Journal Information
Journal ID (publisher-id): chemical
Title: Journal of the Korean Chemical Society
Translated Title (ko): 대한화학회지
ISSN (print): 1017-2548
ISSN (electronic): 2234-8530
Publisher: Korean Chemical Society대한화학회
INTRODUCTION▼
Metallic nanoparticles (NPs) find wide applications in fuel cells, solar energy storage, growth catalysts and bio-medicines, just to name a few.1,2 In particular, the NPs of gold have been extensively utilized in the diagnostics, treatment, catalysis, optical sensing, and biomarkers and building blocks in nanotechnology.3 Various properties of a single NP, such as the optical and electrical properties, sensitively depend on its size.3,4
More often than not, it is an ensemble of NPs of different sizes that is utilized in experiment. It is therefore important to determine the particle size distribution (PSD) of NPs. An ex-situ method, such as the transmission electron5 or scanning electron6 microscopy, can be used to measure the sizes of NPs. Preferably however, a non-destructive in-situ detection of the size distribution is called for. At present, a popular in-situ method is the dynamic light scattering (DLS)7-9 which extracts the PSD by utilizing the Stokes-Einstein relation. DLS technique can be used to measure particle size for particles with a monodisperse size distribution. However, it becomes challenging to measure for particles with polydisperse or large-size distribution because big particles can screen smaller objects.8,10,11
UV-visible spectroscopy is widely used in the study of nanomaterials as a diagnostic of nanoparticle formation. It is a very useful tool employed for the characterization, estimation of the sies of NPs, concentration, and aggregation level. UV-visible spectrometers can be found in many laboratories, the analysis does not change the sample, and the time needed for registration of the spectrum is very short.4,12-15 Several attempts have been made to derive the PSD from a UV-visible extinction spectrum. Kim et al. assumed that the size distribution of gold NPs follows a Gaussian distribution and demonstrated that PSD can be derived solely from UV-visible spectra under this assumption. Calculations considering PSD were more experimentally accurate than calculations that ignored size distribution.3 Haiss et al. demonstrated the predictability of gold NPs sizes using the mean-free-path-corrected Mie theory.4 Doak et al. also utilized Mie theory to predict the size distribution of gold NPs from UV-visible spectra. In this study gold NPs were considered to have a broad size distribution, which resulted in better agreement with experimental data compared to theoretical calculations that only considered a single particle size.2 Karlik et al. showed the multilayered perceptron is efficient for the prediction of the diameter of gold nanospheres. The principal component dimension reduction technique was used during the preprocessing of UV-visible spectra. The training set consisted of 25 spectra calculated according to the Mie theory for nanospheres with diameters from 2 to 50 nm. The approach was validated for theoretical spectra of particles with 5, 7, 10, 15, and 30 nm diameter.13
Herein, we propose a novel alternative method to extract the PSD from the UV-visible extinction spectrum. The present method employs an artificial neural network (ANN) model of the PSD for a colloidal gold NPs. By utilizing the extensive dataset of UV-visible spectra generated from the exact Mie theory, we trained the ANN model to predict the PSD solely from the UV-visible spectra. The present ANN model is favorably tested against a finite set of experimental spectra collected from the literature.
COMPUTATIONAL METHODS▼
Preparation of the UV-Visible Spectra for Training the ANN Model
In order to train the present ANN model, we prepared the UV-visible spectra of gold NPs whose PSDs are known. We generated the discretized distribution of the radii of NPs, P(r), which ranged from r = 1 to 200 nm with an interval of 5 nm (40 nodes in the output layer). P(r) was sampled from the random numbers obeying a Gaussian distribution with a mean μ and standard deviation σ. The mean radius of NP, μ, was varied from 5 nm to 150 nm with an interval of 1 nm. For a given μ, we have chosen various σs ranging from 0.05 μ to 0.125 μ, in order to consider diverse PSDs. To each Gaussian P(r) generated as above, we added a small random noise uniform sampled from the range between -0.2P(r) and 0.2P(r). This addition of random noises was done to mimic the small noises in the experimental size distribution, and the other was to double the number of data for training the present ANN model.
For a given P(r) generated as above, the extinction spectrum Qext(λ) at a specific wavelength of λ was calculated as
where Qext(λ, ri) is the extinction coefficient for a specific radius of NP. We discretized the range of radii by using the bin of Δr = 0.5 nm and N = 200. λ was varied from 400 to 700 nm with an increment 1 nm, in order to include the peak of a typical extinction spectrum (located between 500 and 600 nm). The absorbance, A, is related to Qext(λ) through the following relation,
where the path length of light l varies from equipment to equipment and it is difficult to know exactly the number density N of the samples.2 To solve these, the value of spectrum at each wavelength Qext(λ) was divided by the difference between the maximum and minimum of the spectrum.
The size-dependent extinction coefficient Qext(λ, r) of a gold NP was calculated by using Mie theory,16
where λ is the wavelength of an incident light. an and bn are the scattering coefficients given by
and
In equations (4) and (5), we introduced a dimensionless size parameter x = 2πnm r/λ. Also, Ψn and ξn are the Ricatti-Bessel functions given by
and
where Jn+1/2(x) and
We took the solvent to be water, and it was assumed that the refractive index of the solvent did not depend on the wavelength of light, therefore using nm = 1.333. The refractive index of gold, taken to be dependent on the frequency of an incident light, is taken from the data reported by Johnson and Christy.18 We used a cubic spline interpolation19 to calculate the values of refractive indices intermediate between the discrete raw data reported by Johnson and Christy. We generated total of 1,752 spectra by using an in-house Python code.
Artificial Neural Network Model
We divided the dataset of extinction spectra into the training, validation, and testing sets by randomly taking 60, 20, and 20% of 1,752 spectra in total. We used the validation set to determine the presence of an overfitting. The present ANN model, illustrated in Fig. 1, is made of three hidden layers. The input layer consists of 301 nodes each of which contains the discretized spectrum of the solution of gold NPs where the wavelength of an incident light ranges from 400 to 700 nm with a 1 nm interval. Each hidden layer consists of 256 nodes. The output is discretized PSD of nanoparticles radii, with each node corresponding to the fraction of particles falling into a specific size bin. We used the ReLu function for an activation function, except for the output layer where the identity function was used.
Figure1.
Scheme for the present ANN model to extract the size distribution of NPs from an input extinction spectrum. Each of 301 nodes in the input layer contains the extinction cross-section value for the wavelength of an incident light. Each of 3 hidden layers consists of 256 nodes. Each of 40 nodes in the output layer gives discretized PSD of gold NPs.
We trained the present ANN model by using the back-propagation algorithm and found the optimized values of weights. We employed an Adam optimizer with a learning-rate of 10-6 to minimize cost function defined as the mean-squared error.20 In order to prevent overfitting, we applied a dropout method where the nodes in the hidden layers were dropped out with a rate of 0.7. The present ANN model was trained in 200,000 steps, and the cost values were saved for every 10,000 steps.
We calculated the R2 value, root-mean-square error (RMSE), and mean-absolute error (MAE) as the performance indicators of the ANN model. R2 was defined as
Experimental Data
We further tested the present ANN model against 41 experimental data. The experimental UV-visible spectra were obtained from the nanoComposix22 and Cytodiagnostics23 web pages, and from the paper reported by López-Muñoz, G.A., et al..24 After that, the graphic images of the experimental spectra were converted to digital numbers by using Engauge Digitizer.25 Some experimental data are given size distributions as mean and standard deviation of diameters, mean diameters were calculated from the predicted size distributions to compare the ANN model with experimental value.
RESULTS AND DISCUSSION▼
Fig. 2 shows the variation in the cost function as the training progresses in 200,000 steps. If the error of prediction using the training set is larger than the error of the prediction using the validation set, the learning is stopped. In this study, no learning interruption occurred. The blue line and orange line show the training and validation cost respectively. Training and validation costs decreased from 75.9811 and 75.3383 at 10,000th step to 13.2096 and 8.3396 at 200,000th step, respectively. Since the training cost decreased smoothly during training, it can be seen training of ANN finished well. In addition, the validation cost decreased as smoothly as training cost, and showed lower values during training, so it can be considered that there was no overfitting.
Figure2.
Change in the cost function with the progression of training. Drawn as the broken and dotted lines are the variations in cost for training and validation cost, respectively.
As illustrated in Fig. 3, the present ANN model trained well against the theoretical calculation. The ANN model gave the R2 values of 0.7949 and 0.7945 for the training and test sets, respectively (Table 1). The nearly identical values of R2 for training and test sets show the absence of overfitting. The RMSEs of the ANN model were 2.810 and 2.7639 for the training and test sets, respectively (Table 1). The MAEs were 0.5708 and 0.5740 for the training and test sets, respectively. However, discrepancies were observed at high fraction values. These higher fractions typically occurred in sharp PSDs and corresponded to smaller particle sizes (~30 nm), since the standard deviation σ was expressed as a ratio to the mean diameter during PSD generation. The discrepancies in these regions indicate lower prediction accuracy for smaller particles, which may be attributed to minimal variation in the extinction spectra within this small size range. This problem can be solved by making the intervals of size distribution smaller than 5 nm in a small size section, but in very small cases the prediction may not be as good.
Figure3.
Performance of the present ANN model vs. the original data for the training (a) and test (b) datasets. Each data point indicates the fraction of particles falling into a specific size bin. The training and test datasets consisted of 1050 and 351 data, respectively. As a reference, the dotted lines represent the exact match between the ANN and theoretical values.
Table1.
Training results of the present artificial neural net-work model
| RMSE | MAE | R2 Value | |
|---|---|---|---|
| Train | 2.801 | 0.5708 | 0.7949 |
| Test | 2.7639 | 0.5740 | 0.7945 |
| |Train-Test| | 0.0371 | 0.0032 | 0.0004 |
We investigated whether the ANN model predicts well in the experiment. The RMSE and MAE were 7.944 nm and 5.280 nm, respectively. Table 2 shows error and correlation value between the mean diameter from experiment and the ANN model, and Fig. 4 shows the correlation between them. In Fig. 4, x and y axis represent the mean diameters from experiment and the ANN model, respectively. Overall, they show high correlation, and especially when the average size is 45 nm or more it can be seen that the prediction is very good. However, in the case below size 45 nm it has relatively high RMSE and MAE values although high correlation coefficient.
Figure4.
Mean diameters of experimental NPs plotted vs. those predicted from the ANN model, drawn as diamonds. As a reference, the dotted line represents the exact match between the ANN and theoretical values. The R2 value for the mean diameter is 0.932.
Table2.
RMSE, MAE, R2 values between the calculated mean diameters based on predicted size distribution and the mean diameters from experiment
| RMSE (nm) | MAE (nm) | R2 Value | |
|---|---|---|---|
| Total | 7.944 | 5.280 | 0.932 |
| Size < 45 nm | 11.277 | 8.366 | 0.393 |
| Size > 45 nm | 3.602 | 2.865 | 0.960 |
Table 2 shows the predicted average size by ANN and absolute error and relative error between predicted and experimental values. It can be considered as follows that the prediction results in the section below the size of 45 nm are not good. Note the present ANN model showed lower prediction accuracy for smaller particle sizes. Additionally, in the previous reported study by Haiss et al., when the size was small (below 25 nm), the peak position of the experimental surface plasmon resonance (SPR) was smaller than the theoretical calculation.4 Since the ANN model used in this study predicts the size distribution of gold nanoparticles using only theoretical UV-visible extinction spectra, the difference in the shape between experimental and theoretical spectra can be considered as the reason. It seems that this can be solved by training the experimental spectrum in the case of small sizes.
CONCLUSION▼
In this study, Artificial neural network (ANN) that predicts size distributions of colloidal gold nanoparticles using UV-visible extinction spectrum is introduced. Extinction spectra of colloidal gold nanoparticles which considering size distribution were calculated by using Mie theory. ANN was trained using calculated extinction spectra as input factors and size distribution of gold nanoparticles as out-put factors. For comparison with the experiment, prediction size distributions using 41 experimental data were attempted. The average sizes were calculated from the predicted size distribution and compared with experimentally measured values. Overall, it showed high correlation, especially when the average size was more than 45 nm. However, in the case of size smaller than 45 nm, they showed a high error value although the high correlation coefficient. This can be seen since small particles shows differences in the shape of UV-visible extinction spectrum of theory and experiment, and the size distributions were gathered at 5 nm intervals. If the prediction in small sized section is successfully done, it will lead to the development of a non-destructive measurement method that effectively predicts the particle size using UV-visible extinction spectrum.