. In this paper, we propose a grid search algorithm using response surface methodology (RSM) for optimizing the biosynthesis of silver nanoparticles (AgNPs). A biological method using hot water (HW) Clerodendrum splendens (C. splendens ) leaf extract is adopted for the discussion . The RSM integrated central composite design (CCD) is used to derive a quadratic objective function to model the yield of AgNPs by accounting the effects of the processing parameters, namely, C. splendens concentration , pH, and the reaction time. The optimizati on of the resulting objective function, which is a challenging multi -variate non -convex problem, is solved using a grid search approach. The respective optimum values of the parameters are found to be 54 mL, pH 12, and 23 min and the corresponding AgNPs yi eld is experimentally verified. Keywords: Response surface method, central composite design, silver nanoparticles , Clerodendrum splendens , leaf extract . 2 1 Introduction AgNPs are widely used in many fields of science and technology due to its unique physical and chemical properties including antibacterial, antiviral , and antifungal activities1. Hence, the synthesis and the characterization of AgNPs is an active research topic and there are many recent attempts discussing the optim ality of AgNPs synthesis2-5. These papers address the o ptimization of the yield,2,3 particle size ,4 or the absorbance5 of AgNPs with respect to various process parameters such as silver nitrate concentration, pH, reaction time, etc. The common strategy followed in such works is to adopt a response surface methodology (RSM), which is a statistical approach to optimize the reaction settings by accounting the interaction between the process parameters. This i s often achieved by performing central composite design (CCD) expe riments and then modelling the relationship between the parameters and the response of the reaction using a second order polynomial via regression analysis. The optimization of this regression models is a quadratic programming (QP), which, in the context o f AgNPs synthesis, is a very challenging problem due the non -convex nature of the model. To the best of our knowledge, mos t of the published works3-5 do not put forward a strategy to obtain a global solution for the underlying QP; instead they discuss the optimum interaction between any two process parameters using three -dimensional (3D) response surface graphs, in which the other parameters are fixed to specific value s. Such a strategy fails to address the multi -variate optimization problem in its general form, in which the 3 simultaneous interactions between all the parameters are important . On the other hand, in the paper2, the optimum value is selected f rom the CCD parameter settings; which is a suboptimal method since the search space of QP solver is highly reduced. We remark that since the underlying QP is non -convex in nature any straight forward optimization routine will end up in local optimum points . As a remedy to all the roadblocks, we propose a simple brute force approach based on grid search algorithm. The proposed optimization strategy, with the help of MATLAB programming, maximizes the AgNPs yield over all the possible combinations of the proce ss parameters sampled from a 3D grid. Though the AgNPs can be synthesize d using physical and chemical methods, we adopt a biological approach using plant leaf extract, which has received great attention in the recent years, as it is cost -effective, simple, environmentally friendly, and non -toxic6. Also, since the plant -based nanoparticles synthesis does not require the complicated process of maintaining the fungi or the bacterial cell cultures, it is simple and beneficial over other biological agents. In th is study, we make use of the leaves of Clerodendrum splendens, which is an evergreen climber or running shrub belongin g to the family verbenaceae7. Its common name is Flaming Glorybo wer8. It is native to Western Africa and is widely distributed in the tropical regions throughout the world. The ir leaves contain glycosides, reducing sugars, triterpenoids, unsaturated sterols , and flavonoids9. The plant is used in traditional medicin e to treat wounds and burns10, haemorrho ids, diarrhea and dysentery11. Pre vious studies have reported that the C. splendens leaf extract has antioxidant, antimicrobial and anti-inflammatory properties12. The phytochemicals of the C. splendens leaf extract are vital 4 for the synthesis of non -toxic silver nanoparticles. Our previou s study discussed the synthesis of silver nanoparticles using HW C. splendens leaf extract without a grid search -based RSM13. The aim of the study is to optimize the experimental conditions for the production of silver nanoparticles using C. splendens leaf extract. A grid search -based RSM algorithm is proposed to compute the optimum value of C. splendens concentration, pH, and reaction time. This study may present a better understanding of the role of C. splendens leaf extract for the production of AgNPs. 2 Materials and Methods 2.1 Materials Silver nitrate (AgNO 3) was purchased from Central drug house (CDH), India, and was of analytical grade. C. splendens leaves were collected from the surroundings of Chennai, Tamil Nadu, India. Double distilled water was used throughout the study. 2.2 Preparation of leaf extract The collected C. splendens leaves were t horoughly washed and dried. The clean leaves were cut into fine particles. 10 g of these fine ly cut clean leaves were boiled in 100 mL of distilled water for 2 min at 60 ° C, and then t he extract was filtered twice through Whatman filter paper grade No.1 (Pore size: 11 µm). The obtained leaf extract was stored at ambient temperature (30 °C) for further use. 2.3 Synthesis of silver nanoparticles To estimate the formation of AgNPs , freshly prepared leaf extract was added to 100 mL of 1 mM AgNO 3. The reaction was carried out in a dark room to decrease the photoactivation of 5 AgNO 3 at an ambient temperature of 30 °C. Within a short period, the colour of the resulting solution started changing from pale yellow to colloidal brown indicating the formation of AgNPs. The solution containing AgNPs was centrifuged at 15000 rpm for 15 min to obtain a pellet of AgNPs. The pe llet was washed with distilled water to get rid of biomass residue and then dried in the oven at 60 °C for 24 h, which was used for further studies. 2.4 Optimization of silver nanoparticles production The influence of three process parameters, i.e., the plant extract (C. splendens ) concentration ( A), pH ( B) and the reaction time ( C), on the yield of AgNPs (Y) , was studied using a statistical method called RSM. A CCD experimental matrix consisting of three factors ( A, B and C) at three levels was produced. We choose a Face -centred CCD having 6 centre points, 6 axial (star) points, and 8 factorial points resulting in 20-experimental runs. The yields of AgNPs for all the 20 entries of the CCD matrix were experimentally measured. A second order regres sion equation model of the following form was generated to facilitate the proceeding optimization procedures: 𝑌(𝐴,𝐵,𝐶)=𝛽0+𝛽1𝐴+𝛽2𝐵+𝛽3𝐶+𝛽4𝐴𝐵+𝛽5𝐴𝐶+𝛽6𝐵𝐶+𝛽7𝐴2+ 𝛽8𝐵2+𝛽9𝐶2,(1) Where , 𝜷𝒊,𝒊=𝟏,𝟐,…𝟖 are the regression coefficients and 𝒀(𝑨,𝑩,𝑪)is the predicted response variable as a function of A, B , and C. The development of CCD data and the regression model were done using MATLAB 8.6. We formulate the following optimization problem to calculate the best values of A, B, and C: 6 {𝐴∗,𝐵∗,𝐶∗}= argmax 𝑌(𝐴,𝐵,𝐶), {A,B,C} (2) The optimization problem (2), which is commonly known as QP, is often non -convex for the AgNPs regression model. In order to solve this, a sufficiently large 3D grid representing all the possible discrete values of A, B and C, in a preferred range, is created and an extensive brute force search over the grid is performed. This grid search algorithm is implemented using MATLAB 8.6. 2.5 Statistical analysis The significance of the model coefficient s were statistically analyzed by ANOVA (Analysis of variance). The fit statistics of the quadratic model and ANNOVA table were generated using M ATLAB 8.6. 3 Results and discussions 3.1 Synthesis of AgNPs The present study discusses the synthesis of AgNPs using leaves of C. splendens (Fig.1a) . The AgNPs was synthesized by adding C. splendens extract in 100 mL of 1 mM AgNO 3 solution. The successful production of AgNPs was visually confirmed by the colour change of AgNPs solution from pale yellow to reddish brown (Fig. 1 b). The reddish -brown colour is due to the excitation of the surface pla smon vibrations in the AgNPs 14. 7 Fig.1 Visual photograph of (a) Clerodendrum splendens (b) (i) 1 mM silver nitrate (ii) leaf extract (iii) AgNPs solution 3.2 Optimization of AgNPs formation by RSM The proposed optimization strategy involves two major steps: i) an RSM st ep in which a second -order polynomial model, as per equation (1) , corresponding to a CCD experimental matrix, is computed ; ii) an optimization step in which the QP , described by equation (2) is solved using a grid search algorithm. To design the CCD matrix, we considered three levels of the process parameters, coded as +1, 0, -1, representing high, intermediate, and low values, respectively as shown in Table 1. Table 1. Factors and Levels for Central Composite Design Factors Name Units Levels Low (-1) Intermediate (0) High (+1) A C. splendens concentration mL 2 51 100 B pH 2 8 14 C Reaction time min 0 15 30 8 A 20 -run face -cantered CCD matrix, having various combinations of A, B and C from Table 1, is generated and the corresponding yield ( 𝑌𝑜𝑏𝑠) of AgNPs is experimentally calculated (see Table 2). The regression coefficients of the quadratic equation (1), i.e., 𝛽𝑖,𝑖=1,2,…9, are computed using the values of 𝐴,𝐵, 𝐶, and 𝑌𝑜𝑏𝑠 through a MATLAB regression analysis. Substituting the computed regression coefficients in equation (1) yields Y(A,B,C)= −0.080733 +0.006080 A+0.043438 B+0.064787 C −0.000094 AB−0.0000023 AC+0.000646 BC −0.000046 A2− 0.002271 B2− 0.001552 C2 (3) The predicted values of the response variables (Y) generated according to equation (3), the observed response 𝑌𝑜𝑏𝑠, and CCD matrix are shown in Table 2. Using the regression model (3), the optimization in equation (2) can be rewritten as {𝐴∗,𝐵∗,𝐶∗}= 𝑎𝑟𝑔𝑚𝑎𝑥 {𝐴,𝐵,𝐶}[−0.80733 +0.006080 𝐴+0.043438 𝐵+0.064787 𝐶 −0.000094 𝐴𝐵−0.0000023 𝐴𝐶+0.000646 𝐵𝐶 −0.000046 𝐴2−0.002271 𝐵2−0.001552 𝐶2] (4) 9 Table 2. The central composite design matrix with experimental values and predicted values of AgNPs c oncentration (OD) Runs C. splendens concentration (v/v)% pH Reaction time (min) AgNPs concentration (OD at 420 nm) Experimental value Predicted value A B C (Yobs) (Y) 1 100 2 30 0.7139 0.7067 2 2 14 0 0.0915 0.0918 3 51 8 15 0.9786 0.9723 4 2 14 30 0.9149 0.9116 5 2 2 0 0.009 0.0086 6 100 14 30 0.9186 0.9119 7 100 14 0 0.1067 0.0988 8 51 2 15 0.8149 0.8185 9 51 8 15 0.9786 0.9723 10 51 8 15 0.9786 0.9723 11 51 8 15 0.9786 0.9723 12 51 8 15 0.9786 0.9723 13 51 8 0 0.2734 0.2735 14 100 8 15 0.8867 0.8914 15 51 8 15 0.9786 0.9723 16 51 8 30 0.9582 0.9737 17 2 2 30 0.5954 0.5958 18 51 14 15 0.9516 0.9627 19 2 8 15 0.8248 0.8324 20 100 2 0 0.1298 0.126 2 To solve equation (4), we first construct a 3D grid having all possible combination of A, B and C in the set 𝐴 𝜖 {1,2,...,100 },𝐵 𝜖 {2,3,...,14},𝑎𝑛𝑑 𝐶 𝜖 { 0,1,...,30} as shown in Fig. 2. 10 Fig.2 The 3D grid for the optimization search space Equation (4) is solved by trying all the combination of A, B, and C from the 3D grid using a MATLAB program. The grid search algorithm computed the optimum value as {𝐴∗,𝐵∗,𝐶∗}={54,12,23}. Using equation (3), the optimum yield Y(A*, B*, C*) is calculated to be 1.0926. This means that the optimum values of C. splendens concentration ( A), pH ( B), and the reaction time ( C) for the synthesis of AgNPs are, respectively, 54 mL, 12, and 23 min, and the corresponding AgNPs yield is 1.0926. We repeated the experimental synthesis of AgNPs using the theoretically optimum parameter values of {54 mL, 12, 23 min} and obtained an AgNPs yield of 0.971 5 (Fig. 3). This value is very close to the theoretical yield 1.0926, with a minor error of 0.1211 . 11 Fig.3 UV -Visible spectrum of experimental synthesis of AgNPs under optimum condition 3.3 Interactions among the process parameters The relative interactions among the two parameters by keeping the third one to their optimum value was studied using 3D surf ace pl ot and 2D contour plot in Fig. 4 . Figure 4a shows the interactive effects of C. splendens concentration ( A) and pH ( B) in the production of AgNPs (Y) at the optimum reaction time ( C) of 23 min. The graph shows that the yield of AgNPs is increasing un til the C. splendens concentration ( A) is 54 mL, and beyond 54 mL, the yield of AgNPs starts to decrease . With respect to pH ( B), the yield is gradually increasing until pH12, and then drops slightly. Figure 4b shows the interactive effects of C. splende ns concentration (A) and reaction time (C) in the production of AgNPs ( Y) at the optimum pH ( B) 12. The graph reveals that the reaction time ( C) has a significant effect on the yield of AgNPs, which rapidly increased from 12 0 min to 23 min, and decreases slowly afterwards. However, the effect of C. splendens concentration ( A) on the AgNPs production was not as significant as that of the reaction ti me. Fig.4 The relative interactions between the parameters shown using 3D surface and 2D contour plots 13 Subsequently , in Fig. 4 c, the interactive effects of pH ( B) and reaction time ( C) at the optimum C. splendens concentration ( A) of 54 mL is shown. The production of AgNPs increased significantly within the reaction time ( C) from 0 min to 23 min, and then it begins to decrease a little. The yield increases very slowly with respect to pH until 12, after which, a slight drop is obse rved. The 2D contour plot s corresponding to the Figures 4a,4 b, and 4c are shown respectively in Figures 4d,4e,and 4 f. From these three contour plots, the values of the pairs ( A,B), (A,C), and (B,C), for the best AgNPs yield, are observed to be (54,12), (54 ,23), and (12,23), which are in the good agreement with optimum values computed through the grid search. Table 3. ANOVA (Analysis of Variance) for RSM model Production of AgNPs Sum of squares Df Mean square F-value p-value Model 2.47 9 0.2745 3575.44 <0.0001 significant A 0.0102 1 0.0102 133.47 <0.0001 Significant B 0.0519 1 0.0519 675.84 <0.0001 Significant C 1.22 1 1.22 15871.44 <0.0001 Significant AB 0.0061 1 0.0061 79.10 <0.0001 Significant AC 0.000 1 0.0000 0.3101 0.5899 BC 0.0270 1 0.0270 351.77 <0.0001 Significant A2 0.0328 1 0.0328 427.63 <0.0001 Significant B2 0.0184 1 0.0184 239.45 <0.0001 Significant C2 0.3345 1 0.3354 4368.38 <0.0001 Significant Residual 0.0008 10 0.0001 14 3.4 ANOVA for RSM quadratic equation The ANOVA for RSM quadratic model of AgNPs yield was highly signif icant with an F value of 3575.44 (Table 3) . P-value less than 0.0001 indicate s that the model terms are highly significant. In this case , A, B, C, AB, BC, A2, B2, and C2 are highly significant model terms. P-value greater than 0.1 indicate the model term is not significant. Table 4. Fit statistics Model Parameters Value Standard deviation 0.0088 Mean 0.7030 Coefficient of variation (%) 1.25 R2 0.9997 Adjusted R2 0.9994 Predicted R2 0.9984 Adeq Precision 155.6777 The fit statistics for the qua dratic model is shown in Table 4 . Coefficient of variation (CV) is the ratio of the standard deviation to mean. The lower the CV value of 0.0125 indicates a smaller deviation between the experimental and the predicted value. The coefficient of determination (R2) is a statistical tool to measure the correlation between the experimental and the predicted responses. A high correlation is indic ated by an R2 value close to 1. In the present study, the R2 value is 0.9997, which indicates a very good correlation. For a good model, the difference between the “predicted” and the “adjusted” R2 should be less than 0.2 and in our case, the difference is 0.001. The Adeq Precision measures the signal to noise ratio. 15 An Adeq Precision greater than 4 is desirable and in the present study, the value of the ratio is 155.6777, which indicates an adequate signal level. 4 Conclusions In this paper , we propose a new optimization strategy, in which the biosynthesis of AgNPs is optimized for the best yield. We discuss the synthesis of AgNPs using leaf extract of C. splendens and the optimum combination of the process parameters, i.e., the concentration of the leaf extract, pH, and the reaction time , is calculated. The proposed optimization strategy involves the computation of a second order regression model corresponding to a CCD experimental matrix and the optimization of the regression model using a grid search algorithm. The optimum combination of the process parameters is computed as 54 mL, pH 12, 30 min through a grid search algorithm and this reaction condition is experi mentally verified. The proposed strategy is easily extendable to optimize the reaction conditions in any similar biosynthesis of nanoparticles. Acknowl edgments The authors thank the Department of Chemical engineering, A.C Tech, Anna University