$. In
particular we found, that the direct adsorption efficiency as related to a
single monomer of SAW, is higher for shorter chain length, and reaches the
asymptotic saturation value as $N_{{\rm SAW}}$ increases.

The effective dimension of DLA-SAW agglomerates is given by their gyration
radii $\langle R_{g\,{\rm DLA-SAW}}^{2}\rangle$, it is analyzed in a wide
range of $N_{{\rm SAW}}$ and $p$. A crossover between two characteristic
regimes have been observed at $N_{{\rm DLA}}\gg N_{{\rm SAW}}$. At $N_{{\rm
DLA}}\leq N_{{\rm SAW}}$, the incoming diffusive particles are preferentially
attached to monomers along the perimeter of SAW and form a solvation shell
around the polymer seed; at $N_{{\rm DLA}}\gg N_{{\rm SAW}}$, the underlying
SAW seed is already well “screened” from new incoming particles so that they
mainly coagulate with particles from a solvation shell and the scaling regime
of DLA is retrieved.

Introducing the probabilities for the perimeter sites of underlying SAW
cluster to be encountered by incoming DLA particles, we found the estimates
for multifractal sets of exponents, governing the moments (8) and giving more
subtle characteristics of agglomeration clusters constructed by combination of
two growth processes.

The study contains a wide range of adjustable parameters that can be tuned
towards particular chemical setup, namely: (i) molecular weight and branching
type of a flocculant; (ii) relation between the impurity-flocculant adsorption
and impurity-impurity coagulation; (iii) spatial arrangement of flocculants,
e.g. polymer brush of variable density, etc. Such extensions of this study are
planned for a future.

## Acknowledgements

The work was supported by Academy of Finland, reference number 334244.

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