The Tansig-function model of low-frequency dielectric spectroscopy for the detection of adulterants in the Zamzam water Open Access

complex permittivity

the real part of complex permittivity

the imaginary part of complex permittivity

A

the area of each parallel plate in m²

d

distance between the two parallel plates in m

frequency in Hz

the permittivity of the vacuum

C

capacitance

D

dissipation factor

N

input into the Tansig function

n

output from the Tansig function

Z

impedance

∅

phase angle

R

resistance

x

actual value of the independent variable parameter

y

normalised value of x

y_max

1

y_min

− 1

x_min

minimum value of the independent variable parameter

x_max

maximum value of the independent variable parameter

E

sum of the weighted normalised independent variable

b

bias

W

weight

k

normalised output of the model

k_d

denormalised output of the model

S̅_obs

observed value

S_cal

calculated value

MSE

mean squared error

ZZ

Zamzam water

AC

alternating current

DRS

dielectric relaxation spectroscopy

n

output of the Tansig function

N

input of the Tansig function

L

inductance

ω

2πf

The Zamzam water (ZZ) has been of physical and spiritual significance to global Muslims and, especially, Hajj pilgrims, for ages. The ZZ springs from within the premises of the Holy Mosque in Makkah, Saudi Arabia. The water is routinely transported across many places in Saudi Arabia and overseas. Also, Hajj and Umrah pilgrims, as well as businessmen, transport water in various quantities from the natural source to various places across the world for commercial transactions. So, samples of the water can be found in various packages in the street shops in many cities and towns across the globe, including Nigeria. Muslims use the water for quenching thirst, healing, and spiritual purposes (Shomar 2012; Moni et al. 2022; Islam 2024) because many religious texts specify divine blessings and benefits associated with the water.

Owing to the growing spiritual awareness of the ZZ among Muslims, its commercial value has risen in many places in Nigeria, and the feeling and practice of adulteration have equally been prevalent. The strict legislation on the exportation of the ZZ by Saudi authorities has led to the rise in its adulteration outside the shores of Saudi Arabia (Rasyida et al. 2014). Diluting the water with local water samples or outright packaging of local water as ZZ will definitely alter or replace its physicochemical and other natural characteristics.

According to studies, ZZ is rich in magnesium, calcium, fluoride, and other minerals (Ahmad et al. 2019). While other water samples from local sources may also contain similar minerals, it is believed that the concentrations, polarisability, orientations, and physicochemical characteristics of various local samples in Nigeria, which may be used for the adulteration of the original ZZ, will be chemically and/or physically distinguishable under certain conditions. Thus, this work studies the dielectric characteristics of ZZ purity under various scenarios of adulteration with local water in Nigeria. The technique applied was a non-intrusive and low-cost dielectric spectroscopy, which relies on the unique dielectric property of each material.

Every material has peculiar electrical and magnetic characteristics that can be utilised to estimate its volumetric concentration in a mixture (Gocen & Palandoken 2021). The dielectric permittivity of a material has a relationship with the material's components, which can then be used to monitor changes in the material's contents (Frau et al. 2021). A material is classified as ‘dielectric’ if it has the ability to store energy when an external electric field is applied. Dielectric properties of water can be used to directly monitor water quality in rivers, lakes, and reservoirs after appropriate calibration (Al-Mattarneha & Alwadie 2016). Alternating current (AC) conductivity is extensively used to characterise the electrical properties of various materials to understand the nature of the conduction mechanism of solid materials (Okutan et al. 2005; Darwish et al. 2014). Dielectric relaxation spectroscopy (DRS) of dielectric material provides information about the orientation and translational adjustment of dipoles in the material (El-Nahass et al. 2006). Dielectric permittivity works due to the fact that the electrical energy transferred to a medium is a function of the electric field and the physicochemical properties of the medium (Darwish et al. 2014). The changes displayed by the dielectric permittivity of materials at different frequencies indicate the responses of the different nigrostriatal layers (electronic, atomic, and molecular) (El-Nahass et al. 2006). Thus, the molecular mobility of materials can be investigated using DRS.

Dielectric spectroscopy has been effective, as demonstrated by many authors, in the identification of different kinds and concentrations of organic or inorganic pollutants (Kaya & Fang 1997; Shang et al. 2004). The technique has been applied in many areas of science and engineering. Taylor & Morris (2022) used dielectric spectroscopy to investigate concrete properties, while Kaya & Fang (1997) detected the role of ionic strength and organic liquids in soil using the same technique.

Particularly, the dielectric relaxation losses over a wide frequency range reveal peculiar characteristics of components under analysis. Dielectric spectroscopy has the wide-range frequency over techniques such as light scattering and neutron scattering, providing a large number of dispersion features that are advantageous to modelling (Volkov & Chuchupal 2022). It has also been indicated that combining dielectric permittivity with conductivity provides sensitivity to chemical, physical, and microstructural states of solid and liquid samples (Chelidze et al. 1999; Kyritsis et al. 2000; Saltas et al. 2023).

Parameters of dielectric spectroscopy include dielectric permittivity (ɛ′), dielectric loss factor (ɛ″), dissipation factor (D), resonance frequency (F), impedance (Z), and so on. The ability to measure the parameters ɛ′ and ɛ″ of the response function is one of the edges of dielectric spectroscopy over techniques such as Raman spectroscopy and neutron scattering, which only allow the imaginary part to be determined (Volkov & Chuchupal 2022). Since these parameters can be directly measured using the inductance (L), capacitance (C), and resistance (R) machine, it is particularly desirable to connect these parameters in a mathematical relation with the adulteration or purity level of the ZZ. Thus, this study employs an artificial neural network (ANN) to create a simple mathematical relationship among these dielectrical parameters, which can then be readily employed to determine the purity level of the ZZ at any local outlet.

The function takes/produces input/output between 1 and −1.

Tansig (N) takes one input, N (of S by the Q column vector), and returns each element of N squashed between −1 and 1. The Tan-Sigmoid function was found useful in modelling cardiac abnormality (Adnan et al. 2018); energy system (Tarafdar et al. 2018), and so on.

In the literature, the technique of DRS is usually found where there is a wide gap in the dielectric permittivity values of the samples in the mixture (Ávila et al. 2013). The technique was employed to determine the concentration of phosphate and nitrate in water (Harnsoongnoen 2021), volumetric fractions in water–ethanol mixture (Gocen & Palandoken 2021), glyphosate in water from herbicide (Méndez-Jerónimo et al. 2024), chlorine concentration in water (Abdelgwad & Said 2015), and so on. Most studies involving the technique are presented in the microwave frequency range. Only a very few exceptions are encountered. Among the few radio frequency demonstrations of DRS was the work of Shah et al. (2015). The authors applied DRS to distinguish coconut water and distilled water. However, the authors did not investigate the presence of water adulterants in coconut samples, nor did they develop a simplified equation for generic application. Furthermore, none of these applications of DRS can be found in the area of closely related substances like similar potable water from different sources. This is likely because of the close values of dielectric permittivity among such materials. Apart from the application of the technique in cases of dissimilar materials, the majority of these earlier investigations did not come up with simplified mathematical expressions to aid quick and easy quantification of the material purity.

So, the following questions arise from the above discussions: 1. Can we apply the low-frequency DRS technique in distinguishing similar materials from different sources like ZZ and well water? 2. What roles does the ionic content of ZZ play in the profile of dielectric permittivity? 3. How does the adulteration of ZZ with similar materials affect the dissipation factor profile? 4. Can a simplified technique be developed to capture the multi-parameter system of DRS to detect the purity level of ZZ?

With a view to answering the above research questions, this study presents the application of low-frequency DRS to closely related materials – ZZ and water from different sources – using a combination of experimental investigations with ANN modelling. This work investigates the dielectric properties of pure samples of the ZZ, local well water, and bottled commercial water. Dielectric properties, such as permittivity (ɛ′), loss factor (ɛ″), dissipation factor (D), frequency (F), and impedance (Z), were measured and interpreted in line with the compositional characteristics of each sample. Effects of adulterating ZZ with well water on dielectric properties were analysed and the resulting shifts in dissipation factor were investigated. An ANN-based equation, simpler than that of Abidoye et al. (2018), was developed to relate the dielectric spectroscopy parameters to the ZZ purity. The Tan-Sigmoid transfer function of ANN was employed to establish a mathematical relationship among interrelating variables of the dielectric spectroscopy and the water system.

A sample of the ZZ was obtained from the tap of the Holy Mosque in Makkah, Saudi Arabia, during the Hajj pilgrimage in June–July 2023. The sample was refrigerated for the storage period prior to experimentation. Well water was taken from a local well in Osun State, Nigeria. Also, the bottled water was purchased from the commercial stable of ‘Mr V’ purified water.

Some physicochemical characteristics of the water samples were determined using a hand-held pH, total dissolved solids (TDS), and electrical conductivity metre.

Dielectric characterisation was performed on the different water samples using an impedance analyser LCR instrument (EAST-TESTER, Guangzhou, China). The measuring range of the instrument was 10 Hz–1 MHz, with an accuracy of ±0.05% and an oscillating voltage of 1 V. Sample cell was made of square acrylic plastic with parallel copper plates as electrodes. Parameters like capacitance (C), impedance (Z), dissipation factor (D), and phase angle (⁠⁠) were measured at the frequencies (F). Pure phases of the different water categories listed in Table 1 were tested. Also, blends of the ZZ with well water were tested at various proportions.

where is the real part of the complex permittivity (⁠⁠) called dielectric permittivity. The imaginary part, ⁠, is the dielectric loss factor. ‘A’ is the area of the plate that serves as the electrodes, ‘d’ is the distance between the two parallel plates, ⁠, where is the frequency. is the permittivity of the vacuum. The capacitance at a set frequency is C(ω).

Different ANN configurations were tested to arrive at the best-performing ANN model. But the primary aim is to obtain a simplified ANN-based model using the tansig function. A simplified model will make it easy for users to deploy the equation readily and effectively.

Single-hidden-layer configurations of ANN were designed for tests in this work. The number of neurons at the hidden layer was incrementally tested to arrive at a well-performing and simplified model. Therefore, 1, 2, and 4 neurons were differently tested at the hidden layer. These configurations were ANN[5-1-1], ANN[5-2-1]. and ANN[5-4-1]. The first digit (‘5’) represents the number of input variables (ɛ′, ɛ″, D, F, and Z). The second digit (‘1’, ‘2’, ‘4’) represents the number of neurons at the hidden layer. The last digit (‘1’) represents the number of output variables (ZZ purity)

The performances of the different models were assessed using the following mathematical criteria:

• (1) Pearson product-moment coefficient of correlation (R)

• This criterion measures the strength of linear dependence in the relationship between calculated and observed values of the output variable (ZZ purity). The closeness of ‘R’ to 1 indicates the highness of the prediction accuracy. A value of R = 1 indicates a perfect model. ‘R’ is computed as:

  

  (12)

• are the averages of the observed and calculated output, respectively.

• (2) Mean squared error (MSE)

• This determines the average of the squares of the errors between the observed value (S_obs) and the calculated output value (S_cal). It is expressed mathematically as follows:

  

  (13)

• where N is the number of data points.

The physicochemical characteristics of the different water samples are shown in Table 1. It was interesting to note from the table that ZZ was slightly alkaline (pH of 7.7), signalling the high level of dissolved minerals, as evident from its highest TDS value (317 ppm) in the list. ZZ is rich in magnesium, calcium, fluoride, and other minerals (Ahmad et al. 2019). Moni et al. (2022) reported a pH of 8 for ZZ, which is very close to the value observed in this study. According to Dinka et al. (2015) and Biswas & Mosley (2019), the alkalinity level of water rises with the level of dissolved and particulate components. So, it is interesting that the high level of dissolved minerals in ZZ does not elevate the alkalinity value beyond normal. It can also be observed that ZZ has the highest electrical conductivity value (665 μS/cm). This can also be attributed to the high dissolved minerals in the water.

Moni et al. (2022) reported TDS and electrical conductivity (EC) values of 371 ppm and 718 μS/cm, respectively, for ZZ. These values are very close to the measurements in this study (317 ppm and 665 μS/cm) and point to the accuracy and consistency of the ZZ contents.

In fact, nothing was visible in the case of well water. This pointed to the accuracy of the instrument and the care taken in these measurements.

Figure 2 shows that the dielectric permittivity of all water samples decreases as frequency increases. The values of ɛ′ in ZZ range from 10 Hz to 1 MHz. For the well water, ɛ′ ranges from to for the same frequency range and from to for the bottled water. The results are in agreement with the report of Shah et al. (2015) and Vaja & Rana (2023). Shah et al. (2015) reported ɛ′ for distilled and coconut water from 20 Hz to 2 MHz. They reported ɛ′ for distilled water in the range (of 20 Hz) to (of 2 MHz). Vaja & Rana (2023) reported ɛ′ for distilled water in the range of at 20 Hz to at 1 MHz. Their results agreed largely with those reported in this work. A slight difference may be seen in the water content as a result of the geographical location of their water source (India). Also, the difference of one order of magnitude between their ɛ′ (⁠ at 20 Hz for Vaja & Rana (2023)) and the current result for ZZ (⁠ at 10 Hz) can be traced to the frequency difference at the starting point. It is known that ɛ′ decreases with frequency. So, the difference of 10 Hz (10–20 Hz) might be responsible for the one order of magnitude difference. Furthermore, the results agreed at the higher end of the frequency (1 MHz) as they remained in the same order of magnitude (⁠⁠.

Permittivity indicates the level of energy stored in the medium during an electromagnetic cycle. At lower frequencies, the water dipoles have enough time to orient in line with the applied electromagnetic field (Angulo-Sherman & Mercado-Uribe 2011; Darwish et al. 2014) and thus, are able to store more of the energy as expressed in high values of ɛ′. But, as the frequency increases, there is not enough time for the orientation of dipoles owing to the higher speed of passage. Thus, the materials store less energy at higher frequencies. The dielectric permittivity of the ZZ is higher than that of the well and bottled water at most frequencies. The values virtually coincide for all water samples at higher frequencies. This implies that the ZZ could store more electromagnetic energy than ordinary water. This could be attributed to the presence of vital minerals in the ZZ, giving the water molecular polarisation a tendency to orient in line with the electromagnetic wave passing through it. It could also be observed in the figure that the dielectric permittivities are linear, under a log–log plot, at lower frequencies till 1 kHz. It starts to become nonlinear near 10 kHz. Thus, the permittivity of the water samples is directly proportional to the frequency of the field at lower frequency values.

In the same vein, the lowest loss factor in bottled water can be traced to the low TDS (3 ppm), and low conductivity (μS/cm), which shows the non-conducting nature of its constituents. The bottled water sample does not have many mineral contents and so does not have many conducting molecules, thereby minimising its energy loss value while also affecting its energy storage similarly. This is also connected to the low electrical conductivity in the bottled water sample. As for the ZZ, the high loss factor could be attributed to the presence of various minerals, most of which are polar in nature, leading to high orientation ability under electromagnetic waves, while others are simply ionic, leading to loss of electromagnetic energy. The high TDS of the ZZ further shows magnitudes of various dissolved minerals in the sample that may be of various characteristics, which will give the water samples the observed values of the TDS and electrical conductivity. The loss factor in the well water was in the middle and could be connected to the moderate TDS and electrical conductivity in the sample. This shows that most of the content of the well water samples is relatively easily oriented under an electric field and can save more energy than losing or conducting it away.

According to Jansson et al. (2010), the Maxwell–Wagner–Sillars phenomenon dominates at low frequencies, whereby the dipoles align with the applied field and the ionic pairs migrate to the electrodes. Thus, the loss factor is higher at lower frequencies, especially where minerals and ions are present, as in ZZ. However, the ionic effect reduces at higher frequencies owing to fast-changing electrode polarities under AC. The dipoles also respond, albeit ineffectively, at higher frequencies, obeying Maxwell–Boltzmann statistics (Angulo-Sherman & Mercado-Uribe 2011). Thus, the loss factor reduces with frequencies similar to the dielectric permittivity.

Abdelgwad & Said (2015), though carried out in the microwave frequency range, reported a rise in loss factor with an increase in chlorine concentration in water, attributed to the increase in ion concentration that promotes conduction in the medium. Thus, the loss factor is higher where more conducting ions are present. However, the author shows that the permittivity decreases with an increase in chlorine concentration. Gocen & Palandoken (2021) show that ethanol in water reduces the permittivity of water while the loss factor rises and falls at lower and higher frequencies, respectively. It should be noted that these authors conducted their investigations at microwave frequency ranges for mixtures of different substances in water. However, the pure samples in this work reflect similar patterns of dielectric permittivity and the loss factor.

Particularly, the ZZ could be seen as a liquid of complex ions and molecules, which makes its permittivity and loss factor higher than others. Shah et al. (2015) pointed out that high-mineral solutions with diverse species of components tend to possess high ɛ′ and ɛ″ which agrees with the current results on ZZ, having the highest ɛ′ and ɛ″ among the water samples. At 20 Hz, the authors reported ɛ′ values of 3.36 and 1.02 for coconut water and distilled water, respectively. Coconut water with higher mineral content possesses higher ɛ′. On the other hand, the authors also reported values of loss factor of 7.47 and 2.07 for coconut water and distilled water, respectively, at 20 Hz. This shows coconut having higher ɛ′ and ɛ″ than distilled water because of various dissolved minerals. Thus, the current study found results that are consistent in scope with earlier findings.

ANN was used to model the relationship between the parameters of dielectric spectroscopy. The results of the ANN modelling of the ZZ purity are presented in this section. ANN is efficient in capturing the multivariable complex relationship among water quality parameters (M'nassri et al. 2022) and its supply system (Sörensen et al. 2024).

The resulting model is shown in Equation (14). The equation predicts the purity level of the Zamzam water. The model was built on five parameters of dielectric spectroscopy (ɛ′, ɛ″, D, F, and Z) to predict the purity level of the ZZ. The equation was a much simpler modification to the ANN-based equation espoused in the work of Abidoye et al. (2018). This is because the current ANN modelling approach eliminates tables of parameter values that accompany the model equation in the work of the authors. Hosseini & Baghelani (2021) stated that a simple set of equations, based on the analysis of the shift in resonance frequency, like in this work, could quantify the fractional components of mixtures.

Firouz et al. (2022) and Khaled et al. (2021) also used ANN to model dielectric parameters in detecting adulteration in oil and identification of oil palm disease, respectively. Different ANN configurations were tried by Firouz et al. (2022) to arrive at a satisfactory model for dielectric spectroscopy at radio frequency. However, none of the authors developed a simplified equation that captures relevant parameters of the process that can be easily applied. This work is a great step to fill these gaps in the dielectric spectroscopy method radio frequency for monitoring the quality of potable water samples from different sources.

Overall, it has been shown that a combination of dielectric parameters can reliably be used to determine the purity of liquid samples. The unique source and components of ZZ, with characteristic dielectric properties, prove successful in detecting its purity and adulteration level with similar water but from different sources. While ZZ permittivity and loss factor represent its energy-saving and lossy characteristics, its dissipation characteristic presents the compact behaviour of its component ions and dipole molecules at every frequency as unique and different from similar materials but from other sources. The spectroscopic shift at the crossover point of the dissipation factor with the presence of external components is a unique way to determine the level of adulterants in the ZZ, and the method can be equally applied in material purity characterisation.

The current results and analysis are limited in scope to the frequency range – 10 Hz to 1 MHz. Also, further work is needed to relate the chemical and physical components of ZZ to its observed dielectric characteristics. Since literature reports of ZZ compositions have shown good consistency (see Donia & Mortada 2021; Moni et al. 2022) to the characteristics reported in this work, it is apt to link these constituent compositions and components to its overall dielectrical behaviour and purity. Furthermore, more water samples from rivers, taps, rain, and elsewhere may be investigated as a way of supporting the analysis in this study.

While common investigations utilised dielectric spectroscopy at microwave frequency to analyse mixtures whose components have widely – varying permittivity, this study successfully utilised the technique in distinguishing closely related materials – ZZ from well water, and bottled water samples. The dielectric permittivity (ɛ′), loss factor (ɛ″), dissipation factor (D), frequency (F), and impedance (Z) were measured for ZZ, well water, and bottled water. The dielectric permittivity and loss factor are higher in ZZ than in other water samples. Contrarily, the bottled water had higher impedance than others. The results point to the influence of mineral content as the most significant factor affecting the dielectric properties of the water samples. As indicated by its TDS value, ZZ comprises a high proportion of molecular and ionic components, giving it exceptional polarisability and higher permittivity values than common water samples in Nigeria. Similarly, the loss factor of the ZZ was higher than that of others, which was attributed to the presence of conducting ions in its mineral content. The unique relaxation peak value was detected for the ZZ, at which the characteristic dissipation rate was recorded. This occurred at a relaxation frequency of 10 kHz for the ZZ and well water samples. As for bottled water, the peak shifted leftward, which was attributed to its unnaturally low mineral content. The use of well water to adulterate the ZZ decreases the relaxation peak at the relaxation frequency while also shifting the frequency rightwards under the influence of adulterated water concentration. A simple ANN-based model, based on the Tan-Sigmoid function, captured the relationship between key dielectric variables and ZZ purity. Thus, the level of adulteration in ZZ can be easily measured using the simplified model. Overall, the work presented numerically assisted ultra-low frequency dielectric spectroscopy analysis as a cheaper alternative to the material quality analysis.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.