Simple Method for Optical Detection and Characterization of Surface Agents on Conjugated Gold Nanoparticles

Abstract

In this article, we propose a simple method to calculate electrical permittivity and refractive index of surface agents of gold nanoparticles (Au NPs), in which it is possible to find the refractive index of surface agents shell by using the absorption peak of the gold nano-colloid. One of the usual tests for detection of surface agents is colorimetric methods based on the change of color of Au NPs. The color change is mainly due to the shift of localized surface plasmon resonance which is related to electrical interactions of surface agents. Although there are many mathematical models for simulating the absorption spectrum and calculating the plasmonic peak, using them is not simple and possible for everyone due to the need for programming. Here, the necessary simulations have been performed for different values ​​of refractive index of surface agents and particle size, and absorption peaks have been obtained. Using numerical methods, a simple formula is obtained between the wavelength of plasmonic peak, the ratio of hydrodynamic diameter to Feret size of the particles, and the refractive index of the surface agents. This method can help researchers to obtain the refractive index and consequently the type or concentration of surface agents around Au NPs without the need for programming or complex mathematical operations. It can also open new horizons in analyzing colorimetric diagnosis of biological agents such as viral antibodies, antigens, and other biological agents.

Introduction

In last decades, gold nanoparticles (Au NPs) have been studied widely because of their unique electrical and optical properties which are due to the small size and quantum confinement effects. Exclusive properties of Au NPs are attractive for chemical and biomedical use in biosensing [1, 2], cellular imaging [3], drug delivery [4, 5], and chemical catalysis [6, 7]. Furthermore, Au NPs exhibit strong nonlinear optical and photoacoustic effects which make them good candidates in optoelectronic devices [8] and nonlinear optical processes [9].

One of the practical aspects of these particles is colorimetric detection of a molecule or biological agent in solutions, and they are widely used in biosensors because of chemical stability, easy synthesis, and low cost [10, 11]. This applicable aspect of Au NPs is a relative result of localized surface plasmon resonance (LSPR) and is strongly affected by electrical forces arising from surface agents [12]. By surface modification of Au NPs with agent molecules, electrical and optical properties change, and it can be used in biosensor devices. Changes in electrical properties are widely used in electrochemical biosensors [13], and optical property change is utilized in colorimetric and optical biosensors [14].

LSPR is the resonance in oscillation of free electrons on the surface of Au NPs and is the main factor of absorption peak in the UV–visible absorption spectrum [15, 16]. Any change in electrical charges on the surface of the particles can disturb the plasmonic oscillation and shift the absorption peak. This effect was the subject of many colorimetric assays in colloidal solutions [17]. This method has been used for the detection of many biological factors, and it has been of particular interest for the diagnosis of SARS-CoV-2 virus in the last few years [18]. Many studies have been performed on colorimetric diagnosis and optical biosensing of this virus in which the color of the colloidal Au NPs change in the presence of antibodies, antigens, or protein spikes of SARS-CoV-2 virus [19–22]. Although many works have been done on biological agents, but in all the studies, the lack of numerical and quantitative calculations on the data is very evident.

Theoretical description of LSPR of Au NPs has been studied in some articles [23, 24]. In 2021, a complete set of equations for analytical description of LSPR of these NPs was offered [25] based on the Lorentz-Drude model [26]. According to the existing theories, as the refractive index of the surface agents increase, the LSPR wavelength experiences a further red shift, and this effect has been seen in many experiments which confirms theoretical results [27–29]. By numerical simulation of UV–visible spectrum for different values of the refractive index of the surface agents shell and extracting the value of LSPR and comparing to the experimental spectrum, real value of refractive index can be obtained [25]. Although this method is very accurate, it requires programming and numerical methods, but not everyone can use it. Therefore the need for an already existing equation is inevitable for common users.

In this work, using numerical methods, a simple formula between the wavelength of plasmonic peak (${\lambda }_{\max }$), the ratio of hydrodynamic diameter to Feret size of the particle ($R_0$), and the refractive index of the surface agents shell ($n_2$) was obtained. Experimentalists can easily obtain $n_2$ from the absorption peak and the average diameter of the particles obtained from TEM and DLS measurement, and it can open new sights in colorimetric methods.

Theory and Numerical Models

Lorentz-Drude (LD) model has been accepted as a classical model for describing LSPR in metal nanoparticles such as Au NPs and gives optical properties of individual particles [30]. On the other hand, there are theoretical models to relate optical constants of individual particles to optical constants of colloidal solutions, such as Maxwell Garnett and Bruggeman models [31, 32]. Combining these two sets of equations has been used to simulate the UV–visible spectrum of nanoparticles such as Au NPs [24, 33]. Obtained experimental spectra show that the wavelength of LSPR peak $({\lambda }_{\max })$ of the colloidal solution of Au NPs is longer than theoretical value [25–34] and it has been proven that this red-shift is attributed to the polarizability of the surface agents [25]. Recently, this red-shift mechanism was widely used to detect and evaluate the concentration of biological agents such as glucose and SARS-CoV-2 virus [22, 35, 36].

Since the most common synthesis method of Au NPs is the Turkevich method [37] and in this method citrate ions surround Au NPs, therefore, in most cases, the default surface agents are citrate ions as shown in Fig. 1. In the presence of other organic molecules in the colloid, citrate ions can be replaced, and a new shell with different optical properties would be performed. The effective relative permittivity of the decorated particle would be [25]:

$${\epsilon }_{12}=\frac{({\epsilon }_1+2{\epsilon }_2)({\epsilon }_2+2{\epsilon }_3)-2{(\frac{a}{b})}^3({\epsilon }_3-{\epsilon }_2)({\epsilon }_1-{\epsilon }_2)}{[({\epsilon }_1+2{\epsilon }_2)-{(\frac{a}{b})}^3({\epsilon }_1-{\epsilon }_2)]}-2{\epsilon }_3.$$ 1

where ${\epsilon }_1$, ${\epsilon }_2$ ${\epsilon }_3$, a, and b are the relative permittivity of the particles, the surface agents shell and the host solvent medium, and radiuses of particles and decorated particles, respectively [25]. By the definition:

$$R=\frac{{\epsilon }_1-{\epsilon }_2}{{\epsilon }_1+2{\epsilon }_2},R^{{}^{\prime }}=\frac{{\epsilon }_2-{\epsilon }_3}{{\epsilon }_2+2{\epsilon }_3}$$ 2

the effective permittivity can be reduced to:

$${\epsilon }_{12}={\epsilon }_2(\frac{1+2{(\frac{a}{b})}^{3}R}{1-{(\frac{a}{b})}^{3}R}).$$ 3

Fig. 1.

Au Np synthesized by Turkevich method embedded in citrate ions, and a typical UV–Vis absorption spectrum

Relative permittivity is a complex value and obeys the Lorentz-Drude (LD) model which contains both free and bound electrons [24, 25]:

$${\widehat{\epsilon }}_1={\epsilon }_1+i{{\epsilon }^{{}^{\prime }}}_1=1-\frac{{\omega }_p^2{f}_0}{\omega [\omega +i({\gamma }_{0,bulk}+\frac{v_F}{2a})]}+\sum _{j=1}^k\frac{{\omega }_p^2{f}_j}{{\omega }_{0j}^2-{\omega }^2-i{\gamma }_j\omega }.$$ 4

where ${\omega }_p=\sqrt{\frac{N{e}^2}{m{\epsilon }_0}}$, N, and ${\nu }_F$ are the plasma frequency, density of electrons, and Fermi velocity, respectively. Also, ω, ${\gamma }_0$, ${\gamma }_j$, $f_j=\raisebox{1ex}{$N_j$}\!\left/ \!\raisebox{-1ex}{$N$}\right.$, and ${\omega }_{0i}$ are the angular frequency of the incident light, damping constant of free electrons, damping constant, strength, and natural frequency of oscillator associated with interband transitions of kind j, respectively. Furthermore,$\sqrt{f_j}{\omega }_p$ is the plasma frequency related with transitions of kind j [25, 38, 39]. The parameters related to Au were given in our previous article: Ref. [25] (Table 1). By this definition:

$$R_r=\frac{({\epsilon }_1-{\epsilon }_2)({\epsilon }_1+2{\epsilon }_2)+{{\epsilon }^{{}^{\prime }}}_1^2}{{({\epsilon }_1+2{\epsilon }_2)}^2+{{\epsilon }^{{}^{\prime }}}_1^2},R_i=\frac{3{{\epsilon }^{{}^{\prime }}}_1{\epsilon }_2}{{({\epsilon }_1+2{\epsilon }_2)}^2+{{\epsilon }^{{}^{\prime }}}_1^2}$$ 5

real and imaginary parts of effective relative permittivity of conjugated NPs (${\widehat{\epsilon }}_{12}={\epsilon }_{12}+i{{\epsilon }^{{}^{\prime }}}_{12}$) would be given by:

$${\epsilon }_{12}={\epsilon }_2\frac{(1+{(\frac{a}{b})}^3{R}_r-2{(\frac{a}{b})}^6(R_r^2+R_i^2))}{{(1-{(\frac{a}{b})}^3{R}_r)}^2+{({(\frac{a}{b})}^3{R}_i)}^2},{\epsilon }_{12}^{\prime }={\epsilon }_2\frac{3{(\frac{a}{b})}^3{R}_i}{{(1-{(\frac{a}{b})}^3{R}_r)}^2+{({(\frac{a}{b})}^3{R}_i)}^2}.$$ 6

Table 1.

Shift of LSPR of capped Au NPs used in colorimetric detections and calculated refractive index of capping shell

Capping agents	2a (nm)	2b (nm)	LSPR (nm)	$n_2$ By curve fitting	$n_2$ By Eq.11	Relative error %
Citrate ions [25]	7.5	11.5	529	1.51	1.48	1.96
Citrate ions [40]	25	28	525	1.51	1.45	3.5
Poly ethylene glycol [41]	25	42	526	1.42	1.43	1.28
VHH-122 antibodies [41]	25	45	527	1.45	1.44	1.68
AAP (a triptan-family drug) [42]	10	13.9	544	1.73	1.73	0
Aptamer + phosphate buffer [22]	18	21	530	1.6	1.59	0.6
citrate ions [36]	10	13	527	1.51	1.47	2.6
ASOx [36]	10	55.4	521	1.36	1.41	3.6
SARS-CoV-2AB [43]	18	30	529	1.47	1.47	0

Real and imaginary parts of effective dielectric constant of the nano-colloid (${\widehat{\epsilon }}_{\mathit{colloid}}$) are:

$$\mathrm{Re}{\widehat{\epsilon }}_{\mathit{colloid}}={\epsilon }_3(1+2\varphi \frac{({\epsilon }_{12}-{\epsilon }_3)({\epsilon }_{12}+2{\epsilon }_3)+{{\epsilon }^{{}^{\prime }}}_{12}{}^2}{{({\epsilon }_{12}+2{\epsilon }_3)}^2+{{\epsilon }^{{}^{\prime }}}_{12}{}^2})$$

$$\mathrm{Im}{\widehat{\epsilon }}_{\mathit{colloid}}=\frac{6\varphi {{\epsilon }^{{}^{\prime }}}_{12}{\epsilon }_3^2}{{({\epsilon }_{12}+2{\epsilon }_3)}^2+{{\epsilon }^{{}^{\prime }}}_{12}{}^2}.$$ 7

where φ is the volume fraction of the conjugated Au NPs. The extinction coefficient (κ) of the nano-colloid would be [21]:

$$\kappa =\sqrt{\frac{1}{2}(-\mathrm{Re}{\widehat{\epsilon }}_{\mathit{collod}}+\sqrt{{(\mathrm{Re}{\widehat{\epsilon }}_{\mathit{colloid}})}^2+{(\mathrm{Im}{\widehat{\epsilon }}_{\mathit{colloid}})}^2})}$$ 8

and consequently the absorption coefficient of the colloid would be:

$$\alpha =\frac{4\pi \kappa }{\lambda }.$$ 9

These equations gives the UV–visible spectrum of a nano-colloid of decorated Au NPs and can be used to obtain the permittivity and refractive index ($n_2=\sqrt{{\epsilon }_2}$) of the surface agents shell around Au NPs using a numerical curve fitting procedure.

Results and Discussion

In previous works, the permittivity of the surface agents shell was changed gradually and ${\lambda }_{\max }$ was measured in each step and was compared to real value until the real value of relative permittivity and refractive index of the surface agents was obtained [25]. Here, we are looking for a method to get the refractive index of the surface agents shell by putting the values of ​​${\lambda }_{\max }$, a, and b in a simple equation.

At first, by definition $R_0$ as the ratio of hydrodynamic diameter 2b, to the Feret diameter 2a, Eqs. 1, 3, and 6 can be rewritten by substitution of $\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b$}\right.$ by $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$R_0$}\right.$. The only remaining equation that still depends on Feret diameter itself is Eq. 4. Considering a colloidal solution of Au NPs with the average size of 2a = 8 nm, and putting $R_0=1.1$, the curve of ${\lambda }_{\max }$ versus different values of $n_2$ was obtained. The refractive index $n_2$ changed from 1.33 to 2 in steps of 0.01. The curve which is a linear function was plotted, as shown for some values in Fig. 2. This procedure was repeated for other values of $R_0$ interval of 1.2 to 1.75, and curves were plotted. The best function that can be fitted to the graphs of ${\lambda }_{\max }$ is a linear function in the form of $y=m{n}_2+y_0$. The values of m and $y_0$ depend on $R_0$, as shown in Fig. 3. Based on numerical curve fitting methods, best-fitted polynomial functions were obtained for these coefficients.

Fig. 2.

${\lambda }_{\max }$ versus the refractive index of the surface agents shell for Au NPs with 8 nm diameter

Fig. 3.

Values of m and y₀ versus different values of R₀. The case of Au NPs with 8 nm diameter (2a = 8 nm)

The least-squares method is a mathematical procedure for finding the best-fitting curve to a given set of points. This method assumes that the best-fitted curve is the curve that has the minimal sum of deviations from a given set of data. In other words, the best function would be obtained by minimizing the sum of the squares of the offsets (the residuals) of the points from the function curve [44]. Here, it was used to fitting the curves with a suitable linear function.

The relation of ${\lambda }_{\max }$ versus $n_2$ and $R_0$ was obtained which gives a relationship for $n_2$:

$$n_2=\frac{{\lambda }_{\max }(nm)+y_0(R_0)}{m(R_0)}$$ 10

where $R_0=\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$a$}\right.$ is the ratio of hydrodynamic diameter to Feret diameter and ${\lambda }_{\max }$ is the wavelength of absorption peak in nm. Hydrodynamic and Feret diameters can be easily obtained by dynamic light scattering (DLS) and transmittance electronic microscopy (TEM) measurements, respectively. Coefficients were optimized for different values of a and for particle size at interval of 5 to 40 nm diameters, which has a maximum accuracy. Based on numerical fitting, Eq. 10 can be used to include such variations within a + 5% accuracy for $n_2$ as the form:

$$n_2=\frac{{\lambda }_{\max }(nm)+175.5R_0-678}{124R_0-109}$$ 11

To test Eq. 11, we compared it several times with the absorption curve obtained from Eqs.1–9, and differences were less than 5%. Since the significant figures for the refractive index are up to two decimal places, the present method is an accurate method for measuring the refractive index of the surface agents shell surrounding Au NPs. In Table 1, values of $n_2$ were obtained using two calculation processes: numerical fitting of the LSPR with Eqs. 1–9 and only using Eq. 11. Comparison of these two methods has been expressed in percent error.

The plasmonic peak position depends on both the radius and type (refractive index) of the surface agents. In our previous work, numerical evaluations showed that the sensitivity of the plasmonic peak to the size of particle and hydrodynamic diameter (2a and 2b) is more when these two quantities are close to each other [25]. Here, this can be easily seen through the quantity R₀. But for the macromolecules that form a thick surface agents shell around the particle, the changes of the plasmonic peak rather depend on the refractive index of the agents, not R₀. By increasing the value of b, $b,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$R_0$}\right.$ trends to zero in Eqs. 1–6. It reduces the sensitivity of the equations to the thickness of the surface agent layer. It means that when macromolecules or polymers surround the particles, we don’t have to worry about the exact thickness of the surface agent layer, and the equation gives us the refractive index with good accuracy.

In this article, using the discrete dipole approximation method, we investigate the optical absorption spectra of gold nanospheres. Higher-order approximations such as the quadrupole approximation can help to increase the accuracy of calculations, but where the shape of the nanoparticles is deviated from the spherical state. In these circumstances, quadropole approximation can describe extra LSPR peaks appear in the optical spectra. Also, the contributions of the dipole and the quadrupole components in optical absorption of metal nanoparticles have been studied in literature [45–47]. Based on these studies, in the conditions that ideal spherical particles are considered, the dipole approximation is highly accurate.

It should be noted that this theory deals with ideal conditions and the effect of factors such as polydispersity or distortion in the shape of particles is not included in it. The resulting error can be due to these factors and the error increases with the increase of them.

This method can be proposed as a quick and easy way to estimate refractive index of the surface agents shell, along with other common analysis methods in chemistry that are used to detect the type and amount of surface agents.

Conclusion

In conclusion, colorimetric detection of organic and biological agents using Au NPs was studied from the view point of numerical analyze in order to find a simple formula for calculation of the refractive index of surface agents shell of the particles. Theoretical and numerical calculations are based on evaluation of permittivity of Au NPs and its surface agents and their contribution in UV–Vis absorption spectrum of Au nano-colloid. Using numerical calculations, a simple formula was obtained that researchers can easily obtain the refractive index of the surface agents and it gives more precise data of the type and density of the biological agents such as saliva glucose and antibody/antigens of SARS-CoV-2. This model offers a simple formula which can be applicable for colorimetric assays of detection or evaluation of surface agents using Au NPs.

Author Contribution

Ehsan Koushki: Supervisor 1, writing optical parts, conceptualization, data curation, formal analysis, project administration. Abbas Koushki: Supervisor 2, writing—review and editing, validation, methodology.

Data Availability

The data generated during the current study are available from the corresponding author on reasonable request.

Declarations

Conflict of Interest

The authors declare no competing interests.