3.2 Micro-processes and events
3.2.1 Droplet deposition
Chapter 3 Micro-scale modeling using a Monte Carlo method
equation:
𝑁pos =nint 𝐴p 𝐴contact
. (3.14)
A distribution of the particle size and the corresponding number of positions can be taken into account in the same way as described above.
Layer-wise particle growth implies an enlargement of the surface of single particles. Correspondingly, the number of positions per particle should be adjusted during the layering process. A detailed description of this procedure is given in Section 3.2.4. This adjustment may also be necessary for agglomeration processes if a binder solution is used since undesired layering may take place and lead to growth of the primary particles. However, in previous works [15, 97, 124], this effect has not been taken into account. In this thesis, the focus lies on agglomeration due to glass transition. Since in these processes pure water is sprayed and no solid material is present in the liquid to induce layering, the adjustment of the number of positions per particle is only taken into account in the simulations for layering processes. However, an increasing number of positions due to agglomeration is certainly taken into account and described in Section 3.2.5.
3.2 Micro-processes and events
Table 3.4:Properties and selection probabilities of the set of three particles used in the example to test the particle selection methods in the droplet deposition algorithm.
𝑑p 𝐴p 𝑉p 𝑃sel(𝛬0=1) 𝑃sel(𝛬1=1) 𝑃sel(𝛬2=1) 𝑃sel(𝛬3=1)
No. [mm]
mm2 mm3
[−] [−] [−] [−]
1 0.4 0.5027 0.0335 0.3333 0.3333 0.2857 0.2222
2 0.2 0.1257 0.0042 0.3333 0.1667 0.0714 0.0278
3 0.6 1.1310 0.1131 0.3333 0.5000 0.6429 0.7500
Í 1.2 1.7593 0.1508 1 1 1 1
based on physical properties, e.g., length (diameter), surface area, or volume. The procedure and the influence of the particle selection on the particle formation process is illustrated in the following example.
Consider three particles with different diameters, surface areas and volumes, see Table 3.4. If the particles are selected based on their number, a random integer between unity and three would be necessary in the current example. In this case, each particle has the same probability to be selected. However, the selection probability is not equal if the particles are selected based on physical properties. The procedure is shown in Figure 3.12 for the selection based on the diameter, but is identical for other properties. A random number from the interval(0, 𝑑p,tot)is drawn, where𝑑p,totis the sum of all individual diameters and equals 1.2 mm in this example. Figure 3.12 shows that the size of the intervals corresponds to the value of the diameter. The number of the selected particle can then be obtained by the interval, in which the random number lies:
particle=
1 𝑟 𝑑p,tot ≤0.4, 2 0.4<𝑟 𝑑p,tot ≤0.6, 3 0.6<𝑟 𝑑p,tot ≤1.2,
with 𝑑p,tot =∑︁
𝑗
𝑑p,j. (3.16)
In this case, the selection probability is not equal for each particle. In fact, it depends on the property itself and corresponds to the normalized size of the interval. For the case depicted in Figure 3.12, the interval of the third particle occupies half of the total range and its selection probability is therefore 0.5. Particles 1 and 2 are smaller and their selection probabilities are 1/3 and 1/6, respectively.
Table 3.4 lists the selection probabilities depending on the used property.
In case of layering, the different methods of particle selection for droplet deposition described above directly correspond to the macroscopic growth kinetics given in Equation (2.29). If the particles are selected based on their number, the kinetics correspond to𝐺 =𝛬0𝐺0with𝛬0=1. If the particles are selected according to their length,𝐺 =𝛬1𝐺1with𝛬1 =1 and so on. If values other than unity are used for𝛬j, mixtures of these kinetics are obtained. The selection property in a specific time step can
Chapter 3 Micro-scale modeling using a Monte Carlo method 𝑟 𝑑p,tot
0 0.4 0.6 1.2
𝑑p,1 𝑑p,2 𝑑p,3
Figure 3.12:Schematic representation of the method used to select particles for droplet deposition.
In this example, selection based on particle diameter is shown.
be determined randomly with a uniformly distributed random number𝑟 from the interval(0,1):
property=
number 𝑟 ≤𝛬0,
length 𝛬0<𝑟 ≤ 𝛬0+𝛬1,
surface area 𝛬0+𝛬1<𝑟 ≤𝛬0+𝛬1+𝛬2, volume 𝛬0+𝛬1+𝛬2<𝑟 ≤ 1.
(3.17)
In the simulations presented in this thesis the particles are selected based on their surface area by setting𝛬2to unity.
The described selection algorithm was tested using the set of particles listed in Table 3.4. The particles were selected for droplet deposition based on their number, diameter, surface area, and volume and the number of droplets each particle received was counted. The results are shown in Figure 3.13 for 100 and 106droplets, respectively. Figure 3.13a shows that the probability for each particle to be selected is identical if the selection is based on their number. If the selection property is the particle diameter, particle number 3, being the largest particle in this example, receives half of the droplets, while the other two receive significantly less droplets. This effect increases if the surface area or the volume are used as the selection property. It can also be seen that the relative droplet number each particle receives changes with the number of droplets. For the case of 106droplets, the relative droplet numbers match the selection probabilities in Table 3.4.
The example shown in Figure 3.13 illustrates that the way particles are selected for droplet deposition influences the distribution of droplets within the sample system and therefore the behavior of the particle formation process. The influence of droplet deposition on layering and agglomeration is addressed in the frame of simulation studies in Section 4.2.2 and Section 5.2.1, respectively.
Once a particle has been selected, a position of this particle is randomly chosen and the droplet is deposited. The position is selected based on the number of positions of the specific particle by drawing a random number from the interval(1, 𝑁pos). The label of the position is changed according to Table 3.3. Additionally the time of droplet deposition is stored and later used to calculate the progress of drying during the simulation, see Section 3.2.3.
In the Monte Carlo models presented by Terrazas-Velarde [15] and Hussain [97], particles were selected for droplet deposition based on their surface area as well, which corresponds to the case
3.2 Micro-processes and events
1 2 3
0 0.2 0.4 0.6 0.8 1
0.2800.333 0.3500.333 0.3700.333 property:number
(a)
Number of selected particle[−]
Rel.dropletnumber[−] 100 droplets 106droplets
1 2 3
0 0.2 0.4 0.6 0.8 1
0.310
0.190
0.500 0.333
0.167
0.500 property:length
(b)
Number of selected particle[−]
Rel.dropletnumber[−] 100 droplets 106droplets
1 2 3
0 0.2 0.4 0.6 0.8 1
0.260
0.030
0.710
0.285
0.071
0.643 property:surface area
(c)
Number of selected particle[−]
Rel.dropletnumber[−] 100 droplets 106droplets
1 2 3
0 0.2 0.4 0.6 0.8 1
0.180
0.040
0.780
0.222
0.028
0.750 property:volume
(d)
Number of selected particle[−]
Rel.dropletnumber[−] 100 droplets 106droplets
Figure 3.13:Relative droplet number received by the particles (example of Table 3.4) for different selection properties: number (a), length (b), surface area (c), volume (d).
given in Figure 3.13c and Equation (3.17) (with𝛬2=1). Dernedde [132] picked the particles randomly by their number, which corresponds to the case shown in Figure 3.13a. In comparison to the current thesis, small particles are preferred in the selection for droplet deposition in the model by Dernedde [132].