A of velocity measurement was taken for all 42 samples by taking 20 ml of the sample in the liquid column of Ultrasonic interferometer (F-80, Mittal Enterprises, Delhi, India) (Rubalya et al., 2013). The device was provided with an electric signal connected to a flat plate mounted on a piezo-electric crystal with capacitive reactance that produces 2 MHz frequency. The parameters associated with ultrasonic velocity (U), (i) Acoustic impedance (Z) relate the velocity of sound and density of the medium as given by the equation:

(ii) Newton- Laplace equation links the elastic bulk modulus (K) of liquid with velocity and density of oil by:

(iii) Wood’s equation gives the relation between adiabatic compressibility with velocity and density of sound in liquid:

(iv) Molecular free length L_f, it is the distance covered by the propagating acoustic waves between the surfaces of two neighboring molecules in a liquid. Intermolecular free length is the ratio of available volume to the surface area of the molecule. Jacobson (1953) and Pandey et al. (1979) have shown an empirical relation between ultrasonic velocity U, the density ρ and the intermolecular free length L_f of a liquid as:

where, K is Jacobson constant at room temperatures and β is the adiabatic compressibility of liquid.

The density measurement of density was determined using Pycnometer, following the ASTM standard method D891-09 (Rubalya et al., 2016).

The kinematic viscosity (η) of the oil was measured using a Redwood viscometer (Associated Instrument Manufacturers India Private Limited, New Delhi, India) (Rubalya et al., 2016).

The packing fractions of molecules in corn and peanut oil were analyzed using PY, SPT, and MCSL models (Heying and Corti, 2004; Lebowitz, 1964; Mandell and Reiss, 1975; Mansoori et al., 1971). The analyses were carried out by employing the experimentally measured ultrasound velocity as they are directly related to the thermodynamic behaviour of the system. The PY model proposes an equation of state for hard-sphere fluids, which was analyzed in terms of the viral co-efficient (Mandell and Reiss, 1975). The PY equation mainly involves the radial distribution function of a fluid, which can be generalized to an m-component mixture (Lebowitz, 1964; Percus and Yevick, 1958). SPT explains the feasibility of cavity formation in hard-sphere mixture fluids (Mansoori et al., 1971; Reiss et al., 1959). In addition, the glycerol in the oil leaves a space for the fatty acids to occupy and forms a cavity. A system under study is an m-component mixture and this is possible only because the probability of cavity formation is large (Percus and Yevick, 1958; Reiss et al., 1959; Ravi et al., 2008). An equation relating ultrasonic velocity (U) with the change in the molecular weight using the MCSL model incorporating particle interaction in the PY equation is given as (Wertheim, 1963; Yarnell et al., 1973):

where, R- gas constant, T- temperature, γ- specific heat ratio, M - molecular weight, X^{- l} is the expression that depends on three different theories that define the state of the system (Wertheim, 1963; Yarnell et al., 1973; Yu et al., 2002). The three theories that define these systems are the Percus-Yevick (PY) approach, Scaled Particle Theory (SPT) and Mansoori Carnahan Starling Leland (MCSL) model (Wertheim, 1963; Yu et al., 2002). Based on these models, the equation for X takes the form:

Using equations 6, 7 and 8 packing fraction and hence particle diameter (y) were deduced using an expression with ρ_n which is the number density of the uniform fluid and σ the particle size:

Chemical potential is the energy that can be emitted or absorbed when a particle is added to a system. It is also known as partial molar free energy. In heterogeneous systems, it is the change in internal energy of the system when new particles are added or when the condition of the system is altered like the variation in temperature. The density of a uniform hard-sphere fluid is related to its chemical potential through the Carnahan - Starling equation of state and is given in equation 1, respectively (Wertheim, 1963; Yarnell et al., 1973; Yu et al., 2002; Lennard, 1924):

where, ρ_n is the number density of the uniform fluid.

The Lennard - Jones potential (5- 8 equations) defines the interaction between a pair of neutral atoms or molecules. The potential description includes i) repulsive term r^-12 and ii) long-range attractive term r^-6 and the potential is defined as (Yarnell et al., 1973; Lennard, 1924):

where, ε/k defines inter particle- potential and can be estimated using the following equation (Lennard, 1924):

where, n_s - number of nearest neighboring atoms.

Ultrasound velocity (U) is subjected to intermolecular interfaces and molecular connotations, which enhance the significance of ultrasound velocity measurements for the molecular state, structure, composition and various processes (Izbaim et al., 2010). From the Tables 1 and 2, it can be observed that ultrasonic velocity decreases with cycles of heating and is due to less transmission of ultrasonic wave and scattering produced by molecular clusters in the oil (Rubalya et al., 2013).

Figures 1 (a), (b) and (c) illustrate the disparity in ultrasonic velocity, adiabatic compressibility, and viscosity with heating time, which is observed to be greater in peanut oil compared to corn oil. The formation of polymers and saturated fatty acids due to hydrogenation, oxidation, and hydrolysis accelerate the clustering of molecules that increase density and viscosity in the oil, and in turn, reduce the ultrasonic velocity (Benedito et al., 2007; Rubalya et al., 2013; Izbaim et al., 2010). The reduction in ultrasonic velocity reflects a decrease in intermolecular free length due to the conversion of a double bond in fatty acids to a single bond by the addition of a hydrogen atom that forms saturated fatty acids by which the packing density of molecule increases (Rubalya et al., 2017).

Adiabatic compressibility, Bulk modulus, and molecular free length determine the arrangement of atoms and molecules that influence the inter-atomic forces like London dispersion force, and Van der Waals force, which are seen between molecules in liquid systems (Izbaim et al., 2010; Rubalya et al., 2017; Sankarappa et al., 2005). The nature of forces and their strength between molecules can be understood by determining their internal pressure (Lei et al., 2016). Figure 1 (b) shows an increase in adiabatic compressibility with heating time for peanut oil. However, it remains almost constant for corn oil as the specific molecular interaction between molecules is almost the same.

Tables 1 and 2 elucidate the variation in acoustic impedance with heating time for peanut and corn oil. Acoustic impedance (Z) resists the flow of the ultrasonic wave through the liquid. When bonding between atoms becomes strong, the density of the oil sample will increase with the proliferation in packing density. Unsaturated fatty acids like oleic and linoleic acids get saturated to palmitic and stearic SFA that reduces the flow of sound in oil (Benedito et al., 2007; Benedito et al., 2002). Corn oil shows an experiential rise of Z up to 10.5% after six cycles of heating but in peanut oil, it varies by 33.45%. The Z value of peanut is found to be greater , as the density of the oil increases due to an increase in SFA formed by degradation. Jose Benedito (2007) studied the quality of fried oils using the ultrasonic parameter and correlated it with polar and polymer compounds that are formed by the degradation in oil. Lei Zang et al. (2016) have stated that oil exposed to an ultrasonic oscillatory wave produce cavitation bubbles as it passes through the sample, inducing pressure variation. This interaction of the ultrasonic wave with the oil and the breaking of cavitation bubbles will induce the oxidation effect and accelerate thermal degradation. Hence, from the measured ultrasonic velocity, the quality of oil and its further usage can be evaluated.

Variation in viscosity with respect to heating time for corn and peanut oils is shown in Figure 1 (c). The viscosity of the oil is less when it is composed of a large amount of unsaturated fatty acids and becomes high with an increase in long-chain saturated fatty acids (Adolfo et al., 2006; Fasina et al., 2006; Rubalya et al., 2016). Thus, the viscosity of oil increases with an increase in heating cycles as the quantity of SFA becomes greater. This variation in kinematic viscosity exhibits a steeper increase in peanut oil compared to corn oil, which is due to the molecular structural change in composition and decrease in the degree of unsaturated fatty acids (David, 2008; Everett, 1963). Viscosity ranges from 56.7 to 131.3 ×10^-6 m²/s for corn oil; whereas it varies for peanut oil from 68.51 to 183.5 ×10^-6 m²/s in six cycles of heating. Viscosity increases by 47.6% in corn oil after five hours of heating and is observed to be nearer to twice the times of its unheated value. This is due to the increase in altered double bonds to a single bond in fatty acids. The reason for variation in viscosity is due to the existence of long-chain saturated fatty acids (Rubalya et al., 2016, Rubalya et al., 2017). Similarly, from the rheological study, peanut oil shows a viscosity value 62.7% greater than the first time of heating. This is due to the increase in saturated bonds formed by the degradation of oil and also to the presence of arachidic and behenic acid with C22 long-chain saturated fatty acids (Reidoon, 2005).

Tables 1 and 2 show that the density of corn oil and peanut oil increases with heating cycles. After 6 cycles of heating, it is observed that the density of corn oil increases by 10.9% whereas in peanut oil it increases by 18%. The experimental density of oil was calculated after each cycle of heating at room temperature. The density of oils greatly depends on the molecular weight of its chemical compounds. In general, the molecular weight of oil rises due to the formation of mono, di and triglyceride oligomers and also with an increase in saturated fatty acids. From the measured physical properties, it is pragmatic that changes in molecular structure in the composition of oil influence the changes in the physical property.

Ultrasonic velocity (U) with a viscosity (η) and density (ρ) are related to the empirical equation given by:

The values are fitted to the above equation and the dependency factor (R²) was experiential between 0.922 and 0.966. Constants A= -0.387 and B = 0.403 are correlation coefficients in linear fitting of the above equation (15). The variations are due to the change in the molecular weight of unsaturated fatty acids and other unsaturated chemical composites in oil even after repeated cycles of frying. The oil contains 90% of short and long-chain fatty acids and triglycerides that increase the number of molecules per unit volume, which highly influences the density and viscosity of the oil. Rodenbush et al. (1999) had related density and viscosity of oil as both factors are temperature-dependent by the equation:

Density and viscosity are important quality analysis indices in the oil industry for the pumping, pipelining, designing of the processing system, etc. If the density is known the viscosity of the oil can be determined using equation (15). The dependency of density with viscosity using the Rodenbush equation varies from 0.902 to 913 and was computed using least square fitting, where D = 1.308 and E = -1.302 are the correlation coefficients. The variation in accuracy is due to the heavy molecules polymer: monomer, dimer, and triacylglycerol, which are formed upon heating the oil at the frying temperature in different time cycles.

The backscattering of the ultrasonic wave per unit volume by the ensemble of the molecule in a plane can be theoretically estimated using the PY model (Percus and Yevick, 1958; Wertheim, 1963). The packing fraction of molecules in oil upon degradation with irregular shape in a given volume can be predicted using the PY model. Tables 3 and 4 exemplify that up to sample S4 the packing fraction increases and decreases slightly to 0.002% towards S7 due to the spatially random distribution of molecules in a given volume of oil. The estimated packing fraction by the SPT model envisages the compressibility of hard-sphere in an oil sample which is of medium density. This confirms the cavity formation. As the major constituents of oils are fatty acids, they combine to form a system if their potentials coincide. Tables 3 and 4 illustrate the packing fraction increase with an increase in the duration of heating to up to 1.5 hours. As heating cycles are increased, unsaturated fatty acids in ‘cis’ form are converted to ‘trans’ form of saturated fatty acids (Pushan et al., 2009; Reidoon, 2005; David, 2008; Fasina et al., 2006). Because of this number of particles within the reference area and its size, its packing fraction in a given volume is greater. When heating cycles are increased the heavy molecules may disintegrate to form molecules of lower dimension. Using the MCSL model the bulk phase behavior in a complex liquid like oil under pressure variation of the propagation of the ultrasonic wave will exhibit the density functional theory. The packing fraction of atoms/molecules that are calculated using the models increases till 90 minutes of heating time and then decreases by 0.005% at 5 hours of heating. In this bulk medium liquid the calculated packing fraction varies in the order PY > SPT> MCSL. By comparing the corn and peanut oil using three models, it has been observed that peanut oil has greater packing fraction.

Figure 3 shows the variation in particle diameter/size of hard-sphere with heating time by comparing the behavior of corn and peanut oil. The calculated particle size of peanut oil is greater than corn oil using all three models (PY, SPT, MCSL) due to the presence of a large amount of long-chain saturated fatty acids. These hard-sphere models exemplify the microscopic description of the distribution of molecules in a given volume according to its dimension. The particle size using the PY model for peanut oil starts increasing by 0.008% for 0.5 hours heated sample and 0.001% for 5 hours compared to corn oil in a micrometer range. Using the SPT model, a systematic analytical method, the particle size varies from 0.02% to 0.09% from 1.5 to 5 hours of heating. The changes in the particle size using the fundamental measure theory MCSL model are very significant compared to the other two models.

In colloidal science and liquid state theory, the three models (PY, SPT, MCSL) are very important in the study of the distribution of heavy and low weight molecules that are formed when exposed to different temperatures (Reiss et al., 1959; Ravi et al., 2008). Fatty acid molecules in glycerol get distributed in oil depending on the change in the chemical potential (μ/kT), and also in inter-particle potential. Tables 3 and 4 elucidate the discrepancy of chemical potential (CP) with heating time. It can be observed that CP increases till 1 hour heated sample as more RCOO- and H+ ions are formed by oxidation reaction. When the oil is further exposed to repetitive cycles of heating, Peroxides RCOO- undergo a hydrolysis chemical reaction with glycerol to form monomer, dimer, trimer, etc. Hence the CP value decreases for the sample S7 due to the energy released in the formation of polymers. Using the PY model, the CP value of corn oil is observed to increase by 35.2% compared to peanut oil as it has a smaller quantity of saturated fatty acids and also highly potent antioxidants. Moreover, peanut oil has a fractional amount of C22 long-chain fatty acid. In the SPT model, the CP value for corn oil is 19% greater than peanut oil, whereas it is greater than 12% using the MCSL model. By comparing the computed CP value for the three models PY> SPT>MCSL the interatomic distribution in molecules with the time of heating can be explained.

Corn oil has comparatively greater stability and can withstand even more heating cycles as more external energy is needed to alter the chemical species in oil samples (chemically inert) and this is justified by the observed high values of chemical potential and inter-particle potential. The packing fraction of the corn oil increases from 0.7822 to 0.7846 with an increased duration of heating to up to 1.5 hours and slowly dissociates thereafter and reaches a lower value of 0.7810 after 5 hours of heating. The same trend is observed for peanut oil but with a faster dissociation rate. The packing fraction of peanut oil increases from 0.7817 to 0.7853 and dissociates down to 0.7603. As the packing fractions were estimated from the experimentally obtained ultrasound that traveled through the samples it gives insight about the oil’s status. This stability shows that the formation of the polar products like triglycerides; polymer compounds, free fatty acids, and carbonyl compounds are less in corn oil than in peanut oil. The particle size of corn oil is less than peanut oil as peanut oil contains C22 Arachidic acid; Behenic acid saturated long-chain fatty acid (Reidoon, 2005). Hence the degradation degree of corn oil is comparatively lower than peanut oil after 6 cycles of heating.

The Lennard-Jones (L-J) potential defines the interactive energy, which may be attractive or repulsive between the degraded polar molecules formed with heating cycles (Yarnell et al., 1973, Lennar, 1924). Incorporating the data from Tables 3 and 4, inter-particle potential L-J has been plotted and is shown in Figures 4 and 5 for corn and peanut oil. Figure 4 (a) presents the way in which the minimum potential energy of corn oil upsurge by the increases in inter-particle distance by 8.57% from 0 to 5 hours of heating using the PY hard-sphere model; whereas using the SPT model it increases by 13%. From the graph, the closest possible distance g(r) (also termed radial distribution function) has been observed around 0.12. A shift in internal energy is observed from the comparison plot which varies with heating cycles. The plot is drawn for 1.5-hour heating and 5 hours of heating and a shift was observed in peanut samples due to the breaking of bonds. Changes in the dynamic properties with the propagation of an ultrasonic wave through the sample is due to the presence of irregular long-chain molecules that produce more pressure than short-chain fatty acid in the oil. Using the MCSL hard-sphere model, the lowest potential energy increases by 13.6% from 0.5 to 5 hours of heating with respect to the interatomic distance of atoms in corn oil. Results obtained from the MCSL model almost match the SPT results.

Figure 5 (a) illustrates the variation in potential with a 7.4% increase in the interatomic distance using the PY model in peanut oil. The variation in interatomic distance is 7.69% using SPT and 16.4% using the MCSL model. The results calculated using the PY and SPT models are almost the same; whereas the MCSL model estimation is nearly twice the predicted PY and SPT. Hence to conclude, using this demonstrated L-J potential one can estimate the distribution of molecules using the hard-sphere models with an increase in heating duration.

Particle fraction, particle size, chemical potential, and L-J potential were computed using MATLAB. The low packing fraction and particle size of peanut oil were exemplified and compared to corn oil as it possesses a large amount of saturated and monounsaturated fatty acids.

Simulated output exhibits the distribution of molecules in oil samples that are made up of long and short flexible fatty acid chains and also additive mixtures of glycerol, antioxidants, etc., and agrees with existing data and the standard equations of state. The three hard-sphere models help to understand the chemical reaction taking place in oil exposed to different heating cycles. The CP and L-J potential also states remarkable computational advantages in the simulation to understand the different chemical composition in the oil and their potential variations after each cycle of heating (Stanciu, 2019).