Keywords

3.1 Introduction

Plant nutrients are essential elements for plant growth and metabolism and their slight deficiency causes irregular growth of plants. Plant nutrients have various functions such as structural components of macromolecules and enzymes for reactions (Table 3.1). Plant nutrients are classified into macronutrients and micronutrients with macronutrients at a concentration of 0.1% of dry tissue weight namely: nitrogen (N), phosphorus (P), potassium (K), calcium (Ca), magnesium (Mg) and sulphur (S) and micronutrients found in the concentration of less than 0.01% of dry tissue weight are zinc (Zn), nickel (Ni), manganese (Mn), molybdenum (Mo), iron (Fe), chlorine (Cl), copper (Cu), and boron (B) (Grusak, 2001). Macronutrients are utilized from the germination stage to the ripening stage of plants growth and production and are essential for providing humans with a suitable supply of energy, nutritional value, and promotion of good health (Grusak and DellaPenna 1999). Micronutrients, though present in small amount, play a vital role in various plant processes, photosynthesis and chlorophyll formation (Monreal et al. 2015) but are also toxic to soil and plants at high concentrations (Arnon and Stout 1939).

Table 3.1 Details of selected macro and micronutrients required for plant growth
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Plant nutrients existing in the soil profile are inadequate for crops cultivation. Therefore nutrients are supplied in the form of inorganic fertilizers to suffice the nutrients requirement of crops. Application of fertilizers are inevitable but the excess application and low nutrient use efficiency of the plants lead to 40–75% (Celsia and Mala 2014) lost to the environment through leaching and volatilization causing pollution and toxicity of soil and water bodies; and wastage of fertilizers is an economic loss. The N:P:K ratio for optimal growth is 4:2:1 whereas, in India, it is practice as 10:2.7:1 (Subramanian and Tarafdar 2011); similar increased in fertilizer consumption has been observed in other countries for instances China; where the N used for rice cultivation is 90% more than global average (Guo et al. 2017). Shaviv and Mikkelsen (1993) observed that while the application of nitrogen fertilizers has increased to 15 times, there was only a 3% increase in yield. To overcome the excessive use of fertilizers, without compromising the yield, discovering new and advanced solutions are encouraged. One of the possible solutions is through the application of nanotechnology, i.e., using nano-fertilizers. Since nanomaterials are smaller and more reactive than their bulk materials, it is suggested to have the potential to revolutionize agricultural systems (Singh 2012). Numerous researchers have also reported the possible applications of nanotechnology in agriculture (Subramanian and Tarafdar 2011; Khot et al. 2012; Prasad et al. 2014; Benzon et al. 2015; Monreal et al. 2015; Dubey and Mailapalli 2016; Duhan et al. 2017; and Suppan 2017).

3.2 Nanofertilizers

Nanomaterials refer to any materials having a particle size of 1–100 nm (10−9 m). They have unique physical and chemical properties, which can be more advantageous as compared to their bulk structures (Le Brun et al. 1992). Nanotechnology is an interdisciplinary research field with many practical applications in the field of medicine, electronics, mechanical engineering; however in agriculture, it is still on its research stage (Benzon et al. 2015) where feasibility to large field areas are yet to be ascertained (Khot et al. 2012) but it is also gaining importance gradually (Prasad et al. 2014). Nanotechnology applications in agriculture as nano-fertilizers, nanosensors, nanocapsule, nano-encapsulated flavor enhancer, nanofilms in packaging to prevent spoilage and prevent oxygen absorption, nano-chip use for identification and tracking. Mastronardi et al. (2015) classified three types of nanotechnologies for fertilizer inputs and plant protection, i.e., nano-fertilizer, nano-additives, nano-coatings. Nano-sensors in agriculture acts as specifying agents for the level of fertilizers and pesticides present in the soil, soil physical properties, plant health and toxicity level (Rameshaiah et al. 2015). Numerous reviews and few laboratory studies (Table 3.2), established the advantage of nano-fertilizers over conventional fertilizers in terms of low leaching rate of nutrients, higher nutrient absorption capacity, protect against fungal and bacteria growth and increasing plant biomass and yield. Hence nano-fertilizers, controlled release nano-fertilizers, and nano-pesticides can be used as a substitute for conventional ones without affecting the crop yield (Adhikari et al. 2014) while controlling other unwanted factors such as high leaching, eutrophication, and disease which may cause to humans.

Table 3.2 Effect of different types of nanoparticles and their size on crop growth (Modified after Dubey and Mailapalli 2016); The values in the brackets in column 1 represent sizes of the nanoparticles
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However, efforts arise in synthesizing and stabilizing of the synthesized nano-fertilizers because it involves complex, expensive and sophisticated instruments. Some of the approved commercially available nano-fertilizers are Nano Micro Nutrient by Alert biotech in Maharashtra, India, Nano Ultra-Fertilizer by AB Industries in Taiwan, Nano-Fertilizer by Geetharam Agencies in Kerela, India, Hero Super Nano in Thailand, and Nano Calcium Magic Green-Setia Bersama in Germany (Dimkpa and Bindraban 2017). Nano-zinc, Nano-nitrogen, Nano-phosphorus, Nano-silver, Ultra bio-silver, Nano-sulphur, Nano-Nitrogen and Nano-potassium are some of the other nano-fertilizer products developed in India by Kanak Biotech, New Delhi.

3.3 Synthesis Methods for Nanoparticles

Two types of approaches are available for nanomaterial synthesis; bottom-up and top-down approaches (Fig. 3.1). Bottom-up methods are the chemical approaches for synthesizing nanomaterials with the help of elemental units to combine into larger stable structures Involving building up of the material atom by atom and cluster by cluster until it forms a nanosized material. It involves chemical synthesis, self-assembly, and positional assembly, in which suitable solvents are used for synthesizing ultrafine particles through their dissolved molecular state. Some of the methods involved in the bottom-up approach are plasma arc, rapid solidification, arc discharge, physical vapor deposition, chemical vapor deposition, sol-gel, and inert gas condensation. Bottom-up approach advantages are being able to produce uniform size, shape and distribution of the nanomaterials formed. Bottom-up approaches are common for synthesizing nanoparticles for application to plants (Chalenko 2010; Dutta et al. 2014; Tarafdar 2015; Poopathi et al. 2015; Saha and Gupta 2017). However, the process involved is highly complicated with low yield and expensive machinery are required; it is suitable for highly pure materials only, and it sometimes produces harmful by-products during the synthesis process.

Fig. 3.1
figure 1

Methods for synthesizing nanomaterials. Bulk sizes are being reduced to nano size in the top-down method, whereas, atoms aggregate and form clusters to become nanoparticles in bottom-up method. Modified after Galstyan et al. 2018

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3.3.1 High Energy Ball Milling

The top-down approach is a physical approach for synthesizing nanomaterials from bulk materials to nanosized materials by milling, crushing or grinding (De Castro and Mitchell 2002). Some of the top-down processes are etching technology, high energy ball milling, cold milling or cryo-milling, severe plastic deformation, mechanical polishing, nanoimprint lithography and sliding wear. The most common top-down method is the high energy ball milling; earlier, also known as mechanical milling, which involves breaking down of large-sized particles to nanosized particles through severe plastic deformation to reduce the size of materials and increase the surface area and reactivity of the particles. The high energy ball milling established during an attempt to develop homogeneous composite particles or alloys (Benjamin 1970) uses an attrition mill, followed by the use of other mills such as shaker mill, mixer mill, ball mill, and planetary ball mill. Benjamin’s outcome of the study in the 1970s led to the study of the different stages that occur during the high energy ball milling process (Benjamin and Volin 1974). Subsequently, the method was used by different authors with different mills for obtaining nano-alloys, nanocrystalline and metallic-amorphous materials. The advantage of high energy ball milling is that it is simple, easy handling, the versatility of the process, ability to produce large quantities (Maurice and Courtney 1990) and the method is applicable for different types of materials and scalability of the process and its low cost. The main drawback of the high energy ball milling approach is the non-uniformity of the surface structure formed, i.e., not suitable for preparing uniformly shaped materials.

The high energy ball milling devices are of three types namely: shaker mills, attrition mills, and planetary ball mills (Suryanarayana 2001). Shaker mills have a vial where grinding media, i.e., milling balls and sample are swung vigorously to-and-fro for several times causing an impact of milling ball against each other, with the sample and the wall of the vials causing the size reduction in the end product. Attrition mill is a conventional ball mill with a fixed chamber containing centrally vertical rotating stirrer system; as the stirrer rotates the milling balls drops and grind the material. Fritsch Company in 1962 introduced the first self-developed patented Planetary Mill (Fig. 3.2a). Planetary ball mill includes a disc and vial(s) rotating on top of a disc in planet-like movement, i.e., the disc and the vials rotate in the opposite directions similar to planets rotation around the sun (Fig. 3.2b); as such the centrifugal forces act in like and opposite direction alternately. This total centrifugal force acts on the material and milling balls inside the vials, causing impact between the milling balls, the sample, and the vial and ground the material. Planetary ball milling operates in both dry and wet conditions, with easy handling and moderate costs. High energy ball milling synthesized nanofertilizer by reducing the precursor size to nanoparticle size or by milling two or more types of precursors to develop a nano-carrier or nano-sensor. However, it involves several trial runs to obtain the desired milling parameters, which consume a lot of time and energy; Paul et al. (2007) synthesized a nano-sized fly ash of 148 nm at a milling time of 60 h, Gaffet et al. (1991) synthesized a 350 nm size tungsten carbide at 180 h milling time.

Fig. 3.2
figure 2

Working principle of planetary ball milling; (a) selected planetary ball mill with front panel open: (b) schematic line diagram of rotation of disc and vials. Modified after Chen et al. 2006

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Furthermore, Charkhi et al. (2010) and Guaglianoni et al. (2015) conducted several trials to optimize milling speed, milling time, ball to powder ratio and milling medium for synthesizing nanoparticles. Thus, the use of modeling comes in to picture, where optimization of the milling parameters is possible by providing basic material properties. This paper attempts to briefly study the top-down method of nanomaterial synthesis using planetary ball milling generally adopted in material science/engineering and review different mathematical models available in dealing with planetary ball milling for possible application in nano-fertilizer synthesis.

3.3.2 Synthesis of Nanoparticles Using a Planetary Ball Mill

Table 3.3 presents some of the studies conducted using planetary ball mills during the past three decades. The table indicates the different input parameters involved during the High energy ball milling process in a planetary ball mill and also the final size obtained during the synthesis process. Earlier studies are more towards obtaining alloys or homogeneous amorphous products and details of the milling parameters were not explicitly mentioned, unlike the latter studies which are more size oriented. Table 3.3 highlights the importance of different milling parameters viz., milling speed, milling time, ball to powder ratio, material type, milling medium and their effect on the final size of the end product. Since planetary ball mills’ s has been used for synthesizing of non-metallic materials such as fly ash (Paul et al. 2007), biochar (Peterson et al. 2012) and zeolite (Charkhi et al. 2010; Mukhtar et al. 2014), hence planetary ball mills are capable of synthesizing the nanofertilizer material. Milling parameters information gathered from previous studies (Paul et al. 2007; Rao et al., 2010; Patil and Anandhan 2012; Raghavendra et al. 2014; Patil and Anandhan 2015; Rajak et al. 2017) for fly ash material were used for studying the interaction effect of milling speed, milling time and ball to powder ratio on fly ash particle size (Figs. 3.3 and 3.4).

Table 3.3 Selected planetary ball mills and the values of milling parameters used in the synthesis of nanoparticles from different bulk materials
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Fig. 3.3
figure 3

Effect of milling speed and milling time on the particle size of bulk fly ash. Note the decrease in particle size with an increase in milling time and milling speed

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Fig. 3.4
figure 4

Effect of milling speed and ball to powder ratio on the particle size of bulk fly ash. Note the decrease in the final size of fly ash particles with increasing milling speed and ball to powder ratio

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3.3.2.1 Milling Speed

Table 3.3 shows the vast range of milling speed from 60 to 2400 rpm used by authors for achieving the nanosized materials. Increasing the milling speed reduced the size of the bulk material as seen in Table 3.3. The higher milling speed of greater than 500 rpm evidently reduced the milling time (Canakci et al. 2013a; Feng et al. 2007; Le Brun et al. 1992; Lee et al. 2017; Wakihara et al. 2011) while the lower milling speeds have been compromised with higher milling time (Kong et al. 2000; Patil and Anandhan, 2012) to attain nanoparticles of 50–1000 nm. Figure 3.3 shows the interaction effect of milling speed and milling time on the size of the fly ash material; it also indicates rapid decreases in the size of the fly ash particle with an increase in milling speed. However, continuous increase in milling speed leads to an increase in temperature and may cause welding of nanoparticles.

3.3.2.2 Milling Time

The milling time used in most of the studies is more than 10 h (Table 3.3), while some studies have used milling time up to 180 h (Gaffet and Harmelin 1990) to obtain nano-sized products 20–350 nm. Lim et al. (2003) used a milling time of 10 min only to obtain an end product of 100 nm with a mixer mill, but it was adjusted with the higher milling speed of 920 rpm, indicating an interrelationship between milling speeds and milling time. Figure 3.3 indicates a gradual decrease in fly ash particle size with increasing milling time. However, extended milling time decomposed or agglomerate the product and also cause contamination (Suryanarayana 2001; Burmeister and Kwade 2013; Mukhtar et al., 2014; Malayathodi et al. 2018) and an increase in temperature inside the mill and welding of particle take place. The interaction effect of milling speed and milling time on the final particle size (Fig. 3.3) indicate that milling time and milling speed parameters are interdependent hence; the interaction plays a role together to reduce the overall size of the material.

3.3.2.3 Ball to Powder Ratio

Optimum ball to powder ratio is an essential factor because with less ball to powder ratio there will not be enough impact to be able to reduce the size of the material. Table 3.3 indicates ball to powder ratio of 10:1 is the most commonly used ratio and it can go as high as 100:1 and as low as 1:10. Increase in a ball to powder ratio decreases the particle size; Guaglianoni et al. (2015) reported the crystallite size of 53.6 and 48.3 nm at the ball to powder ratio of 5:1 and 20:1 respectively. A higher ball to powder ratio generally reduces the milling time for a particular material (Lee et al. 2017; Zakeri et al. 2012); but the higher ball to powder ratio also reduces the amount of initial input material. Figure 3.4 shows the slight decrease of final size of fly ash to the increase in ball to powder ratio. However increase in ball to powder ratio leads to collision of balls against each other and cause restricted movement and it also increases impurities in nanoparticles products (Li et al. 2018).

3.3.2.4 Milling Medium

Milling medium in planetary ball mills acts as surfactant or process control agent and a medium to avoid contamination during milling which may cause due to the reaction of powder materials with the surrounding particle such as the formation of oxides. Milling medium is one of the parameters governing the size of the end product it can be dry milling, wet milling or salt assisted milling. Numerous studies (Charkhi et al. 2010; Mukhtar et al. 2014; Munkhbayar et al. 2013) suggested that wet milling is efficient than dry milling, because it creates higher efficiency, lowers the enthalpy, and eliminates dust formation. Salt assisted milling is also a milling option more effective than wet milling according to Peterson et al. 2012. Argon is found to be the best milling medium (Table 3.3) since it is widely used by most of the researchers due to its inert properties.

Apart from the above milling parameters, type of milling balls, milling ball diameter, temperature and also material parameters such as the initial size of bulk material, types of material, material physical and chemical properties are also considered as factors affecting the size of the end product. An in-depth study is required to understand the effect of the milling parameters, which may be possible through mathematical models.

3.4 Modeling of Planetary Ball Milling

Generally, a number of experimental trials are conducted to determine the different milling parameters in planetary ball mills but, trial and error methods are time-consuming, inefficient and not practical. Real-time ball milling is a very slow process as compared to a discrete element method procedure which takes only a few seconds to simulate the process (Feng et al. 2004). For economical and time-saving synthesis, few models (Tables 3.4, 3.5, and 3.6) were developed to determine the optimum milling parameters in a planetary ball mill. Zhang (2004) stated that modeling and mathematical analysis of high energy ball milling process are still lacking and deserve the attention of researchers. Since large-scale production of nanomaterials is much more feasible in case of the top-down method, modeling of planetary ball milling process is essential to facilitate industrial-scale nanofertilizer production. High energy ball milling is a dynamic process, and it is a challenge to develop mathematical models to represent the description of the process. The models cannot represent the exact process but, they are still capable of providing valuable insights into the behavior of nanoscale materials. Thus, understanding the process with the help of different simulation models is an excellent deal in the synthesis of nano-fertilizers, and it is urgently needed to assist effective, efficient and economical results. The model’s aid in optimizing the milling parameters for nanofertilizer synthesis by providing the material properties as input parameters; based on the nanofertilizer size required, the model optimized the different milling parameters for direct synthesis in planetary ball mills without the labor of trial and error. However, not all models required the material properties as input parameters; some models required a large trials data as inputs for calibration and validation so based on the available inputs the model can be selected for nano-fertilizers synthesis.

Table 3.4 Analytical models used in the simulation of the planetary ball milling process
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Table 3.5 Statistical models used in simulating planetary ball milling process
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Table 3.6 Numerical models used to simulate planetary ball milling process
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3.4.1 Analytical Models

Table 3.4 presents some of the analytical models developed on planetary ball mills with different types of equations used and research highlights, during the last two and a half decades. The first attempt to model the underlying geometry, mechanics, and physics of the process of mechanical alloying was established by Maurice et al. (1990), using the concept of Hertzian contacts between the grinding media; the study used three types of mill viz., attrition mill, vibratory mill, and horizontal mill. In planetary ball mills, the arrangement of the vial and disc exerted centrifugal force on the balls in the milling container towards the center of the disc and the center of the milling container, resulting in frictions between the balls, the milling container, and the material. Burgio et al. (1991) originally proposed a theoretical-empirical model based on the kinematic equation of the velocity and accelerations of a ball in the vial of a planetary ball mill. The total power, P, transferred from mill to system during collisions, can be obtained from the kinetic energy (E), which is a function of velocity (V), which is a function of speed (ω) and properties (r) of the vial and disc of the mill.

$$ \left.\begin{array}{c}P=C\times E\\ {}E=\frac{1}{2}\;m{(V)}^2\\ {}V=f\left(r,\omega \right)\end{array}\right\} $$
(3.1)

Abdellaoui and Gaffet (1994) established a mathematical model and claimed a better geometrical description than Burgio’s model and concluded that the end product depends not only on the kinetic energy but also on the shock power. Correspondingly, other authors follow the Burgio’s model which is established based on the kinematics of ball movement inside the vial with few modifications (Magini et al. 1996; Iasonna and Magini 1996; Chattopadhyay et al. 2001; Radune et al. 2014; Ghayour et al. 2016; Lee et al. 2017). Further study was conducted by Abdellaoui and Gaffet (1995, 1996) for planetary ball mills and horizontal rod mill in subsequent years to follow up in detail on their previous work and to compare model results with experimental data, and concluded that the injected shock power ‘f’ is responsible for milling the material to nanoparticles. Chattopadhyay et al. (2001) include a detachment parameter to ensure that the detachment angle does not assume any value for a ball vial impact and calculate the dissipated energy for various conditions of milling. Gusev and Kurlov (2008) deduced an analytical expression to calculate the final size based on the milling parameters and material properties.

Subsequently, other analytical models (Le Brun et al. 1993) calculate the angle of impact of the milling balls on the vial surface but fail to obtain a good correlation between theory and in-situ observations. Alkebro et al. (2002) derived the model based on the Hertzian concept by Maurice and Courtney (1990), to describes the frequency distribution of impact energies and account the energy loss between balls in the vial. Ghayour et al. (2016) studied the effects of the vial to plate spinning rate, ball size distribution and type of balls on the performance of the mill using the Burgio’s model. Most of the literature has not been able to correctly simulate the process involved in ball milling since it is a highly complex process, Rosenkranz et al. (2011) investigated the ball motion in a planetary ball mill with a high-speed video camera to recorded grinding ball motion during the process. The ball motion observed from ball trajectories, contradict earlier theoretical calculations, since previous theories do not account for friction and slip which, occur during ball milling. The model proposes by Burgio et al. (1991) becomes the foundation for most of the latter study of planetary ball mill modeling. These models have been able to reduce the time and cost of synthesizing nanoparticle by providing insight into possible parameter values.

Furthermore, understanding and representation obtained through video recording, or transparent mills can provide in-depth knowledge of the process. Analytical models presented in Table 3.4 could be a valuable tool for synthesizing of nano-fertilizers. The precursor for nano plant nutrients or nano-fertilizers are mainly ceramics and nonmetals viz., C, H, O, N, P, S; which are more brittle and soft as compared to the typical materials or metals. Hence the power required for synthesizing these nano size might be much lesser as compared to typical material. The nonmetal materials such as zeolite and biochar have lower milling time as compared to other typical metals (Table 3.3). Gusev and Kurlov (2008) derived a relationship between the milling parameters and material properties to obtain a nano-sized particle. However, most of the models do not incorporate the material properties; therefore for the synthesis of nanofertilizer; it is necessary to incorporate the material properties of the precursor.

3.4.2 Statistical Modeling

Statistical models including artificial neural network with mathematical equations are presented in Table 3.5. The statistical models cannot deliver the factors of high energy ball milling, but only provide guidelines on the reliability and validate the experimental results. They measure the error and the confidence level of the experimental results. Fixed model, artificial neural network, and regression model are some of the models used for simulating the milling process.

Regression model

$$ {y}_{ij}={\beta}_0+{\beta}_1{x}_1+{\beta}_2{x}_2+{\beta}_{12}{x}_1{x}_2+{\beta}_{11}{x}_1^2+{\beta}_{22}{x}_2^2+\varepsilon $$
(3.2)

Where yijis the response variable including crystallite size/lattice strain, and the mean particle size of nanocomposite powders, x1and x2 are the variables that represent factors/parameters, and ε represents the random error term, β0is the constant of the model, β1 and β2represent the main effects of parameters e.g., milling time milling speed ball to powder ratio and other parameters, β11 and β22represent the square effect of the parameters, and β12 represents the interaction effect of parameters.

The artificial neural network is similar to the biological neuron network. It is one of the most potent modeling techniques, in all fields of sciences, using the statistical approach. It can provide suitable and logical results with acceptable accuracy and faster prediction. Dashtbayazi et al. (2007) used artificial neural network by using two types of neural network architectures, i.e. multi-layer perceptron and radial basis function for modeling the effects of milling parameters of planetary ball mills on the characteristics of aluminium silicon nanocomposite powders; taking into consideration the crystallite size and lattice strain of the aluminum matrix. Regression analysis confirmed that the developed artificial neural network agreed with experimental data. However, Sha (2008) commented on the inappropriate extrapolation of artificial neural network modeling results due to over-emphasis on modeling. So in 2012, Dashtbayazi (2012) study the mechanical alloying process for synthesizing of aluminum silicon nanocomposite powders in a planetary ball mill through artificial neural network and established that low milling speed, low milling time and low ball to powder ratio could produce better nanocomposite structure.

Similarly, other authors (Canakci et al. 2012, 2013b; Hamzaoui et al. 2009; Lemine and Louly 2014; Ma et al. 2009) have used artificial neural network model for studying the process of milling in a planetary ball mills, by modifying the parameters, by providing weighting factor, using different sigmoid functions such as log-sigmoid and hyperbolic tangent sigmoid (Ma et al. 2009). Varol et al. (2013) suggested the artificial neural network model as a powerful tool for modeling of high energy ball milling. Canakci et al. (2012) claim that artificial neural network was successful in predicting the apparent density, particle size and microhardness of the composite powders with a mean percentage error of 4.93%. Dashtbayazi and Shokuhfar (2007) suggested a statistical approach for milling of nanocomposite powders through problem description; identifying the response variables; setting of factors, levels, and range; selection of experimental design; conducting the experiment; statistically analyzing the data and obtaining conclusions. Hou et al. (2007) integrated three methods: Taguchi model, response surface method and genetic algorithm for optimization of milling parameters of a planetary ball mill. The Taguchi method determines the proper working levels of the design factors and analyses the most significant factors of input parameters. Response surface method determines the optimized parameters that produce a maximum or minimum value of the response by developing the first and second order mathematical models. Genetic algorithm approach was applied to optimize the milling parameters using the response function of the response surface method model as the fitness function (Table 3.5). Parameter optimization using Taguchi method showed a good representation of the planetary ball mills process and provided an understanding of the most significant parameters (Su and Hou 2008; Zhang et al. 2008; Canakci et al. 2013b). The statistical models require large initial input data for modeling nanomaterial synthesis efficiently, which involved lots of trial synthesis. Combining and utilizing the results of different studies conducted for a particular material, e.g., zeolite (Charkhi et al. 2010; Mukhtar et al. 2014) can be an option, but since limited studies are available, it is difficult to conclude whether the model is suitable or not for milling parameter optimization.

3.4.3 Numerical Models

Modeling of granular particles and understanding of macroscopic particulate behavior are mostly discussed using the discrete element methods incorporating the particle properties, and interaction forces (Luding 2008). The properties of particles are calculated at every time step by integrating the translational and rotational displacements of Newton’s second law of motion, while the contact forces between particles are calculated using different models, e.g., Kevin model, Maxwell model Dallimore, P.G. McCormick Linear normal contact model (Luding 2008). The expression for translational and rotational motions of a single particle is as follow (Zhao 2017);

$$ {m}_i\frac{d^2}{d{t}^2}\overrightarrow{x_i}={m}_i\overrightarrow{g}+\sum \limits_{Nc}\left(\overrightarrow{f_n}+\overrightarrow{f_t}\right)+\overrightarrow{f_f} $$
(3.3)
$$ {I}_i\frac{d}{dt}\overrightarrow{\omega_i}=\sum \limits_{Nc}\left(\overrightarrow{r_c}\times \overrightarrow{f_t}+\overrightarrow{R_r}\right) $$
(3.4)

Where mi is the mass of a particle i; \( \overrightarrow{x_i} \) is the position of its centroid; \( \overrightarrow{g} \)is the acceleration due to gravity; \( \overrightarrow{f_n} \) and \( \overrightarrow{f_t} \) are the normal and tangential forces exerted among the particles and the wall; Nc is the summation of the contact forces or over all the contacts; \( \overrightarrow{f_f} \)is the interaction forces between fluid and particles; Ii is the moment of inertia about the grain centroid; \( \overrightarrow{\omega_i} \) is the angular velocity;\( \overrightarrow{r_c} \) is the vector from the particle mass centre to the contact point; \( \overrightarrow{R_r} \)is the rolling resistant moment, which inhibits particle rotation over other particles.

Few studies used discrete element method to simulate the planetary ball milling process for the production of nanosized materials are presented in Table 3.6. For obtaining the tangential and normal force, different authors have used different contact models (Ashrafizadeh and Ashrafizaadeh 2012; Dallimore and McCormick 1996; Feng et al. 2004; Kano et al., 2000; Kano and Saito 1998; Mio et al. 2002, 2004a, 2004b; Sato et al. 2010). Kevin model, Modified Kevin model, the Maxwell model, and Elastic/Plastic yield (Dallimore and McCormick 1996) model simulate actual milling impacts. Kano and Saito (1998) simulate the ball movement in a planetary ball mill using the particle element method and established that the rate of size reduction and rate of amorphization increased with a decrease in ball diameter while controlling the milling speed of the mill and also conducted the same study on different types of mills (Kano et al. 2000). When rotation to revolution speed ratio increases there is an increase in the impact energy of balls as calculated from the computer simulation based on discrete element method. Mio et al. (2004b, 2002) conducted another study on a scale-up method using discrete element method and established that the impact energy is proportional to the cube of the vial diameter, the depth of the vial and the revolution radius of the disk, while the scale-up ratio of planetary ball mills is of the power of 4.87. Feng et al. (2004) stated that discrete element method simulation of the planetary ball milling dynamics is a better digital approach as compared to analytical based modeling procedures since it can easily stimulate and investigate any change of operation conditions on the dynamics of the system.

Sato et al. (2010) analyzed the abrasion mechanism of grinding media in a planetary mill using discrete element method simulation and observed the relation between impact energy and wear rate constant suggesting that the wear rate constant might be able to simulate using discrete element method. Ashrafizadeh and Ashrafizaadeh (2012) also study the simulation of planetary ball milling using discrete element method regarding effects on impact energy through the rotational speed of the disk and the ball to powder ratio. The frequency of the impacts, the abrasion of the balls and the dissipated energy, and results indicate the suitability of the method for calculating the improved efficiency of grinding operation regarding the required grinding time and reduced abrasion.

Broseghini et al. (2016a) developed a numerical dynamic-mechanical model and used an MSC ADAMS software to solve the model for planetary ball mill. The study is on the effect of milling parameters: – ball size and number, jar geometry and milling speed on the efficiency of milling in ceramic powders. Broseghini et al. (2016b) developed a new vial design for a planetary ball mill using a similar model as Broseghini et al. (2016a) to simulate the ball milling process. The new design was found to give more uniform and more reliable results. Since planetary ball mill involves a dynamic process, and numerical models are capable of handling large systems of equations and nonlinearities, it can represent the dynamic process, which is often impossible to solve analytically. The different material properties, ball motion, and impact force and other processes involve inside the planetary ball mill can be simulated iteratively to obtain actual milling parameters. Numerical models are considered as the optimal option for synthesizing nanofertilizer since the material properties can be provided as input parameters to synthesized nanofertilizer. The main issues are understanding and simulating the process inside the mill, the effect of force developed inside the mill and representing it mathematically and also obtaining the detailed material properties.

3.5 Discussion

High energy ball milling as one of the possible top-down approach for synthesizing of nano-fertilizers and can counteract the limitations of bottom-up methods; since the method is simple and easy and also suitable for large scale of production (Lam et al. 2000). However, selection of synthesis methods depend on the type of product or the kind of results desired; such as, if uniformity of product is more desired then one can opt for the bottom-up method, but the top-down method is feasible for low cost and high production. As observed in Table 3.3, most of the nanoparticles are produced using planetary ball mill; however, limited studies are available for the synthesis of nano-fertilizers. The production of nano-fertilizers in a planetary ball mill can follow the same procedure as of the commonly used materials but due to the difference in material property, retesting of the methods for a particular material type is required. Increase in milling speed and milling time decreases the material size, however prolonged milling time in some materials causes clustering of the particles (Burmeister and Kwade 2013; Mukhtar et al. 2014) or does not decrease in size any further (Kong et al. 2000; Biyika and Aydind 2014). Milling medium or process control agent aids in the milling of particles by reducing cold welding and also regulating the temperature inside the mill (Kleiner et al. 2005; Canakci et al. 2013a). Wet milling is considered more effective than salt assisted milling and dry milling (Peterson et al. 2012). Increase in a ball to powder ratio decreases the particle size (Zakeri et al. 2012) but introduce impurities (Li et al. 2018) to the end product. Milling parameters such as milling speed, milling time, milling medium and ball to powder ratio of some studies conducted for non-metals or ceramics materials viz., fly-ash (Paul et al. 2007), biochar (Peterson et al. 2012) zeolite (Charkhi et al. 2010) can be used as insights for synthesizing of nanofertilizer.

Modeling the planetary ball milling process is essential because real-time ball milling is time-consuming, inefficient and not practical. The mathematical representation of the process assists the use of models for different types of materials such as bulk fertilizer materials. Modeling tools developed for common materials such as metals can be used for modeling the synthesis of nanofertilizer. Analytical, numerical and statistical models are developed to represent the dynamic process of milling in a planetary ball mill (Tables 3.4, 3.5, and 3.6). Statistical models such as artificial neural network model, Taguchi model, regression model, response surface method and genetic algorithm are suitable methods for optimization of the milling parameters, as they provide the most and least significant parameter details; however, they required a large initial data for reliable optimization. Numerical and analytical models follow the basic principle of centrifugal force and momentum of the mill’s disc and vial to predict the milling parameters regarding energy and equations involving material properties where the properties of bulk fertilizer can be incorporated to obtain desired nano-sized particles of nanofertilizer. However, the analytical model involves simplifying assumptions, due to inherent complexity combine with dynamic, non-linear behavior of the process. While numerical models can be alternative tools to simulate the planetary ball milling process efficiently, it includes multi parameters and complicated processes which are difficult to understand and incorporate mathematically. Most of the models estimate the milling energy or power as the primary output and concluded that with more power/energy more size reduction takes place because the principle of size reduction in high energy ball milling devices depends on the energy imparted to the sample during impacts between the milling media. Higher energy generates more impact and hence lead to more size reduction. Most of the models do not consider the effect of the material or precursor properties; these models have been mostly stimulated for common materials but not for the synthesis of nanofertilizer. Few models have linked the kinematic equation along with the milling energy equations (Gusev and Kurlov 2008; Choi et al. 2001). Unlike other studies whose aim is to achieved amorphous state or some stable state of the materials, the main aim of synthesizing nanofertilizer is to obtain the nanometer size these models can provide the direct relation of the impact of milling energy and material properties to the milling size of the nanofertilizer. Hence the present models need to incorporate and link the material properties with the kinematic equation of planetary ball mill for better results. Analytical and numerical models can be used for synthesizing of nano-fertilizers. An analytical model developed by Gusev and Kurlov (2008) relates the material properties with the planetary ball milling equation, where the particle size is model as the function of milling time, can be a suitable model for synthesizing the nanofertilizer.

3.6 Conclusion

Farmers have been excessively supplying the essential nutrients to plants through the application of inorganic fertilizers to obtain the desired amount of yields. Conversely, the fertilizers applied have low nutrient use efficiency and cause pollution to the environment and affect the health of the soil. An alternative method to increase nutrient use efficiency and reduce the loss of fertilizers is through the use of nano-fertilizer (Dhewa 2015). The biological, physical, and chemical properties of nanoparticles are more enhanced than their bulk material (Dasgupta et al. 2017; Gruère 2012). Synthesis of nano-fertilizers is still in its initial stage with maximum of the methods used are of bottom-up approaches, involving sophisticated, costly instruments and low production. Simplified synthesizing methods of nano-fertilizer are essential to be able to replace conventional fertilizer and minimize the risk of environmental pollution. Nano-fertilizer can supply nutrients efficiently to plants and increase the yield while minimizing nutrient losses and controlling pollution in the environment. Understanding the milling parameters impact on the milling of nanofertilizer is essential to optimize the milling parameters to obtain desired nanosized fertilizer at low cost and less time. This review reveals a few of the laboratory studies conducted on a planetary ball mill for synthesizing of nanoparticles and also the modeling techniques developed to represent its dynamic process; analytically, numerically and statistically. It is evident from the study that planetary ball mills have been used for decades to synthesize nanoparticles and also various types of models have been successfully developed to simulate the milling process. Thus, nano-fertilizer can be economically and timely synthesized, through planetary ball mill with the help of available modeling tools. The numerical model is the best method to simulate the planetary ball milling process and material properties. However it required clear understanding and representation of the process mathematically which is a complicated process and it is not practical. Analytical models are suitable for nanofertilizer synthesis practically as it involves simplification and assumption of the process and can simulate the size up to 6–60% error (Gusev and Kurlov 2008). Since over-simplification of many of the models developed so far; further study conducted on planetary ball mill by using a transparent mill and video recorder to understand the movement and working process of the mill (Rosenkranz et al. 2011) is necessary; also further studies on linking the kinematic equation to the final size of the nanoparticle are required. More focused research is essential on the development and synthesis of nano-fertilizers using the planetary ball mill along with the modeling tools for efficient synthesis of nano plant nutrients. At the present stage of understanding, it can be speculated that the analytical models can provide an idea of the optimized milling parameters while numerical models are also an efficient method, but a better understanding of the planetary ball mill internal process is still required. Hence, models are presently best solution for understanding and simulating significant parameters of planetary ball mills by way of reducing energy, cost and time.