Optimization of Nanosized Silver Particle Synthesis via Experimental

Jul 24, 2007 - In this work, the optimization of the synthesis of nanosized silver particles by chemical reduction using formaldehyde in aqueous solut...
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Ind. Eng. Chem. Res. 2007, 46, 5591-5599

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Optimization of Nanosized Silver Particle Synthesis via Experimental Design Jyh-Shyong Chang,* Yuan-Ping Lee, and Rong-Chi Wang Department of Chemical Engineering, Tatung UniVersity, 40 Chungshan North Road, 3rd Sec., Taipei, Taiwan, ROC

Highly dispersed silver nanoparticles could be used as catalysts, as staining pigments for glasses and ceramics, as antimicrobial materials, in surface-enhanced Raman spectroscopy, as transparent conductive coatings, in electronics, etc. Consequently, the versatility of such particles provides strong incentives to the development of a systematic way to produce their dispersions. In this work, the optimization of the synthesis of nanosized silver particles by chemical reduction using formaldehyde in aqueous solution was studied. Effects of the possible processing variables such as the reaction temperature T, the mole ratio of [formaldehyde]/[AgNO3], [NaOH]/[AgNO3], the weight ratio of PVP/AgNO3, and the molecular weight (MW) of protective agent PVP (Polyvinyl-pyrrolidone) were considered. The data-driven model on the basis of the 44 designed experimental runs provided us the optimal conditions for closely achieving the product with the specified mean particle size and conversion of silver nitrate. Introduction In the competitive market, product life cycles including the new product development, market introduction, growth, mature, and decline or stability stages are usually very short. The testing resources during the product development stage could be very expensive with no sales revenue at hand. Therefore, speeding up the product or process development is an important issue. This target may be achievable through a reliable mechanistic process model or a data-driven model for traditional chemical processes.1-3 On the other hand, as the complex physical and chemical behavior usually occurs in nanoparticle production, the data-driven model approach would appear to be more appropriate in obtaining the optimal recipe or operating conditions effectively.4-7 Silver powders having ultrafine and uniformly distributed sizes are currently of considerable use in electronics, the chemical industry, and medicine. They display unique properties normally associated with the noble metals (chemical stability, excellent electrical conductivity, and catalytic activity), along with other more specific ones (antibacteriostatic effects, nonlinear optical behavior, etc.). In this regard, nanosized powders and colloidal dispersions of silver have attracted a great deal of attention in recent years.8,9 Silver nanoparticles are produced physically or chemically. The former method is to grind the big silver grains into finer ones or to cool them after heating so as to obtain finer particles. Nevertheless, physically produced particles are usually of larger size, uneven particle diameter, and irregular shape. The disadvantages of residual stresses and lattice structure changes make it difficult to apply nanoparticles to the precision industry. As a result, chemical methods are usually adopted. Reviews of the research work on syntheses of silver nanoparticles or silver colloid dispersions5,8-9 found that the application of a one-factor-at-a-time method was commonly used to find the optimal recipe and operating conditions. That means that only one factor is changed at a time and the others are fixed. This method requires a great quantity of experiments, and the interactions between the factors are not taken into consideration so that the apparent optimum experimental result * To whom correspondence should be addressed. Telephone number: +886-2-5925252-2561-13. Fax number: +886-2-5861939. E-mail: [email protected].

Figure 1. Procedure for carrying out the synthesis of nanosized silver particles.

is not the actual one. Hence, in order to overcome this disadvantage, application of the data-driven models coupled with the effective design of experiments (DOE)3,10-12 to optimize nanosized silver particles synthesis would be quite inviting. In addition to the traditional regression model, artificial neural networks (ANNs) which offer a data-driven modeling approach are well suited for the above-mentioned purpose. Presently, the most commonly used ANN type is a multilayer artificial feed forward neural network (FNN), which is trained by the back propagation (BP) algorithm. The application of FNNs to the experimental data is claimed to be one of the powerful datadriven models, since they have a capability to learn and extract

10.1021/ie061355f CCC: $37.00 © 2007 American Chemical Society Published on Web 07/24/2007

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Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007 Table 2. Measurements of Mean Particle Size and Conversion for the U10(103) and SPUD9(193) Experiments run 1 2 3 4 5 6 7 8

Figure 2. Half normal probability plot of effects.

9

Table 1. Factors and Levels Considered in the Synthesis of Nanosized Silver Particles

10

level

11

factor

physical term

-

+

A B C D E

T (°C) [HCHO]/[AgNO3] (mol/mol) [NaOH]/[AgNO3] (mol/mol) PVP/AgNO3 (g/g) PVP molecular weight (g/mol)

20 1 0.5 5 10000

60 10 1.5 10 40000

the input/output relationships from the presentation of a set of training samples.13-14 Their flexibility has been a valuable quality compared with parametric techniques that require the assumption of a specific hard model form. Therefore, if the datadriven model approach can be applied in the design and operation of a new complex system, the expenditure of time would be greatly reduced in practical application. To build these FNNs, learning and validation procedures are, in general, required. The interpolation and/or extrapolation capabilities of an FNN are directly related to the training data. To achieve a reliable performance model, data must be distributed across all of the regions of the input space that are of interest. The developed sequential pseudo-uniform design (SPUD) method3 could meet the need, because it can guide the experimenter to locate limited but sufficient experiments for collecting the experimental data. The developed SPUD method is an extended version of the uniform design (UD) method11 with a special feature that could locate experiments uniformly and sequentially in the investigated domain. If one first applies the UD method, the UD tables11 can be adopted accordingly. It is expressed as Un(qs), where U stands for uniform design, n is for the number of experimental trials, q is for the number of levels, and s is for the maximum number of factors. In essence, one system has s factors and each factor is divided into q levels by the UD method that requires q experiments (n e q). The number of experiments based on the UD method is limited. If the experimental design based on a first selection of the number of levels q0 from a UD table cannot provide adequate information to build an accurate network model, then we may need to interpolate sequentially another q1 levels of each factor in the range of the original q0 levels to another q1 experiments.3 If qt ) (q0 + q1) experiments cannot adequately provide adequate information to create a reliable model, then the process discussed above can be repeated again. The proposed algorithm for determining the optimum operating conditions for producing a product or a process via the SPUD method is as follows:

12 13 14 15 16 17 18 19

T (°C)

[NaOH]/[AgNO3] (mol/mol)

PVP/AgNO3 (g/g)

ζ (nm)

X (%)

42 42 24 24 38 38 20 20 46 46 55 55 33 33 60 60 51 51 29 29 31 31 57 57 36 36 49 49 26 26 53 53 45 45 40 40 22 22

1.16 1.16 1.27 1.27 0.72 0.72 0.94 0.94 0.5 0.5 1.06 1.06 1.5 1.5 0.83 0.83 1.38 1.38 0.61 0.61 1.00 1.00 0.67 0.67 1.10 1.10 1.44 1.44 1.22 1.22 0.56 0.56 0.89 0.89 1.3 1.3 0.78 0.78

4.99 4.99 8.87 8.87 9.97 9.97 6.67 6.67 8.3 8.3 9.41 9.41 7.78 7.78 7.29 7.29 6.09 6.09 5.57 5.57 6.95 6.95 9.1 9.1 8.06 8.06 7.51 7.51 5.28 5.28 5.84 5.84 6.39 6.39 9.72 9.72 8.61 8.61

44.4 47.1 30.4 32.0 36.8 35.8 30.3 31.4 40.2 42.5 47.5 48.6 40.3 39.2 61.3 63.8 50.9 51.0 46.6 43.9 39.4 42.4 56.32 55.06 35.46 39.94 45.12 43.1 39.3 39.38 51.56 54.6 47.84 46.22 44.2 42.42 30.4 29.58

82.4 81.1 82.4 79.3 56.4 57.6 61.3 60.8 52.0 53.2 84.2 82.9 100.0 100.0 70.6 74.3 98.4 99.8 44.8 44.6 67.2 68.8 61.8 63.6 79.1 80.6 99.8 99.6 78.8 79.6 51.7 50.7 68.3 70.0 88.9 88.8 57.6 54.9

Table 3. Comparison Experiments for Different Molecular Weights of PVP T (°C)

[HCHO]/ [AgNO3] (mol/mol)

[NaOH]/ [AgNO3] (mol/mol)

PVP/AgNO3 (g/g)

PVP MW (g/mol)

mean particle size (nm)

20 20 20 20

3 3 3 3

0.5 0.5 0.5 0.5

5 5 5 5

40000 40000 10000 10000

66.50 69.20 30.20 31.04

Step 1: Choose a suitable level q0 for each control factor using the available UD table. Step 2: Build an identified model based on the experimental data from step 1. Step 3: Check the adequacy of the identified model by the interpolated experiments (another q1 level of each factor among the original level q0) chosen according to the SPUD method (recycle from step 4) or the UD method (recycled from step 6). Step 4: Find the optimal conditions by the random search method if the identified model is adequate; go to step 5. Otherwise, perform the chosen experiments provided by the SPUD method and identify a new identified model from the available experiment data; go to step 3. Step 5: Check the adequacy of the optimal conditions experimentally; if the optimal conditions determined are reliable, stop the procedure. Step 6: If the condition given in step 5 cannot be met, carry out more experiments around the assumed optimal conditions from the UD table again and identify a new identified model; go to step 3.

Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007 5593 Table 4. Identified Regression Models and FNN Models for ζ and X model DOE

identified model Regression Model (Second-Order) ζˆ ) 2750.98 - 41.45x1 - 932.91x2 - 442.41x3 +0.52x12 + Xˆ ) - 536.47 + 8.48x1 + 232.73x2 + 88.71x3 - 0.1x12 729.50x22 + 29.54x32 - 5.35x1x2 + 0.74x1x3 - 36.70x2x3 (6) 131.25x22 - 5.95x32 + 0.82x1x2 - 0.11x1x3 + 5.98x2x3 (7) ζˆ ) 72.42 + 0.62x1 - 18.83x2 - 9.79x3 + 0.007x12 Xˆ ) -39.99 + 0.99x1 + 66.86x2 + 6.62x3 - 0.005x12 + 0.52x22 + 0.22x32 - 0.41x1x2 + 0.007x1x3 + 4.41x2x3 (8) 1.52x22 - 0.37x32 - 0.19x1x2 + 0.006x1x3 - 1.05x2x3 (9) ζˆ ) 39.82 + 0.42x1 + 1.24x2 - 2.75x3 + 0.0009x12 Xˆ ) -50.88 + 0.75x1 + 73.65x2 + 10.06x3 - 5.54x12 6.3x22 - 0.1732 - 0.21x1x2 + 0.04x1x3 + 2.45x2x3 (10) 3.6x22 - 0.36x32 + 0.01x1x2 - 0.05x1x3 - 2.34x2x3 (11)

(S3) (S5) (S7)

hidden nodes

hidden nodes FNN Model

(S7) (S7) (S7) (S9) (S11)

3 4 5 4 4

3 4 5 4 4

Table 5. Comparison of the Performance Indices of the Identified Models for ζand X

Table 6. Comparison of the Measurements and the FNN Model Outputs (ζ and X) for Achieving Minimum Mean Particle Size (S8) run

T (°C)

[NaOH]/[AgNO3] (mol/mol)

PVP/AgNO3 (g/g)

ζ (nm)

X (%)

exp 1 exp 2 model output

20 20 20

0.5 0.5 0.5

7.64 7.64 7.64

31.06 30.18 19.5

45.8 45.8 45.07

performance ζ

model DOE

hidden nodes

γE,I (nm)

X

γE,T (nm)

γE,I

a

γE,T

a

2.88 × 101 2.41 1.99

-

(S4) (S6) (S8)

Regression Model (Second-Order) 0.91 1.55 × 104 + 9.87 × 10-1 2.57 5.35 - 1.42 2.93 4.59 - 2.02

(S8)

2.53

6.32

FNN Model - 1.74

1.21

-

2.02 1.78 2.41 2.51

3.97 1.66 4.12 4.85

+ -

1.41 2.51 2.21 2.21

+ -

(S8) (S8) (S10) (S11) a

3 4 5 4 4

1.28 1.18 1.96 1.96

Model fitted (-) or overfitted (+).

Performance Indices of the Identified Models In this work, an identified model is considered to be a reliable one if the performance indices are achieved, including (a) a smallest root-mean-square value due to residual for both the experiments for identification and testing γE,I, γE,T and (b) the identified model being suitably fitted (denoted by “s”) as estimated by the model outputs on the basis of the testing inputs chosen randomly from the operating region. If some of the model outputs exceed the physical limit, the identified model is estimated to be overfitted. Optimization on the Identified Model by the Random Search Method Two optimization methods are usually available to find the optimum of an objective function constrained by a regression model or nonlinear functional model such as FNN. The first method is a gradient-based optimization technique such as the Newton method. The other method is a nongradient-based optimization technique such as the random search method.16 The random search method17 explores sequentially the parameter space of an objective function in a seemingly random fashion to find the optimal condition for minimizing (or maximizing) the objective function. Although the random search is a direct application method not calling for differential information, the optimal point obtained is quite dependent on the initial guess. Therefore, a large amount of initial guesses will be made to locate the most representative candidate points for the best

performance. The calculation strategy of the random search method proposed by Solis and Wets17 was adopted as follows: Let f(x) be the objective function to be minimized and x0 be the best point currently under consideration.

minimize f(x)

(1)

subject to -∞ e x e ∞ Where, x is the vector of n variables (x1, x2, ..., xn). The random search method is tried to find the best point x* by iterating the following six steps: Step 1: Choose a starting point x0 as the current point. Set initial bias b equal to a zero vector. Step 2: Add a bias b and a random vector dx to the current point x0 in the input space, and evaluate the value of the objective function at the new point x0 + b + dx. Step 3: If f(x0 + b + dx) < f(x0), then set the current point x0 equal to x0 + b + dx, set the bias b equal to 0.2b + 0.4dx, and go to step 6. Otherwise, go to the next step. Step 4: If f(x0 + b - dx) < f(x0), then set the current point x0 equal to x0 + b - dx, set the bias b equal to b - 0.4dx, and go to step 6. Otherwise, go to the next step. Step 5: Set the bias b equal to 0.5b, and go to the next step. Step 6: Stop if the maximum number of function evaluations is reached. Otherwise, go back to step 2 to find a new point. Experimental System The chemical reduction method for synthesizing silver nanoparticles and colloid dispersions is usually to choose an appropriate metal as the precursor and the other reactant as the reducing agent. Both then will be well mixed by controlling the reaction rate and the reaction conditions. After the processes of nucleation and growth, ultimate silver nanoparticles are produced. As far as the materials are concerned, metals of nitrate or chloride salt are the major choices. Nevertheless, they are not necessarily dissolved but probably suspended. The reduction reaction now is supposed to cause the coexistence of liquid and solids. It differs from liquid reaction. The property of solid particles that affects the products of liquid-solid reaction stems

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Figure 3. Experiments performed at the designed locations.

Figure 4. Calculated outputs of the identified FNN model with five hidden nodes (S7) at the testing points (1 × 104) chosen randomly in the domain of the control factors. (a) Distribution of ζ. (b) Distribution of X. (c-e) Distribution of the control factors.

from the influences of particle size and shape. Colloidal syntheses of silver nanoparticles require a stabilizing agent in an aqueous solution. Various stabilizing agents were used to produce stable silver nanoparticles, among them being the nonionic [polyoxyethylene sorbitan monolaurate, polyoxyethylene nonylphenyl ether], cationic [cetyltrimethylammoniumbromide (CTBA)], and anionic [sodium dodecyl sulfate (SDS)], alkyl ether sulfonate, sodium salts of naphthalene sulfonate formaldehyde condensate (Daxad 11 and Daxad 19), and

Table 7. Comparison of the Measurements and the FNN Model Outputs (ζ and X) for Achieving Minimum Mean Particle Size (S10) run

T (°C)

[NaOH]/[AgNO3] (mol/mol)

PVP/AgNO3 (g/g)

ζ (nm)

X (%)

exp 1 exp 2 model output

20 20 20

0.5 0.5 0.5

9.92 9.92 9.92

31.86 31.04 26.9

47.6 48.5 49.86

polyvinyl pyrrolidone (PVP). As for the reducing agents, there are many choices, such as formaldehyde, glucose, ascorbic acid,

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Figure 5. Calculated outputs of the identified FNN model with four hidden nodes (S11) at the testing points (1 × 104) chosen randomly in the domain of the control factors. (a) Distribution of ζ. (b) Distribution of X. (c-e) Distribution of the control factors. Table 8. Comparison of the Measurements and the Model Outputs (ζ and X) for Achieving the Desired Objectives run

T (°C)

[NaOH]/[AgNO3] (mol/mol)

PVP/AgNO3 (g/g)

Minimum Mean Particle Size 0.5 8.842 0.5 8.842 0.5 8.842

exp 1 exp 2 model output

20 20 20

exp 1 exp 2 model output

Assigned Desirability Function 30 1.5 7.97 30 1.5 7.97 29.6 1.5 7.97

ζ (nm)

X (%)

28.96 29.42 28.63

47.6 48.5 47.94

36.72 37.68 38.8

97.3 97.9 97.41

Minimum Particle Size at an Assigned Conversion exp 1 20 1.43 5.76 32.76 exp 2 20 1.43 5.76 31.78 model output 20 1.43 5.76 32.66

86.9 87 85

sodium citrate, hydrazine hydrate, etc.5,8-9,18-19 In the cited literature, the nanosized silver particle synthesis procedure suggested in the work of Chou and Ren5 was adopted for this study. A silver nitrate (Showa) solution of 0.01 M was used as the precursor. A predetermined quantity of the protecting agent PVP (molecular weight (MW) ) 10 000 or 40 000, Sigma) was then added to AgNO3, using formaldehyde (37% solution, Acros) as a reducing agent. However, if only formaldehyde was used, the reduction rate would be too slow at room-temperature due to low pH. Suitable quantities of an alkaline solution consisting of sodium hydroxide (Showa) were added. In this study, the formation of silver colloids can be achieved in a suitable course of time as shown in Figure 1. Products of silver colloids were characterized mainly by their particle size distribution using the dynamic light scattering submicron particle size distribution analyzer (LB-500, Horiba).

Transmission electron microscopy (TEM; H-7100, HITACHI) was also applied to obtain the morphology of sample colloids. Variations in size distribution and the averaged particle size ζ were measured in the course of the reaction. In order to analyze the conversion of silver nitrate defined as

X(conversion)% )

Ci - Cf × 100 Ci

(2)

the flame atomic absorption (AA) spectrometer (SpectrAA 220, Varian) was used to measure the concentration of the silver ion in the reacting medium. X-ray diffraction (XRD; XRD-6000, Shimadzu) was also applied to obtain the information on the structure of silver particles. UV-visible spectroscopy (V-500, Jasco) was used to measure the absorbance of the reacting medium, with small volumes of samples taken at different times and immediately diluted with deionized water. Effects of the possible processing variables shown in Table 1 on the particle size and the conversion of silver nitrate were considered in this study. In this table, the symbols A-E represent the control factors T, [HCHO]/[AgNO3], [NaOH]/[AgNO3], PVP/AgNO, and PVP molecular weight, respectively. The following experiments were carried out heuristically. In this work, two replicates on each experiment were performed. The notation Un(ns) denotes that there are s control factors with n levels partitioned and a total of n experiments are designed. Following Un(ns) designed experiments, the notation SPUDm((m + n)s) denotes that there are s control factors with m + n levels partitioned and a total of m experiments are designed. (S1) Make a survey of the possible control factors affecting the process.

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Figure 8. XRD pattern of silver particles sampled at the optimal condition for achieving the minimum particle size at an assigned conversion given in Table 8.

Figure 6. (a) Sampled particle size distribution. (b) Sampled TEM image produced under the optimal condition for achieving the minimum mean particle size given in Table 8 (particle size ranges from 5 to 35 nm, estimated ζ ) 14.1 nm from the TEM image).

Figure 7. UV-vis absorption spectra of silver particles at different reaction times: (a) 2; (b) 30; (c) 60; (d) 120; (e) 180 min.

(S2) Screen out experiments using 25-1 fractional factorial design (FFD). (S3) Identify a second-order regression model based on the initial U10(103) experiments. (S4) Test the identified model from S3 using the partial experimental data provided by the SPUD9(193) experiments (NI ) 10, NT ) 4). (S5) Identify a second-order regression model based on the based on the initial U10(103) experimental data and the testing data given in S4.

(S6) Test the identified model from S5 using the remaining experimental data provided by the SPUD9(193) experiments (NI ) 14, NT ) 5). (S7) Identify a second-order regression model and FNN models with different nodes based on the experimental data gathered from the U10(103), SPUD9(193) experiments, and partial SPUD18(373) experiments. (S8) Test the identified models from S7 using the remaining experimental data provided by the SPUD18(373) experiments. The identified FNN model with 4 hidden nodes was assumed to be reliable. The optimal condition for achieving minimum mean particle size was estimated. However, there still existed some mismatch between the measurements and the predicted model outputs at this optimal condition (NI ) 32, NT ) 5). (S9) Identify the FNN model with four hidden nodes based on the experimental data provided by U10(103) experiments, partial SPUD18(373) experiments, and the additional boundary experiments provided by the 23 factorial design (FD). (S10) Test the identified models from S9 using the remaining experimental data provided by the SPUD18(373) experiments. The testing results were acceptable; however, the mismatches between the measurements collected under the assumed optimal condition and the model outputs still existed (NI ) 40, NT ) 5). (S11) Identify the FNN model with four hidden nodes based on the experimental data gathered in S9 and the additional U4(43) experiments near the assumed optimal condition determined in S8. At this time, the testing points provided by the remaining SPUD18(373) experiments and the measurements collected under the assumed optimal condition determined in S8 were very close to the model outputs identified in S11. The identified FNN model at this time was checked to be acceptable (NI ) 44, NT ) 5). (S12) Achieve the optimal operating conditions for different objective functions on the basis of the identified FNN model given in S11. Following the heuristic steps S1-S12 shown above, the experimental measurements of mean particle size and conversion for the designed U10(103) and SPUD9(193) experiments are summarized in Table 2. In step S2, we applied the 25-1 fractional factorial design by selecting I ) ABCDE as the generator and, then, set the levels of the fifth factor E ) ABCD to examine the possibility of factor screening. Consequently, every main effect is aliased with a four-factor interaction. The experiment was replicated. Figure 2 depicts a half normal probability plot of the effects estimated from this experiment. The effects A, C, D, E, AC, and CE seem large. Remember that, because of

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aliasing, these effects are in fact A + BCDE, C + ABDE, D + ABCE, E + ABCD, AC + BDE, and CE + ABD. However, because it seems plausible that three-factor and higher interactions are negligible, we feel safe in concluding that only A, C, D, E, AC, and CE are important effects. From the factor screening experiments described in S2, the mole ratio of [formaldehyde]/[AgNO3] (factor B) was set at 3 because factor B is not an important factor. Furthermore, by the comparison experiments done at the near-optimal condition for achieving the minimum particle size of the product, the PVP with the MW (10 000) was chosen, and the experimental results are shown in Table 3. The steps S3-S12 were carried out, and the experimental results and analyses are summarized in Table 4. In this table, it gives the identified models (regression model and FNN model) for the averaged particle size ζ and the conversion of silver nitrate X. Comparisons of the performance indices including γE,I and γE,T and fitted or overfitted status of the identified models for ζ and X are summarized in Table 5. In this table, a smaller value of γE,I denotes a better model output for the corresponding experimental data for identification of the model. The same deduction can be applied to the experimental data adopting for testing the identified model. Among the experiments, the uniform design U10(103) was used initially to locate the experiments (symbol “o” in Figure 3). Ten levels were distributed uniformly along the ranges of the three control factors. The results of the designed experiments (S3) are shown in Table 2 (runs 1-10). The collected experimental data were used to obtain the second-order regression models of mean particle size and conversion. Although acceptable second-order regression models (eqs 6 and 7 in Table 4) were identified based on the collected data, no testing data were available from the designed experiments (U10(103)). Therefore, the four experiments (runs 11-14 in Table 2) on the basis of SPUD9(193) provided us with the additional data (symbol “9” in Figure 3) for testing the identified models (eqs 6 and 7) and further modeling. On the basis of eqs 6, 7, and S4 in Table 5, one can find that the values of λE,I for mean particle size ζ and conversion X are small, but large values of γE,T are predicted by the identified models (eqs 6 and 7). Furthermore, the identified regression model for ζ is overfitted. For model improvement (S5 and S7), the new second-order regression models (eqs 8-9 and eqs 10-11 in Table 4) were identified. At the same time, the available experimental data were also used to identify an FNN model with optimal hidden nodes (S7 in Table 4). On the basis of the criterion given in the previous section for judging which model identified was most reliable, one could conclude that the identified FNN model with four hidden nodes was the most reliable as shown in Tables 5 (S8). On the other hand, the calculated outputs ζ and X corresponding to the testing points (1 × 104) chosen randomly in the domain of the control factors of the identified FNN model with five hidden nodes are shown in Figure 4. This result signifies that the identified FNN model with five hidden nodes is overfitted. In this figure, the model outputs ζ and X are too large to be acceptable. Therefore, in S8, the optimal conditions for achieving minimum mean particle size was estimated; however, there still existed some mismatch between the measurements and the predicted model outputs under this optimal condition as shown in Table 6. This mismatch could be attributed to the sparse experiments arranged around the boundary of the control factor (at the lower bound of the operating temperature T ) 20 °C in this case). Therefore, steps S9 and S10 were carried out and the optimal conditions for achieving minimum mean particle

size were estimated again. The results for the testing points were acceptable; however, the measurements collected under the assumed optimal condition still revealed some mismatch compared to the model outputs as shown in Table 7. Improvement of the results obtained in step S10 was carried out following step S11, and the identified FNN model with four hidden nodes was confirmed to be acceptable. The calculated outputs ζ and X corresponding to the testing points (1 × 104) chosen randomly in the domain of the control factors are shown in Figure 5. In plotting Figures 4 and 5, the calculated outputs ζ and X were sorted in ascending order and their corresponding independent variables were determined by the control factors T, [NaOH]/[AgNO3], and PVP/AgNO which were relocated according to the sorted outputs ζ and X. From this figure, one can observe that the mean particle size ζ and the conversion X seem to form a linear relationship with respect to the regressor T and the model is well identified. Process Optimization In S11, the identified FNN model with four hidden nodes can give a good prediction of the experimental data for achieving the objective of minimizing the mean particle size of the silver particles as shown in Table 8. The particle size distribution and TEM image of the sample are shown in Figure 6. As shown in Figure 6a, silver particle diameters range from 15 to 25 nm; however, it is estimated to be 14.1 nm from the sampled TEM image as shown in Figure 6b. The formation of nanosized silver particles was followed by absorbance measurements. Figure 7 shows the evolution of the absorption spectra of silver colloids during the reduction reaction (2, 30, 60, 120, and 180 min). One could find that when the reaction proceeded, in addition to the increase of the absorbance, the plasmon band of silver colloids shifted from 416 nm (2 min) to 422 nm (180 min) because the concentration of silver particles increased. Given the indices of individual desirability functions (eqs 3 and 4), the composed desirability function for these two objectives D ) (dζ‚dX)1/2 can be determined:

dξ ) dX )

( (

ζˆ - C Bξ - C

) )

Xˆ - A BX - A

D ) (dξ‚dX)1/2

t

s

(3) (4) (5)

Where, Bξ is the minimal value of ζ () 28.42 nm), C is the maximal value of ζ () 73.4 nm), t is the weight for ζ () 8), Bx is the maximal value of X () 100%), A is the minimal value of X () 44.6%), and s is the weight for X () 8). The identified FNN model with four hidden nodes (S11) can give a good prediction to the experimental data sampled under the condition for achieving a maximum value of D (the case of achieving the assigned desirability function in Table 8). If the minimum particle size is to be achieved at a preassigned conversion, a close prediction for the experimental data sampled under the optimal condition provided by the identified FNN model with four hidden nodes was verified (the case of achieving the minimum particle size at an assigned conversion in Table 8). The corresponding XRD analysis shown in Figure 8 indicates that the silver particles are well crystallized, with sharp peaks of (1, 1, 1), (2, 0, 0), (2, 2, 0), and (3, 1, 1). Conclusions Demonstration of the applicability of the SPUD method mixed with the FD method in the interior and at the boundary of the

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operating region for effectively identifying a reliable model for a real complex system was accomplished in this work. For the synthesis of nonosized silver particle, effects of the possible processing variables such as the reaction temperature T, the mole ratios of [formaldehyde]/[AgNO3] and [NaOH]/[AgNO3], PVP/ AgNO3, and the molecular weight of the protective agent PVP (polyvinyl pyrrolidone) were considered. The 25-1 fractional factorial design was first applied to screen out the insignificant factor [formaldehyde]/[AgNO3]. By the comparison experiment done under the near-optimal condition for achieving the minimum particle size of the product, the PVP with a MW of 10 000 was then chosen. Three significant factors for this reaction system were determined to be the reaction temperature T, the mole ratios of [NaOH]/[AgNO3], and the weight ratio of PVP/AgNO3. In this work, the experimental data collected via the mixed experimental methods including UD, SPUD, and FD were used to identify a second-order regression model and FNN models with different hidden nodes. An identified model is thought to be most acceptable if the performance indices are achieved including (a) the smallest root-mean-square due to residual for both the experiments for identification and testing γE,I and γE,T and (b) the identified model being suitably fitted (s), a result examined by model output calculations on the basis of a random testing set. On the basis of the given criterion, the FNN model with four hidden nodes was chosen to be the most acceptable model for mean particle size; another FNN model with four hidden nodes was also chosen to be the most acceptable model for conversion. Adopting the identified models, the following three objectives sought for the product were successfully achieved: (a) minimum mean particle size, (b) the assigned desirability function which was a function of minimum mean particle size and maximum conversion, and (c) minimum mean particle size at an assigned conversion. Acknowledgment We thank the National Science Council (Grant NSC 93-2214E-036-001) and Tatung University, Taipei, Taiwan, for all the support conducive to the completion of this work.

S ) weight adopted in the desirability function for conversion; control factors or regressors SS ) sum of squares t ) weight adopted in the desirability function for mean particle size T ) temperature, °C U) uniform design x ) variables X ) conversion of silver nitrate Greek Symbols γ ) root-mean-square of SS ζ ) mean particle size, nm Supscripts t, s ) weight for the individual desirability function * ) optimal 0 ) initial Subscripts E ) error I ) identified T ) testing t ) total Acronyms AA ) atomic absorption ANN ) artificial neural network BP ) back propagation CTBA ) cetyltrimethylammonium-bromide DOE ) design of experiments FNN ) feedforward neural network FD ) factorial design FFD ) fractional factorial design SDS ) sodium dodecyl sulfate SPUD ) sequential pseudo-uniform design TEM ) transmission electron microscopy MW ) molecular weight PVP ) polyvinyl pyrrolidone UD ) uniform design XRD ) X-ray diffraction

Nomenclature A, B, C, D, E ) control factors or regressors A ) the minimal value of X Bx ) the maximal value of X Bξ) the minimal value of ζ C ) the maximal value of ζ b ) bias vector Ci ) initial silver ion concentration in the sample, ppm Cf ) final silver ion concentration in the sample, ppm dX ) individual desirability function for X dζ ) individual desirability function for ζ D ) composed desirability function f ) objective function I ) the generator of the FFD m ) number of regressor levels or observations n ) number of experimental trials; number of levels NI ) number of experimental data for identifying a model NT ) number of experimental data for testing an identified model q ) number of levels based on the UD method q0 ) initial number of levels based on the UD method q1 ) additional number of levels based on the SPUD method R2 ) R-squared

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ReceiVed for reView October 23, 2006 ReVised manuscript receiVed May 24, 2007 Accepted June 11, 2007 IE061355F