Since the 1970s microwave rapid synthesis of organic materials (Gedye et al. (1988) [1] , Gedye et al. (1991) [2] ) and inorganic materials (Rinaldi et al. (2015) [3] ) has become a well-established chemical synthesis route. The emergence of Green Chemistry with its twelve principles has embedded itself within this microwave synthesis approach (Grewal et al. (2013) [4] . In the case of metal precursor and reactants within a colloid form transition-metal nanostructure products, the synthesis process feeds into the linear economy model of nanoscience and nanotechnology (i.e., energy, food, sensor, medicine, and medical-diagnostics (Priecel and Lopez-Sanchezl (2018) [5] . It is generally thought that the microwave-assisted reaction pathways within the colloid involve both dielectric volume heating and ionic conduction within the precursors and reactants and possible resistive heating of the metal atoms themselves within the nucleation, growth, and subsequent capping stages. An understanding of these selective heating pathways in terms of the sixth principle of Green Chemistry (reduction in energy usage and an increase in energy efficiency that are recognized for environmental and economic impact) and its implementation through process intensification (minimization of reactor size, conversion from batch to continuous flow and the utilization of microwave technology) is of growing interest and importance.

In 2022 it was shown that for microwave-assisted processes operating at a running frequency f_o = 2.45 ± 0.05 GHz (~12.2 cm), energy phase-space projection that maps the process energy budget (kJ) and energy density (kJ∙m⁻¹) can be used to define the sixth principle (Law and Dowling (2022) [6] ). In the following year (2023), the energy mapping technique was applied to historical “non-thermal” microwave-assisted microorganism inactivation experiments and their reconstruction using pseudo microorganism suspension, in this case, the eco-friendly and benign solvent “water” (H₂O) (Law and Dowling (2023) [7] ). More recently it has been shown that using log-log coordinates, historical microwave-assisted synthesis of gold (Au), and silver (Ag) nanostructure data (n = 30) may be visualized (Law and Dowling (2023) [8] ) with the addition of fitting in a general power-law function (Equation (1)) where small and large events are common over four orders of magnitude in both x- and y-axes.

y = c x n (1)

where y is the parameter of interest, c and n are the coefficient and the exponent of the equation. In this case, without recourse binning or Pareto cumulative distribution function (Hanel et al. (2017) [9] ), a fit of y = 0.3293x^0.846 is obtained. The goodness of fit obtained through linear regression (R²) = 0.7923, however it is not used as an indicator of accuracy Figure 1.

In 1936, J.S. Huxley and G. Teissier [10] first coined the word Allometry for the measuring relative growth in biological systems, since then allometric scaling has expanding in to the scientific of physiology and bibilometic studies (W.A. Calder. III (1981) [11] , Stumpf and Porter (2012) [12] , Dong et al. (2017) [13] ).

In all cases the exponent defines the type of scaling relationship between y and x; < 1 means the energy transfer increases slower (negative, or, hypoallomerty) than the applied process energy increase; an exponent = 1 would mean that relationship is isometry; and if the exponent > 1 then the energy transfer increase faster (positive, or hyperallometry) than the applied process energy increase. Thus, the mathematical response model (y = 0.3293x^0.846, R² = 1) implies that the energy transferred to the suspension volume is subcritically proportional to the process energy budget and is Scale Free throughout the process energy budget range therefore preserving some underlying invariant microwave-assisted synthesis.

From a statistics and probability perspective the power-law signature is a 0-dimensional (0-D) representation of the underlying microwave synthesis processes. However, the power-law signature encompasses different microwave applicators-types used (domestic microwave oven, temperature controlled microwave chemistry (TCMC) applicator, digestion applicator, and the axial field helical antenna applicator (Discover2), metal precursors, and synthesis chemistries [8] . Therefore it is reasonable to postulate that the power-law signature may contain phase-transitions (PT) arising from the applicator-type and the mode that it is used (batch, or, continuous flow) or the way the microwave power is applied. For example, commercial Digestion applicators use proprietary software to vary the pulse width modulation (PWM) of the cavity-magnetron voltage. In practices this requires the cavity-magnetron instantaneous or continuous wave (CW, 100% duty cycle) to be applied in the temperature ramp-up period, and < 100% duty cycle in the temperature hold period. The lack of reporting of ramp-up period and in the hold period power [14] [15] [16] [17] [18] has lead to the use of an estimated 75% of the cavity-magnetron CW rated power when calculating the process energy budget (power multiplied by process time) [8] .

The uncertainty in the actual value of process energy budget raises a number of questions. First, are PTs present, and second are outliers’ presents in original database. Both PT and outliers are problematic as they may be natural deviations in the database, or due to poor recording and reporting of experimental conditions.

Given these questions, the aim of this work is threefold. First to identify and classify PTs within the original microwave-assisted synthesis database (n = 30) [8] . This database is now referred to as Database A. Second to apply the same tests to a larger database (n = 50) to establish if PTs are an artifact of sample size. This new database is referred to as Database B (n = 30 + 20 = 50), where the additional data pertains to microwave-assisted synthesis of palladium (Pd) and platinum (Pt), and zinc oxide (ZnO) nanostructures, plus an additional microwave applicator (ERTEC, Wrocław, Poland). The transition-metals Pd and Pt are chosen as they are in the same Group (V and VI) and electron orbital filling (10 and 11) as Ag and Au, hence they would be expected to have similar energy phase-space properties. To extend this hypothesis, Zn is chosen as it is a close neighbor (Group IV with 12 electrons in its d-orbital). Zinc oxide is also chosen as it is of great interest due to its n-type wide band gap (~3.37 eV at room temperature) semiconductor property [19] . Figure 2 shows the relative d-block position and atomic number for these five transition metals within the modern Periodic Table. Third, two application are compared directly for real comparison purposes. The applicator is the Samsung microwave oven (CE2877) that is used for the synthesis of Pt [20] and Ag nanostructures [21] , and the Discover2 applicator used in both batch and continuous and mode.

The building of Database B (n = 50) follows the same procedure as Database A (n = 30) as set out in [8] : that is the data extracted from each new published paper is represented by a new data entry into an Excel spread sheet. The data includes: first author and year of publication, applicator-type, metal precursor, Claimed Green Chemistry, Non-Green Chemistry, calculated process energy budget, and process energy density. To calculate the process energy budget and

process energy density, each paper must provide information on the estimation of microwave power, microwave irradiation time, and reactant quantities. A total of thirty new publications pertaining to microwave-assisted synthesis of Pd, Pt, and ZnO nanostructures were investigated, of which only twenty fitted the criteria. The ten papers not used [22] - [31] were published between 2004 and 2023 thus indicting that full power conditions are still not being reported. Secondary data, temperature, and working pressure were recorded. Where the primary and second data was ambiguous in the publication text, private communication with the publication authors were sought and is given in the text. In addition, on-line microwave applicator manufactures manuals are used to refine the information within the accepted twenty publications.

The remainder of this work is constructed as follows. Section 2 details the analysis methodology used. The test results for Database A are given in Section 3. The building of Dataset B and test results is given in Section 4. Section 5 provides an estimation of the global (average of Database, A, B and B_adjusted) microwave-assisted synthesis power law. Section 6 is a summary and outlook of this work. To add the reader, only one of the applicators is operated in the continuous flow (Discover2) mode is heighted in the text, the remaining applicators are operated in the batch mode. In addition, full details of Database A may be found in [8] .

It is generally regarded that classical linear regression analysis is based on the assumption that the dataset under investigation has a normal distribution (bell-shaped) where the sum of data points either side of the arithmetic mean is equal zero (Σ = 0). In the real-world, however, outliers distort the normal distribution, resulting in the summation of all data points either side of the arithmetic mean not equal to zero (Σ ¹ 0) in most cases. Moreover where a dataset exhibits a power-law signature the distribution is non-linear (for example a Pareto distribution that contains a long and flat tail), meaning that summation of the data points will never equate to zero. Indeed it is well documented that outliers distort both group variance and group arithmetic mean [9] [10] [11] , and Andriani and McKelvey (2009) [32] , particularly when attempting to verify a power-law over a limited magnitude range (< 2 to 2.5; in both x and y axes) due to either limited sample size, or the complex nature of physiological, chemical, and geophysics datasets. For example in the plasma etching of semiconductors it s known that competing reactive ion etching and decomposition processes limit the power-law magnitude range < 2 [33] [34] . Whereas in the field of high density pulse laser ablation, where energy transfer processes dominate, rather than physiological and chemical processes, it is found that the propulsion velocity of tin microdroplets has a power-law dependency over 3 orders of magnitude with laser power [35] . Simulations performed with the RALEF-2D radiation-hydrodynamic code indicate the magnitude is possibly limited by the applied laser pulse width (10 nanoseconds). Lastly, bibilometric studies within the scientific fields of physics, mathematics and econometric have produced a scale range of 8 to 10 orders of magnitude [13] . Hence a linear regression analysis approach to experimental datasets exhibiting a power-law signature is nontrivial as complex processes within the dataset has the potential to influence variance, and produce an unstable arithmetic mean.

Acknowledging this complexity, this study attempts to identify and verify deterministic features within the power-law signature of Database A and the enlarged sample size of Database B. To this end Microsoft Excel spread sheet software is used to collate and display the datasets within a XY scatter plot, flowered by fitting a linear regression power function trend-line to the data. The trend-line is then modeled using the Excel power function (=POWER (A1, 2, 3, $Y$1) * $Z$2), where A1, 2, and 3 are the cell coordinates, and the cells $Y$1 and $Z$2 are the values of the c and n constants of Equation (1). This power function, straighten-out the power function into a horizontal 0-D model identity-line that represents the process energy density prediction at each process energy budget value (Figure 3).

Two evaluation tests are used to detect PT and outliers from the characteristics of normal objects within each database. The tests are as follows:

1) 1-dimesitional (1-D) test is made on the disagreement, or difference, between the observed process energy density values and the 0-D model data points. The tests are performed on Database A, and B in high-dimensional space of a residual plot, where the vertical axes represent the observed difference and the horizontal axes represents the 0-D model identity-line. The disagreement (± from the identity line) is quantified using the ± 3 Sigma rule of thumb (68%, 95%, and 99.7%) where data points close to, or beyond, the ± 3 Sigma limit are

classed as problematic outliers and require further examination as outliers strongly depends on a detailed understanding of their experimental origins, see Sections 3, 4, and 5.

2) 2-demesional (2-D) non-parametric cluster analysis as described by Tynan et al. (2009) [36] , and Law et al. (2010) [37] is used to quantify group residual errors. The basic assumption used here is that outliers are few in number and their x, y coordinate distance from a cluster is some multiples of the normal distance between data points (as measured by the metric) within a cluster. However, in the LabVIEW Vision builder software was used to threshold the Microsoft Excel data points into cluster and separate out outliers using their x, y pixel connectivity on a graph 2-D plane. This was then followed by computing the information-based entropy of the cluster. In this work, the 2-D parametric cluster analysis using three metrics is performed on the Microsoft Excel data points presented in a high-dimensional of the residual plot, but this time the data points are pre-assigned into N groups (applicator-type, water-MO, NS-metal precursor, claimed Green Chemistry, and non-Green Chemistry).

The information gain from these tests along with detailed knowledge of the PT and outliers origins is used to decide if the PT and outliers are part of the data natural variation, or are due to errors in the reporting of experiment data. In the latter case this problematic outliers are cleaned (filtered) to produce an adjusted Database B (B_adjusted). The power-law signatures of Database A and B and B_adjusted are compared and global a microwave-assisted synthesis power-law is calculated.

Figure 4 shows the residual plot for Database A. In this plot the data points are colored coded into five applicator-types (water-MO, NS-Discover2, NS-MO, NS-TCMC, and NS-digestion, MO refers to microwave oven and NS refers to nanostructures). Visually it is observed that the data points have a non-null, distribution around the model identity-line. For the water-MO (five open circles) and the NS-Discover2 that is operating in the continuous mode (one yellow circle) all exhibit a similar disagreement value from the model identity-line. Whereas, the NS-MO (thirteen green diamonds) and the NS-TCMC (five blue triangles) form clusters around the model identity-line, with the TCMC-102 applicator [38] having the greatest negative disagreement (1440 kJ, -46.2 kJ∙ml⁻¹). It is worth noting that the remaining four tightly grouped applicators are Shikoku Keisoku SMW-064 applicators synthesizing Ag nanostructures. Lastly the Digestion applicators (six red squares) generate an open cluster with the MARS-6 applicator synthesizing Au nanostructures [18] having the greatest disagreement (+90.5 kJ∙ml⁻¹) with the model identity-line. The MARS-5 applicator [14] forms the second greatest disagreement (+50 a kJ∙ml⁻¹). The remaining four applicators (Aton Paar Multiwave, Aton Parr Pro, and two μSnth form an open cluster in the range of 280 to 360 kJ, +1 to +34 kJ kJ∙ml⁻¹.

Figure 5 shows a residual plot of Database A, but this time color coded in to three groups (water-MO (five open circles), NS-Au (fourteen yellow circles) and NS-Ag (blue circles). The notation MO, and NS have the same meaning as in Figure 4. Visually it is observed that the thermal microwave-assisted processing of water-MO group has the same disagreement values close to the model identity-line. The NS-Au and NS-Ag synthesis groups are spread either side of the model identity-line with two of NS-Ag cluster having the greatest positive disagreement (+90.59 kJ∙ml⁻¹) for the MARS-6 applicator, and the TCMC-102 applicator having greatest negative disagreement (−46.2 kJ∙ml⁻¹).

Figure 6 shows a residual plot for Database A, the data is divided into the

following: claimed Green Chemistry (green circles) and non-Green Chemistry (red squares). Water is considered as a universal polar solvent that is nontoxic and environmentally benign is included in the Green Chemistry group. Visually it is observed that the non-Green Chemistry data is spread along the model indentify-line up to approximately 200 kJ process energy budget, from where the disagreement diverges strongly to −46.2 kJ∙ml⁻¹ kJ for the TCMC-102, and up to +90.5 kJ∙ml⁻¹ for the MARS-6 applicator.

A visual comparison of the residual error data points in Figures 4-6 suggests two possible PTs. The first (PT₁) is between the water-MO plus Discover2 (continuous flow) groups and NS-MO applicator-type group. The second PT₂ is located between the NS-TCMC and NS-Digestion group boundaries.

From a Sigma test point of view, Table 1, reveals twenty six of the data points fall within ± sigma, three within the ± 2 to ± 3 Sigma limit and one at the ± 3 Sigma limit indicating that the power-law residual error has a uniform distribution with the problematic MARS-6 applicator outlier at the ± 3 Sigma limit. Further consideration of PT₁ and PT₂, and the problematic outlier is given in Section 4.

For completeness this section provides a description of the reactants, microwave

power, and process time given in the incorporated eighteen publications within Database B. No attempt is given here to quality the quality of the nanostructure product outcome.

Euclidean and Manhattan distance function are two of the commonly used metric to define a cluster of data points and identify problematic outlier [57] [58] and [59] . In this section a simple method of cluster analysis is proposed that uses the Excel spread sheet format (row and columns) populated by data points, and where formula functions (=SUM (MAX, MIN, AVERAGE, and STDEV)) are used to calculate the following three cluster metrics;

1) Energy area (kJ × kJ∙ml⁻¹ = kJ²∙ml⁻¹)

2) Group (N) density (energy area divided by the number data points, n)

3) Group (N) x, y arithmetic mean

As PT₁ has been shown to diminish with sample size, only Database B is explored for problematic outliers using the three cluster metrics. Figure 11 shows the results of three cluster metrics as applied to the Database B residual error plot. The six semi-transparent (60%) color areas (clear, yellow, green, blue black and red are the applicator type) represent the respective applicator-type group energy areas and the bold plus signs denote the arithmetic mean of each group area. Also shown is the +3 sigma line and region of possible PT₂. The metrics numeric values are given are given in Table 3.

The first feature of note within Figure 11, as with Figure 7, Figure 8, and Figure 10, is the water-MO group separated from the other 5 applicator-type groups along the model identity-line. The second feature of note is that the data point falls within ± sigma. Now consider the new NS-ERTEC applicator area and arithmetic mean data, both of which contaminate the NS-TCMC applicator group and the NS-Digestion application group in the 300 to 900 kJ range. This mixing of datasets and the possible PT₂ location where the NS-Digestion applicator group exhibits a marked increase in residual error suggest two scenarios. First the underlying energy transfers mechanism (first-order to second-order reaction kinetics) is changing between applicator-types, or, poor reporting of the real Digestion applicator energy profile that is due to a lack of knowledge of how the preparatory software controls the Digestion applicator cavity-magnetron.

Now consider the NS-Digestion applicator-type group that contains the

problematic MARS-5 applicator. Before adjustment the group has a area density of 101,485.5 kJ²∙ml⁻¹ per data point, when the group is adjusted by removing the problematic MARS-5 applicator data point, the area deceases by factor 2.15 to 47,178.9 kJ²∙ml⁻¹ per data point. The corresponding arithmetic mean also moves from x = 1172.1 kJ, y = +66.1 kJ∙ml⁻¹ to x = 876.1 kJ, y = +36.4 kJ∙ml⁻¹. In this cleaning/filtering process, the filtered MARS-5 applicator now has a notional area density of 481,636.6 kJ²∙ml⁻¹ per data point, an increase in area of ten times per data point.

Assuming that the microwave-assisted synthesis power-laws are Scale Free, an estimation of allometric scaling is now performed Table 4. In this table, the average of the three databases are computed and yield an allomteric scaling of 0.7172 ± 0.3214 kJ∙ml⁻¹ at x = 0.001 kJ, and exponent = 0.791 ± 0.055, or approximately 3/4. This computed average value for the exponent has a hypoallometric trait that falls within close to the bibliometric scaling laws for scientific fields (0.87 to 1.01 [13] and within reported physiological (relative body mass)

scaling laws (0.7 to 0.92 [10] [11] ).

This paper is the second in the series of microwave-assisted synthesis power-law signature studies over 4 order of magnitude relating to process energy budget (kJ) relationship and energy density (kJ∙ml⁻¹). Two historical databases are investigated, Database A (n = 30) relating Au and Ag nanostructure synthesis [8] , Database B (n = 50) containing n = 20 new datasets relating Pd, Pt and ZnO nanostructures. Microsoft Excel Linear regression analysis is applied to Database A and B to generate power-law 0-D model (y = cxⁿ) to provide a horizontal 0-D model identity-line from which the residual error (measured in ± kJ×ml⁻¹) is obtained. The residual error is explored in null and rank plot format as a function six microwave applicator–types (domestic microwave oven, TCMC oven, Digester, axial field helical antenna (disover2, and ERTEC). In addition residual error analysis is performed on the five metal precursors, along with their claimed Green Chemistry and non-Green Chemistry.

The findings of this study are summed as follows:

1) The original Au and Ag database (n = 30) [8] and new database containing Pd, Pt and ZnO (n = 50) and (adjusted Database (n = 49) yield a global microwave-assisted synthesis 0-D model over four order of magnitude with constants of c = 0.7172 ± 0.3214 kJ∙ml⁻¹ at x-axes = 0.001 kJ, with an exponent = 0.791 ± 0.055, approximately 3/4.

2) It is noted the constant c within the 0-D model is a mathematical construct that does not exist in reality due to the warm-up time of the cavity-magnetron (approximately 3 seconds [7] ). Thus for a cavity-magnetron rated at 800 W operating in CW mode, the estimated process energy budget lower limit (x_min) equates to 800 multiplied by 3 = 2.4 kJ. This is an important observation when allometric scaling principles are employed as c has generally been ill defined [10] [11] [12] and [13] .

3) Phase-transitions are observed to be reduced with sample number (30 to 50) to leave one permanent PT₂ that is aligned to excess positive process energy density residual error valves associated with the Digestion applicator-type group. This group of applicator have inbuilt software to control temperature and pressure targets. But their operation has lead to poor reporting of the applied power and energy profile over process time, from which it is thought an error in the initial energy calculation, carries though to the observed residual error. The high positive residual error values associated with the loaded turntable and its sealed reaction vessels where cumulated heat produces a second-order effect.

4) Statistical analysis and cluster analysis is used to detect outliers. This knowledge is used with a detailed understanding of the outlier origins, to direct cleaning/filtering of the problematic outliers from the database.

5) A comparison of Claimed Green Chemistry and non-Green Chemistry has been performed, and found that Claimed Green Chemistry process (which here includes the non-thermal microwave-assisted dielectric heating of water) have a generally greatly reduced process energy budgets. The non-Green Chemistry synthesis has a higher process energy budget ranging from 3.2 kJ to 3.3 MJ.

The statistical analyses and parametric cluster analyses performed on the historical microwave-assisted database provides a new outlook in the field of microwave-assisted synthesis of transition metal-nanostructures. This work gives researchers a wider and better understanding of the energy process involved, all of which are hidden within the historical published microwave-assisted synthesis records. The use of a power-law signature calibration allows researchers to scale-out working microwave-assisted processes with focus on process intensification; i.e., choosing a microwave applicator with one cavity-magnetron rather than two cavity-magnetrons, or by reducing the applicator size; thereby reducing energy use and increasing energy efficiency. The power-law signature also allows new synthesis to be developed with the six principle of Green Chemistry embedded at the start of the design process. In addition, with focus on the understanding and reporting of the process energy profile (not just temperature and pressure targets) when using software controlled microwave applicators. Finally in relation to process intensification, only one of the 50 microwave applicators mentioned in this study is operated in the continuous mode. Clearly more attention to continuous flow mode syntheses is required.

We would like to thank the delegates of the 16^th Chaos 2023 International Conference 13^th to 16^th June 2023 (Hellenic Mediterranean University. Heraklion, Crete, Greece) for their discussions and support in the preparation of this work.

The authors declare no conflicts of interest regarding the publication of this paper.

Law, V.J. and Dowling, D.P. (2023) Microwave-Assisted Transition Metal Nanostructure Synthesis: Power-Law Signature Verification. American Journal of Analytical Chemistry, 14, 326-349. https://doi.org/10.4236/ajac.2023.148018