Dynamical aspects of nanoparticle formation by wire explosion process

Copper nanoparticles (NPs) were produced by wire explosion process (WEP) and it was noted that the amount of energy (E) deposited on the wire and the ambient pressure play a major role on the size of particles formed. Dynamic diffusion and condensation processes of NPs formation by WEP were modelled. Calculations of critical size of embryo, activation energy and nucleation rate of the formation of NPs in WEP were made considering classical homogeneous nucleation theory. Decrease in critical size of nuclei and activation energy, increase in nucleation rate with high E (540 J) and low operating pressure (10 kPa) confirm the formation of small size NPs (26 nm). Different cooling rates due to unsymmetrical shape of the vapour cloud has been identified as the cause for generating mixed particle sizes. The qualitative analysis conducted in this work validates the obtained experimental results and can be used as a design tool for industrial apparatus to produce NPs in bulk.


Introduction
Nanotechnology provides a systematic control on the morphology of functional nanomaterials and its unique characteristics have potential applications in catalysis, nano-manufacturing, photonics, sensor technology, environmental remediation etc (Rogers et al 2015). The desired properties of nanomaterials could be achieved by specifying its shape, size, phase and its composition (Aliofkhazraei 2016). Precious metal nanoparticles of different sizes and colours were produced by exploiting the property of quantum confinement (Fatti et al 2000).
Various techniques were employed worldwide to synthesize NPs of controlled morphology and phase. Solid state top down methods including ball milling need a long processing time (Koch 1997). Many chemical methods like sol-gel, hydrothermal etc require a number of steps and precursors to obtain pure NPs (Yu et al 2008). Gas/vapour phase methods which were based on the condensation of vapour of materials to get its NPs, e.g. thermal plasma torch, pulsed laser deposition, arc plasma synthesis etc (Kruis et al 1998) has low energy conversion. The principle of wire explosion process (WEP) is based on the pulsed current injection through the conducting wire (mostly metal) to vaporise it after the joule heating (Kotov 2003). It utilizes the wide difference in temperature and density of the vapour and that of the ambient fluid which acts as the cooling medium. The process of solidification of supersaturated vapour in proper inert atmosphere form the metallic nanoparticles (Sindhu et al 2007).
Tokoi et al 2013 used high speed photography to calculate the density (Den Vap ) of the vapour/plasma of Copper (Cu) produced after the explosion and found that for a wire with constant mass (M Wire ), its volume of vapour cloud, Vol Vap increases with an increase in deposited energy and/or a decrease in pressure, P; which leads to decrease in Den Vap . Shikoda et al 2009 calculated the expanding diameter, d Vap and the temperature (T) profile of Cu vapour for different P and found that d Vap and the cooling speed of the vapour increase with decrease in P. Thermodynamic calculation of nucleation rate and the activation energy using the classical Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. homogeneous nucleation theory for the formation of embryo in WEP also leads to the same conclusion correlating it with high temperature and supersaturated vapour (Ranjan et al 2017).
In the preparation methods of NPs from vapour/plasma of metal, one of the possible steps before the condensation is the diffusion/effusion of the metal vapour into ambience. Bora et al 2014 used ICCD (intensified charge coupled device) images to see the diffusion of Cu vapour during the heating process in WEP and explained the partial reheating of the vapour/plasma during arc discharge. Recently, Zhao et al 2017 described the WEP using the diffusion of generated Aluminium plasma into Argon ambience. Latest work published by Han et al 2020 describes the quenching of Cu vapour in WEP as diffusion of discharge channel and the ambient. These works talk about the basic physics of WEP but the discussion on the diffusion for dynamics of particle formation is missing. The speed of diffusion depends on the concentration/Den Vap , temperature gradient and pressure difference of the two gases (vapour/plasma of metal and the ambient). The kinetics would lead to the different morphology of formed NPs as shown by Simchi et al 2007 for Silver and Cu-Tin alloy in vapour phase condensation process. Hence, it is necessary to have a qualitative evaluation of the manufacturing process to design the apparatus for large scale industrial production of NPs. However, theoretical computation for the kinetics of NPs formation in WEP has not been investigated.
Based on the known facts, the present study aims to calculate through Fick's diffusion, the kinetics and concentration profile of the vapour/plasma generated in WEP as a function of time with the radial distance of the wire vapour. The process of NPs formation in WEP with different energy and P are discussed. Deposited energy to the wire was calculated and expansion of vapour was observed with high speed camera. SEM was used to see the morphology. The critical embryo size and its activation energy were calculated as a function of temperature and saturation ratio. The variation in nucleation rate considering the homogeneous nucleation in vapour phase is elaborated in the present work. Figure 1 shows the schematic of WEP experimental setup. In the present study, nano Cu powder is produced in Ar ambience. The specifications of Cu conductor used in the present study is provided in table 1. Charging circuit consists of a high voltage DC supply which charges the capacitor bank of total capacitance, Cap to the required voltage, V c . Energy stored in Cap (E Cap =0.5CapV c 2 ), is discharged through the wire using gap switch and its controller leading to joule heating of the wire. E Cap is always greater than the vaporisation energy, E Vap , which vaporises the wire. Ratio of E Cap and E Vap is defined as the relative energy, or reheating factor or energy ratio/factor, K. Formed NPs were collected on the membrane filter (Millipore, 0.1 μm pore size) placed between the chamber and the vacuum pump.

Experimental details
The voltages at each electrode, V 1 and V 2 ; and current, I during the explosion were measured using two voltage probes (Tektronix P6015A) and a current transformer (C.T., Pearson 101) respectively as shown in figure 1. Voltage outside the chamber, V as shown in figure 2, V=V 1 −V 2 . The deposited energy on the wire during WEP, E Wire is the time integral of instantaneous power (V Wire * I) and is expressed as equation (1), and V R (R C * I) and V L (L * dI/dt) are voltage drops across the circuit resistance (sum of resistance of electrodes and the wire), R C and inductance L respectively. The values of R C and L were measured experimentally to be 1 mΩ and 250 nH respectively.
The explosion of wire followed by formation of Cu vapour cloud and its interaction with ambient gas during WEP were observed using a high speed camera (Photron Fastcam SA4) seeing through the quartz window of the chamber. In the present study, the explosion process was captured at 150,000 frames per second. A scanning electron microscope (SEM; FEI Quanta FEG 200) was used to observe the morphology of WEP synthesized NPs. Sizes (Heywood diameter, diameter of circle with area equal of 2D projection of 3D object) of about 500 NPs were measured using ImageJ software which follows log normal distribution as per equations (3) where, f (d) represents the log-normal distribution, d and D 50 are the particle and geometric mean diameter, respectively, n i and d i are the number of particles and its diameter respectively, and σ g is the geometrical standard deviation.

Results and analysis
In this section, we will go through the energy deposition to the wire and effect of different pressure and energy on the vapour cloud expansion and particle size. Calculation of vapour concentration profile will be carried out considering its effusion in the ambience. Finally, calculated activation energy and nucleation rate will be correlated qualitatively with vapour concentration and different size NPs produced. 10 (P10) 100 (P100) Figure 2. Electric discharge circuit inside the explosion chamber.

Energy deposition in WEP
Energy deposited to the wire during WEP was calculated using equation (1), and plotted in figure 3 along with corresponding voltages and currents. First jump in voltage is due to sudden application of voltage across the wire, which causes the wire to melt and the point of evaporation happens at the peak of the voltage curve. Oscillations in current and voltage waveforms are due to inductance being present in the circuit (Ranjan et al 2019). There is no dwell time observed in the present work. Amplitude of the voltage peak increases and instant of its occurrence decreases with increase in K because of the high energy deposition. Total and rate of deposition of energy to the wire also increases with an increase in K. Rate of deposition increases due to fast melting of the wire. There are two parts of energy deposition: (a) wire heating occurs before evaporation and (b) arc discharge takes place after evaporation. Most of the energy is deposited in the process of arc discharge as evident from the literature (Ranjan et al 2019). The formation and morphology of vapour cloud from the wire depends on the energy deposited during arc discharge. Figure 4 shows the high-speed photographs of vapour cloud during WEP. The area of the light emission (volume of the Cu vapour cloud) increases with increase in K and/or decrease in P. For increasing K, the area increase is attributed to higher energy deposition, whilst the change in area due to decreasing P is because of the fact that, vapour cloud expands more and expansion rate will be faster if the pressure gradient is higher. Luminescence time decreases with decreasing P for K3 due to fast expansion i.e. volume is more which leads to less dense vapour/plasma cloud subsequently resulting in faster cooling of vapour and yielding smaller dimensioned NPs. Vapour cloud expands (figure 4) even after the instant of voltage peak (less than 7 μs) and current peak (less than 10 μs) as shown in figure 3. 3.3. SEM and particle size distribution studies Figure 5 shows the SEM micrographs of NPs produced at different K and P values. Shape of the NPs is spherical in all cases in the present work. Size distribution follows log normal distribution as per equations (3)-(5). Figure 6 shows the particle size distribution of NPs produced by WEP and   is due to less dense vapour cloud and fast cooling when wire is subjected to high K in low pressure ambient as discussed in section 4.2. . Fick's second law (Crank 1975) was used to describe the kinetics and profile of effusion of metal vapour into inert ambience. In WEP, on deposition of energy to the conductor, joule heating occurs followed with formation of highly conductive plasma/vapour. High speed photographs measuring the expansion diameter of the vapour cloud was reported by Tokoi et al 2013. The first instant of the maximum expansion (formation of the metal vapour cloud) is the first step of interest (time, t=0) for our problem. After this, effusion takes place due to high difference in the density of metal vapour, which is transient and varies with radial distance. Fick's second law is chosen to model the transport of the metal vapour. Figure 7 shows the schematic representation of expanded vapour/plasma, which is considered as sphere of radius R and r denotes the radial distance from the centre of the sphere.

High speed imaging
For spherical geometry, Fick's law can be written as (Crank 1975) as equation (6), where, r is the radial distance from the centre/core of the vapour cloud, C is the concentration of vapour cloud and D is the diffusion coefficient, which is defined using equation (7) as follows for binary gases A and B (Hirschfelder et al 1964); where, M is molar mass of gas in kg/mol, n is the number density, T is the temperature in K, k B is Boltzmann's constant (1.3807×10 −23 J K −1 ), Ω AB is a collision integral for the interaction between species A and B.  Formation of nanoparticles depends on the diffusion of vapour into ambient which in turn is a function of concentration of the metal vapour generated. Hence, it is quite imperative to have the concentration profile of the vapour as function of t and T. The solutions of equation (6) with D=0.008 m 2 s −1 (Murphy 1996), initial and boundary conditions as given by equations (8) and (9) respectively, were obtained numerically. Initial conditions are the concentration, C(r, t=0) defined from equation (8) for different value of capacitor energy and ambient pressure as provided in table 3 (calculated through equations (11) and (12)).

= =
Initial condition, C r, t 0 constant value from Table 3 8 Boundary conditions: C r 0, t constant value from Table 3 and C r R, 0 9 The diameter of the vapour cloud was calculated using equation (12) for different values of P and E C keeping M Wire constant. From table 3, it is observed that as E C increases and/or P decreases, C decreases. All calculations were done for Cu vapour.
The concentration of the vapour can be defined using mass density of the vapour (Den Vap_Expt ) experimentally as the ratio of the mass of wire and the volume of expansion of vapour, Vol Vap_Expt : Den Vap_Expt =M Wire /Vol Vap_Expt where, Vol Vap_Expt =4πR 3 /3 and R=0.5d Vap . As reported in previous literature (Kotov 2003, Shikoda et al 2009, Tokoi et al 2013, Han et al 2017 on WEP, the expansion of vapour depends on P and K. K plays the role of increasing the temperature as the heating of the wire in WEP is so fast and energy efficient that it gets converted completely to thermal energy if we neglect other losses of energy. Expansion of the vapour/plasma is volumetric (Tokoi et al 2013) and it increases as P decreases, so with ideal gas law, we can define the relationship of theoretical volume of vapour cloud, Vol Vap_Ther :

Vol _ E P 10
Vap Ther C 1 where E C =0.5Cap V , C 2 Cap and V C are the capacitance and the charging voltage of the capacitor, respectively. The number of particles in vapour cloud is not constant due to simultaneous nucleation and NPs formation. In the present work, ideal gas law was adopted to provide qualitative understanding on the impact of different parameters like cooling rate, ambient pressure and energy on NPs formation dynamics.
We can define theoretical vapour density as,   Table 3. Value of C(r, t=0) in g m −3 and diameter of vapour cloud for different E C and P. P (kPa)

10
50 100  Figure 8 shows variation in C with time (the variation for total time period of t=10 ms) and radial distance (maximum expansion depends on the value of E C and P). It is observed that C gets reduced with increase in K and/or decrease in P. Cooling rate of the vapour relies on magnitude of diffusion coefficient. The concentration profile of vapour/plasma also depends on the value of diffusion coefficient D between the metal vapour produced and the ambience gas. In addition, the diffusion coefficient is a function of temperature and it varies much in the range of 7-8 kiloKelvin, the ionisation of the vapour/gas starts which has their own collision integral value (Murphy 1996). In WEP, the local T is as high as 10 kiloKelvin and increases with increase in E C (Tokoi et al 2013). With increase in D, the diffusion is faster and the vapour/plasma gets depleted faster with time and for all the radial distances as observed from figures 8(c), (d). It is due to the more penetration of ambience in the metal vapour/plasma produced due to faster cooling.

Nucleation and condensation of the vapour for NPs formation
Wide difference in T and C of metal vapour/plasma and ambience leads to diffusion and consequently to the formation of stable nuclei (embryo). Embryo of equal to or more than the critical size (stable) nuclei will lead to the growth otherwise it gets dissolved again to vapour. The C at which the nucleation starts is C at time, t m at which temperature of the vapour/plasma reaches its boiling point, which was experimentally calculated as about 0.1-0.2 ms for a particular energy and different P (Shikoda et al 2009). Cooling rate is higher in case of lower P. C is calculated at t m through equation (6). Vapour pressure of metal increases tremendously if heated to high temperature. CRC databook (Haynes 2014) provides the vapour pressure of metals as a function of temperature, T. equation (13) shows the corresponding equation for pressure of Cu (P Cu ). The vapour pressure of Cu is plotted in figure 9 which increases with increase in T. Increase in partial pressure of the vapour leads to the super-saturation which drives the vapour for the phase change. The phase transformation from vapour to solid takes place through the formation of stable nuclei. Critical size of the nucleus, r C on which growth occurs is given as equation (14) as (Wyslouzil and Wolk 2016), where, s Metal is the surface tension of solid vapour interface, v Metal is the molar volume of the molten metal, m D is chemical potential given by -RTln(S), T is temperature and S is saturation ratio. Activation energy (DG rc ) for the formation of r C in classical homogeneous nucleation is given as (Wyslouzil and Wolk 2016), Variation of critical size embryo and corresponding activation energy for Cu vapour (for calculations, all the values were taken from Haynes 2014) is shown in figure 10 as a function of S and T. With the increase in S and/or T, there is a decrease in the size of r C and the corresponding activation energy. Increase in saturation ratio for WEP is not easily conceivable. It can be visualised through the high speed photographs taken during WEP illustrated in figure 4, where it takes more time to quench the vapour cloud in the case of higher P. For P= 10 kPa and K=51.9 (Tokoi et al 2008), the gradient has increased two folds than that of the 100 kPa case. Similarly, for P=10 kPa and K=1 and 3 as illustrated in the present work. Higher temperature gradient can be correlated to higher saturation ratio achieved in WEP. Even as the S value is doubled, it will leads to lower r C and consequently results in low-dimension NPs.
Homogeneous nucleation is considered in WEP and is assumed that the formation of the NPs occurs from the gas phase. The expression for the nucleation rate is given by equation (16),  Figure 9. Partial pressure of Cu as a function of temperature.
where, k B is Boltzmann constant and P Total is the sum of pressure in the ambience/explosion chamber and vapour pressure of metal (given by equation (13)). Figure 11 shows the nucleation rate for Cu embryo varying with S and T. It is observed that nucleation rate is always high for all values of S at higher temperatures. With increase in S also, it leads to higher nucleation rate for any temperature. The nucleation and growth of the embryo after the diffusion of the vapour/cloud of the metal in ambience will lead to the formation of NPs. Particle size distribution studies clearly indicate that the particles formed by WEP are of mixed size and it follows log normal distribution. The formation of mixed size NPs is due to the generation of different size embryos, as the temperature and/or saturation ratio of the vapour/plasma generated is not the same in the reaction chamber due to non-uniform heating of the wire and non-uniform energy deposition during arc discharge in the experimental conditions. As the cooling rate is not the same on the boundary of the vapour cloud, the WEP yields mixed size NPs under different experimental conditions. However, smaller sized NPs are obtained for high K and/or low P scenario due to the different diffusion and cooling rate, faster nucleation and slower growth of the small-sized embryos.

Conclusions
A detailed study on the formation mechanism of metallic nanoparticles in wire explosion process was made. Diffusion of metal vapour/plasma of spherical geometry formed after the explosion was calculated by Fick's second law. The following conclusions can be drawn based on the present study:  • Higher the value of deposited energy and/or lower the value of ambient pressure (P), concentration of vapour will be low due to more vapour expansion which leads to formation of small sized NPs. It also shows that diffusion is faster for higher cooling rate which can be achieved with ambient of appropriate thermal conductivity.
• Critical sizes and activation energy of embryo were lower and the nucleation rate was higher for high S and T values.
• In WEP, higher T and S values were achieved by increasing energy ratio, K and/or decreasing P. So, low concentrated vapour/plasma, higher cooling rate, high T and S for high K and low P leads to small stable embryo, low activation energy and fast nucleation rate; forming smaller nanoparticles.
• Difference in cooling rate due to non-symmetrical geometry of the metal vapour/plasma, different concentration gradient in experimental conditions of WEP leads to mixed size nanoparticle (narrow size distribution, skewed to low dimension NPs) in WEP as observed in the present work.