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Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model Dongjing Liu 1, *, Weiguo Zhou 1 , Zumin Qiu 2 and Xu Song 1 1 2

*

College of Mechanical Engineering, Tongji University, Shanghai 200092, China; [email protected] (W.Z.); [email protected] (X.S.) School of Resources Environmental & Chemical Engineering, Nanchang University, Nanchang 330031, China; [email protected] Correspondence: [email protected]; Tel./Fax: +86-021-6598-1482

Received: 25 October 2017; Accepted: 15 November 2017; Published: 18 November 2017

Abstract: In flocculation processes, particulates randomly collide and coagulate with each other, leading to the formation and sedimention of aggregates exhibiting fractal characteristics. The diffusion-limited aggregation (DLA) model is extensively employed to describe and study flocculation processes. To more accurately simulate flocculation processes with the DLA model, the effects of particle number (denoting flocculation time), motion step length (denoting water temperature), launch radius (representing initial particulate concentration), and finite motion step (representing the motion energy of the particles) on the morphology and structure of the two-dimensional (2D) as well as three-dimensional (3D) DLA aggregates are studied. The results show that the 2D DLA aggregates possess conspicuous fractal features when the particle number is above 1000, motion step length is 1.5–3.5, launch radius is 1–10, and finite motion step is more than 3000; the 3D DLA aggregates present clear fractal characteristics when the particle number is above 500, the motion step length is 1.5–3.5, the launch radius is 1–10, and the finite motion step exceeds 200. The fractal dimensions of 3D DLA aggregates are appreciably higher than those of 2D DLA aggregates. Keywords: fractal; flocculation; diffusion-limited aggregation; DLA; finite motion step

1. Introduction Flocculation is an important method in the field of water and wastewater treatment [1–3]; it can assist in migrating, transforming, and removing particulate matter in water. In flocculation processes, particulates in colloidal suspension randomly collide and coagulate with each other, resulting in the formation and sedimention of fractal-like flocs (aggregates). It is well known that the structure of flocs (aggregates) has a vital impact on the properties of the flocculated colloidal suspension; a few research works have contributed towards the understanding of this impact. Jiang and Logan [4] examined the fractal dimensions of fluorescent microsphere aggregates, and reported that the fractal dimensions of colloidal aggregates are 1.8 and 2.2 when the particles are highly destabilized and more stable, respectively. Oles [5] found that the fractal dimensions increased from 2.1 to 2.5, resulting from the compaction of the aggregate structure which was attributed to the selective breakup that removed the weaker (and more porous) parts of the colloidal aggregates. Adachi et al. [6,7] conducted many valuable works on the structure properties of colloidal flocs, such as effect of ionic strength on the initial flocculation dynamics [8], shear viscosity of coagulated suspensions [9], effect of floc structure on the rate of shear coagulation [10], and the restructuring behavior of small flocs [11]. The diffusion-limited aggregation (DLA) model is an idealization of the process by which matter irreversibly combines to form random objects in the case where the rate-limiting step is the diffusion of matter to the aggregate [12,13]. Therefore, the DLA model is often used to describe flocculation processes, and study of the DLA model could help us to better understand flocculation processes and Fractal Fract. 2017, 1, 12; doi:10.3390/fractalfract1010012

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contribute to the development of flocculants and flocculation methods. To date, the DLA model has been extensively studied [14–16] and applied in many fields, such as fluid displacement in porous media [17,18], electroless deposition dynamics of silver nanoparticle clusters [19], and wax/asphaltene aggregation in crude oil [20]. Novotny et al. [21] assumed that the DLA clusters grew with the inclusion of varying amounts of surface tension, and found that the DLA cluster rapidly breaks up into many disconnected, smaller clusters but the system fractal dimension is essentially unchanged. Tan et al. [22] conducted two-dimensional DLA simulations with different particle sizes, and found that when large particles were in the minority, the patterns of clusters appear asymmetrical and nonuniform, and their fractal dimensions increase. Braga et al. [23] studied the influence of particle size on DLA clusters, and proposed an angle selection mechanism on dendritic growth that influences the shielding effect of the DLA edge and affects the fractal dimension of the clusters. Deng [24] performed two-dimensional DLA simulations with various particle shapes (polygons), and reported that different particle shapes only change the local structure but have no impact on the global structure of the aggregates, although the local compactness decreases as the number of polygon edges increases. Xu [25] studied the effects of varying particle motion step length, maximum motion step number, and collision probability on the morphology and structure of the flocculation process by using the two-dimensional DLA model. Though many studies on the DLA model have been conducted, most of the researchers only focused on one or few factors and their influences on the fractal characteristics of the two-dimensional DLA aggregates. In this work, the we address the impacts of four key model parameters—particle number (denoting flocculation time, N), motion step length (denoting water temperature, Ns ), launch radius (representing initial particulate concentration, Re ), and finite motion step (representing the motion energy of the particles, Ms )—on the morphology, fractal dimensions (Df ), and porosities (Vf ) of the two-dimensional (2D) as well as three-dimensional (3D) DLA aggregates, in order to get a deeper understanding of the fractal features of the DLA aggregates, which would assist in describing flocculation processes more exactly. 2. Model The classical DLA simulation is carried out in the following steps [26,27], and depicted in Figure 1a. First, place a fixed “seed” particle (the seed radius is R0 ) in the center of a square divided into an L × L grid (or a sphere divided into an L × L × L grid) and draw a large circle (or a big sphere) centering around the “seed” particle; the radius of the circle (or the sphere) is the so-called launch radius (Re ). Then, release a free particle (not the “seed” particle) from any site on the perimeter of the circle (or the sphere). The launched particle moves randomly, following a Brownian path, which means that it walks in a random direction at each step, with no correlation to the previous direction, in which the new position of the free particle is given by xn = xn−1 + Re cos α,

(1)

yn = yn−1 + Re sin α,

(2)

zn = zn−1 + Re cos β,

(3)

where α, β are angles randomly allocated at each motion step. Assume the free particle movement has two possible results: One is that it arrives at one of the lattice sites adjacent to the occupied sites, then stops and generates another particle; another is that it leaves the circle (or the sphere) and disappears. The killing radius (Rk ) is a limiting radius that is used to identify a disposal event when the walking particle moves beyond the maximum circle (or sphere) region. Another particle is then launched and halted when adjacent to the two occupied sites, and so forth. As this process repeats a large number of times, a fractal-like DLA aggregate can be formed. The “seed” particle and other free particles are considered to have circular shapes (or spherical shapes) and their radius is fixed at 0.5. The fractal dimensions of the DLA aggregates are obtained

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function gyration radius of method the aggregates (depicted in Figure 1b), which obeys theparticles following by using of thethe gyration calculation [7,28,29]. Particularly, the number of primary is form: ascaling function of the gyration radius of the aggregates (depicted in Figure 1b), which obeys the following Df scaling form:  Rg  D f N = k Rg  (4) N = k00  R0  (4) R0 where N N is is the the number number of of primary primary particles enclosed in a circle (or sphere) centered at the aggregate where center,kk00 is is the the prefactor, prefactor,RRg gisisthe thegyration gyrationradius radiusdenoting denotingthe themaximum maximum radius aggregate, center, radius of of thethe aggregate, R0 Ris0 is the radius of the primary particle, and D f is the Hausdorff dimension (fractal dimension). Equation the radius of the primary particle, and Df is the Hausdorff dimension (fractal dimension). Equation (4) (4)logarithmic in logarithmic form be expressed as follows: in form cancan be expressed as follows:  R   log( N log((kk0 0) )++ D log  Rgg  .. Dff log ( N) )==log log R0  R

(5) (5)

If we we plot plot log(N) log(N) against against log log (R (Rgg/R /R00),),aastraight straightline line can can be be obtained, obtained, and and the slope of the fitting curve fractal dimension dimension D Dff. is equal to the fractal

Figure Figure 1. 1. Sketch of particle aggregation of DLA model (a) and fractal dimension calculation by using gyration method method (b). (b).

The porosity porosity can can be be determined determined using using the the following following formulas: formulas: The 2

 r  2 2πN r2N πr p p , = − = 1 − N  pr p ( 2 D()2D 1 V Vf = 1f − 2 = 1− N ),  R R πRπ2g Rg g g   4 43 3 r N 3 πr pπN Vf = 1 − 4 3 3 p V f = 1 −πR g 3 4 3

rp = 1 − N rp  = 1 − N  Rg R   g π Rg 

(6) (6)

3

3

, ), ( 3D()3D

(7) (7)

where rp is the radius of the primary particle 3 and Vf is the porosity of the DLA aggregate.

where rp is the radius of the primary particle and Vf is the porosity of the DLA aggregate. 3. Results and Discussion 3. Results Discussion 3.1. Effect ofand Particle Number The fractal simulation 3.1. Effect of Particle Number was conducted with a motion step length of 2 (Ns = 2), a launch radius of 2 (Re = 2), and different particle numbers; the effect of particle number on the fractal features and The fractal simulation was conducted with a motion step length of 2 (Ns = 2), a launch radius of structures of the DLA aggregates is shown in Figures 2 and 3. Generally, the particle number relates 2 (Re = 2), and different particle numbers; the effect of particle number on the fractal features and to the flocculation time: the higher the number of particles, the longer the flocculation time. For the structures of the DLA aggregates is shown in Figures 2 and 3. Generally, the particle number relates 2D DLA simulation, when the particle number is fairly small (N < 1000), the aggregates are fairly to the flocculation time: the higher the number of particles, the longer the flocculation time. For the small and only show local fractal features, and their scale-invariances as well as self-similarities are 2D DLA simulation, when the particle number is fairly small (N < 1000), the aggregates are fairly insignificant. The aggregate gradually grows with rising particle numbers, and it presents typical small and only show local fractal features, and their scale-invariances as well as self-similarities are insignificant. The aggregate gradually grows with rising particle numbers, and it presents typical

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fractal morphology with properties of global scale-invariances when the particle number reaches fractal morphology with properties of global scale-invariances when the particle number reaches a a certain value (N ≥ 1000). The fractal dimension increases rapidly as the particle number rises from certain value (N ≥ 1000). The fractal dimension increases rapidly as the particle number rises from 50 50 to 1000, and then changes little as the particle number increases further (Figure 4). The fractal to 1000, and then changes little as the particle number increases further (Figure 4). The fractal dimension slightly fluctuates in the range 1.66–1.74, which coincides with the fractal dimension value dimension slightly fluctuates in the range 1.66–1.74, which coincides with the fractal dimension value range of typical 2D DLA aggregates [12]. The porosity is smaller (around 0.8) when the particle number range of typical 2D DLA aggregates [12]. The porosity is smaller (around 0.8) when the particle is less than 500, as it is easier to form a compact DLA aggregate when the particle numbers are lower, number is less than 500, as it is easier to form a compact DLA aggregate when the particle numbers and it becomes compact and grows and to a two-dimensional branched aggregate rising are lower, and itincreasingly becomes increasingly compact grows to a two-dimensional branchedwith aggregate particle numbers, to the increase in the fractalindimension porosity. Therefore, optimal with rising particleleading numbers, leading to the increase the fractaland dimension and porosity.the Therefore, particle number for 2D DLA simulation is 2000. For 3D DLA simulation, when the particle number is the optimal particle number for 2D DLA simulation is 2000. For 3D DLA simulation, when the particle very small (N < 400), the aggregates are also small with lower fractal dimensions (Figures 3 and 4). number is very small (N < 400), the aggregates are also small with lower fractal dimensions (Figures The aggregate grows to agrows highlytoassembled three-dimensional aggregation structure with the rising 3 and 4). The aggregate a highly assembled three-dimensional aggregation structure with particle numbers; the fractal dimension rises sharply as the particle number increases from 50 tofrom 400, the rising particle numbers; the fractal dimension rises sharply as the particle number increases and then it varies little as the particle number continues to increase. When the particle number is above 50 to 400, and then it varies little as the particle number continues to increase. When the particle 1000, theisfractal generally stays at generally around 2.20, which agrees2.20, withwhich the fractal dimension number abovedimension 1000, the fractal dimension stays at around agrees with the values of representative 3D DLA aggregates [30]. However, the porosity only slightly fluctuates fractal dimension values of representative 3D DLA aggregates [30]. However, the porosity with only varying particle numbers because the 3D DLA aggregate has a much bigger packing volume compared slightly fluctuates with varying particle numbers because the 3D DLA aggregate has a much bigger with the 2D DLA compared aggregate. with Thus,the the2D optimal number 3Doptimal DLA simulation is 1000. For packing volume DLA particle aggregate. Thus,forthe particle number for 2D 3D and 3D DLA aggregates, the fractal dimensions generally increase with flocculation time elapsed DLA simulation is 1000. For 2D and 3D DLA aggregates, the fractal dimensions generally increase (rising particle numbers) when(rising Ns < 1000, and numbers) then fluctuate with flocculation time elapsed particle whenlittle. Ns < 1000, and then fluctuate little.

Figure 2. 2D with versatile versatile particle particle numbers. numbers. Figure 2. 2D DLA DLA aggregates aggregates with

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Figure 3. 3D DLA aggregates with versatile particle numbers. Figure 3D DLA DLA aggregates aggregates with Figure 3. 3. 3D with versatile versatile particle particle numbers. numbers.

Figure 4. Effect of particle number on the fractal dimension and porosity of 2D and 3D DLA aggregates. Figure 4. Effect of particle number on the fractal dimension and porosity of 2D and 3D DLA Figure 4. Effect of particle number on the fractal dimension and porosity of 2D and 3D DLA aggregates.

aggregates.

3.2. Effect of Motion Step Length 3.2. Effect offractal Motionsimulation Step Length The was implemented with particle numbers of 2000 (2D DLA) and 1000 (3D

3.2. Effect of Motion Step Length DLA), a launch radius ofwas 2 (Reimplemented = 2), and motion lengths varying 10; the effect of(3D motion The fractal simulation withstep particle numbers offrom 20000.5 (2DtoDLA) and 1000 DLA), The fractal simulation was implemented with particle numbers of 2000 (2D DLA) and 1000 (3D step length on of the2fractal structures of the DLA aggregates is illustrated Figures and a launch radius (Re = features 2), and and motion step lengths varying from 0.5 to 10; theineffect of 5motion DLA), a launch radius of 2 (R e = 2), and motion step lengths varying from 0.5 to 10; the effect of motion 6. length It is well thatfeatures water temperature has aofbig on flocculation processes: High water step onknown the fractal and structures theinfluence DLA aggregates is illustrated in Figures 5 and 6. step length on the fractal features and structures of the DLA aggregates is illustrated in Figures 5 and 6. It is well known that water temperature has a big influence on flocculation processes: High water

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It is well known that water temperature has a big influence on flocculation processes: High water temperature can accelerate the movement of colloidal particles and hydrolytic processes of flocculants, Fractal Fract. 2017, 1, 12 6 of 13 which contributes to the enhancement of flocculation efficiency. Thus, the motion step length is adopted to temperature can accelerateofthe of colloidalprocesses: particles the andbigger hydrolytic processes denote the water temperature the movement practical flocculation the motion stepof length, flocculants, which contributes to the enhancement of flocculation efficiency. Thus, the motion step the higher the water temperature. When the motion step length is small (Ns < 4), the 2D DLA aggregates length is adopted to denote the water temperature of the practical flocculation processes: the bigger present two-dimensional slender dendritic planar structures of typical fractal features with scale-invariant the motion step length, the higher the water temperature. When the motion step length is small (Ns < properties, and the 3D DLA aggregates show fairly loose three-dimensional spatial structures. The DLA 4), the 2D DLA aggregates present two-dimensional slender dendritic planar structures of typical aggregate grows to an increasingly compact as the motion step 4 to 10: fractalgradually features with scale-invariant properties, andstructure the 3D DLA aggregates showlength fairly rises loosefrom threeThe 2D DLA aggregate a dense cake-like structure of inconspicuous features with higher dimensional spatialgenerates structures. The DLA aggregate gradually grows to anfractal increasingly compact fractal dimension (Dmotion and the 3D DLA forms a highly compact ball-like structure structure as the step length rises fromaggregate 4 to 10: The 2D DLA aggregate generates a dense cake- with f = 1.92), like structure of inconspicuous fractal higher fractal dimension (Df =increasing 1.92), and motion the 3D step higher fractal dimension of 2.40 when thefeatures motion with step length approaches 10. With DLA forms a highly compact ball-likewhile structure with higher fractal dimension of 2.40 when length, theaggregate fractal dimensions gradually increase the porosities decrease correspondingly (Figure 7). the motion stepislength approaches 10.motion With increasing motion step length, the fractal motion dimensions The reason for this that the smaller the step length is, the smaller the particle region is, gradually increase while the porosities decrease correspondingly (Figure 7). The reason for this is and the released particles can be more easily captured by the external branches of the aggregates, leading that the smaller the motion step length is, the smaller the particle motion region is, and the released to the rapider growth of the aggregate branches with highly slender structures, which contributes to the particles can be more easily captured by the external branches of the aggregates, leading to the decrease in fractal and increase porosities. Conversely, when the contributes motion step is rapider growthdimensions of the aggregate branches in with highly slender structures, which to length the bigger, the motion region of the particle becomes larger, and it is more difficult for the aggregate external decrease in fractal dimensions and increase in porosities. Conversely, when the motion step length is branches to the capture theregion launched can more interior the aggregate to bigger, motion of theparticles particle that becomes larger,easily and itgo is into morethe difficult forof the aggregate collide and coagulate the “seed” particle or clusters. This eliminate theinto shielding effectofand external brancheswith to capture the launched particles that cancan more easily go the interior themake aggregate grow to collide coagulate withbranches the “seed” particle or clusters. Thisstructure, can eliminate theto the the aggregate moreand slowly with less to form a highly compact leading shielding effect and make the aggregate grow more slowly with less branches to form a highly increase in fractal dimension and decrease in porosity. Thus, the DLA aggerates will grow into increasingly compact structure, leading to the increase in fractalasdimension decrease in(the porosity. Thus, compact objects with inconspicuous fractal features the waterand temperature motion stepthe length) DLA aggerates will grow into increasingly compact objects with inconspicuous fractal features as the increases. It should be noted that when the motion step length is equal to 2 (Ns = 2), the DLA aggregate water temperature (the motion step length) increases. It should be noted that when the motion step presents prominent fractal structure characteristics. length is equal to 2 (Ns = 2), the DLA aggregate presents prominent fractal structure characteristics.

Figure DLAaggregates aggregates with step lengths. Figure 5. 5. 2D2DDLA withdifferent differentmotion motion step lengths.

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Figure 6. 3D DLA aggregates with different motion step lengths. Figure 6. 3D DLA aggregates with different motion step lengths.

Figure 7. Effect of motion step length on the fractal dimension and porosity of 2D and 3D DLA Figure 7. Effect of motion step length on the fractal dimension and porosity of 2D and 3D DLA aggregates. Figure 7. Effect of motion step length on the fractal dimension and porosity of 2D and 3D DLA aggregates. aggregates.

3.3. Effect 3.3. Effect of of Launch Launch Radius Radius 3.3. Effect of Launch Radius The fractal simulation simulation wasperformed performed with particle numbers of 2000 (2D DLA) and (3D 1000 (3D The with particle numbers of 2000 (2D DLA) and 1000 DLA), The fractal fractal simulationwas was performed with particle numbers of 2000 (2D DLA) and 1000 (3D DLA), a motion step of length of 2 (N s = 2), and a versatile launch radius ranging from 1 to 50. The effect aDLA), motion step length 2 (N = 2), and a versatile launch radius ranging from 1 to 50. The effect of launch s a motion step length of 2 (Ns = 2), and a versatile launch radius ranging from 1 to 50. The effect of launch radius on features the fractal and of the DLA aggregatesinisFigures displayed radius on the fractal andfeatures structures of structures the DLA aggregates is displayed 8 andin 9. of launch radius on the fractal features and structures of the DLA aggregates is displayed in Figures 8 andradius 9. Thecan launch radiusthe caninitial represent the initial particulate concentration of the practical The launch represent particulate concentration of the practical flocculation Figures 8 and 9. The launch radius can represent the initial particulate concentration of the practical flocculation A bigger radius that the initial particulates in are the wastewater processes: A processes: bigger launch radiuslaunch means that themeans initial particulates in the wastewater away flocculation processes: A bigger launch radius means that the initial particulates in thefurther wastewater are further away from each other; that is, the initial particulate concentration is lower. As we can see, are further away from each other; that is, the initial particulate concentration is lower. As we can see, the aggregates gradually grow outward with the increasing launch radius. When the launch radius the aggregates gradually grow outward with the increasing launch radius. When the launch radius rises from 1 to 50, the fractal dimensions of the 2D DLA aggregates generally decrease from 1.69 to rises from 1 to 50, the fractal dimensions of the 2D DLA aggregates generally decrease from 1.69 to

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Fractalfrom Fract.each 2017, other; 1, 12 that is, the initial particulate concentration is lower. As we can see, the aggregates 8 of 13

gradually grow outward with the increasing launch radius. When the launch radius rises from 1 to fractal dimensions of the 2D DLA aggregates from 1.69 to 1.53, and3D theDLA 1.53, 50, andthethe porosities slightly increase from 0.82 togenerally 0.89; thedecrease fractal dimensions of the porosities slightly increase from 0.82 to 0.89; the fractal dimensions of the 3D DLA aggregates generally aggregates generally drop from 2.24 to 2.01, and the porosities slowly rise from 0.94 to 0.98 (Figure drop from 2.24 to 2.01, andexceeds the porosities slowly risedimensions from 0.94 toof 0.98 10). aggregates When the launch 10). When the launch radius 10, the fractal the(Figure 2D DLA fluctuate radius exceeds 10, the fractal dimensions of the 2D DLA aggregates fluctuate between 1.53 and between 1.53 and 1.71, and the porosities stay at around 0.82–0.90; the fractal dimensions of1.71, the 3D and the porosities stay at around 0.82–0.90; the fractal dimensions of the 3D DLA aggregates fluctuate DLA aggregates fluctuate between 2.01 and 2.16, and the porosities stay at 0.96–0.98. The bigger the between 2.01 and 2.16, and the porosities stay at 0.96–0.98. The bigger the launch radius, the larger the launch radius, the larger the motion radius of the particle and, thus, it is easier for the released motion radius of the particle and, thus, it is easier for the released particles to go outward, resulting particles tooutward go outward, resulting in the outward of theinaggregate as well and as the decrease in the growth of the aggregate as well asgrowth the decrease fractal dimension increase in in fractal dimension and in porosity. DLA aggregates outward porosity. Thus, theincrease DLA aggregates willThus, grow the increasingly outward will withgrow risingincreasingly initial particulate with concentration rising initial(launch particulate concentration radius) accompanying of fractal radius) accompanying(launch the reduction of fractal dimensionsthe andreduction rise of porosities. dimensions rise of porosities. It aggregates can be concluded that the DLA aggregates conspicuous It can beand concluded that the DLA display conspicuous fractal featuresdisplay when the launch radius is between andlaunch 10. fractal features when 1the radius is between 1 and 10.

Figure 8. 2D DLA variouslaunch launchradii. radii. Figure 8. 2D DLAaggregates aggregates with with various

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Figure 8. 2D DLA aggregates with various launch radii.

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Figure 9.9.3D withvarious variouslaunch launch radii. Figure 3DDLA DLAaggregates aggregates with radii. Figure 9. 3D DLA aggregates with various launch radii.

Figure 10. Effect of launch radius on the fractal dimension and porosity of 2D and 3D DLA aggregates.

Figure 10. Effect of launch radius thefractal fractaldimension dimension and of of 2D2D and 3D 3D DLA aggregates. Figure 10. Effect of launch radius ononthe andporosity porosity and DLA aggregates.

3.4. Effect of Finite Motion Step

3.4. Effect of Finite Motion Step

In practical flocculation processes, the movement energy of colloidal particles is limited, and the In practical movementTherefore, energy offinite colloidal particles is limited, and the particles will flocculation stop moving processes, if the energythe is dissipated. motion steps were introduced into the DLA model to describe the flocculation processes more accurately. We assume that the particles will stop moving if the energy is dissipated. Therefore, finite motion steps were introduced walking wouldthe cease and become new “seed” particles when their into random the DLA modelparticles to describe flocculation processes more accurately. We motion assumesteps that the reach certain given values. The fractal simulation was carried out at particle numbers of 2000 (2Dsteps random walking particles would cease and become new “seed” particles when their motion DLA) and 1000 (3D DLA), motion step length of 2 (Ns = 2), and launch radius of 2 (Re = 2) with various

reach certain given values. The fractal simulation was carried out at particle numbers of 2000 (2D

Figure 10. Effect of launch radius on the fractal dimension and porosity of 2D and 3D DLA aggregates. Fract. 2017, 1, 12 3.4. Effect of Fractal Finite Motion Step

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In practical flocculation processes, the movement energy of colloidal particles is limited, and the 3.4. Effect of Finite Motion Step particles will stop moving if the energy is dissipated. Therefore, finite motion steps were introduced In practical flocculation processes, the movement energy of colloidal particles is limited, and the into the DLA model to describe the flocculation processes more accurately. We assume that the particles will stop moving if the energy is dissipated. Therefore, finite motion steps were introduced random walking cease and become new particles when motion steps into the particles DLA model would to describe the flocculation processes more“seed” accurately. We assume that thetheir random walking particles would cease and become new “seed” particles when theiratmotion steps numbers reach certainof 2000 (2D reach certain given values. The fractal simulation was carried out particle given values. The fractal simulation was carried out at particle numbers of 2000 (2D DLA) and 1000 DLA) and 1000 (3D DLA), motion step length of 2 (Ns = 2), and launch radius of 2 (Re = 2) with various (3D DLA), motion step length of 2 (Ns = 2), and launch radius of 2 (Re = 2) with various finite motion finite motion steps. The of effect of the finite motion step on the and structures of the steps. The effect the finite motion step on the fractal features and fractal structuresfeatures of the DLA aggregates is presented in Figures 11 12. DLA aggregates is presented in and Figures 11 and 12.

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Figure 2D DLA aggregateswith with diverse finite motion step numbers. Figure 11. 2D 11. DLA aggregates diverse finite motion step numbers.

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Figure 11. 2D DLA aggregates with diverse finite motion step numbers.

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Figure 12.12. 3D3D DLA diversefinite finitemotion motion step numbers. Figure DLAaggregates aggregates with with diverse step numbers.

For the 2D DLA simulation, when the finite motion step numbers are below 800, the aggregates present a highly discrete morphology consisting of numerous separated tiny clusters with lower fractal dimensions (Df < 1.60) and larger porosity (Vf > 0.87). They grow increasingly discrete and the fractal dimension reduces rapidly with the gradual decrease of the finite motion step numbers. When Ms exceeds 800 (but is less than 3000), the aggregates grow to increasingly compact branched structures with typical scale-invariant features, and fractal dimension between 1.62 and 1.69 (Figure 13). However, the fractal dimension slightly increases to 1.64–1.74 when the finite motion step numbers rise from 3000 to 5000. For the 3D simulation, when the finite motion step numbers are less than 200, the aggregates exhibit a highly scattered spatial structure composed of a few isolated small clusters with lower fractal dimensions (Df < 1.95) and larger porosity (Vf > 0.88). They also grow increasingly scattered and the fractal dimension drops sharply with decreasing finite motion step numbers. When Ms exceeds 200, the aggregates produce increasingly compact spatial structures, and the fractal dimension stays at 2.00–2.30.

13). However, the fractal dimension slightly increases to 1.64–1.74 when the finite motion step numbers rise from 3000 to 5000. For the 3D simulation, when the finite motion step numbers are less than 200, the aggregates exhibit a highly scattered spatial structure composed of a few isolated small clusters with lower fractal dimensions (Df < 1.95) and larger porosity (Vf > 0.88). They also grow Fractal Fract. 2017, 1, 12 of 14 increasingly scattered and the fractal dimension drops sharply with decreasing finite motion12step numbers. When Ms exceeds 200, the aggregates produce increasingly compact spatial structures, and the fractal dimension stays at 2.00–2.30. The smaller the finite motion step is, the lower the particle motion energy is; thus, it is harder The smaller the finite motion step is, the lower the particle motion energy is; thus, it is harder for the particles to collide into the “seed” particle or clusters. With the motion energy dissipating for the particles to collide into the “seed” particle or clusters. With the motion energy dissipating away (reaching the maximum motion step), the particles would stop moving and become new “seed” away (reaching the maximum motion step), the particles would stop moving and become new “seed” particles, and would generate new tiny clusters after colliding and sticking with other randomly particles, and would generate new tiny clusters after colliding and sticking with other randomly walking particles. This prevents the particles from going into the interior of the aggregates, contributing walking particles. This prevents the particles from going into the interior of the aggregates, to the formation of highly discrete or scattered aggregates with lower fractal dimensions and larger contributing to the formation of highly discrete or scattered aggregates with lower fractal dimensions porosities. On the contrary, the motion energy of the particles gradually increases with the rising finite and larger porosities. On the contrary, the motion energy of the particles gradually increases with the motion step and, therefore, it is easier for the particles to collide into the “seed” particle or clusters. rising finite motion step and, therefore, it is easier for the particles to collide into the “seed” particle The energy of the motion particles may not entirely dissipate away before they coagulate, contributing or clusters. The energy of the motion particles may not entirely dissipate away before they coagulate, to the generation of more compact aggregates, leading to the increase in fractal dimensions and the contributing to the generation of more compact aggregates, leading to the increase in fractal decrease in porosities [26,27]. dimensions and the decrease in porosities [26,27].

Figure 13. Effect of finite motion step numbers on the fractal dimension and porosity of 2D and 3D Figure 13. Effect of finite motion step numbers on the fractal dimension and porosity of 2D and 3D DLA aggregates. DLA aggregates.

4. Conclusions 4. Conclusions A fractal simulation of flocculation processes using the DLA model with varying particle A fractal simulation of flocculation processes using the DLA model with varying particle numbers, numbers, motion step lengths, launch radii, and finite motion steps was conducted. The abovemotion step lengths, launch radii, and finite motion steps was conducted. The above-mentioned four mentioned four model parameters all have great influence on the morphologies and structures of the model parameters all have great influence on the morphologies and structures of the DLA aggregates. DLA aggregates. For the 2D DLA model, the simulation with too few particles (10), and too small a finite motion step (4), too large a launch radius (>10), and too small a finite motion step (1000, motion step length of 1.5–3.5, launch with the particle number of >1000, motion step length of 1.5–3.5, launch radius of 1–10, and finite radius of 1–10, and finite motion step of >3000 can form 2D DLA aggregates possessing conspicuous motion step of >3000 can form 2D DLA aggregates possessing conspicuous fractal features with fractal fractal features with fractal dimensions of around 1.70, which is close to the experimental result of dimensions of around 1.70, which is close to the experimental result of 1.66 reported in [10]. For the 3D 1.66 reported in [10]. For the 3D DLA model, the simulation with too few particles (4), too large a launch radius (>10), and too small a finite motion step (10), and too small a finite motion step (500, motion step the particle number of >500, motion step length of 1.5–3.5, launch radius of 1–10, and finite motion length of 1.5–3.5, launch radius of 1–10, and finite motion step of >200 can form 3D DLA aggregates having apparent fractal features with fractal dimensions of around 2.20, which agrees well with the experimental values of 1.8–2.20 reported in [30] and is considerably higher than that of the 2D DLA aggregates, probably due to the excluded volume effect [7] which means that the centers of the primary particles in three-dimensional space cannot approach each other more closely than the length of the diameter of the primary particle. As reported, the diffusion-limited cluster–cluster aggregation model (DLCA) with fractal dimension of 1.75–1.85 [31–33] can reveal the real flocculation processes better than the DLA model; we intend to conduct further research on fractal simulations of the flocculation processes by using the DLCA model as well. Author Contributions: Dongjing Liu conducted the simulations and wrote the paper; Weiguo Zhou, Zumin Qiu, and Xu Song contributed valuable ideas and discussions.

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Conflicts of Interest: The authors declare no conflict of interest.

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