G Mohanty et al./ Materials Research Bulletin 43(2008)1814-1828 2. 4. Design matrix A full 2* factorial design with addition of four center point experiments was chosen to model particle deposition Design Expert v 7 statistical software(Stat Ease Inc )was used for the analysis of the experimental data and for statistical modeling. The four factors investigated were: (i) concentration, (ii) electrode separation, (iii) applied voltage, and (iv)deposition time by A, B, C and D, respectively. Each factor was run at two levels and the intermediate response was assumed to be linear, which is necessary for 2 designs. The possibility of non-linearity within the design space has been accounted for through the introduction of center points and model augmentation. The center points are essentially used to test for evidence of pure second-order or quadratic effects in the response region of exploration. The high and low levels for each factor as given in Table I were chosen on the basis of preliminary trials. These high and low levels are expressed in coded form as -l and +l, respectively to convert the absolute quantity into a dimensionless quantity making the handling of the experimental data convenient. Also since all variables used in the model are normalized to vary in this way, the relative change of a variable is directly related to the size of its regression coefficient. The weight of the alumina deposit was studied as the response for the different combinations of factor levels. Table 2 shows the experimental matrix in actual and coded factors along with the weight of alumina deposited as response. The regression equation for the matrix is then represented by the following expression Y=b0+b1X1+b2X2+b3X3+ b4X4+ b12X1X2+b13X1X3+ b14X1X4+ b23X2X3+b34X2X4+ b34X3X4 +b123X1X2X3+b124X1X2X4+b134X1X3X4+b234X2X3X4+b1234X1X2X3X4 where y is the response (alumina deposited weight), bo a constant, i.e., response at the zero level(center point) experiment, b1, b2, b3 and ba the linear coefficients(independent parameters), b12, b13, b14, b23, b24, b34, b123, b134, b234, b1234 are interaction coefficients representing the parameters in their coded form. The relationship between the actual and coded values are given below XI X,、2 and X The regression coefficients were estimated by the following expression b (XXk…Xn)Y j…,i=1,2,3,…n The experimental order for obtaining the responses were done by a completely randomized design in which the allocations of the experimental parameters as well as the order in which the individual runs or trials of the experiment are to be performed are randomly determined. Such randomization of the order of experiments tend to average out the effect of any uncontrolled variables and validate the usual normality assumptions. All the factors except concentration have been randomly selected Randomization with respect to concentration was not possible because each alumina suspension was used for three deposition experiments before replenishing it with a fresh one. Table 3 shows weight deposited for same suspension as well for second suspension under similar deposition conditions. It was found that the difference in weight of alumina deposited varies by about 7%(max) for the replicated center points and by 4%(max) for runs replicated from the same suspension. The center point placement was non-random as this is a well-known process. Two center points were front loaded in the run order and the remaining two were run at the end. Transformation of the response data was necessary in our analysis since the ratio of maximum to minimum value of response obtained in the design matrix was very large. Transformations apply a mathematical function to all the Table I Actual vis-a-vis coded values of parameters Concentration(A)(wt/100 ml) Electrode separation(B)(cm) Applied potential (C)(V) Deposition time(D)( Actual(xu) Code(X1) Actual(x Code(X,) Actual (x3) Code(X3) Actual (x4) Code(X4) Max level 30 3 Min level 10 Zero level 20 0 02.4. Design matrix A full 24 factorial design with addition of four center point experiments was chosen to model particle deposition Design Expert v.7 statistical software (Stat Ease Inc.) was used for the analysis of the experimental data and for statistical modeling. The four factors investigated were: (i) concentration, (ii) electrode separation, (iii) applied voltage, and (iv) deposition time by A, B, C and D, respectively. Each factor was run at two levels and the intermediate response was assumed to be linear, which is necessary for 2k designs. The possibility of non-linearity within the design space has been accounted for through the introduction of center points and model augmentation. The center points are essentially used to test for evidence of pure second-order or quadratic effects in the response region of exploration. The high and low levels for each factor as given in Table 1 were chosen on the basis of preliminary trials. These high and low levels are expressed in coded form as 1 and +1, respectively to convert the absolute quantity into a dimensionless quantity making the handling of the experimental data convenient. Also since all variables used in the model are normalized to vary in this way, the relative change of a variable is directly related to the size of its regression coefficient. The weight of the alumina deposit was studied as the response for the different combinations of factor levels. Table 2 shows the experimental matrix in actual and coded factors along with the weight of alumina deposited as response. The regression equation for the matrix is then represented by the following expression: Y ¼ b0 þ b1X1 þ b2X2 þ b3X3 þ b4X4 þ b12X1X2 þ b13X1X3 þ b14X1X4 þ b23X2X3 þ b24X2X4 þ b34X3X4 þ b123X1X2X3 þ b124X1X2X4 þ b134X1X3X4 þ b234X2X3X4 þ b1234X1X2X3X4 where Y is the response (alumina deposited weight), b0 a constant, i.e., response at the zero level (center point) experiment, b1, b2, b3 and b4 the linear coefficients (independent parameters), b12, b13, b14, b23, b24, b34, b123, b134, b234, b1234 are interaction coefficients representing the parameters in their coded form. The relationship between the actual and coded values are given below: X1 ¼ x1 20 10 ; X2 ¼ x2 2 1 ; X3 ¼ x3 225 75 ; and X4 ¼ x4 2 1 The regression coefficients were estimated by the following expression: bo ¼ X i¼1;2;3;  ;n Yi N ; bj ¼ X j;i¼1;2;3;  ;n X jYi N ; bjk  n ¼ X j;k;...;i¼1;2;3;  ;n ðX jXk ... XnÞYi N The experimental order for obtaining the responses were done by a completely randomized design in which the allocations of the experimental parameters as well as the order in which the individual runs or trials of the experiment are to be performed are randomly determined. Such randomization of the order of experiments tend to average out the effect of any uncontrolled variables and validate the usual normality assumptions. All the factors except concentration have been randomly selected. Randomization with respect to concentration was not possible because each alumina suspension was used for three deposition experiments before replenishing it with a fresh one. Table 3 shows weight deposited for same suspension as well for second suspension under similar deposition conditions. It was found that the difference in weight of alumina deposited varies by about 7% (max) for the replicated center points and by 4% (max) for runs replicated from the same suspension. The center point placement was non-random as this is a well-known process. Two center points were front loaded in the run order and the remaining two were run at the end. Transformation of the response data was necessary in our analysis since the ratio of maximum to minimum value of response obtained in the design matrix was very large. Transformations apply a mathematical function to all the G. Mohanty et al. / Materials Research Bulletin 43 (2008) 1814–1828 1817 Table 1 Actual vis-a`-vis coded values of parameters Level Concentratioin (A) (wt/100 ml) Electrode separation (B) (cm) Applied potential (C) (V) Deposition time (D) (min) Actual (x1) Code (X1) Actual (x2) Code (X2) Actual (x3) Code (X3) Actual (x4) Code (X4) Max level 30 + 3 + 300 + 3 + Min level 10 1 150 1 Zero level 20 0 2 0 225 0 2 0