«Dissertation zur Erlangung des akademischen Grades Doktoringenieurin (Dr.-Ing.) von: Yashodhan Pramod Gokhale geboren am: 05. October 1981 in Pune, ...»
5.29 are the simplified equations that can only be used to conserve numbers and mass and will, hence, yield identical results as the ones derived by (Hounslow, Ryall et al. 1988).
The advantage of the fixed pivot technique is its generality in terms of the properties to be conserved and the grid choice. The predictions for the cases involving simultaneous agglomeration and breakage suffered from severe over predictions in the large particle size range.
5.5.2 Cell Average Technique- CAT
This section summarizes the newly developed numerical method cell average technique (CAT) for solving population balances (Kumar 2006). This method approximates the number density in terms of Dirac point masses and is based on an exact prediction of some selected moments to solve the population balance equation. The objective behind the cell average technique is to divide the entire size domain into a finite number of cells. The lower and upper boundaries of the i th cell are denoted by x i 1/ 2 and xi 1/ 2, respectively. All particles belonging to a cell are identified by a representative size in the cell, also called grid point. The representative size of a cell can be chosen at any position between the lower and upper boundaries of the cell. A typical discretized size domain is shown in Figure 5-3.The representative of the i th cell is represented by x i (x i 1/ 2 x i 1/ 2 ) / 2 and the width of the i th cell is denoted by xi xi 1/ 2 xi 1/ 2. The size of a cell can be fixed arbitrarily depending upon the process of application. In most applications, however, geometric type grids are preferred.
Figure 5-3 Averaging and rearrangement of newly formed particles: Cell Average Technique The method works in two steps: first we calculate the average of all new born particles due to particulate processes in the i th cell. Then we distribute the particles between two neighboring nodes in such a way that the total number and the total mass of the system remain conserved.
It should be noted that the basic difference between the cell average technique and the fixed pivot method is, the averaging of the new born particles in cell average technique.
We wish to transform the general continuous population balance equation into a set of I ordinary differential equations (ODEs) that can be solved using any standard ODE solver.
The processing events that may change the number concentration of particles include disintegration, aggregation, growth, nucleation etc. However, here we consider only aggregation and disintegration. Note that this general formulation is not similar to the traditional sectional formulation where birth terms corresponding to each process are summed up to determine the total birth. Here all particulate events will be considered in a similar fashion as we treat individual discrete processes. The first step is to compute particle birth and death in each cell. Consideration of all possible events that lead to the formation of new particles in a cell provides the birth term. Similarly all possible events that lead to the loss of a particle from a cell give the death rate of particles.
The new particles in the cell may either appear at some discrete positions or they may be distributed continuously in accordance with the distribution function. For example, in a binary aggregation process particles appear at discrete points in the cell whereas in the
Since we know the positions of the newborn particles inside the cell, it is easy to calculate the average birth of newborn particles vi. It is given by the following formula
Now we may assume that Bi particles are sitting at the position vi. It should be noted that the averaging process still maintains consistency with respect to the first two moments. If the average volume vi matches with the representative size x i then the total birth Bi can be
Here a1 (vi, x i ) and a 2 (vi, x i 1 ) are the number fractions of the birth Bi to be assigned at x i and x i 1, respectively. Solving the above equations we obtain
There are 4 possible birth fractions that may add a birth contribution at the node x i : two from the neighboring cells and two from the i th cell. Collecting all the birth contributions, the birth term for the cell average technique is given by
Substituting the values of BiCA and D iCA into the Eq. 5.31, we obtain a set of ordinary differential equation. It will be then solved by any higher order ODE solver. Note that there is no need to modify the death term since particles are just removed from the grid points and therefore the formulation remains consistent with all moments due to the discrete death. As a result the death term D iCA in the cell average formulation is equal to the sum of total death in the i th cell. For the detailed description of the scheme, readers are referred to (Kumar 2006;
Kumar, Peglow et al. 2008). The next chapter explains the numerical simulation by using the cell average technique and comparing it with the experimental results.
6 Experimental and Modeling Results In this chapter, we discuss: first the synthesis of silver nanoparticles by double reduction method and second the agglomeration and disintegration process of titanium dioxide nanoparticles synthesized by sol-gel process.
The prime goal is the optimization of nanoparticles formation process in the liquid phase with different conditions. Silver and titanium dioxide nanoparticles are produced in the batch reactor. They are investigated both by experimentally as well as by simulations based on the population balance equations. The population balance models for agglomeration and disintegration leads to a system of integro-partial differential equations, which can be numerically solved by several numerical schemes. Here the cell average technique is used to solve PBEs and predict the particle size distributions and moments.
6.1 Experimental results of silver nanoparticles
S ynthesis of silver nanoparticles is done by double reduction method. In this process silver particles are capped with citrate ions and then it is reduced by sodium formaldehyde sulphoxylate. In general surface capped silver powder can be effectively converted to colloidal state via re-dispersion. Here, the agglomeration process is caused by rapid collision of the particles and their afterward bonding. Depending on their interactions, this collision results in the agglomeration or redispersion of particles. During the time of the process, after the drop wise addition of reducing agent the redispersion begins and then the particle size distribution develops rapidly. The size of the particles distribute into the wide and varied range. Particle size distribution and zeta potential were measured using Dynamic Light scattering method (DLS). Results obtained with all the experiments performed are summarized in tables and graphical representations in this section.
6.1.1 Effect of Capping Agent
In this work, the capping of silver particles by tri-sodium citrate is investigated under different conditions. Citrate is an efficient stabilizer. All the solutions were clear and stable for weeks in absence of air. Capping agents when present inhibit the growth of nanoparticles by passivating their surfaces. The synthesis of almost all the nanoparticles is done in the presence of capping agents in order to stabilize the size of nanoparticles for a desired application.
The Table 6-1 shows the average particle size (d50,0) of silver particles synthesized with capping agent and reducing agents of different molar ratios. Here 0.58 M silver nitrate (25ml AgNO3) with 0.85 M Tri sodium citrate (150ml) as capping agent and 0.45 M Sodium formaldehyde sulphoxylate-SFS (25ml) as reducing agent under different shear rates at 500 C is given.
Figure 6-1 Particle size distribution, Q0(d) for shear rate of 120 s-1 with different molar ratios of capping agent and T=50ºC, reaction time t= 3 hrs.
It shows the influence of the ratios of capping agent on the particle size distribution. In the meanwhile Figure 6-2 and Figure 6-3 illustrate the relationship between the particle sizes and the capping agent at different stirrer speed. Silver nitrate reacts slowly, almost since the beginning of the reaction. Then it reacts with tri-sodium citrate at optimum temperature and shear rates.
Figure 6-2 Particle size distribution, Q0(d) for shear rate of 370 s-1 with different molar ratios of capping agent and T=50ºC, reaction time t= 3 hrs.
Figure 6-3 Particle size distribution, Q0(d) for shear rate of 623 s-1 with different molar ratios of capping agent and T=50ºC, reaction time t= 3 hrs.
The formation of silver nanoparticles in aqueous medium proceeds rapidly and their stabilization is primarily the result of the adsorption of negatively charged citrate ions. As citrate plays an important role as a stabilizer; a clear yellow solution is obtained. The Table 6-1 shows that the zeta potential value asserts that a higher concentration of capping agent has more stability on the particle charge surface than others. It makes the colloidal suspension from the smallest particles. It is observed that particle size in the range of 14-30 nm due to higher concentration of capping agent.
6.1.2 Effect of Reducing Agent
This study is based on the effect of reducing agent on the particle size distribution. Sodium formaldehyde sulphoxylate (SFS) is used as mild reducing agent for reduction of silver from Ag+ to Ag0. The conversion of the bigger particles to smaller ones is normally done by means of physical processes such as ball milling or mechanical grinding.
Table 6-2 shows the average particle size d50,0 of silver particles synthesized with different molar ratios of reducing agents. Here 0.58 M silver nitrate (25ml AgNO3) with 0.45 M Tri sodium citrate (150ml) as capping agent and 0.42 M Sodium formaldehyde sulphoxylate-SFS (25ml) as reducing agent under different shear rates at 500 C is given.
This sulphoxylate group helps to terminate the particle growth. The use of this particle growth in the formation of silver powder has been demonstrated.
Figure 6-5 shows that with decrease in the particle size, also presents the molar ratio of reducing agent increases at different shear rates. Table 6-2 also presents the particle size at different shear rates with the variation of the reducing agent (SFS) concentration in the solution.
Figure 6-4 Particle size distribution, Q0(d) for shear rate of 120 s-1 with different molar ratios of reducing agent and T=50ºC, reaction time t= 3 hrs.
Figure 6-5 Particle size distribution, Q0(d) for shear rate of 370 s-1 with different molar ratios of reducing agent and T=50ºC, reaction time t= 3 hrs.
Figure 6-6 Particle size distribution, Q0(d) for shear rate of 623 s-1 with different molar ratios of reducing agent and T=50ºC, reaction time t= 3 hrs.
The values of zeta potential also assert that at the lowest concentration of reducing agent, these values are fall into a range which has the better stability behavior than other concentrations. Both the figures and the table make show the results that the lower the concentration of reducing agent (SFS) in the suspension, the higher the shear rate to get smaller particles as shown in Figure 6-6 and Figure 6-4.
6.1.3 Effect of Shear Rate on the particle size distribution
In this section the influence of shear rate on the silver nanoparticles formation is investigated by varying the concentration of reactants.
Table 6-3 shows the influence of the different shear rates on the mean particle diameter (d50,0) under different molar ratios of capping agent and reducing agents. The shear rate also varies from 120 to 623s-1by using 6 blade stirrer. The zeta potential values assert that a higher shear rate has more stability on the particle charge surface than others. Figure 6-7 and Figure 6-8 shows that the formation of the small agglomerates was a result of the reducing agent at different molar concentrations, and at a shear rate ranging from 120 to 623s-1.
1 120 0.28 65.1 -28.5 1:2:1 2 370 0.58 25.2 -38.5 3 623 0.86 22.1 -41.5 4 120 0.28 59.1 -27.6 1:2:0.5 5 370 0.58 21.1 -31.5 6 623 0.86 18.2 -37.2
Figure 6-7 Particle size distribution, Q0(d) for different shear rates with molar ratios of 1:2:1
and T=50ºC, reaction time t= 3 hrs.
As observed from the experimental results, the growth in aggregates size is faster at higher shear rate. The formation of the bigger agglomerates occurs at a low stirrer speed from starting process 120s-1.It apparently means that shear rates close to the impeller are too high to cause agglomeration. Thus, the higher shear rates, which have greater shear stress, would lead to more collisions and also make faster disintegration process.
Figure 6-8 Particle size distribution, Q0(d) for different shear rates with molar ratios of 1:2:0.5 and T=50ºC, reaction time t= 3 hrs.
Also it affects the zeta potential of silver citrate colloids are stable in a much wider range of pH values, extending from pH 1.9 to 4. By the lowest concentration of reducing agent at pH
1.9 shows an appreciable decrease in intensity, related to an increase in the surface charge of the nanoparticles and consequently to a increase in their stability, gives rise to decrease in zeta potential values at higher shear rate.
6.1.4 Morphology and Particle Size Distribution 184.108.40.206 Scanning Electron Microscopy (SEM) The morphology of the silver particles, which are synthesized for different molar ratios of capping agent and reducing agents, was observed using second electron images from scanning electron microscope.
Figure 6-9 shows the morphology of the sample prepared with the ratio of silver nitrate to sodium citrate to SFS 1:3:1 to 1:2:1.