Plackett Burman Design and ANOVA data analysis
The impact of the three considered factors on dispersion stability was experimentally conducted and statistically analyzed using Plackett Burman Design as shown in Table 3. The identified factors were further optimized with Central Composite Design.
Table 3
Plackett Burman Design for SnO2 dispersing in DI water and NMP (actual and predicted).
| | Factor 1 | Factor 2 | Factor 3 | Drop in Absorbance for 7 Days |
Solvent | Run | A: Concentration of Nanoparticles | B: Sonication Time | C: Addition of Surfactant | Actual Value | Predicted Value |
DI Water | Unit | mg/ml | hours | mg/ml | | |
1 | 0.5 | 4 | 3 | -4.749 | -4.97 |
2 | 0.5 | 1 | 0.5 | -5.528 | -5.47 |
3 | 2 | 1 | 3 | -18.343 | -17.91 |
4 | 0.5 | 1 | 3 | -5.544 | -4.16 |
5 | 0.5 | 1 | 0.5 | -4.054 | -5.47 |
6 | 2 | 4 | 0.5 | -21.177 | -20.03 |
7 | 2 | 1 | 3 | -17.217 | -17.91 |
8 | 0.5 | 4 | 0.5 | -7.049 | -6.28 |
9 | 2 | 4 | 3 | -18.378 | -18.72 |
10 | 2 | 1 | 0.5 | -19.466 | -19.23 |
11 | 2 | 4 | 0.5 | -19.251 | -20.03 |
12 | 0.5 | 4 | 3 | -4.4 | -4.97 |
NMP | 1 | 0.5 | 4 | 3 | -1.74 | -0.5917 |
2 | 0.5 | 1 | 0.5 | -1.19 | -2.15 |
3 | 2 | 1 | 3 | -5.44 | -8.44 |
4 | 0.5 | 1 | 3 | -1.39 | 0.5833 |
5 | 0.5 | 1 | 0.5 | -1.74 | -2.15 |
6 | 2 | 4 | 0.5 | -12.96 | -12.34 |
7 | 2 | 1 | 3 | -9.21 | -8.44 |
8 | 0.5 | 4 | 0.5 | -0.84 | -3.32 |
9 | 2 | 4 | 3 | -8.01 | -9.62 |
10 | 2 | 1 | 0.5 | -12.79 | -11.17 |
11 | 2 | 4 | 0.5 | -13.95 | -12.34 |
12 | 0.5 | 4 | 3 | -1.31 | -0.5917 |
Table 4
ANOVA analysis for SnO2 in DI water (Plackett Burman Design).
Source | Sum of Squares | df | Mean Square | F-value | p-value | |
Model | 574.45 | 3 | 191.48 | 200.07 | < 0.0001 | significant |
A-Concentration of Nanoparticles | 567.30 | 1 | 567.30 | 592.75 | < 0.0001 | |
B-Sonication Time | 1.96 | 1 | 1.96 | 2.05 | 0.1901 | |
C-Addition of Surfactant | 5.19 | 1 | 5.19 | 5.43 | 0.0482 | |
Residual | 7.66 | 8 | 0.9571 | | | |
Lack of Fit | 4.02 | 4 | 1.01 | 1.11 | 0.4623 | not significant |
Pure Error | 3.64 | 4 | 0.9090 | | | |
Cor Total | 582.11 | 11 | | | | |
R2 −0.9868; Adjusted-R2 −0.9819; Predicted-R2 −0.9704; |
Table 5
ANOVA analysis for SnO2 dispersed in NMP (Plackett Burman Design).
Source | Sum of Squares | df | Mean Square | F-value | p-value | |
Model | 270.83 | 3 | 90.28 | 23.51 | 0.0003 | significant |
A-Concentration of Nanoparticles | 244.35 | 1 | 244.35 | 63.64 | < 0.0001 | |
B-Sonication Time | 4.14 | 1 | 4.14 | 1.08 | 0.3293 | |
C-Addition of Surfactant | 22.33 | 1 | 22.33 | 5.82 | 0.0424 | |
Residual | 30.72 | 8 | 3.84 | | | |
Lack of Fit | 22.88 | 4 | 5.72 | 2.92 | 0.1622 | not significant |
Pure Error | 7.84 | 4 | 1.96 | | | |
Cor Total | 301.54 | 11 | | | | |
R2 −0.8981; Adjusted-R2 − 0.8599; Predicted-R2 −0.7708; |
As depicted in Tables 4 and 5, The results of the analysis of variance (ANOVA) for SnO2 dispersion in both DI water and NMP highlight the significance of the models in elucidating variations in the response variable. For SnO2 in DI Water as shown in Table 4, the model proves highly significant (p < 0.0001), with a notable F-value of 200.07. Factor A (Concentration of Nanoparticles) significantly influences the response, as evidenced by a low p-value (< 0.0001) and a high F-value of 592.75. Factors B (Sonication Time) and C (Addition of Surfactant) show lower significance, with p-values of 0.1901 and 0.0482, respectively. Hence, Factors A and C are considered significant factors as their p-values are less than 0.05. Lack of Fit is not significant (p = 0.4623), indicating a well-fitting model. The coefficients of determination (R²) denote a strong overall fit (R² = 0.9868), robustness after adjusting for predictors (Adjusted-R² = 0.9819), and reliable predictive capability (Predicted-R² = 0.9704).
For SnO2 in NMP as shown in Table 5, the model is significant (p = 0.0003) with an F-value of 23.51. Similar to DI water, Factor A (Concentration of Nanoparticles) significantly influences the response, with a low p-value (< 0.0001) and a high F-value of 63.64. Factors B (Sonication Time) and C (Addition of Surfactant) have shown p-values of 0.3293 and 0.0424, respectively. Hence, Factors A and C are considered significant factors as their p-values are less than 0.05. Lack of Fit is not significant (p = 0.1622), affirming the model's adequacy. The model exhibits a robust overall fit (R² = 0.8981), with adjustments for predictors (Adjusted-R² = 0.8599) and reliable predictive capability (Predicted-R² = 0.7708), considering the difference is smaller than 2. In summary, both models demonstrate high significance, with the concentration of nanoparticles and the addition of surfactant being the key influential factors, while the lack of fit is not a concern, and the models exhibit strong overall fits and predictive capabilities.
Central Composite Design and ANOVA data analysis
The significant factors identified from Plackett Burman Design, namely, the concentration of nanoparticles and the addition of surfactant, were subjected to further optimization using Central Composite Design. The levels assigned to these factors were determined based on the preceding Plackett Burman Design analysis. Each variable was investigated at five coded levels (-α, -1, 0, 1, α), with all variables set at the central point with a coded value of zero. The Central Composite Design matrix, along with the experimental and predicted results, for SnO2 dispersed in different solvents, are detailed in Table 6.
Table 6
Central Composite Design for SnO2 dispersing in DI water and NMP (actual and predicted).
| | Factor 1 | Factor 2 | Drop in Absorbance for 7 Days |
Solvent | Run | A: Concentration of Nanoparticles | C: Addition of Surfactant | Actual Value | Predicted Value |
| Unit | mg/ml | mg/ml | | |
DI Water | 1 | 1.25 | 1.75 | -13.32 | -11.87 |
2 | 1.25 | -0.01777 | -11.13 | -12 |
3 | 1.25 | 1.75 | -11.56 | -11.87 |
4 | 1.25 | 1.75 | -12.33 | -11.87 |
5 | 0.5 | 0.5 | -5.528 | -5.25 |
6 | 1.25 | 1.75 | -12.56 | -11.87 |
7 | 2 | 3 | -18.343 | -18.49 |
8 | 1.25 | 1.75 | -11.49 | -11.87 |
9 | 0.5 | 3 | -5.544 | -5.06 |
10 | 1.25 | 3.51777 | -11.38 | -11.73 |
11 | 2.31066 | 1.75 | -20.35 | -21.36 |
12 | 0.18934 | 1.75 | -1.274 | -2.37 |
13 | 2 | 0.5 | -19.466 | -18.68 |
NMP | 1 | 1.25 | 1.75 | -4.24 | -6.91 |
2 | 1.25 | -0.01777 | -6.31 | -6.9 |
3 | 1.25 | 1.75 | -6.87 | -6.91 |
4 | 1.25 | 1.75 | -6.89 | -6.91 |
5 | 0.5 | 0.5 | -1.74 | -1.35 |
6 | 1.25 | 1.75 | -6.74 | -6.91 |
7 | 2 | 3 | -12.79 | -12.46 |
8 | 1.25 | 1.75 | -6.91 | -6.91 |
9 | 0.5 | 3 | -1.39 | -1.36 |
10 | 1.25 | 3.51777 | -6.82 | -6.91 |
11 | 2.31066 | 1.75 | -15.62 | -14.75 |
12 | 0.18934 | 1.75 | -0.34 | 0.9408 |
13 | 2 | 0.5 | -13.12 | -12.45 |
Table 7
Model recommendation for SnO2 dispersing in DI water.
Source | Sequential p-value | Lack of Fit p-value | Adjusted R² | Predicted R² | |
Linear | < 0.0001 | 0.3898 | 0.9768 | 0.9659 | Suggested |
2FI | 0.5285 | 0.3435 | 0.9754 | 0.9620 | |
Quadratic | 0.4461 | 0.2882 | 0.9749 | 0.9305 | |
Cubic | 0.8778 | 0.0913 | 0.9666 | 0.5001 | Aliased |
Table 8
Model recommendation for SnO2 dispersing in NMP.
Source | Sequential p-value | Lack of Fit p-value | Adjusted R² | Predicted R² | |
Linear | < 0.0001 | 0.7126 | 0.9505 | 0.9377 | Suggested |
2FI | 0.9928 | 0.6284 | 0.9450 | 0.9284 | |
Quadratic | 0.1285 | 0.9549 | 0.9607 | 0.9551 | |
Cubic | 0.8325 | 0.9887 | 0.9488 | 0.9666 | Aliased |
The model recommendations for SnO2 dispersion in both DI water and NMP are provided in Tables 7 and 8, respectively. For SnO2 in DI water, the sequential p-values indicate the significance of different model terms. The linear model, with a p-value < 0.0001, is highly significant, suggesting a strong linear relationship. The lack of fit is not significant (p = 0.3898), affirming the model's adequacy. The adjusted R² (0.9768) and predicted R² (0.9659) signify a robust fit and predictive capability.
Similarly, for SnO2 in NMP, the linear model is highly significant with a sequential p-value < 0.0001. The lack of fit is not significant (p = 0.7126), indicating the model's appropriateness. The adjusted R² (0.9505) and predicted R² (0.9377) demonstrate a strong fit and predictive accuracy.
In summary, both models, whether for SnO2 in DI water or NMP, exhibit high significance in their linear forms, suggesting a strong linear relationship with the response variable. The lack of fit is not a concern in either case, and the models demonstrate strong adjusted and predicted R² values, indicating their reliability in explaining and predicting the variability in the response.
Table 9
ANOVA analysis of Central Composite Design for SnO2 dispersing in DI water.
Source | Sum of Squares | df | Mean Square | F-value | p-value | |
Model | 360.73 | 2 | 180.36 | 253.51 | < 0.0001 | significant |
A- Concentration of Nanoparticles | 360.66 | 1 | 360.66 | 506.92 | < 0.0001 | |
C- Addition of Surfactant | 0.0710 | 1 | 0.0710 | 0.0997 | 0.7586 | |
Residual | 7.11 | 10 | 0.7115 | | | |
Lack of Fit | 4.81 | 6 | 0.8023 | 1.39 | 0.3898 | not significant |
Pure Error | 2.30 | 4 | 0.5753 | | | |
Cor Total | 367.84 | 12 | | | | |
R2 −0.9807; Adjusted-R2 −0.9768; Predicted-R2 −0.9659; |
Table 10
ANOVA analysis of Central Composite Design for SnO2 dispersing in NMP.
Source | Sum of Squares | df | Mean Square | F-value | p-value | |
Model | 246.30 | 2 | 123.15 | 116.24 | < 0.0001 | significant |
A- Concentration of Nanoparticles | 246.30 | 1 | 246.30 | 232.48 | < 0.0001 | |
C- Addition of Surfactant | 0.0002 | 1 | 0.0002 | 0.0002 | 0.9890 | |
Residual | 10.59 | 10 | 1.06 | | | |
Lack of Fit | 5.12 | 6 | 0.8528 | 0.6227 | 0.7126 | not significant |
Pure Error | 5.48 | 4 | 1.37 | | | |
Cor Total | 256.89 | 12 | | | | |
R2 −0.9588; Adjusted-R2 − 0.9505; Predicted-R2 −0.9377; |
Tables 9 and 10 present the analysis of variance (ANOVA) for SnO2 dispersion in DI water and NMP using Central Composite Design. In both cases, the models are highly significant (p < 0.0001), indicating their effectiveness in explaining the variability in the response variable. The concentration of nanoparticles (Factor A) significantly influences the response, as evidenced by the low p-value (< 0.0001) and the substantial F-value of 506.92 and 232.48, respectively. However, the addition of surfactant (Factor C) has a negligible effect, with a p-value of 0.7586 and 0.9890, respectively, indicating its lack of significance.
The lack of fit are not significant in both cases, with p-values of 0.3898 and 0.7126, respectively, suggesting that the models fit the data adequately. The coefficients of determination (R²) for DI water and NMP are 0.9807 and 0.9588, respectively, indicating strong overall fits. The adjusted R² values (0.9768 for DI water and 0.9505 for NMP) account for the number of predictors, and the predicted R² values (0.9659 for DI water and 0.9377 for NMP) reflect the models' predictive capabilities.
Other than results from ANOVA analysis, the Central Composite Design results for SnO2 dispersion in DI water and have described the correlations between the factors through coded and actual equations given in Design Expert 13. In the case of SnO2 in DI water, the coded equation:
Drop in absorbance for 7 days = − 11.869–6.710*A + 0.094*C (3)
and the actual equation:
Drop in absorbance for 7 days = – 0.808–8.952*(Concentration of Nanoparticles) + 0.075*(Addition of Surfactant) (4)
allow predictions based on coded levels of factors A and C, or with factors specified in their original units, respectively.
Similarly, for SnO2 in NMP, the coded equation:
Drop in absorbance for 7 days = – 6.906–5.549*A – 0.005*C (5)
and the actual equation:
Drop in absorbance for 7 days = 2.349–7.398*(Concentration of Nanoparticles) + 0.004*(Addition of Surfactant) (6)
serve analogous purposes. These equations provide a mathematical understanding of the relationship between factors (concentration of nanoparticles and addition of surfactant) and the response variable (Drop in absorbance for 7 days) in both DI water and NMP, with coded equations aiding in the comparison of factor impacts and actual equations facilitating predictions in original units.
The application of RSM facilitated the quantification of the impact of nanoparticle concentration and the addition of surfactant (SDC) on SnO2 dispersion stability. The effects were expressed as coefficients in the respective equations for SnO2 dispersing in DI water and NMP, which were – 6.710 and 0.094, − 7.398 and 0.004, respectively. The equations revealed that higher nanoparticle concentration in both solvents led to lower dispersion stability, indicating a propensity for agglomeration in smaller mediums. Notably, NMP exhibited a slightly greater adverse effect than DI water.
Other than that, the 3D contour plot depicted in Figs. 1 and 2 provides a visual representation of the dispersion stability of SnO2 in DI water and NMP. Notably, lower concentrations of nanoparticles contribute to improved dispersion, mitigating sedimentation and agglomeration and consequently resulting in a more gradual decrease in absorbance over time. The introduction of a surfactant, while showing a marginal positive influence, does not significantly enhance dispersion stability.